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Connection between nuclear moments of inertia and collective variables
Volume 84B, number 4
PHYSICS LETTERS
16 July 1979
CONNECTION BETWEEN NUCLEAR MOMENTS OF INERTIA AND COLLECTIVE VARIABLES
Giinter WUNNER and Heinz HEROLD lnstitut ffir Theoretische Physik, Universitdt Erlangen-Narnberg, D.8520 Erlangen, West Germany Received 16 April 1979
Several frequently used formulae for calculating nuclear moments of inertia are shown to be associated with specific choices of collectivevariables.
In this note we report results of an investigation of various frequently used formulae for calculating nuclear moments of inertia from the point of view of their connection with collective variables. The theoretical tool for this is the method of specific decoupling (SD) which is set forth in detail in refs. [1 - 3 ] where also references to the relevant literature may be found. The basic idea of our method is to introduce a body-fixed frame in such a way that the best possible decoupling between internal motion and collective rotation is obtained (extremum principle of SD). The internal motion is described by body-fixed cartesian coordinates x', yet at any step in the theory their redundancy is fully taken into account, which is expressed, e.g., by the fact that the hamiltonian H intr (x') which governs the intrinsic wave function ¢~ntr(x') contains terms which depend explicitly on the definition of the body-fixed frame. L e t us ask for the set of collective variables a, 15,3' (Euler angles) which for any configuration of the particles determine the orientation of the optimal bodyfixed frame relative to some laboratory frame. Although our method can be applied equally well to triaxial deformations, we shall concentrate here on the collective rotations of axially symmetric even nuclei with deformations not too small and with K = 0. Then the collective variable 3' does not enter our considerations, and if we substitute S 1 = -15 sin a , S 2 = 15COSt~, the extremum principle of SD requires that, for a given intrinsic state ~0(x'), the reciprocal effective moment of inertia
1/O(S1, S 2) 1 t~. ((V}S1) 2 + (V[SE)E)lq~0(x,)) = %(x')l ~m
(1)
(eL eq. (33) in ref. [1 ]) should be rninimised, where the functions S are restricted by the corotation condition for the body-fixed frame [1]. The Euler-Lagrange equations of this variational problem can be written as [ndef(X'), S1, 2(x')] ~b0(x') = (ifli)(1/0) Jx,yg~O(X'),
(2)
where Hdef(X' ) is the deformed intrinsic hamiltonian of which 4~0(x') is an eigenfunction. Denoting the excited eigenstates by ~bn (x') (eigenvalues E n), the rigorous solution of eq. (2) can be expressed in the form
S 1(x') ¢o(X') = ' ~ Cn¢n(X' ) ,
(3)
n
where
Cn 0¢ (¢)nIJx ICkO>/(En - EO)
(4)
(for S2J x is replaced by J~.). We remark that the equations S 1(x') ~0(x') = S2(x' ) ~b0(x') = 0 implicitly define the collective variables which characterise the optimal body-fixed frame. The corresponding moment of inertia turns out to be given by the cranking formula [4]
calculated with the eigenstates Of Hdef(X' )
377
Volume 84B, number 4 0=2~
' 2/ ( g n - E O ) . I(~nlJxl¢0)l
PHYSICS LETTERS (5)
Thus by our method the cranking moment of inertia is reduced to a specific definition of a body-fixed frame. There is also a relation between our method and the self-consistent (or Thouless-Valatin) moment of inertia [5]. In the Thouless-Valatin method ~0 is a Hartree-Fock-Bogolyubov state, and the cranked wave function is taken to be ~ = exp(koF) ¢0, where F is a one-body operator. If we identify F with 0 • S, and if, moreover, we replace Hdef(X') by the model hamiltonian of the HFB calculation, the projection of (2) on the two-quasi-particle states leads immediately to the Thouless equations for F, and thus we end up with the self-consistent moment of inertia. Apparently the more intuitive self-consistent cranking approach provides an approximate solution to the extremum principle of SD which avoids the use of our intrinsic hamiltonian. Numerical