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Nuclear reactions
1) Introduction
2) Nuclear reaction yield
3) Conservation laws
4) Nuclear reaction mechanism and models
5) Elastic scattering
6) The principle of detailed balance
7) Compound nucleus reactions
8) Resonances
9) Optical model
10) Direct reactions
Fission of 252Cf nucleus
(taken from WWW pages of
group studying fission at LBL)
Introduction
Incident particle a collides with a target nucleus A → different processes:
1) Elastic scattering – (n,n), (p,p), …
2) Inelastic scattering – (n,n‘), (p,p‘), …
3) Nuclear reactions:
a) creation of new nucleus and
particle - A(a,b)B
b) creation of new nucleus and
more particles - A(a,b1b2b3…)B
c) nuclear fission – (n,f)
d) nuclear spallation
from point of view of used projectile:
e) photonuclear reactions - (γ,n), (γ,α), …
f) radiative capture – (n, γ), (p, γ), …
g) reactions with neutrons – (n,p), (n, α) …
h) reactions with protons – (p,α), …
i) reactions with deuterons – (d,t), (d,p), (d,n) …
j) reactions with alpha particles – (α,n), (α,p), …
k) heavy ion reactions
Reaction can be described in the form A(a,b)B, for example:
27Al(n,α)24Na
or 27Al + n → 24Na + α
input channel - particles (nuclei) enter into reaction and their characteristics (energies,
momenta, spins, …)
output channel – particles (nuclei) get off reaction and their characteristics
Cross section σ depends on energies, momenta, spins, charges … of involved particles
Dependency of cross section on energy σ (E) – excitation function.
Threshold reactions – occur only for energy higher than some value.
Reaction yield – number of reactions divided by number of incident particles.
Thin target – does not changed intensity and energy of beam particles
Thick target – intensity and energy of beam particles are changed
Nuclear reaction yield
Reaction yield – number of reactions ΔN divided by number of incident particles N0: w = ΔN /N0
Depends on specific target
Thin target – does not changed intensity and energy of beam particles → reaction yield:
w = ΔN /N0 = σnx
where n – number of target nuclei in volume unit, x is target thickness → nx is surface target
density.
Thick target – intensity and energy of beam particles are changed. Process depends on type of
particles:
1) Reactions with charged particles – energy losses by ionization and excitation of target atoms.
Reactions occur for different energies of incident particles. Number of particle is changed by
nuclear reactions (can be neglected for some cases). Thick target (thickness d > range R):
dN = N(x)nσ(x)dx ≈ N0nσ(x)dx
(reaction with nuclei are neglected N(x) ≈ N0)
R
Reaction yield is (d > R):
ΔN
w
 n   (x)dx  n
N0
0
E KINa

0
 (E KIN )
dE
 KIN
dx
dE KIN
Higher energies of incident particle and smaller ionization losses → higher range and yield
w=w(EKIN) – excitation function
R
Mean cross section:
1
    (x)dx
R0
→
w  n  R
2) Neutron reactions – no interaction with atomic shell, only scattering and absorption on nuclei.
Number of neutrons is decreasing but their energy is not changed significantly. Beam of
monoenergy neutrons with yield intensity N0. Number of reactions dN in a target layer dx for
deepness x is:
dN = -N(x)nσdx
where N(x) is intensity of neutron yield in place x and σ is total cross section σ = σpr + σnepr + σabs + …
We integrate equation: N(x) = N0e-nσx
for
0≤x≤d
Number of interacting neutrons from N0 in target with thickness d is: ΔN = N0(1 – e-nσd)
N  R  R
σ – total cross section


(1  e  n d )
Reaction yield is: w 
N0 

σR – cross section of given reaction
N  R
w

 n R d
For thin target nσd << 1 and yield is:
N0 
1
N(d)
Total cross section can be determined by transmission method
   ln
→ attenuation measurement:
nd
N0
3) Photon reactions – photons interact with nuclei and electrons → scattering and absorption →
decreasing of photon yield intensity:
I(x) = I0e-μx
where μ is linear attenuation coefficient (μ = μan, where μa is atomic attenuation coefficient and n is
number of target atoms in volume unit).
For thin target (attenuation can be neglected) reaction yield is:
where ΔI is total number of reactions and from this
reactions.
We obtain or thick target with thickness d:
w
I

