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Dirac phenomenology and nuclear single-particle states
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A585 (1995) 641-656
Dirac phenomenology and nuclear single-particle states Y. N e d j a d i 1, J.R. R o o k Nuclear Physics Laboratory, University of Oxford, Keble Road, Oxford OX1 3RH, UK Received 11 August 1994; revised 24 November 1994
Abstract
This paper is concerned with the application of a relativistic shell model to the study of nuclear single-particle states. This relativistic model is applied to medium and heavy nuclei and shown to reproduce satisfactorily the single-particle energies and the rms radii. Energy-independent potentials give a reasonable account of the experimental binding energies, but the empirical analysis of the energy dependence of the potentials reveals a Fermi-surface anomaly. The parameters of the model are also analysed and found to vary systematically with both energy and mass number. The relativistic model is then reformulated in terms of folded-model potentials. The single-particle spectrum and charge rms radius of 4°Ca are reasonably reproduced, with an energy- and density-independent nucleon-nucleon force, and their sensitivity to the model parameters is studied.
1. Introduction
Over the past years, Dirac phenomenology has been extensively used to determine global optical potentials over a wide range of energies and nuclei [1]. In this paper, we are concerned with extending Dirac phenomenology to include the study of nuclear single-particle states. We use a relativistic shell model, whose single-particle interaction is the sum of a Lorentz scalar and time-like vector real potentials, to analyse the bound single-particle states of some spherical nuclei. While the empirical energy dependence of the potentials at positive energies has been thoroughly studied, at negative energies it has not. We therefore investigate the empirical energy dependence of the shell-model relativistic potentials.
i Present address: Department of Physics, University of Surrey, Guildford GU2 5XH, UK. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 7 8 8 - 8
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We also study the systematics of the potential parameters and discuss an alternative formulation of the single-particle potentials in terms of folded-model interactions. The phenomenological shell-model potentials are found to be consistent with the microscopic ones [2]. This work is presented as follows: In Section 2 the formalism required for this study is briefly discussed. The Dirac equation with Lorentz scalar and time-like vector potentials is reduced to a Schr6dinger equation and the main features of the Schr6dinger-equivalent potential are identified. Section 3 is devoted to the comparison between the experimental binding energies and the calculated ones for 4°Ca, 48Ca, 5SNi, 9°Zr, ll6Sn and 2°spb. A good description of the empirical data on binding energies is achieved. A study of the energy dependence of the relativistic potentials is then carried out. For surface states, the relativistic potentials are found to present the Fermi-surface anomaly. The systematics of the parameters of the potentials are also considered and it is shown that they meet the conditions to be satisfied by a phenomenological model, i.e. they vary smoothly with energy and mass number. In Section 4 the scalar and time-like vector interactions are reformulated in terms of folded-model potentials. It is found that using effective nucleonnucleon interactions that are density- and energy-independent is satisfactory. The sensitivity of the calcium single-particle spectrum to the parameters of the foldedmodel potentials is investigated. Finally, the main results are summarised in Section 5.
2. A relativistic shell model
In the non-relativistic approach the single-particle states of a nucleus are obtained by solving a standard single-particle Schr6dinger equation. Here, by contrast the single-particle states are obtained using the Dirac equation as the relevant wave equation. The essential feature of this model is the Lorentz character of the interaction which is assumed to be the sum of a scalar Us and a time vector potential U°. These single-particle potentials are viewed as the average meson fields generated by the other nucleons in the target. Both Us, and U° are assumed to be static, local and spherically symmetric. The Dirac equation describing the dynamical state ~0(r) of the nucleon is {•-p +/3[m + U~(r)] + where r is the of the target. matrices, E is We use the
U°(r)}qJ(r) = E 0 ( r ) ,
(2.1)
coordinate of the nucleon considered relative to the centre of mass In Eq. (2.1), ~ and /3 are the internal variables known as Dirac the energy of the nucleon of rest mass m, and p is its momentum. representation
o(0 o) (0 °i1
Y. Nedjadi, J.R. Rook~NuclearPhysicsA585 (1995)641-656
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where tr is the 2 × 2 Pauli spin vector and I is the 2 × 2 unit matrix. We define ,(r)
=
[ ~b(r) ) x(r)
(2.3)
'
where ~b(r) and x ( r ) are each 2-dimensional column matrices. From Eqs. (2.1)(2.3) we obtain o'.p (2.4) x(r) E + m + U s - Uv°qb(r)" It is possible without approximation to obtain a Schr6dinger-type equation for a function related to $. We define $ ( r ) = K ( r)c~(r),
(2.5)
where K is a function of I rl only such that K(r) =~(E+m
+ U~- U ° ) / ( E + m) .
(2.6)
Thus from Eqs. (2.1), (2.4)-(2.6) we obtain V2~b(r) + ~
e 1 + ~m
- Vefr- Vs'°" 4;(r) = 0,
(2.7)
where (2.8)
e=E-m, Vcf~=
1 + Us
Voar~n=2m h2 1
+
4~l-V
1+
0_ +
m
US ) + VDarwin,
2rn
(2.9)
(2.10)
V~
Vs.°. 2m r 1 - I 1 o * "L'
(2.11)
uo-u V°
2m + e
(2.12)
Here L is the orbital angular-momentum operator of the particle, and the prime denotes differentiation with respect to r. From Eq. (2.7) we see that the function 4; satisfies a Schr6dinger-like equation with V~ff as the central potential, Vs.o. as the spin-orbit potential and where the energy term involves a relativistic expression for the energy. The central potential Ven contains an explicit linear energy dependence. For protons we replace U° by U° + Vo where Vc is the Coulomb potential, and this gives a Coulomb correction term Uv° + V c / m in Eq. (2.9). This cross term Uv°Vc/m is of the type considered in the nonrelativistic optical potential for energy-dependent potentials [3]. The spin-orbit term in Eq. (2.7) arises naturally. Note that the scalar and vector potentials contribute with equal
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Y. Nedjadi, J.R. Rook ~Nuclear Physics A585 (1995) 641-656
sign to central The system
the spin-orbit interaction, whereas they enter with opposite sign into the potential. calculation of x(r) once ~b(r) is known is quite straightforward. Since the is assumed spherically symmetric each single-particle state can be written
[ ~(r) ~17(~"~))
~b(r) = l i x ( r )
(2.13)
~rmj(a ) ,
where ,g,~(f2) is the normalised spin angular function formed by the combination of the Pauli spinor with the spherical harmonics of order l, i.e. j = L + s. j is the total angular momentum, m its projection on the z-axis, l and l' are the orbital angular momenta of the large and small components respectively and they satisfy l = 1'+ 1 as appropriate. The single-particle states are labeled by the quantum numbers n, 1, j associated with the large components to keep the non-relativistic spectroscopic notation.
