Chapter 2

Nuclear Reaction Theories

In a nuclear reaction an atomic nucleus interacts with the nuclear projectile, emitting

nuclear particles and/or radiations leaving behind the residual nucleus. A nuclear

reaction is typically represented as

X(a, b)Y

in which an incident particle a interacts with the target nucleus X emits an outgoing

particle b leaving behind the residual nucleus Y . Macroscopically we know the system

before and after the reaction but what exactly happens during the reaction process is

not known. Since it is not possible to look into the reaction process directly, models

for reaction mechanism is proposed to explain the yield and angular and energy distri-

butions of the reaction products. The first attempt to model a reaction was made by

Niels Bohr [1] in 1936. According to him a nuclear reaction takesplace in two steps.

The projectile is absorbed by the target nucleus and a compound nucleus is formed

and a thermodynamic equilibrium is established. Then it decays by emitting particles

and/or radiations. It is assumed that the decay of the compound nucleus depends

only on the excitation energy and other good quantum numbers of the compound nu-

cleus and is totally independent of its mode of formation. This is called ’independent

hypothesis’. Compound nucleus mechanism accounts well the isotropic distributions

of emitted particles in the center of mass frame at lower excitation energies. How-

ever, at relatively higher excitation energies, the forward peaked angular distribution

of particles indicates the presence of direct reaction mechanism [2, 3]. In compound

nucleus mechanism the whole system is involved while in direct reaction mechanism

only a few nucleons take part in the evolution of the reaction processes. The results of

recent measurements indicates the presence of reaction process which is intermediate

between these two extreme reaction mechanisms. It is assumed that the nuclear reac-

9

Figure 2.1: A pictorial representation of the progress of a typical nuclear reactioninduced by an intermediate energy proton.

tion develops as a result of successive interaction of the projectile and the nucleons of

the target nucleus and particle emission may takesplace at any stage even before the

establishment of thermodynamic equilibrium. In reactions initiated by a few tens of

MeV the continuous particle spectra indicate the presence of such multistep process.

The particles which are emitted during the equilibrium are called pre - compound or

pre - equilibrium particles and the process is called pre - equilibrium nuclear reaction

[4, 5]. A pictorial representation of the progress of a typical nuclear reaction induced by

an intermediate energy proton is given in Fig. 2. 1. A brief descriptions of compound

nucleus and pre - equilibrium mechanisms are presented in the following sections.

2.1 Compound nucleus model

Weisskopf and Ewing [6] developed theoretical calculations of reaction cross - sections

according to the Bohr’s model [1] using partial wave analysis. In this model the conser-

vation of angular momentum and parity for each partial wave is not taken into account.

Nevertheless, it provides a good estimate for the magnitude of the cross - section. On

the other hand Hauser and Feshbach [7] treated the problem in a more detailed way and

have explicitly taken into account the conservation of angular momentum and parity.

2.2 Hauser-Feshbach theory

Hauser-Feshbach theory describes reaction cross section that involve a large number of

compound nuclear states. The Bohr independence hypothesis allows the cross section

10

to be written as the product of two factors, the cross section σa for formation of the

compound nucleus and the probability Pb will be decay in to the channel b:

σab = σaPb (2.1)

The compound nucleus formation cross section at a particular orbital angular momen-

tum ℓ is given by

σa = πλ2a(2ℓ + 1)Ta, (2.2)

where λa is the reduced wavelength in the incident channel (inverse of the wave

number k) and the transmission coefficient Ta for the entrance channel a is given by

Ta = 1 − |Saa|2 [8] where Saa is average S - matrix. The reciprocity theorem relates a

cross section to its inverse:

λ2bσab = λ2

aσba, (2.3)

where a and b refer to time-reversed states. Combining these equations imply that

TaPb = TbPa. (2.4)

Since this is true for all channels a and b,

Pa

Ta

=Pb

Tb

= ξ, (2.5)

where ξ is a constant. Since the P ′s are probabilities

∑

a

P a = ξ∑

a

Ta = 1 (2.6)

Therefore

Pb = ξT b =T b

∑

a Ta

. (2.7)

Thus we get

σab = πλ2a(2ℓ + 1)

TaTb∑

c Tc

(2.8)

The transmission coefficients Ta can be calculated from the appropriate optical model

potentials. This Hauser - Fesbach formula provides the cross section of a compound

nuclear reaction from a single incident channel a to a single outgoing channel b, in a

particular angular momentum state ℓ in the absence of spin. If the interacting particles

11

have spin, the expressions for the cross section must be include in the appropriate spin

weighting factors. Suppose that the spins and angular momentum for the reaction

X(a, b)Y are defined as follows

ai + XI

︸ ︷︷ ︸

s=i+I

ℓ−→ (CN)J∗ ℓ′−→ bi′ + Y I′

︸ ︷︷ ︸

s′=i′+I′

(2.9)

The spin i of the incident particle and the spin I of the target combine to form the

channel spin s, which in turn combines with the orbital angular momentum ℓ to form

the total nucleus angular momentum J , and similarly for the outgoing channel. The

probability that the spins i and I combine to give a particular s is

P (s) =2s + 1

(2i + 1)(2I + 1)(2.10)

and similarly the probability of s combining with ℓ to give J is

P (J) =2J + 1

(2s + 1)(2ℓ + 1)(2.11)

Weighting the partial waves by (2ℓ + 1) the cross section becomes

σHFab (E) = πλ2

a

∑

JΠ

2J + 1

(2i + 1)(2I + 1)

TaTb∑

c Tc

. (2.12)

where Π is the parity of the compound state. The transmission coefficients depend on

the angular momenta ℓ, s and J . The transmission to the final states depends upon the

possible final states and is expressed in terms of level density parameters.