a
w
I 

 n  d
I0 a
is number of studied photonuclear
I 



(1  e  a nd )
I0 a a
Conservation laws
Energy conservation law and momenta conservation law:
Described in the part about kinematics. Directions of fly out and possible energies of reaction
products can be determined by these laws. Vector momentum diagram can be again used for
determination of possible directions of reaction products fly out. Diagram is not dependent on
reaction type and it is valid only in the case of nonrelativistic approximation.
Type of interaction must be known for determination of angular distribution.
Angular momentum conservation law – orbital angular momentum given by relative motion of
two particles can have only discrete values l = 0, 1, 2, 3, … [ħ]. → For low energies and short range
of forces → reaction possible only for limited small number l. Semiclasical (orbital angular
momentum is product of momentum and impact parameter): pb = lħ → l ≤ pb /ħ = 2πR/ λ
max
where λ is de Broglie wave length of particle and R is interaction range. Accurate quantum
mechanic analysis → reaction is possible also for higher orbital momentum l, but cross section
rapidly decreases. Total cross section can be split:    l
l
Charge conservation law – sum of electric charges before reaction and after it are conserved.
Baryon number conservation law – for low energy (E < mnc2) → nucleon number conservation
law
Parity conservation law – parity of initial state is not changed during reaction. Because during
change of relative orbital angular momentum by Δl, initial parity Πi is changed into Πf = (-1)ΔlΠi →
for example, change of orbital angular momentum by Δl = odd is not possible during elastic
scattering, even if change is allowed from point of view of conservation of angular momentum in
the case of spin orientation change.
Mechanisms and models of nuclear reactions
Different reaction mechanism:
1) Direct reactions (also elastic and inelastic scattering) - reactions insistent very briefly τ ≈ 10-22s →
wide levels, slow changes of σ with projectile energy
2) Reactions through compound nucleus – nucleus with lifetime τ ≈ 10-16s is created → narrow levels
→ sharp changes of σ with projectile energy (resonance character), decay to different channels
Models of reactions are created for reaction description, which describe different reaction types.
Mean nuclear potential is created by nucleons of target nucleus. Projectile fly into nucleus → it is in
mean field influence → mean field can be changed by projectile energy.
Necessity of inclusion of electromagnetic interaction and Coulomb field influence – photonuclear
and electronuclear reactions, reactions of Coulomb excitation. Electromagnetic part of interaction
can be calculated accurately.
Optical model – nucleus is continuous environment – refracts and absorbs de Broglie waves
connected with incident particle
Statistical model – in reactions through compound nucleus many intermediate states → large
number of degree of freedom → we are working only with mean values of quantities.
Cascade models – high (relativistic) energies → short wave length of nucleons → nucleons are
localized → reaction (spallation) as sequence of single nucleon collisions.
Nuclear reaction is described fully – we know σ for measurable parameters (energies, angles,
particle types …). Direct reaction models are near to this, can not be for statistical model.
Elastic scattering – angular distribution of particles
We study scattering produced by nuclear forces.
Assumptions:
1) We have local central potential → forces acts in the direction to force centre
2) Potential has short range (decreasing is faster than 1/r)
3) Beam of particles is moving in the direction of z axe.
Simplification for limiting case: Accurately defined energy → indeterminate time → accurately
defined momentum → from uncertainty relation large scale in direction
 of z axe → process is
practically stationary. We have described particle with momentum p :

1 2

p    k where k  
 
Plane waves impact on scattering center and stationary spherical waves fly out from it.
Incidence wave function is in the form of plane wave and it can be modified for our station case:
 ~e

i k r t
Wave moving in opposite direction:
 ~e
 ~ eikr ~ eikz

-ik r

~ e-ikz
Fly out (scattered) spherical wave is described by function:
 ~e

ik r
r
(part 1/r causes decrease of density 1/r2 → number of particles is conserved). The same sign
of exponent for incidence and fly out waves. Amplitude of fly out spherical waves depends generally
only on angle  (axial symmetry is valid) → we append amplitude factor f(). Total wave function is
sum of incidence plane waves and fly out spherical waves:

  A[e ikz  f   eikr r]
Relation between amplitude factor
and cross section:
Particle density is: P = ψ*ψ
Flow density j of incident particles with
velocity vd: jd = vd∙P
Incidence
wave
For incidence wave: P = |Aeikz|2 = A2
and then: jd
= A2v
Scattered
wave
Axe Z
d
Flow density of fly out spherical waves is
labeled as jv. Particle flow dI coming
through area dS is then:
dI = jvdS = vv|ψv|2dS = vv|Af()eikr/r|2dS
= vvA2|f()|2dS/r2 [s-1]
It is valid for area dS: dS = r2dΩ and then: dI = vvA2|f()|2dΩ
Differential cross section is obtained using division by flow density of incident particles (for elastic
scattering vd = vv):
dσ = dI/jd = |f()|2dΩ
and then
2
 d 

  f  
 d 
Amplitude f() must be calculated by Schrődinger equation and we obtain cross section from
given equation, which can be compared with experiment.
The principle of detailed balance
Low energy reactions → energy of interaction Hint << energy of whole system → we can use for
determination of transition probability Pif from state φi to state φf Fermi´s golden role of
perturbation theory:
2
2 d
Pif 