3. Phenomenological analysis
3.1. Single-particle spectra
ll6Sn
We consider the single-particle states in 4°Ca, 48Ca, 58Ni, 9°Zr, and 2°spb. 4°Ca, 48Ca and 58Ni are sufficiently light nuclei for the energies of their proton single particle energies to have been determined experimentally, while 9°Zr, 116Sn and 2°8pb have many single-particle states. For ll6Sn and 2°spb the energies of the deeper bound states are not measured experimentally but are determined by extrapolation of the potential as described in Ref. [4]. We assume that Us and U° in Eq. (2.1) have the Woods-Saxon form with respective radii and diffuseness parameters Rs, Rv, as, a v and strengths V~, V°. For protons we usually take additionally a Coulomb potential Vc corresponding to a uniformly charged sphere of radius R c = ~/(5/3)(r~2h) where \,/,.2-eh!31/2 is the experimental charge radius of the nucleus of interest. The parameters of the relativistic potentials are obtained by constraining them to reproduce a given doublet in the particular nucleus considered. In this procedure the geometrical factors are first fixed to some optimum values and then the strengths are adjusted. The single-particle Dirac equation is then solved for all the single particle states of the nucleus using these parameters. Tables 1, 2 show the comparison between the calculated proton- and neutronbinding energies and the experimental data for 58Ni and 116Sn; the results for the other nuclei are analogous. The parameters used for the potentials are given in each table. Most noticeable is the good agreement between the theoretical and the experimental results, especially when we note that the relativistic potentials do not depend on energy. All the shell gaps are well reproduced. In Table 3 we present experimental and theoretical charge rms radii. The potentials reproduce the charge radii of all nuclei and there is no problem to reconcile the single-particle
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Table 1
(a) 5SNi single-particle spectra. Energies are in MeV Protons
lSl/2 1P3/2 1Pw2 1D5"/2 1D~/2 251/2 1F7/2 2])3/2 2P1/2 1Fs/2
Neutrons
RM
Expt [4]
RM
Expt [4]
50.74 35.77 31.76 21.21 14.20 15.57 7.60 2.70 0.76 -
55 :i: 7 42 + 7 38:1:5 23:1:5 14 5:2 11.16 7.60 2.69 -
57.95 42.87 38.39 28.34 20.60 22.48 14.91 9.52 7.40 5.89
19.06 14.70 9.51 7.33 8.90
(b) Model parameters
Protons Neutrons
R s (fro)
R v (fm)
a s (fm)
a v (fm)
Vs (MeV)
Vv° (MeV)
4.208 4.208
4.208 4.208
0.65 0.65
0.6 0.6
- 481.63 - 507.68
391.78 420.38
Table 2 (a) 116Sn single-particle spectra. Energies are in M e V Protons
151/2 1P3/2 1Px/2 1D5/2 1D3/2 281/2 1F7/2 1F5/2 21)3"/2 2P1"/2 1G9/2 1G7/2 2D5/2 2D3/2 1Hll/2
Neutrons
RM
Expt [5]
RM
Expt [5]
60.86 48.41 45.24 35.50 28.87 28.59 22.78 12.82 14.27 11.90 10.68 2.71 -
58.30 47.60 45.10 35.20 30.00 27.20 21.94 14.26 13.21 11.79 11.05 -
66.08 54.06 51.71 41.66 36.59 36.33 29.49 21.62 23.25 20.91 17.90 8.30 11.68 8.82 6.99
72.1 59.8 57.5 46.2 41.2 39.6 32.1 24.0 23.4 20.9 17.9 10.3 11.2 8.92 -
(b) Model parameters
Protons Neutrons
R s (fm)
R v (fln)
a s (fm)
a v (fin)
V~ (MeV)
V° (MeV)
5.365 5.365
5.365 5.365
0.65 0.65
0.59 0.59
- 511.10 - 511.10
414.00 422.16
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Y. Nedjadi, J.l~ Rook ~Nuclear PhysicsA585 (1995) 641-656
Table 3 Charge rms radii in fro. The rms radii are calculated using the potentials given in the previous tables, r , refers to the point-neutron distribution.
4°Ca 48Ca 58Ni 9°Zr 116Sn 2°spb
rc¢~p
rcC~Ic
rntalc
3.481 a 3.48 b 3.84 c 4.30 b 4.62 a 5.503 b
3.482 3.48 3.85 4.32 4.58 5.502
3.32 3.59 3.63 4.22 4.59 5.64
a Ref. [7]. b Ref. [8]. Ref. [4].
spectra and charge radii. All levels are correctly ordered apart from the 1175/2(for all nuclei) and the 1G7/2 for 116Sn. These features being determined by the spin-orbit part of the non-relativistic nucleon-nucleus potential, these results show that the model does reproduce the phenomenological spin-orbit splitting of the shell model.
3.2. Energy dependence of the relativisticpotentials As already mentioned in Section 2, ~ff presents a linear energy dependence with a slope U°/m if U° and Us do not depend on energy. Note that the non-linear energy variation of the rather small Darwin term does not bear on this dependence. Although this prescription is seen from the results of the previous section to be satisfactory it is of interest to investigate the extent to which the potentials Us and Uv° may be energy-dependent. In the non-relativistic analysis of the energy dependence of the central potential it is usual to consider the energy variation of the depth of this potential, its geometrical factors being kept fLxed. The energy dependence of the potential well is assumed to be entirely contained in the potential depth. However, in our case, the Schr6dinger-equivalent central potential is not a simple function of Us, Uv° and so that it is not possible to fix its geometry and vary the depth. Therefore we consider the energy dependence of the integrated potential strength per nucleon feff(E) which is defined as 1 fefe= ~
fv~ff d3r.
(3.1)
Within each nucleus, the experimental single-particle energies of all the available doublets up to the Fermi surface are used to determine the associated depths of the potentials Us and Uv°. The overall trends of the energy variation of Jeff are quite similar for all the nuclei considered. Figs. 1, 2 show the energy variation of feff for ll6Sn and 2°spb respectively_ There is a sharp difference between the surface and deep states. Whereas Jeff decreases linearly for deep states, this
Y.. Nedjadi, J.R. Rook/Nuclear PhysicsA585 (1995) 641-656
647
55.0
~SSn 50.0
e
Neutron on
E
LL
=> 45.0
v
40.0
n
35'-"50.0
i
r
-40.0
-30.0 e (MeV)
i
i
-20.0
-10.0
Fig. 1. Dependence upon single-particle energy of the volume integrals per nucleon of I/ele that yields the experimental proton- and neutron-binding energies in ll6Sn.
linearity is lost at the surface. This behaviour around the Fermi energy is known as the Fermi-surface anomaly [9,10]. This implies that the Fermi-surface anomaly does not arise as a relativistic effect and must be included explicitly. This conclusion can be expected from the interpretation of the Fermi-surface anomaly given in Ref. [9]. For the deeper states which are coupled to an "inert" core, the non-relativistic single-particle potential varies linearly with energy. However, near the Fermi surface many particle-hole excitations take place more readily, thus allowing the possibility of the single-particle states to couple to core-polarised
50.0 2oSpb 45.0
8 >
40.0
35.0
30.0
-60.0
",=n .... -50.0
-40.0
-30.0
-20.0
-10.0
0.0
e (MeV) Fig. 2. Dependence upon single-particle energy of the volume integrals per nucleon of I/ett that yields the experimental proton- and neutron-binding energies in 2°apb.
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Table 4 Slopes of the proton and neutron potentials
4SCa n6Sn 2°8pb
Proton
Neutron
0.14 0.09 0.06
0.25 0.25 0.30
states. As shown in Ref. [9], including the possibility of exciting low-lying corepolarised states introduces an explicit energy dependence in agreement with the departure from linearity. The question remains open as to whether a fully relativistic calculation which includes these effects would reproduce the behaviour at the Fermi surface. We now examine just the linear region away from the Fermi-surface anomaly. We find the slope of the line obtained when Jeff is plotted against the single-particle energy E. These slopes are given in Table 4. For neutrons the average slope is 0.27 whereas for protons it is smaller and decreases with increasing Z. In the non-relativistic case the slopes are found to be 0.24 on average, for both protons and neutrons [11]. Such a result is obtained with a central potential assumed to have a fixed Woods-Saxon shape and a depth adjusted to reproduce the singleparticle state in question. For neutrons the energy dependence of the Schr6dinger-equivalent central potential is almost equal to the non-relativistic result. We interpret the large difference for the proton slopes as due to the fact that fell(e) is sensitive to the choice of the geometry [12]. This is supported by Fig. 3 which shows that the effect of adding terms involving the Coulomb potential
55.0 11SSn 500
> 45.0 v
40.0 • With Coulomb 35"760.0
~
II
i
r
i
I
i
-50.0
-40.0
-30.0
-20.0
-10.0
¢ (MeV)
Fig. 3. Effect of the Coulomb potentialon the energy dependence of Jefffor protons in 11asn. Without the Coulomb cross term in Ezi. (2.9) the slope is about 0.23 in the linear region away from the Fermi-surface anomaly. Keeping the Coulomb cross term gives a slope of 0.10.
