Nuclear reactions

1. 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)

2. 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, forexample: 27Al(n,α)24Na or27Al + n → 24Na + α input channel- particles (nuclei) enter into reactionand 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

3. 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) Reaction yield is (d > R): Higher energies of incident particle and smaller ionization losses → higher range and yield w=w(EKIN) – excitation function Mean cross section: →

4. 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-nσxfor 0 ≤ x ≤ d Number of interacting neutrons from N0in target with thickness d is: ΔN = N0(1 – e-nσd) σ – total cross section σR – cross section of given reaction Reaction yield is: For thin target nσd << 1 and yield is: Total cross section can be determined bytransmission 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 isnumber 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 is number of studied photonuclear reactions. We obtain or thick target with thickness d:

5. 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 ≤ pbmax/ħ = 2πR/ λ where λ is deBroglie 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: 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 Πiis 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.

6. 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.

7. 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 : where 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. 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: Wave moving in opposite direction: Fly out (scattered) spherical wave is described by function: (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:

8. Scattered wave Incidence wave Axe Z 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 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 = r2dΩand 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.

9. It is valid for final state: → only one independent momentum (we choose pb). 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: where Hfiis transition matrix element: In volumeV numberdof states (elementary cells with single particle with momentump  p+Δp) is: and then: we further discuss reaction A(a,b)B in the centre of mass system: IfdE0 = dEb +dEB: We substitute dE=(p/m)dp: where mfis reduced mass of final state. Then we obtain:

10. 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: We substitute to expression for probability: Relation between differential cross section and transition probability: where (Pif)=(1/4)Pifis 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 where mi is initial reduced mass (nucleus is in the rest and then ví is relative velocity). We substitute toPif: 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|.

11. We derive similar equation for inverse process. If: |Hif|2 = |Hfi|2 we calculate ratio of both cross sections: This relation is named as principle of detailed balanceof nuclear reaction. If |Hfi|2is constant in small energy range, we obtain: Let us discuss different reaction types: a) Elastic scattering of neutral particles→va = vb→σ = const→ independent on velocityva b) Exotermic reactions excited by thermal neutrons→ Q ≈ 1 MeVand 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|2in the case of charged particles c) Exothermic reactions with charged particles – dependency on factorexp(-Ga) predominates. d) Inelastic neutron scattering– endothermic, vbstrongly depends on energy → above thresholdva ≈ const. Product energy is given by excess of energy above the thresholdEb ≈ Ea - Es→ → e) Endothermic reaction with charged particle  production– memberexp(-Ga) predominates

12. Reaction through compound nucleus Reactions during which projectileenergy is distributed to more nucleons of target nucleus → excited compound nucleus is created → energy cumulating → single or more nucleonsfly 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: where σaCis cross section for compound nucleus creation and Pb is probability of compound nucleus decay to channel b. σab = σaCPb Sum through all final channels: Partial level width Γb– width against decay to channel b: Relation between ΓbandPb: Pb=Γb/Γwhere

13. Cross section [barn] Energy [eV] Resonances Element of transition matrix|Hfi|2and thus also cross sectionσabdo not change only slowly. Reactions proceedthrough compound nucleus→ expect slow changes presence of fluctuations -resonance structuresin the behavior of|Hfi|2andσab 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 byBreit-Wigner formule: Example of resonance character of spectra of reaction through compound nucleus (typical example of reaction with slows neutrons)

14. Cross section [barn] Energy [eV] Sum through all final channels (also elastic scattering) → total cross section of compound nucleus creation: It is valid: Thus independency of creation and decay of compound nucleus. ForE = Eresit is valid (we assume elasticσaaand one inelasticσabchannel →Γ = Γa + Γb): Maximum for elastic part (Γb = 0, Γa = Γ): 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

15. 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

16. 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)