MICROSCOPIC MODEL FOR THE NUCLEAR INERTIA TENSOR

MICROSCOPIC MODEL FOR THE NUCLEAR INERTIA TENSOR
D. N. POENARU1,2, R. A. GHERGHESCU1, W. GREINER2
1
IFIN-HH, PO Box MG-6, 077125 Bucharest, Romania
Frankfurt Institute for Advanced Studies, J. W. Goethe
University, Pf 111932, D-60054 Frankfurt am Main, Germany
2
Received December 10, 2005
The wave function of a spheroidal harmonic oscillator without spin-orbit
interaction is expressed in terms of associated Laguerre and Hermite polynomials.
The pairing gap and Fermi energy are found by solving the BCS system of two
equations. Analytical relationships for the matrix elements of inertia are obtained
function of the main quantum numbers and potential derivative. The results given for
144Nd and 252Cf are compared with a hydrodynamical model.
1. INTRODUCTION
The potential energy surface [1, 2] in a multi-dimensional hyperspace of
deformation parameters β1, β2, …, βn gives the generalized forces acting on the
nucleus. Information concerning how the system reacts to these forces is
contained in a tensor of inertial coefficients {Bij}. Unlike the potential energy
E = E (β) which only depends on the nuclear shape, the kinetic energy is
determined by the contribution of the shape change
Ek = 1
2
n
dβ dβ j
dt
∑ Bij (β) dti
i , j =1
(1)
where Bij is the inertia (or the effective mass) tensor.
In a phenomenological approach based on incompressible irrotational flow,
the value of an effective mass Bir is usually close to the reduced mass
µ = ( A1 A2 /A) M in the exit channel of the binary system. Here M is the nucleon
mass. One may use the Werner–Wheeler approximation [3]. The microscopic
(cranking) model introduced by Inglis [4] leads to much larger values of the
inertia. By assuming the adiabatic approximation the shape variations are slower
than the single-particle motion. According to the cranking model, after including
the BCS pairing correlations [5, 6], the inertia tensor is given by [7, 8]:
Bij = 2 =2
∑
νµ
ν ∂H /∂βi µ µ ∂H /∂β j ν
( Eν + Eµ )3
(uν υµ + uµ υ ν )2 + Pij
Rom. Journ. Phys., Vol. 50, Nos. 1– 2 , P. 187–197, Bucharest, 2005
(2)
188
D. N. Poenaru, R. A. Gherghescu, W. Greiner
2
where H is the single-particle Hamiltonian allowing to determine the energy
levels and the wave functions ν ; uν and υν are the BCS occupation
probabilities, Eν is the quasiparticle energy, and Pij gives the contribution of the
occupation number variation when the deformation is changed. When the shape
is determined by one deformation parameter, ε:
Pε = 2 =
8
2
∑
ν
( )
1 ⎡ ∆ dλ
Eν5 ⎢⎣ dε
2
( ) + 2∆(ε − λ) ddλε dd∆ε −
2
+ (εν − λ )2 d∆
dε
ν
−2 ∆ 2 dλ ν dH /dε ν −2∆(εν − λ ) d∆ ν dH /dε ν ⎤⎥
dε
dε
⎦
Similar to the shell correction energy, the total inertia is the sum of
contributions given by protons and neutrons B = Bp + Bn. The denominator is
minimum for the levels in the neighbourhood of the Fermi energy. A large value
of inertia is the result of a large density of levels at the Fermi surface. In the
present work we consider a single-particle model of a spheroidal harmonic
oscillator without spin-orbit interaction for which the cranking approach allows
to obtain analytical relationships of the nuclear inertia. The result may be used to
test complex computer codes one should develop in a realistic treatment of the
fission dynamics based on the deformed two center shell model [9, 10].
2. NUCLEAR SHAPE PARAMETRIZATION
The shape of a spheroid with semiaxes a, c (c is the semiaxis along
symmetry) expressed in units of the spherical radius R0 = r0 A1/ 3 may
determined by a single deformation coordinate which can be one of
following quantities: the semiaxes ratio c/a; the eccentricity e = (1 − a2 /c 2 );
the
be
the
the
deformation δ = 1.5(c 2 − a2 )/ (2c 2 + a 2 ), or the quadrupolar deformation [11]
ε = 3(c − a)/ (2c + a) which will be used in the following, and according to
which the two oscillator frequencies are expressed as:
( )
(
ω⊥ (ε) = ω0 1 + ε ; ωz (ε) = ω0 1 − 2ε
3
3
)
(3)
and by taking into account the condition of the volume conservation
ω2⊥ ωz = (ω00 )3 where =ω00 = 41A−1/ 3 MeV, one has
(
)
ω0 = ω00 ⎡1 − ε2 1 + 2ε ⎤
3 27 ⎦⎥
⎣⎢
−1/ 3
(4)
A particularly interesting value is ε = 0.6 of a superdeformed spheroid
with the ratio c/a = 2 .