Y. Nedjadi, J.R. Rook /Nuclear Physics A585 (1995) 641-656
649
300.0
250.0
200.0
150.0
• " Js ~oe?Pb
•Jv( Pb) • " Js "esn • Jv ( lleSn )
100..060.0
-40.0
i
-20.0
0.0
Fig. 4. Dependence upon single-particle energy of the volume integrals per nucleon of Us and Uv° that yield the experimental proton- and neutron-binding energies in 116Sn and 2°spb.
introduces a drastic change in the slope. Note that without Coulomb cross terms the slope is the same as that of the neutrons. An alternative way of looking at the energy dependence of the relativistic potentials is through the volume-integrated potentials of Us and U° defined as
47rA
d3r,
(3.2)
1
fv=4rcAfU°d 3r,
(3.3)
where A is the mass number. Fig. 4 shows the results for 116Sn and 2°spb. For deeper states, .~ and fv are almost energy-independent as most of the energy dependence of ~ is found to arise as a relativistic effect. At the Fermi surface, where is energy-independent, the empirical relativistic potentials present a pronounced energy variation.
fen
Table 5 Proton single-particle levels of 4aCa. Potentials are energy-dependent. Energies are in MeV 1S1/2 1P3/2 1P1/2 1Ds/2 2S1/2 1D3/2 1F7/2 2P3/2 2P1/2
RM
Expt
61.25 43.57 37.36 26.34 18.18 16.67 10.29 3.58 1.22
55.0 + 10.0 35.0:1:10.0 35.0 :t: 10.0 23.5 + 2.0 16.45 16.17 9.62 3.6+2.0 0.0
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We now consider an intrinsic linear energy dependence for Us and U° in order to establish limits for any possible energy dependence. Hence we take
Uv°(e) = U ° ( 0 ) - ae,
(3.4)
US(0) (0)
US(e) = Us(O) - a,-Tb-7-~,x e,
(3.5)
where a = 0.5 and e is the binding energy. 4SCa is chosen as an example and the relevant geometrical factors and strengths of U°(0) and US(0) are given in Table 2. The results are shown on Table 5. There is no major improvement because of the large quoted errors. Note, however, that the deeper states are more affected than those at the Fermi surface. Further calculations showed that the parameter a cannot be greater than 1.5. This figure is proportionally less than would be expected for the energy dependence in a non-relativistic treatment.
3.3. Systematics o f the model The radii of the potentials Us and U° were constrained to vary as 1.1 A 1/3 (frn) where A is the mass number. The calculations have indicated a constraint between the values a s and a v. The relative positions of 1D3/2 and 2S1/2, for both neutrons and protons, can be fitted only if a s is greater than av. This inequality is expected on general grounds since Us and U° correspond, in a first approximation, to scalarand vector-meson exchanges respectively and the mass of the vector meson is greater. Using a fixed value of a s (0.65 fm) for all the nuclei considered, a~ is found to have values that decrease with increasing mass number. This is due to the fact that the matter density is less diffuse with increasing A. Table 6 shows the integrated strengths per nucleon of Us and U° for the different nuclei considered. Js and J~ are independent of A to within 13%. The last two columns refer to the ratios of the r.m.s, radii of the vector and scalar potential. They are almost the same for neutrons as well as protons in all nuclei. Among the requirements that a phenomenological model must satisfy is the condition that its parameters have to vary in a systematic way with energy and mass number. For bound states and the range of nuclei we consider, this shell model satisfies this condition. The volume integrals of Us and U° are the same for
Table 6 V o l u m e - i n t e g r a t e d p o t e n t i a l p e r n u c l e o n f (in M e v . f m 3 u n i t s ) a n d r a t i o s o f t h e r m s r a d i i o f t h e v e c t o r and scalar potentials A
.~ (p)
.~ (n)
fv (P)
fv (n)
(R v/Rs) p a
(Rv/Rs)n a
48 56 90 116
-
-
231.99 208.63 204.33 205.64
239.76 223.86 206.80 209.69
-
-
291.70 263.98 257.74 259.65
291.70 278.20 257.74 259.65
0.795 0.791 0.793 0.792
0.822 0.805 0.802 0.808
a R v is t h e r m s r a d i u s o f t h e v e c t o r p o t e n t i a l . R s is t h e r m s r a d i u s o f t h e s c a l a r p o t e n t i a l . T h e i n d i c e s p and n refer to protons and neutrons respectively.
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651
protons and neutrons and they are independent of A. If the potentials Us and U° are viewed as potentials obtained from a folding procedure where a zero-range force is used, the above result suggests that the single-particle spectra of these nuclei would be reproduced from the same nucleon-nucleon force.
4. A folding model An alternative to assuming Woods-Saxon forms for the potentials is to take the folding-model approach which relates the potentials to the underlying nucleonnucleon force and the properties of the nucleus. The Lorentz scalar and vector potentials are defined as
Us(r ) = fp~(r')Vs(Ir-r'])
dr',
U°(r) = fp°(r')Vv( I r - r' I) dr'.
(4.1) (4.2)
The nucleon-nucleon interactions vs and vv are effective interactions for point nucleons interacting via exchange of scalar and vector point mesons respectively. These interactions are chosen to be simple local functions of I r - r'l that are
g2 e-rest vs(r) = 4~r - - ,r g2 e-mvr Vv(r) = 47r - - ,r
(4.3) (4.4)
with m s and m v as the masses of the vector and scalar mesons exchanged and gs and gv as the scalar-meson-nucleon and vector-meson-nucleon coupling constants respectively. The density pv° is the time-like component of the current density and represents the nuclear-matter density of the nucleus. The scalar density Ps is empirically undetermined. However, since the relativistic shell model prescribes a relationship between the time-like vector and scalar densities, i.e Ps(r) =
E
21"+1
[~bf(r) - x 2 ( r ) ]
(4.5)
occupied j shells 4~r
p°(r)=
~_,
2j+ 1
4---~[~b2(r)+x~(r)],
(4.6)
occupied j shells
where dpi(r) and xi(r) are the large- and small-component wave functions as defined by Eq. (2.3), this property can be used to calculate a "semi-empirical" scalar density. This is done by constraining the "semi-empirical" density p~¢ to satisfy pS.e
pscalc
p~xp
p~alc'
(4.7)
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Y. Nedjadi, J..R.Rook ~Nuclear PhysicsA585 (1995)641-656
Table 7 (a) 4°Ca single-particle spectra. Energies are in MeV. rch talc= 3.451 fin, rc~p = 3.480 fin. Protons Neutrons FM a Expt [5] NR [13] b FM a Expt [5] NR [13] b 1S1/2 1P3/2 1PI/2 1D5/2 281/2 1D3/2 1F7/2
43.99 30.20 25.18 15.60 7.93 8.44 1.28
48.5 + 5.0 36.0 + 3.0 31.5+3.5 16.0 + 2.0 12.0 + 1.0 8.5 + 2.0 1.36
42.36 29.08 25.69 16.68 9.99 10.58 4.11
56.32 41.30 36.26 25.58 17.79 18.13 10.06
(b) Model parameters Mass (MeV)
g 2/4~"
Scalar meson Vector meson
8.50 12.90
a
b
540 780
18.1 15.6 8.36
50.0 30.0 27.0 21.9 18.2 15.6 8.4
FM: Folding model. NR: Non-relativistic calculation using Hartree-Fock theory (DDHF).
where pCvalCand p~l¢ are calculated using a model that reproduces the single-particle energies and the r.m.s, radius of the nucleus in question. These distributions are spherically symmetric and hence U~(r) and Uv°(r) are reduced to one-dimensional integrals. Once U~(r) and U°(r) are determined the single-particle Dirac equation (3.1) is solved for the particular nucleus considered.
4.1. Calcium single-particle spectrum We consider the doubly-closed-shell nucleus 4°Ca. The potentials Us and U° are calculated from Eqs. (4.1), (4.2). For protons the Coulomb potential Vc corresponds to a uniformly charged sphere. Table 7 shows the comparison between the calculated binding energies and the experimental data. The calculated charge rms radius and the model p a r a m e t e r s are given in the table. T h e single-particle energies and the s p i n - o r b i t splitting agree reasonably with the data. Note that the charge r.m.s, radius is also well reproduced. T h e ordering of the 2S1/2 relative to the 1D3/2 is inverted. T h e calculations using W o o d s - S a x o n potentials for Us and U° have shown that this inversion arises if the diffuseness of the scalar potential is lower than (or close to) the diffuseness of the vector potential. In this folding model the potentials found have almost the same diffuseness too. The spectra obtained are very similar to the one obtained by Campi and Sprung [13] using the density-dependent H a r t r e e - F o c k theory. T h e problem of the inversion of the 2S1/2 relative to the 1D3/2 is also encountered and they relate it to the energy dependence of the n u c l e o n - n u c l e o n force. In the folding model we use, the n u c l e o n - n u c l e o n force is neither energy-dependent nor density-dependent. It is interesting to note that it is possible to obtain the right ordering of the 2S1/2 relative to the 1D3/2 but unfortunately at the expense of the charge r.m.s, radius.
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Table 8 (a) 4°Ca single-particle spectra. Energies are in MeV. r ~ e = 3.638 fro, r~hw= 3.480 fro. Protons
1S1/2 1P3/z 1P1/2 1Ds/2 2S~/2 1D3/2 1F7/2
Neutrons
FM a
Expt [5]
FM a
Expt [51
41.38 27.82 22.11 14.32 7.84 6.86 1.45
48.5 + 5.0 36.0 + 3.0 31.5 + 3.5 16.0 + 2.0 12.0+ 1.0 8.5 + 2.0 1.36
54.33 39.13 33.28 24.20 17.85 16.20 9.97
18.1 15.6 8.36
(b) Model parameters
Scalar meson Vector meson
Mass (MeV)
g 2 / 4 ~r
450 780
6.48 14.21
a FM." Folding model.
