# The nuclear moment of inertia and spin distribution of nuclear levels

###### Abstract

We introduce a simple model to calculate the nuclear moment of inertia at finite temperature. This moment of inertia describes the spin distribution of nuclear levels in the framework of the spin-cutoff model. Our model is based on a deformed single-particle Hamiltonian with pairing interaction and takes into account fluctuations in the pairing gap. We derive a formula for the moment of inertia at finite temperature that generalizes the Belyaev formula for zero temperature. We show that a number-parity projection explains the strong odd-even effects observed in shell model Monte Carlo studies of the nuclear moment of inertia in the iron region.

###### pacs:

21.60.-n, 21.60.Cs, 21.10.Hw, 05.30.-d## I Introduction

The shell model Monte Carlo (SMMC) method has proven to be quite accurate for calculating the nuclear level density in the range of excitation energies up to several tens of MeV na97 ; al99 ; al00 . The advantage of the SMMC method is that it can be used to calculate thermal observables in model spaces that are orders of magnitude larger than can be treated by conventional diagonalization methods. In practice, most of the SMMC calculations are carried out in truncated spaces (e.g., one major shell), hence the limitation on the excitation energy. Correlations become less important at higher temperatures, and the results of the truncated SMMC calculations can be extended to higher temperatures or excitation energies by taking into account the effects of a larger space in the independent-particle model al03 .

The SMMC method can also be used to calculate the distribution of nuclear spins at finite temperature spin-proj . However, this requires spin projection, and the associated computational effort is rather large. For general purposes such as constructing tables for large numbers of nuclei, simplified models are thus invaluable. The aim of this paper is to construct and study such a simple model that can reproduce well the spin distributions of the microscopic SMMC method.

A common assumption il92 ; ra97 in global parameterizations of nuclear level densities is that the spin distribution follows the spin-cutoff model

(1) |

Here is the total level density counting all states in a multiplet, while is the density of spin- levels without the degeneracy factor. Thus . The parameter is known as the spin cutoff parameter. The quantities and are all functions of excitation energy.

The model can be derived assuming that the individual nucleon spins add up as independent random vectors er60 , , leading to a Gaussian distribution of the spin vector . Integrating over the orientation of , we have , where J is the magnitude of the angular momentum (the pre-exponential factor comes from the Jacobian in spherical coordinates). We recover Eq. (1) by making the semiclassical substitution ; the spin cutoff parameter is then given by

(2) |

In thermal ensembles it is common to define an effective moment of inertia by the relation between and temperature , which we can write as

(3) |

In many of the empirical parameterizations, is determined by this formula using for the rigid-body moment of inertia, , where fm is the usual nuclear radius parameter, is the mass number and is the nucleon mass. Other treatments of based on the independent-particle model have also been proposed go96 . SMMC calculations of nuclei in the mass region show that the assumption of a rigid-body moment of inertia breaks down at low excitation energies starting somewhat below the neutron separation energy, especially in even-even nuclei. The effect has a clear odd-even mass dependence. Furthermore, at the lowest excitations, deviations are observed from the spin-cutoff model itself, and odd-even staggering effects (in spin) can be seen. Here we will show that a fairly simple model based on a fixed deformation and a fluctuating pairing field reproduces very well the detailed SMMC results for the effective moment of inertia at finite temperature. In particular, odd-even effects observed in the microscopic SMMC calculations are nicely reproduced by a number-parity projection method go81 ; ro98 ; ba99 ; fl01 . We would therefore advocate this model for global calculations of the spin distributions below the neutron separation energy. Such distributions are needed for theoretical estimates of nucleosynthesis reaction rates ra97 , among other applications.

Our model is based on the static path approximation mu72 ; al84 ; la88 to the BCS Hamiltonian bcs . BCS theory is valid in the limit when the mean level spacing is much smaller than the pairing gap. However, this condition does not hold in the finite nucleus, in which case fluctuations must be taken into account. A similar situation occurs in ultra-small metallic particles whose linear size is smaller than nm nano01 . Theoretical studies have indicated that pairing correlations in the crossover from BCS to the fluctuation-dominated regime are manifested through their number-parity dependence. Odd-even effects that originate in pairing correlations were found in the SMMC heat capacity of nuclei li01 . Such effects were also observed in the heat capacities of rare-earth nuclei that were extracted from level density measurements sc01 ; gu03 . Finite-temperature pairing correlations at a fixed number of particles were also studied in Ref. fr03 .

