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Influence of non-central interactions on 1s-0d nuclei
~ I
NuclearPhysics A306 (1978) 2 0 1 - 2 2 0 : ( ~ North.HollandPublishino Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
I N F L U E N C E O F N O N - C E N T R A L I N T E R A C T I O N S ON lsd)d N U C L E I N. A N Y A S - W E I S S
Ontario Hydro Research Dieision, 800 Kipling Avenue, Toronto, Ontario, Canada and D. S T R O T T M A N *" **
Department o[' Physics, State University of New York, Stony Brook, N Y 11794 Received 1 March 1978 (Revised 3 May 1978) Abstraet: The influence of non-central interactions is investigated in mass 18 to 22 nuclei. The effect of non-central components in the realistic matrix elements of K u o is evaluated in mass 18, 19 and 20 nuclei. A phenomenological interaction is constructed to reproduce the results obtained with the K u o Hamiltonian and compared with an interaction fitted to experimental levels. The fit to experimental levels requires a repulsive triplet even tensor force.
1. Introduction The shell model with residual interaction has been very successful in describing a wide range of properties of 0p and ls-0d shell nuclei. However, an embarassing feature of this model is the fact that many diverse residual interactions apparently enjoy equally good success in obtaining agreement with experiment. The two-body interaction was historically taken to be a phenomenological interaction with only central components having either Yukawa or Gaussian form factors and with the strength in each spin-isospin channel taken as a free parameter 1-3). Recently, "realistic" matrix elements 4.5) by which we mean matrix elements obtained from potentials which fit nucleon-nucleon scattering data up to about 350 MeV [ref. 6)] have found favour amongst theorists. These two approaches to the residual interaction differ in that, besides the latter having in principle no free parameters (although not in practice), the "realistic" matrix elements contain non-central forces as well as the usual central forces. The implications are that the total orbital angular momentum L will not be diagonal for the two-body matrix elements (2BME). The 2BME of Kuo 5) also do not have S, the total spin, diagonal, but those matrix elements for which S is not diagonal are an order of magnitude smaller ~) than those for which S is diagonal and may be ignored in the present discussion. Their effects are calculated later in this paper. Other approaches to the residual interaction have been to take a number of the 2BME themselves as free parameters which are then chosen so as to reproduce the * Supported in part by U S D O E contract EY-76-5-02-3001. ** Present address: Theoretical Division, MS452, LASL, Los Alamos, N M 87545. 201
202
N. ANYAS-WEISS AND D. STROTTMAN
observed energy spectra 8) or, in an attempt to avoid the use of potentials entirely, to relate the 2BME directly to observed nucleon-nucleon phase shifts 9). In reality a rather less definite division exists than indicated above. There have been a few attempts to include non-central forces in a phenomenological potential, and to use it in the two- and three-body system outside 160 [-refs. 10,11)] but such a program has not previously been carried out for more complex nuclei. Although it might appear desirable to use exclusively "realistic" matrix elements in calculations, and discard the phenomenological potentials as no longer useful, it may be contended that for a number of reasons such a decision would be premature. First, it is suspected that the G-matrix expansion in terms of the number of interactions does not converge 13), and that one reason for this failure is the strong tensor force. The 2BME now available may not adequately represent the effects of the non-central forces. Second, by systematically studying the effects of changing parameters, one may obtain information on what aspects of nuclear structure are sensitive to noncentral forces. The properties of many s-d shell nuclei have been calculated within a complete sd basis using the realistic 2BME 12, 14, 15) of Kuo 5) and a number of discrepancies exist between the theoretical predictions and experiment, even in nuclei with six or fewer nucleons outside a closed 160 core. Examples are the positions of the K = ½+ excited bands in 19F and 21Ne which come too low in the calculations, and the ground state and first few excited states in 2°F, 21F and 22Na are usually predicted in the wrong sequence when compared with experiment. The spectrum of 22Na is of particular embarrassment; using only central forces, two 1 + states are usually calculated as having lower energies than the 3 + ground state. The realistic interactions do somewhat better; often only one 1 + state lies beneath the experimental ground state. In all cases mentioned the total spin S is non-zero, allowing two-body spin-orbit and tensor forces to directly contribute to the energy. An indication that the non-central forces are partly responsible is the fact that disagreement is more pronounced when only central forces are used 3). The solutions to other discrepancies such as the position of the K = 2 + bands in even-even nuclei presumably lie elsewhere, inasmuch as the low-lying states in these nuclei have S = 0. Before sweeping all these difficulties under the rug of "core-excitation", the effects of small modifications in the two-body interaction should first be investigated. Third, all calculations of realistic matrix elements have assumed that the intermediate states are always nucleons, whereas it is possible to have excited states of the nucleon, particularly the A in intermediate states: such contributions might give rise to a very different spin dependence in the interaction. Fourth, three-body forces are assuredly present, the effects of which have not been estimated. The success of calculations assuming only a two-body interaction indicates that their effects on energy spectra may be small for low-lying states, although the configuration mixing due to such forces is clearly different than that already included in the realistic matrix elements. The latter two effects may be partially compensated over a small range of mass number by a suitably chosen phenomenological interaction.
NON-CENTRAL INTERACTIONS
203
Finally, the phenomenological interactions are easier to work with since the total spin S, and possibly also L, is diagonal. Further, and perhaps most important, phenomenological interactions may provide physical insight into the results of the shell-model calculations, obviously a non-trivial advantage when the possible matrix dimensions may run into the thousands. With only a very few parameters - each of which has an obvious physical interpretation - one may more easily determine to which aspect of the interaction nuclear structure is sensitive, and the possible size of unitary symmetry breaking terms in the Hamiltonian. It is the aim of the present paper to investigate the importance of non-central interactions on the energy spectra of nuclei near the beginning of the ls-0d shell. This is done by first investigating the importance of such terms in the Kuo interaction 5) and then constructing a phenomenological Hamiltonian to allow a systematic variation of the strengths of these components. In sect. 2 the interaction will be defined. In sect. 3 the interaction of Kuo 5) will be investigated and in sect. 4 the strengths of a phenomenological Hamiltonian will be fitted to experimental levels. 2. The model In this paper the effects of non-central forces on the spectra of nuclei with less than eight particles outside an 160 core is investigated. The phenomenological Hamiltonian is assumed to be of the form
H = ~l" s+~e,(6-12)+ ~ V(ri~), i
(1)
where V(ri~) is the sum of the central, tensor, spin-orbit and quadratic spin-orbit terms. The individual terms are defined as follows. The central potential is taken as usual to be g = {13 g p 1 3 +31 gp31 +11 Vp11 + 33 VcP33 }f(r/pc)"
(2)
In eq. (2) Pzr+ 1.2s+ 1 is a projection operator onto a state with spin S and isospin T, with 2r+ 1.2s+1V being the strength. The other terms are defined similarly: VT = [13 VTP13 .+_33 VTP33}f(r/itv)S12 '
(31
where $12 = 3(# l • ~'12)(0"2 " ~'12)--(0"1 " 0"2),
VLS = {13 VLsP13 q_ 33 VP33 }f(r/llLs)L. S,
(4)
VQL = {13VQLPI3-l-31VQLP31+11%LPll + 33VQLPz3}f(r/[AQL)L12.
(5)
In eq. (5) the definition of the quadratic spin-orbit operator is the same as that used by Hamada and Johnston 6), L I 2 = {fiLJ+ (~1 "0'2)} L2 -- (L" S) 2.
(6)
204
N. ANYAS-WEISS AND D. STROTTMAN
The form factors are all assumed to be Yukawa with the range parameter defined by 2 = x/21t/b.
