Report No. 3
Submitted to Prof. Dr. Mahdi H. Jasim and Dr. Shafik S. Almola
as a study report during their supervision on my Ph.D. Thesis.
Level Density Calculations for the Nuclear Exciton States
Ahmed A. Selman,
Department of Physics, College of Science, University of Baghdad
June 22nd 2007
Final correction made: November 24th 2007
Table of Contents
Table of Contents..................................................................................1
Introduction ..........................................................................................2
I. The Level Density of the Precompound Nucleus, ω(n,E) ...............2
II. Level Densities with no Corrections ..............................................6
III. Correction due to Pauli Principle ...............................................11
IV. Corrections due to Bound – States and the Nuclear Well Finite
Depth Correction ................................................................................16
V. Pairing Correction and Fermi Gas Formula...............................19
VI. Numerical Calculations ...............................................................20
VII. Results and Discussions..............................................................23
VII.A. Level density calculations with no corrections ..........................23
VII.B. Level density calculations with Pauli correction ......................26
VII.C. Level density calculations with corrections due to Pauli
principle and the nuclear potential well with finite depth ...................29
VII.D. A final comparison .......................................................................34
VIII. Suggestions ................................................................................36
IX. The Program.................................................................................36
IX.A. Description ......................................................................................36
IX.B. Testing the Program.......................................................................38
3-1
REFERENCES ........................................................................................39
Introduction
The present report focuses on level state density calculations for three
cases: uncorrected Ericson’s formula, Williams’ formula corrected to Pauli
blocking energy, and Williams’ formula corrected to Pauli blocking energy
and finite well depth of the nuclear potential. The systems considered here
are based on the ESM approximation.
An over review is presented for the different formulae used for this
purpose. The specific formulae are listed for one- and two-component Fermi
gas system. Numerical calculations are performed based on these formulae
for each system and the results are briefly discussed.
I. The Level Density of the Precompound Nucleus, ω(n,E)
The level density of the precompound nucleus, ω(n,E), is defined as [1]
the number of states per unit energy for a given state described by the
exciton number, n, and excitation energy E. In the exciton model, the level
density is the most important quantity where the main calculations of the
differential cross section and double differential cross section are based on.
Also the transition rates of the system during the equilibration process is
based on knowledge of the nuclear level density. This quantity is sometimes
called “partial level density, PLD” in order to distinguish it from the “total
level density”, ω(E). The total level density is the sum of the level density
for all possible exciton number, n, at a given excitation energy, E, i.e.,
).1(),()( EnEn
ωω ∑=
3-2
In the present report we use ω(n,E) or ω(p,h,E) to give the same
meaning. ω(n,E) is used herein for simplicity, but when it is needed, the
other term, i.e., ω(p,h,E) is used. The quantity ω(n,E) is a nuclear property
that is not easy to measure experimentally. This measurement difficulty
rises from the instability of the nuclear excited states. The theoretical
formulations can thus only approach the description of the nuclear level
density by approximate calculations. This is mainly due to the lack of the
knowledge about the exact nuclear structure [2]. Therefore, all the theoretical
calculations of the nuclear level density are limited to some certain
conditions.
However it was confirmed that the nuclear level density, ω(n,E),
increases rapidly with n and E. Theoretical treatment must also show this
property of the nuclear level density when using the statistical models to
describe the population of states. Therefore, it is acceptable to give the
relation between ω(n,E) and E as being an exponential relation with n.
Indeed, almost all experimental calculations of the total level density,
ω(E), can be fitted to an exponential form [2]. Examples are,
[ ][ ]
[ ])2(
2exp)(
2exp)(
2exp)(
2
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
=
=
=
EaECE
EaECE
aECE
ω
ω
ω
,
3-3
where C and a are fitting constants. So, one can see that there are many
formulae used for level density calculations. However, we will focus only
on Ericson’s and Williams’ formulae for equally-spaced nuclear levels. This
equal-spacing assumption of nuclear level is described by the “Equi-distant
Spacing Model”, ESM. In the ESM, all nuclear energy levels are assumed to
be separated by an equal spacing. In other words, if one defines the single
particle state density per each MeV by g, then one can write the energy
difference between any two successive energy levels , En, as,
∆En = En+1 – En = 1/g = constant,
and the idea is illustrated in Fig.(1) schematically.
Fig.(1). Schematic representation of real (a) and ESM approximation (b) of the energy level structure of the nucleus. Note that at high energy both representations will be approximately the same. This justifies the validity of the ESM approximation for high excitation energies.
Although the ESM sounds like a crude approximation to describe the
nuclear level density, it gives acceptable results [1] when the interest is about
calculating the level density of the excited nucleus, ω(n,E). Later in the
3-4
∆Εn
Εn+1
Εn
(a) (b)
present report, we will see under what conditions the ESM fails to apply. In
general, the ESM approximation is valid for calculations performed on
nuclei with mass number A≥40 and excitation energy E≥15 MeV [1].
It is usual to use the expression “Fermi Gas system” to describe the
mechanism of nuclear system equilibration. This expression means that the
constituents of the nuclear matter are basically non interactive with each
other.
In the exciton model, one is mainly interested in defined calculations of
the level density ω(n,E). This quantity is specified by n and E. The exciton
number, n, is defined as
n=p + h (for one-component Fermi gas system) (3),
where p and h stand for the number of particles and holes, respectively. The
eq. (3) is for one-component Fermi gas system, i.e., it assumes that protons
and neutrons are indistinguishable particles.