In this paper, we first discuss in Section II general aspects of calculating the thermal moment of inertia and projection on number parity. We work in a grand canonical ensemble, but the odd-even effects can be extracted by the number-parity projection operator. In Section III, we apply the formalism to a model Hamiltonian that includes a deformed single-particle field and a pairing interaction treated in the static path approximation. This yields a formula for the moment of inertia that is a generalization of the Belyaev formula be61 for zero temperature, explaining the suppression of the inertia at low temperature. In Section IV, we further generalize the moment-of-inertia formula to take into account odd-even differences, making use of the number-parity projection operator. In Section V, we apply the model to nuclei in the iron region using the shell with single-particle energies and wave functions determined from a deformed Woods-Saxon potential. The calculated moments of inertia are found to be in good agreement with the SMMC calculations.

## Ii Formal aspects

In general, the SMMC method la93 ; al94 can be used to calculate thermal expectation values of observables

(4) |

where

(5) |

is the nuclear partition function. is the nuclear Hamiltonian, containing rotational invariant one-body and two-body terms. In Ref. la93 ; al94 exact particle-number projection was performed to calculate the traces in Eq. (4) at fixed number of protons and neutrons.

In Ref. spin-proj , the spin distribution was calculated using spin projection techniques. For temperatures that are not too low, it was found that the spin-cutoff model (1) describes rather well the spin distribution but with an energy-dependent moment of inertia. The purpose of the present work is to understand the temperature dependence of the moment of inertia in terms of a simple model. We note, however, that at the lowest temperatures the SMMC calculations reveal deviations from the spin-cutoff model (1), which are beyond the scope of the model discussed here.

### ii.1 Moment of inertia

In this work we shall assume that the spin distribution can be described by Eq. (1), and we therefore only need to calculate the variance . The obvious way to do this is to evaluate the expectation value of the operator directly from (4), as is done in SMMC. However, our model in Section III is based on a deformed Hamiltonian , and for such Hamiltonians it is useful to define a moment of inertia tensor as the response of the nucleus to a rotational field .

We shall work in the grand-canonical ensemble, replacing by in Eqs. (5) and (4). In the presence of a rotational field, the Hamiltonian is given by and its free energy is

(6) |

The moment of inertia is defined by the expansion of to second order in , , where are the components of . Equivalently al87

(7) |

where

(8) |

is the spin response function in imaginary time. For a rotationally invariant Hamiltonian, , and with , in agreement with Eqs. (2) and (3). Choosing the cranking axis along the fixed -axis of the laboratory frame, we can calculate from

(9) |

where^{1}^{1}1Here we use a different notation for the angular
momentum in the laboratory frame to distinguish it from the angular
momentum in the intrinsic frame.

(10) |

and the angular momentum component along is denoted by .

A non-rotational invariant effective Hamiltonian arises in the mean-field approximation when the single-particle potential is deformed. In such a case describes the Hamiltonian in the intrinsic frame of the nucleus. The quantity in Eq. (7) is then the moment of inertia tensor in this intrinsic frame, where are the intrinsic components of the angular momentum . To recover the moment of inertia in (3), it is necessary to integrate over all orientations of the intrinsic frame and then use (9). One obtains the result (see Appendix B)

(11) |

Eq. (11) expresses the effective moment of inertia in terms of the intrinsic principal moments .

### ii.2 Number-parity projection

The calculations in the previous Section II.1 were described in the grand-canonical ensemble. While this allows the average number of particles to be specified, it is not precise enough to reproduce odd-even effects. We note that the behavior of odd and even nuclei at low temperatures is quite different; the spin goes to zero for even nuclei due to pairing, but remains finite at zero temperature for odd nuclei. Exact particle-number projection can be done using the projection operator as in the SMMC, but leads to cumbersome expressions. In order to capture the main odd-even effects, it is often sufficient to use a number-parity projection go81 ; ro98 ; ba99 ; fl01 that distinguishes only between even and odd number of particles. The number-parity projection is defined by

(12) |

where or -1 describes the projection on an even or odd number of particles, respectively. Thus, the number-parity projected partition function is

(13) |

where the bracket denotes a thermal trace, . We can also calculate number-parity projected expectation values of observables

(14) |

Using (12), we find

(15) |

where we have used the notation

(16) |

The number-parity projected moment of inertia is defined from the second-order expansion (in ) of the number-parity projected free energy

(17) |

We find

(18) |

where is defined as in (16).

