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A Recursive Method
to Calculate
Nuclear Level Densities
Piet Van Isacker
GANIL, France
• Models for nuclear level densities
• Level density for a harmonic oscillator potential
• Simple illustrations
• Extension to general potentials
Models for nuclear level densities
• « An Attempt to Calculate the Number of Energy
Levels of a Heavy Nucleus » (Bethe 1936): Statistical
analysis of Fermi gas of independent particles.
• Numerous extensions: eg back shift.
• « Theory of Nuclear Level Density » (Bloch 1953);
« Influence of Shell Structure on the Level Density of a
Highly Excited Nucleus » (Rosenzweig 1957): ‘Exact’
counting methods in single-particle shell model.
• Numerous extensions (Zuker, Paar, Pezer,... ).
• « Nuclear Level Densities and Partition Functions
with Interactions » (French & Kota 1983): Effects of
residual interaction via spectral distribution method.
• « [] Level Densities [] in Monte Carlo Shell Model »
(Nakada & Alhassid 1997); « Estimating the Nuclear
Level Density with the Monte Carlo Shell Model »
(Ormand 1997): ‘Exact’ shell-model calculations.
Level density in a harmonic oscillator
• Question: How many (antisymmetric) states with an
energy Et exist for A particles in an isotropic HO?
• Answer: Given by the number of solutions of



k
 n1n2n3  A
n 1 n 2 n 3 0 

  n
1
n 1 n 2 n 3 0 
 n2  n3 kn 1 n 2 n3

3
 Et /    Q
2
• Solution: c3(A,Q) calculated recursively through
cd A,Q   c d 1 A, Qcd A  A,Q  Q A  A 
A Q
with initial values
cd A  0,Q   Q 0
cd A,Q  0, if
c0 A,Q 
Q  Qdmin A
2s  1! 
Q0
A! 2s  1 A !
Solution method
• We need the number of solutions of


 kn1n2n3  A,

n 1 n 2 n 3 0 



n

n

n

k
  1 2 3 nnn
1 2 3
n1 n 2 n 3  0 
Q
• Rewrite as


   k

n 1 n 2 0 n 3 1

n 1n 2 n 3

   n

n 1 n 2 0 n 3 1
1
 A  A'
 n2  n3 kn1 n2 n 3  Q  Q'
with

 k
n 1 n 2 0 

n1n2 0
 A,

  n
n 1 n 2 0 



n
k
 Q

2
n
1
1 n2 0

• Introduce new unknowns k n1 n 2 n 3  kn 1 n 2 n 3 1



k

 n1n2n3  A  A
n 1 n 2 n 3 0 


n

n

n
k



  1 2 3 nnn
1 2 3
n 1 n 2 n 3 0 
 Q  Q A  A 
• Hence we find the recurrence relation:
cd A,Q   c d 1A, Qcd A  A,Q  Q A  A
A Q
Harmonic oscillator with spin
• Simple numerical implementation:
spin=1/2; deg=2*spin+1;
c[d_,aa_,qq_]:=c[d,aa,qq]=
Sum[c[d,aa-aap,qq-qqp-aa+aap]*c[d-1,aap,qqp],
{aap,0,aa},{qqp,qqmin[d-1,aap],qq-aa+aap-qqmin[d,aa-aap]}];
c[d_,aa_,qq_]:=Binomial[deg,aa]/; d==0 && qq==0;
c[d_,aa_,qq_]:=1/; aa==0 && qq==0;
c[d_,aa_,qq_]:=0/; aa==0 && qq!=0;
c[d_,aa_,qq_]:=0/; qq<qqmin[d,aa];
• c3(A,Q) can be calculated to very high excitation.
• Example: The number of independent Slater
determinants for A=70 (s=1/2) particles at an
excitation energy of 30 hw is
c 3 70,240  896647829312727644544457613187541
Comparison with Fermi-gas estimate
• Fermi-gas estimate (Bethe; cfr Bohr & Mottelson):
 A,E 
1
2
exp 2 g F E / 3
48E
• Correspondence:
 A,E   c3 A,Q  Q3mi n  E /  / 
g   N  1N  2,   N 
Leonhard Euler
• L Euler in Novi Commentarii
Academiae Scientiarum
Petropolitanae 3 (1753) 125:
Tables for the ‘one-dimensional
oscillator’ problem.
 A,E 
1
2
exp 2 E / 3
48E
Enumeration of spurious states
• Only states that are in the ground configuration with
respect to the centre-of-mass excitation are of interest.
• c3(A,Q) includes all solutions. Let us denote the
physical solutions as
c˜ 3 A,Qe , Qe  Q  Q3mi n A
• This is found by substracting from c3(A,Q) those
states that can be constructed by acting with the stepup operator for the centre-of-mass motion. Hence:
c˜ 3 A,Qe   c3 A,Qe  
Qe
1
 2 Qe 1Qe 2c˜3 A,Qe  Qe
Qe 1
Harmonic oscillator with isospin
• Question: How many states with an energy E exist
for N neutrons and Z protons in a HO?
• Answer: Given by the number of solutions of




