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Effects of nuclear size and curvature on spin-orbit splitting
Volume
35B, number 6
PHYSICS
EFFECTS
OF
NUCLEAR
ON
SPIN-ORBIT
LETTERS
SIZE
5 July 1971
AND
CURVATURE
SPLITTING
W. STOCKER Sektion Physik
dev Universitiit,
Received
Munich,
Germany
20 May 1971
The A -dependence of spin-orbit potentials and splittings following from two-body spin-orbit potentials is investigated taking into account higher derivatives of the nuclear density. The specific curvature effects are shown to be small.
The spin-orbit splittings of the Id-, If- and 2p-levels show a characteristic increase forzs/yall mass numbers A < 50; the 2d-, 3p- and 2f-splittings decrease for A > 100. Scheerbaum’s A -approximation [l] is a good description for this falloff. The purpose of the present letter is to investigate theoretically the overall behaviour of spin-orbit splittings. In shell model or optical model calculations the spin-orbit potential ULs is usually of the form
(1) f is taken to be the central part UC of the shell model potential or the nuclear density n, X is a parameter to be fitted. Scheerbaum [l] in his comprehensive work on spin-orbit splitting gives an expansion of the direct part of the single-particle potential following from two-body spin-orbit potentials for spherical nuclei.
u LS
=
ldn_E -xK4 7dr
4n 1 v=2 2v+l m
l d K2v+2 rdr
(2)
( Y dY2v-2
K4 and Ki are moments of the spatial part of the two-body spin-orbit potential. In the first approximation the exchange contribution is included by changing the coefficient K4; the structure of relation (2) remains unaltered [l]. The first term in eq. (2) is formally identical with the ansatz (1). The expression (2) can be rewritten as (2V -3) ULS’
- ,gl
h4v(n(2V-&
- (2v-2,$(+9))
5 I* s .
For the case of a plane surface the direct spin-orbit potential is given by [2] us0
=
part of the spin-orbit
(3) potential
following
from a two-body
v,lNV n(2v-& Cuxky - uykx) .
Comparing eqs. (3) and (4) the second term in formula (3) is identified as a specific curvature contribution being of the order l/r of the first term; it does therefore not enter into the spin-orbit potential (4) of the one-dimensional nucleus. In the approximation up to the moment KG the spin-orbit potential is given by ULS =
1 dn -rK4rdr-
%K Ld3, 15 6rdr3
_ g
2.K6+&$~)/,s Y
(5)
By the inclusion of the two additional terms the effectiveness of the potential form (1) can be tested, and the influence of curvature on spin-orbit splitting can be estimated. The effective moments K4 andK6 have been calculated using Wang’s correlation function [3] for a series of realistic spin-orbit potentials [l]. In the present paper the Gammel-Thaler (GT) 143 and the 472
Volume
35B, number
6
PHYSICS
3
LETTERS
5 July 1971
I i
2
3
4
6
7
Fig. 1. Spatial parts of the R spin-orbit potentials for the nuclei A ~20, 50, 90. (Solid line: potential according to formula (5)) broken line: .without curvature correction, dotted line: first term of (5) .)
Reid (R) [5] potentials are considered; K4= -51.51 MeV fm5, Kg = -40.87 MeV fm7 for GT, K4= -27.04, K6 = 78.62 for R. The moments K6 differ in sign from one another. So characteristic differences in the results for the GT and R potentials are expected. For the nuclear density n the Fermi distribution with the parametrization of Elton [6] 1s used. The proton and neutron densities are set proportional to one another. R Gl
h
!d f
Fig. 2. Spin-orbit splitting according for Reids potential. (solid line: splitting according to formula (5)) broken line: without curvature correction).
50
Fig. 3. Spin-orbit
150
splittings
200
for GT potential.
