- Email: [email protected]
Thomas-Fermi description of nuclear systems including spin-orbit interactions
(1972) 38-48;
Nuclear Physics Al87
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THOMAS-FERMI
DESCRIPTION
INCLUDING
OF NUCLEAR
SPIN-ORBIT
W. STOCKER,
INTERACTIONS
G. SUSSMANN
Sektion Physik der Universitiit,
SYSTEMS
and S. KNAAK Munich, Germany
Received 17 January 1972 Abstract: The Thomas-Fermi
statistical method is generalized to include spin-orbit interactions. The momentum distributions are given by toroids, different for two particle spin orientations. A system of two coupled differential equations is derived by a variational procedure for the densities of the two populations. From these equations the polarization at the surface of nuclear matter is calculated, as well as the change of the nuclear surface tension due to spin-orbit coupling. Within the statistical framework the coupling strength of the spin-orbit potential is found to be in reasonable agreement with experiment by using only the experimental singleparticle level order of the shell model which implies an excess of states with spin parallel to the orbital angular momentum.
1. Introduction The Thomas-Fermi (TF) and related statistical methods have been proved successful in nuclear physics. However, the generalization of the method to include a spinorbit potential is an open question. The present paper deals with this problem. In sect. 2 the two spin orientations of nucleons at the nuclear surface are discussed. In sect. 3 the momentum distribution for the two nucleon populations is investigated including a spin-orbit potential. It is shown that in this case the Fermi sphere in momentum space is distorted into two toroids different for the two spin orientations. In sect. 4 the statistical method is used to establish the energy density for systems with two-particle spin-orbit interactions. The minimalization condition for the surface tension yields a system of two coupled differential equations of TF structure for the particle densities n, and n_. After a linearization of these equations a simple expression is obtained for the polarization effect at the nuclear surface which results because one spin orientation is energetically favoured at the surface. The change of the Weizsacker surface energy coefficient induced by spin-orbit coupling is also calculated. Because of the polarization of the nuclear surface it is expected that there is an excess of one kind of particles which is of the order of A*. Indeed, the shell-model ordering of states implies an excess of states with spin parallel to the orbital angular momentum such that in nuclei particles with spin +predominate on the average. Considering the shell-model level ordering no smooth curve is obtained for the excess of spin+particles in nuclei as a function of the nucleon number A. But the realistic curve may be approximated by the smooth statistical curve. Thus it is possible to 38
THOMAS-FERMI
39
DESCRIPTION
determine, by fitting the statistical curve to the realistic one, the value of the spin-orbit coupling. This concept was outlined by Swiatecki in an earlier investigation ‘). In sect. 5 his procedure is refined to lead to a coupling constant which is comparable with experiment.
2. The spin orientation in a semi-infinite system Since curvature effects arising from spin-orbit potentials are small due to the short range of the two-particle spin-orbit interaction “) the study of the simple system of semi-infinite nuclear matter with its plane surface provides a good picture of the influence of spin-orbit coupling in finite nuclei. In the presence of a one-particle spinorbit potential two kinds of nucleons with different spin orientations z = + 1 and 1 = - 1 have to be distinguished “). As illustrated in fig. 1, for the population 1= + 1 or i = - 1, respectively) three vectors form a right-(left-) hand coordinate system: (
_________ Fig. 1. The spin orientation in a semi-infinite system (left-hand part). The vector k, is the projection of the wave vector k on the surface; e, is the surface normal. The case of the plane surface is obtained as the limit of a large spherical nucleus (right-hand part).
the surface normal, the projection k, of the wave pseudo-vector s = $a which lies in the surface. nucleus given in the second part of fig. 1 the spin is seen to correspond in finite nuclei to a particle parallel) to the orbital angular momentum. From
vector k on the surface and the spin From the limiting case of a large of a particle with 1 = + 1 (z = - 1) the spin of which is parallel (antifig. 1 the relations “)
(1) are evident. The most simple expression for a single-particle fulfills the invariance requirements is of the form vii,, = 2a[k, s, VB],
spin-orbit
potential
V,
which
(2)
40
W. STOCKER ef al.
where B describes the inhomogeneity of the nuclear surface and a is the coupling constant. The function B is often set proportional to the nuclear density n or to the shell-model central potential. In the case of a semi-infinite nuclear system the potential V,, is specified to v,, = l+)(k, ay - k, %,>, (3) if the z-direction is chosen to coincide with the surface normal. The value of the potential YLSin the spin state z is then given by
6)
the function I-SlweV-fm Wz) = \ () being a possible approximation
06zS3fm, otherwise
for realistic nuclear systems.