## Iii Model

We now ask, starting from the independent-particle shell model, what is the minimal model that will include the most relevant interaction effects for calculating the spin distribution. Clearly, the most important correlations are those associated with the quadrupole deformation and the pairing field. Both of these can be treated in a mean-field approximation, but the mean-field equations predict sharp transitions that are not supported by more detailed theories. Thus we go one step further in the finite-temperature theory, using the static path approximation (SPA) mu72 ; al84 ; la88 of the partition function to include time-independent fluctuations of the order parameters.

We consider an Hamiltonian composed of an axially deformed Woods-Saxon well for the single-particle potential and orbital-independent pairing for the interaction. We denote by the single-particle eigenstates in the deformed potential with energies . They can be divided into degenerate time-reversed pairs . For an axially symmetric potential, where is the projection of the angular momentum on the symmetry axis and are other labels of the states. The time-reversed states are defined by (known as the BCS phase convention), and we adopt the convention . The Hamiltonian may then be expressed in the form

(19) |

where is the pair creation operator, , and is the pairing strength.

### iii.1 Static path approximation

The Hamiltonian (19) contains a pairing interaction. Using the Hubbard-Stratonovich transformation, the imaginary-time propagator can be written as a functional integral over pairing fields of propagators that describe non-interacting quasi-particles. Here we shall use the SPA, which takes into account only time-independent pairing fields. The functional integral then reduces to an ordinary integral over a complex pairing field mu72

(20) |

where

(21) |

Our model (19) describes nucleons moving in a deformed well, but it could have been derived from a rotationally invariant Hamiltonian that included quadrupolar two-body interaction. This would introduce five additional integration variables in the SPA integral (20), two representing the intrinsic deformation and three representing the orientation of the deformed field la88 . This integration over the Euler angles of the intrinsic frame is equivalent to the symmetry restoration described by (82).

#### iii.1.1 Partition function

Using (20), we can represent the grand-canonical partition function in the form

(22) |

Here we used , where is the trace evaluated in the Fock space of the orbital pair , i.e. in the 4-dimensional space spanned by {}. In this representation, is the matrix

(23) |

The traces in (22) are easily evaluated by diagonalizing each in the corresponding 4-dimensional space. The four eigenvalues are , where

(24) |

are the familiar quasiparticle energies^{2}^{2}2One usually denotes the
self-consistent as in BCS formulation. but now defined
for an arbitrary complex pairing field . The trace in the subspace
() is then easily evaluated as

(25) |

The last algebraic form is convenient when dividing by in the evaluation of expectation values, as the reciprocal is proportional to where are the quasiparticle occupation probabilities.

An alternative way of calculating the trace is to write where is the matrix

(26) |

and use the identity fl01

(27) |

for the matrix . The eigenvalues of are just , leading again to Eq. (25).

The complete grand-canonical partition function is given by

(28) |

It can also be written in the form

(29) |

where

(30) |

is the free energy for a complex pairing field .

The canonical partition function can be calculated by a Fourier transform of the grand-canonical partition

(31) |

The integrals in (31) can be evaluated in the saddle point approximation in both and . The corresponding saddle-point equations, and , will give the usual finite-temperature BCS equations

(32) |

and

(33) |

where the quasi-particle energies are given by (24). The solutions of (32) and (33) determine the pairing gap and the chemical potential as a function of and particle number (the phase of is undetermined).

A better estimate of the canonical partition function can be obtained by a saddle-point integration in (31) over for every , but keeping the integration over intact. We find

(34) |

where is determined from , i.e., Eq. (33), and

(35) |

In Eq. (34) we have carried out explicitly the integral over the phase of the pairing field since the integrand was only a function of .

#### iii.1.2 Moment of inertia

Our model (19) describes a non-rotationally invariant Hamiltonian, and we can use the formalism of Section II.1 to estimate the moment of inertia in terms of the intrinsic moments (see Eq. (11)). Rather then using Eqs. (7) and (8) with the full Hamiltonian , we first apply an SPA representation similar to (20) but for the cranked Hamiltonian in (6), and then calculate the intrinsic moments from . If this is done starting from the canonical partition function in (6), we obtain the following expression

(36) |

where is the free energy (30) and

(37) |

where and is the mean-field Hamiltonian in a pairing field . Expression (37) is analogous to (7) and (8), except that the Hamiltonian in those expressions is now replaced by .