For a two-component Fermi gas system, we have,
n = nπ + nν = pπ + hπ + pν + hν (for two-component Fermi gas system) (4),
where the subscripts ν and π stand for neutron and proton, respectively. We
shall denote the level density for one-component Fermi gas system by
ω1(n,E) and for two-component system by ω2(n,E).
3-5
In the following sections, the different expressions used to find ω(n,E)
are given for one- and two-component Fermi gas systems. At the end of this
report, several numerical calculations are performed for each case described
by the subsequent sections. To make the theory clear, a comparison is made
between examples of one- and two-component Fermi gas systems. Each
case is discussed briefly.
II. Level Densities with no Corrections
The simplest way to describe ω(n,E) is to exclude all possible (and
real) corrections during the calculations. The corrections of the level density
are,
1. Correction due to Pauli principle.
2. Spin, angular and linear momentum corrections.
3. Charge effects.
4. Finite depth of the nuclear potential well.
5. Nucleon-nucleon pairing effect.
6. Non-ESM (i.e., real) energy level spacing.
Using non of the above corrections, Ericson showed that the nuclear
level density of the excited state is given for one-component Fermi gas
system by the following relation, (Ericson’s formula for one-component
system)
)5()!1(!!
),(1
1 −=
−
nhpEgEn
nnω ,
where g is the single particle density, it corresponds to eq. (5) itself but with
exciton number set to one, i.e.,
g= ω1(1,E)= ω 1(1,0,Ep)= ω1(0,1,Eh) (6)
3-6
and here Ep (and Eh) stand for the particle’s (and hole’s) excitation energy
for a single particle (or hole), respectively.
By using successive application of the recursion relations one can get
eq.(5) for any number of particles or holes [1],
)7(
),,0(),0,(),(),,(
),1,0(1),,0(
),0,1(1),0,(
110
11
10
1
10
1
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
−==
−=
−=
∫
∫
∫
dUUEhUpEnEhp
dUUhgh
Eh
dUUpgp
Ep
E
hE
h
pE
p
ωωωω
ωω
ωω
.
As an example, we have,
,22
1),0,1(21),0,2(
,),0,1(2
0
21
01
1
EgdUgdUUgE
gEEE
===
=
∫∫ ωω
ω
etcEgdUUgdUUgEEE
,...123
1),0,2(31),0,3(
22
0
31
01 === ∫∫ ωω
g in eq.(5) is given approximately by the relation,
)8()(23 1−= MeV
FAg ,
3-7
where F is the Fermi energy given as, 2
23/2
289
omrF h
⎟⎠⎞
⎜⎝⎛=
π (m is the nucleon
mass and ro is its radius). If A is the mass number of the nucleus then this
relation is usually approximated by the following phenomenological
approximation equation, g1513AtoA
≅ , which is used in most of the practical
present numerical calculations that include nucleon induced reactions.
For two-component Fermi gas system, the nuclear level density is
given by the following relation [3] (Ericson’s formula for two-component)
)9()!1(!!!!
),(1
2 −=
−
nhphpEgEn
vv
nn
ππω ,
where is the total exciton number. Also, the exciton
number of protons is
,vv hphpn +++= ππ
,πππ hpn += and that of neutrons is so,
again we have . Eq.(9) was written when one assumes that the
single particle density g is the same for protons and neutrons. If, however,
one assumes that the single particle density for the neutrons, gν , differs from
that of the protons, gπ, then eq.(9) will be written as [3],
,vvv hpn +=
vnnn += π
( ) ( ) )10()!1(!!!!
),(1
2 −=
−
nhphpEggEn
vv
nvnv
n
ππ
ππω ,
and in this case we define
)11()(, gANg
AZAgg
AZg v =
−≅≅π .
The total level density is given by eq.(1) in both cases of one- and two-
component Fermi gas systems. Also note that in both cases, the difference of
the exciton number is given as ∆n=2, i.e., we have:
ω(E)= ω(1,E)+ ω(3,E)+ ω(5,E)+ ω(7,E)+ ...etc.
3-8
This behavior of the nuclear level density was based originally on the
idea of the exciton model due to Griffin [4]. The exciton model is a semi-
classical model that depends on the states at which particles and holes can
occupy. As stated earlier in this report, the number of (particles + holes)
gives the exciton number, n. Excitons are created when the projectile
incident on the target nucleus excites one particle (or more), and the process
cascades from the initial configuration to higher configurations. The
development of the configurations is schematically shown in Fig.(2) below.
From Fig.(2) below, one can see the importance of some of the realistic
corrections that are needed to describe the nuclear reaction in a clear manner.
For example, and since nuclear reactions deal mainly with fermions, then the
correction due to Pauli principle prohibits the existence of two (or more)
identical particles in the same state. Therefore, this will force the
configuration of Fig.(2-c), for example, to be rearranged so that Pauli
principle is not violated. One also expects that for closed shells the particles
excitation energy would be higher than for particles in non-closed shells.
Another expected behavior is that paired nucleons would also require higher
energy to be excited than un-paired nucleons,…etc. Note that in Fig.(2) one
assumed that the nucleus is treated as an entity made of one type of particles.
When one is considering two-component system, the situation becomes more
complicated because then one needs to add other restrictions on the system
such as the difference in the binding energies and pairing correction terms,
added to those corrections mentioned above.
3-9
Another restriction is seen from the figure is that the maximum number
of excitons is limited to (2A+Aa), where Aa is the mass number of the
incident projectile. However, this is considered as a severe limit because it
requires that all the nucleon particles inside the nucleus are moved to
excited levels, which requires an extreme kinetic energy for the incident
projectile. Since there are many possibilities for the nucleus excited to this
high extent to go through spallation or disassociation emitting lighter
particles (evaporation process), then the limit of total excitation of the
nucleus is of very small probability.
