Radioactivity

1) Introduction

2) Decay law

3) Alpha decay

4) Beta decay

5) Gamma decay

6) Fission

7) Decay series

8) Exotic forms of decay

Detection system GAMASPHERE for study rays from gamma decay

IntroductionTransmutation of nuclei accompanied by radiation emissions was observed - radioactivity Discovery of radioactivity was made by H Becquerel (1896)

Three basic types of radioactivity and nuclear decay 1) Alpha decay2) Beta decay3) Gamma decay

and nuclear fission (spontaneous or induced) and further more exotic types of decay

Transmutation of one nucleus to another proceed during decay (nucleus is not changed during gamma decay ndash only energy of excited nucleus is decreased)

Mother nucleus ndash decaying nucleus Daughter nucleus ndash nucleus incurred by decay

Sequence of follow up decays ndash decay series

Decay of nucleus is not dependent on chemical and physical properties of nucleus neighboring (exception is for example gamma decay through conversion electrons which is influenced by chemical binding)

Electrostatic apparatus of P Curie for radioactivity measurement (left) and present complex for measurement of conversion electrons (right)

Radioactivity decay law

Activity (radioactivity) Adt

dNA

where N is number of nuclei at given time in sample [Bq = s-1 Ci =371010Bq]

Constant probability λ of decay of each nucleus per time unit is assumed Number dN of nuclei decayed per time dt

dN = -Nλdt dtN

dN

t

0

N

N

dtN

dN

0

Both sides are integrated

ln N ndash ln N0 = -λt t0NN e

Then for radioactivity we obtaint

0t

0 eAeNdt

dNA where A0 equiv -λN0

Probability of decay λ is named decay constant Time of decreasing from N to N2 is decay half-life T12 We introduce N = N02 21T

00 N

2

N e

2lnT 21

Mean lifetime τ 1

For t = τ activity decreases to 1e = 036788

Heisenberg uncertainty principle ΔEΔt asymp ħ rarr Γ τ asymp ħ where Γ is decay width of unstable state Γ = ħ τ = ħ λ

Total probability λ for more different alternative possibilities with decay constants λ1λ2λ3 hellip λM

M

1kk

M

1kk

Sequence of decay we have for decay series λ1N1 rarr λ2N2 rarr λ3N3 rarr hellip rarr λiNi rarr hellip rarr λMNM

Time change of Ni for isotope i in series dNidt = λi-1Ni-1 - λiNi

We solve system of differential equations and assume

t111

1CN et

22t

21221 CCN ee

hellipt

MMt

M1M2M1 CCN ee

For coefficients Cij it is valid i ne j ji

1ij1iij CC

Coefficients with i = j can be obtained from boundary conditions in time t = 0 Ni(0) = Ci1 + Ci2 + Ci3 + hellip + Cii

Special case for τ1 gtgt τ2τ3 hellip τM each following member has the same

number of decays per time unit as first Number of existed nuclei is inversely dependent on its λ rarr decay series is in radioactive equilibrium

Creation of radioactive nuclei with constant velocity ndash irradiation using reactor or accelerator Velocity of radioactive nuclei creation is P

Development of activity during homogenous irradiation

dNdt = - λN + P

Solution of equation (N0 = 0) λN(t) = A(t) = P(1 ndash e-λt)

It is efficient to irradiate number of half-lives but not so long time ndash saturation starts

Alpha decay

HeYX 42

4A2Z

AZ

High value of alpha particle binding energy rarr EKIN sufficient for escape from nucleus rarr

Relation between decay energy and kinetic energy of alpha particles

Decay energy Q = (mi ndash mf ndashmα)c2

Kinetic energies of nuclei after decay (nonrelativistic approximation)EKIN f = (12)mfvf

2 EKIN α = (12) mαvα2

v

m

mv

ff From momentum conservation law mfvf = mαvα rarr ( mf gtgt mα rarr vf ltlt vα)

From energy conservation law EKIN f + EKIN α = Q (12) mαvα2 + (12)mfvf

2 = Q

We modify equation and we introduce

Qm

mmE1

m

mvm

2

1vm

2

1v

m

mm

2

1

f

fKIN

f

22

2

ff

Kinetic energy of alpha particle QA

4AQ

mm

mE

f

fKIN

Typical value of kinetic energy is 5 MeV For example for 222Rn Q = 5587 MeV and EKIN α= 5486 MeV

Barrier penetration

Particle (ZA) impacts on nucleus (ZA) ndash necessity of potential barrier overcoming

For Coulomb barier is the highest point in the place where nuclear forces start to act R