In their calculations on s-d shell nuclei Arima and his collaborators 23) have found 2 c = ~ to be satisfactory and we shall adopt the same value as well. In the one-pion-exchange potential the tensor force is also of long range and we therefore choose 2c = 2 T. The values for 2rs and 2QL are more uncertain; their precise value is not of great importance as changes in the interaction radius may be largely compensated by appropriate changes in the strength of the force. In the HamadaJohnston potential the spin-orbit and quadratic spin-orbit forces are of short range; we shall here use '~QL = )'~s = 2 2 -- ~. There has been little investigation into the need for introducing a quadratic spin-orbit force into the phenomenological interaction. To some extent its effects may be mocked up by an appropriate redefinition of the remaining three components. As pointed out by Hamada and Johnston 6), this is particularly simple in those partial waves which are coupled by non-central forces, and is achieved by replacing VLs + VQt" by Vl.s. The I/QLterm is more important in the singlet even channel as the nuclear spectra are almost totally insensitive to the singlet odd potential. Further, as the quadratic spin-orbit force enters more strongly in high relative l states, positions of high spin states may be particularly sensitive to the quadratic spin-orbit interaction. By virtue of the interaction defined above being translationally invariant and parity conserving, all two-body matrix elements will be diagonal in the spin S. As an example of realistic matrix elements, the 2BME of Kuo 5) were used. These matrix elements contain in addition to central, vector and tensor forces, interactions which do not conserve S. Such matrix elements may be non-zero for a variety of reasons, among which are the presence of non-local potentials or spin-orbit partners having different radial wave functions. Further, such effects may enter via the renormalization corrections to G2p or G2h through the use of partially spurious intermediate states or allowing one of spin-orbit pair to contribute and not the other. The matrix elements arising from G3plh need not conserve S. The matrix elements which do not preserve S are important because only they connect the spatially symmetric two-particle state with the spatially antisymmetric state and therefore directly contribute to the destruction of SU(4) symmetry. Other components of the two-body interaction may connect different SU(4) representations in a many-particle configuration, although in many cases there exist selection rules which tend to diminish the admixing. For example, the Bartlett interaction may in the four-particle system connect the SU(4) representation [4] with [-22]; the centroid of [22] is 15 MeV higher than that of [4] while the centroid of [31] is only 10 MeV higher than that of [4], but the Bartlett interaction will not connect [4] with [31]. The central, vector and tensor components of any two-body interaction may he
N O N - C E N T R A L INTERACTIONS
205
found by transforming the interaction from jj coupling to LS coupling and then summing over the total angular momentum of the two-body matrix elements with a weighting factor provided by angular momentum algebra; if V(r) is written as 2
(7)
V(r) = ~ (Y~¢{k}(r)" f/){k)(tr l, O'2)), k-O
then
x (LST~flJ[VIIZS'T~'ffJ).
(8)
In eq. (8) the additional labels a and /3 needed to uniquely specify the two-body matrix elements are provided by the irreducible representations of SU(3) and SU(4). Only for k equal to one need S not equal S'. A quantitative measure of the size of such matrix elements may be obtained from the quantity ~rr(S ~ S') defined as the ratio of the root mean S non-conserving matrix element to the root mean S conserving matrix element:
{!
DT =
~
}1
~ ((21~)LSTJIGI()o'ff)I2S'= S T J ) 2 LL'JJ ()4~)(2'u')
O.r =
{9)
((2/OLSTJIGI(2' ff)I2S' ¢ S T J ) 2
,j
,
(;4n(~'u') OT(S :F S') = OT/D T.
(10)
In eqs. (9) and (10) n o and n o are the number of terms contributing to each sum. In table 1 the value of a T for the various terms contributing to G in the case of Kuo interaction are shown. For the isospin-zero case the contributions are negligibly small save for the G3plh contribution where the matrix elements are nearly as large as those which do conserve S. In the isospin-one case there are also appreciable TABLE 1 The quantity oT(S ~ S') as defined by eq. (10), which is a measure of the SU(4) symmetry breaking part of the Kuo realistic matrix elements
Gb~c
G3pih G2p
Gzh G,o~
ao(S =~ S')
%{S ~ S')
0.003 0.623 0.034 0.004 0.051
0.008 0.886 0.186 0.262 0.039
206
N. A N Y A S - W E I S S A N D D. S T R O T T M A N
values of al from both G2p and G2h. Such contributions will arise from partially spurious intermediate states. The expectation value of the Hamiltonian will be written as
E= ( C > + ( V ) + ( T ) + ( S S ' ) + ( S P ) .
(11)
where the meaning of the first three terms is evident; (SS') signifies the contribution of those matrix elements which do not preserve S, and (SP> is the contril~ution of the one-body operators appearing in eq. (1). The values of ¢ and ~: are determined from the spectrum of ~O. Although the matrix elements which do not preserve S are only 5 ,'~, of the magnitude of those which do, it will presently be seen that they contribute as much to the energy of a several-particle system as do the tensor or the two-body spin-orbit interactions.
3. Fitting the Ku0 interaction The realistic matrix elements of Kuo provide a reasonable description of the spectra of nuclei near the beginning of the l s-0d shell. The importance of the non-central components will here be estimated in three ways. In the first the Hamiltonian of eq. (1) is alternately diagonalized with various two-body terms of eq. (1) omitted; the results for 19F are shown in fig. 1. The spectrum is in all cases shifted slightly so that the ½+ level lies at zero energy. With only the central part of the two-body interaction contributing, the gross spectrum is apparent with both the correct number of states and their approximate positions. The energy of the ground state is within 200 keV of the answer obtained with the complete Hamiltonian. With the addition of the rank-two interaction, the position of all levels is essentially correct. The contributions of ( V ) and (SS') have little effect on the final spectrum. A more quantitativ e method is to calculate the expectation value of each of the terms appearing in eq. (11); the wave functions used were obtained by the diagonalization of the complete Hamilt.onian. The results are given in table 2 for an example in which the dominant components of the low-lying states have S = 0 (2°Ne), or S ½ (t 9F) and a case in which there is an appreciable admixture ( ~ 20 °4) of S - 3 into an otherwise S = ½ wave function (21F). The expectation values of the one-body terms, I. S and ~ / 2 - 6 ) , are also given; to obtain their contribution to the energy one must multiply by - 2 . 0 2 and - 1 . 1 6 MeV, respectively. One may note that the occupation probability of the 0d~, 0d~ and ls~ orbits may be now calculated using the information in table 2: n(~) =
t f H - 3 (g( 1 12 - 6 ) ) + 2 ( / . S ) } , .~,3
n(~) = z{5n - (~(/2 _ 6)> - (1' S>}, n(~) = (g(1 l: - 6 ) > .
NON-CENTRAL INTERACTIONS Kut~ +
19 F
~ ~
207
170
3
3
~
? S
"as7
3
3
5
5
7
7
7
13
13
13
7
5 7
9
5
1
7
:3 5
g"
----~5
7
~
1
1
6
7
7
~ 1 3
~5.g, 13
7
7
S
S
5
4
3
2 ~
3
3 ~
3
~
3 1
~ I
5
~ 5
I
C*V*T*SS
,~
C
5
~
5 ~
I
I
C÷V*T
I
I
C*T
5
I
S I
C÷V
I
C*V÷SS
Fig. I. T h e s p e c t r u m o f ~gF w i t h the full K u o i n t e r a c t i o n a n d w i t h v a r i o u s t e r m s o f the i n t e r a c t i o n o m i t t e d ; C = r a n k 0, V = r a n k 1, T = r a n k 2 a n d SS = r a n k I (S n o t d i a g o n a l ) .