This is shown on Table 8. Combining this new model together with including some energy dependence would reconcile the ordering of the 2Sx/2 relative to the 1 D 3 / 2 and the charge r.m.s, radius. As seen from Section 3.2, including an intrinsic energy dependence does not affect the states near the Fermi surface but increases the binding energy of the deeply bound states, hence reducing the r.m.s, radius. However, it is not possible to estimate the latter, as the orthogonality between the single-particle state wave functions is then lost.
4. 2. Sensitivity of the model parameters In order to study the sensitivity to the parameters, the masses and the coupling constants of the exchanged mesons are handled as free parameters. This makes it possible to study the sensitivity to different regions of the nucleon-nucleon force independently from one another. The scalar-meson mass was varied from 400 to 700 MeV with the ratio gs/ms 2 2 fixed at the value corresponding to the set of parameters given in Table 7. The dependence of the single-particle energies and the r.m.s, radii upon the scalar-meson mass (in pion-mass unit) is shown in Fig. 5 for protons and Fig. 6 for neutrons. We note that the single-particle energies and r.m.s, radii are very sensitive to the scalar-meson mass (ms). As m s increases, the attractive part of the nucleon-nucleon potential becomes stronger at shorter internucleon distances. As a result the binding energy of nucleons increases and the r.m.s, radius becomes smaller. The scalar density is fixed and hence Eq. (4.1) shows that increasing m s yields a scalar potential that is deeper and less diffuse. Figs. 5, 6 also illustrate the inversion of the ordering of the 2S1/2 relative to the 1D3/2. For heavier scalar mesons the diffuseness of the scalar potential decreases and yields a value close to the vector-potential diffuseness. This results in the inversion of the 2S1/2 as expected from the systematics discussed in Section 3.3.
Y. Nedjad~ J.K Rook/Nuclear PhysicsA585 (1995) 641-656
654
0.0
,
i
,
I
-10.0
-20.0
~o= -30.0
-40.0
,
-50.0
~
3.60
^
3.40
d
V
3.20
I
3.0
I
3.5
4.0 mjm~
4.5
5.0
Fig. 5. Effect of the scalar-meson mass m s upon the proton single-particle spectrum and the rms radius of the proton distribution in 4°Ca.
It is interesting to remark (see Figs. 5, 6) that although the single-particle energies are very sensitive to the scalar-meson mass, almost no change has occurred in the spin-orbit splitting. Further calculations have shown that this feature recurs even if the ratio gs2~ms2 is not constrained to a fixed value. For both protons and neutrons, increasing the scalar mass from 3m~ to 5m~ shifts the P-states by about 13 MeV and the D-states by 8 MeV. The spin-orbit splittings are increased by 0.63 and 2 MeV for the P-shell and the D-shell respectively. This
0,0
,
'i
••
,i
..
F
•10•O
,
1F7/2
-20.0 1D3/2
-40.0
J
1P3/2
-50.0 -60.0
1S1/2 -70.0
*
I
3.40
A
3.20 3.00
30
i
410 mJl'n s
Fig. 6. Effect of the scalar meson mass m s upon the neutron single-particle spectrum and the rms radius of the neutron distribution in 4°Ca.
Y. Nedjad~ J.R. Rook/Nuclear PhysicsA585 (1995) 641-656
655
Table 9 (a) 4°Ca single-particle spectra. Energies are in MeV. rch talc 3.527 fro, rce~v = 3.480 fm. Protons Neutrons FM a Expt [5] FM a Expt [5] =
1S1/2 1P3/2 1P1/2 1D5/2 2Si/2 1D3/2 1FT/2
43.30 29.60 24.60 15.11 7.51 8.00 0.91
48.5 + 5.0 36.0 _+3.0 31.5 5:3.5 16.0 _+2.0 12.0+ 1.0 8.5 + 2.0 1.36
55.62 40.69 35.67 25.07 17.33 17.66 9.66
18.1 15.6 8.36
(b) Model parameters Scalar meson Vector meson a FM:
Mass (MeV)
g 2/4 w
450 1000
8.50 21.20
Folding model.
consideration together with the Fermi-surface anomaly has an interesting consequence from a practical point of view. W h e n the shell model with energy-independent potentials is used, the choice of the doublet to which the potentials are constrained is ambiguous. F r o m Figs. 5, 6 and the results of Section 3.2 we see that the doublet must not be too deep so that V~n is undetermined and not loosely bound because of the Fermi-surface anomaly. C o m p a r a b l e changes to the vector-meson mass (from 780 to 1000 MeV) with a fixed ratio of g v2/ m v2 and the p a r a m e t e r s of the scalar-meson mass given in Table 7 yield only a very small change (less than 5%) for both the single-particle energies and the r.m.s, radius (see Table 9). This arises because the range of the repulsive force is much smaller than the m e a n internucleon distance, or in other words the nucleon's motion is insensitive to these changes in the diffuseness of the vector potential obtained from Eq. (4.2). This result also explains the systematics found for the vector diffuseness (see Section 3.3), that is to say the diffuseness of the vector potential decreases with increasing mass number. The vector potential U° is mainly determined by the nuclear density p0 and hence this behaviour occurs as a result of the reduction of the diffuseness of nuclear matter as the mass number increases. Finally, this study of the sensitivity to the model p a r a m e t e r s has shown that Uv° is better determined than Us.
5. C o n c l u s i o n
A phenomenological relativistic shell model was applied to many nuclei and was shown to be successful in reproducing the single-particle motion, spin-orbit splitting and r.m.s, radii. To gain some empirical insight into this shell model, a study of the energy d e p e n d e n c e of the relativistic potentials was made. Whereas
656
Y. Nedjad~ J.l~ Rook ~Nuclear PhysicsA585 (1995) 641-656
for deep states the potentials were found to be roughly energy-independent, for loosely bound states at the surface, their behaviour is of the Fermi-surface-anomaly type. The analysis of the parameters of the model showed that they vary in a systematic way with both mass number and energy. Reformulating this SV model in terms of folded-model potentials showed another feature of the Schr6dingerequivalent central potential, implicit in the use of a relativistic approach, that is the density dependence. This density dependence is sensitive to the relationship between the vector and scalar densities. The scalar density being empirically undetermined, a "semi-empirical" scalar density was used instead. Using an energy- and density-independent nucleon-nucleon force, the single-particle spectra and charge r.m.s, radius of 4°Ca were reasonably reproduced. The study of the sensitivity to the model parameters showed that Uv° is better defined than Us.
References [1] E.D. Cooper, S. Hama, B.C. Clark and R.L. Mercer, Phys. Rev. C 47 (1993) 297, and references therein. [2] B.B. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [3] D.H. Wilkinson, Isospin in nuclear physics, (North-Holland, Amsterdam, 1969) p. 389. [4] F. Malaguti, A. Uguzzoni, E. Verondini and P.E. Hodgson, Nuovo Cimento 5 (1982) 38. [5] L. Ray and P.E. Hodgson, Phys. Rev. C 20 (1979) 2403. [6] J.W. Negele, Phys. Rev. C 1 (1970) 1260. [7] B.A. Brow, C. Bronk and P.E. Hodgson, J. of Phys. G 10 (1984) 1683. [8] At. Data Nucl. Data Tables 36 (1987) 495. [9] B. Day, Proc. Int. School of Physics (North-Holland, Amsterdam, 1982) p. 1. [10] G.F. Bertsch and T.T. Kuo, Nucl. Phys. A 112 (1968) 204; Proc. 2nd La Rabida Summer School on Nuclear physics (1985). [11] M. Bauer, E. Heruandez-Saldana, P.E. Hodgson and J. Quintanilla, J. of Phys. G 8 (1982) 525. [12] C. Mahaux and R. Sartor, Theory of nuclear structure and reactions. [13] X. Campi and D.W. Sprung, Nucl. Phys. A 194 (1972) 401.