2.3 Nuclear level densities

Nuclear level densities play an important role in estimating nuclear reaction cross sec-

tions in general. The nuclear level density is the number of levels per unit energy

interval. The nuclear energy state may be divide into two regions. Up to a certain

excitation energy (typically between 1 and 2 MeV) the number of levels is limited,

and they are relatively well separated. In this energy region the levels are relatively

simple in structure, and often can be understood on the basis of nuclear models. With

increasing excitation energy, the number of levels increases, the spacing between them

reduces, and the nature of excitation becomes very complicated. Therefore at high ex-

citation energies the only way to describe them is using a statistical procedure. Bethe

12

[9] introduced the concept of level density function ρ in the form

ρ(E) =dN(E)

dE, (2.13)

where N(E) is the cumulative number of levels up to the excitation energy E. From ob-

servation of the distribution of low-energy levels, an empirical formula can be obtained.

Based on Fermi gas model of nucleus the level density is given as,

ρ(E, J) =f(J)

Texp

(E − E0

T

)

, (2.14)

where E0 and T (the nuclear temperature) can be adjusted to fit into experimental

data; f(E, J) is the spin distribution factor given as

f(E, J) =2J + 1

2σ2c

exp

(−(J + 1/2)2

2σ2c

)

(2.15)

where σc is the spin cut-off parameter. In the simplest approach, the nucleus can be

considered as a system of fermions which can occupy levels equidistant in energy. The

density of excited states is given as [9]

ρ(E) =exp(2

√aE)

4√

3E(2.16)

where the parameter a is known as the level density parameter. A more realistic ex-

tension of this model considers the fact that fermions have the tendency to form pairs,

and that it takes an extra amount of energy to separate them. This can be taken into

account by introducing a shift E1 in the excitation energy (E1 is considered as an ad-

justable parameter), which leads to the Back - Shifted Fermi Gas model (BSFG) [9].

Thus we can write,

ρ(E, J) = f(J)e2

√

a(E − E1)

σc12√

2a1/4(E − E1)5/4. (2.17)

and the spin cut-off factor has the form [9]

σ2c = 0.0888A2/3

√

a(E − E1). (2.18)

The parameters a and E1 differ from nucleus to nucleus and can be adjusted in or-

der to agree with the values of the level density at lower energies and at the nucleon

separation energy. This model can describe experiments in a narrow energy interval

around the nucleon binding energy, where most experimental data are obtained from

resonance measurements. However, the BSFG model does not permit extrapolation

13

to higher-energy regions and shell effects are not properly taken into account. Gilbert

and Cameron [10] extended this model by incorporating CTF model. According to this

model excitation energy is splitted into two regions. Different functional forms of level

densities are applied in each of them. At low excitation energies (below the matching

point Ux) the constant temperature formula is used

ρT (E) =1

Texp[(E − ∆ − E0)/T ], (2.19)

where T is the nuclear temperature, E is the excitation energy (E = U + ∆ with ∆

being the pairing correction), and E0 is an adjustable energy shift. Above Ux the Fermi

gas formula is applied

ρF (E) =exp(2

√aU)

12√

2σ(U)a1/4U5/4(2.20)

The level density parameter a is assumed to be energy independent. The spin cut-off

factor σ(U) is given by

σ2(U) = 0.146A2/3√

aU. (2.21)

Three model parameters, T, Ux, and E0, are determined by the requirement of the level

density and its derivative are continuous at the matching point Ux.

2.4 Pre - equilibrium models

Nuclear reaction models based on semiclassical and totally quantum mechanical the-

ories are proposed for pre - equilibrium emission mechanism. Some of the important

semiclassical models are Intranuclear Cascade Model, Harp Miller and Berne Model,

Exciton model, Geometry Dependent Hybrid model and Index model. Most commonly

used quantum mechanical models are Feshbach, Kerman and Koonin (FKK) model

and Nishioka, Verbaarshot, Weidenmuller and Yoshida (NVWY) model or Hydelberg

model. Brief outline of these models are described in the following sections.

2.4.1 Intra-Nuclear-Cascade (INC) model

Intra-Nuclear Cascade (INC) model proposed by Serber [11] is the first nuclear re-

action model that incorporated pre - equilibrium emission. The first calculation of

pre-equilibrium angular distributions was performed by this model using the quasi-free

scattering inside the nucleus. A graphical presentation of the model is given in Fig. 2.

2.

14

Figure 2.2: Schematic diagram of intra nuclear cascade model.

The projectile enters the target nucleus with a given impact parameter ”b”, after trav-

eling a certain distance inside the nucleus it interacts with a target nucleon and excites

it above Fermi Sea. Each scattered particles then travel through the nucleus interact-

ing with the other nucleons. The Intra-nuclear cascade model traces the individual

nucleon trajectories in three-dimensional geometry. The trajectory of an excited par-

ticle is followed until some arbitrary energy, generally considerably above the average

equilibrium value, has been attained by the nucleon. Particles reaching the nuclear

surface with sufficient energy are assumed to be emitted. When all particles of a given

cascade have been traced, the total energy of the residual nucleus, its density, and the

energies and angles of the emitted particles are shared, and a new cascade with new

impact parameter is calculated. With the help of such an approach, the time evolution

of the reaction can be generated. However, after few collisions the actual calculations

becomes too much complicated. The intra-cascade model is a realistic model but in

general, the predictions are not satisfactory at back ward angles and in some forward

angles also.

2.4.2 Harp-Miller-Berne (HMB) model

The Harp-Miller-Berne [12] proposed a different approach for describing the nuclear

interaction mechanisms. The nuclear single particle states are classified according to

their energies in-groups or ’bins’ whose size ∆ε is chosen to be of some convenient

15

Figure 2.3: Schematic diagram of Harp Miller Berne model.

dimension as indicated in Fig. 2. 3.