F r o m these quantities the F(M1) strength may be estimated using the Kurath 17) sum rule. For XgF and 2°Ne the average value of I(V)I is approximately 75 keV; the contribution in the 21F case is somewhat larger, although even here it is less than 1 ~o of the contribution in magnitude of ( C ) . Since the low-lying states of 2°Ne are nearly pure S = 0, one would expect that the contribution of {V) to be less than for 19F for which rank-one potentials may contribute even in the diagonal matrix elements. Surprisingly, this is not observed.
208
N. ANYAS-WEISS A N D D. S T R O T T M A N Kuo +170
ZONe
aV
/¢
~
l//
••
i
2
I
I
C+V+T÷SS
m
I
I
C*V+T
l
I
C+T
2
I
I
C+Y"
I
C+V+SS
Fig. 2. As fig. 1, save for 2°Ne.
The contribution from the tensor force is larger than that from ( V ) , averaging from 200 to 350 keV. The largest contribution in /°Ne comes from (SS') and is always attractive. Its omission would not seriously affect the appearance of the spectrum since its contribution is nearly the same for all levels, although it would result in an underestimate of the binding energy by 250 keV; see fig. 2. In conclusion, it appears that non-central components of the Kuo interaction merely shift levels by a few hundred keV and the relative energy shift is much less. Hence, to produce a correct energy spectrum it is essential to get the central interaction correct.
NON-CENTRAL
209
INTERACTIONS
TABLE 2
Expectation value of the various rank interactions for eigenstates of the Kuo interaction Nucleus
Spin
Ex (calc.)
19F
½ 1
13 ~3 z
0 6,75 1,25 6.45 0.36 4.97 5.12 5.68 6.49 10.09 5.26 9.97
0 2 4 6 8 0 2 2 4
0 1.45 3.86 8.19 12.36 6.81 8.43 10.37 10.18
1
11.78
3
10.69
11 2
2°Ne
z~ F
61(12 - 6)
(C)
(v)
(T)
(SS')
1.08 0.53 0.82 0.79 1.53 1.67 0.51 2.51 2.19 2.57 2.37 1.13
0.98 2.19 0.53 0.95 0.52 0.90 0.60 0.17 0.55 0.29 0.00 0.00
- 14.54 -7.58 - 13.88 -8.33 - 14.05 -8.67 - 10.27 -7.34 - 5.92 -2.20 -8.13 - 6.33
-0.02 0.02 0.03 0.03 0.04 -0.07 -0.04 -0.15 0.00 -0.06 0.12 0.06
-0.07 -0.18 - 0.53 -0.55 0.14 0.14 -0.80 0.58 - 0.60 -0.04 0.21 0.18
-0.17 0.00 - 0.21 -0.11 -0.16 -0.11 -0.15 -0.21 - 0.02 -0.17 -0.24 0.25
1.30 1.35 1.47 1.53 1.90 1.42 2.22 1.50 1.67 1.13 2.18
0.89 0.99 0.66 0.58 0.00 0.12 0.41 1.66 0.67 0.94 1.04
-29.1 l -27.51 - 25.23 - 20.90 - 16.90 -22.01 - 19.89 - 17.57 - 18.59 - 17.02 - 16.60
0.03 0.01 0.06 0.03 0.10 0.08 0.09 -0.15 0.02 0.37 -0.12
-0.14 -0.02 -0.07 0.04 0.28 0.06 0.23 0.10 0.01 - 1.12 0.08
-0.26 -0.29 -0.31 - 0.35 -0.41 -0.19 0.16 -0.18 - 0.23 -0.20 -0.18
1.22 1.26 1.34 1.25 1.50 2.19
-
-0.12 -0.17 -0.24 - 0.15 -0.20 -0.04
0.08 0.16 0.19 -0.33 -0.45 -0.03
-0.18 -0.18 -0.19 -0.15 -0.14 -0.12
(/
S)
25
0
1.78
3 1
0.62 0.20 1.04 0.34
1.76 2.11 1.61 1.59
1z
0.75
1.10
17.35 16.76 16.32 16.21 16.50 16.89
The calculations for ~*F and 2°Ne are for a complete s-d basis; those for 2iF are in an SU(3) truncated basis. The expectation values of the two one-body operators must be multiplied by - 2 . 0 2 a n d - 1.16 to get the contribution to the energy.
The third technique to investigate the nature of the non-central components in a realistic interaction is to fit such 2BME with matrix elements obtained using the phenomenological interaction of eq. (1). However, since the energy eigenvalues of a many-particle system depend on the 2BME in a non-linear fashion, it is more reliable to fit the phenomenological interaction to eigenvalues obtained by diagonalizing the Kuo interaction in a several-particle system. Effects ofnon-linearity which would appear as an A-dependent interaction were investigated by fitting the parameters of the phenomenological interaction first to eigenvalues obtained by diagonalizing the Kuo interaction for all possible values of J and T in mass 18 and
210
N. A N Y A S - W E I S S
AND
D. S T R O T T M A N
19, and thirdly, only even values of J for the T = 0 states of mass 20. All eigenvalues of mass 18 and the six lowest eigenvalues for each J and T in mass 19 were included in the fit. The 2°Ne fit included the first 18 eigenvalues of angular momentum 0, 2 and 4 and all the angular momentum 6 and 8 eigenvalues. The resulting strengths for these fits to results of calculations using the Kuo interaction are given in table 3. TABLE 3 T h e strengths o f the p h e n o m e n o l o g i c a l interactions which provide the best fit to positions o f energy
eigenvalues as calculated using the K u o interaction Interaction
18
18,19
20
13 V~ 31 V~ ~ 1I.
37.85 - 29.16 84.38 -4.51 13.33 -- 71.65 -- 28.63 6.07 2.70 2.14 9.76 0.49
- 37.95 (0.01) - 29.42 (0.00) - 49.29 (82.83) 1.46 (0.22) - - 3 . 2 4 (11.10) --40.50 (2.05) -- 35.07 (0.62) 7.04 (0.06) 4.71 (0.03) 1.40 (0.00) -- 10.92 (2.08) 1.02 (0.04)
30.04 - 37.84 - 337.88 12.01 --21.48 -- 116.84 -- 70.96 19.07 15.66 -- 0.06 -- 58.25 3.26
33~ c
13VLS 331% s
l 3 Vl 33 VT
13I/QL 31 l/eL
11VQL 33
VQL
N
cr2/N
16 0.19
82 0,21
62 0.26
The n u m b e r s in parenthesis are the statistical uncertainties. The quantity N is the n u m b e r o f degrees 12) a n d a2/N is the error per degree o f freedom.
o f f r e e d o m ( N = n u m b e r o f levels
For the combined mass 18 and 19 fit, an estimate of the uncertainty in the parameters is also given. The uncertainty is here defined as the diagonal element of the covariance matrix (see appendix) multiplied by cr2/N where
a2
= ~ (E~i' - - Ecale}2i " i
and N is the number of degrees of freedom. In contrast to fits which include only the first one or two levels of each angular momentum, the fits described here should adequately represent the odd state interaction. When comparing the mass 18 and the mass 18 and 19 fit, there is little change in the values of the parameters, save for those which have a large uncertainty. Amongst the latter are the singlet odd interactions, a 1 V (the value of which changes by 134 MeV) and 1 t VQL,and the two-body spin-orbit interaction. Otherwise the interaction has the expected properties: the singlet and triplet even central and quadratic spin-orbit interactions are well determined. The even tensor interaction is attractive.