3. Statistical description of particles in a single-particle
spin-orbit potential
Following Thomas-Fermi we suppose the system to be locally homogeneous. Then the single-particle energy of a nucleon with spin I, moving in a single-particle central potential U,(z) and a single-particle spin-orbit potential of the form (3), is given by &,(k, 9; z) = E
+ U,(z) f I W(z)k sin 9,
9 = 4 (k, e,).
(5)
The condition of minimal energy requires the following. First of all, for each point z and for a fixed direction 9 the absolute values k of the momentum vector k are occupied up to a quantity k,(9; z); secondly, all these values k,(9; z) have to be adjusted in such a way that the energy &r of the most energetic particle is independent of the S-value; finally, eF has to be independent of z. Thus, the Thomas-Fermi condition e,@,(‘% z), 9; z, =
&F,
(6)
is obtained which is a special case of the principle of communi~ting vessels well known from hydrostatics. This defines the surface of the region in momentum space which is occupied by the particles with spin z, explicitely given by k;2(9; z)+2z;l(z)k,($; n(Z) = $ hk,(z) =
$
z) sin 9 = k:(z), W(z),
,h&‘,, -
(7)
u,(z).
It is the surface of the toroid which is obtained by the rotation of of the circle kf + (k, + ~1)~ = k; + A2 in the (k,, k,) plane around Thus, by the inclusion of a spin-orbit potential the Fermi sphere distorted to the two toroids T, belonging to the spin o~entations
the upper-half part the k, axis (fig. 2). S with radius kF is 1.
THOMAS-FERMI
The density n, of the particles momentum distribution:
qz> =
41
DESCRIPTION
with spin z is now obtained
kF(Z)JT -1
using the Fermi toroid
n(Z)+ O@‘).
(8)
67~’
The first term is the density obtained without the spin-orbit potential, the second is the perturbation An of the density after a spin-orbit interaction has been included. The total density n = n, +n_ remains unchanged if higher-order terms in W are neglected.
4. Thomas-Fermi
4.1. THE SPHERES
equations for semi-infinite nuclear matter including spin-orbit coupling S, EQUIVALENT
TO THE TOROIDS
T,
It is possible to replace, in a first-order approximation with respect to the spin-orbit potential W, the toroids T, by equivalent spheres S, of radius kt;. The sphere S, is determined in such a way that it is centered about the origin, possessing the same volume in momentum space as the corresponding toroid T,, From this requirement, which ensures that S, and T, lead, in first order, to the same densities n,, it follows for the radius kk of the sphere S, that kk3 = k;-z&ilk;+
. . ..
(9)
In Thomas-Fermi theory the average value of the kinetic energy taken over the Fermi distribution is of importance. If only first-order terms in the spin-orbit potential are retained the average value of the kinetic energy is the same if taken over the toroid T, or the sphere S,. The same holds, of course, for the potential energy. For the toroids we obtain the relations (k’), I
= $(k;-+dk,+
. . .),
= &~kFi-z(2-&~2)~2+
. . ..
If, instead, we average over the spheres S,, small second-order deviations occur. By this simplification the second-order results derived below will be influenced a little. Starting from the two-nucleon interaction we treat the TF problem by a variational procedure in which the ground state of the nucleus is described semi-classically by a phase-space distribution f(r, p) consistent with Pauli’s exclusion principle. It is taken for granted by symmetry considerations that the one-particle spin-orbit potential has the form assumed in eq. (2). Then the energy density q is established in a statistical approximation using the Fermi spheres S, as momentum distributions. The two Fermi radii kk depending on z are taken as variational parameters.
42
W. STOCKER et al.
4.2. THE TF SINGLE-PARTICLE ORBIT I~RAC~ON
POTENTIAL RESULTING
FROM THE TWO-BODY
From the nuclear two-body spin-orbit interaction V$’ Vii’)
= 2V(r,,)(v,-rJx(k,-k,)*(s,+s,),
SPIN-
defined by r12 = lr,--4,
the single-particle potential Vii’ is obtained in the TF approximation sion -k,) d3r2 V(ri 2)(rI - r2) x (2:)s s k*LXLFtzzP3kz(k1
w
by the expres-
* (UI+ (4~Ad).
This expression can be reduced to the form
If the difference of the densities n, and n- is small compared with the total density only the first term in relation (12) contributes to the single-particle spin-orbit potential. Its form is the usual one given already in eq. (4). If the density n(z2) in the integral of the first term in eq. (12) is expanded into its Taylor series about the point z1 the orbital part W(z) of the single-particle spin-orbit potential is obtained as W(z) = y1 n’(z)+ ys n”‘(z)+ . * ..