The integrals over in (36) can be done in the saddle-point as before to obtain the final expression for the intrinsic moments

(38) |

It remains to calculate the moments . The operator leaves the subspace invariant, but the operators and connect different subspaces and , so the trace in (37) is to be evaluated in a 16-dimensional space. This is most conveniently done in the quasiparticle representation. The transformation from deformed single-particle states to the quasi-particle states is achieved through a Bogoliubov transformation

(39) |

where is real and is complex, and to preserve the fermionic commutation relations. Relations (39) imply

(40) |

The parameters are chosen such that in Eq. (21) is diagonal in the quasi-particle representation, i.e.,

(41) |

where is the matrix (26) and are the eigenvalues (24) of . The solution is

(42) | |||||

and . The Hamiltonian is now given by , and

(43) |

Expressing in the quasi-particle representation, and using (43), we can calculate the intrinsic moments in closed form (see Appendix A). The final result is

(44) |

where

(45) |

are the quasi-particle occupations. and are still given by (42), except that now we have chosen

(46) |

and are still given by (40). Eq. (44) is the finite-temperature generalization of the Belyaev formula be61 .

Eq. (44) can be rewritten by separating out the contribution from the terms in the sum. Using , we obtain (for )

(47) |

In particular, for , and . Therefore, the moment of inertia around an axis parallel to the symmetry axis (non-collective rotation) is given by

(48) |

while the moment of inertia around an axis perpendicular to the symmetry axis (collective rotation) is given by

(49) |

## Iv Number-Parity Projection

In the relations we derived in the previous Section for the partition function and moment of inertia, the number of particles is fixed only on average, and odd-even effects cannot be reproduced. Here we go through the same derivation steps but include now the number-parity projection operator . The resulting formulas will exhibit explicit terms depending on the number parity.

### iv.1 Partition function

The projected partition function (13) introduces the operator . In the SPA

(51) |

Within each subspace , the operator changes the sign of the two vectors , but leaves the sign of unchanged. The matrix representing is then

(52) |

The transformation that diagonalizes in Eq. (23) leaves this matrix invariant, and therefore the trace is now given by

(53) |

The projected grand-canonical partition function is now calculated from (25) and (IV.1) to be

(54) |

Notice that the integrand in (54) has the form of Eq. (13) when applied to the Hamiltonian at a fixed pairing field . Indeed

(55) |

This projected partition can also be written as where

(56) |

is the number-parity projected free energy. Proceeding as in Section III.1.1, we can derive (in the saddle point approximation) number-parity projected BCS equations

(57) |

and

(58) |

where

(59) |

It is interesting to take the limit for the above equations. For , they simply become the usual BCS equations. However, for the odd projection , we find (assuming there are no degeneracies and )

(60) |

where is given by (42) and is the lowest quasi-particle energy (corresponding to closest to ). In deriving (60), we have used the limit . Eqs. (60) reproduce what is known as the blocking effect, since one level is “blocked” and does not contribute to the sum over .

In Fig. 1 we display the solution of the number-parity projected BCS equations for a Hamiltonian corresponding to the nucleus Fe. The pairing gap is shown as the solid line as a function of temperature . For the particle-number projected BCS equations (57), the solutions for even and odd particle numbers are shown as the dot-dashed and dashed lines, respectively. The proton gap and the neutron gap are shown in the left and right panels. Note the strong suppression of the gap for the odd projection. Our results for the projected gap are qualitatively similar to Ref. ba99 ; fl01 , but the formulas are quite different.

The number-parity projected partition function at a fixed average number of particles is given by an equation similar to (34) except that is replaced by .

### iv.2 Moment of inertia

The number-parity projected moment of inertia can be calculated as in Section III.1.2, but now starting from the number-parity projected free energy in the presence of a rotational field . The result is

(61) |

where

(62) |

Thus we need to calculate the projected value . The odd-even number projection can be carried out for any operator using (15) at a fixed pairing field together with (55)

(63) |

In general, will have the same form as but with replaced by .

For the intrinsic moment of inertia we find

(64) |

where is given by (44) and is obtained from the expression for by the substitution

(65) |