In the calculations the fractional occupation of each bin is taken as a function of

time. This model calculates the occupation probability of the average state in the

’ith’ bin as a function of time using Fermi gas distribution. At the initiation of the

reaction, at the time τ0, all the levels below Fermi energy are filled up (as the target

is in ground state), and the projectile is in an excited state. This gives the fractional

occupation probability at time τ = τ0. Two body interaction leads to a redistribution of

probabilities. Harp and Miller [13] suggested minor modification in the HMB model and

they considered the nucleus to be composed of independent proton and neutron Fermi

gases. Therefore, the proton and neutron occupation numbers for the single particle

states of these gases completely specifies the internal configuration of the nucleus at any

time. Further it is also assumed that the mechanism for the equilibrium of gases takes

place through binary nucleon-nucleon collisions. Correspondingly a new set of master

equation is obtained, the solution of which gives the proton and neutron occupation

numbers. This model predicts the absolute spectral yield satisfactorily.

16

2.4.3 Exciton model

Still modified and logically simple model is presented by Griffin [14]. In this model it is

assumed that the incoming projectile, by interacting with the target nucleus, gives rise

to a simple initial configuration characterized by a small number of excited particles and

holes called excitons (n=p+h). Successive two-body residual interactions give rise to an

intranuclear cascade through which a sequence of states characterized by an increasing

exciton number, eventually leads to a fully equilibrated residual nucleus. Restriction

to two-body residual interactions leads to the following selection rules concerning the

possible variation of the number n of excitons, of particles p, and holes h:

∆n = 0,±2; ∆p = 0,±1; ∆h = 0,±1. (2.22)

The states which are excited in this interaction cascade are very unstable. The

possible sequence of events considered in the exciton model are shown in Fig. 2. 4. At

each stage of this equilibrium process there is a competition between two decay modes

of the composite nucleus: the decay by exciton-exciton interactions are more complex

configurations and the decay by emission of particles into the continuum.

The exciton model assumes that

1. At each stage of the cascade all of the states with the same configuration and the

same total energy are equiprobable.

2. At each stage of the cascade all the processes which may occur are also equiprob-

able.

The first assumption gives the energy distribution of the excitons. The number

dNp(p, h, E, ε) of the excitons with energy between ε and ε + dε in a configuration of

p particles and h holes with total energy E is given by the ratio between the number

of states in which one particle has energy between ε and ε + dε and the remaining

p − 1 particles and h holes have the energy E − ε and the number of states with p

particle, and h hole configuration with energy E. If ρp,h(E) is the density of states of

this configuration. ρp−1,h(E − ε) is the density of states of the configuration of p − 1

particles and h holes with energy E − ε, then dNp(p, h, E, ε) is given by

dNp(p, h, E, ε) =ρp−1,h(E − ε)gdε

ρp,h(E)(2.23)

where g = 1/d the density of single-particle states(for which one usually assumes the

Fermi gas model g=3A/2εF ), d is the spacing between the states.

17

Figure 2.4: Schematic representation of the first few stages of a nucleon-induced reactionin the exciton model. The horizontal lines indicate equally spaced single-particle statesin the potential well. The particles are shown as solid circles. E is the initial excitationenergy. B is the average nucleon binding energy. Part (a) shows nucleon-nucleoninteractions leading to the equilibrium process, in this case all particles are bound.Part (b) shows interactions leading to configurations in which at least one particle isunbound and thus may be emitted into the continuum with energy ǫ leaving a residuewith energy U = E − ǫ.

18

The second assumption gives the simplification of the evaluation of the cross sections of

the various reactions in terms of escape width and spreading width defined as follows.

Γ(ν, εν |E, p, h)dεν as escape width for emission into the continuum of one unbound

particle ν with energy εν , Γ↓n,n′ as spreading width for nucleon-nucleon interaction, in

which most cases spread the excitation energy among an increasing number of excitons.

These widths are related to the probabilities per unit time (or decay rates) for the

corresponding processes, W ↑ and W ↓, by the relation Γ = W~ (W = W ↑ + W ↓),

When a particle is emitted, a residual nucleus with p − 1 excited particles and h holes

is created. If ρph(E) and ρp−1,h(U) are the state densities for the composite and the

residual nuclei. The escape width for emission of a particle ν with energy between εν

and εν + dεν is given by

Γ↑(ν, εν |E, p, h)dεν = W~(ν, εν |E, p, h)dεν (2.24)

Γ↑(ν, εν |E, p, h)dεν =~

ρph(U)

(υνσinv(εν)

V

)

ρp−1, h(U)ρc(εν)dεν (2.25)

where υν is the emitted nucleon velocity, σinv(εν is the inverse process cross section,

and

ρc(εnu) =1

π2~3

(2sν + 1)mνενV

υν

(2.26)

is the density of the translational continuum states of the emitted particle as predicted

by the Fermi gas model. V is the laboratory volume. The estimate of the spreading

width Γ↓n,n′ is obtained by using time-dependent perturbation theory which gives

Γ↓n,n′(E) = 2π|M |2ρf , (2.27)

for a transition from the states of a configuration of n excitons to a state of a con-

figuration of n′ excitons. ρf is the density of states which may be excited in the

nucleon-nucleon interaction and M is the transition matrix element. Considering only

two-body interactions, by selection rules n′ = n or n ± 2 at low energies,

Γ↓n,n+2 ≥ Γ↓

n,n ≥ Γ↓n,n−2 (2.28)

The dominant n → n + 2 transition spreading width is given by

Γ↓n,n+2(E) = 2π|M |2 g3E2

2(n + 1)= 2π|M |2 (3A/2εF )3E2

2(n + 1). (2.29)

19

The average squared matrix for two-body interaction is given by

|M |2 = KA−3E−1 (2.30)

where the value of K varies between 100 and 700 MeV. The total escape width for stage

N is

Γ↑N =

∑

ν

∫ εν

0

Γ↑N(ν, εν |E, p, h)dεν , (2.31)

and, since

PN(εν)dεν =Γ↑

N(ν, εν |E, p, h)dεν

(Γ↑N + Γ↓

N)(2.32)

PN(εν)dεν =Γ↑

N(ν, εν |E, p, h)dεν

ΓN

(2.33)

where Γ↓N ≈ Γ↓

N+2 and the total widths ΓN equals Γ↑N + Γ↓

N and

λk−1,k =Γ↓

k−1

Γk−1

(2.34)

it follows that

σDi + PE(εν)dεν = σC

∑

N

Γ↑N(ν, εν |E, p, h)dεν

ΓN

N∏

k=1

Γ↓k−1

Γk−1

(2.35)

with Γ↑0/Γ0 ≡1. Usually after a few stages the pre - equilibrium emission becomes

negligible and further interactions lead to an equilibrated compound nucleus.

