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) 

IntroductionIncident 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

input channel - particles (nuclei) enter into reaction and their characteristics (energies, momenta, spins, …) output channel – particles (nuclei) get off reaction and their characteristics

Reaction can be described in the form A(a,b)B, for example: 27Al(n,α)24Na or 27Al + n → 24Na + α

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

Thin target – does not changed intensity and energy of beam particlesThick target – intensity and energy of beam particles are changed

Reaction yield – number of reactions divided by number of incident particles.

Threshold reactions – occur only for energy higher than some value.

Cross section σ depends on energies, momenta, spins, charges … of involved particles

Dependency of cross section on energy σ (E) – excitation function.

Nuclear reaction yieldReaction yield – number of reactions ΔN divided by number of incident particles N0: w = ΔN /N0

Thin target – does not changed intensity and energy of beam particles → reaction yield: w = ΔN /N0 = σnx

Depends on specific target

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)

Reaction yield is (d > R): KIN

R

0

E

0 KIN

KIN

0

dE

dx

dE)(E

n(x)dxnN

ΔNw

KINa

Higher energies of incident particle and smaller ionization losses → higher range and yieldw=w(EKIN) – excitation function

Mean cross section: R

0

(x)dxR

1 Rnw →

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)nσdxwhere 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)

Reaction yield is: )e1(N

Nw dnRR

0

For thin target nσd << 1 and yield is: dnN

Nw R

R

0

0N

N(d)ln

nd

1Total cross section can be determined by transmission method

→ attenuation measurement: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).

dnI

Iw

a0

For thin target (attenuation can be neglected) reaction yield is:

where ΔI is total number of reactions and from this is number of studied photonuclear reactions. a

I

We obtain or thick target with thickness d: )e1(I

Iw nd

aa0

a

σ – total cross sectionσR – cross section of given reaction

Conservation lawsEnergy 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 ≤ pbmax/ħ = 2πR/ λ

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:

ll

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 lawParity 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 centre2) Potential has short range (decreasing is faster than 1/r)3) Beam of particles is moving in the direction of z axe.

p

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 :

kp

where 21

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:

ikzrkitrki e~e~e~

Wave moving in opposite direction:-ikzrk-i e~e~

Fly out (scattered) spherical wave is described by function: re~ rki

(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:

]refA[e rkiikz

Relation between amplitude factor and cross section:

Particle density is: P = ψ*ψFlow density j of incident particles with velocity vd: jd = vd∙P

For incidence wave: P = |Aeikz|2 = A2

and then: jd = A2vd

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

Amplitude f() must be calculated by Schrődinger equation and we obtain cross section from given equation, which can be compared with experiment.

2f

d

d

Axe Z

Incidencewave

Scattered wave

The principle of detailed balanceLow 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:

0

2

fiif dE

dH

2P

where Hfi is transition matrix element: dVHHH int*

intfi ifif

32

3

2

2

dpVp4

h

dpp4Vd

In volume V number d of states (elementary cells with single particle with momentum p p+Δp ) is:

and then: 32

00 2

dpVp4

dE

1

dE

d

we further discuss reaction A(a,b)B in the centre of mass system:

Bb ppIt is valid for final state: → only one independent momentum (we choose pb).

3b

2b

Bb0 2

dpVp4

dEdE

1

dE

d

If dE0 = dEb +dEB:

We substitute dE=(p/m)dp: bbf

bbBb

BB

Bb

b

bBb dpp

m

1dpp

m

1

m

1dp

m

pdp

m

pdEdE

where mf is reduced mass of final state.

Then we obtain: bf30

pm2

V4

dE

d

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:

bfBb30

p1)m1)(2I(2I2

V4

dE

d

We substitute to expression for probability:

bf

2

fiBb42bfBb3

2

fiif pmH1)1)(2I(2I2

V4p1)m1)(2I(2I

2

V4H

2P

j

P

j

P ifif

d

dRelation between differential cross section and transition probability:

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

ifi

i

i

if Pp4

Vm

v4

VP

d

dThen

where mi is initial reduced mass (nucleus is in the rest and then ví is relative velocity). We substitute to Pif:

i

ffi

2

normfi42Bb

i

ffi

2

fi42Bb

2

p

pmmH

2

12I12I

p

pmmH

2

12I12IV

dd

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: |Hif|2 = |Hfi|2

we calculate ratio of both cross sections: 2

iAa

2fBb

if

fi

p12I12I

p12I12I

This relation is named as principle of detailed balance of nuclear reaction.

If |Hfi|2 is constant in small energy range, we obtain:i

f

p

pkonst

Let us discuss different reaction types:

a) Elastic scattering of neutral particles → va = vb → σ = const → independent on velocity va

)G{G2

fibae~H

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

c) Exothermic reactions with charged particles – dependency on factor exp(-Ga) predominates.

Sabbb EE2mp~v Sa EE

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 →

→

e) Endothermic reaction with charged particle production – member exp(-Ga) predominates

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 character2) 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 dropletenergy decreasing by nucleon escape – cooling by molecule evaporating → evaporation models

Two independent processes: Compound nucleus creation Compound nucleus decayCross 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: 1Pb

b Partial level width Γb – width against decay to channel b:

b

bRelation between Γb and Pb: Pb=Γb/Γ where

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)

For range about 1 – 20 MeV resonances are densely nearly and they are broad → they can not be distinguished → continuum is created (statistical range)

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:

22res

ba2a

ab

41

EEk

π

Example of resonance character of spectra of reaction through compound nucleus (typical example of reaction with slows neutrons) Energy [eV]

Cro

ss s

ecti

on [

barn

]

Sum through all final channels (also elastic scattering) → total cross section of compound nucleus creation:

22res

a2a

aC

41

EEk

π

It is valid:

bb

2res

a2a

22res

ba2a

ab

41

EEk4

1EEk aC π

Thus independency of creation and decay of compound nucleus.

2ab

2a

ab k4

For E = Eres it is valid (we assume elastic σaa and one inelastic σab channel → Γ = Γa + Γb):

2

2a

2a

aa k4

Maximum for elastic part (Γb = 0, Γa = Γ):2a

aamax k4

2a

abmax k

Maximum for inelastic part (Γb = Γa = Γ/2):

Resonance fast changes are given by reactions through compound nucleus, slow changes are given by direct reactions

Energy [eV]C

ross

sec

tion

[b

arn

]

Optical modelReflection 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 directionb) Excitation function has not resonance characterc) Larger ratio of flying out particles with higher energyd) 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)