3
Microscopic model for the nuclear inertia tensor
189
3. SPHEROIDAL HARMONIC OSCILLATOR
The eigenvalues [1] in units of =ω0 are given by:
E = =ω⊥ (n⊥ + 1) + =ωz (nz + 1/ 2) = =ω0 [ N + 3/ 2 + ε(n⊥ − 2 N / 3)]
(5)
where the quantum numbers n⊥ and nz are nonnegative integers. Their
summation gives the main quantum number N = n⊥ + nz . In units of =ω0 one
has a linear variation of the energy levels in function of deformation ε. By
including the variation of ω0 with ε, one obtains the analytical relationship
εi = Ei /=ω00 = [ N + 3/ 2 + ε(n⊥ − 2 N / 3)][1 − ε 2 (1/ 3 + 2ε/ 27)]−1/ 3
(6)
in units of =ω00 . Due to the Pauli principle, each energy level εi , with quantum
numbers n⊥ and N, can accomodate g = 2 n⊥ + 2 nucleons. One has a number of
( N + 1)( N + 2) nucleons in a completely occupied shell characterized by the
main quantum number N, and the total number of states for the lowest N + 1
shells is
∑ N =0 ( N + 1)( N + 2) = ( N + 1)( N + 2)( N + 3)/3.
N
For each value of N
there are N + 1 levels with n⊥ = 0, 1, 2, …, N. When the deformation ε > 0
increases, a level with n⊥ = 0 decreases in energy and the one with n⊥ = N
increases. For some particular values of the deformation parameter there is a
crossing of several levels in the same point leading to a degeneracy followed by
an empty gap. If no spin-orbit coupling is considered for the vanishing
deformation parameter, ε = 0, one has the following sequence of magic numbers:
2, 8, 20, 40, 70, 112, 168, 240, … If now ε = 0.6 they become 2, 4, 10, 16, 28,
40, 60, 80, 110, 140, 182, … The known experimental values can be obtained
only with a spin-orbit coupling included.
The spin Σ contributes with pozitive or negative values (up or down) for
every state with quantum numbers nz, nr = 0, 1, 2, ... n⊥ and m = n⊥ − 2 nr , hence
in a system of cylindrical coordinates (ρ, ϕ, z) the wave function [12, 13] can be
written as
Ψ = nr mnz Σ = ψ m
(7)
n (ρ)Φ m (ϕ)ψ nz ( z )χ(Σ )
r
The eigenfunctions are given by
η
2 m m2 − 2 |m|
2 m
ψm
nr (ρ) = α N nr η e Lnr (η) = α ψ nr (η)
⊥
⊥
(8)
Φ m (ϕ) = 1 eimϕ
2π
(9)
190
D. N. Poenaru, R. A. Gherghescu, W. Greiner
4
ξ2
−
ψ nz ( z ) = 1 N nz e 2 H nz (ξ) = 1 ψ nz (ξ)
αz
αz
(10)
where L|nm| are the associated (or generalized) Laguerre polynomials and H nz are
r
the Hermite polynomials. The new dimension-less variables η and ξ are defined by
η=
ρ2
,
α2⊥
ξ= z
αz
(11)
where the quantities
0
= ≈ A1/ 6 ω0
M ω⊥
ω⊥
α⊥ =
0
= ≈ A1/ 6 ω0
M ωz
ωz
αz =
(12)
which depend on the nucleon mass, M, posses a dimension of a length. Their
numerical values, in fm, can be estimated by knowing that
=2 /M ≈ 41.5 MeV ⋅ fm2 and =ω00 = 41A−1/ 3 MeV. The normalization constants
( N nm )2 =
r
nr !
1
, ( N nz )2 =
(nr + | m |)!
π 2 nz nz!
(13)
are obtained from the orthonormalization conditions
∞
∫0
m
ψm
n (ρ)ψ n (ρ)ρ dρ = δ nr nr ,
r
r
2π
∫0
∞
∫ −∞ ψ n′ (z)ψ n (z)dz = δn′n
z
z
z z
Φ∗m′ (ϕ)Φ m (ϕ)dϕ = δm′m
(14)
(15)
One should take into account that the factorial 0! = 1! = 1. We shall
substitute the wave functions ψ m
n (η) and ψ nz (ξ) in the equation (2) of the
r
nuclear inertia.