(131
Here the constants yi are moments of the orbital part W(r,J spin-orbit interaction (10). 4.3. THE TF POTENTIAL-ENERGY POTENTIAL
DENSITY
RESULTING
of the two-particle
FROM
THE
SPIN-ORBIT
From the single-particle spin-orbit potential I’&) given by eq. (12) we obtain for the spin-orbit potential energy per unit surface area the expression
Both terms in eq. (12) contribute the same amount to the potential energy per unit surface area. With the TF assumption the energy density qLs for a spin-orbit potential may be written in terms of the densities n, and n_ as
r&z) = &~(3n')fCn~3(zl)-n!3(zl)]~d3r~
V(rj2)(zI
-z&(q).
(15)
THOMAS-FERMI
43
DESCRIPTION
If the expansion (13) is made for the orbital part of the spin-orbit potential eq. (15) takes on the form q&z) = nn’(z)fn~3(z)- n?!3(i)],
A.= +c(3n2)f~l,
w
if only the first term of the expression (13) is considered. 4.4. DERIVATION
OF THE TF EQUATIONS
INCLUDING
SPIN-ORBIT COUPLING
The energy density q(z) for semi-infinite nuclear matter is now written including spin-orbit coupling:
i =
$3”‘)%
(17)
For the spin-independent part of the potential energy density, assumed to be a functional of the total density n = n, +n_, a form is chosen similar or identical with expressions usual in literature “). By use of eq, (17) the surface tension of nuclear matter can be calculated according to u =
s
sr = - 15.9 MeV.
dz(q(t)-aFn(z)),
The minimalization conditions da
-=-.-.-..~O
dn,
do
an-
yield two coupled differential equations for the densities II, and n _ :
This system of equations may be considered as a generalization of the TF equation in the presence of spin-orbit coupling. Starting from the expressions (12) and (13) for the single-particle spin-orbit potential, the average spin-orbit potential energy of the most energetic particle with spin 1 amounts to (Y,>:
= EyIn’&r(3n2)3n$
(21)
where the index F indicates averaging over the Fermi surface. This is easily seen to be identical with the last term of the TF eq. (20) if only first-order terms of the spin-orbit potential W are considered. Thus the TF equations (20) are compatible with the idea that theenergy of the most energetic particle with spin 1 is independent of its position z.
44
et al.
W. STOCKER
5. Applications 5.1. POLARIZATION
OF THE
By subtracting
NUCLEAR
SURFACE
the two eqs. (20) from one another @I3, -n?)
= -$(fl$
the following
relation
is obtained
+ .k)rl’.
(22)
The central part of the potential which is a functional of the total density n does not enter. From the structure of the TF equations (20) one has the symmetry relation n_(z; 1) = n+(z; -A) for the dependence of the densities on the coupling constant 1. It implies the following expansion n,(z) = no(z)+ r&(z)
+ A2n2(z) + O(A”).
(23)
Here no is the solution of the TF equations without the spin-orbit term, whereas and n2 are concentrated at the nuclear surface. The first-order approximation for total density remains unchanged if a spin-orbit interaction is included. By using (23), we linearize eq. (22) thus obtaining for the first-order change of the density
n, the eq. the
Fig. 2. The Fermi domains in k-space in the presence of a spin-orbit potential W (< 0). The lefthand toroid is occupied by the particles with spin L = + 1, the right-hand one by particles with spin L = -1.
n(fmw3) . 0.1-
0.05 -
I
Fig.
I
I
r -1
* 0
1
z(fm)
3. The densities n, and n_ according to formula (24) for a realistic choice coupling (rI = 90 MeV fm5) taking for no a Fermi distribution.
of the spin-orbit
THOMAS-FERMI
45
DESCRIPTION
result An = in,
2A
=
- 7 n;5(2no)‘* This relation is consistent with eq. (8) if here the isospin degree of freedom is considered. Eq. (24) describes a polarization effect caused by the accumulation of the particle kind energetically favoured at the nuclear surface. In fig. 2 the densities n, are given in first order of A as calculated from a Fermi distribution for n, with 1, chosen as the experimental value in the region of 180 MeV * fm’.
5.2. THE CHANGE COUPLING
OF THE NUCLEAR
SURFACE
TENSION
DUE
TO SPIN-ORBIT
For a given nuclear density n, an energy density qc consisting of the kinetic energy and the central potential part, and an energy density qLS from the spin-orbit part, the nuclear surface tension c(n) is given by
If a one-parametric
class of trial densities
n,(p) is chosen conform . . .,
n,(p) = no+~pn1+p2n2+ the surface tension
can be expanded
with respect to the parameter
0,(p) = ~,+p%;+ 6&i)
to eq. (23)
p to give
. . .,
= L(pcJy+p3a,Ls+
. . .).
Using eq. (17) one obtains LS 62
The minimum of the surface the two relations
s
dz$z$(2n,j’n,.