k

A
 n1n2n3
n 1 n 2 n 3 0 
A

 N, A   Z 


n

n

n
k


  1 2 3 nnn
1 2 3
n 1 n 2 n 3 0 
= Q
• Solution: c3(N,Z,Q) can be calculated recursively or
through
c 3N,Z,Q   c 3N,Q- Qc3 Z,Q
Q
Shell effects
• Fermi-gas estimate (Bethe; cfr Bohr & Mottelson):
N, Z,E  
4
np
9g
gn  nF gp  Fp 
g E
5 / 4
np
12
exp 2 gnp E / 3
2
gnp  gn  Fn  gp  Fp 
• The quantity c3(N,Z,Q) can be evaluated for closed
as well as open shells => effects of shell structure on
level densities.
• Example: Comparison of 16O and 28Si.
Anisotropic harmonic oscillator
• So far: independent particles in a spherical HO =>
interaction effects (eg deformation) are not included.
• The analysis can be repeated for an anisotropic HO
with different frequencies w1, w2 and w3.
• Example: Axial symmetry with  1   2   12   3
• Energy is determined by Q12 and Q3:
Et  Q12  1 12  Q3  2  3
1
• Number of configurations c3(N,Z,Q12,Q3) from:




k

A
 n1n2n3
n 1 n 2 n 3 0 

 n1  n2  k n n n
n 1 n 2 n 3 0


1 2 3
A

 N, A   Z 

= Q12 ,

n
k
 3 n1n2n3 = Q3
n 1 n2 n 3  0 
• Calculated recursively from:
c 3 N,Z,Q12 ,Q3  
 c N , Z , Q 
N Z Q12
2
12
 c 3 N  N ,Z  Z,Q12  Q
12,Q3  N  N  Z  Z 
Anisotropic harmonic oscillator
• Cumulative number of levels up to energy E:
E
FE    E dE 
0
• Example: Prolate & oblate. Normal & superdeformed.
  3 1223
Anisotropic harmonic oscillator
• Example 1: 38Ar for 2=0.2.
• Example 2: 56Fe for 2=0.2.
E
FE     E dE 
0
 12   1  13  ,  3   1 23  ,   45/ 16  2

3
 122 3  41A 1/ 3 MeV
Extension to general potentials
• Assume single-particle levels with energies n and
degeneracies n with n=1,2,…
• Question: How many A-particle states with energy E?
• Answer: Given by the number c(A,E) of solutions of

i
 k
n 1 m 1
nm
 A,
i

 k
n 1
n
m 1
nm
E
• Solution: c(A,E)c(0,A,E) with c(i,A,E) calculated
recursively through
 i 


ci, A,E   
ci  1, A  A, E   i A
 
A  A
with initial values
ci, A  0,E    E 0
ci, A,E   0, if
E  HF energy
Conclusions
• Versatile approach to compute level densities of
particles in a harmonic oscillator potential which
includes spin, isospin, deformation... (but without
residual interactions).
• Extension to a general potential [cfr. (micro)canonical
partition function for Fermi systems, S.Pratt, PRL 84
(2000) 4255].
Perspectives (general potential)
• Systematic use in combination with Hartree-Fock
calculations (eg for astrophysics).
• Spurious fraction of states can be estimated.
• Effects of the continuum can be included.
• Inclusion of interaction effects?