Abstract
Full Text
I. N. MIKHAILOV
ON THE ACCURACY OF CALCULATING THE MOMENT OF INERTIA OF NUCLEI BY THE METHOD OF FORCED ROTATION
(Presented by Academician N. N. Bogolyubov, 10 VIII 1963)
This paper gives an estimate of the accuracy of determining the moment of inertia of a nucleus by means of the model of forced rotation. The estimate is based on the use of the method of approximate projection of functions, developed in [1].
The moment of inertia is the coefficient of \(I(I+1)\) in the expansion of the energy of the system in powers of the angular momentum \(I\):
\[ E_I = E_0 + I(I+1)/2J+\cdots \tag{1} \]
As is well known (see [2]), the Inglis formula
\[ J = 2 \sum_i \frac{|\langle i|\hat I_x|0\rangle|^2}{E_i-E_0} \tag{2} \]
for the moment of inertia exactly describes the term quadratic in \(I\) in the expression for the energy of a rotating system characterized by states of internal motion \(|i\rangle\) and the corresponding energies \(E_i\). However, the very separation of the rotational and internal motions of a closed system is an approximation*. Additional errors arise because the form of the functions \(|i\rangle\) and the values \(E_i\) are chosen more or less arbitrarily. On the other hand, determining the moment of inertia by means of formula (1) is not connected with the approximations mentioned above, and therefore the question of the accuracy of the forced-rotation model is meaningful.
Let \(\hat H\) be the Hamiltonian of the system and \(\hat I_x\) one of the projections of the angular-momentum operator, having in the second-quantization representation the form
\[ \hat H=\sum_i \varepsilon_i a_i^+ a_i +\frac{1}{2}\sum_{ijkl}\langle ij|G|lk\rangle a_i^+a_j^+a_k a_l, \qquad \hat I_x=\sum_{ij}\langle i|j_x|j\rangle a_i^+a_j. \tag{3} \]
Taking into account the increase of the energy with \(I\), one may assert that the problem
\[ \delta\langle \Psi,\hat H\Psi\rangle=0,\qquad \langle \Psi,\Psi\rangle=1,\qquad \hat I_x\Psi=I\Psi, \tag{4} \]
with no restrictions on the form of the functions \(\Psi\), exactly determines the lowest of the states of the system with quantum numbers \(I\), \(I(I+1)\), corresponding to the operators \(\hat I_x\) and \(\hat I^2=\hat I_x^2+I_y^2+I_z^2\).
In the model of forced rotation, instead of problem (4) one solves the problem
\[ \delta\langle \Psi,\hat H\Psi\rangle=0,\qquad \langle \Psi,\Psi\rangle=1,\qquad \langle \Psi,\hat I_x\Psi\rangle=I, \tag{5} \]
in which \(\Psi\) are taken from some restricted class of functions (see [3]). These functions are not, generally speaking, eigenfunctions of the ope-
* The fact that the states \(|i\rangle\) and energies \(E_i\) describe the internal motion of the system and not the motion of the system as a whole is most easily seen from the fact that the energy operator of a closed system \(\hat H\) commutes with the operator \(\hat I_x\), and consequently the operator \(\hat I_x\) can connect only states with one and the same energy.
operators $\hat I^2$, $\hat I_x$. The solution of problem (5) can be improved if, instead of the functions $\Psi$, one takes the trial functions
\[ \Phi_I=\frac{1}{2\pi i}\oint \frac{dz}{z}\, z^{(\hat I_x-I)\frac{1}{\hbar}}\,\psi, \tag{6} \]
which are projections of the functions $\Psi$ onto the space of eigenfunctions of the operator $\hat I_x$ with eigenvalue $I$ (see, for example, (4)). The use of projected functions does not guarantee the elimination of all errors in the solution of problem (5), but it makes it possible to estimate the accuracy of its solution using the functions $\Psi$.
We shall assume, as in work (1), that the mean values over the functions $\Phi$ can be calculated by the saddle-point method. Then, in accordance with the results of work (1), one may assert that the correction to the solution of problem (5), associated with the use of projected functions, is determined by an additional term in the energy of the system
\[ \Delta E=-\frac{1}{2\Delta \hat I_x^2}\,[H\Delta \hat I_x^2-\bar H \Delta \hat I^2]. \tag{7} \]
In formula (7) the averaging is performed over the function $\Psi$, which is the solution of the variational problem (5), and the operator $\Delta \hat I_x=(\hat I_x-I)^2$. Formula (7) is valid if $\Delta \hat I_x^2/\hbar^2 \gg 1$. Let us recall that projection operators were used for the analysis of rotational motions of nuclei in papers (5, 6); however, the form of the projection operator chosen by the authors of these papers did not make it possible to solve the variational problem (4) in the class of projected functions. In fact, in papers (5, 6) the internal inconsistency of the deformed-nucleus model was shown for sufficiently large values of the deformation parameter (and, consequently, of the quantity $\Delta \hat I_x^2/\hbar^2$).