Nuclear Physics A585 (1995) 641-656
Dirac phenomenology and nuclear single-particle states Y. N e d j a d i 1, J.R. R o o k Nuclear Physics Laboratory, University of Oxford, Keble Road, Oxford OX1 3RH, UK Received 11 August 1994; revised 24 November 1994
Abstract
This paper is concerned with the application of a relativistic shell model to the study of nuclear single-particle states. This relativistic model is applied to medium and heavy nuclei and shown to reproduce satisfactorily the single-particle energies and the rms radii. Energy-independent potentials give a reasonable account of the experimental binding energies, but the empirical analysis of the energy dependence of the potentials reveals a Fermi-surface anomaly. The parameters of the model are also analysed and found to vary systematically with both energy and mass number. The relativistic model is then reformulated in terms of folded-model potentials. The single-particle spectrum and charge rms radius of 4°Ca are reasonably reproduced, with an energy- and density-independent nucleon-nucleon force, and their sensitivity to the model parameters is studied.
1. Introduction
Over the past years, Dirac phenomenology has been extensively used to determine global optical potentials over a wide range of energies and nuclei [1]. In this paper, we are concerned with extending Dirac phenomenology to include the study of nuclear single-particle states. We use a relativistic shell model, whose single-particle interaction is the sum of a Lorentz scalar and time-like vector real potentials, to analyse the bound single-particle states of some spherical nuclei. While the empirical energy dependence of the potentials at positive energies has been thoroughly studied, at negative energies it has not. We therefore investigate the empirical energy dependence of the shell-model relativistic potentials.
i Present address: Department of Physics, University of Surrey, Guildford GU2 5XH, UK. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 7 8 8 - 8
642
Y. Nedjadi, ZR. Rook ~Nuclear Physics A585 (1995) 641-656
We also study the systematics of the potential parameters and discuss an alternative formulation of the single-particle potentials in terms of folded-model interactions. The phenomenological shell-model potentials are found to be consistent with the microscopic ones [2]. This work is presented as follows: In Section 2 the formalism required for this study is briefly discussed. The Dirac equation with Lorentz scalar and time-like vector potentials is reduced to a Schr6dinger equation and the main features of the Schr6dinger-equivalent potential are identified. Section 3 is devoted to the comparison between the experimental binding energies and the calculated ones for 4°Ca, 48Ca, 5SNi, 9°Zr, ll6Sn and 2°spb. A good description of the empirical data on binding energies is achieved. A study of the energy dependence of the relativistic potentials is then carried out. For surface states, the relativistic potentials are found to present the Fermi-surface anomaly. The systematics of the parameters of the potentials are also considered and it is shown that they meet the conditions to be satisfied by a phenomenological model, i.e. they vary smoothly with energy and mass number. In Section 4 the scalar and time-like vector interactions are reformulated in terms of folded-model potentials. It is found that using effective nucleonnucleon interactions that are density- and energy-independent is satisfactory. The sensitivity of the calcium single-particle spectrum to the parameters of the foldedmodel potentials is investigated. Finally, the main results are summarised in Section 5.
2. A relativistic shell model
In the non-relativistic approach the single-particle states of a nucleus are obtained by solving a standard single-particle Schr6dinger equation. Here, by contrast the single-particle states are obtained using the Dirac equation as the relevant wave equation. The essential feature of this model is the Lorentz character of the interaction which is assumed to be the sum of a scalar Us and a time vector potential U°. These single-particle potentials are viewed as the average meson fields generated by the other nucleons in the target. Both Us, and U° are assumed to be static, local and spherically symmetric. The Dirac equation describing the dynamical state ~0(r) of the nucleon is {•-p +/3[m + U~(r)] + where r is the of the target. matrices, E is We use the
U°(r)}qJ(r) = E 0 ( r ) ,
(2.1)
coordinate of the nucleon considered relative to the centre of mass In Eq. (2.1), ~ and /3 are the internal variables known as Dirac the energy of the nucleon of rest mass m, and p is its momentum. representation
o(0 o) (0 °i1
Y. Nedjadi, J.R. Rook~NuclearPhysicsA585 (1995)641-656
643
where tr is the 2 × 2 Pauli spin vector and I is the 2 × 2 unit matrix. We define ,(r)
=
[ ~b(r) ) x(r)
(2.3)
'
where ~b(r) and x ( r ) are each 2-dimensional column matrices. From Eqs. (2.1)(2.3) we obtain o'.p (2.4) x(r) E + m + U s - Uv°qb(r)" It is possible without approximation to obtain a Schr6dinger-type equation for a function related to $. We define $ ( r ) = K ( r)c~(r),
(2.5)
where K is a function of I rl only such that K(r) =~(E+m
+ U~- U ° ) / ( E + m) .
(2.6)
Thus from Eqs. (2.1), (2.4)-(2.6) we obtain V2~b(r) + ~
e 1 + ~m
- Vefr- Vs'°" 4;(r) = 0,
(2.7)
where (2.8)
e=E-m, Vcf~=
1 + Us
Voar~n=2m h2 1
+
4~l-V
1+
0_ +
m
US ) + VDarwin,
2rn
(2.9)
(2.10)
V~
Vs.°. 2m r 1 - I 1 o * "L'
(2.11)
uo-u V°
2m + e
(2.12)
Here L is the orbital angular-momentum operator of the particle, and the prime denotes differentiation with respect to r. From Eq. (2.7) we see that the function 4; satisfies a Schr6dinger-like equation with V~ff as the central potential, Vs.o. as the spin-orbit potential and where the energy term involves a relativistic expression for the energy. The central potential Ven contains an explicit linear energy dependence. For protons we replace U° by U° + Vo where Vc is the Coulomb potential, and this gives a Coulomb correction term Uv° + V c / m in Eq. (2.9). This cross term Uv°Vc/m is of the type considered in the nonrelativistic optical potential for energy-dependent potentials [3]. The spin-orbit term in Eq. (2.7) arises naturally. Note that the scalar and vector potentials contribute with equal
644
Y. Nedjadi, J.R. Rook ~Nuclear Physics A585 (1995) 641-656
sign to central The system
the spin-orbit interaction, whereas they enter with opposite sign into the potential. calculation of x(r) once ~b(r) is known is quite straightforward. Since the is assumed spherically symmetric each single-particle state can be written
[ ~(r) ~17(~"~))
~b(r) = l i x ( r )
(2.13)
~rmj(a ) ,
where ,g,~(f2) is the normalised spin angular function formed by the combination of the Pauli spinor with the spherical harmonics of order l, i.e. j = L + s. j is the total angular momentum, m its projection on the z-axis, l and l' are the orbital angular momenta of the large and small components respectively and they satisfy l = 1'+ 1 as appropriate. The single-particle states are labeled by the quantum numbers n, 1, j associated with the large components to keep the non-relativistic spectroscopic notation.