ZeZ

4

1

)A(Ar

ZZ

4

1V

2

03131

0

2

0C

α

e

Barrier height is VCB asymp 25 MeV for nuclei with A=200

Problem of penetration of α particle from nucleus through potential barrier rarr it is possible only because of quantum physics

Assumptions of theory of α particle penetration

1) Alpha particle can exist at nuclei separately2) Particle is constantly moving and it is bonded at nucleus by potential barrier3) It exists some (relatively very small) probability of barrier penetration

Probability of decay λ per time unit λ = νP

where ν is number of impacts on barrier per time unit and P probability of barrier penetration

We assumed that α particle is oscillating along diameter of nucleus

212

0

2KI

2KIN 10

R2E

cE

Rm2

E

2R

v

N

Probability P = f(EKINαVCB) Quantum physics is needed for its derivation

Centrifugal barrier depends on angular momentum of emitted or incident particle

classical r

V

rm

LrmF l

3

22

2

2

l rm2

L(r)V

quantum L2 rarr l(l+1)ħ2 rarr2

2

l rm2

1)l(lV

bound statequasistationary state

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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IntroductionTransmutation of nuclei accompanied by radiation emissions was observed - radioactivity Discovery of radioactivity was made by H Becquerel (1896)

Three basic types of radioactivity and nuclear decay 1) Alpha decay2) Beta decay3) Gamma decay

and nuclear fission (spontaneous or induced) and further more exotic types of decay

Transmutation of one nucleus to another proceed during decay (nucleus is not changed during gamma decay ndash only energy of excited nucleus is decreased)

Mother nucleus ndash decaying nucleus Daughter nucleus ndash nucleus incurred by decay

Sequence of follow up decays ndash decay series

Decay of nucleus is not dependent on chemical and physical properties of nucleus neighboring (exception is for example gamma decay through conversion electrons which is influenced by chemical binding)

Electrostatic apparatus of P Curie for radioactivity measurement (left) and present complex for measurement of conversion electrons (right)

Radioactivity decay law

Activity (radioactivity) Adt

dNA

where N is number of nuclei at given time in sample [Bq = s-1 Ci =371010Bq]

Constant probability λ of decay of each nucleus per time unit is assumed Number dN of nuclei decayed per time dt

dN = -Nλdt dtN

dN

t

0

N

N

dtN

dN

0

Both sides are integrated

ln N ndash ln N0 = -λt t0NN e

Then for radioactivity we obtaint

0t

0 eAeNdt

dNA where A0 equiv -λN0

Probability of decay λ is named decay constant Time of decreasing from N to N2 is decay half-life T12 We introduce N = N02 21T

00 N

2

N e

2lnT 21

Mean lifetime τ 1

For t = τ activity decreases to 1e = 036788

Heisenberg uncertainty principle ΔEΔt asymp ħ rarr Γ τ asymp ħ where Γ is decay width of unstable state Γ = ħ τ = ħ λ

Total probability λ for more different alternative possibilities with decay constants λ1λ2λ3 hellip λM

M

1kk

M

1kk

Sequence of decay we have for decay series λ1N1 rarr λ2N2 rarr λ3N3 rarr hellip rarr λiNi rarr hellip rarr λMNM

Time change of Ni for isotope i in series dNidt = λi-1Ni-1 - λiNi

We solve system of differential equations and assume

t111

1CN et

22t

21221 CCN ee

hellipt

MMt

M1M2M1 CCN ee

For coefficients Cij it is valid i ne j ji

1ij1iij CC

Coefficients with i = j can be obtained from boundary conditions in time t = 0 Ni(0) = Ci1 + Ci2 + Ci3 + hellip + Cii

Special case for τ1 gtgt τ2τ3 hellip τM each following member has the same

number of decays per time unit as first Number of existed nuclei is inversely dependent on its λ rarr decay series is in radioactive equilibrium

Creation of radioactive nuclei with constant velocity ndash irradiation using reactor or accelerator Velocity of radioactive nuclei creation is P

Development of activity during homogenous irradiation

dNdt = - λN + P

Solution of equation (N0 = 0) λN(t) = A(t) = P(1 ndash e-λt)

It is efficient to irradiate number of half-lives but not so long time ndash saturation starts

Alpha decay

HeYX 42

4A2Z

AZ

High value of alpha particle binding energy rarr EKIN sufficient for escape from nucleus rarr

Relation between decay energy and kinetic energy of alpha particles

Decay energy Q = (mi ndash mf ndashmα)c2

Kinetic energies of nuclei after decay (nonrelativistic approximation)EKIN f = (12)mfvf

