Kinetic energy in the collective quadrupole Hamiltonian from the experimental data

Dependence of the kinetic energy term of the collective nuclear Hamiltonian on collective momentum is considered. It is shown that the fourth order in collective momentum term of the collective quadrupole Hamiltonian generates a sizable effect on the excitation energies and the matrix elements of the quadrupole moment operator. It is demonstrated that the results of calculation are sensitive to the values of some matrix elements of the quadrupole moment. It stresses the importance for a concrete nucleus to have the experimental data for the reduced matrix elements of the quadrupole moment operator taken between all low lying states with the angular momenta not exceeding 4.

The Hamiltonian of the collective nuclear model introduced by A.Bohr [1] is a sum of the kinetic and the potential energy terms. Significant progress has been achieved in understanding of the functional dependence of the potential energy on the collective coordinates.
There are several methods of calculation of the potential energy based on the microscopic nuclear models. The potential energy plays a principal role in description of nuclear shape evolution, i.e. transition from spherical to deformed shapes. Generally, the potential energy is a complicated function of two invariants constructed using the collective quadrupole coordinates α 2µ . Namely, (α 2 , α 2 ) 0 and (α 2 , α 2 , α 2 ) 0 .
At the same time, the kinetic energy term plays as important role in description of collective nuclear dynamics as the potential energy. However, much less progress has been achieved in understanding of a role of different terms which can be presented in the kinetic energy. As for expression for kinetic energy its important ingredient is a mass tensor. It was known from the general expressions obtained within the Generator coordinate method [2][3][4] and Adiabatic Time Dependent Hartree Fock method [5][6][7] that the mass tensor has a complicated dependence on collective coordinates. Nevertheless, in practice, it was often assumed that the mass coefficient can be considered as a constant in the analysis of properties of the low-lying excited states. We should mention, however, that a deformation dependent mass tensor has been considered in [8][9][10][11][12][13]. Only in description of nuclear fission where parameters of a nuclear shape undergo considerable variations, dependence of mass coefficient on the deformation parameter was taken into account from the beginning (see [14] and refs. therein). Some years ago it was shown in [15,16] based on the experimental data for the excitation energies of the low-lying nuclear excited states and E2 transition probabilities and used in the theoretical analysis [17][18][19][20] that the mass parameter in the Bohr Hamiltonian is, in fact, a tensor depending on the shape variables.
The other important feature of the kinetic energy term is its dependence on the collective momentum. It is usually assumed [4,21] that it is possible to be limited only by the terms quadratic in collective momentum. However, using the Generator coordinate method [2][3][4] or the Generalized density matrix method [22][23][24][25] it is possible to show that, generally, the expression of the collective Hamiltonian contains all degrees of the square of the collective momentum. Usually terms of the order higher than the square assuming adiabaticity of the collective motion with respect to the single particle one are neglected. Nevertheless, it is interesting to obtain some information on the importance of the neglected terms basing on the experimental data.
In order to stress ones more the important role of the potential energy we should mention that a satisfactory description of the collective features of the spherical, transitional and deformed nuclei has been obtained with the potential energy taken as a complicated function of the collective coordinates (in some cases with several minima) but with a simplest form of the kinetic energy. The simplest form means that only a quadratic in collective momentum term with a constant mass coefficient is taken into account. Therefore, realizing that potential energy is a very complicated function of collective coordinates, and aiming to obtain information about that part of the Hamiltonian that depends on the collective moment we need the relations independent on the potential energy. As it is shown below these relations can be obtained by considering the ground state averages of a double and a fourth order commutators of the collective Hamiltonian with the quadrupole moment operator. These expressions do not contain the potential energy. The aim of the present paper is to estimate the effect of the fourth order in the collective momentum term in the Hamiltonian. To our knowledge this problem was not yet analyzed in the literature.
The method which is used below to achieve the aim of the paper is a continuation of the method applied in [15,16]  values of these quantities can be extracted from the experimental data. We can also calculate the fourth order commutator which in the simplest case can be taken aŝ If the kinetic energy contains only term quadratic in collective momentum the last commutator is equal identically to zero. Thus, taking into account the matrix elements of the fourth order commutator of the Hamiltonian with the operator of the quadrupole moment and expressing it through experimental data we obtain information on that term of collective Hamiltonian which contains the operator of the collective momentum in the fourth order.
For shortness, we denote this term below asT 4 .
Using the full set of intermediate states we obtain the following expressions for the ground state average ofŜ This expression contains the sums over the eigenstates of the Hamiltonian with angular momenta I = 0, 2, 3, 4. It is clear that in order to perform calculations of S for concrete nuclei we have to restrict the summations in (2)  Below we restrict summation in (2) by the following collective positive parity states: 0 + 1 , 0 + 2 , 2 + 1 , 2 + 2 , 2 + 3 and 4 + 1 . Only these states contribute into S in the spherical and rotor limits of the Bohr Hamiltonian.