=
tension
0 is obtained
(27) for p = A.. This condition
yields
2u; + cri” = 0, a; > 0. From the relations (26) and (28) the change do of the surface tension adding of a spin-orbit interaction is calculated to be ACT= +i2t$
(28) induced
by the
(29)
46
W. STOCKER et al.
This becomes da = -i
:
dzn0nr2,
(30)
s
if the explicit expression (24) for n, is used. The value for the decrease of the WeizsSicker surface energy coefficient following from eq. (30) is in the range of 2 MeV for a realistic choice of the occurring parameters. 5.3. ESTIMATION OF THE NUCLEAR SPIN-ORBIT COUPLING ORDER OF SHELL-MODEL NEUTRON LEVELS
CONSTANT
FROM THE
At the nuclear surface the reahstic spin-orbit potential energetically favours particles with spin E = + 1. Therefore a polarization of the nuclear surface is obtained by an excess of spin~particles. The amount of this excess is given in statistical approximation by formula (24). This formula holds also - due to the possibility of neglecting curvature effects - for finite nuclei. Since the polarization is a surface effect the ratio AN/N of the excess of spin+neutrons to the total neutron number N is statistically proportional to N -*. For real nuclei AN/N is no smooth function of N; but it can be approximated by the smooth curve predicted by the statistical theory. By a fitting procedure the coupling constant of the spin-orbit potential can thus be estimated. This concept was established by Swiatecki ‘). In the present subsection the idea of Swiatecki is taken up again; refinements of his procedure lead to a coupling constant which is comparable with the experimental one. The polarization of neutrons in a nucleus with N = Z is easily calculated to be onehalf of An of formula (24). For spherical nuclei, we then have
s a2
AN
N+-N-
_N = N++N_
=-_- 2;1 () ’
dr r”n;3n’
mdrr2n
(31) 0
’
s 0
n being the total and no half the nuclear density. For an estimation of this expression we choose the trapezoidal density n =
rs;;a a 5 r S a+3 otherwise,
(32)
with n, = 0.17 fmm3 and r and a are in units of fm. The parameter a can be calculated as a function of the neutron number N assuming a nucleus with N = 2. Then the polarization AN/N is calculated. (33) Since for great vaIues of N the extension a of the nucleus is roughly proportional N* the polarization is proportional to N-j for large values of N.
to
THOMAS-FERMI
DESCRIPTION
47
To find the experimental equivalent of AN/N the shell-model ordering of levels is now considered. First of all we have to ask what the spin-orientation number I should mean for a particle in a finite nucleus. In a rough approximation the alternatives z = f 1 correspond to i = If 3, as is seen from fig. 1. Due to spin-orbit coupling in each shell a = (n, I) the spin+levels are occupied first. In some cases two spin+ levels follow each other. This ordering of occupation implies that nuclei with an excess of spin + neutrons predominate on the average. In the explicit evaluation of the shell-model level order two difficulties arise. The first is the question of how to treat the neutrons in s-orbitals where the quantization
Fig. 4. The excess ON/N of spin+neutrons as a function of the neutron number N. The jagged line are the experimental vafues taken from she&modei level ordering [given e.g. in ref. 5)1, the continuous line is the statistical curve fitted to the experimental one.
axis 1 disappears. Secondly, in each closed E-shell the number N&, = 2Zf2 of neutrons with j = 2-i-i is greater by 2 than the number N&, = 21 of neutrons in the j = l-3 level. But the difference of 2 does not appear in the statistical picture. In order to eliminate this bias we replace NC& by NC:,, = N&,7 1 = 2Z+ 1. This leads to the following simple definition of AN/N for finite nuclei with closed levels CC= (n, /,j)
(34) where x; denotes summation over the open shells a only. This definition leads to no resultant excess of one spin sort for a closed-shell configuration. In fig. 4 the excess AN/N of spin+neutrons is given as a function of the neutron number N. The coupling constant y1 is adjusted in such a way that the statistical curve for AN/N given by formula (33) has the same area with the N-axis as the experimental curve for AN/N following from the shell-model level ordering. By this procedure a value of y1 = 67 MeV . fm’ is obtained. It compares well with the experimental one which is in the region of 85 MeV * fm5.