N O N - C E N T R A L INTERACTIONS
211
However, the triplet odd central interaction in the fit is attractive, whereas in the investigations of Arima and collaborators 33V is repulsive, but since 33VQL is repulsive, the net effect on the triplet odd channel may still be one of repulsion. When restricting the fit to the levels of Z°Ne, there are changes in the strengths of the fitted interaction, even for those parameters which were seemingly well determined in the mass 18 and 19 fit. Of particular note are the values of the even central interaction; for mass 20 the singlet even is stronger than the triplet even, whereas in the mass 18 and 19 fit the opposite was true. However, the average central interaction strength (to which ~-particle nuclei are most sensitive), Vym = !~3t 2, V+13V~ c,,
(12)
is unchanged. A comparison between the two fits is instructive and sobering in one other respect. Even when the interaction is greatly overdetermined (92 levels were fit in the mass 18 and 19 fit), changing one's choice of fitted levels changes the nature of the interaction. One must therefore be extremely cautious in the interpretation of the results of calculations in which the number of degrees of freedom is much less. 4. The fit to A = 21 and 22 nuclei
From the results of the previous section one has an idea of the nature of a phenomenological interaction which simulates the realistic matrix elements of Kuo as well as the effects of non-central interactions on the spectra. As already mentioned in the introduction, the positions of several levels of nuclei near the beginning of the s-d shell are not reproduced by the realistic matrix elements; in this section we investigate what changes in the interaction need be made to reproduce the experimental level ordering. As many of the most notable disagreements occur for the mass 21 and 22 nuclei, it was decided to attempt a fit to the levels of 21Ne, 2ZNe and 22Na; the structure of the latter nucleus should be particularly sensitive to the tensor interaction. Difficulties inherent in the selection of a proper set of levels to include in a fit have been extensively discussed by MacFarlane 16); the two of most relevance here are firstly, that the spins of many levels are unknown (or perhaps incorrectly assigned) and secondly, the possibility that a level is an intruder state and should be described by a configuration lying outside the model space. The former difficulty is obviated by a circumspect selection of levels and restricts in most cases the number of levels fitted per angular momentum to one. Systematics based on the weak coupling model suggest there are no positive-parity intruder states in the energy region of interest in mass 21 and 22; the 7.20 MeV 0 + state in 2°Ne must, however, be excluded. A set of 19 levels from 21Ne, 22Ne and 22Na was selected to be included in the fit, and these levels are listed in table4. Additionally, some fits also included the members of the ground-state band and the excited 0 + level at 6.70 MeV of 2°Ne.
212
N. ANYAS-WEISS A N D D. S T R O T T M A N TABI,~i 4
Experimental energies of levels of mass 21 and 22 included in the fit Nucleus 21Ne
22Ne
22N[t
Spin ~ ",~
Ex(exp) 3.735
11 ~z
2.796 0 0.350 1.747 2.867 4.433 6.450
0 "~ 3 4 6
0.00 1.28 5.63 3.35 6.30
6.24 4.46
1 2 3 4 5 6 7
0.583 3.059 0.000 0.891 1.528 3.708 4.522
1.937
3.944
1.984
2.969
5.36
5.52
The bases for masses 21 and 22 were truncated to a maximum of 60 states by including only certain SU(3) and SU(4) irreducible representations. In order to facilitate comparison with earlier work, the truncation chosen was identical to that of Akiyama et al. 3). A comparison between calculations with the Kuo interaction in this basis and calculations in a complete s-d basis by the Darmstadt group 15) shows that for 2~Ne and 22Ne the omitted components account for less than 1 ')J,i of the complete wave function. The low-lying states of 22Na are dominantly S = 1 and the truncation is somewhat worse, although in most cases less than 2 oj/,, of the wave function is omitted. The many-particle matrix elements were calculated and stored on tape; the strengths of the interaction were then fitted using linear regression. Because of the success of the SU(3) model, the calculation usually converged after only three iterations unless a particularly "bad" set of initial choice of parameters was made in order to investigate the possibility of secondary minima. The low-lying levels of ZZNa are dominated by the SU(4) representation [42] and should therefore be particularly sensitive to the triplet-even tensor interaction. As a preliminary attempt to understand the poor agreement in 22Na between calculation and experiment, the triplet-even tensor interaction was varied whilst aH other parameters were fixed at the values given in table 3 as determined by the fit to the Kuo interaction for mass 18 and 19. The best fit was obtained by using a repulsive
213
NON-CENTRAL INTERACTIONS 21 He
5
3 ....
7
-
5,11
-
3
9
- - - 3 - - 1 1
3 -
-
5
3
3
3
- - . 5
~ ~
3 5
-
-
5
- - ' 5 - - 9 3
_
7
1
1
7
- - 5 7
~
7
- - 1
~
5
5
- - 1
- - 5
-----E---3
5
c------7--3c,--ET~-,r3 ~ 3 °OL
3
~ 3
5
E-E~--.3
*OI.°SP
Fig. 3. The spectrum of 2iNe calculated with the phenomenological interaction indicated which provides the best fit to levels of mass 21 and 22. The experimental levels shown are only those included in the fit.
t e n s o r i n t e r a c t i o n r a t h e r t h a n an a t t r a c t i v e i n t e r a c t i o n as r e q u i r e d by the K u o interaction. As this i n d i c a t e d t h a t a small a l t e r a t i o n in the K u o i n t e r a c t i o n was i n a d e q u a t e to fit the e x p e r i m e n t a l spectra, a general v a r i a t i o n of the H a m i l t o n i a n was c o n d u c t e d . T h e initial fits were restricted to the 19 levels of t a b l e 4. T h e singlet o d d i n t e r a c t i o n t e r m s of H were set to zero a n d n o t varied since it was a n t i c i p a t e d t h a t the levels w o u l d be insensitive to them. This was certainly true in the m a s s 18, 19 a n d 20 fit as well as the w o r k of A k i y a m a et al. 3). Several fits were p e r f o r m e d by successively
214
N. ANYAS-WEISS AND D. STROTTMAN Na -5
- - 2 7 - - ' 3
- - 5 27 ~
3
2 -
-
I
2 -
-
3
7 14
- - 5 2
~
-
2
. ~
7
1
4 3,1 2
- - 5
1
1 - - 6
6
-
-
"4
6
5
3
3
7
36
1,4
~
-
2
6
~
2
3
3
~
2 3
3
- - 3 - - 4 - - 1 5
- - 1 ~
1
5
5
-
-
5
2
3
3 ~
2
~
3
- - 1 - - 3
1 3
-
4
5
- - 2 ~
4
5
1
~
1
-
4
4
1 4
14
- - 1
- - 1
- - 1
- - 3
3
C
3
C *T
3
C * V*T
3
C *V*T • Or
3
C -V*T ~(~..*SP
-
-
3
0
EXPI".
Fig. 4. T h e s p e c t r u m o f 2 2 N a , o t h e r w i s e a s f o r fig. 3.
adding various terms of the interaction Hamiltonian. The calculated results obtained with the optimum strengths are shown in figs. 3-5 and the resulting strengths are given in table 5. It may be seen that when only the central components of the interaction are used, the results are very similar to the results ofAkiyama et al. 3) even though the strengths are not; in particular the lowest state of 22Na is still 1 + rather than the observed 3 +. Also, the ½+ level is calculated to be much to low. The error per degree of freedom, a2/N, is correspondingly large and hence, allowing only a central interaction, it is difficult to improve upon the results of ref. 3).
NON-CENTRAL 'INTERACTIONS
215
22 Ne
-
6 2
-
4
-
~
'6
4
0
2
6
~
2 -6
0 ~
2
~
0
O4 4
~ N
m
3
~ 4 - - 2
~ 3
~ ~
3
~
4
~
2
~
2
2
~
3 4 •
2
2
-5 M
2
4
4 4 ~
~ 2
4
-3
-2 ~
2
2 ~
~
0
0 c.T
~
0 C.V.T
2
0 C,V.T. QL
2
0 C.V..T *QL*SP
-0 EXP.T
Fig. 5. The spectrum of 2ZNe, otherwise as for fig. 3.