calculations to confirm this observation are being performed presently. Eq. (1) makes it possible to also associate specific definitions of the internal frame with other moments of inertia which are customary in the literature. As an example we mention the Peierls-Yoccoz moment of inertia [6], which is usually derived by angular momentum projection techniques. We find that the definition of the collective variables corresponding to PeierlsYoccoz can be represented in the form (3) with coefficients c n o: (¢n IJxl~0)" It must be noted, however, that this is not an exact solution to the extremum principle of SD. If we compare with (4) we see that the two
378
16 July 1979
definitions of the body-fixed flame coincide if there is one state ~n which dominates the expansion (3), which is equivalent to saying that Jx~b0 is an (approximate) eigenfunction Of Hdef(X' ). The conclusion that under these circumstances the cranking and the Peierls-Yoccoz moment of inertia coincide [7] is not novel. But what is novel in our approach to the problem of collective rotations is that this result is traced back to specific definitions of an internal frame, and that the usual formulae for calculating nuclear moments of inertia are found to be indeed associated with distinct collective variables, something the inventors of these formulae probably never thought of. A detailed account of the investigations reported in this note, including a discussion of the case of triaxial deformation, will appear elsewhere. The authors gratefully acknowledge stimulating discussions with Prof. Dr. H. Ruder. References
[1 ] H. Herold and H. Ruder, J. Phys. G5 (1979) 341. [2] H. Herold, J. Phys. G5 (1979) 351. [3] H. Herold, M. Reinecke and H. Ruder, J. Phys. G, to be published. [4] D.R. Inglis, Phys. Rev. 103 (1956) 1786. [5] D.J. Thouless and J.G. Valatin, Nuel. Phys. 21 (1960) 225. [6] R.E. Peierls and J.G. Yoeeoz, Prec. Phys. Soc. A70 (1957) 388. [7] M.K. Banerjee, D. d'Oliveira and G.J. Stephenson Jr., Phys. Roy. 181 (1969) 1404; W.A. Friedman and L. Wilets, Phys. Roy. C2 (1970) 892.
PHYSICS LETTERS
16 July 1979
CONNECTION BETWEEN NUCLEAR MOMENTS OF INERTIA AND COLLECTIVE VARIABLES
Giinter WUNNER and Heinz HEROLD lnstitut ffir Theoretische Physik, Universitdt Erlangen-Narnberg, D.8520 Erlangen, West Germany Received 16 April 1979
Several frequently used formulae for calculating nuclear moments of inertia are shown to be associated with specific choices of collectivevariables.
In this note we report results of an investigation of various frequently used formulae for calculating nuclear moments of inertia from the point of view of their connection with collective variables. The theoretical tool for this is the method of specific decoupling (SD) which is set forth in detail in refs. [1 - 3 ] where also references to the relevant literature may be found. The basic idea of our method is to introduce a body-fixed frame in such a way that the best possible decoupling between internal motion and collective rotation is obtained (extremum principle of SD). The internal motion is described by body-fixed cartesian coordinates x', yet at any step in the theory their redundancy is fully taken into account, which is expressed, e.g., by the fact that the hamiltonian H intr (x') which governs the intrinsic wave function ¢~ntr(x') contains terms which depend explicitly on the definition of the body-fixed frame. L e t us ask for the set of collective variables a, 15,3' (Euler angles) which for any configuration of the particles determine the orientation of the optimal bodyfixed frame relative to some laboratory frame. Although our method can be applied equally well to triaxial deformations, we shall concentrate here on the collective rotations of axially symmetric even nuclei with deformations not too small and with K = 0. Then the collective variable 3' does not enter our considerations, and if we substitute S 1 = -15 sin a , S 2 = 15COSt~, the extremum principle of SD requires that, for a given intrinsic state ~0(x'), the reciprocal effective moment of inertia
