- Email: [email protected]
Nuclear moments of inertia in the absence of pairing
Volume 72B, number 4
NUCLEAR
PHYSICS LETTERS
MOMENTS OF INERTIA
16 January 1978
IN THE ABSENCE OF PAIRING
B.L. BIRBRAIR
Academy of Sciences of the USSR, Leningrad Nuclear Physics Institute, Gatehina 188.]50, Leningrad District, USSR Received 9 November 1977 The nuclear moments of inertia in the absence of pairing are not equal to the rigid-body values because of quantal corrections due to the finite size of the system. The problem o f nuclear moments of inertia at the A = 0 limit is of practical interest since this limit can be reached at large angular momenta because o f the Coriolis anti-pairing effect [ 1 ]. In 1959, Rockmore [2] demonstrated for the infinite non-superfluid fermi system that the interaction between quasiparticles exactly compensates the difference between the effective and real mass thus leading to the rigid-body moment of inertia. Since this work the statement J ( A = 0) = Jrig is known as the Rockmore theorem. However the practical calculations do not give the rigid-body values at A = 0. Unfortunately none o f the calculations is exact. Therefore the above discrepancy is usually attributed to some inconsistencies in the approximations (the use of a restricted single-particle basis, an inadequate choice o f the equilibrium deformation, the neglect of the interaction between quasiparticles, etc). F o r these reasons it is not clear whether there is a physical reason for the difference between J ( A = 0) and Jrig" Here we shall show that such a reason does exist. Let us consider the nucleus as a system o f quasiparticles moving in the average nuclear potential depending on space variables only: p2 h = ~-~ + V(r),
h~, x = ex4, x ,
(1)
e x and Ox being eigenenergies and eigenfunctions o f the single-particle hamiltonian h. In the self-consistent treatment of the problem this corresponds to the case when the interaction between quasiparticles is both velocity- and spin-independent. Such a situation is the most favourable from the viewpoint o f the Rockmore theorem. Indeed there is no difference between the effective and real mass and thus one may believe the moment o f inertia to be a rigid-body one.
The moment o f inertia in the non-superfluid nucleus is o f the form (nucleon spins do not contribute in the case under consideration)
J= ~ n ~ - n a x x X,v ex - e~---~lvalxv = 2 ~
h ,v
X
nv
X
l~xl~,~ ~
(2)
-- e v
Let us now introduce the angle ~x -- arctg (z/v), which is conjugate to the angular momentum lx and use the two following relations:
[lx,Ox] = - i ,
lx=iM[h, (~xl(y2+z2).
(3)
Taking the matrix element of the latter and separating the diagonal element o f the transverse coordinate y2 + z 2 we get
l~v = iM(e x - ev) qS~v(y2 + z2)v v (e x - eu) q~,~(y2+ zZ),v.
+ iM ~
(4)
,u
(u ~v) Putting eq. (4) into eq. (2) we see that the moment o f inertia splits into two terms. The first one which is due to the diagonal elements o f y 2 + z 2 is equal to the rigid-body value. Indeed x x 2 +z2)~,v J1 = 2 i M ~ n ,,l,,x4~x,,(y 2~,p
= iM ~
n v [lx, Ox] vv(Y 2 + z2)uv
v
=M f (y 2 + z2)fl d V - Jrig •
(5)
The second one is the quantal correction
425
Volume 72B, number 4 ex - e u Jqc = 2iM x,u,v ~ nv ~ lXx(~[u(Y2 + z2)uv'
PHYSICS LETTERS
(6)
(u~v) since it arises only from nondiagonal elements o f y 2 + z 2. Clearly such a correction always exists in finite nuclei because the quasi-particle motion is quantized. So the Rockmore theorem is valid only for the infinite system. Indeed the summation over ~ goes into the integration over the spectrum and Jqc vanishes since the spectrum is symmetric with respect to the fermisurface in this case. The theorem can also be valid for strongly heated nuclei: here n v = (1 + exp [(ev - eF)/ kT])- 1 and Jqc ~ 0 since the summation over v covers a wide enough region about the fermi-surface. The Jqc value is strongly dependent on the nuclear individuality. For instance in spherical nuclei it exactly cancels the rigid-body moment. In deformed nuclei c has been calculated by Pashkevich and Frauendorf 3] using the Strutinsky shell correction method. What they call the "shell correction" is in fact the quantal correction. This is seen from the fact that the Strutinsky average of eq. (2) is nearly equal to the rigid-body value (see ref. [3] for details). The calculations in ref. [3] show that Jqc can be of different sign
~q
426
16 January 1978
and magnitude and thus the comparison between the nuclear moment of inertia and the rigid-body one has no direct physical meaning. The same qualitative conclusion has been obtained by Kolomiets [4] in the quasiclassical approximation for the rotation around the symmetry axis. Our consideration concerns the "normal" rotation and does not include any approximations. The author is indebted to Professors Ya.A. Smorodinsky and V.G. Soloviev and to Drs. I.A. Mitropolsky, R.V. Jolos, V.M. Kotomiets, I.N. Mikhailov, N.I. Pyatov, G. Vagradov and F. Gareev for valuable discussions.
References
[1] B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5 (1960) 511. [2] R.M. Rockmore, Phys. Rev. 116 (1959)469. [3] V.V. Pashkevich and S. Frauendorf, Soviet J. Nucl. Phys. 20 (1974) 1122. [4] V.M. Kolomiets, Questions of atomic science and technique (Kharkov, 1975) p. 37.