This extremely excited state limit is usually not considered in the
practical uses of the exciton model [1]. The density of the final accessible
states is given by the following equations for the case of one-component
system [1],
3-10
incident particle
empty levels
occupied levels
(a) before excitation (1p, 0h)
(b) first excitation (2p, 1h)
(c) second excitation (3p, 2h)
…etc…
Fig.(2) Schematic representation of the excitation process (for one component Fermi gas system) and exciton creation during nucleon-induced reaction.
)12(
121),(
)2(21),(
22
⎪⎪⎭
⎪⎪⎬
⎫
+=
−=
+
−
nEggEn
nhpgEn
f
f
ω
ω,
where the superscripts (+ and -) sand for the change in the exciton number,
∆n=±2.
III. Correction due to Pauli Principle
This correction is a very necessary one and should be taken into
account to perform accurate calculations. This correction applies for both
systems in the exciton model one- and two-component systems); and for
both particles and holes in each case. The inclusion of Pauli principle
correction complicates the calculations performed to find the level density
of the excited nucleus.
The effect of Pauli principle correction on the level density is that the
magnitude of the excitation will be lowered because there are several states
that will be blocked, thus decrease the contribution of such states in the
amount of the excitation energy. A net effect will be as if the excitation
energy E is shown to have the value of [E-Ap,h(p,h,E)], where Ap,h(p,h,E) is
the Pauli correction term. By including this term in the level density
calculations one can write [1,3],
)13()!1(!!)(
),(1
,1 −
−=
−
nhpAEg
nEn
hpn
ω ,
3-11
where the correction term, Ap,h is the Pauli blocking energy, given as [3]
).14(4
)3()1(, g
hhppA hp−++
=
Eq.(13) is Williams’ formula for finding the level density for one-
component Fermi system. For two-component system we have [3],
,)15()!1(!!!!
)(),(
1,,,
2 −
−=
−
nhphpAEgg
Envv
nvhvphp
vnv
n
ππ
πππ
πω
with the correction term given in this case by the following formula which
takes into account the different types of excitons participated in the amount
of blocked energy due to Pauli principle [3], (Pauli blocking energy)
),16(4
)3()1(4
)3()1(,,,
v
vvvvvhvphp g
hhppg
hhppA
−+++
−++=
π
ππππππ
where gπ and gν are given by eq.(11) above. The most difficulties in eq.(14
and 16) is that they are not symmetric in p and h, and not corrected for
energy, which may add some uncertainty in the calculations. Even though,
we will restrict ourselves on using these formulae and a slightly modified
formula (see below) in the present calculations because these formulae are
oftenly used in many papers [1,3 and references therein]. Eq.(15) is
Williams’ formula for two-component Fermi system.
When applying eq.(1) on both of eq.(13 and 15), one gets the total level
density of excitons as [1],
[ ] ),17(482exp),()( 11 E
aEEnEn
== ∑ ωω
3-12
for one-component and,
[ ] ),18()(
2exp12
),()( 4/1522 EaaEEnE
n
πωω == ∑
for two-component. The constant a in these equations is 6
2 ga π= .
The total level density can also be given by the renormalization of level
density for one-component [3]. This is done by calculating the ratio
ω1(U)/ω2(U), -see eq.(19) below- where U is the effective excitation energy,
[ ]
[ ]( )
)19(
)(
2exp12
)(
482exp)(
4/152
1
⎪⎪
⎭
⎪⎪
⎬
⎫
+=
=
tUa
aUU
UaUU
πω
ω
where the nuclear temperature t is related to U by the following [3],
U = a t2- t (20).
sometimes, the effective excitation energy U is related to the pairing energy.
This will be shown in report no.4.
Then the renormalized density for one-component will be given as [3],
( )).22(
)(3)(
),21(),()(),(
4/15
11
tUa
UUf
EnUfEnR
+=
=
π
ωω
3-13
Another form of equations (13 and 15) is to include a slight correction
due to energy symmetry [1], that is to use the formula
)23()()!1(!!)(
),( ,
1,
1 hp
nhp
n
EnhpAEg
nE αω −Θ−
−=
−
where Θ(E-αp,h) is the Heaviside step function defined as,
⎪⎩
⎪⎨
⎧
>−
≤−
=−Θ01
)24(
00
)(
,
,
,
hp
hp
hp
E
E
Eα
α
α ,
and the correction term αp,h is given as [1],
)25(2
)1()1(, g
hhpphp
−++=α ,
for one-component and,
,)26()()!1(!!!!