2.4.4 Geometry Dependent Hybrid model

Hybrid model [15] was proposed by Blann. It maintains the physical transparency and

simplicity of the exciton model while permitting the calculation absolute spectral yield

as in the HMB model. The continuum decay rates are computed from the partial state

densities while the intranuclear transition rates are calculated from the mean free path

(MFP) of the nucleons in the nuclear matter. The MFP, in turn, may be evaluated

either from free nucleon - nucleon scattering cross - sections or from the imaginary

part of the optical potential [15]. The particle emission probability in a given range

of channel energy ε and ε + dε may be given as the sum over the contribution of the

intermediate states. The probability of a particle of type ν with channel energy ε to

20

ε + dε is given by the expression [15],

P (ε)dε =n∑

n=n0

nPν

[ρn(U, ε)

ρn(E)

]

gdε

[λc(ε)

λc(ε) + λn+2(ε)

]

Dn (2.36)

Where, nPν is the number of particles of type ν in an n exciton state. one of which has

an energy such that if emitted, the residual nucleus would have excitation energy U (=

E-Bν − ε) and the particle would have channel energy ε. Bν is the binding energy of

the emitted particle and ρn(E) is the state density of n - exciton state with excitation

energy E. λ(ε) is the decay constant for transition into the continuum for a particle at

excitation energy (Bν + ε) above Fermi energy and λn+2(ε) is the corresponding decay

rate for creating another particle - hole pair leading to the final state of (n+2) excitons.

Dn is the population surviving the particle emission. The emission rate λ(ε) into the

continuum is given as,

λ(ε) = σ(ε)

[2E

M

]1/2 [ρc(ε)

gΩ

]

(2.37)

where, σ(ε) is the inverse cross - section, ρc(ε) is the density of transitional state of

a particle in the continuum and Ω is the volume in which the free phase space is

normalized.

The non - uniform distribution of nucleons in the nucleus may affect the decay rates

as the mean free path in the diffused surface region will be larger as compared to the

mean free path in the interior of the nucleus. This nuclear geometry effect is taken into

account by taking Fermi density distribution as follows. The nuclear density at radius

R is given by

d(R) = ds[exp(Rl − C)/Z + 1]−1 (2.38)

where, Z = 0.55fm and ds is the saturation density of the nuclear matter in the interior

of the nucleus, C is the nuclear charge radius (1.07A1/3 fm), the radius of lth partial

wave is defined by

Rl(l +1

2) (2.39)

The dependence of Fermi energy and single particle level density (gx) on nuclear matter

density are related as,

Ef (R1) = EF [< d(R1) > /ds]2/3MeV (2.40)

g(x)(R1) = [EF (R1)](A/28) (2.41)

21

where, Ef is the Fermi energy at the saturation density and x represents particle type.

The HMS model has a number of attractive features. First of all, there are no physical

limits on a number of pre - equilibrium emissions (apart from energy conservation).

With the addition of linear momentum conservation by M. Chadwick and P. Oblozinsky,

the model provides a nearly complete set of observables. This include cross section

for the production of residual, light-particle double-differential spectra and spectra of

recoils. Spin and excitation-energy dependent populations of residual nuclei can also

be obtained, an essential feature for coupling the pre - equilibrium mechanism to the

subsequent Compound Nucleus decay. The binding energies in the HMS model are

thermodynamically correct. This is a clear improvement over the intranuclear cascade

model, although exact account, typical of the Compound Nucleus model, is still out of

reach.

2.4.5 Index Model

Ernst et. al. [16] developed index model, of independently interacting excitons, for

pre - equilibrium emission which unifies the exciton and hybrid models. The basic

assumption of the index [16] model is that all excited particles which survive emission

undergoes two - body collisions and create further particle - hole pairs independently

from each other. Thus the energy of each exciton is shared by the three excitons of the

following stage. The average nucleon - nucleon collisions rates and the internal energy

in this model are taken from the interaction rate of nucleons inside the nuclear matter.

In Index model it is possible to include the multiparticle emission and it is shown that

for light ion induced reactions below 100 MeV, emission of more than three PE nucleons

is not important [16]. This model is not commonly used as compared to exciton and

GDH model.

2.5 Quantum mechanical theories

Several quantum mechanical theories of pre - equilibrium reactions are also proposed

[17 - 20]. Some of the most commonly used models are FKK [17] and NVWY [21]

models. These models are based on the assumptions similar to that of exciton model.

The reaction is considered to proceed through stages of increasing complexity. At each

interaction stage it is useful to consider separately the states with at least one particle

in the continuum and the states with all particles are bound, termed respectively as

the MultiStep Direct (MSD) and MultiStep Compound (MSC) reactions. In multistep

compound reaction all the particles remain bound during the equilibration cascade,

22

Figure 2.5: Multistep description of a nuclear reaction, according to FKK model withfinite possibilities of shifting from one chain into another at any stage.

while in multistep direct reactions at least one particle is always in the continuum.

Brief description of these models are given in the following sections:

2.5.1 Feshbach, Kerman, and Koonin theory

In FKK [17] model the MSC and MSD processes involve are represented in terms of Q

and P chains respectively as shown in Fig. 2. 5. These may be formally described by

the projections P and Q acting on the total wave function ψ, with total probabilities

P+Q=1. The set of states Pψ contributes to the multistep direct process and the

complementary set of states Qψ contributes to the multistep compound process. Two

assumptions are made as follows: the first, the chaining hypothesis, assumes that the

residual interaction can induce transitions from the nth stage to the (n ± 1)th stages

only and the second is that the relative phases of certain matrix elements are assumed

to be random. In multistep compound reactions the phases of the competing processes

interfere so that multistep compound reactions have energy-averaged cross sections that

are symmetric around 900. Multistep direct reactions takesplace rapidly and have cross

sections that are generally peaked in the forward direction. Pre - equilibrium reactions

can takesplace directly from each stage of the P chain, or indirectly from the Q chain.