NUCLEAR
PHYSICS LETTERS
MOMENTS OF INERTIA
16 January 1978
IN THE ABSENCE OF PAIRING
B.L. BIRBRAIR
Academy of Sciences of the USSR, Leningrad Nuclear Physics Institute, Gatehina 188.]50, Leningrad District, USSR Received 9 November 1977 The nuclear moments of inertia in the absence of pairing are not equal to the rigid-body values because of quantal corrections due to the finite size of the system. The problem o f nuclear moments of inertia at the A = 0 limit is of practical interest since this limit can be reached at large angular momenta because o f the Coriolis anti-pairing effect [ 1 ]. In 1959, Rockmore [2] demonstrated for the infinite non-superfluid fermi system that the interaction between quasiparticles exactly compensates the difference between the effective and real mass thus leading to the rigid-body moment of inertia. Since this work the statement J ( A = 0) = Jrig is known as the Rockmore theorem. However the practical calculations do not give the rigid-body values at A = 0. Unfortunately none o f the calculations is exact. Therefore the above discrepancy is usually attributed to some inconsistencies in the approximations (the use of a restricted single-particle basis, an inadequate choice o f the equilibrium deformation, the neglect of the interaction between quasiparticles, etc). F o r these reasons it is not clear whether there is a physical reason for the difference between J ( A = 0) and Jrig" Here we shall show that such a reason does exist. Let us consider the nucleus as a system o f quasiparticles moving in the average nuclear potential depending on space variables only: p2 h = ~-~ + V(r),
h~, x = ex4, x ,
(1)
e x and Ox being eigenenergies and eigenfunctions o f the single-particle hamiltonian h. In the self-consistent treatment of the problem this corresponds to the case when the interaction between quasiparticles is both velocity- and spin-independent. Such a situation is the most favourable from the viewpoint o f the Rockmore theorem. Indeed there is no difference between the effective and real mass and thus one may believe the moment o f inertia to be a rigid-body one.
The moment o f inertia in the non-superfluid nucleus is o f the form (nucleon spins do not contribute in the case under consideration)
J= ~ n ~ - n a x x X,v ex - e~---~lvalxv = 2 ~
h ,v
X
nv
X
l~xl~,~ ~
(2)
-- e v
Let us now introduce the angle ~x -- arctg (z/v), which is conjugate to the angular momentum lx and use the two following relations:
[lx,Ox] = - i ,
lx=iM[h, (~xl(y2+z2).
(3)
Taking the matrix element of the latter and separating the diagonal element o f the transverse coordinate y2 + z 2 we get
l~v = iM(e x - ev) qS~v(y2 + z2)v v (e x - eu) q~,~(y2+ zZ),v.
+ iM ~
(4)
,u
(u ~v) Putting eq. (4) into eq. (2) we see that the moment o f inertia splits into two terms. The first one which is due to the diagonal elements o f y 2 + z 2 is equal to the rigid-body value. Indeed x x 2 +z2)~,v J1 = 2 i M ~ n ,,l,,x4~x,,(y 2~,p
= iM ~
n v [lx, Ox] vv(Y 2 + z2)uv
v
=M f (y 2 + z2)fl d V - Jrig •
(5)
The second one is the quantal correction
425
Volume 72B, number 4 ex - e u Jqc = 2iM x,u,v ~ nv ~ lXx(~[u(Y2 + z2)uv'
PHYSICS LETTERS
(6)
(u~v) since it arises only from nondiagonal elements o f y 2 + z 2. Clearly such a correction always exists in finite nuclei because the quasi-particle motion is quantized. So the Rockmore theorem is valid only for the infinite system. Indeed the summation over ~ goes into the integration over the spectrum and Jqc vanishes since the spectrum is symmetric with respect to the fermisurface in this case. The theorem can also be valid for strongly heated nuclei: here n v = (1 + exp [(ev - eF)/ kT])- 1 and Jqc ~ 0 since the summation over v covers a wide enough region about the fermi-surface. The Jqc value is strongly dependent on the nuclear individuality. For instance in spherical nuclei it exactly cancels the rigid-body moment. In deformed nuclei c has been calculated by Pashkevich and Frauendorf 3] using the Strutinsky shell correction method. What they call the "shell correction" is in fact the quantal correction. This is seen from the fact that the Strutinsky average of eq. (2) is nearly equal to the rigid-body value (see ref. [3] for details). The calculations in ref. [3] show that Jqc can be of different sign
~q
426
16 January 1978
and magnitude and thus the comparison between the nuclear moment of inertia and the rigid-body one has no direct physical meaning. The same qualitative conclusion has been obtained by Kolomiets [4] in the quasiclassical approximation for the rotation around the symmetry axis. Our consideration concerns the "normal" rotation and does not include any approximations. The author is indebted to Professors Ya.A. Smorodinsky and V.G. Soloviev and to Drs. I.A. Mitropolsky, R.V. Jolos, V.M. Kotomiets, I.N. Mikhailov, N.I. Pyatov, G. Vagradov and F. Gareev for valuable discussions.
References
[1] B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5 (1960) 511. [2] R.M. Rockmore, Phys. Rev. 116 (1959)469. [3] V.V. Pashkevich and S. Frauendorf, Soviet J. Nucl. Phys. 20 (1974) 1122. [4] V.M. Kolomiets, Questions of atomic science and technique (Kharkov, 1975) p. 37.