When the tensor interaction is included, the error, 62/N, drops to 0.37 and the level ordering in 22Na is much improved. The even state central interaction, eq. (12) remains almost unchanged, but a strongly repulsive triplet even tensor interaction is required to produce a good fit. A d ~ ~g the two-body spin-orbit interaction causes the error per degree of freedom to increase slightly indicating the vector interaction is statistically irrelevant. Finally, the addition of the quadratic spin-ortrit term and variation of the singleparticle energies reduced a2/N to only 0.09. Again, the triplet even tensor interaction
216
N. A N Y A S - W E I S S A N D D. S T R O T T M A N 2ZNo !
i
w
I
E(MeV)
BE(MeV)
2.0
-35.5
1.0
-37.0 s
~
3
0.0
~
7---I.0
.
-38.5
~ I 5
'
l
I
-15
5
I
25
laVr Fig. 6. The spectrum of 2ZNa as a function of the triplet even tensor interaction ~3 Vf. The strengths of other interactions are those of table 5, column five.
TABLE 5 Values of the interaction parameters which give a best fit to experimental energy levels
C 13V~ 31V~
-12.7 -36.4
33V c
0.2
C+ T -20.4 -41.7 22.5
'3VLs 33VLs 13V r 33VT 13VQL 31VQL 33VQ1~ e Vsy m
96.3 --1.6
-2.03 -1.16 -28.1
-2.03 -1.16 -31.1
C+ T+ V+QL+SP -28.8 -36.9 -110.0 232.9 --242.9 65.3 31.2 --3.2 --2.0 62.7 - 1.36 -0,81 -32.6
(0.1) (0.1) (45) (106.6) (13.3) (1.8) (6.4) (0.2) (0.0) (l.1) (0.39) (0.12)
C+ T+ V+QL -31.6 -35.0 -97.2 326.5 --303.6 53.0 11.2 --7.0 --6.6 53.7 -2.03 -1.16 -33.3
(0.2) (0.1) (14.8) (147.2) (72.0) (11.2) (7.8) (0.5) (0.1) (3.8)
C o l u m n s two through four are fits to mass 21 and 22 levels; the fifth column also includes levels m 2°Ne. For reasons explained in the text ~~V, and ~~VQL were set to zero.
is repulsive contrary to expectation. By diagonalizing the error matrix, one may determine the linear combination of parameters which are best determined; Vsy m is invariably determined best as expected. The triplet even tensor interaction is also well determined, indicating the repulsive nature of 13 VTis not a statistical aberration. The two-body spin-orbit interaction is always very poorly determined and, as mentioned earlier, does not improve the goodness of the fit.
217
NON-CENTRAL INTERACTIONS
T o investigate the origin of the a p p a r e n t l y a n o m a l o u s value of 13 VT' levels were selectively omitted from the fit. First, all levels of ZZNa were omitted and only the levels of the three isotopes of Ne were included in the fit. Although good agreement was obtained with the experimental level ordering, the strength of the tensor interaction was very poorly determined. Thus, the tensor interaction is essentially determined by 22Na, as m a y be seen from fig. 6. T h e small calculated splitting between the K = 0 and the K = 2 bands in 2/Ne has been attributed to a variety of reasons ~8), including the presence of excitations to the lp-0f shell 19). T o check that these levels did not unduly affect the nature of the tensor interaction, all Z2Ne levels were omitted from the fit. Although the values of the fitted strengths changed, ~3V.r remained positive, or repulsive. Hence, we conclude that fitting the levels of 22Na forces the triplet even tensor interaction to be repulsive.
5. Discussion In sect. 4 it was d e m o n s t r a t e d that it is quite possible to obtain agreement with the k n o w n spectra of the mass 21 and 22 nuclei, although substantial modifications a p p e a r necessary in the interaction. The interaction fitted to experiment is s o m e w h a t closer to that fitted to the calculated levels of 2°Ne. T h e most substantial modification necessary was to m a k e the triplet even tensor force repulsive; slight changes were also necessary in the quadratic spin-orbit force. T h e role the quadratic spin-orbit force plays is interesting. F r o m figs. 3 and 5 it is seen that VQL acts to shift the K = 2 b a n d of 2:Ne u p w a r d and (since the I + of 2~Ne is d o m i n a n t l y (62)) it also separates different SU(3) representations. Further, it was seen that VQL tends to i m p r o v e the goodness of SU(3). Each of the latter two TABLE 6
Percentage of the leading SU(3) representations present in eigenstates of the phenomenological Hamiltonians with the parameters of table 5, columns two and four Nucleus
Rep.
Spin
C
C + NC
2tNe
(81)
~+ 2~+ 25+
73.2 25.2 68.4
89.2 9.5 83.4
22Ne
(82)
0+ 2+ 2~ 4+
60.0 65.4 60.6 66.5
73.3 74.4 58.7 68.4
22Na
(82)
3+ I1 1~2+
71.6 51.2 50.7 51.2
86.3 58.8 47.6 57.2
218
N. ANYAS-WEISS A N D D. S T R O T T M A N
characteristics is not unlike a quadrupole-quadrupole force; this perhaps should be anticipated from the definition of L12 of eq. (6): the first term is proportional to L(L + 1) as is the quadrupole-quadrupole interaction. Thus, the importance of the quadratic spin-orbit interaction may be suggestive that a longer range interaction should be included into the interaction. This possibility has already been investigated in 22Ne [ref. 18)]. A possible source of worry is that the truncation scheme which was based on SU(4) and SU(3) might break down in the presence of the strong non-central forces, with the result that the basis used in the work is inadequate. Although a definitive answer is impossible without diagonalizing the Hamiltonian within a complete basis, there are two items which suggest that the truncation scheme is still valid. An essential requirement for the validity of SU(3) in that the ls level lies below the 0d centroid. This is true for the single-particle energies of ~70 and if the singleparticle energies are allowed to vary, it remains true. Further, the variation of the one-body spin-orbit strength suggests a value smaller than that indicated by the l vO spectra which would help restore the SU(4) symmetry. As another check, one may simply examine the wave functions in the two cases of only central and central plus non-central interaction to note the degree of SU(3) and SU(4) breaking. In table 5 are listed the percentage of the leading SU(3) symmetry for the lowest few states. Surprisingly, when all the non-central interactions are included, SU(3) is a better truncation scheme contrary to common belief which would have SU(3) a less valid symmetry in the presence of non-central interactions. The simplest explanation is that the central interaction produces a too compact spectrum; the non-central interactions - particularly the quadratic spin-orbit interaction tend to expand the spectrum thus making large admixtures of lower SU(3) representations more difficult. Indeed, in the case of only central plus tensor, the amount of SU(3) breaking is somewhat greater than for the pure central interaction. This is opposite to the effect found by Manakos and collaborators 20) who investigated the effects of those two-body matrix elements which do not conserve S. One may note the anomalously small amount of (81) for the ½+ level of 21Ne. This is simply explained: the wave function is dominated by (62) L = 0 which is favoured by the attractive nucleon-nucleon interaction over the L = 1 state of the (81) representation. A similar situation occurs in 23Mg. With the possible exception of some states of 2°F, nuclei with fewer nucleons than six are not particularly sensitive to the tensor force, and therefore a repulsive triplet even tensor interaction may not cause undue difficulty. The interesting question remains, why should the effective triplet even tensor interaction be negative? Possible origins may lie in the convergence problems in calculating G as already mentioned, or the presence of three-body forces. The importance of three-body forces increases as (g), n = A - 16, and are thus more important in mass 22 than mass 19 and 20. Further, as much of the contribution to the three-body interaction arises from Pauli corrections, it may be anticipated that they are repulsive and further,
NON-CENTRAL INTERACTIONS
219
their effects must show up first in the even interaction. Another possibility is from the so-called Vary-Sauer-Wong effect: the contributions to G3p~h from intermediate states of up to 20h~o excitation are found to be repulsive and large 2 ~. 22). They arise primarily from the tensor interaction and could therefore explain some of the needed repulsion. Other investigations 15.2o) have been conducted to obtain an interaction which successfully reproduces the s-d spectra. One of the more successful is that of Preedom and Wildenthal 23) who varied several of the 2BME which, although l;roviding little physical insight into the interaction, did produce an interaction which fits levels remarkably well. Although an exhaustive investigation, such as that of sect. 3, into the nature of the effective interaction was not made, an examination of the transformed two-body matrix elements does provide some information. The matrix elements of the Preedom-Wildenthal interaction which violate S are larger than those of Kuo; this may be an effect of the fitting procedure used in ref. 23) in that matrix elements containing 0d~ particles were varied, but not those with 0d~, this producing essentially a non-local interaction. The effects of such matrix elements were investigated by the Darmstadt group 15, 2 0 ) and were not found to be essential. The authors wish to thank Sir Denys Wilkinson and Dr. Peter Hodgson for the hospitality of the Nuclear Physics Laboratory, Oxford where much of this work was done. Thanks also are extended to members of the experimental groups of Oxford, particularly Dave Rogers, for their ceaseless inquiring into the accuracy of the parameters of the levels and to Prof. Manakos and other members of the Darmstadt group for numerous discussions concerning their exhaustive calculations.