1/O(S1, S 2) 1 t~. ((V}S1) 2 + (V[SE)E)lq~0(x,)) = %(x')l ~m
(1)
(eL eq. (33) in ref. [1 ]) should be rninimised, where the functions S are restricted by the corotation condition for the body-fixed frame [1]. The Euler-Lagrange equations of this variational problem can be written as [ndef(X'), S1, 2(x')] ~b0(x') = (ifli)(1/0) Jx,yg~O(X'),
(2)
where Hdef(X' ) is the deformed intrinsic hamiltonian of which 4~0(x') is an eigenfunction. Denoting the excited eigenstates by ~bn (x') (eigenvalues E n), the rigorous solution of eq. (2) can be expressed in the form
S 1(x') ¢o(X') = ' ~ Cn¢n(X' ) ,
(3)
n
where
Cn 0¢ (¢)nIJx ICkO>/(En - EO)
(4)
(for S2J x is replaced by J~.). We remark that the equations S 1(x') ~0(x') = S2(x' ) ~b0(x') = 0 implicitly define the collective variables which characterise the optimal body-fixed frame. The corresponding moment of inertia turns out to be given by the cranking formula [4]
calculated with the eigenstates Of Hdef(X' )
377
Volume 84B, number 4 0=2~
' 2/ ( g n - E O ) . I(~nlJxl¢0)l
PHYSICS LETTERS (5)
Thus by our method the cranking moment of inertia is reduced to a specific definition of a body-fixed frame. There is also a relation between our method and the self-consistent (or Thouless-Valatin) moment of inertia [5]. In the Thouless-Valatin method ~0 is a Hartree-Fock-Bogolyubov state, and the cranked wave function is taken to be ~ = exp(koF) ¢0, where F is a one-body operator. If we identify F with 0 • S, and if, moreover, we replace Hdef(X') by the model hamiltonian of the HFB calculation, the projection of (2) on the two-quasi-particle states leads immediately to the Thouless equations for F, and thus we end up with the self-consistent moment of inertia. Apparently the more intuitive self-consistent cranking approach provides an approximate solution to the extremum principle of SD which avoids the use of our intrinsic hamiltonian. Numerical calculations to confirm this observation are being performed presently. Eq. (1) makes it possible to also associate specific definitions of the internal frame with other moments of inertia which are customary in the literature. As an example we mention the Peierls-Yoccoz moment of inertia [6], which is usually derived by angular momentum projection techniques. We find that the definition of the collective variables corresponding to PeierlsYoccoz can be represented in the form (3) with coefficients c n o: (¢n IJxl~0)" It must be noted, however, that this is not an exact solution to the extremum principle of SD. If we compare with (4) we see that the two
378
16 July 1979
definitions of the body-fixed flame coincide if there is one state ~n which dominates the expansion (3), which is equivalent to saying that Jx~b0 is an (approximate) eigenfunction Of Hdef(X' ). The conclusion that under these circumstances the cranking and the Peierls-Yoccoz moment of inertia coincide [7] is not novel. But what is novel in our approach to the problem of collective rotations is that this result is traced back to specific definitions of an internal frame, and that the usual formulae for calculating nuclear moments of inertia are found to be indeed associated with distinct collective variables, something the inventors of these formulae probably never thought of. A detailed account of the investigations reported in this note, including a discussion of the case of triaxial deformation, will appear elsewhere. The authors gratefully acknowledge stimulating discussions with Prof. Dr. H. Ruder. References
[1 ] H. Herold and H. Ruder, J. Phys. G5 (1979) 341. [2] H. Herold, J. Phys. G5 (1979) 351. [3] H. Herold, M. Reinecke and H. Ruder, J. Phys. G, to be published. [4] D.R. Inglis, Phys. Rev. 103 (1956) 1786. [5] D.J. Thouless and J.G. Valatin, Nuel. Phys. 21 (1960) 225. [6] R.E. Peierls and J.G. Yoeeoz, Prec. Phys. Soc. A70 (1957) 388. [7] M.K. Banerjee, D. d'Oliveira and G.J. Stephenson Jr., Phys. Roy. 181 (1969) 1404; W.A. Friedman and L. Wilets, Phys. Roy. C2 (1970) 892.