A
Volume 35B, number
6
PHYSICS
LETTERS
5 July 1971
In fig. 1 the spatial part of the spin-orbit potential is given for the Reid potentiai. The form (5) (solid line) is deeper and of smaller range than the first term of (5) (dotted line). For the GT potential the opposite behaviour is obtained. The curvature effect shown by the difference between broken and solid line is small. The inclusion of higher terms to the form (1) produces positive parts in the R spinorbit potential. For the investigation of the mass number and curvature dependence of spin-orbit splitting of neutron levels in a spin-saturated core in first order perturbation theory the mean value of the spin-orbit potential is calculated between oscillator wave functions. The oscillator parameter is taken to be &I = 45 A-1/3 - 25 Aw2j3 (MeV). Fig. 2 shows the splittings of the lowest Z-levels as functions of A up to A=90 for the Reid potential. The solid lines are the splittings following from the potential (5), the broken lines are the splittings without inclusion of the curvature term. The n= 1 - splittings have a maximum shifted to larger mass numbers if the angular momentum 1 increases. The r-value of the peak of the wave function is proportional to Ali , that of the spin-orbit potential to A1/3. The maximum of the splitting is obtained when the square of the wave function peaks at the same point as the spinorbit potential. For the %p-splitting a minimum occurs when the node region of the 2p-wave function coincides with the peak of the spin-orbit potential. In fig. 3 the splittings following from formula (5) for the GT spin-orbit potential are shown up to A= 300. The specific curvature effects on spin-orbit potentials and splittings are small; as a consequence of the short range of the two-body spin-orbit potential the particles do not experience the curvature. The characteristic maximum of the n= 1 -splittings in the range of small mass numbers can be explained. For positive moments K3 the spin-orbit interaction differs remarkably from the simple form (1). More detailed investigations of the single-particle spin-orbit potential may fix the sign of K3 and therefore lead to a better determination of the two-body spin-orbit interaction.
References [l] R.R. Scheerbaum, thesis, Cornell University (1969). [2] W.Stocker, Nucl. Phys. Al59 (1970) 222. [3] C.W.Wong, Nucl. Phys. A108 (1968) 481. [4] J. L.Gammel and R. M. Thaler, Phys. Rev. 107 (1957) 291. [5] R.V.Reid, Ann, Phys. (N.Y.) 50 (1968) 411. [6] L. R. B. Elton, Nuclear Radii in Landoldt-Bornstein l/2 (1967).
474
35B, number 6
PHYSICS
EFFECTS
OF
NUCLEAR
ON
SPIN-ORBIT
LETTERS
SIZE
5 July 1971
AND
CURVATURE
SPLITTING
W. STOCKER Sektion Physik
dev Universitiit,
Received
Munich,
Germany
20 May 1971
The A -dependence of spin-orbit potentials and splittings following from two-body spin-orbit potentials is investigated taking into account higher derivatives of the nuclear density. The specific curvature effects are shown to be small.
The spin-orbit splittings of the Id-, If- and 2p-levels show a characteristic increase forzs/yall mass numbers A < 50; the 2d-, 3p- and 2f-splittings decrease for A > 100. Scheerbaum’s A -approximation [l] is a good description for this falloff. The purpose of the present letter is to investigate theoretically the overall behaviour of spin-orbit splittings. In shell model or optical model calculations the spin-orbit potential ULs is usually of the form
(1) f is taken to be the central part UC of the shell model potential or the nuclear density n, X is a parameter to be fitted. Scheerbaum [l] in his comprehensive work on spin-orbit splitting gives an expansion of the direct part of the single-particle potential following from two-body spin-orbit potentials for spherical nuclei.
u LS
=
ldn_E -xK4 7dr
4n 1 v=2 2v+l m
l d K2v+2 rdr
(2)
( Y dY2v-2
K4 and Ki are moments of the spatial part of the two-body spin-orbit potential. In the first approximation the exchange contribution is included by changing the coefficient K4; the structure of relation (2) remains unaltered [l]. The first term in eq. (2) is formally identical with the ansatz (1). The expression (2) can be rewritten as (2V -3) ULS’
- ,gl
h4v(n(2V-&
- (2v-2,$(+9))
5 I* s .
For the case of a plane surface the direct spin-orbit potential is given by [2] us0
=
part of the spin-orbit
(3) potential
following
from a two-body
v,lNV n(2v-& Cuxky - uykx) .