48
W. STOCKER et al.
If the prescription given above would be used to determine the coupling constant y1 from a level ordering obtained in the limiting case y -+ O-i- a finite value for y1 would be obtained. It is not expected that the value above for y1 must be corrected due to this zero effect since the level ordering of the shell model ensures a selection of states which is at random, whereas in the limiting case y --f 0 + the states with f neutrons have too much weight in the procedure. References 1) W. J. Swiatecki, hectographed manuscript, 1954 2) W. Stocker, Phys. Lett. 35B (1971) 472. 3) W. Stocker, Nucl. Phys. Al59 (1970) 222 4) R. A. Berg and L. Wilets, Phys. Rev. lOl(1956) 201; K. A. Brueckner, J. R. Buchler, S. Jorna and R. J. Lombard, Phys. Rev. 171 (1968) 1188 5) E. McCarthy, Introduction to nuclear theory (Wiley, New York, 1968)
Nuclear Physics Al87
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THOMAS-FERMI
DESCRIPTION
INCLUDING
OF NUCLEAR
SPIN-ORBIT
W. STOCKER,
INTERACTIONS
G. SUSSMANN
Sektion Physik der Universitiit,
SYSTEMS
and S. KNAAK Munich, Germany
Received 17 January 1972 Abstract: The Thomas-Fermi
statistical method is generalized to include spin-orbit interactions. The momentum distributions are given by toroids, different for two particle spin orientations. A system of two coupled differential equations is derived by a variational procedure for the densities of the two populations. From these equations the polarization at the surface of nuclear matter is calculated, as well as the change of the nuclear surface tension due to spin-orbit coupling. Within the statistical framework the coupling strength of the spin-orbit potential is found to be in reasonable agreement with experiment by using only the experimental singleparticle level order of the shell model which implies an excess of states with spin parallel to the orbital angular momentum.
1. Introduction The Thomas-Fermi (TF) and related statistical methods have been proved successful in nuclear physics. However, the generalization of the method to include a spinorbit potential is an open question. The present paper deals with this problem. In sect. 2 the two spin orientations of nucleons at the nuclear surface are discussed. In sect. 3 the momentum distribution for the two nucleon populations is investigated including a spin-orbit potential. It is shown that in this case the Fermi sphere in momentum space is distorted into two toroids different for the two spin orientations. In sect. 4 the statistical method is used to establish the energy density for systems with two-particle spin-orbit interactions. The minimalization condition for the surface tension yields a system of two coupled differential equations of TF structure for the particle densities n, and n_. After a linearization of these equations a simple expression is obtained for the polarization effect at the nuclear surface which results because one spin orientation is energetically favoured at the surface. The change of the Weizsacker surface energy coefficient induced by spin-orbit coupling is also calculated. Because of the polarization of the nuclear surface it is expected that there is an excess of one kind of particles which is of the order of A*. Indeed, the shell-model ordering of states implies an excess of states with spin parallel to the orbital angular momentum such that in nuclei particles with spin +predominate on the average. Considering the shell-model level ordering no smooth curve is obtained for the excess of spin+particles in nuclei as a function of the nucleon number A. But the realistic curve may be approximated by the smooth statistical curve. Thus it is possible to 38
THOMAS-FERMI
39
DESCRIPTION
determine, by fitting the statistical curve to the realistic one, the value of the spin-orbit coupling. This concept was outlined by Swiatecki in an earlier investigation ‘). In sect. 5 his procedure is refined to lead to a coupling constant which is comparable with experiment.
2. The spin orientation in a semi-infinite system Since curvature effects arising from spin-orbit potentials are small due to the short range of the two-particle spin-orbit interaction “) the study of the simple system of semi-infinite nuclear matter with its plane surface provides a good picture of the influence of spin-orbit coupling in finite nuclei. In the presence of a one-particle spinorbit potential two kinds of nucleons with different spin orientations z = + 1 and 1 = - 1 have to be distinguished “). As illustrated in fig. 1, for the population 1= + 1 or i = - 1, respectively) three vectors form a right-(left-) hand coordinate system: (
_________ Fig. 1. The spin orientation in a semi-infinite system (left-hand part). The vector k, is the projection of the wave vector k on the surface; e, is the surface normal. The case of the plane surface is obtained as the limit of a large spherical nucleus (right-hand part).
the surface normal, the projection k, of the wave pseudo-vector s = $a which lies in the surface. nucleus given in the second part of fig. 1 the spin is seen to correspond in finite nuclei to a particle parallel) to the orbital angular momentum. From
vector k on the surface and the spin From the limiting case of a large of a particle with 1 = + 1 (z = - 1) the spin of which is parallel (antifig. 1 the relations “)
(1) are evident. The most simple expression for a single-particle fulfills the invariance requirements is of the form vii,, = 2a[k, s, VB],
spin-orbit
potential
V,
which
(2)
40
W. STOCKER ef al.
where B describes the inhomogeneity of the nuclear surface and a is the coupling constant. The function B is often set proportional to the nuclear density n or to the shell-model central potential. In the case of a semi-infinite nuclear system the potential V,, is specified to v,, = l+)(k, ay - k, %,>, (3) if the z-direction is chosen to coincide with the surface normal. The value of the potential YLSin the spin state z is then given by
6)
the function I-SlweV-fm Wz) = \ () being a possible approximation
06zS3fm, otherwise
for realistic nuclear systems.
3. Statistical description of particles in a single-particle
spin-orbit potential
Following Thomas-Fermi we suppose the system to be locally homogeneous. Then the single-particle energy of a nucleon with spin I, moving in a single-particle central potential U,(z) and a single-particle spin-orbit potential of the form (3), is given by &,(k, 9; z) = E
+ U,(z) f I W(z)k sin 9,
9 = 4 (k, e,).