)(),( ,,,
1,,,
2 vhvphpvv
nvhvphp
vnv
n
Enhphp
AEggEn ππ
ππ
πππ
π αω −Θ−
−=
−
for two-component with,
).27(2
)1()1(2
)1()1(,,,
v
vvvvvhvphp g
hhppg
hhpp −+++
−++=
π
ππππππα
When using the Pauli principle correction to the level density, the
density of the final accessible states will be given as [1], (for one-
component)
3-14
),28(
),(/),2(),(
),(/),(),(
),(/),(),(
1
100
1
⎪⎪⎭
⎪⎪⎬
⎫
−=
=
=
+−
++
EnEnCEn
EnEnCEn
EnEnCEn
f
f
f
ωω
ωω
ωω
where the constants C+ and C0 are the numbers of realizations of the final
states. An explicit form of eq.(28) at the highest powers of E, is given by
[1],
( )
)29()()1(4
)(2)2()1()1(
)(2)2()1()1(
2),(
)29()2)(1()2)(1()()2(8
)1(1
2)2(),(
,,
,
,0
,
bAEnphnph
AEhppnpp
AEpppnpp
nAE
En
ahhppAEn
n
nhpEn
hphp
hp
hpf
hp
f
⎥⎥⎦
⎤
−−
−+−
+−−−
⎢⎢⎣
⎡+
−+−
−−×−
≅
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−+−−×
−−−
−
×−
≅−
ω
ω
and finally,
( )
)29(2
)1(85)1(
85
)(11
)1(2),(
,
2,
cnhphhpp
AEnn
nAE
Enhp
hpf
⎥⎦
⎤⎟⎠⎞
⎜⎝⎛ ++−+−
⎢⎢⎣
⎡×
−+
−+
−≅+ω
The level density of final accessible states is important for calculating
the transition rates of the system in its course to equilibration. This subject is
important to our research, therefore, this subject should be considered in the
future. However, eq.(29) is not frequently. Instead a further approximation
is made by ignoring the information of Pauli principle on the ∆n=-2
transitions, then [1]
[ ][ ] )30(
)(
)()1(2
),( 1,
11,1
−
++++
−
−
+= n
hp
nhp
fAEg
AEgngEnω
3-15
which is comparable to eq.(12).
IV. Corrections due to Bound – States and the Nuclear Well Finite Depth Correction
When a pre-equilibrium state occurs, we know that exciton theory
assumes that there are certain decay probabilities to the continuum and to
other states (which we call internal system transition rates) from that exciton
state. The potential-type affect both of these transitions but it is important to
study this effect on emission rates rather than the internal transition rates for
simplicity. If one seeks the ideal case, the effect of the potential well should
be considered for all transition rates, i.e., decay and internal transition rates.
The energy of the emitted particles depends on many properties. The
most important effect in this case is the type of the nuclear potential that the
particle was located in. When we assume a nuclear potential with finite well
depth, the emitted particles will be expected to have more discriminated
energies in the continuum, corresponding to well separated (non-overlapped)
states. In general, when emission occurs from an excited nucleus having a
finite well potential properly defined, there a good probability for some
particles to have higher emission energy.
3-16
There is also some quantum effect that is called “Continuum Effect”,
which leads to decreasing of the level density as the particle emission energy
increases. This phenomenon occurs because there is energy sharing between
nucleons in the nucleus with the emitted particle [5]. The continuum effect
may change the emission rates significantly in some cases when the energy
of the emitted particles is of order of the excitation energy of the nucleus,
which may be due to better overlapping probability between states of the
emitted particles and the nucleons inside the nucleus. Then, in order to
correct the level density for bound-states and finite depth of the nuclear
potential well, one will need to define the shape of the nuclear potential. If
the potential well was properly defined, one will have more accurate and
reliable calculations.
This important feature of correcting the calculations performed in the
exciton model was first pointed out by Blann [6,7]. Many types of potential
shapes are used to achieve this, of which the Woods-Saxon is the most
popular one [5].
For a system described by one-component configuration, the finite well
depth and bound state correction on the level density will be given as [3],
[ ] )31()(
)1()!1(!!
),(
,1
,
001
jFiBEjFiBAE
CCnhp
gEn
hpn
hp
ip
jh
jih
j
p
i
n
−−−Θ−−−
×−−
=
−
+
==∑∑
α
ω
where Chj and Cp
i are numerical coefficients, Cpi =p!/i!(p-i)! and similar
equation for Chj. F and B are Fermi energy and the nucleon binding energy,
respectively. Pauli correction Ap,h and energy symmetry αp,h terms are given
by reference [3] in the following forms
).33(2
),32(4
)1()1(
22
,
,
ghp
ghhppA
hp
hp
+=
−+−=
α
We shall use eq.(31) in the present report because it gives reliable
results that are adopted widely [3]. Some other different formulae are given,
for example ref. [1]
3-17
( )[ ] ( )( ) )34(
)1(),(
,,
,
001
jihgiFEjihgiFAE
Cgg
En
hphp
jiih
j
p
i
hh
pp
−+−−Θ−+−−
×−= ∑∑==
ππ α
ω
where gp and gh are the single particle level density for particles and holes,
respectively, and Ci,j are numerical coefficients that differ from Cip and Cj
h .
the new coefficients are given as [1],
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+≤≤
−−
==
= ∑<<<<
=+++
otherwise
ijiiihfor
jifor
C
hkkjkkk
jih
j
0
),35(2
)11(2
)1(1
01
...0...
,
21
21
which can be approximated as [1]
).36()!(!
!
00
,
ihih
ih
Ch
j
h
j
jih −
=⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑∑
==
Using this, and neglecting the terms corresponding to [gπ (ih-j)], then
the eq.(34) above can be written in the approximated form [1] (assuming that
gp=gh=g, and n=p+h),
( ) ).37()()!1(!!
)1(),(1
01 jFE
nhpjFE
jh
gEnn
jh
j
n −Θ−
−−⎟⎟
⎠
⎞⎜⎜⎝
⎛≅
−
=∑ω
Yet there is another form to include the finite depth and bound state
correction which is [1] (we give these formulae for better review of the
theory, although not used in the present calculations performed here)
( )
).38()(
)!1(!!)1(
)!1(!!),(
,
1,
001
jFiBE
nhpjFiBAE
ip
jh
nhpgg
En
hp
nhpj
p
i
h
j
hh
pp
−−−Θ
×−
−−−−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
=−
==∑∑
α
ω
3-18
For a two-component Fermi gas system we have the massive formula [3],
[ ]
( ) ),39(
)1()!1(!!!!