3. Phenomenological analysis
3.1. Single-particle spectra
ll6Sn
We consider the single-particle states in 4°Ca, 48Ca, 58Ni, 9°Zr, and 2°spb. 4°Ca, 48Ca and 58Ni are sufficiently light nuclei for the energies of their proton single particle energies to have been determined experimentally, while 9°Zr, 116Sn and 2°8pb have many single-particle states. For ll6Sn and 2°spb the energies of the deeper bound states are not measured experimentally but are determined by extrapolation of the potential as described in Ref. [4]. We assume that Us and U° in Eq. (2.1) have the Woods-Saxon form with respective radii and diffuseness parameters Rs, Rv, as, a v and strengths V~, V°. For protons we usually take additionally a Coulomb potential Vc corresponding to a uniformly charged sphere of radius R c = ~/(5/3)(r~2h) where \,/,.2-eh!31/2 is the experimental charge radius of the nucleus of interest. The parameters of the relativistic potentials are obtained by constraining them to reproduce a given doublet in the particular nucleus considered. In this procedure the geometrical factors are first fixed to some optimum values and then the strengths are adjusted. The single-particle Dirac equation is then solved for all the single particle states of the nucleus using these parameters. Tables 1, 2 show the comparison between the calculated proton- and neutronbinding energies and the experimental data for 58Ni and 116Sn; the results for the other nuclei are analogous. The parameters used for the potentials are given in each table. Most noticeable is the good agreement between the theoretical and the experimental results, especially when we note that the relativistic potentials do not depend on energy. All the shell gaps are well reproduced. In Table 3 we present experimental and theoretical charge rms radii. The potentials reproduce the charge radii of all nuclei and there is no problem to reconcile the single-particle
Y. Nedjadi, J.l~ Rook ~Nuclear Physics A585 (1995) 641-656
645
Table 1
(a) 5SNi single-particle spectra. Energies are in MeV Protons
lSl/2 1P3/2 1Pw2 1D5"/2 1D~/2 251/2 1F7/2 2])3/2 2P1/2 1Fs/2
Neutrons
RM
Expt [4]
RM
Expt [4]
50.74 35.77 31.76 21.21 14.20 15.57 7.60 2.70 0.76 -
55 :i: 7 42 + 7 38:1:5 23:1:5 14 5:2 11.16 7.60 2.69 -
57.95 42.87 38.39 28.34 20.60 22.48 14.91 9.52 7.40 5.89
19.06 14.70 9.51 7.33 8.90
(b) Model parameters
Protons Neutrons
R s (fro)
R v (fm)
a s (fm)
a v (fm)
Vs (MeV)
Vv° (MeV)
4.208 4.208
4.208 4.208
0.65 0.65
0.6 0.6
- 481.63 - 507.68
391.78 420.38
Table 2 (a) 116Sn single-particle spectra. Energies are in M e V Protons
151/2 1P3/2 1Px/2 1D5/2 1D3/2 281/2 1F7/2 1F5/2 21)3"/2 2P1"/2 1G9/2 1G7/2 2D5/2 2D3/2 1Hll/2
Neutrons
RM
Expt [5]
RM
Expt [5]
60.86 48.41 45.24 35.50 28.87 28.59 22.78 12.82 14.27 11.90 10.68 2.71 -
58.30 47.60 45.10 35.20 30.00 27.20 21.94 14.26 13.21 11.79 11.05 -
66.08 54.06 51.71 41.66 36.59 36.33 29.49 21.62 23.25 20.91 17.90 8.30 11.68 8.82 6.99
72.1 59.8 57.5 46.2 41.2 39.6 32.1 24.0 23.4 20.9 17.9 10.3 11.2 8.92 -
(b) Model parameters
Protons Neutrons
R s (fm)
R v (fln)
a s (fm)
a v (fin)
V~ (MeV)
V° (MeV)
5.365 5.365
5.365 5.365
0.65 0.65
0.59 0.59
- 511.10 - 511.10
414.00 422.16
646
Y. Nedjadi, J.l~ Rook ~Nuclear PhysicsA585 (1995) 641-656
Table 3 Charge rms radii in fro. The rms radii are calculated using the potentials given in the previous tables, r , refers to the point-neutron distribution.
4°Ca 48Ca 58Ni 9°Zr 116Sn 2°spb
rc¢~p
rcC~Ic
rntalc
3.481 a 3.48 b 3.84 c 4.30 b 4.62 a 5.503 b
3.482 3.48 3.85 4.32 4.58 5.502
3.32 3.59 3.63 4.22 4.59 5.64
a Ref. [7]. b Ref. [8]. Ref. [4].
spectra and charge radii. All levels are correctly ordered apart from the 1175/2(for all nuclei) and the 1G7/2 for 116Sn. These features being determined by the spin-orbit part of the non-relativistic nucleon-nucleus potential, these results show that the model does reproduce the phenomenological spin-orbit splitting of the shell model.
3.2. Energy dependence of the relativisticpotentials As already mentioned in Section 2, ~ff presents a linear energy dependence with a slope U°/m if U° and Us do not depend on energy. Note that the non-linear energy variation of the rather small Darwin term does not bear on this dependence. Although this prescription is seen from the results of the previous section to be satisfactory it is of interest to investigate the extent to which the potentials Us and Uv° may be energy-dependent. In the non-relativistic analysis of the energy dependence of the central potential it is usual to consider the energy variation of the depth of this potential, its geometrical factors being kept fLxed. The energy dependence of the potential well is assumed to be entirely contained in the potential depth. However, in our case, the Schr6dinger-equivalent central potential is not a simple function of Us, Uv° and so that it is not possible to fix its geometry and vary the depth. Therefore we consider the energy dependence of the integrated potential strength per nucleon feff(E) which is defined as 1 fefe= ~
fv~ff d3r.
(3.1)
Within each nucleus, the experimental single-particle energies of all the available doublets up to the Fermi surface are used to determine the associated depths of the potentials Us and Uv°. The overall trends of the energy variation of Jeff are quite similar for all the nuclei considered. Figs. 1, 2 show the energy variation of feff for ll6Sn and 2°spb respectively_ There is a sharp difference between the surface and deep states. Whereas Jeff decreases linearly for deep states, this
Y.. Nedjadi, J.R. Rook/Nuclear PhysicsA585 (1995) 641-656
647
55.0
~SSn 50.0
e
Neutron on
E
LL
=> 45.0
v
40.0
n
35'-"50.0
i
r
-40.0
-30.0 e (MeV)
i
i
-20.0
-10.0
Fig. 1. Dependence upon single-particle energy of the volume integrals per nucleon of I/ele that yields the experimental proton- and neutron-binding energies in ll6Sn.
linearity is lost at the surface. This behaviour around the Fermi energy is known as the Fermi-surface anomaly [9,10]. This implies that the Fermi-surface anomaly does not arise as a relativistic effect and must be included explicitly. This conclusion can be expected from the interpretation of the Fermi-surface anomaly given in Ref. [9]. For the deeper states which are coupled to an "inert" core, the non-relativistic single-particle potential varies linearly with energy. However, near the Fermi surface many particle-hole excitations take place more readily, thus allowing the possibility of the single-particle states to couple to core-polarised
50.0 2oSpb 45.0
8 >
40.0
35.0
30.0
-60.0
",=n .... -50.0
-40.0
-30.0
-20.0
-10.0
0.0
e (MeV) Fig. 2. Dependence upon single-particle energy of the volume integrals per nucleon of I/ett that yields the experimental proton- and neutron-binding energies in 2°apb.
648
Y.. NedjadL J.K Rook/Nuclear PhysicsA585 (1995) 641-656
Table 4 Slopes of the proton and neutron potentials
4SCa n6Sn 2°8pb
Proton
Neutron
0.14 0.09 0.06
0.25 0.25 0.30
states. As shown in Ref. [9], including the possibility of exciting low-lying corepolarised states introduces an explicit energy dependence in agreement with the departure from linearity. The question remains open as to whether a fully relativistic calculation which includes these effects would reproduce the behaviour at the Fermi surface. We now examine just the linear region away from the Fermi-surface anomaly. We find the slope of the line obtained when Jeff is plotted against the single-particle energy E. These slopes are given in Table 4. For neutrons the average slope is 0.27 whereas for protons it is smaller and decreases with increasing Z. In the non-relativistic case the slopes are found to be 0.24 on average, for both protons and neutrons [11]. Such a result is obtained with a central potential assumed to have a fixed Woods-Saxon shape and a depth adjusted to reproduce the singleparticle state in question. For neutrons the energy dependence of the Schr6dinger-equivalent central potential is almost equal to the non-relativistic result. We interpret the large difference for the proton slopes as due to the fact that fell(e) is sensitive to the choice of the geometry [12]. This is supported by Fig. 3 which shows that the effect of adding terms involving the Coulomb potential
55.0 11SSn 500
> 45.0 v
40.0 • With Coulomb 35"760.0
~
II
i
r
i
I
i
-50.0
-40.0
-30.0
-20.0
-10.0
¢ (MeV)
Fig. 3. Effect of the Coulomb potentialon the energy dependence of Jefffor protons in 11asn. Without the Coulomb cross term in Ezi. (2.9) the slope is about 0.23 in the linear region away from the Fermi-surface anomaly. Keeping the Coulomb cross term gives a slope of 0.10.