(5)
The condition of minimal energy requires the following. First of all, for each point z and for a fixed direction 9 the absolute values k of the momentum vector k are occupied up to a quantity k,(9; z); secondly, all these values k,(9; z) have to be adjusted in such a way that the energy &r of the most energetic particle is independent of the S-value; finally, eF has to be independent of z. Thus, the Thomas-Fermi condition e,@,(‘% z), 9; z, =
&F,
(6)
is obtained which is a special case of the principle of communi~ting vessels well known from hydrostatics. This defines the surface of the region in momentum space which is occupied by the particles with spin z, explicitely given by k;2(9; z)+2z;l(z)k,($; n(Z) = $ hk,(z) =
$
z) sin 9 = k:(z), W(z),
,h&‘,, -
(7)
u,(z).
It is the surface of the toroid which is obtained by the rotation of of the circle kf + (k, + ~1)~ = k; + A2 in the (k,, k,) plane around Thus, by the inclusion of a spin-orbit potential the Fermi sphere distorted to the two toroids T, belonging to the spin o~entations
the upper-half part the k, axis (fig. 2). S with radius kF is 1.
THOMAS-FERMI
The density n, of the particles momentum distribution:
qz> =
41
DESCRIPTION
with spin z is now obtained
kF(Z)JT -1
using the Fermi toroid
n(Z)+ O@‘).
(8)
67~’
The first term is the density obtained without the spin-orbit potential, the second is the perturbation An of the density after a spin-orbit interaction has been included. The total density n = n, +n_ remains unchanged if higher-order terms in W are neglected.
4. Thomas-Fermi
4.1. THE SPHERES
equations for semi-infinite nuclear matter including spin-orbit coupling S, EQUIVALENT
TO THE TOROIDS
T,
It is possible to replace, in a first-order approximation with respect to the spin-orbit potential W, the toroids T, by equivalent spheres S, of radius kt;. The sphere S, is determined in such a way that it is centered about the origin, possessing the same volume in momentum space as the corresponding toroid T,, From this requirement, which ensures that S, and T, lead, in first order, to the same densities n,, it follows for the radius kk of the sphere S, that kk3 = k;-z&ilk;+
. . ..
(9)
In Thomas-Fermi theory the average value of the kinetic energy taken over the Fermi distribution is of importance. If only first-order terms in the spin-orbit potential are retained the average value of the kinetic energy is the same if taken over the toroid T, or the sphere S,. The same holds, of course, for the potential energy. For the toroids we obtain the relations (k’), I
= $(k;-+dk,+
. . .),
= &~kFi-z(2-&~2)~2+
. . ..
If, instead, we average over the spheres S,, small second-order deviations occur. By this simplification the second-order results derived below will be influenced a little. Starting from the two-nucleon interaction we treat the TF problem by a variational procedure in which the ground state of the nucleus is described semi-classically by a phase-space distribution f(r, p) consistent with Pauli’s exclusion principle. It is taken for granted by symmetry considerations that the one-particle spin-orbit potential has the form assumed in eq. (2). Then the energy density q is established in a statistical approximation using the Fermi spheres S, as momentum distributions. The two Fermi radii kk depending on z are taken as variational parameters.
42
W. STOCKER et al.
4.2. THE TF SINGLE-PARTICLE ORBIT I~RAC~ON
POTENTIAL RESULTING
FROM THE TWO-BODY
From the nuclear two-body spin-orbit interaction V$’ Vii’)
= 2V(r,,)(v,-rJx(k,-k,)*(s,+s,),
SPIN-
defined by r12 = lr,--4,
the single-particle potential Vii’ is obtained in the TF approximation sion -k,) d3r2 V(ri 2)(rI - r2) x (2:)s s k*LXLFtzzP3kz(k1
w
by the expres-
* (UI+ (4~Ad).
This expression can be reduced to the form
If the difference of the densities n, and n- is small compared with the total density only the first term in relation (12) contributes to the single-particle spin-orbit potential. Its form is the usual one given already in eq. (4). If the density n(z2) in the integral of the first term in eq. (12) is expanded into its Taylor series about the point z1 the orbital part W(z) of the single-particle spin-orbit potential is obtained as W(z) = y1 n’(z)+ ys n”‘(z)+ . * ..
(131
Here the constants yi are moments of the orbital part W(r,J spin-orbit interaction (10). 4.3. THE TF POTENTIAL-ENERGY POTENTIAL
DENSITY
RESULTING
of the two-particle
FROM
THE
SPIN-ORBIT
From the single-particle spin-orbit potential I’&) given by eq. (12) we obtain for the spin-orbit potential energy per unit surface area the expression
Both terms in eq. (12) contribute the same amount to the potential energy per unit surface area. With the TF assumption the energy density qLs for a spin-orbit potential may be written in terms of the densities n, and n_ as
r&z) = &~(3n')fCn~3(zl)-n!3(zl)]~d3r~
V(rj2)(zI
-z&(q).