2 EKIN α = (12) mαvα2

v

m

mv

ff From momentum conservation law mfvf = mαvα rarr ( mf gtgt mα rarr vf ltlt vα)

From energy conservation law EKIN f + EKIN α = Q (12) mαvα2 + (12)mfvf

2 = Q

We modify equation and we introduce

Qm

mmE1

m

mvm

2

1vm

2

1v

m

mm

2

1

f

fKIN

f

22

2

ff

Kinetic energy of alpha particle QA

4AQ

mm

mE

f

fKIN

Typical value of kinetic energy is 5 MeV For example for 222Rn Q = 5587 MeV and EKIN α= 5486 MeV

Barrier penetration

Particle (ZA) impacts on nucleus (ZA) ndash necessity of potential barrier overcoming

For Coulomb barier is the highest point in the place where nuclear forces start to act R

ZeZ

4

1

)A(Ar

ZZ

4

1V

2

03131

0

2

0C

α

e

Barrier height is VCB asymp 25 MeV for nuclei with A=200

Problem of penetration of α particle from nucleus through potential barrier rarr it is possible only because of quantum physics

Assumptions of theory of α particle penetration

1) Alpha particle can exist at nuclei separately2) Particle is constantly moving and it is bonded at nucleus by potential barrier3) It exists some (relatively very small) probability of barrier penetration

Probability of decay λ per time unit λ = νP

where ν is number of impacts on barrier per time unit and P probability of barrier penetration

We assumed that α particle is oscillating along diameter of nucleus

212

0

2KI

2KIN 10

R2E

cE

Rm2

E

2R

v

N

Probability P = f(EKINαVCB) Quantum physics is needed for its derivation

Centrifugal barrier depends on angular momentum of emitted or incident particle

classical r

V

rm

LrmF l

3

22

2

2

l rm2

L(r)V

quantum L2 rarr l(l+1)ħ2 rarr2

2

l rm2

1)l(lV

bound statequasistationary state

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
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Radioactivity decay law

Activity (radioactivity) Adt

dNA

where N is number of nuclei at given time in sample [Bq = s-1 Ci =371010Bq]

Constant probability λ of decay of each nucleus per time unit is assumed Number dN of nuclei decayed per time dt

dN = -Nλdt dtN

dN

t

0

N

N

dtN

dN

0

Both sides are integrated

ln N ndash ln N0 = -λt t0NN e

Then for radioactivity we obtaint

0t

0 eAeNdt

dNA where A0 equiv -λN0

Probability of decay λ is named decay constant Time of decreasing from N to N2 is decay half-life T12 We introduce N = N02 21T

00 N

2

N e

2lnT 21

Mean lifetime τ 1

For t = τ activity decreases to 1e = 036788

Heisenberg uncertainty principle ΔEΔt asymp ħ rarr Γ τ asymp ħ where Γ is decay width of unstable state Γ = ħ τ = ħ λ

Total probability λ for more different alternative possibilities with decay constants λ1λ2λ3 hellip λM

M

1kk

M

1kk

Sequence of decay we have for decay series λ1N1 rarr λ2N2 rarr λ3N3 rarr hellip rarr λiNi rarr hellip rarr λMNM

Time change of Ni for isotope i in series dNidt = λi-1Ni-1 - λiNi

We solve system of differential equations and assume

t111

1CN et

22t

21221 CCN ee

hellipt

MMt

M1M2M1 CCN ee

For coefficients Cij it is valid i ne j ji

1ij1iij CC

Coefficients with i = j can be obtained from boundary conditions in time t = 0 Ni(0) = Ci1 + Ci2 + Ci3 + hellip + Cii

Special case for τ1 gtgt τ2τ3 hellip τM each following member has the same

number of decays per time unit as first Number of existed nuclei is inversely dependent on its λ rarr decay series is in radioactive equilibrium

Creation of radioactive nuclei with constant velocity ndash irradiation using reactor or accelerator Velocity of radioactive nuclei creation is P

Development of activity during homogenous irradiation

dNdt = - λN + P

Solution of equation (N0 = 0) λN(t) = A(t) = P(1 ndash e-λt)

It is efficient to irradiate number of half-lives but not so long time ndash saturation starts

Alpha decay

HeYX 42

4A2Z

AZ

High value of alpha particle binding energy rarr EKIN sufficient for escape from nucleus rarr

Relation between decay energy and kinetic energy of alpha particles

Decay energy Q = (mi ndash mf ndashmα)c2

Kinetic energies of nuclei after decay (nonrelativistic approximation)EKIN f = (12)mfvf