The emission from the Q chain takesplace through states in the P chain, and this can

takesplace in three ways. The more energetic particles from the early stages of the

chains and the less energetic from the later stages.

2.5.2 Multistep compound reactions

In the multistep compound theory the cross section for emission from each stage is

expressed, as in the exciton model, by product of three factors:

23

1. The cross section for the formation of the composite nucleus.

2. The probability of reaching the nth stage.

3. The probability of pre - equilibrium emission from the nth stage.

The total cross section for the pre - equilibrium emission is then given by the sum of

these products over all stages before the formation of the fully equilibrated compound

nucleus. The cross section for the formation of the composite nucleus is

σa = πλ2∑

J

(2J + 1)2π < Γ1J >

< D1J >(2.42)

where < Γ1J > is the level width and < D1J > is the level spacing in the first stage ie.,

the compound system. The term 2π < Γ1J > / < D1J > gives the strength function for

the initial reaction stage. The probability of reaching the nth stage through a particular

sequence of stages without pre - equilibrium emission is given by the product of the

probabilities of passing through the intervening n − 1 stages. Assuming the chaining

and the never-come-back approximations, the total probability of reaching the nth stage

is given by

n−1∏

k=1

< Γ↑kJ >

ΓkJ

, (2.43)

where the total width < ΓkJ> is the sum of the total escape width Γ↑kJ and the damping

width Γ↓kJ , referring to emission and internal transitions respectively:

< ΓkJ >=< Γ↑kJ > + < Γ↓

kJ > . (2.44)

The probability of emission of a particle into the continuum from the nth stage is given

by the sum of the decay rates for all possible emission processes divided by the total

decay rate. The sum of the decay rates for all emission processes is proportional to

the products of the emission widths Γ↑ℓsυn,J and the densities ρυ

s (U) of the final states at

excitation energy U which may be reached from a compound state of spin J , and the

total decay rate is proportional to the width ΓnJ .

Here υ labels the three exit modes corresponding to ∆n = 0,±2. l and s labels

orbital angular momentum and channel spin of incoming particle.

Thus the required probability of emission of a particle into the continuum from the nth

stage is given by

∑

υ

< Γ↑ℓsυn,J (U)ρυ

s (U) >

< ΓnJ >(2.45)

24

Now using random phase approximation for all channel quantum numbers, the double-

differential cross section for pre - equilibrium emission is given as

d2σ

dΩdε= πλ2

∑

J

(2J + 1)

[r∑

n=1

∑

ℓsλ

CλℓsJPλ(cosθ)

]

×n+1∑

ν=n−1

< Γ↑ℓsυn,J (U)ρυ

s (U) >

< ΓnJ >

n−1∏

k=1

< Γ↓kJ >

< ΓkJ >

2π < Γ1J >

D1J

(2.46)

where CλℓsJ is the angular momentum coupling coefficient and equals

CλℓsJ = (−)s

(2λ+14π

)1/2

(

J J λ

0 0 0

)

Z(ℓJℓJ ; sλ)

with λ=0,2,4....., ensuring the symmetric emission characteristic of quasi - equilib-

rium processes. Integration over all angles gives the energy spectrum

dσ

dε= πλ2

∑

J

(2J + 1)r∑

n=1

∑

ℓsλ

< Γ↑ℓsυn,J ρυ

s (U) >

< ΓnJ >

n−1∏

k=1

< Γ↓kJ >

< ΓkJ >

2π < Γ1J >

D1J

(2.47)

The cross section for pre - equilibrium emission falls quite rapidly from stage to stage.

As the excitation spreads through the whole compound nucleus, the rate of inverse

in level density with the stage number falls so that the rate of backward transitions

increases. When these rates become the same, full statistical equilibrium has been

established. The total cross section for emission after the truncation of the pre - equi-

librium chain is called the residual or rth- stage cross section and is given by

dσ(f)r

dε= πλ2

∑

J

(2J + 1)π < Γ1J >

< D1J >

(n−1∏

k=1

Γ↓kJ

ΓkJ

)

Γ(f)rJ

ΓrJ

(2.48)

Since the rth-stage is the last stage in the chain,

ΓrJ = Γ↑rJ =

∑

c

Γ(c)rJ (2.49)

and the rth-stage cross section becomes

στ = πλ2∑

J

(2J + 1)TiTf∑

c Tc

, (2.50)

25

where

Ti =2π < Γ1 >

D1

(k=r−1∏

k=1

ΓkJ

ΓkJ

)

(2.51)

and

Tf =2πΓ

(f)rJ

Dr

(2.52)

Eq (2.47) is similar to the Hauser-Feshbach formula, but there are important difference.

All of the factors in the expressions are calculated quantum-mechanically or, as in

the case of the level density function, obtained from known systematic of nuclear

properties.

Pre - equilibrium emission occurs as a result of nucleon-nucleon interactions during

the early stages of a nuclear reaction. At each stage there are three possibilities (a)

excitation of an additional particle-hole pairing, (b) De-excitation of an particle-hole

pairing, (c) Emission into continuum state.

In order to evaluate the cross section for the emission of particles by the multistep

compound (MSC) process it is necessary to evaluate the escape and damping widths.