(15)
THOMAS-FERMI
43
DESCRIPTION
If the expansion (13) is made for the orbital part of the spin-orbit potential eq. (15) takes on the form q&z) = nn’(z)fn~3(z)- n?!3(i)],
A.= +c(3n2)f~l,
w
if only the first term of the expression (13) is considered. 4.4. DERIVATION
OF THE TF EQUATIONS
INCLUDING
SPIN-ORBIT COUPLING
The energy density q(z) for semi-infinite nuclear matter is now written including spin-orbit coupling:
i =
$3”‘)%
(17)
For the spin-independent part of the potential energy density, assumed to be a functional of the total density n = n, +n_, a form is chosen similar or identical with expressions usual in literature “). By use of eq, (17) the surface tension of nuclear matter can be calculated according to u =
s
sr = - 15.9 MeV.
dz(q(t)-aFn(z)),
The minimalization conditions da
-=-.-.-..~O
dn,
do
an-
yield two coupled differential equations for the densities II, and n _ :
This system of equations may be considered as a generalization of the TF equation in the presence of spin-orbit coupling. Starting from the expressions (12) and (13) for the single-particle spin-orbit potential, the average spin-orbit potential energy of the most energetic particle with spin 1 amounts to (Y,>:
= EyIn’&r(3n2)3n$
(21)
where the index F indicates averaging over the Fermi surface. This is easily seen to be identical with the last term of the TF eq. (20) if only first-order terms of the spin-orbit potential W are considered. Thus the TF equations (20) are compatible with the idea that theenergy of the most energetic particle with spin 1 is independent of its position z.
44
et al.
W. STOCKER
5. Applications 5.1. POLARIZATION
OF THE
By subtracting
NUCLEAR
SURFACE
the two eqs. (20) from one another @I3, -n?)
= -$(fl$
the following
relation
is obtained
+ .k)rl’.
(22)
The central part of the potential which is a functional of the total density n does not enter. From the structure of the TF equations (20) one has the symmetry relation n_(z; 1) = n+(z; -A) for the dependence of the densities on the coupling constant 1. It implies the following expansion n,(z) = no(z)+ r&(z)
+ A2n2(z) + O(A”).
(23)
Here no is the solution of the TF equations without the spin-orbit term, whereas and n2 are concentrated at the nuclear surface. The first-order approximation for total density remains unchanged if a spin-orbit interaction is included. By using (23), we linearize eq. (22) thus obtaining for the first-order change of the density
n, the eq. the
Fig. 2. The Fermi domains in k-space in the presence of a spin-orbit potential W (< 0). The lefthand toroid is occupied by the particles with spin L = + 1, the right-hand one by particles with spin L = -1.
n(fmw3) . 0.1-
0.05 -
I
Fig.
I
I
r -1
* 0
1
z(fm)
3. The densities n, and n_ according to formula (24) for a realistic choice coupling (rI = 90 MeV fm5) taking for no a Fermi distribution.
of the spin-orbit
THOMAS-FERMI
45
DESCRIPTION
result An = in,
2A
=
- 7 n;5(2no)‘* This relation is consistent with eq. (8) if here the isospin degree of freedom is considered. Eq. (24) describes a polarization effect caused by the accumulation of the particle kind energetically favoured at the nuclear surface. In fig. 2 the densities n, are given in first order of A as calculated from a Fermi distribution for n, with 1, chosen as the experimental value in the region of 180 MeV * fm’.
5.2. THE CHANGE COUPLING
OF THE NUCLEAR
SURFACE
TENSION
DUE
TO SPIN-ORBIT
For a given nuclear density n, an energy density qc consisting of the kinetic energy and the central potential part, and an energy density qLS from the spin-orbit part, the nuclear surface tension c(n) is given by
If a one-parametric
class of trial densities
n,(p) is chosen conform . . .,
n,(p) = no+~pn1+p2n2+ the surface tension
can be expanded
with respect to the parameter
0,(p) = ~,+p%;+ 6&i)
to eq. (23)
p to give
. . .,
= L(pcJy+p3a,Ls+
. . .).
Using eq. (17) one obtains LS 62
The minimum of the surface the two relations
s
dz$z$(2n,j’n,.