Appendix We briefly discuss here the fitting procedure and certain results which are very useful in the interpretation of the parameters resulting from the fit. The Hamiltonian is defined here as M
H = ~hH
+Ho,
(A.1)
~=1
where the ha are the parameters to be determined and H o is that part of the Hamiltonian left unchanged. The parameters {hi} are found by varying the quantity m
c~~ = ~ ( E I - - ( H ) , ) z i=1 m
M
= ~ ( E , -
a=l
(A.2)
220
N. ANYAS-WEISS AND D. STROTTMAN
with respect to the h,. In eq. (A.2) the E i are the experimental energies, m is the number of levels and N = m - M is the number of degrees of freedom. The hessian matrix is the M x M matrix consisting of second derivatives of 0.2. ('320.2
m
(A.3) Let K be the inverse of G; then an unbiased estimate of the variance ofg~ is 0-2
var,q~=~K
.
(A.4)
In table 5 the values of var.G are given in parentheses. References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12)
J. P. Elliott and B. H. Flowers, Proc. Roy. Soc. A229 (1955) 536 T. Inoue, T. Sebe, H. Hagiware and A. Arima, Nucl. Phys. 59 (1964) 1; 85 (1966) 184 Y. Akiyama, A. Arima and T. Sebe, Nucl. Phys. A138 (1969) 273 T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 T. T. S. Kuo, Nucl. Phys. AI03 (1967) 71 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 D. Strottman, Particles and Nuclei 5 (1973) A. Arima, S. Cohen, R. Lawson and M. Macfarlane, Nucl. Phys. AI08 (1968) 94 J. P. Elliott, H. A. Mavromatis and E. A. Sanderson, Nucl. Phys. AI21 (1968) 241 M. E. Fosada and K. B. Wolf, Rev. Mex. Fis. 14 (1965) 29 T. lnoue, T. Sebe, K. K. Huang and A. Arima, Nucl. Phys. 99 (1967) 365 E. C. Halbert, J. B. McGrory, B. H. Wildenthal and S. P. Pandya, in Advances in nuclear physics, vol. 4, ed, M. Baranger and E, Vogt (Plenum, NY, 1971) 13) B. R. Barrett and M. W. Kirson, Phys. Lett. 30B (1969) 8 14) R. R. Whitehead, A. Watt, B. J. Cole and 1. Morrison, Advances in nuclear physcis, vol. 9 (Plenum, NY, 1977) p. 123, and references therein 15) R. Manakos, M. Conze and H. Feldmeier, private communication 16) M. MacFarlane, in Proc. Int. School of Physics Enrico Fermi, course 40, Varenna 1969, ed. M. Jean (Academic Press, NY, 1969) p. 457 17) D. Kurath, Phys. Rev. 130 (1963) 1525 18) C. Abulaffio, Nucl. Phys. A144 (1970) 225 19) A. Arima and D. Strottman, Nucl. Phys. A162 (1971) 605 20) M. Conze, H. Feldmeier and P. Manakos, Phys. Lett. 43B (1973) 101; H. Feldmeier, Dissertation, Technische Hochschule Darmstadt, 1974 21) J. P. Vary, P. U. Sauer and C. W. Wong, Phys. Rev. C7 (1973) 1776 22) C. L. Kung, private communication 23) B. M. Preedom and B. H. Wildenthal, Phys. Rev. C6 (1972) 1632
NuclearPhysics A306 (1978) 2 0 1 - 2 2 0 : ( ~ North.HollandPublishino Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
I N F L U E N C E O F N O N - C E N T R A L I N T E R A C T I O N S ON lsd)d N U C L E I N. A N Y A S - W E I S S
Ontario Hydro Research Dieision, 800 Kipling Avenue, Toronto, Ontario, Canada and D. S T R O T T M A N *" **
Department o[' Physics, State University of New York, Stony Brook, N Y 11794 Received 1 March 1978 (Revised 3 May 1978) Abstraet: The influence of non-central interactions is investigated in mass 18 to 22 nuclei. The effect of non-central components in the realistic matrix elements of K u o is evaluated in mass 18, 19 and 20 nuclei. A phenomenological interaction is constructed to reproduce the results obtained with the K u o Hamiltonian and compared with an interaction fitted to experimental levels. The fit to experimental levels requires a repulsive triplet even tensor force.