In the limit of the spherical harmonic oscillator of the Bohr-Mottelson model only the following matrix elements are not equal to zero. In the units of 0 1 ||Q 2 ||2 1 they are In the rigid rotor limit of this model only the following nonzero matrix elements of Q 2 contribute to the expression for S: In both cases after substitution of these matrix elements into S we obtain that S = 0. This is an expected result since Bohr Hamiltonian contains only quadratic in collective momentum terms.
It is shown below that the values of S obtained for some nuclei for which there is a relatively sufficient set of the experimental data and in the dynamical symmetry limits of IBM are not equal to zero. In order to be able to estimate the effect ofT 4 we derive below some useful expressions. First of all, let us assume that the kinetic energy termT of the collective Hamiltonian has the form where π 2µ = −ih ∂ ∂α 2µ and α 2µ is the collective coordinate which is proportional to the quadrupole moment operator in the Bohr-Mottelson model Thus, in our caseT 4 = D( µ π + 2µ π 2µ ) 2 . Our task is to estimate the coefficient D. To obtain a dimensionless relation let us find an average value ofT over the ground state An effect ofT 4 is characterized by the value of quantity To express 0 1 | µ π + 2µ π 2µ |0 1 through 0 1 | µ Q + 2µ Q 2µ |0 1 we apply the procedure which is used in [26] to derive the uncertainty relation.
Consider the positively determined quantity where ξ is a real auxiliary variable. This expression can be rewritten as Since this expression is non-negative (for real ξ), this means that the roots of J(ξ) are complex. This is possible only if Using the relation between Q 2µ and α 2µ we obtain where For further consideration it is convenient to separate in (2) a dimensional factor. Then the quantity S will be presented as where s depends only on the ratios of the excitation energies and the E(2) reduced transition matrix elements. The last quantities are expressed in terms of the corresponding B(E2)'s with additional information about their signs.
To estimate X we should know the values of the parameters B and D. To find both quantities we consider together with S the following quantity By analogy with S this quantity can be expressed through B(E2)'s and the excitation energies of the collective states. With a good accuracy At the same time we can express t and S through the parameters B and D substituting into (1) and (13) the expressions for the kinetic energy (3) and the quadrupole moment operator (4). As a result we obtain where lower boundary of 0 1 | µ π + µ π µ |0 1 is given by (10). Equating (15) to (14) and (16) to (2) and taking into account that summation in (2) is restricted by the states listed above we obtain the equations for B and D. Finally, we get that and X ≥ 25 224 Equation (2) for S and (14) for t contains the reduced matrix elements of Q 2 . The absolute values of these matrix elements are equal to the square root of the corresponding B(E2)'s.
However, signs of these matrix elements generally are not known from the experiment. The exceptions are the quadrupole moments of the first 2 + states.
We have determined the signs of I ′ m ||Q 2 ||I n calculating them in the rigid rotor and the spherical harmonic oscillator limits. The results obtained in both limits are in a consent.
For the intermediate situation the signs of some matrix elements can be determined using the consistent-Q formalism [27]. There is also known the following relation between the signs of the matrix elements [28] sign( 2 1 ||Q 2 ||0 1 ) = −sign( 0 1 ||Q 2 ||2 1 2 1 ||Q 2 ||2 2 2 2 ||Q 2 ||0 1 ). (20) Before consideration of the concrete nuclei let us calculate the value of s in the dynamical symmetry limits of IBM [29] for which all necessary reduced matrix elements of Q 2 are known. In the SU (5)  Let us apply the consideration outlined above to nuclei for which there is a relatively large set of the experimental data. These are so called X(5) nuclei [31,32] -150 Nd, 152 Sm and 154 Gd. We also consider 110 Pd. There is no an experimental information on the spectroscopic quadrupole moments of the 2 2 and 2 3 states. Following the rigid rotor model results we have assumed that The results obtained are presented in Table I. We can see from this table that the correction ofT 4 to the total kinetic energy lies between 2% and 9%, i.e. is restricted by 10%.
It is also interesting to estimate a contribution ofT 4 into the energy weighted sum rule determined by (13). This contribution is given by the second term in the circle brackets of (15). We denote it by F Substituting (10), (17) and (18) into (21) we obtain F = 5 16 The results for F are also presented in Table I In conclusion, it is shown that the fourth order in collective momentum term of the collective quadrupole Hamiltonian generates a sizable effect on the excitation energies, and especially, on the matrix elements of the quadrupole moment operator. Its contribution to the collective kinetic energy can achieve 10% and to the energy weighted sum rule -26%.
These estimates are obtained basing on the experimental data on the excitation energies of the collective states and the E2 transitions matrix elements. It is demonstrated that the results of calculation are sensitive to the values of some matrix elements of the quadrupole moment operator. It stresses the importance for a concrete nucleus to have the experimental data for the reduced matrix elements of the quadrupole moment operator taken between all low lying states with the angular momenta not exceeding 4.
Authors acknowledge the partial support from the Heisenberg-Landau Program and the Russian Foundation for Basic Research.