2 EKIN α = (12) mαvα2

v

m

mv

ff From momentum conservation law mfvf = mαvα rarr ( mf gtgt mα rarr vf ltlt vα)

From energy conservation law EKIN f + EKIN α = Q (12) mαvα2 + (12)mfvf

2 = Q

We modify equation and we introduce

Qm

mmE1

m

mvm

2

1vm

2

1v

m

mm

2

1

f

fKIN

f

22

2

ff

Kinetic energy of alpha particle QA

4AQ

mm

mE

f

fKIN

Typical value of kinetic energy is 5 MeV For example for 222Rn Q = 5587 MeV and EKIN α= 5486 MeV

Barrier penetration

Particle (ZA) impacts on nucleus (ZA) ndash necessity of potential barrier overcoming

For Coulomb barier is the highest point in the place where nuclear forces start to act R

ZeZ

4

1

)A(Ar

ZZ

4

1V

2

03131

0

2

0C

α

e

Barrier height is VCB asymp 25 MeV for nuclei with A=200

Problem of penetration of α particle from nucleus through potential barrier rarr it is possible only because of quantum physics

Assumptions of theory of α particle penetration

1) Alpha particle can exist at nuclei separately2) Particle is constantly moving and it is bonded at nucleus by potential barrier3) It exists some (relatively very small) probability of barrier penetration

Probability of decay λ per time unit λ = νP

where ν is number of impacts on barrier per time unit and P probability of barrier penetration

We assumed that α particle is oscillating along diameter of nucleus

212

0

2KI

2KIN 10

R2E

cE

Rm2

E

2R

v

N

Probability P = f(EKINαVCB) Quantum physics is needed for its derivation

Centrifugal barrier depends on angular momentum of emitted or incident particle

classical r

V

rm

LrmF l

3

22

2

2

l rm2

L(r)V

quantum L2 rarr l(l+1)ħ2 rarr2

2

l rm2

1)l(lV

bound statequasistationary state

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
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Total probability λ for more different alternative possibilities with decay constants λ1λ2λ3 hellip λM

M

1kk

M

1kk

Sequence of decay we have for decay series λ1N1 rarr λ2N2 rarr λ3N3 rarr hellip rarr λiNi rarr hellip rarr λMNM

Time change of Ni for isotope i in series dNidt = λi-1Ni-1 - λiNi

We solve system of differential equations and assume

t111

1CN et

22t

21221 CCN ee

hellipt

MMt

M1M2M1 CCN ee

For coefficients Cij it is valid i ne j ji

1ij1iij CC

Coefficients with i = j can be obtained from boundary conditions in time t = 0 Ni(0) = Ci1 + Ci2 + Ci3 + hellip + Cii

Special case for τ1 gtgt τ2τ3 hellip τM each following member has the same

number of decays per time unit as first Number of existed nuclei is inversely dependent on its λ rarr decay series is in radioactive equilibrium

Creation of radioactive nuclei with constant velocity ndash irradiation using reactor or accelerator Velocity of radioactive nuclei creation is P

Development of activity during homogenous irradiation

dNdt = - λN + P

Solution of equation (N0 = 0) λN(t) = A(t) = P(1 ndash e-λt)

It is efficient to irradiate number of half-lives but not so long time ndash saturation starts

Alpha decay

HeYX 42

4A2Z

AZ

High value of alpha particle binding energy rarr EKIN sufficient for escape from nucleus rarr

Relation between decay energy and kinetic energy of alpha particles

Decay energy Q = (mi ndash mf ndashmα)c2

Kinetic energies of nuclei after decay (nonrelativistic approximation)EKIN f = (12)mfvf

2 EKIN α = (12) mαvα2

v

m

mv

ff From momentum conservation law mfvf = mαvα rarr ( mf gtgt mα rarr vf ltlt vα)

From energy conservation law EKIN f + EKIN α = Q (12) mαvα2 + (12)mfvf

2 = Q

We modify equation and we introduce

Qm

mmE1

m

mvm

2

1vm

2

1v

m

mm

2

1

f

fKIN

f

22

2

ff

Kinetic energy of alpha particle QA

4AQ

mm

mE

f

fKIN

Typical value of kinetic energy is 5 MeV For example for 222Rn Q = 5587 MeV and EKIN α= 5486 MeV

Barrier penetration

Particle (ZA) impacts on nucleus (ZA) ndash necessity of potential barrier overcoming

For Coulomb barier is the highest point in the place where nuclear forces start to act R