4. NUCLEAR INERTIA
By ignoring the spin-orbit coupling the Hamiltonian of the harmonic
spheroidal oscillator contains the kinetic energy and the potential energy term, V:
=ω00 ⎡⎣⎢(3 + ε)η + (3 − 2ε)ξ2 ⎤⎦⎥
V (η, ξ; ε) = 1 =ω⊥ η + 1 =ωz ξ2 =
2
2
2[27 − ε2 (9 + 2ε)]1/ 3
(16)
5
Microscopic model for the nuclear inertia tensor
191
Now we are making some changes in the equation (2), by denoting the
deformation β = ε, by assuming [8, 12] that the derivative ∂H /∂ε = ∂V /∂ε , and
that the term Pij can be neglected.
3=ω00
dV=
{[ε(ε + 6) + 9]η + 2[ε(2ε + 3) − 9]ξ2 }
dε
2[27 − ε2 (9 + 2ε)]4 / 3
(17)
1 dV = 3 ⎡ f (ε)η + f (ε)ξ2 ⎤
1
2
⎦⎥
=ω00 dε 2 ⎣⎢
(18)
or
where
f1 =
ε(ε + 6) + 9
;
[27 − ε2 (9 + 2ε)]4 / 3
f2 = 2
ε(2ε + 3) − 9
[27 − ε 2 (9 + 2ε)]4 / 3
(19)
For a single deformation parameter the inertia tensor becomes a scalar
Bε = 2 =2
∑
νµ
ν ∂V /∂ε µ µ ∂V /∂ε ν
(uν υµ + uµ υ ν )2
( Eν + Eµ )3
(20)
where the summation is performed for all states ν, µ taken into consideration in
the pairing interaction.
4.1. PAIRING INTERACTION
We consider a set of doubly degenerate energy levels {εi } expressed in
units of =ω00 . Calculations for neutrons are similar with those for protons, hence
for the moment we shall consider only protons. In the absence of a pairing field,
the first Z/2 levels are occupied, from a total number of nt levels available. Only
few levels below (n) and above (n′) the Fermi energy are contributing to the
pairing correlations. Usually n′ = n . If g s is the density of states at Fermi energy
obtained from the shell correction calculation gs = dZ /dε, expressed in number
of levels per =ω00 spacing, the level density is half of this quantity: gn = gs / 2.
We can choose as computing parameter, the cut-off energy (in units of
0
=ω0 ), Ω 1 ∆ . Let us take the integer part of the following expression
Ωg s / 2 = n = n′
(21)
When from calculation we get n > Z / 2 we shall take n = Z / 2 and similarly if
n′ > nt − Z / 2 we consider n′ = nt − Z / 2.
192
D. N. Poenaru, R. A. Gherghescu, W. Greiner
The gap parameter ∆ =| G |
∑ k uk υk
6
and the Fermi energy with pairing
corellations λ (both in units of =ω00 ) are obtained as solutions of a nonlinear
system of two BCS equations
n′ − n =
kf
∑
k = ki
2 =
G
kf
∑
k = ki
εk − λ
(εk − λ )2 + ∆ 2
1
(εk − λ )2 + ∆ 2
(22)
(23)
where ki = Z / 2 − n + 1; k f = Z / 2 + n′.
The pairing interaction G is calculated from a continuous distribution of
levels
g (ε )dε
2 = λ+Ω
(24)
G
λ−Ω
(ε − λ )2 + ∆ 2
∫
where λ is the Fermi energy deduced from the shell correction calculations and
∆ is the gap parameter, obtained from a fit to experimental data, usually taken
as ∆ = 12 / A=ω0 . Both ∆ and ∆ decrease with increasing asymmetry
0
p
n
( N − Z )/A . From the above integral we get
2 2 g (λ ) ln ⎛ 2Ω ⎞
⎜ ⎟
G
⎝ ∆ ⎠
(25)
Real positive solutions of BCS equations are allowed if
G
2
∑ | εk 1− λ | > 1
(26)
k
i.e. for a pairing force (G-parameter) large enough at a given distribution of
levels. The system can be solved numerically by Newton-Raphson method
refining an initial guess
λ 0 = (nsεd + nd εs )/ (ns + nd ) + G(ns − nd )/ 2
∆ 20 = ns nd G 2 − (εd − εs )/ 4
(27)
where εs , ns are the energy and degeneracy of the last occupied level and εd , nd
are the same quantities for the next level. Solutions around magic numbers, when
∆ → 0, have been derived by Kumar et al. [14].
