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