ZeZ

4

1

)A(Ar

ZZ

4

1V

2

03131

0

2

0C

α

e

Barrier height is VCB asymp 25 MeV for nuclei with A=200

Problem of penetration of α particle from nucleus through potential barrier rarr it is possible only because of quantum physics

Assumptions of theory of α particle penetration

1) Alpha particle can exist at nuclei separately2) Particle is constantly moving and it is bonded at nucleus by potential barrier3) It exists some (relatively very small) probability of barrier penetration

Probability of decay λ per time unit λ = νP

where ν is number of impacts on barrier per time unit and P probability of barrier penetration

We assumed that α particle is oscillating along diameter of nucleus

212

0

2KI

2KIN 10

R2E

cE

Rm2

E

2R

v

N

Probability P = f(EKINαVCB) Quantum physics is needed for its derivation

Centrifugal barrier depends on angular momentum of emitted or incident particle

classical r

V

rm

LrmF l

3

22

2

2

l rm2

L(r)V

quantum L2 rarr l(l+1)ħ2 rarr2

2

l rm2

1)l(lV

bound statequasistationary state

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
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• Slide 12
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Alpha decay

HeYX 42

4A2Z

AZ

High value of alpha particle binding energy rarr EKIN sufficient for escape from nucleus rarr

Relation between decay energy and kinetic energy of alpha particles

Decay energy Q = (mi ndash mf ndashmα)c2

Kinetic energies of nuclei after decay (nonrelativistic approximation)EKIN f = (12)mfvf

2 EKIN α = (12) mαvα2

v

m

mv

ff From momentum conservation law mfvf = mαvα rarr ( mf gtgt mα rarr vf ltlt vα)

From energy conservation law EKIN f + EKIN α = Q (12) mαvα2 + (12)mfvf

2 = Q

We modify equation and we introduce

Qm

mmE1

m

mvm

2

1vm

2

1v

m

mm

2

1

f

fKIN

f

22

2

ff

Kinetic energy of alpha particle QA

4AQ

mm

mE

f

fKIN

Typical value of kinetic energy is 5 MeV For example for 222Rn Q = 5587 MeV and EKIN α= 5486 MeV

Barrier penetration

Particle (ZA) impacts on nucleus (ZA) ndash necessity of potential barrier overcoming

For Coulomb barier is the highest point in the place where nuclear forces start to act R

ZeZ

4

1

)A(Ar

ZZ

4

1V

2

03131

0

2

0C

α

e

Barrier height is VCB asymp 25 MeV for nuclei with A=200

Problem of penetration of α particle from nucleus through potential barrier rarr it is possible only because of quantum physics

Assumptions of theory of α particle penetration

1) Alpha particle can exist at nuclei separately2) Particle is constantly moving and it is bonded at nucleus by potential barrier3) It exists some (relatively very small) probability of barrier penetration

Probability of decay λ per time unit λ = νP

where ν is number of impacts on barrier per time unit and P probability of barrier penetration

We assumed that α particle is oscillating along diameter of nucleus

212

0

2KI

2KIN 10

R2E

cE

Rm2

E

2R

v

N

Probability P = f(EKINαVCB) Quantum physics is needed for its derivation

Centrifugal barrier depends on angular momentum of emitted or incident particle

classical r

V

rm

LrmF l

3

22

2

2

l rm2

L(r)V

quantum L2 rarr l(l+1)ħ2 rarr2

2

l rm2

1)l(lV

bound statequasistationary state

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

Barrier height is VCB asymp 25 MeV for nuclei with A=200

Problem of penetration of α particle from nucleus through potential barrier rarr it is possible only because of quantum physics

Assumptions of theory of α particle penetration

1) Alpha particle can exist at nuclei separately2) Particle is constantly moving and it is bonded at nucleus by potential barrier3) It exists some (relatively very small) probability of barrier penetration

Probability of decay λ per time unit λ = νP

where ν is number of impacts on barrier per time unit and P probability of barrier penetration

We assumed that α particle is oscillating along diameter of nucleus

212

0

2KI

2KIN 10

R2E

cE

Rm2

E

2R

v

N

Probability P = f(EKINαVCB) Quantum physics is needed for its derivation

Centrifugal barrier depends on angular momentum of emitted or incident particle

classical r

V

rm

LrmF l

3

22

2

2

l rm2

L(r)V

quantum L2 rarr l(l+1)ħ2 rarr2

2

l rm2

1)l(lV

bound statequasistationary state

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

Beta decay

Nuclei emits electrons

1) Continuous distribution of electron energy (discrete was assumed ndash discrete values of energy (masses) difference between mother and daughter nuclei) Maximal EEKIN = (Mi ndash Mf ndash me)c2