H fi
dE 0
H fi   f H int i    *f H inti dV
where Hfi is transition matrix element:
In volume V number d of states (elementary cells with single particle with momentum p  p+Δp ) is:
d
1 4  Vp 2dp

and then:
dE 0 dE 0 2   3
V  4  p 2 dp 4  Vp 2 dp
d 

h3
2   3
we further discuss reaction A(a,b)B in the centre of mass system:
It is valid for final state:
If dE0 = dEb +dEB:


p b  p B
→ only one independent momentum (we choose pb).
4  Vp 2b dp b
d
1

dE 0 dE b  dE B 2   3
We substitute dE=(p/m)dp: dE b  dE B 
 1
pb
p
1 
1
p b dp b 
dp b  B dp B  

p b dp b
mb
mB
m
m
m
B 
f
 b
where mf is reduced mass of final state.
Then we obtain:
d
4  V

mf pb
dE 0 2   3
If particle (fermion) have spin I, against Pauli principle it can be 2I+1 particles in every state. It is
valid for both reaction products:
d
4  V
dE 0

2   3
(2I b  1)(2I B  1)m f p b
We substitute to expression for probability:
4  V
4  V
2
(2I b  1)(2I B  1)m f p b 
(2I b  1)(2I B  1) H fi m f p b
3
2 4
2   
2  
Pif  Pif
 d 


 
Relation between differential cross section and transition probability:
j
j
 d 
Pif 
2
H fi

2
where (Pif)=(1/4)Pif is probability per solid angle unit. Flow density of incident particles:
j = Nvi
where vi is velocity of incident particles and N is their number per volume unit. We normalize it on
single incident particle:
N=1/V → j=vi/V
Then
PV
Vm i
 d 

Pif

  if
d

4


v
4


p


i
i
where mi is initial reduced mass (nucleus is in the rest and then ví is relative velocity). We
substitute to Pif:
2I b  12I B  1 H
V 2 2I b  12I B  1
p
2
 d 
H fi mi m f f 

 
fi
2 4
pi
2  
2 2  4
 d 
2
mi m f
norm
pf
pi
where member V2 was multiplied by factor 1/V2, which appears before member |Hfi| in the case of
normalization of wave functions by factor 1/√V. Angular dependency is fully given by |Hfi|.
We derive similar equation for inverse process. If:
we calculate ratio of both cross sections:
|Hif|2 = |Hfi|2
 if 2I b  12I B  1p f2

 f i 2I a  12I A  1pi2
This relation is named as principle of detailed balance of nuclear reaction.
If |Hfi|2 is constant in small energy range, we obtain:
  konst
pf
pi
Let us discuss different reaction types:
a) Elastic scattering of neutral particles → va = vb → σ = const → independent on velocity va
b) Exotermic reactions excited by thermal neutrons → Q ≈ 1 MeV and neutron energies
Ea ≈ 1eV → vb = const → σ = const/va. It is valid only for neutral fly out particles. Penetration
factors of type of Gamow factor are in |Hfi|2 in the case of charged particles H 2 ~ e {G  G )
fi
c) Exothermic reactions with charged particles –
dependency on factor exp(-Ga) predominates.
d) Inelastic neutron scattering – endothermic, vb
strongly depends on energy → above threshold va
≈ const. Product energy is given by excess of
energy above the threshold Eb ≈ Ea - Es →
v b ~ p b  2m b E a  E S 
→

