One of the most practical applications of nuclear reactions occurs with the compound nucleus resulting from A>230 nuclei absorbing neutrons.

Often split into two medium mass nuclear fragments plus additional neutrons.

Alpha particle energy

Cro

ss S

ectio

n

,n

,2n

,3n,4n

Total

NUCLEAR FISSION

We’ve noted a collision reaction that produces free neutrons:

1930 Bothe & BeckerStudying -rays bombarding berylliumproduced a very penetrating non-ionizing form of radiation

-rays?

Irène and Frédéric Joliot-Curie

knocked protons free from paraffin targetsthe proton energy range revealed the uncharged radiation from Be to carry 5.3 MeV

1932 James Chadwickin discussions with Rutherford

became convinced could not be s

since assuming Compton Scattering to be the mechanism, E>52 MeV!

Neutron chamber

ionization(cloud)

chamber

Replacing the paraffin with other light substances, even beryllium, the protons were still produced.

Nature, February 27, 1932

Chadwick developed the theory explaining the phenomena as due to a 5.3 MeV neutral particle

with mass identical to the protonundergoing head-on collisions with nucleons in the target.

1935 Nobel Prize in Physics

9Be has a loosely bound neutron (1.7 MeV binding energy)

above a closed shell:

nCBeHe 1294

5-6 MeV from someother decay

Q=5.7 MeV

Neutrons produced by many nuclear reactions (but can’t be steered, focused or accelerated!)

Natural sources of neutrons

Mixtures of 226Ra ( source) and 9Be ~constant rate of neutron production

also strong source

so often replaced by 210Po, 230Pu or 241Am

Spontaneous fission, e.g. 252Cf ( ½ = 2.65 yr)

only 3% of its decays are through fission 97% -decaysYield is still 2.31012 neutrons/gramsec !

PdU 11946

23892 2

A possible (and observed) spontaneous fission reaction

238U119Pd

8.5 MeV/A

7.5 MeV/A

Gains ~1 MeV per nucleon!2119 MeV = 238 MeV

released by splitting

PdU 11946

23892 2

Yet

is a rare decay: ½ = 1016 yr

not as probable as the much more common -decay ½ = 4.510

9 yr

Atomic (chemical) processes ~few eV

Fission involves 108 as much energy as chemical reactions!

From the curve of binding energy per nucleon the most stable form of nuclear matter is as medium mass nuclei.

),(),(),( 2211 ZAMZAMZAM Consider:

),(),(),( 2211 ZABZABZABQ The Q value (energy release) of this process is

The mass differences cancel since the total number of constituents remains unchanged.

For simplicity, if we assume the protons and neutrons divide in the same ratio as the total nucleons:

111 // yZZAA

222 // yZZAA

121 yy

The difference in binding energy comes from the surface and coulomb terms

so the energy released can then be expressed in terms of the surface energy Es and the coulomb energy Ec

of the original nucleus (A,Z).

3/2AaEss

3/1)1( AZZaEcc

3/22

3/21

3/2 ][][ AyaAyaAasss

]1[ 3/22

3/21

3/2 yyAas

3/1222

3/1111

3/1 ])[1(])[1()1( AyZyZyaAyZyZyaAZZaccc

]1[)1( 3/1222

3/1111

3/1 yyyyyyAZZac

]1[ 3/52

3/51

yyEc

))()(1())()(1( 3/52

3/51

3/22

3/21 yyEyyEQ CS

Expressing the energy released in terms of the surface energy Es and the coulomb energy Ec

of the original nucleus (A,Z).

maximum Q is found by setting dQ/dy1 = 0

121 yyNote: 1/ 12 dydy

maximum occurs when y1 = y2 = 1/2.

SC EEQ 26.037.0

0)1()1( 3/223

53/213

53/123

23/113

2

1

yEyEyEyEdydQ

CCSS

])1([])1([ 3/21

3/213

53/11

3/113

2 yyEyyECS

Fission into two equal nuclei (symmetric fission) produces the largest energy output or Q value

The process is exothermic (Q > 0) if Ec/Es > 0.7.

in terms of the fission parameter, x

50)/()/2)(/()2/(

22 AZaaAZEEx CSSC >0.35

Suggesting all nuclei with (Z2/A) > 18 (i.e. heavier than 90Zr) should spontaneously release energy

by undergoing symmetric fission.