Y. Nedjadi, J.R. Rook /Nuclear Physics A585 (1995) 641-656
649
300.0
250.0
200.0
150.0
• " Js ~oe?Pb
•Jv( Pb) • " Js "esn • Jv ( lleSn )
100..060.0
-40.0
i
-20.0
0.0
Fig. 4. Dependence upon single-particle energy of the volume integrals per nucleon of Us and Uv° that yield the experimental proton- and neutron-binding energies in 116Sn and 2°spb.
introduces a drastic change in the slope. Note that without Coulomb cross terms the slope is the same as that of the neutrons. An alternative way of looking at the energy dependence of the relativistic potentials is through the volume-integrated potentials of Us and U° defined as
47rA
d3r,
(3.2)
1
fv=4rcAfU°d 3r,
(3.3)
where A is the mass number. Fig. 4 shows the results for 116Sn and 2°spb. For deeper states, .~ and fv are almost energy-independent as most of the energy dependence of ~ is found to arise as a relativistic effect. At the Fermi surface, where is energy-independent, the empirical relativistic potentials present a pronounced energy variation.
fen
Table 5 Proton single-particle levels of 4aCa. Potentials are energy-dependent. Energies are in MeV 1S1/2 1P3/2 1P1/2 1Ds/2 2S1/2 1D3/2 1F7/2 2P3/2 2P1/2
RM
Expt
61.25 43.57 37.36 26.34 18.18 16.67 10.29 3.58 1.22
55.0 + 10.0 35.0:1:10.0 35.0 :t: 10.0 23.5 + 2.0 16.45 16.17 9.62 3.6+2.0 0.0
Y. Nedjadi, J.R. Rook/Nuclear Physics A585 (1995) 641-656
650
We now consider an intrinsic linear energy dependence for Us and U° in order to establish limits for any possible energy dependence. Hence we take
Uv°(e) = U ° ( 0 ) - ae,
(3.4)
US(0) (0)
US(e) = Us(O) - a,-Tb-7-~,x e,
(3.5)
where a = 0.5 and e is the binding energy. 4SCa is chosen as an example and the relevant geometrical factors and strengths of U°(0) and US(0) are given in Table 2. The results are shown on Table 5. There is no major improvement because of the large quoted errors. Note, however, that the deeper states are more affected than those at the Fermi surface. Further calculations showed that the parameter a cannot be greater than 1.5. This figure is proportionally less than would be expected for the energy dependence in a non-relativistic treatment.
3.3. Systematics o f the model The radii of the potentials Us and U° were constrained to vary as 1.1 A 1/3 (frn) where A is the mass number. The calculations have indicated a constraint between the values a s and a v. The relative positions of 1D3/2 and 2S1/2, for both neutrons and protons, can be fitted only if a s is greater than av. This inequality is expected on general grounds since Us and U° correspond, in a first approximation, to scalarand vector-meson exchanges respectively and the mass of the vector meson is greater. Using a fixed value of a s (0.65 fm) for all the nuclei considered, a~ is found to have values that decrease with increasing mass number. This is due to the fact that the matter density is less diffuse with increasing A. Table 6 shows the integrated strengths per nucleon of Us and U° for the different nuclei considered. Js and J~ are independent of A to within 13%. The last two columns refer to the ratios of the r.m.s, radii of the vector and scalar potential. They are almost the same for neutrons as well as protons in all nuclei. Among the requirements that a phenomenological model must satisfy is the condition that its parameters have to vary in a systematic way with energy and mass number. For bound states and the range of nuclei we consider, this shell model satisfies this condition. The volume integrals of Us and U° are the same for
Table 6 V o l u m e - i n t e g r a t e d p o t e n t i a l p e r n u c l e o n f (in M e v . f m 3 u n i t s ) a n d r a t i o s o f t h e r m s r a d i i o f t h e v e c t o r and scalar potentials A
.~ (p)
.~ (n)
fv (P)
fv (n)
(R v/Rs) p a
(Rv/Rs)n a
48 56 90 116
-
-
231.99 208.63 204.33 205.64
239.76 223.86 206.80 209.69
-
-
291.70 263.98 257.74 259.65
291.70 278.20 257.74 259.65
0.795 0.791 0.793 0.792
0.822 0.805 0.802 0.808
a R v is t h e r m s r a d i u s o f t h e v e c t o r p o t e n t i a l . R s is t h e r m s r a d i u s o f t h e s c a l a r p o t e n t i a l . T h e i n d i c e s p and n refer to protons and neutrons respectively.
Y. Nedjad~ J.l~ Rook/Nuclear PhysicsA585 (1995) 641-656
651
protons and neutrons and they are independent of A. If the potentials Us and U° are viewed as potentials obtained from a folding procedure where a zero-range force is used, the above result suggests that the single-particle spectra of these nuclei would be reproduced from the same nucleon-nucleon force.
4. A folding model An alternative to assuming Woods-Saxon forms for the potentials is to take the folding-model approach which relates the potentials to the underlying nucleonnucleon force and the properties of the nucleus. The Lorentz scalar and vector potentials are defined as
Us(r ) = fp~(r')Vs(Ir-r'])
dr',
U°(r) = fp°(r')Vv( I r - r' I) dr'.
(4.1) (4.2)
The nucleon-nucleon interactions vs and vv are effective interactions for point nucleons interacting via exchange of scalar and vector point mesons respectively. These interactions are chosen to be simple local functions of I r - r'l that are
g2 e-rest vs(r) = 4~r - - ,r g2 e-mvr Vv(r) = 47r - - ,r
(4.3) (4.4)
with m s and m v as the masses of the vector and scalar mesons exchanged and gs and gv as the scalar-meson-nucleon and vector-meson-nucleon coupling constants respectively. The density pv° is the time-like component of the current density and represents the nuclear-matter density of the nucleus. The scalar density Ps is empirically undetermined. However, since the relativistic shell model prescribes a relationship between the time-like vector and scalar densities, i.e Ps(r) =
E
21"+1
[~bf(r) - x 2 ( r ) ]
(4.5)
occupied j shells 4~r
p°(r)=
~_,
2j+ 1
4---~[~b2(r)+x~(r)],
(4.6)
occupied j shells
where dpi(r) and xi(r) are the large- and small-component wave functions as defined by Eq. (2.3), this property can be used to calculate a "semi-empirical" scalar density. This is done by constraining the "semi-empirical" density p~¢ to satisfy pS.e
pscalc
p~xp
p~alc'
(4.7)
652
Y. Nedjadi, J..R.Rook ~Nuclear PhysicsA585 (1995)641-656
Table 7 (a) 4°Ca single-particle spectra. Energies are in MeV. rch talc= 3.451 fin, rc~p = 3.480 fin. Protons Neutrons FM a Expt [5] NR [13] b FM a Expt [5] NR [13] b 1S1/2 1P3/2 1PI/2 1D5/2 281/2 1D3/2 1F7/2
43.99 30.20 25.18 15.60 7.93 8.44 1.28
48.5 + 5.0 36.0 + 3.0 31.5+3.5 16.0 + 2.0 12.0 + 1.0 8.5 + 2.0 1.36
42.36 29.08 25.69 16.68 9.99 10.58 4.11
56.32 41.30 36.26 25.58 17.79 18.13 10.06
(b) Model parameters Mass (MeV)
g 2/4~"
Scalar meson Vector meson
8.50 12.90
a
b
540 780
18.1 15.6 8.36
50.0 30.0 27.0 21.9 18.2 15.6 8.4
FM: Folding model. NR: Non-relativistic calculation using Hartree-Fock theory (DDHF).
where pCvalCand p~l¢ are calculated using a model that reproduces the single-particle energies and the r.m.s, radius of the nucleus in question. These distributions are spherically symmetric and hence U~(r) and Uv°(r) are reduced to one-dimensional integrals. Once U~(r) and U°(r) are determined the single-particle Dirac equation (3.1) is solved for the particular nucleus considered.
4.1. Calcium single-particle spectrum We consider the doubly-closed-shell nucleus 4°Ca. The potentials Us and U° are calculated from Eqs. (4.1), (4.2). For protons the Coulomb potential Vc corresponds to a uniformly charged sphere. Table 7 shows the comparison between the calculated binding energies and the experimental data. The calculated charge rms radius and the model p a r a m e t e r s are given in the table. T h e single-particle energies and the s p i n - o r b i t splitting agree reasonably with the data. Note that the charge r.m.s, radius is also well reproduced. T h e ordering of the 2S1/2 relative to the 1D3/2 is inverted. T h e calculations using W o o d s - S a x o n potentials for Us and U° have shown that this inversion arises if the diffuseness of the scalar potential is lower than (or close to) the diffuseness of the vector potential. In this folding model the potentials found have almost the same diffuseness too. The spectra obtained are very similar to the one obtained by Campi and Sprung [13] using the density-dependent H a r t r e e - F o c k theory. T h e problem of the inversion of the 2S1/2 relative to the 1D3/2 is also encountered and they relate it to the energy dependence of the n u c l e o n - n u c l e o n force. In the folding model we use, the n u c l e o n - n u c l e o n force is neither energy-dependent nor density-dependent. It is interesting to note that it is possible to obtain the right ordering of the 2S1/2 relative to the 1D3/2 but unfortunately at the expense of the charge r.m.s, radius.