E a  ES 
e) Endothermic reaction with charged particle
production – member exp(-Ga) predominates
a
b
Reaction through compound nucleus
Reactions during which projectile energy is distributed to more nucleons of target nucleus → excited
compound nucleus is created → energy cumulating → single or more nucleons fly out.
Compound nucleus decay 10-16s.
Different excited levels of compound nucleus – level live time connected with their width by
Heisenberg uncertainty principle
Γτ ≈ h
Division of reactions through compound nucleus:
1) Resonance – level distance ΔE >>Γ → σ(E) resonance character
2) Nonresonance - ΔE << Γ → σ(E) nonresonance character – statistic way of description
Possible interpretation of reaction through compound nucleus in the frame of drop model:
excited compound nucleus
– heating water droplet
energy decreasing by nucleon escape – cooling by molecule evaporating → evaporation models
Two independent processes: Compound nucleus creation
Compound nucleus decay
Cross section σab reaction from incident channel a and final b through compound nucleus C:
σab = σaCPb
where σaC is cross section
for compound nucleus creation and Pb is probability of compound nucleus decay to channel b.
Sum through all final channels:
P
b
1
b
Partial level width Γb – width against decay to channel b:
Relation between Γb and Pb: Pb=Γb/Γ where
   b
b
Resonances
Element of transition matrix |Hfi|2 and thus also cross section σab do not change only slowly.
Reactions proceed through compound nucleus → expect slow changes presence of fluctuations resonance structures in the behavior of |Hfi|2 and σab
Resonance are given by reactions through compound nucleus:
a + A → C* → b + B
(reaction a + A → C* → γ + C is also shown)
Resonance maximum in behavior of cross section in the place of
isolated (separated from other levels) level Eres. We can derive using
quantum mechanics, that shape of resonance can be described by
Breit-Wigner formule:
a  b
π
 ab  2
k a E  E 2  1  2
res
4
Example of resonance character of spectra of reaction through
compound nucleus (typical example of reaction with slows
neutrons)
Cross section [barn]
For range about 1 – 20 MeV resonances are densely nearly and they
are broad → they can not be distinguished → continuum is created
(statistical range)
Energy [eV]
Sum through all final channels (also elastic scattering) → total cross section of compound
nucleus creation:
a  
π
 aC  2
k a E  E 2  1  2
res
4
a  b
a 
b
b
π






It is valid:  ab  2
aC
2
2
k a E  E res   1  2 k a E  E 2  1  

4
res
4
Thus independency of creation and decay of compound nucleus.
For E = Eres it is valid (we assume elastic σaa and one inelastic σab channel → Γ = Γa + Γb):
k 
2
a
2
Maximum for elastic part (Γb = 0, Γa = Γ):
Maximum for inelastic part (Γb = Γa = Γ/2):
 ab  4
 b a
k a2  2
 aamax  4
 abmax 

k a2

k a2
Resonance fast changes are given by reactions through
compound nucleus, slow changes are given by direct
reactions
Cross section [barn]
 aa  4
 a2
Energy [eV]
Optical model
Reflection maxima in impact direction are seen in rough averaged excitation function → potential
scattering. Expect potential scattering, absorption of incident particle (creation of compound
nucleus) must be described.
It can be described by optical model:
Assumption: nucleus is continuous environment , which reflects and absorbs de Broglie waves of
incident particles.
Limit case is black body model → nucleus absorbs all incident particles
Simplification: reaction of incident particle with nucleus is approximated by scattering and
absorption of particle by force centre
Problem of A1 + A2 particles → two particle problem
We search form of mean field (optical potential) U(r) produced by force center, which after
substitution to Schrődinger equation and fulfillment of boundary conditions gives directly mean
value of scattering amplitude.
Optical potential is involved as empirical potential. Choice of parameters → calculations of
differential cross section → comparison with experimental angular distribution.
Presence of absorption → complex part → U(r) = V(r) + iW(r)
Real part V(r) has shape of shell model potential (most often Woods-Saxon form with inclusion
of spin-orbital interaction)
Imaginary part: Low energies → predominance of absorption on surface
Higher energy ( ≥ 80 MeV) → predominance of absorption in volume
Influence of Coulomb potential and centrifugal potential can be included during
calculations of particular processes
Direct reactions
Direct reactions (also elastic and inelastic scattering) - reactions continuing very short 10-22s
Stripping reactions – target nucleus takes away one or more nucleons from projectile, rest of
projectile flies further without significant change of momentum - (d,p) reactions.
Pickup reactions – extracting of nucleons from nucleus by projectile
Transfer reactions – generally transfer of nucleons between target and projectile.
Diferences in comparison with reactions through compound nucleus:
a) Angular distribution is asymmetric – strong increasing of intensity in impact direction
b) Excitation function has not resonance character
c) Larger ratio of flying out particles with higher energy
d) Relative ratios of cross sections of different processes do not agree with compound nucleus model
Fundamentally we can calculate element of transition matrix Hfi → we can calculate σ. Cross
section can be divided to two components: σ = S· σDWBA
Part σDWBA has kinematics character – it determines angular distribution dependent on
transferred angular momentum.
Spectroscopic factor S contains wave functions of initial and final states of nucleus – it is
determined by experiment and then it is compared with model calculation.
We need to know σDWBA. In the simplest case we proceed from approximation of wave functions of
incident and fly out particles by plane waves – Born approximation.
It is not accurate enough for particles in the influence of nucleus potential → for wave function we
take solution from scattering by optical potential – Born approximation with distorted wave (DWBA
– Distorted Wave Born Approximation)