1. Introduction The shell model with residual interaction has been very successful in describing a wide range of properties of 0p and ls-0d shell nuclei. However, an embarassing feature of this model is the fact that many diverse residual interactions apparently enjoy equally good success in obtaining agreement with experiment. The two-body interaction was historically taken to be a phenomenological interaction with only central components having either Yukawa or Gaussian form factors and with the strength in each spin-isospin channel taken as a free parameter 1-3). Recently, "realistic" matrix elements 4.5) by which we mean matrix elements obtained from potentials which fit nucleon-nucleon scattering data up to about 350 MeV [ref. 6)] have found favour amongst theorists. These two approaches to the residual interaction differ in that, besides the latter having in principle no free parameters (although not in practice), the "realistic" matrix elements contain non-central forces as well as the usual central forces. The implications are that the total orbital angular momentum L will not be diagonal for the two-body matrix elements (2BME). The 2BME of Kuo 5) also do not have S, the total spin, diagonal, but those matrix elements for which S is not diagonal are an order of magnitude smaller ~) than those for which S is diagonal and may be ignored in the present discussion. Their effects are calculated later in this paper. Other approaches to the residual interaction have been to take a number of the 2BME themselves as free parameters which are then chosen so as to reproduce the * Supported in part by U S D O E contract EY-76-5-02-3001. ** Present address: Theoretical Division, MS452, LASL, Los Alamos, N M 87545. 201
202
N. ANYAS-WEISS AND D. STROTTMAN
observed energy spectra 8) or, in an attempt to avoid the use of potentials entirely, to relate the 2BME directly to observed nucleon-nucleon phase shifts 9). In reality a rather less definite division exists than indicated above. There have been a few attempts to include non-central forces in a phenomenological potential, and to use it in the two- and three-body system outside 160 [-refs. 10,11)] but such a program has not previously been carried out for more complex nuclei. Although it might appear desirable to use exclusively "realistic" matrix elements in calculations, and discard the phenomenological potentials as no longer useful, it may be contended that for a number of reasons such a decision would be premature. First, it is suspected that the G-matrix expansion in terms of the number of interactions does not converge 13), and that one reason for this failure is the strong tensor force. The 2BME now available may not adequately represent the effects of the non-central forces. Second, by systematically studying the effects of changing parameters, one may obtain information on what aspects of nuclear structure are sensitive to noncentral forces. The properties of many s-d shell nuclei have been calculated within a complete sd basis using the realistic 2BME 12, 14, 15) of Kuo 5) and a number of discrepancies exist between the theoretical predictions and experiment, even in nuclei with six or fewer nucleons outside a closed 160 core. Examples are the positions of the K = ½+ excited bands in 19F and 21Ne which come too low in the calculations, and the ground state and first few excited states in 2°F, 21F and 22Na are usually predicted in the wrong sequence when compared with experiment. The spectrum of 22Na is of particular embarrassment; using only central forces, two 1 + states are usually calculated as having lower energies than the 3 + ground state. The realistic interactions do somewhat better; often only one 1 + state lies beneath the experimental ground state. In all cases mentioned the total spin S is non-zero, allowing two-body spin-orbit and tensor forces to directly contribute to the energy. An indication that the non-central forces are partly responsible is the fact that disagreement is more pronounced when only central forces are used 3). The solutions to other discrepancies such as the position of the K = 2 + bands in even-even nuclei presumably lie elsewhere, inasmuch as the low-lying states in these nuclei have S = 0. Before sweeping all these difficulties under the rug of "core-excitation", the effects of small modifications in the two-body interaction should first be investigated. Third, all calculations of realistic matrix elements have assumed that the intermediate states are always nucleons, whereas it is possible to have excited states of the nucleon, particularly the A in intermediate states: such contributions might give rise to a very different spin dependence in the interaction. Fourth, three-body forces are assuredly present, the effects of which have not been estimated. The success of calculations assuming only a two-body interaction indicates that their effects on energy spectra may be small for low-lying states, although the configuration mixing due to such forces is clearly different than that already included in the realistic matrix elements. The latter two effects may be partially compensated over a small range of mass number by a suitably chosen phenomenological interaction.
NON-CENTRAL INTERACTIONS
203
Finally, the phenomenological interactions are easier to work with since the total spin S, and possibly also L, is diagonal. Further, and perhaps most important, phenomenological interactions may provide physical insight into the results of the shell-model calculations, obviously a non-trivial advantage when the possible matrix dimensions may run into the thousands. With only a very few parameters - each of which has an obvious physical interpretation - one may more easily determine to which aspect of the interaction nuclear structure is sensitive, and the possible size of unitary symmetry breaking terms in the Hamiltonian. It is the aim of the present paper to investigate the importance of non-central interactions on the energy spectra of nuclei near the beginning of the ls-0d shell. This is done by first investigating the importance of such terms in the Kuo interaction 5) and then constructing a phenomenological Hamiltonian to allow a systematic variation of the strengths of these components. In sect. 2 the interaction will be defined. In sect. 3 the interaction of Kuo 5) will be investigated and in sect. 4 the strengths of a phenomenological Hamiltonian will be fitted to experimental levels. 2. The model In this paper the effects of non-central forces on the spectra of nuclei with less than eight particles outside an 160 core is investigated. The phenomenological Hamiltonian is assumed to be of the form
H = ~l" s+~e,(6-12)+ ~ V(ri~), i
(1)
where V(ri~) is the sum of the central, tensor, spin-orbit and quadratic spin-orbit terms. The individual terms are defined as follows. The central potential is taken as usual to be g = {13 g p 1 3 +31 gp31 +11 Vp11 + 33 VcP33 }f(r/pc)"
(2)
In eq. (2) Pzr+ 1.2s+ 1 is a projection operator onto a state with spin S and isospin T, with 2r+ 1.2s+1V being the strength. The other terms are defined similarly: VT = [13 VTP13 .+_33 VTP33}f(r/itv)S12 '
(31
where $12 = 3(# l • ~'12)(0"2 " ~'12)--(0"1 " 0"2),
VLS = {13 VLsP13 q_ 33 VP33 }f(r/llLs)L. S,
(4)
VQL = {13VQLPI3-l-31VQLP31+11%LPll + 33VQLPz3}f(r/[AQL)L12.
(5)
In eq. (5) the definition of the quadratic spin-orbit operator is the same as that used by Hamada and Johnston 6), L I 2 = {fiLJ+ (~1 "0'2)} L2 -- (L" S) 2.
(6)
204
N. ANYAS-WEISS AND D. STROTTMAN
The form factors are all assumed to be Yukawa with the range parameter defined by 2 = x/21t/b.
In their calculations on s-d shell nuclei Arima and his collaborators 23) have found 2 c = ~ to be satisfactory and we shall adopt the same value as well. In the one-pion-exchange potential the tensor force is also of long range and we therefore choose 2c = 2 T. The values for 2rs and 2QL are more uncertain; their precise value is not of great importance as changes in the interaction radius may be largely compensated by appropriate changes in the strength of the force. In the HamadaJohnston potential the spin-orbit and quadratic spin-orbit forces are of short range; we shall here use '~QL = )'~s = 2 2 -- ~. There has been little investigation into the need for introducing a quadratic spin-orbit force into the phenomenological interaction. To some extent its effects may be mocked up by an appropriate redefinition of the remaining three components. As pointed out by Hamada and Johnston 6), this is particularly simple in those partial waves which are coupled by non-central forces, and is achieved by replacing VLs + VQt" by Vl.s. The I/QLterm is more important in the singlet even channel as the nuclear spectra are almost totally insensitive to the singlet odd potential. Further, as the quadratic spin-orbit force enters more strongly in high relative l states, positions of high spin states may be particularly sensitive to the quadratic spin-orbit interaction. By virtue of the interaction defined above being translationally invariant and parity conserving, all two-body matrix elements will be diagonal in the spin S. As an example of realistic matrix elements, the 2BME of Kuo 5) were used. These matrix elements contain in addition to central, vector and tensor forces, interactions which do not conserve S. Such matrix elements may be non-zero for a variety of reasons, among which are the presence of non-local potentials or spin-orbit partners having different radial wave functions. Further, such effects may enter via the renormalization corrections to G2p or G2h through the use of partially spurious intermediate states or allowing one of spin-orbit pair to contribute and not the other. The matrix elements arising from G3plh need not conserve S. The matrix elements which do not preserve S are important because only they connect the spatially symmetric two-particle state with the spatially antisymmetric state and therefore directly contribute to the destruction of SU(4) symmetry. Other components of the two-body interaction may connect different SU(4) representations in a many-particle configuration, although in many cases there exist selection rules which tend to diminish the admixing. For example, the Bartlett interaction may in the four-particle system connect the SU(4) representation [4] with [-22]; the centroid of [22] is 15 MeV higher than that of [4] while the centroid of [31] is only 10 MeV higher than that of [4], but the Bartlett interaction will not connect [4] with [31]. The central, vector and tensor components of any two-body interaction may he
N O N - C E N T R A L INTERACTIONS
205
found by transforming the interaction from jj coupling to LS coupling and then summing over the total angular momentum of the two-body matrix elements with a weighting factor provided by angular momentum algebra; if V(r) is written as 2
(7)
V(r) = ~ (Y~¢{k}(r)" f/){k)(tr l, O'2)), k-O
then
x (LST~flJ[VIIZS'T~'ffJ).