),(
,,,
1,,,
2
vvvvvhvphp
nvvvvvhvphp
vjvh
vivp
jh
ip
vjvijivh
vj
vp
vi
h
j
p
ivv
vnv
n
BjBiFjBiE
BjBiFjBiAECCCC
nhphpgg
En
−−−−−Θ
×−−−−−
×−−
=
−
+++∑∑∑∑
ππππππ
ππππππππ
ππ
πππ
π
π
πππ
ππ
α
ω
with Pauli term given as,
),40()1()1()1()1(41
,,, ⎥⎦
⎤⎢⎣
⎡ −+−+
−+−=
v
vvvvvhvphp g
hhppg
hhppAπ
ππππππ
and finally the energy symmetry term
).41(22
2222
,,,v
vvvhvphp g
hpg
hp ++
+=
π
ππππ
α
It is common to sum over all possible values of the exciton number, n, to see
the behavior of the eq.(39) above.
V. Pairing Correction and Fermi Gas Formula
In order to have a more realistic frame to describe nuclear level
density, one should introduce as many as possible of the realistic corrections.
So far we are dealing here with the ESM concept. Although this
approximation gives good agreement with experiment, it still has some
difficulties.
3-19
The most important defect in the ESM is that it ignores the
considerable shift in the energy levels due to nucleon-nucleon interaction.
This shift of energy is not essentially included in the idea of the ESM, so one
needs to add some correction factor that deals with energy shift due to
nucleon-nucleon interaction.
Experimentally, it is confirmed that nucleons of the same type tend to
couple together in pairs. This is seen from the higher nucleon separation
energy for even Z and even N nuclei. Moreover, the even-even nuclei have
their position on the top when concerning nucleon separation energy or
excitation energy. This is a natural phenomenon that characterizes the
nuclear forces. To break this pairing between nucleons, one needs an extra
energy about 1~2 MeV.
Pairing energy also affects the level density, because the amount of
energy added to the excitation energy will highly influence the excitation
process and exciton formation, thus level density of the excited nucleus. In
experiments dealing with level density calculations it is very useful to add
the pairing energy as a leading parameter [2].
The correction of pairing energy will be left for the next report
(Repot No.4) because it possesses a somewhat prolonged formalism used for
the practical calculations.
VI. Numerical Calculations
As an application for the purpose of better understanding the present
approximations, three numerical examples are performed here to calculate
the level density of the precompound nucleus. The present report presents
different equations used to calculate the level density according to the theory
of the exciton model with three corrections for the cases of one- and two-
component Fermi systems.
3-20
The present corrections include:
1-Pauli principle effect,
2- Finite depth of the nuclear potential well and bound state effects.
In order to make more meaningful use of the presentation above, three
examples were studied numerically using codes written for this purpose.
The code used is named “PEESM”-see paragraph IX at the end of the present
report-, and it consists of many subroutines that are written in (MATLAB
R2006b), and are compatible to work on any other (older or newer) version
of MATLAB. The present programs can be compiled via the mcc MATLAB
compiler and distributed as stand-alone application programs (just as any
code written in FORTRAN language); so that users who do no have
MATLAB program installed on their computers can also operate the present
codes. The programs written here, however, also include direct plotting of
the results by the MATLAB GUI (Graphical-User Interface). However,
stand-alone versions can be made to produce numerical output that are saved
in text files.
Using the MATLAB program; therefore, guarantees the two most
important conditions and specifications met by FORTRAN and C/C++
languages, which are compatibility (work on any computer machine) and
portability (application with extension of .exe which have a relatively small
size and can be saved on floppy diskettes).
The examples which are considered in the present report include the
following three main cases:
1- Level density calculations for one- and two-component systems with no corrections,
3-21
2- Level density calculations for one- and two-component systems with Pauli principle corrections,
3- Level density calculations for one- and two-component systems with corrections due to finite depth of the nuclear potential well and bound states.
and the target nucleus and other specifications of the numerical study of
level density calculations are listed in Table (1) below.
Table(1). The parameters used in the present numerical study of level density calculations
Target nucleus under investigation 28
5426
Fe
Mass number, A 54 Atomic number, Z 26 Maximum excitation energy, E 100 MeV Exciton number 5
Exciton number configurations one-component: (3,2) [= p,h] two-component: (3,2,0,0) [=pπ, hπ, pv, hv]
Single particle density, g A/13 (MeV-1)
Nucleon’s binding energy Proton: 8 (MeV) Neuron: 10 (MeV)
Finite depth of the nuclear potential well (this equals to the Fermi Energy)
Proton:38 (MeV) Neutron: 40 MeV
3-22
The results of each example are given in the next paragraph. Future
reports will include further corrections of the nuclear level density such as
pairing and charge corrections.
VII. Results and Discussions
VII.A. Level density calculations with no corrections
The one- and two-component nuclear level density calculated using
Ericson’s formula (with no corrections) is shown in Fig.(3), based on eq.(5) -
for one-component) and on eq.(10) -for two component-. The specific
calculation parameters are those taken from Table(1).
Fig.(3). The results of the level density for one- and two-component Fermi gas system based on Ericson’s formula. The two-component configuration is (3,2,0,0).