For these, one has to consider the interaction involved in the emission process, and

the density of states available in the residual nucleus in detailed. The escape width

is expressed as a product of the energy dependent width ΓℓsνnJ and the final state level

density ρνs , and this product may itself be factorized into a term Xℓsν

nJ (U) containing

the angular momentum dependence due to the nucleon-nucleon interaction and the

spin distribution of single - particle states and the term Y νn (U) containing all the U

dependence due to final state level density,

Γn =n+1∑

ν=n−1

Γnν =n+1∑

ν=n−1

< ΓℓsνnJ (U)ρνs(U) >= Xℓsν

nJ (U)Y νn (U) (2.53)

The damping width is also given by

< Γ↓n,J >= X↓n+2

nJ Y ↓n+2n (E). (2.54)

In order to calculate the X and Y functions we need to know the density of a p particle,

h hole configuration at an excitation energy E. This is obtained from the equidistant

spacing model. It takes into account the Pauli principle correction and the limitation

of the particle energy to bound states and the hole energy to the depth of the potential.

26

The X functions contain angular momentum coupling factors and the radial overlap

integrals in the matrix element for the transition. The Y function gives the accessible

phase space and is obtained when the matrix element for a certain transition squared,

averaged over initial states, and summed over final states [21]. It is computed by

considering the state densities of the initial and final particle-hole configurations, and

angular momentum is independent.

2.5.3 Multistep direct reactions

At lower incident energies all the particles in the intra-nucleus cascade remain bound,

and the reaction may be described by the multistep compound theory discussed in

the previous section. As the incident energy increases it becomes more likely that

one particle remains in the continuum and so retains a strong memory of the original

direction of the projectile. The time scale of the reaction is much shorter than in the

multistep compound reaction.

The interaction of a projectile with a target nucleus takesplace by a series of nucleon-

nucleon interactions, and at each of the interactions pre - equilibrium emission can take

place. The total double differential cross section for pre - equilibrium emission is thus

the sum of the cross sections for emission from each stage

d2σ

dUdΩ=

(d2σ

dUdΩ

)

1

+

(d2σ

dUdΩ

)

M

(2.55)

where the subscript 1 indicates the first stage, and M the subsequent multistep stages.

The cross section for the first stage is evaluated using the distorted wave theory, as-

suming the spectroscopic factors to be unity, and is given by

d2σ

dUdΩ1=

∑

ℓ

(2ℓ + 1)ω(U, ℓ)

[(d2σ1

dUdΩ

)

DW

]

ℓ

(2.56)

where < (d2σ/dUdΩ)DW > is the first order distorted wave Born approximation differ-

ential cross section averaged over all energetically possible 1p1h states in the residual

nucleus that corresponds to a particular angular momentum transfer and ω(U, ℓ) is the

density of 1p1h levels in the residual nucleus.

The multistep cross section is the sum of the cross sections from all the subsequent

27

stages, and each of these is given by the folding integral of transition probabilities:

d2σ

dUdΩM=

∑

n

n+1∑

m=n−1

∫dk1

(2π)3....

∫dkn

(2π)3

d2Wmn(kf , kn)

dUdΩ

×d2Wn,n−1(kn, kn−1)

dUndΩn

....d2W21(k2, k1)

dU2dΩ2

(d2σ(k1, ki)

dUdΩ

)

1

, (2.57)

where m labels the exit mode and n the stage. The transition probability for the

(n − 1)th to nth stage, when the particle momentum changes from kn−1 to kn is

d2Wn,n−1(kn, kn−1)

dUndΩn

= 2π2ρ(kn)ρn(U) < |νn,n−1(kn, kn−1)|2 >, (2.58)

where ρ(kn) = mkn/(2π)3~

2 is the density of states of the particle in the contin-

uum, ρ(U) is the level density of the residual nucleus at excitation energy U , and

νn,n−1(kn, kn−1) is the matrix element for the transition from a state n − 1 to a state

n when the particle in the continuum changes its momentum from kn−1 to kn. The

matrix element can be evaluated by the distorted wave Born approximation expression

νab(ki, kf ) =

∫ ∫

χ(−)∗b < ψf |V (r)|ψi > χ(+)

a dradrb (2.59)

where V (r) is the effective interaction for the transition, χ(+)a and χ

(−)∗b are incoming

and outgoing distorted waves, and ψi and ψf the wave functions of the initial and final

nuclear states. In the FKK theory it is assumed that the interference terms cancel

and the different partial waves contribute incoherently, so that the average value of the

square matrix element becomes,

< |υ(ki, kf )|2 >=∑

ℓ

(2ℓ + 1) < |υ(ki, kf )|2 > R(ℓ) (2.60)

where R(ℓ) is the spin distribution function of the residual nuclear levels. This is same

for all stages in the reaction.

To calculate the cross section for the first stage the inelastic cross section (dσdΩ)DW

is evaluated microscopically as a function of angle for each transferred ℓ value for all

possible pairs of initial and final bounds states compatible with energy conservation.

The single particle shell model is used to describe the nuclear states. To obtain the

first stage cross section, the density of levels

ω(U, ℓ) = ρ(U)Rnℓ (2.61)

28

can be obtained from the Erickson formula [22,23]

ρ(U) =g(gU)n−1

p!h!(n − 1)!(2.62)

and the spin distribution function

Rnℓ =2ℓ + 1√π23/2σ3

exp

(

−(ℓ + 1/2)2

2σ2

)

, (2.63)

where σ2 is the spin cut-off parameter.

2.5.4 Spin effects of the pre - equilibrium reactions

Chadwick et al. [24] incorporated the effect of nonzero intrinsic spin in MSD calcula-

tions by treating the 1p1h states excited in the interaction as absorbing the transferred

angular momentum, after which their angular momentum couples with the intrinsic

“core” spin of the target. For projectile spin i and a target spin I, leaving a resid-

ual nucleus with spin J after inelastic scattering with an orbital angular momentum

transfer l, the 1-step MSD cross section is given by

d2σ(E, Ω ← E0, Ω0)

dΩdE 1step=

∑

J

2J + 1

(2I + 1)(2i + 1)

×I+Sf∑

S=|I−Sf |

J+S∑

ℓ=|J−S|

ρ(1p, 1h,E0 − E, ℓ)

[d2σ(E, Ω ← E0, Ω0)

dΩ

]DWBA

ℓ

(2.64)

where Sf is the spin flip. In the limit of zero intrinsic spins this expression reduces the

usual FKK 1-step MSD result.