2) Angular momentum ndash spins of mother and daughter nuclei differ mostly by 0 or by 1 Spin of electron is but 12 rarr half-integral change

Schematic dependence Ne = f(Ee) at beta decay

rarr postulation of new particle existence ndash neutrino

mn gt mp + mν rarr spontaneous process

epn

neutron decay τ asymp 900 s (strong asymp 10-23 s elmg asymp 10-16 s) rarr decay is made by weak interaction

inverse process proceeds spontaneously only inside nucleus

Process of beta decay ndash creation of electron (positron) or capture of electron from atomic shell accompanied by antineutrino (neutrino) creation inside nucleus Z is changed by one A is not changed

Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
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• Slide 7
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• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

According to mass of atom with charge Z we obtain three cases

1) Mass is larger than mass of atom with charge Z+1 rarr electron decay ndash decay energy is split between electron a antineutrino neutron is transformed to proton

eYY A1Z

AZ

2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 ndash 2mec2 rarr electron capture ndash energy is split between neutrino energy and electron binding energy Proton is transformed to neutron YeY A

Z-A

1Z

3) Mass is smaller than mZ+1 ndash 2mec2 rarr positron decay ndash part of decay energy higher than 2mec2 is split between kinetic energies of neutrino and positron Proton changes to neutron

eYY A

ZA

1ZDiscrete lines on continuous spectrum

1) Auger electrons ndash vacancy after electron capture is filled by electron from outer electron shell and obtained energy is realized through Roumlntgen photon Its energy is only a few keV rarr it is very easy absorbed rarr complicated detection

2) Conversion electrons ndash direct transfer of energy of excited nucleus to electron from atom

Beta decay can goes on different levels of daughter nucleus not only on ground but also on excited Excited daughter nucleus then realized energy by gamma decay

Some mother nuclei can decay by two different ways either by electron decay or electron capture to two different nuclei

During studies of beta decay discovery of parity conservation violation in the processes connected with weak interaction was done

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 13
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• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

Neutrino ndash particle interacted only by weak interaction very small cross-section Detection by inverse beta decay

enp epn

Determination of neutrino mass from form of end of electron spectra

We can express function connected to dependency of number of electrons on their energy

)E(Ekonst

EZF

ENeMAX

e

e

where N(Ee) ndash number of electrons F(ZEe) ndash Fermi function containing of correction on Coulomb field of nucleus and electron cloud In the case on nonzero neutrino mass EMAX=Q - mνc2 (Q ndash decay energy) The graph of dependency is named as

Fermi graph ndash possibility of accurate determination of maximal energy (decay energy) ndash eventually neutrino mass The neutrino mass determined by tritium decay is 2 eV at present time

Fermi graph for decay of tritium 3HTests and transport of main vacuumchamber of KATRIN spectrometer Electron energy

Rel

ativ

e el

ectr

on in

ten

sity

Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
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Gamma decay

After alpha or beta decay rarr daughter nuclei at excited state rarr emission of gamma quantum rarr gamma decay

Eγ = hν = Ei - Ef

More accurate (inclusion of recoil energy)

Momenta conservation law rarr hνc = Mjv

Energy conservation law rarr 2

j

2jfi c

h

2M

1hvM

2

1hEE

Rfi2j

22

fi EEEcM2

hEEhE

where ΔER is recoil energy

Excited nucleus unloads energy by photon irradiation

Different transition multipolarities Electric EJ rarr spin I = min J parity π = (-1)I

Magnetic MJ rarr spin I = min J parity π = (-1)I+1

Multipole expansion and simple selective rules

Energy of emitted gamma quantum

Transition between levels with spins Ii and If and parities πi and πf

I = |Ii ndash If| pro Ii ne If I = 1 for Ii = If gt 0 π = (-1)I+K = πiπf K=0 for E and K=1 pro MElectromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

Moumlssbauer phenomena makes possible very accurate measurements of level energy and width We have

1) Source of gamma quanta2) Moving absorber3) Detector of gamma rays

Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE = E∙vc we can measure intensity of absorption rarr form of Moumlssbauer lines is visible

Level width Γ is connected with its live time by Heisenberg uncertainty principle Γτ ħ

And then Γ ħτ ~ uncertainty of (Ei ndash Ef)

Nucleus can be excited by the same energy Eγ which can emit Nucleus recoil must be included (recoil is created also during absorption)

Γ ge 2∙ ΔER

for possibility of resonance absorption This is right for free atom

Transition Eγ = 14 keV for isotope 57Fe For level τ ~ 10-7 s rarr Γ ~ 10-8 eV and ΔER ~ 10-3 eV rarr Γ ltlt ΔER