3-23
From Fig.(3) one can see clearly that the two-component results are
less than those of the one-component. This behavior is expected physically
because, as the nature of calculations suggests, two-component system will
have to share the energy with more entities. The extra entities are those due
to the neutron particle and holes. The effect is apparent from the concept of
the single-particle density, g, where for one-component only one entity
(indistinguishable particles or holes) will share the excitation energy. For
two-components, on the other hand, to have the ability to distinguish the type
of the exciton particle or hole, we need to give different population of states
for each type of fermions (neutrons or protons), thus we define gπ and gν.
Although pν and hν are set to zero, the effect is still clear. Let’s take an
example here. In the present calculations, the system under study was taken
as, n=5 for both cases, in the configurations (3,2)-for one-component- and
(3,2,0,0)-for two-component-. Now since -see eq.( 11)-,
( )( )
( )38.6
10.02587/
14814.05426
5
5
5
1
2 ≈=⎟⎠⎞
⎜⎝⎛===⇒
<===⇒≅
AZ
gAZg
gg
AZ
gg
gAZg
n
nππ
ππ
ωω
3-24
so even if we set pν and hν equal zero, Ericson’s formula for two-component
system will still give less value of the level density than that of one-
component system. The ratio between ω2/ω1 is large, but this difference is
accepted [1,3]. Some researchers try to overcome this by defining a single-
particle density for each exciton individually. This means that we define (gp,
gh) for one-component system and (gpπ, ghπ, gpν, ghν) for two-component
system [1]. However, this will add extra effort in programming and is not
improving the calculations. It is known that Ericson’s formulae for one- and
two-component systems actually overestimates the level density in a
considerable manner [1,3]. So practical calculations should give as much
less of these formulae as possible. We shall see when adding Pauli and the
finite depth corrections that the level density will be decreased as we add
more corrections.
Fig.(4) shows a comparison between different configurations for Ericson’s
formula for two-component system, all have n=5. The configurations
(3,2,0,0) and (2,3,0,0) give the same results so they are undistinguished in
Fig.(4). The configuration (4,1,0,0) is the lowest while the configuration
(2,1,1,1) is the highest.
Fig.(4). A comparison of the level density for two-component Fermi gas system based on Ericson’s formula, eq.(9). Different configurations are considered. The exciton number is n=5, and the case of n=3 is also added for comparison.
The plot for one-component for n=3 was also added for comparison,
where one see that n=3 is higher than any configuration of n=5. The case of
one-component with n=5 is even higher that n=3, indicating the effect of
level density reduction for the two-component than that of one-component in
a very clear manner.
3-25
n=3
n=5 26
54Fe28
VII.B. Level density calculations with Pauli correction
The results are shown in Fig.(5). These results are based on,
respectively for one- and two-component, eq.(23) and eq.(26), using Pauli
factor from eq.(14) and (16) and the energy symmetry term from eq.(25) and
eq.(27).
Fig.(5). The results of the level density for one- and two-component Fermi gas system based on Williams’ formula, with Pauli correction added. The two-component configuration is (3,2,0,0).
Here the other form of Williams’ formulae, eq.(13) and eq.(15) were
not used because they are not corrected for energy symmetry term, αp, h.
3-26
In this case one can also see that for a Fermi gas system made of one-
component the results of the nuclear level density are higher than those for
two-component, and this is similar to the case of Ericson’s results. A
comparison between the two lines shown in Fig.(5) also show that the
maximum value for one-component is to be somewhat higher than that of
Fig.(4) above, which is as expected since in Williams’ formulae corrected to
Pauli principle, the correction will apply for the energy. Pauli energy, Apπ,, hπ,
pν, hν, for the case (pν=hν=0) of the two-component, eq.(15), is actually not
the same for one-component. The effect here is, again, to the single-particle
density difference, gπ and g. Consider the example:
1
1
,
,,,
1
2
,
,,,
4)3()1(
4)3()1(
!!!!!!
0
4)3()1(
,4
)3()1(4
)3()1(
−
−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−++−
−++−
=⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−
×=∴
⇒==
−++=
−+++
−++=
n
n
vnv
nn
hp
vhvphp
vvn
vnv
n
vv
hp
v
vvvvvhvphp
ghhppE
ghhpp
E
ggg
AEAE
hphphp
ggg
hp
ghhppA
ghhpp
ghhpp
A
π
ππππ
ππππ
ππ
ππ
π
ππππππ
ωω
Q
where hv = pv = nv= 0. Then we have, 1
1
2
4)3()1(
4)3()1( −
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−++−
−++−
=
n
n
n
ghhppE
ghhpp
E
gg π
πππππ
π
ωω
1
1
)3()1(4/
)3()1(4
4)3()1(4
4)3()1(4
−
−
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−+−
−−+−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−+−
−−+−
=
n
nn
nn
n
n
n
ghhppgE
AgZhhppEg
gAgZ
ghhppgE
ghhppEg
gg
πππππ
π
πππππ
ππ
3-27
1
))3()1(4()3()1(4 −
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−
−−+−=
n
nn
nn
hhppgEhhppEg
ZA
gAgZ πππππ
Putting:
4
4
1
2
10410/4
))32(2)13(34()32(2)13(3/4,2,3
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−
−−+−=⇒====
gEAZgE
AZ
gEAZgE
AZhhpp
ωω
ππ
and if we used: A=54, Z=26, E=100, g=A/15, we’ll have,
( ) 2
1
2 1020521418.05426 −×≈=
ωω
.
which explains the point.
A comparison of this case for different exciton configurations is shown
below in Fig.(6). The same behavior seen before in Fig.(4) is repeated here,
for the same reasons.
Fig.(6). A comparison of the level density for two-component Fermi gas system based on Williams’ formula with Pauli correction for energy. Different configurations are considered. The exciton number is n=5, and the case of n=3 is also added for comparison.