Comparing eqs. (3) and (4) the second term in formula (3) is identified as a specific curvature contribution being of the order l/r of the first term; it does therefore not enter into the spin-orbit potential (4) of the one-dimensional nucleus. In the approximation up to the moment KG the spin-orbit potential is given by ULS =
1 dn -rK4rdr-
%K Ld3, 15 6rdr3
_ g
2.K6+&$~)/,s Y
(5)
By the inclusion of the two additional terms the effectiveness of the potential form (1) can be tested, and the influence of curvature on spin-orbit splitting can be estimated. The effective moments K4 andK6 have been calculated using Wang’s correlation function [3] for a series of realistic spin-orbit potentials [l]. In the present paper the Gammel-Thaler (GT) 143 and the 472
Volume
35B, number
6
PHYSICS
3
LETTERS
5 July 1971
I i
2
3
4
6
7
Fig. 1. Spatial parts of the R spin-orbit potentials for the nuclei A ~20, 50, 90. (Solid line: potential according to formula (5)) broken line: .without curvature correction, dotted line: first term of (5) .)
Reid (R) [5] potentials are considered; K4= -51.51 MeV fm5, Kg = -40.87 MeV fm7 for GT, K4= -27.04, K6 = 78.62 for R. The moments K6 differ in sign from one another. So characteristic differences in the results for the GT and R potentials are expected. For the nuclear density n the Fermi distribution with the parametrization of Elton [6] 1s used. The proton and neutron densities are set proportional to one another. R Gl
h
!d f
Fig. 2. Spin-orbit splitting according for Reids potential. (solid line: splitting according to formula (5)) broken line: without curvature correction).
50
Fig. 3. Spin-orbit
150
splittings
200
for GT potential.
A
Volume 35B, number
6
PHYSICS
LETTERS
5 July 1971
In fig. 1 the spatial part of the spin-orbit potential is given for the Reid potentiai. The form (5) (solid line) is deeper and of smaller range than the first term of (5) (dotted line). For the GT potential the opposite behaviour is obtained. The curvature effect shown by the difference between broken and solid line is small. The inclusion of higher terms to the form (1) produces positive parts in the R spinorbit potential. For the investigation of the mass number and curvature dependence of spin-orbit splitting of neutron levels in a spin-saturated core in first order perturbation theory the mean value of the spin-orbit potential is calculated between oscillator wave functions. The oscillator parameter is taken to be &I = 45 A-1/3 - 25 Aw2j3 (MeV). Fig. 2 shows the splittings of the lowest Z-levels as functions of A up to A=90 for the Reid potential. The solid lines are the splittings following from the potential (5), the broken lines are the splittings without inclusion of the curvature term. The n= 1 - splittings have a maximum shifted to larger mass numbers if the angular momentum 1 increases. The r-value of the peak of the wave function is proportional to Ali , that of the spin-orbit potential to A1/3. The maximum of the splitting is obtained when the square of the wave function peaks at the same point as the spinorbit potential. For the %p-splitting a minimum occurs when the node region of the 2p-wave function coincides with the peak of the spin-orbit potential. In fig. 3 the splittings following from formula (5) for the GT spin-orbit potential are shown up to A= 300. The specific curvature effects on spin-orbit potentials and splittings are small; as a consequence of the short range of the two-body spin-orbit potential the particles do not experience the curvature. The characteristic maximum of the n= 1 -splittings in the range of small mass numbers can be explained. For positive moments K3 the spin-orbit interaction differs remarkably from the simple form (1). More detailed investigations of the single-particle spin-orbit potential may fix the sign of K3 and therefore lead to a better determination of the two-body spin-orbit interaction.
References [l] R.R. Scheerbaum, thesis, Cornell University (1969). [2] W.Stocker, Nucl. Phys. Al59 (1970) 222. [3] C.W.Wong, Nucl. Phys. A108 (1968) 481. [4] J. L.Gammel and R. M. Thaler, Phys. Rev. 107 (1957) 291. [5] R.V.Reid, Ann, Phys. (N.Y.) 50 (1968) 411. [6] L. R. B. Elton, Nuclear Radii in Landoldt-Bornstein l/2 (1967).
474