Atom bounded in crystal lattice rarr momentum is transferred to whole crystal lattice rarr little energy transfer rarr possibility of resonance absorption ndash Moumlssbauer phenomena

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 17
• Slide 18
• Slide 19
• Slide 20

Mean lifetimes of levels are mostly very short ( lt 10-7s ndash electromagnetic interaction is much stronger than weak) rarr life time of previous beta or alpha decays are longer rarr time behavior of gamma decay reproduces time behavior of previous decay

They exist longer and even very long life times of excited levels - isomer states

Probability (intensity) of transition between energy levels depends on spins and parities of initial and end states Usually transitions for which change of spin is smaller are more intensive

System of excited states transitions between them and their characteristics are shown by decay schema

Example of part of gamma ray spectra from source 169Yb rarr 169Tm

Decay schema of 169Yb rarr 169Tm

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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• Slide 20

Internal conversion

Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb interaction between nucleus and electrons) Energy of emitted electron Ee = Eγ ndash Be

where Eγ is excitation energy of nucleus Be is binding energy of electron

Alternative process to gamma emission Total transition probability λ is λ = λγ + λe

The conversion coefficients α are introduced It is valid dNedt = λeN and dNγdt = λγNand then NeNγ = λeλγ and λ = λγ (1 + α) where α = NeNγ

We label αK αL αM αN hellip conversion coefficients of corresponding electron shell K L M N hellip α = αK + αL + αM + αN + hellip

The conversion coefficients decrease with Eγ and increase with Z of nucleus

Transitions Ii = 0 rarr If = 0 only internal conversion not gamma transition

The place freed by electron emitted during internal conversion is filled by other electron and Roumlntgen ray is emitted with energy Eγ = Bef - Bei

characteristic Roumlntgen rays of correspondent shell

Energy released by filling of free place by electron can be again transferred directly to another electron and the Auger electron is emitted instead of Roumlntgen photon

Pair internal conversion ndash Eγ gt 2mec2 rarr pair electron and positron pair can be created rarr it is not connected to electron shell rarr probability increases with Eγ

Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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Nuclear fissionThe dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two nuclei (fragments) with masses in the range of half mass of mother nucleus

2AZ1

AZ

AZ YYX 2

2

1

1

Penetration of Coulomb barrier is for fragments more difficult than for alpha particles (Z1 Z2 gt Zα = 2) rarr the lightest nucleus with spontaneous fission is 232Th Example of fission of 236U

Energy released by fission Ef ge VC rarr spontaneous fission

31

2

310

22

0C A

ZC

2A2r

e2Z

4

1V

We assume A1=A2=A2 a Z1=Z2=Z2 Then magnitude of

Coulomb potential barrier is

For fission energy is valid Efc2 = m(ZA) ndash 2m(Z2A2)

After substitution from Weizsaumlker formula

Ef = (1-213) c2aSA23+(1-2-23) c2aCZ2A-13 = aSlsquoA23+aClsquoZ2A-13 = A23 (aSlsquo+aClsquoZ2A)

Ef c2 = aSA23(1-213)+aCZ2A-13(1-2-23) rarr

From this Ef gt 0 rarr Z2A gt -aSlsquoaClsquo~ 18

Ef ge VC 51aC

a

A

Z

C

S2

Ratio Z2A (fission parameter) is critical for stability against spontaneous fission

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
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• Slide 20

After supplial of energy ndash induced fission ndash energy supplied by photon (photofission) by neutron hellip

Induced fission can be described by comparison of surface and Coulomb energy of symmetrical sphere and deformed ellipsoid with half-axe a and b with the same volume V ELIPSOID

23KOULE Vab

3

4R

3

4V

The same volume rarr the same volume energy for sphere and ellipsoid We express a = R(1- ε) and b = R(1-ε)-12

where ε is ellipsoid eccentricity Ellipsoid surface is )5

21(R4S 22

ELIPSOID

Surface energy in Weizsaumlker formula then is )5

2(1AaE 232

SS

Coulomb energy of charged ellipsoid )5

11(

R

eZ

4

1

5

3E 2

22

0C

Then from Weizsaumlker formula )5

1(1AZaE 2312

CC

Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0)

5

1AZa

5

2AaEEE 2312

C232

SCSD

After substitution of constants from Weizsaumlker formula

ED = ε2(734∙A23 ndash014∙Z2A-13) = ε2∙A23 (734ndash014∙ Z2A) [MeV]