Let problem (5) be solved by the method of a canonical transformation. In this case the function $\Psi$ is the vacuum with respect to the operators $\alpha$, related to the operators $a$ by the formula
\[ a_\nu=\sum_{\nu'}(u_{\nu\nu'}\alpha_{\nu'}+v_{\nu\nu'}\alpha_{\tilde{\nu}'}^+). \tag{8} \]
(The state $\tilde{\nu}$ is time-conjugate to the state $\nu$.) The mean values over the function $\Psi$ are determined with the aid of the following matrix elements*:
\[ \langle a_i^+a_i\rangle=v_i^2+(u_i^2-v_i^2)\sum_j |f_{ij}|^2, \]
\[ \langle a_i^+a_{\bar i}^+\rangle=\langle a_{\bar i}a_i\rangle^* =u_iv_i-2u_iv_i\sum_j |f_{ii}|^2; \tag{9} \]
\[ \langle a_i^+a_j\rangle=(u_iv_j-v_iu_j)f_{ij},\qquad \langle a_j^+a_{\bar i}^+\rangle=\langle a_{\bar j}a_i\rangle^* =(u_iv_j+v_iu_j)f_{ij}^*. \tag{10} \]
The coefficients $f_{ij}$ satisfy the relations
\[ f_{ij}=-f_{\bar j \bar i},\qquad f_{ij}+f_{ji}^*=0,\qquad f_{ii}=0 \tag{11} \]
and are determined from the variational equations, which have the form
\[ (E_i+E_j)f_{ij} -\sum_{i'j'}\langle \tilde{ij}|G|\tilde{j'i'}\rangle (u_iu_j+v_iv_j)(u_{i'}u_{j'}+v_{i'}v_{j'})f_{i'j'} + \]
\[ +\sum_{i',j'} (\langle i\tilde{i'}|G|\tilde{j}j'\rangle-\langle i\tilde{i'}|G|\tilde{j'}j\rangle) (u_iv_j-v_iu_j)(u_{i'}v_{j'}-v_{i'}u_{j'}) = \]
\[ =\omega (u_iv_j-v_iu_j)\langle i|j_x|j\rangle, \tag{12} \]
\[ \sum_{it}\langle i|j_x|j\rangle (u_iv_j-v_iu_j)f_{ij}^*=I. \tag{13} \]
\[ \text{* The notation is the same as in work (3).} \]
Here
\[ E_i=\sqrt{(\tilde{\varepsilon}_i-\lambda)^2+\Delta_i^2},\qquad \Delta_i=\sum_j \langle i\tilde{i}\mid G\mid j\tilde{j}\rangle u_jv_j, \tag{14} \]
and the quantities \(u_i, v_i\) are connected with the parameters \(\lambda, \Delta_i, E_i\) by the usual equations of the \(u, v\)-transformation method.
The quantities in formula (7) have the form
\[ \overline{\Delta \hat I_x^2} = \sum_{\substack{\nu\nu'\\ \mu\mu'}} \langle \nu\mid j_x\mid \nu'\rangle \langle \mu\mid j_x\mid \mu'\rangle \left[ \langle a_\nu^+ a_{\mu'}\rangle \langle a_{\nu'} a_\mu^+\rangle - \langle a_\nu a_{\mu'}\rangle \langle a_{\nu'}^+ a_\mu^+\rangle \right] = \]
\[ = \sum_{\nu,\mu} \left|\langle \nu\mid j_x\mid \mu\rangle\right|^2 \left[v_\nu^2u_\mu^2-(u_\nu v_\nu)(u_\mu v_\mu)\right], \tag{15} \]
\[ \Delta\hat I_x^2-\overline{\Delta\hat I_x^2} = \sum_{\substack{\nu\nu'\\ \eta\eta'}} \sum_{\substack{\mu\mu'\\ \rho\rho'}} \langle \nu\mid j_x\mid \nu'\rangle \langle \mu\mid j_x\mid \mu'\rangle u_{\nu\eta}^{*}v_{\nu'\eta'}u_{\mu\rho}v_{\mu'\rho'}^{*} \alpha_\eta^+\alpha_{\eta'}^+\alpha_\rho\alpha_{\rho'} + \]
\[ +\ \text{Hermitian conjugate}. \tag{16} \]
It follows from formula (16) that the corrections determined by formula (7) are connected with the interaction Hamiltonian, since only it contains terms that can compensate the 4 creation operators \(\alpha^+\) in formula (16). This is connected with the possibility of exact diagonalization of the Hamiltonian of noninteracting particles by the method of the canonical transformation (8). It should be remembered, however, that the Bardin function, constructed on the single-particle functions of an anisotropic potential, in principle cannot be an exact eigenfunction of the Hamiltonian of a closed system.