3-28
n=3
n=5
VII.C. Level density calculations with corrections due to Pauli principle and the nuclear potential well with finite depth
The results of this case are shown in Fig.(7), and they are based on,
respectively for one- and two-component, eq.(31) and eq.(39), using the
accompanied set of equations.(32) and (33) for one-component and the
eqs.(40) and (41) for two-component systems. A plot with linear y-axis of
the same figure one will get the Fig.(8). Also a comparison is made in this
case but not for different exciton configurations as before, but for different
values of the system’s Fermi energy, F, and the nucleons’ binding energy, B.
The results are shown in Fig.(9) for different values of F and in Fig.(10) for
different values of B.
Fig.(7). The results of the level density for one- and two-component Fermi
gas systems based on Williams’ formula, with both of Pauli and the finite well depth corrections added. The two-component configuration is (3,2,0,0). Values for Fermi energy and binding energy for each case are taken from the Table(1).
3-29
Fig.(8). Same as Fig.(7) with linear y-axis.
3-30
Fig.(9). A comparison of the level density for one-component Fermi gas system based on Williams’ formula with Pauli and finite depth corrections. Different Fermi energies, F, are considered.
Fig.(10). A comparison of the level density for one-component Fermi gas
system based on Williams’ formula with Pauli and finite depth corrections. Different Binding energies, B are considered.
An interesting behavior of the system seen from Fig.(7) is that at
energy E~90 MeV, the two-component level density results are higher than
the one-component level density results. From Fig.(8), one sees the perfect
behavior of the two-component system which continuously increases with
the excitation energy, E. One-component reduction as the energy increases
strongly suggests that if one wishes to correct the nuclear level density to the
finite depth of the nuclear well potential, two-component system should be
used. However, we used one-component system for further comparison of
the system’s Fermi energy, F, and nucleon’s binding energy, B, because
from the one-component system one can see exactly how the result would be
when changing F or B for one time. These results are shown in Fig.(9) and
Fig.(10).
3-31
From changing the Fermi energy of the system -Fig.(9)- one can notice
that as the value of F increases the level density of the system increases for
the same excitation energy E and for the same binding energy, B. This is a
very interesting because if we remember the definition of the Fermi energy,
it is the energy value that lays half the way between the last filled and first
unfilled energy level. Fermi energy is, therefore, considered as a proper
indication for the shape of the nuclear potential. We also have:
1- if one looks at eq.(31), we see there is the term (E-iF-jB)n-1.
Lets consider(E-iF-jB) as the effective excitation energy.
When E=iF+jB, we’ll have the above term equals to zero!
This means that the effective excitation energy is zero, leading
thus to no level density value.
2- if F or B increases, we have (E-iF-jB), and not (E-F-B). The
summation indices i and j, have values from zero to p and h,
respectively -see eq.(31)-. So the effective excitation energy
can has a negative value, but this is given to the power (n-1),
since n in the exciton model reads an odd value (usually
always), we have (n-1) = even, so (-ve)n-1=+ve ..always!
From the above, when the value of Fermi energy F increases one expects
that the term (effective energy)n-1 will increase. Let’s consider a numerical
example to explain this point clearly:
Let p=3, h=2, E=10 MeV, F=38 MeV, B=10 MeV.
We have n=5, and the effective excitation energy in this case is
Eeff=[10-i (38) - j (10) ] for i =0 to 3 and j=0 to 2. The minimum effective
excitation energy is at i=j=0, leading to Eeff=10 MeV, so
[Eeff]5-1=[10]4 =104.
3-32
The maximum of Eeff is at p=3 and h=2, then Eeff =10 - 3x38 - 2x10 = 10-
114-20=-124. Then
[Eeff]4=[-124]4= + 236421376= 2.3 x 108..!
now let’s do the same calculations with F=50 MeV, one will get minimum of the Eeff=10 MeV and maximum Eeff=-160 MeV thus, we will have the following value
[Eeff]4=655360000=6.5x108.
The same considerations apply completely for the binding energy, B, as
seen from Fig.(10), where as the binding energy increases the value of the
level density also increases.
Fig.(11) shows the comparison of this correction (Williams’ with Pauli
and finite depth) for two-component system with different exciton
configurations. The values of F and B are those taken from Table(1). The
configuration (1,0,2,2) represents an anomaly.
Fig.(11). A comparison of the level density for two-component Fermi gas
system based on Williams’ formula with Pauli and finite depth corrections. Different exciton configurations are considered.
3-33
Log
ω2(n
,E)
VII.D. A final comparison
Finally we compare the three cases for one- and two-component
systems.
Fig.(12). A comparison of the level density for one-component Fermi gas
system based on Ericson’s, Williams’ formula with Pauli and with Pauli and finite depth corrections.
3-34
Fig.(12) shows this comparison for one-component. The curves for
Williams’ with Pauli correction only interferes with the other curves so we
plot the same figure as log-log plot shown in Fig.(13). From Fig.(13) one
can see that Williams’ with Pauli correction starts near Williams’ with Pauli
and finite depth corrections and terminates near Ericson’s results. Also one
can see that as we add more corrections the level density values decrease
which is expected since these corrections will minimize the effective
excitation energy. The last figure, Fig.(14) shows the same comparison but
for two-component system. The same behavior is seen from these results.
Fig.(13). Same as Fig.(12) with both axes as logarithmic.
Fig.(14). Same as Fig.(12) for two-component system, and with both axes as
logarithmic.
This concludes the present calculations.