However

Half-life of spontaneous fission as a function of x

where

criticalAZAZx

)/(/

2

2

and

49)/( 2 criticalAZ

R.Vandenbosch and

J.R.Huizenga.Nuclear Fusion,Academic Press,New York, 1973.

There isa competition between

the nuclear force binding the nucleus together

and the coulomb repulsion

trying to tear it apart

Induced fission as nuclear reaction

nBrLaUUn 29535

13957

23692

23592

nCsRbUUn 214155

9337

23692

23592

suggests the absorption of the neutron (and its energy)may induce such distortions/vibrations in the nucleus.

The surface if any arbitrary figure can be expanded as

00 ]),(1[

l

l

lm

mlm lYRR

If lm time-independent: permanent deformation of the nucleusIf lm time-dependent: an oscillation of the nucleus

The Spherical Harmonics Y ,ℓ m(,)

ℓ = 0

ℓ = 1

ℓ = 2

ℓ = 34

100 Y

ieY sin

83

11

cos43

10

Y

ieY 2

2sin215

41

22

ieY cossin

815

21

212cos

23

415

20

Y

ieY 3

3sin435

41

33

ieY 2

cos2sin2105

41

32

ieY 12cos5sin

421

41

31

cos

233cos

25

47

30Y

ℓ = 0

ℓ = 1

sin1~R

cos1~R

z Nuclear Charge Density

ℓ = 2Lowest order to be considered:

quadrupole deformation

For which we write the nuclear radius

]),(1[2

2220

m

mmYRR

The l=2, m=0 mode:

]1[)( )12

cos3(

2/1

16

5200

RtR

]1[)( )12

cos3(

2/1

16

5200

RtR

Z

Example of a vibrational spectrum (levels denoted by the number of phonons, N)O.Nathan and S.G.Nilsson, Alpha- Beta- and Gamma-Ray Spectroscopy,

Vol.1, (K. Siegbahn, ed.) North Holland, Amsterdam, 1965.

Nuclei do show spectra for such vibrational modes

We can approximate any small elongation from a spherical shape by

)1(21

0 Rb

)1(0 Ra

3/12123/2 )( AZaANZaAaAaB CsymSV

The semi-empirical mass formula

3/42

2

23/1 )(

AZa

ANZaAaa

AB

CsymSV

semi-major axis

semi-minor axis

)1( 23/152 AaE SS

)1( 23/1251 AZaE CC

From which:

ee

eba

11ln

22 2

)/(1 22 abe

surface of spheroid

)()( 23/223/1252

51 AaAZa

EEE

SC

SC

With the surface energy (strong nuclear binding force) proportional to area

2Ewhich we can write in the form

where ][ 3/23/12 2 51 AaAZa SC

Notice > 0 (so the Coulomb force wins out) for:

.492 2

C

S

aa

AZ Same fission parameter

introduced when estimating available Qin symmetric fission

Coulomb force deforming nucleus

surface tension holding spherical

shape

2E

r

comes from considering small perturbations from a sphere.

As long as these disturbances are slight, the Separation, r, of distinct fragments linearly follows

2r

for small r

2

04

)(

RrQrV

separation r

V(r

)

At zero separation the potential just equals the release energy Q

For Z2/A<49, is negative.

r

for small r reZZrV

221)(

separation r

V(r

)

r r

While for large r, after the fragments have been scissioned

for large r

For such quadrupole distortions the figure shows the energy of

deformation (as a factorof the original sphere’s

surface energy Es)plotted against

for different values of the fission parameter x.

When x > 1 (Z2/A>49)

the nuclei are completely unstable to such distortions.

The potential energy V(r) = constant-B as a function of the separation, r, between fragments.

Z2/A=49

Z2/A=36

such unstable statesdecay in characteristicnuclear times ~10-22 sec

Tunneling does allow spontaneous fission, but it must compete with

other decay mechanisms (-decay)

No stable stateswith Z2/A>49!

Tunnelingprobabilitydrops as

Z2/A drops(half-life

increases).