ρ(1p, 1h,E0 − E, ℓ) is the density of 1p1h states with energy E0 − E and angular mo-

mentum ℓ. The density of states for a p particle, h hole system can be partitioned into

the energy dependent density multiplied by a spin distribution,

ρ(p, h, E, ℓ) = ω(p, h, E)Rn(ℓ), (2.65)

where, using the finite-well-depth restricted Williams [25,26]expression,

ω(p, h, E) =gn

p!h!(n − 1)!

h∑

j=0

(

h

j

)

(−1)j

×(E − Aph − jεF )n−1Θ(E − Aph − jεF ) (2.66)

with n = p + h, the single particle spacing g = A/13, Pauli-blocking factor Aph =

[p2 + h2 + p − 3h]/4g and the Fermi energy ε. The function Θ is unity if its argument

29

is greater than zero, and zero otherwise.

MSD multistep cross sections obtained from a convolution of normal DWBA matrix

elements, which give a contribution from the N th stage,

d2σ(E, Ω ← E0, Ω0)

dΩdE=

m

4π2~2

∫

dΩN−1

∫

dEN−1EN−1

×d2σ1(E, Ω ← EN−1, ΩN−1)

dΩdE

d2σN−1(E, Ω ← EN−1, ΩN−1 ← E0, Ω)

dΩN−1dEN−1

(2.67)

At above 10 MeV energies of the incident protons pre - equilibrium is dominated by 1-

step scattering, which from phase space considerations, contain a large MSD component.

2.5.5 NVWY MSC Model

The NVWY theory describes the equilibration of the composite nucleus as a series of

transitions along the chain of classes of closed channels of increasing complexity [21].

The classes are defined in terms of the number of excited particle-hole pairs (N) plus

the incoming nucleon, i.e. excitons. Thus the exciton number is n = 2N +1 for nucleon

induced reactions. Assuming that the residual interaction is a two-body force only

neighboring classes are coupled (∆n = ±1). The average MSC cross section leading

from the incident channel a to the exit channel b is given by

dσab

dE= (1 + δab)

∑

n,m

T anΠn,mT b

m, (2.68)

which also has to be summed over spins and parities of the intermediate states. The

transmission coefficients T an describing the coupling between channel a and class n are

given as

T an =

4π2Uan

(1 + π2∑

m Uam)2

, (2.69)

where Uan = ρb

n < Wn,a > is microscopically defined in terms of the average bound

level density ρbn of class n, and in terms of the average matrix elements Wn,a connecting

channel a with the states in class n. The probability transport matrix Πm,n is defined

via its inverse as,

(Π−1)n,m = δn,m(2πρbn)(Γ↓

n + Γextn ) − (1 − δn,m)2πρb

nV2n,m2πρb

m. (2.70)

The mean squared matrix element V 2n,m couples states in classes n and m. The Γ↓

n is

the spreading width of states in class n and Γextn in class n is the average total decay

width. The spreading width Γ↓n is again related to the mean squared matrix element

30

V 2n,m as

Γ↓n = 2π

∑

m

V 2n,mρb

m. (2.71)

V 2n,m couples only neighboring classes (V 2

n,m = 0 unless |n − m| = 1). The decay width

Γextn is determined by the sum of the transmission coefficients T a

n over all open channels.

Γextn = (2πρb

n)−1∑

a

T an . (2.72)

More explicitly Γextn may be expressed through the energy integrals of the product of

transmission coefficients and level densities

Γextn = (2πρb

n)−1∑

α

m=n+1∑

m=n−1

∫

Tαn (ε)ρb

m(E − Qp − ε)dε. (2.73)

Here, ε stands for the ejecitle particle energy, Qp for its binding in a composite system,

and α symbolically accounts for the angular momentum coupling of the residual spin,

ejectile spin and orbital angular momentum to the composite nucleus spin. Again, due

to the chaining hypothesis, only those emissions which change class number by |n−m| ≤1 are allowed. Unlike the FKK [17] formulation, in the NVWY theory transmission

coefficients Tn→m carry two class indexes. Following Ref. [27] the microscopic quantities

< Wn,a > and V 2n,m are expressed in terms of optical model potentials. The matrix

element V 2n,m is related to the imaginary part of the optical model potential W (ε).

From Eq no. 2.71 we get Γ↓n = 2W (ε). To evaluate Γ↓

n,W (ε) has to be averaged over

the probability distribution of particles and holes. Once the matrix Π−1 is determined

it is inverted numerically and used in Eq no. 2.68 to calculate MSC emission spectra.

2.5.6 NVWY MSD approach

The approach to statistical Multi-step Direct reactions, in NVWY model, is based on

the Multi-step Direct (MSD) theory of pre - equilibrium scattering to the continuum

originally proposed by Tamura, Udagawa and Lenske [19].

The evolution of the projectile-target system, from small to large energy losses, in

the open channel space is described in the MSD theory with a combination of direct

reaction (DR), microscopic nuclear structure and statistical methods. As typical for the

DR approach, it is assumed that the closed channel space, i.e., the MSC contributions

have been projected out and can be treated separately within the Multi-step Compound

mechanism.

In the NVWY MSD theory the effective Hamiltonian in the open channel space is

31

divided into an energy averaged optical model part Hopt, describing the relative motion

of the projectile a and target A, the intrinsic H intr of the asymptotically separated

nuclei and the residual effective projectile-target interaction V res leading to non-elastic

processes

H = Hopt + H intr + V res (2.74)

Both H intr and V res are non-hermitian operators. To a large extent the imaginary parts

are related to the flux absorbed into the closed channels, but those open channels which

are not treated explicitly are also contributing.

