A Recursive Methodto Calculate

Nuclear Level Densities

• Models for nuclear level densities

• Level density for a harmonic oscillator potential

• Simple illustrations

• Extension to general potentials

Piet Van IsackerGANIL, France

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

• Solution: c3(A,Q) calculated recursively through

n1n2n3=0

∞

∑ kn1n2n3

σ

σ∑ =A

n1n2n3=0

∞

∑ n1 +n2 +n3( )kn1n2n3

σ

σ∑ =Et /hω −

32

≡Q

cd A,Q( ) = cd−1 ′ A , ′ Q ( )cd A− ′ A ,Q− ′ Q −A+ ′ A ( )′ A ′ Q ∑

withinitialvalues

cd A =0,Q( )=δQ0

cd A,Q( ) =0, if Q<Qdmin A( )

c0 A,Q( )=2s+1( )!

A! 2s+1−A( )!δQ0

Solution method

• We need the number of solutions of

• Rewrite as

• Introduce new unknowns

• Hence we find the recurrence relation:

n1n2n3=0

∞

∑ kn1n2n3

σ

σ∑ =A,

n1n2n3=0

∞

∑ n1 +n2 +n3( )kn1n2n3

σ

σ∑ =Q

n1n2=0

∞

∑n3=1

∞

∑ kn1n2n3

σ

σ∑ =A−A'

n1n2=0

∞

∑n3=1

∞

∑ n1 +n2 +n3( )kn1n2n3

σ

σ∑ =Q−Q'

with

n1n2=0

∞

∑ kn1n20σ

σ∑ = ′ A ,

n1n2=0

∞

∑ n1 +n2( )kn1n20σ

σ∑ = ′ Q

n1n2n3=0

∞

∑ ′ k n1n2n3

σ

σ∑ =A− ′ A

n1n2n3=0

∞

∑ n1 +n2 +n3( ) ′ k n1n2n3

σ

σ∑ =Q− ′ Q −A+ ′ A

′ k n1n2n3

σ ≡kn1n2n3+1σ

cd A,Q( ) = cd−1 ′ A , ′ Q ( )cd A− ′ A ,Q− ′ Q −A+ ′ A ( )′ A ′ Q ∑

Harmonic oscillator with spin

• Simple numerical implementation:

• 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

c3 70,240( ) =896647829312727644544457613187541

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];

Comparison with Fermi-gas estimate

• Fermi-gas estimate (Bethe; cfr Bohr & Mottelson):

• Correspondence:

ρ A,E( ) =148E

exp 2π 2g εF( )E /3

ρ A,E( )⇔ c3 A,Q=Q3min+E/hω( ) /hω

g ε( )= N +1( ) N +2( ), ε =Nhω

Leonhard Euler

• L Euler in Novi Commentarii Academiae Scientiarum Petropolitanae 3 (1753) 125: Tables for the ‘one-dimensional oscillator’ problem.

ρ A,E( ) =148E

exp 2π 2E/3

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

• This is found by substracting from c3(A,Q) those states that can be constructed by acting with the step-up operator for the centre-of-mass motion. Hence:

˜ c 3 A,Qe( ), Qe =Q−Q3min A( )

˜ c 3 A,Qe( ) =c3 A,Qe( )−12′ Q e=1

Qe

∑ ′ Q e +1( ) ′ Q e +2( )˜ c 3 A,Qe − ′ Q e( )

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

• Solution: c3(N,Z,Q) can be calculated recursively or through

n1n2n3=0

∞

∑ kn1n2n3

στ

στ∑ =Aτ A+ =N,A− =Z( )

n1n2n3=0

∞

∑ n1 +n2 +n3( )kn1n2n3

στ

στ∑ = Q

c3 N,Z,Q( )= c3 N,Q- ′ Q ( )c3 Z, ′ Q ( )′ Q

∑

Shell effects

• Fermi-gas estimate (Bethe; cfr Bohr & Mottelson):

• 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.

ρ N,Z,E( ) =9gnp

4

gn εFn( )gp εF

p( )

gnpE( )−5/4

12exp 2π2gnpE/ 3

gnp =gn εFn

( )+gp εFp

( )

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 ω1 =ω2 ≡ω12≠ω3

• Energy is determined by Q12 and Q3:

• Number of configurations c3(N,Z,Q12,Q3) from:

• Calculated recursively from:

Et = Q12 +1( )hω12 + Q3 +1

2( )hω3

c3 N,Z,Q12,Q3( ) = c2 ′ N , ′ Z , ′ Q 12( )′ N ′ Z ′ Q 12

∑

×c3 N − ′ N ,Z− ′ Z ,Q12 − ′ Q 12,Q3 −N + ′ N −Z+ ′ Z ( )

n1n2n3=0

∞

∑ kn1n2n3

στ

στ∑ =Aτ A+ =N,A− =Z( )

n1 +n2( )n1n2n3=0

∞

∑ kn1n2n3

στ

στ∑ = Q12, n3

n1n2n3=0

∞

∑ kn1n2n3

στ

στ∑ = Q3

Anisotropic harmonic oscillator

• Cumulative number of levels up to energy E:

• Example: Prolate & oblate. Normal & superdeformed.

F E( ) = ρ ′ E ( )0

E

∫ d ′ E

ω = ω122ω3

3

Anisotropic harmonic oscillator

• Example 1: 38Ar for 2=0.2.

• Example 2: 56Fe for 2=0.2.

F E( ) = ρ ′ E ( )0

E

∫ d ′ E

ω12 =ω 1+13δ( ), ω3 =ω 1−2

3δ( ), δ = 45/16πβ2

hω ≈h ω122ω3

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

• Solution: c(A,E)c(0,A,E) with c(i,A,E) calculated recursively through

n=1

∞

∑ knmm=1

Ω i

∑ =A, εnn=1

∞

∑ knmm=1

Ω i

∑ =E

c i,A,E( ) =Ω i

′ A ⎛ ⎝ ⎜ ⎞

⎠ c i +1,A− ′ A ,E −εi ′ A ( )

′ A ∑

withinitialvalues

c i,A=0,E( ) =δE0

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?