At smaller values of x, fission by barrier penetration can occur, However recall that the transmission factor (e.g., for -decay) is

eXwhere

drh

ErVm ])([22 m

while for particles (m~4u)this gave reasonable, observable probabilities for tunneling/decay

for the masses of the nuclear fragments we’re talking about, can become huge and X negligible.

nBrLaUUn 2* 9535

13957

23692

23592

nRbCsUUn 2* 9337

14155

23692

23592

Neutron absorption by heavy nuclei can create a compound nucleus in an excited state

above the activation energy barrier.As we have seen, compound nuclei have many final states into which they can decay:

nYXUUn AZ

AZ 2

211

23692

23592 *

where Z1+Z2=92, A1+A2+=236

...in general:

Experimentally find the average A1/A2 peaks at 3/2

PROMPTNEUTRONS

nSrXeUUn 2* 9538

13954

23692

23592

The incident neutron itself need not be of high energy.

Thermal neutrons E< 1 eVSlow neutrons E ~ 1 keVFast neutrons E ~ 100 keV – 10 MeV

Typicalof decayProducts& nuclearreactions

“Thermal neutrons” (slowed by interactions with any material they pass through) have been demonstrated to be particularly effective.

This merely reflects the general ~1/v behavior we have noted for all cross sections!

Cro

ss se

ctio

n

incident particle velocity, v

At such low excitation there may be barely enough available energy to drive the two fragments of the nucleus apart.

Thus the individual nucleons settle into the lowest possible energy configurations

Division can only proceed if as much binding energy as possible

is transformed into the kinetic energy separating them out.

involving the most tightly bound final states.

(so MOST of the available Q goes into the kinetic energy of the fragments!)

There is a strong tendency to produce a heavy fragment of A ~ 140 (with double magic numbers N = 82 and Z = 50).

PdU 11946

23892 2

A possible (and observed) spontaneous fission reaction

238U119Pd

8.5 MeV/A

7.5 MeV/A

Gains ~1 MeV per nucleon!2119 MeV = 238 MeV

released by splitting

238 MeV represented an estimate of the maximum available energyfor symmetric fission.

For the observed distribution

of final statesthe typical average is

~200 MeV per fission.

Fragment kinetic energy 165 MeVPrompt neutrons 5 MeVPrompt gamma rays 7 MeVRadioactive decay fragments 25 MeV

This 200 MeV is distributed approximately as:

235U

Isobars off the valley of stability (dark squares on preceding slide)-decay to a more stable state.

and decays can leave a daughter in an excited nuclear state

187W 1/2

5/2

187Re

0.13425

0.20625

0.618900.68610

198Au 2

0198Hg

0.412 MeV

1.088 MeV

nKrBaUUn 3* 9036

14356

23692

23592

With the fission fragments radioactive, a decay sequence to stable nuclei must follow

14357

14356 eLaBa

neZrNdUUn 388* 9040

14360

23692

23592

eCe 14358

14359 ePr

14359 edN

9037

9036 eRbKr

eSr 9038

9039 eY

9040 eZr

nRbCsUUn 2* 9337

14155

23692

23592

With the fission fragments radioactive, a decay sequence to stable nuclei must follow

Pr 14159

14158

14157

14156

14155 CeLaBaCs

CeLaBaCs 14058

14057

14056

14055 n

0.03%

25 sec

18 min

4 hr

33 d

65 sec

13 d

40 h

NbZrYSrRb 9341

9340

9339

9338

9337

ZrYSrRb 9240

9239

9238

9237 n

1.40%

6 sec

7 min

10 hr

106 yr

5 sec

3 hr

4 h

nePrCeUUn 2888* 14159

14058

23692

23592

n3 n4sometimes or

nBrLaUUn 2* 9535

13957

23692

23592

nRbCsUUn 2* 9337

14155

23692

23592

nSrXeUUn 2* 9538

13954

23692

23592

nKrCsUUn 3* 9036

14356

23692

23592

For 235U fission, average number of prompt neutrons ~ 2.5

with a small number of additional delayed neutrons.

with every neutron freed comes the possibility of additional fission events

This avalanche is the chain reaction.

235U will fission (n,f) at all energies of the absorbed neutron.

It is a FISSILE material.

However such a reaction cannot occur in natural uranium (0.7% 235U, 99.3% 238U)

Total (t) and fission (f) cross sections of 235U.

1 b = 10-24 cm2