In order to describe the statistical content of pre - equilibrium spectra the real states

are expanded into n-particle and n-hole model states c. Explicitly, H intr is chosen as

H intr = H intr0 + V intr (2.75)

The states c are eigenstates of unperturbed nuclear Hamiltonian H intr0 and the residual

interaction V intr couples states from different particle-hole classes only. It is assumed

that the configuration mixing between np−nh classes is stochastic in nature and leads

to a random distribution of amplitudes with mean value zero [28]. When the density

matrix is averaged over a finite energy interval, e.g. with a Lorentzian or Gaussian g(x)

of full width ∆ large compared to the mean level spacing,

ρ(E) =

∫

dE ′g(E − E ′)ρmicro(E) (2.76)

the coherence of the basis state is lost and the density matrix becomes a statistical

matrix summed over all particle classes

ρ(E) =∑

n

ρn(E). (2.77)

with

ρn(E) =∑

c=[npnh]

|c)Pc(E)(c| (2.78)

and the probability per energy to find the system in the configuration c is given by the

spectroscopic densities,

Pc(E) = − 1

πIm

[∫

dE ′g(E − E ′)(c|Gintr(E ′)|c)]

(2.79)

with Gintr(E) being the intrinsic Green’s function. Integrating Pc(E) over an interval

32

∆E one obtains the spectroscopic factor for the configuration c in ∆E. The partial

level density of np − nh states are determined by tr(ρn) (Eq. 2.78). Now the cross

section becomes an incoherent super-position of n-step contributions

d2σ

dΩdE=

∑

n

d2σn

dΩdE, (2.80)

where the multi-step cross sections are defined as

d2σn

dΩdE=

∑

c=[npnh]

Pc(E)|T nc0|2. (2.81)

Expanding V res into multi-poles Vλ and noting that only 1p − 1h configurations are

directly excited in a one-step process. Accordingly the first step cross section σ1 is

determined by an average over transitions into the 1p − 1h states c around excitation

energy E with form factors

F c0λ = (c|Vλ|0). (2.82)

Rather than treating each transition separately it is sufficient to consider averages over

the microscopic form factors. Thus, V res is represented in terms of state independent

multipole form factors Fλ and nuclear transition operators Oλ as

V res(r, ξ) =∑

λ

Fλ(r)Oλ(ξ). (2.83)

Here, r denotes the relative motion coordinate and ξ=(ξa, ξA) are the intrinsic coordi-

nates including spin and isospin respectively. The multipole form factors Fλ in turn are

related to Oλ. In a self-consistent approach they are obtained by averaging V res over

Oλ:

Fλ(r) = (c|O†λρ(E)V res|c)/Sλ(E, c). (2.84)

Here, the general case of a transition starting from an arbitrary state c is considered

which appears in the intermediate steps of higher order multi-step processes. For one

step relations the initial state is the ground state c=0. By normalization to the transi-

tion strength Sλ

(c|O†λ, ρ(E)Oλ|c) = δλλ′Sλ(E, c) (2.85)

the dependence of the form factor on the internal state is removed to a large extent. Sλ

is the nuclear response function for the external operator Oλ describing the transition

33

rate per unit energy from the state c into the ensemble of state c′ centered at energy

E. The above relations are appropriate for one-step reactions where c is the ground

state. However, in higher steps c is an arbitrary intermediate np − nh state which is

summed over in the cross section. Thus, for multi-step scattering in the form factor, Eq.

2.84 actually utilizes a too microscopic picture. The statistical aspects in multi-step

transitions are taken fully into account by the average multipole form factors

Fλ =tr(ρO†

λρV res)

tr(ρO†λρOλ)

(2.86)

which are independent of the initial state and the multi-step order respectively. In the

applications of the theory these global form factors together with the response functions

of Eq. 2.85 are used. With the above results the one-step cross section is expressed as

d2σ1

dΩdE=

∑

λ

(E)dσ1dΩ|λ, (2.87)

where σ1 is a reduced DWBA cross section calculated with the average form factors

(Eq. 2.84). The multi-step part of the theory is discussed here for two-step interactions

only. The state-independent and slowly varying two-step amplitudes read

T 2λ1,λ2 =< χ−

E|Fλ2GoptFλ1|χ+

α >, (2.88)

with Gopt being Green’s function for the optical model potential. The nuclear structure

information is now contained completely in

(0|O†λ′1, G

(intr)†(E ′1)O

†λ′2ρ(E)Oλ2G

(intr)(E1)Oλ1|0). (2.89)

By definition, the exit channel configurations are 2p− 2h states which are excited from

1p−1h states c1. Also in the first step only 1p−1h states a’s are excited. Therefore, we

only have to consider the 1p− 1h reduced parts of the two Green functions. To a good

approximation the dependence of Sλ2(E, c1) on c1 can be replaced by a dependence on

E1 by considering that the spectroscopic strength usually is located in the vicinity of

the unperturbed energy. Theoretically, this is achieved by taking the average over the

response functions belonging to states c1 at energy E1

Sλ(E,E1) =

∑

c1 Pc1(E1)Sλ(E, c1)∑

c1 Pc1(E1)(2.90)

Sλ(E,E1) =tr(ρ1(E1)O

†λρ(E)Oλ)

tr(ρ1(E1))(2.91)

34

The final result for the two-step MSD cross section is of very intuitive structure

d2σ1

dΩdE=

∑

λ1λ2

∫

dE1Sλ2(E,E1)Sλ1(E1, 0)dσ2dΩ(E,E1)|λ1λ2. (2.92)

σ2 is an averaged cross section defined in terms of the T 2-matrix elements (Eq.2.88)

which describes two-step scattering wave- mechanically as a coherent quantal process.

The total response of the intrinsic system at energy loss E is contained in the first and

second step transition strength functions. In most cases it is a good approximation to

use the ground state response functions also for the second step but at an energy shifted

by the amount of the total energy loss in the first step. The statistical treatment is

introduced in a minimal way, namely referring only to the intrinsic systems while multi-

step scattering is described quantum-mechanically as a coherent process at all steps.

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36