3-35
Log
Log
ω2(n
,E)
VIII. Suggestions
The present treatment needs to be improved for other corrections. The
improvement due to other corrections, such as pairing, shell structure,
angular and linear momentum…etc, shall be considered in coming reports.
Also it is important to perform these calculations with non-ESM approach.
IX. The Program
IX.A. Description
The program used here is called “PEESM1”, written in MALAB
language. The name “PEESM” stands for “Pre-Equilibrium with Equidistant
Spacing Model”.
The program PEESM1 is the main (calling) routine. It calls eight
subroutines that are classified as below:
3-36
1- strt.m subroutine: which starts (initializes) the variables
required for the entire program. This routine actually reads an
external input file named peesm.inp where all the input
parameters required are fed into the code. The shape of
peesm.inp is given as blow. Currently this is being read by
strt.m as a matrix of 5x3, in the future this will be modified to
be read line by line in order to include detailed (text)
description for each case. The letters in this input program
refer to:
p(pi)=pp, h(pi)=hp, p(nu)=pn, h(nu)=hn, Emax=maximum excitation energy, Ef(pi)=Fermi energy for protons, Ef(nu)=Fermi energy for neurons, B(pi)=binding energy for protons, B(nu)=binding energy for neutrons, A=mass number of the specified nucleus, Z=its atomic number, d=spacing of the ESM (d=13 or 15 used in g=A/d)
When dealing with one-component of any case, then the values of
p(pi)=p, h(pi)=h, and p(nu) and h(nu) are ignored, Ef(pi)=Fermi energy
for particles, Ef(nu)=binding energy, B(pi) and B(nu) are ignored, and Z
is ignored. At least one space is required between adjacent columns.
The last two zeros can be ignored because they are not read in the code.
n p(pi) h(pi) p(nu) h(nu) Emax Ef(pi) Ef(nu) B(pi) B(nu) A Z d 0 0
(the shape of peesm.inp input file to PEESM1.m program)
2- Ericson1.m subroutine: this calculates the one-component
Ericson’s formula.
3- Ericson2.m subroutine: this calculates the two-component
Ericson’s formula.
4- Williams1.m subroutine: this calculates the one-component
Williams’ formula (with Pauli correction only).
5- Williams2.m subroutine: this calculates the two-component
Williams’ formula(with Pauli correction only).
6- WilliamsF1.m subroutine: this calculates the one-component
improved Williams’ formula (with Pauli and finite well depth
corrections).
7- WilliamsF2.m subroutine: this calculates the two-component
improved Williams’ formula (with Pauli and finite well depth
corrections).
3-37
8- DP4PEESM.m subroutine: it is a plotting subroutine.
IX.B. Testing the Program
To program for scientific purpose is to write a sequence of logical steps
that follow a given equation. We first learned that: “One can write anything
in any programming language and name it -a program-, and it might work!..
But.. who says it is correct?” So in order to check that the present code
“works and correct,” some check is needed for proving is consistency. I refer
to Table(1) of ref.[1] where some numerical values for two specified
examples are given, calculated by Ericson’s formula for one-component -
eq.(5)-. In there, the first example is by setting:
n=3 (p=2, h=1), A=50, Emax=15 MeV and d=13 MeV-1. The result of [1]
for ω(n,E) was 3200. If we make these values as input to PEESM1.m, we
find that our value is ω(n,E) =3200.4, with percentage error ~ 0.0125%.
n=9 (p=5, h=4), A=50, Emax=15, d=13, the results of [1] is 4.1x106, and our
result is 4.065x106 with percentage error 0.85%.
The second example in [1] is:
n=3 (p=2, h=1), A=200, Emax=100 MeV and d=13 MeV-1. The result of [1]
for ω(n,E) was 9.1x106. If we make these values as input to PEESM1.m, we
find that our value is ω(n,E) =9.103x106, with percentage error ~ 0.003%.
n=43 (p=22, h=21), A=200, Emax=100 MeV and d=13 MeV-1. The result of
[1] was 1.4x1043, and our result is 1.373x1043, the percentage error is 1.8%.
3-38
I also refer to Table(2) of reference [1], where Williams formula for
one-component with Pauli correction is given. The values given there were
also checked with PEESM1.m results and the percentage errors were also
about ~ 1-2 %. So our program seems to be “works and correct”! A. A. Selman.
REFERENCES
[1] E. Bĕták and P. E. Hodgson, “ Particle-Hole State Density in Pre-equilibrium Nuclear Reactions”, University of Oxford, available from CERN Libraries, Geneva, report ref. OUNP-98-02 (1998).
[2] P. E. Hodgson, “Nuclear Reactions and Nuclear Structure”, Clarendon Press, Oxford, (1971).
[3] M. Avrigeanu and V. Avrigeanu, “Partial Level Density for Nuclear Data Calculations”, Comp. Phys. Comm. 112(1998)191. also available from: Cornell University online publications, ref. arXiv:physics/9805002.
[4] J. J. Griffin, “Statistical Model of Intermediate Structure”, Phys. Rev. Lett.,17(1966)478.
[5] Ye. A. Bogila, V. M. Kolomietz, A. I. Sanzhur, and S. Shlomo, “Preequilibrium Decay in the Exciton Model for Nuclear Potential with a Finite Depth”, Phys. Rev. C53(1996)855.
[6] M. Blann, “Importance of Nuclear Density Distribution on Pre-Equilibrium
Decay”, Phys. Rev. Lett., 28(1972)757. [7] M. Blann, “Preequilibrium Decay”, Ann. Rev. Nucl. Sci. 25(1975)123.
3-39