Y. Nedjadi, J.R. Rook/Nuclear Physics A585 (1995) 641-656
653
Table 8 (a) 4°Ca single-particle spectra. Energies are in MeV. r ~ e = 3.638 fro, r~hw= 3.480 fro. Protons
1S1/2 1P3/z 1P1/2 1Ds/2 2S~/2 1D3/2 1F7/2
Neutrons
FM a
Expt [5]
FM a
Expt [51
41.38 27.82 22.11 14.32 7.84 6.86 1.45
48.5 + 5.0 36.0 + 3.0 31.5 + 3.5 16.0 + 2.0 12.0+ 1.0 8.5 + 2.0 1.36
54.33 39.13 33.28 24.20 17.85 16.20 9.97
18.1 15.6 8.36
(b) Model parameters
Scalar meson Vector meson
Mass (MeV)
g 2 / 4 ~r
450 780
6.48 14.21
a FM." Folding model.
This is shown on Table 8. Combining this new model together with including some energy dependence would reconcile the ordering of the 2Sx/2 relative to the 1 D 3 / 2 and the charge r.m.s, radius. As seen from Section 3.2, including an intrinsic energy dependence does not affect the states near the Fermi surface but increases the binding energy of the deeply bound states, hence reducing the r.m.s, radius. However, it is not possible to estimate the latter, as the orthogonality between the single-particle state wave functions is then lost.
4. 2. Sensitivity of the model parameters In order to study the sensitivity to the parameters, the masses and the coupling constants of the exchanged mesons are handled as free parameters. This makes it possible to study the sensitivity to different regions of the nucleon-nucleon force independently from one another. The scalar-meson mass was varied from 400 to 700 MeV with the ratio gs/ms 2 2 fixed at the value corresponding to the set of parameters given in Table 7. The dependence of the single-particle energies and the r.m.s, radii upon the scalar-meson mass (in pion-mass unit) is shown in Fig. 5 for protons and Fig. 6 for neutrons. We note that the single-particle energies and r.m.s, radii are very sensitive to the scalar-meson mass (ms). As m s increases, the attractive part of the nucleon-nucleon potential becomes stronger at shorter internucleon distances. As a result the binding energy of nucleons increases and the r.m.s, radius becomes smaller. The scalar density is fixed and hence Eq. (4.1) shows that increasing m s yields a scalar potential that is deeper and less diffuse. Figs. 5, 6 also illustrate the inversion of the ordering of the 2S1/2 relative to the 1D3/2. For heavier scalar mesons the diffuseness of the scalar potential decreases and yields a value close to the vector-potential diffuseness. This results in the inversion of the 2S1/2 as expected from the systematics discussed in Section 3.3.
Y. Nedjad~ J.K Rook/Nuclear PhysicsA585 (1995) 641-656
654
0.0
,
i
,
I
-10.0
-20.0
~o= -30.0
-40.0
,
-50.0
~
3.60
^
3.40
d
V
3.20
I
3.0
I
3.5
4.0 mjm~
4.5
5.0
Fig. 5. Effect of the scalar-meson mass m s upon the proton single-particle spectrum and the rms radius of the proton distribution in 4°Ca.
It is interesting to remark (see Figs. 5, 6) that although the single-particle energies are very sensitive to the scalar-meson mass, almost no change has occurred in the spin-orbit splitting. Further calculations have shown that this feature recurs even if the ratio gs2~ms2 is not constrained to a fixed value. For both protons and neutrons, increasing the scalar mass from 3m~ to 5m~ shifts the P-states by about 13 MeV and the D-states by 8 MeV. The spin-orbit splittings are increased by 0.63 and 2 MeV for the P-shell and the D-shell respectively. This
0,0
,
'i
••
,i
..
F
•10•O
,
1F7/2
-20.0 1D3/2
-40.0
J
1P3/2
-50.0 -60.0
1S1/2 -70.0
*
I
3.40
A
3.20 3.00
30
i
410 mJl'n s
Fig. 6. Effect of the scalar meson mass m s upon the neutron single-particle spectrum and the rms radius of the neutron distribution in 4°Ca.
Y. Nedjad~ J.R. Rook/Nuclear PhysicsA585 (1995) 641-656
655
Table 9 (a) 4°Ca single-particle spectra. Energies are in MeV. rch talc 3.527 fro, rce~v = 3.480 fm. Protons Neutrons FM a Expt [5] FM a Expt [5] =
1S1/2 1P3/2 1P1/2 1D5/2 2Si/2 1D3/2 1FT/2
43.30 29.60 24.60 15.11 7.51 8.00 0.91
48.5 + 5.0 36.0 _+3.0 31.5 5:3.5 16.0 _+2.0 12.0+ 1.0 8.5 + 2.0 1.36
55.62 40.69 35.67 25.07 17.33 17.66 9.66
18.1 15.6 8.36
(b) Model parameters Scalar meson Vector meson a FM:
Mass (MeV)
g 2/4 w
450 1000
8.50 21.20
Folding model.
consideration together with the Fermi-surface anomaly has an interesting consequence from a practical point of view. W h e n the shell model with energy-independent potentials is used, the choice of the doublet to which the potentials are constrained is ambiguous. F r o m Figs. 5, 6 and the results of Section 3.2 we see that the doublet must not be too deep so that V~n is undetermined and not loosely bound because of the Fermi-surface anomaly. C o m p a r a b l e changes to the vector-meson mass (from 780 to 1000 MeV) with a fixed ratio of g v2/ m v2 and the p a r a m e t e r s of the scalar-meson mass given in Table 7 yield only a very small change (less than 5%) for both the single-particle energies and the r.m.s, radius (see Table 9). This arises because the range of the repulsive force is much smaller than the m e a n internucleon distance, or in other words the nucleon's motion is insensitive to these changes in the diffuseness of the vector potential obtained from Eq. (4.2). This result also explains the systematics found for the vector diffuseness (see Section 3.3), that is to say the diffuseness of the vector potential decreases with increasing mass number. The vector potential U° is mainly determined by the nuclear density p0 and hence this behaviour occurs as a result of the reduction of the diffuseness of nuclear matter as the mass number increases. Finally, this study of the sensitivity to the model p a r a m e t e r s has shown that Uv° is better determined than Us.
5. C o n c l u s i o n
A phenomenological relativistic shell model was applied to many nuclei and was shown to be successful in reproducing the single-particle motion, spin-orbit splitting and r.m.s, radii. To gain some empirical insight into this shell model, a study of the energy d e p e n d e n c e of the relativistic potentials was made. Whereas
656
Y. Nedjad~ J.l~ Rook ~Nuclear PhysicsA585 (1995) 641-656
for deep states the potentials were found to be roughly energy-independent, for loosely bound states at the surface, their behaviour is of the Fermi-surface-anomaly type. The analysis of the parameters of the model showed that they vary in a systematic way with both mass number and energy. Reformulating this SV model in terms of folded-model potentials showed another feature of the Schr6dingerequivalent central potential, implicit in the use of a relativistic approach, that is the density dependence. This density dependence is sensitive to the relationship between the vector and scalar densities. The scalar density being empirically undetermined, a "semi-empirical" scalar density was used instead. Using an energy- and density-independent nucleon-nucleon force, the single-particle spectra and charge r.m.s, radius of 4°Ca were reasonably reproduced. The study of the sensitivity to the model parameters showed that Uv° is better defined than Us.
References [1] E.D. Cooper, S. Hama, B.C. Clark and R.L. Mercer, Phys. Rev. C 47 (1993) 297, and references therein. [2] B.B. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [3] D.H. Wilkinson, Isospin in nuclear physics, (North-Holland, Amsterdam, 1969) p. 389. [4] F. Malaguti, A. Uguzzoni, E. Verondini and P.E. Hodgson, Nuovo Cimento 5 (1982) 38. [5] L. Ray and P.E. Hodgson, Phys. Rev. C 20 (1979) 2403. [6] J.W. Negele, Phys. Rev. C 1 (1970) 1260. [7] B.A. Brow, C. Bronk and P.E. Hodgson, J. of Phys. G 10 (1984) 1683. [8] At. Data Nucl. Data Tables 36 (1987) 495. [9] B. Day, Proc. Int. School of Physics (North-Holland, Amsterdam, 1982) p. 1. [10] G.F. Bertsch and T.T. Kuo, Nucl. Phys. A 112 (1968) 204; Proc. 2nd La Rabida Summer School on Nuclear physics (1985). [11] M. Bauer, E. Heruandez-Saldana, P.E. Hodgson and J. Quintanilla, J. of Phys. G 8 (1982) 525. [12] C. Mahaux and R. Sartor, Theory of nuclear structure and reactions. [13] X. Campi and D.W. Sprung, Nucl. Phys. A 194 (1972) 401.