7
Microscopic model for the nuclear inertia tensor
193
As a consequence of the pairing correlation, the levels situated below the
Fermi energy are only partially filled, while those above the Fermi energy are
partially empty; there is a given probability for each level to be occupied by a
quasiparticle
⎡
⎤
εk − λ
⎥
υ2k = 1 ⎢1 −
(28)
2⎢
2 + ∆2 ⎥
(
)
ε
−
λ
k
⎣
⎦
or a hole uk2 = 1 − υ2k . Only the levels in the near vicinity of the Fermi energy (in
a range of the order of ∆ around it) are influenced by the pairing correlations.
For this reason, it is sufficient for the value of the cut-off parameter to exceed a
given limit Ω ∆ , the value in itself having no significance.
4.2. VARIATION WITH DEFORMATION
The following relationship allows to calculate the effective mass in units of
=2 / (=ω00 )
=ω00
Bε = 9
2
=2
∑
νµ
ν f1η + f2 ξ2 µ µ f1η + f2 ξ2 ν
(uν υµ + uµ υ ν )2
( Eν + Eµ )3
(29)
Matrix elements are calculated by performing some integrals
nz′ nr′ m′ f1 (ε)η + f2 (ε)ξ2 nz nr m = δm′m N nm N nm N nz′ N nz ⋅
r
⋅ ⎡⎢ f1
⎣
+ f2
∞
∫0
∞
∫0
dηη|m|+1e −η L|nm| (η)L|nm| (η)
r
r
dηη|m| e −η L|nm| (η)L|nm| (η)
r
r
∞
r
∫ −∞ dξe−ξ H n′ (ξ)Hn (ξ) +
2
z
∞
z
⎤
∫ −∞ dξξ2 e−ξ H n′ (ξ)Hn (ξ)⎥⎦
2
z
z
where
N n′z N nz
N nm′ N nm
r
r
∞
∫ −∞ dξe−ξ Hn′ (ξ)H n (ξ) = δn′n
2
z
∞
∫0
z
(30)
z z
dηη|m| e −η L|nm′ | (η)L|nm| (η) = δnr′ nr
r
(31)
r
so that
nz′ nr′ m′ f1 (ε)η + f2 (ε)ξ2 nz nr m = δ m′m N nm′ N nm N n′z N nz .
r
⎡
⎢⎣ δnz′ nz f1
∞
∫0
r
dηη|m|+1e −η L|nm′ | (η)L|nm| (η) + δnr′ nr f2
r
r
∞
⎤
∫ −∞ dξξ2 e−ξ H n′ (ξ)H n (ξ)⎥⎦
2
z
z
194
D. N. Poenaru, R. A. Gherghescu, W. Greiner
8
Next we can use the relationships [16]:
∞
∫0
∞
dηη|m|+1e −η L|nm′ | (η)L|nm| (η) = δ nr′ nr (2 nr + | m | +1)
r
r
(nr + | m |)!
nr !
(32)
∫ −∞ dξξ2 e−ξ Hn′ (ξ)H n (ξ) =
2
z
z
(33)
)
(
= πnz!2 ⎡δnz′ nz nz + 1 + δnz′ nz + 2 (nz + 1)(nz + 2) + δ nz′ nz −2 1 ⎤
⎢⎣
2
4 ⎥⎦
nz
which were obtained by using the recurrence relations and orthonormalization
conditions for Hermite polynomials and associated Laguerre polynomials.
Lkn ( x ) = Lkn+1 ( x ) − Lkn+−11 ( x );
(34)
(n + 1)Lkn +1 ( x ) = [(2 n + k + 1) − x ]Lkn ( x ) − (n + k )Lkn−1 ( x )
H n +1 ( x ) = 2 xH n ( x ) − 2 nH n −1 ( x )
(35)
with particular values Lk0 ( x ) = 1, L1k ( x ) = 1 + k − x , H 0 ( x ) = 1, H1 ( x ) = 2 x .