=
tension
0 is obtained
(27) for p = A.. This condition
yields
2u; + cri” = 0, a; > 0. From the relations (26) and (28) the change do of the surface tension adding of a spin-orbit interaction is calculated to be ACT= +i2t$
(28) induced
by the
(29)
46
W. STOCKER et al.
This becomes da = -i
:
dzn0nr2,
(30)
s
if the explicit expression (24) for n, is used. The value for the decrease of the WeizsSicker surface energy coefficient following from eq. (30) is in the range of 2 MeV for a realistic choice of the occurring parameters. 5.3. ESTIMATION OF THE NUCLEAR SPIN-ORBIT COUPLING ORDER OF SHELL-MODEL NEUTRON LEVELS
CONSTANT
FROM THE
At the nuclear surface the reahstic spin-orbit potential energetically favours particles with spin E = + 1. Therefore a polarization of the nuclear surface is obtained by an excess of spin~particles. The amount of this excess is given in statistical approximation by formula (24). This formula holds also - due to the possibility of neglecting curvature effects - for finite nuclei. Since the polarization is a surface effect the ratio AN/N of the excess of spin+neutrons to the total neutron number N is statistically proportional to N -*. For real nuclei AN/N is no smooth function of N; but it can be approximated by the smooth curve predicted by the statistical theory. By a fitting procedure the coupling constant of the spin-orbit potential can thus be estimated. This concept was established by Swiatecki ‘). In the present subsection the idea of Swiatecki is taken up again; refinements of his procedure lead to a coupling constant which is comparable with the experimental one. The polarization of neutrons in a nucleus with N = Z is easily calculated to be onehalf of An of formula (24). For spherical nuclei, we then have
s a2
AN
N+-N-
_N = N++N_
=-_- 2;1 () ’
dr r”n;3n’
mdrr2n
(31) 0
’
s 0
n being the total and no half the nuclear density. For an estimation of this expression we choose the trapezoidal density n =
rs;;a a 5 r S a+3 otherwise,
(32)
with n, = 0.17 fmm3 and r and a are in units of fm. The parameter a can be calculated as a function of the neutron number N assuming a nucleus with N = 2. Then the polarization AN/N is calculated. (33) Since for great vaIues of N the extension a of the nucleus is roughly proportional N* the polarization is proportional to N-j for large values of N.
to
THOMAS-FERMI
DESCRIPTION
47
To find the experimental equivalent of AN/N the shell-model ordering of levels is now considered. First of all we have to ask what the spin-orientation number I should mean for a particle in a finite nucleus. In a rough approximation the alternatives z = f 1 correspond to i = If 3, as is seen from fig. 1. Due to spin-orbit coupling in each shell a = (n, I) the spin+levels are occupied first. In some cases two spin+ levels follow each other. This ordering of occupation implies that nuclei with an excess of spin + neutrons predominate on the average. In the explicit evaluation of the shell-model level order two difficulties arise. The first is the question of how to treat the neutrons in s-orbitals where the quantization
Fig. 4. The excess ON/N of spin+neutrons as a function of the neutron number N. The jagged line are the experimental vafues taken from she&modei level ordering [given e.g. in ref. 5)1, the continuous line is the statistical curve fitted to the experimental one.
axis 1 disappears. Secondly, in each closed E-shell the number N&, = 2Zf2 of neutrons with j = 2-i-i is greater by 2 than the number N&, = 21 of neutrons in the j = l-3 level. But the difference of 2 does not appear in the statistical picture. In order to eliminate this bias we replace NC& by NC:,, = N&,7 1 = 2Z+ 1. This leads to the following simple definition of AN/N for finite nuclei with closed levels CC= (n, /,j)
(34) where x; denotes summation over the open shells a only. This definition leads to no resultant excess of one spin sort for a closed-shell configuration. In fig. 4 the excess AN/N of spin+neutrons is given as a function of the neutron number N. The coupling constant y1 is adjusted in such a way that the statistical curve for AN/N given by formula (33) has the same area with the N-axis as the experimental curve for AN/N following from the shell-model level ordering. By this procedure a value of y1 = 67 MeV . fm’ is obtained. It compares well with the experimental one which is in the region of 85 MeV * fm5.
48
W. STOCKER et al.
If the prescription given above would be used to determine the coupling constant y1 from a level ordering obtained in the limiting case y -+ O-i- a finite value for y1 would be obtained. It is not expected that the value above for y1 must be corrected due to this zero effect since the level ordering of the shell model ensures a selection of states which is at random, whereas in the limiting case y --f 0 + the states with f neutrons have too much weight in the procedure. References 1) W. J. Swiatecki, hectographed manuscript, 1954 2) W. Stocker, Phys. Lett. 35B (1971) 472. 3) W. Stocker, Nucl. Phys. Al59 (1970) 222 4) R. A. Berg and L. Wilets, Phys. Rev. lOl(1956) 201; K. A. Brueckner, J. R. Buchler, S. Jorna and R. J. Lombard, Phys. Rev. 171 (1968) 1188 5) E. McCarthy, Introduction to nuclear theory (Wiley, New York, 1968)