Rather laborious calculations make it possible to write the following expression for the quantity \(\Delta E\):
\[ \Delta E = -\frac{1}{4\Delta\hat I_x^2} \sum_{ijkl} \langle ij\mid G\mid lk\rangle \langle kl\mid \Delta\hat I_x^2-\overline{\Delta\hat I_x^2}\mid ji\rangle + \text{complex conjugate}, \tag{17} \]
where
\[ \langle kl\mid \Delta\hat I_x^2-\overline{\Delta\hat I_x^2}\mid ji\rangle = \]
\[ = 8\sum_{\nu\nu'} \langle \nu\mid j_x\mid \nu'\rangle \langle a_k a_\nu^+\rangle \langle a_l a_{\nu'}\rangle \sum_{\mu\mu'} \langle \mu\mid j_x\mid \mu'\rangle \langle a_\mu a_j\rangle^{*} \langle a_i^+a_{\mu'}\rangle - \]
\[ - 4\sum_{\nu\nu'}\sum_{\mu\mu'} \langle \nu\mid j_x\mid \nu'\rangle \langle \mu\mid j_x\mid \mu'\rangle \left[ \langle a_k a_\nu^+\rangle \langle a_j^+ a_{\nu'}\rangle \langle a_l a_{\mu'}^+\rangle \langle a_i^+ a_\mu\rangle + \right. \]
\[ \left. + \langle a_\nu a_j\rangle^{*} \langle a_k a_{\nu'}\rangle \langle a_\mu a_i\rangle^{*} \langle a_l a_{\mu'}\rangle \right]. \tag{18} \]
The determination of the corrections to the moment of inertia is easily carried out by separating out the part quadratic in \(I\) from expression (18) with the aid of formulas (10). The resulting expressions contain a large number of different terms. If the interaction Hamiltonian has the form
\[ \langle ij\mid G\mid lk\rangle = -\frac{G}{2}\lambda(i)\lambda(l)\delta_{i\tilde{j}}\delta_{l\tilde{k}}, \tag{19} \]
where
\[ \lambda(i)=-\lambda(\tilde{i}),\qquad |\lambda(i)|=1, \tag{20} \]
then the magnitude of the correction to the moment of inertia can be estimated by means of only the first term on the right-hand side of formula (18)
\[ \Delta E = \frac{I^2G}{2\Delta\hat I_x^2}\xi_+\xi_- \simeq \frac{1}{2}\omega^2J_0 \frac{J_0G}{\Delta I_x^2}\xi_+\xi_- . \tag{21} \]
In formula (21) the quantities
\[ \xi_{\pm}= \left\{ \sum_{i\mu} \frac{|\langle i|j_x|\mu\rangle|^2}{E_i+E_\mu} (u_i v_{\mu'}-v_i u_{\mu'})\lambda(i) \left[(1\pm(u_i^2-v_i^2))(u_i u_{\mu'}+v_i v_{\mu'}) +2u_i v_i(u_i v_{\mu'}-v_i u_{\mu'})\right] \right\} \left\{ \sum_{i\mu} \frac{|\langle i|j_x|\mu\rangle|^2}{E_i+E_\mu} (u_i v_{\mu'}-v_i u_{\mu'})^2 \right\}^{-1} \tag{22} \]
are, in order of magnitude, quantities of order unity; \(\omega=I/J_0\) is the rotation frequency corresponding to the angular momentum \(I\), and
\[ J_0=\sum_{i\mu}\frac{|\langle i|j_x|\mu\rangle|^2}{E_i+E_\mu} (u_i u_\mu-v_i u_\mu)^2 \tag{23} \]
is the value of the moment of inertia found without projection. From formula (21) it follows that the correction to the moment of inertia is equal to
\[ |\Delta J/J_0|\simeq GJ_0/\Delta \hat I_x^2. \tag{24} \]
The character of the expression on the right-hand side of formula (24) is easiest to see if one uses formula (2) for \(J_0\) and takes into account that the average energy of the excited states \(E_{\mathrm{av}}\) contributing to the moment of inertia is, in order of magnitude, equal to twice the gap \(\Delta\). But, on the other hand:
\[ J_0=2\sum_i\frac{|\langle i|j_x|0\rangle|^2}{E_i-E_0} =\frac{2}{E_{\mathrm{av}}}\sum_i|\langle i|j_x|0\rangle|^2 =\frac{2\Delta \hat I_x^2}{E_{\mathrm{av}}}. \tag{25} \]
Therefore formula (24) can be rewritten in the form
\[ |\Delta J/J_0|\sim G/\Delta. \]
It is thus seen that the order of magnitude of the correction is \(10\%\). Formula (25) also makes it possible to estimate, from known data on the magnitude of single-particle excitations and from the values of the moment of inertia, the value of the parameter \(\Delta \hat I_x^2/\hbar^2\), which is a measure of the accuracy of the cranking method. For nuclei in the rare-earth group this quantity is of order \(20\)—\(30\), which attests to the applicability of the method.
In conclusion I express my sincere gratitude to E. Bang for numerous discussions, and also to V. G. Solov'ev for his constant interest in the work.
Joint Institute
for Nuclear Research
Received
12 VII 1963
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