Eventually from the sum of equation (29) one is left with an important
diagonal contribution and two nondiagonal terms
k
f
2
=ω00
2 (uν υ ν )
9δ δ
=
+
|
|
+
+
+
/
δnz′ nz
B
f
(2
n
m
1)
f
(
n
1
2)
⎡
⎤
′
′
1
1
2
ε
n
n
m
m
r
z
⎣
⎦
4 r r
=2
Eν3
ν= k
(36)
(uν υµ + uµ υ ν )2
=ω00
f22
9δ δ
=
+
+
δ n′z nz + 2
B
(
n
1)(
n
2)
z
ε2
4 nr′ nr m′m
2 z
=2
( Eν + Eµ )3
ν≠µ
(37)
(uν υµ + uµ υ ν )2
=ω00
f22
9δ δ
=
−
δnz′ nz −2
B
(
n
1)
n
z
ε3
4 nr′ nr m′m
2 z
=2
( Eν + Eµ )3
ν≠µ
(38)
∑
i
∑
∑
where ki and kf have been defined above. In order to perform the summations of
the nondiagonal terms for a state with a certain ν (specified quantum numbers
nz nr m) one has to consider only the states with µ ≠ ν and nr = nr ; m′ = m for
which nz = nz + 2 or nz = nz − 2 respectively. Finally one arrives at the nuclear
inertia in units of =2 / MeV by adding the three terms and dividing by =ω00 .
There are several hydrodynamical formulae [15] of the mass parameters. For a
spherical liquid drop with a radius R0 = 1.2249 A1/ 3 fm one has Birr (0) =
=2
1.22492 A5 / 3 fm 2 = 0.0048205 A5 / 3
= 2 MAR02 = 2
15
15 41.5MeV ⋅ fm 2
When the spheroidal deformation is switched on it becomes
in
=2
MeV
9
Microscopic model for the nuclear inertia tensor
Bεir (ε) = Birr (0)
81
[27 − ε2 (9 + 2ε)]4 / 3
9 + 2ε 2
(3 − 2ε)2
195
(39)
Good results for the fission halflives of actinides have been obtained by
using an inertia larger by about an order of magnitude.
5. RESULTS
The main result of the present study is represented by the equations (36–38),
which could be used to test complex computer codes developed for realistic
single-particle levels, for which it is not possible to obtain analytical
relationships.
Fig. 1. – Top: comparison of the effective mass (in units of =2 / MeV) calculated by using the
cranking model for the proton plus neutron level schemes, only for neutrons, as well as for the
irrotational spheroidal and spherical shapes. Bottom: shell corrections for neutrons and protons, only
for neutrons, pairing corrections, and shell plus pairing corrections. On the left hand side 144Nd, on
the right hand side 252Cf.
Nuclear inertia of 144Nd and 252Cf calculated with the hydrodynamical
equations for a spherical liquid drop and spheroidal shapes are illustrated in
Fig. 1. One can see how Birr (0) increases when the mass number of the nucleus
196
D. N. Poenaru, R. A. Gherghescu, W. Greiner
10
is increased. The irrotational value Bεir (ε) monotonously increases with the
spheroidal deformation parameter ε. Due to the fact that in this single center
model the nucleus only became longer without developing a neck and never
arriving to a scission configuration when the deformation is increased, the
reduced mass is not reached as it should be in a two center model [3].
As expected, the cranking inertia calculated by using the analytical
relationships (36–38) of the spheroidal harmonic oscillator shows very pronounced
fluctuations which are correlated to the shell corrections (calculated with the
macroscopic-microscopic method [17]) plotted at the bottom of Fig. 1.
In conclusion, by using the wave functions of the spheroidal harmonic
oscillator (the simplest variant of the Nilsson model) without spin-orbit coupling
one can obtain analytical relationships for the cranking inertia. Consequently the
result may be conveniently used to test complex computer codes developed for
realistic two center shell models. This single center oscillator is not able to
describe fission processes reaching the scission configuration or ground states
with necked in or diamond shapes. When the deformation parameter is increased
the nucleus became longer and longer without developing a neck and reaching
the touching point configuration. In this way it is not possible to obtain at the
limit a nuclear inertia equal to the reduced mass of the final fragments in a
process of fission, alpha decay or cluster radioactivity. The cranking inertia is
larger than the hydrodynamical one for a spheroidal shape which is higher than
that of a spherical nucleus.
Acknowledgments. This work was partly supported by the Centre of Excellence IDRANAP
under contract ICA1-CT-2000-70023 with European Commission, Brussels, by Deutsche
Forschungsgemeinschaft, Bonn, by Gesellschaft für Schwerionenforschung (GSI), Darmstadt, and
by the Ministry of Education and Research, Bucharest.
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