Z2A ge 51 rarr ED le 0 rarr spontaneous fission

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

Energy Ea needed for overcoming of potential barrier ndash activation energy ndash for heavy nuclei is small ( ~ MeV) rarr energy released by neutron capture is enough (high for nuclei with odd N)

Certain number of neutrons is released after neutron capture during each fission (nuclei with average A have relatively smaller neutron excess than nucley with large A) rarr further fissions are induced rarr chain reaction

235U + n rarr 236U rarr fission rarr Y1 + Y2 + ν∙n

Average number η of neutrons emitted during one act of fission is important - 236U (ν = 247) or per one neutron capture for 235U (η = 208) (only 85 of 236U makes fission 15 makes gamma decay)

How many of created neutrons produced further fission depends on arrangement of setup with fission material

Ratio between neutron numbers in n and n+1 generations of fission is named as multiplication factor kIts magnitude is split to three cases

k lt 1 ndash subcritical ndash without external neutron source reactions stop rarr accelerator driven transmutors ndash external neutron sourcek = 1 ndash critical ndash can take place controlled chain reaction rarr nuclear reactorsk gt 1 ndash supercritical ndash uncontrolled (runaway) chain reaction rarr nuclear bombs

Fission products of uranium 235U Dependency of their production on mass number A (taken from A Beiser Concepts of modern physics)

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20

Decay series

Different radioactive isotope were produced during elements synthesis (before 5 ndash 7 miliards years)

Some survive 40K 87Rb 144Nd 147Hf The heaviest from them 232Th 235U a 238U

Beta decay A is not changed Alpha decay A rarr A - 4

Summary of decay series A Series Mother nucleus

T 12 [years]

4n Thorium 232Th 1391010

4n + 1 Neptunium 237Np 214106

4n + 2 Uranium 238U 451109

4n + 3 Actinium 235U 71108

Decay life time of mother nucleus of neptunium series is shorter than time of Earth existenceAlso all furthers rarr must be produced artificially rarr with lower A by neutron bombarding with higher A by heavy ion bombarding

Some isotopes in decay series must decay by alpha as well as beta decays rarr branching

Possibilities of radioactive element usage 1) Dating (archeology geology)2) Medicine application (diagnostic ndash radioisotope cancer irradiation)3) Measurement of tracer element contents (activation analysis)4) Defectology Roumlntgen devices

Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

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Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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Exotic forms of decay

Proton emission ndash protons must penetrate Coulomb barrier rarr life time (also in μs and ms range) is longer than characteristic nuclear time (time of nucleon transit through nucleus ndash 10 -

21s) rarr existence of proton radioactivity It is possible only for exotic light nuclei with large excess of protons (for example 9B) ndash decay has sufficiently short decay time and hence it is not suppressed by competitive positron beta decay

Emission of couple of protons ndash made by coupling (maybe also in form of 2He) - year 2000 at Oak Ridge laboratory for nucleus 18Ne

Delayed proton emission ndash emission of protons following after proton decay rarr nuclei with large excess of protons rarr created nucleus at highly excited state emits proton

Neutron emission ndash life time of nuclei with big neutron excess if neutron decay is energy possible is comparable with characteristic nuclear time ndash it can not be named as neutron radioactivity

Delayed neutron emission following after beta decay Nucleus with big neutron excess rarr beta decay with longer decay time rarr consecutive quick neutron emission during time comparable with characteristic nuclear time

Emission of hevier nuclei ndash 12C 16O hellip rarr fragmentation of highly excited nuclei

We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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We have potentially 35 (ββ 2ν) ndash emitters Nine was observed up to now (48Ca 76Ge 82Se 100Mo hellip) Very long decay time T12 = 1019 ndash 1024 years

Studied using underground experiments (main problem is background) For example new device NEMO-3 (10 kg of 100Mo Qββ = 3038 MeV) Next possibility ndash geochemical measurements

Double beta decay (ββ2ν) ndash is possible if double decay is allowed by energy conservation and single beta decay is not allowed

e22eYX A2Z

AZ

Examples of nuclei decayed by double beta decay

Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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Device NEMO-3 for double beta studies Device NEMO-3 in underground tunnel at Alps

Neutrinoless double beta decay (ββ0ν) ndash possible only in the case of nonzero neutrino rest mass and if neutrino is Majorana particle type (antiparticle is identical with particle ndash difference of lepton number is still there) In this case two neutrons can change neutrino and antineutrino in the process violated lepton number conservation and only pair of electrons is emitted This decay was not observed up to now Limit is in the range of 1025 years measured with 76Ge rarr limit on mass ~ 045 eV

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