(8)
In eq. (8) the additional labels a and /3 needed to uniquely specify the two-body matrix elements are provided by the irreducible representations of SU(3) and SU(4). Only for k equal to one need S not equal S'. A quantitative measure of the size of such matrix elements may be obtained from the quantity ~rr(S ~ S') defined as the ratio of the root mean S non-conserving matrix element to the root mean S conserving matrix element:
{!
DT =
~
}1
~ ((21~)LSTJIGI()o'ff)I2S'= S T J ) 2 LL'JJ ()4~)(2'u')
O.r =
{9)
((2/OLSTJIGI(2' ff)I2S' ¢ S T J ) 2
,j
,
(;4n(~'u') OT(S :F S') = OT/D T.
(10)
In eqs. (9) and (10) n o and n o are the number of terms contributing to each sum. In table 1 the value of a T for the various terms contributing to G in the case of Kuo interaction are shown. For the isospin-zero case the contributions are negligibly small save for the G3plh contribution where the matrix elements are nearly as large as those which do conserve S. In the isospin-one case there are also appreciable TABLE 1 The quantity oT(S ~ S') as defined by eq. (10), which is a measure of the SU(4) symmetry breaking part of the Kuo realistic matrix elements
Gb~c
G3pih G2p
Gzh G,o~
ao(S =~ S')
%{S ~ S')
0.003 0.623 0.034 0.004 0.051
0.008 0.886 0.186 0.262 0.039
206
N. A N Y A S - W E I S S A N D D. S T R O T T M A N
values of al from both G2p and G2h. Such contributions will arise from partially spurious intermediate states. The expectation value of the Hamiltonian will be written as
E= ( C > + ( V ) + ( T ) + ( S S ' ) + ( S P ) .
(11)
where the meaning of the first three terms is evident; (SS') signifies the contribution of those matrix elements which do not preserve S, and (SP> is the contril~ution of the one-body operators appearing in eq. (1). The values of ¢ and ~: are determined from the spectrum of ~O. Although the matrix elements which do not preserve S are only 5 ,'~, of the magnitude of those which do, it will presently be seen that they contribute as much to the energy of a several-particle system as do the tensor or the two-body spin-orbit interactions.
3. Fitting the Ku0 interaction The realistic matrix elements of Kuo provide a reasonable description of the spectra of nuclei near the beginning of the l s-0d shell. The importance of the non-central components will here be estimated in three ways. In the first the Hamiltonian of eq. (1) is alternately diagonalized with various two-body terms of eq. (1) omitted; the results for 19F are shown in fig. 1. The spectrum is in all cases shifted slightly so that the ½+ level lies at zero energy. With only the central part of the two-body interaction contributing, the gross spectrum is apparent with both the correct number of states and their approximate positions. The energy of the ground state is within 200 keV of the answer obtained with the complete Hamiltonian. With the addition of the rank-two interaction, the position of all levels is essentially correct. The contributions of ( V ) and (SS') have little effect on the final spectrum. A more quantitativ e method is to calculate the expectation value of each of the terms appearing in eq. (11); the wave functions used were obtained by the diagonalization of the complete Hamilt.onian. The results are given in table 2 for an example in which the dominant components of the low-lying states have S = 0 (2°Ne), or S ½ (t 9F) and a case in which there is an appreciable admixture ( ~ 20 °4) of S - 3 into an otherwise S = ½ wave function (21F). The expectation values of the one-body terms, I. S and ~ / 2 - 6 ) , are also given; to obtain their contribution to the energy one must multiply by - 2 . 0 2 and - 1 . 1 6 MeV, respectively. One may note that the occupation probability of the 0d~, 0d~ and ls~ orbits may be now calculated using the information in table 2: n(~) =
t f H - 3 (g( 1 12 - 6 ) ) + 2 ( / . S ) } , .~,3
n(~) = z{5n - (~(/2 _ 6)> - (1' S>}, n(~) = (g(1 l: - 6 ) > .
NON-CENTRAL INTERACTIONS Kut~ +
19 F
~ ~
207
170
3
3
~
? S
"as7
3
3
5
5
7
7
7
13
13
13
7
5 7
9
5
1
7
:3 5
g"
----~5
7
~
1
1
6
7
7
~ 1 3
~5.g, 13
7
7
S
S
5
4
3
2 ~
3
3 ~
3
~
3 1
~ I
5
~ 5
I
C*V*T*SS
,~
C
5
~
5 ~
I
I
C÷V*T
I
I
C*T
5
I
S I
C÷V
I
C*V÷SS
Fig. I. T h e s p e c t r u m o f ~gF w i t h the full K u o i n t e r a c t i o n a n d w i t h v a r i o u s t e r m s o f the i n t e r a c t i o n o m i t t e d ; C = r a n k 0, V = r a n k 1, T = r a n k 2 a n d SS = r a n k I (S n o t d i a g o n a l ) .
F r o m these quantities the F(M1) strength may be estimated using the Kurath 17) sum rule. For XgF and 2°Ne the average value of I(V)I is approximately 75 keV; the contribution in the 21F case is somewhat larger, although even here it is less than 1 ~o of the contribution in magnitude of ( C ) . Since the low-lying states of 2°Ne are nearly pure S = 0, one would expect that the contribution of {V) to be less than for 19F for which rank-one potentials may contribute even in the diagonal matrix elements. Surprisingly, this is not observed.
208
N. ANYAS-WEISS A N D D. S T R O T T M A N Kuo +170
ZONe
aV
/¢
~
l//
••
i
2
I
I
C+V+T÷SS
m
I
I
C*V+T
l
I
C+T
2
I
I
C+Y"
I
C+V+SS
Fig. 2. As fig. 1, save for 2°Ne.
The contribution from the tensor force is larger than that from ( V ) , averaging from 200 to 350 keV. The largest contribution in /°Ne comes from (SS') and is always attractive. Its omission would not seriously affect the appearance of the spectrum since its contribution is nearly the same for all levels, although it would result in an underestimate of the binding energy by 250 keV; see fig. 2. In conclusion, it appears that non-central components of the Kuo interaction merely shift levels by a few hundred keV and the relative energy shift is much less. Hence, to produce a correct energy spectrum it is essential to get the central interaction correct.
NON-CENTRAL
209
INTERACTIONS
TABLE 2
Expectation value of the various rank interactions for eigenstates of the Kuo interaction Nucleus
Spin
Ex (calc.)
19F
½ 1
13 ~3 z
0 6,75 1,25 6.45 0.36 4.97 5.12 5.68 6.49 10.09 5.26 9.97
0 2 4 6 8 0 2 2 4
0 1.45 3.86 8.19 12.36 6.81 8.43 10.37 10.18
1
11.78
3
10.69
11 2
2°Ne
z~ F
61(12 - 6)
(C)
(v)
(T)
(SS')
1.08 0.53 0.82 0.79 1.53 1.67 0.51 2.51 2.19 2.57 2.37 1.13
0.98 2.19 0.53 0.95 0.52 0.90 0.60 0.17 0.55 0.29 0.00 0.00
- 14.54 -7.58 - 13.88 -8.33 - 14.05 -8.67 - 10.27 -7.34 - 5.92 -2.20 -8.13 - 6.33
-0.02 0.02 0.03 0.03 0.04 -0.07 -0.04 -0.15 0.00 -0.06 0.12 0.06
-0.07 -0.18 - 0.53 -0.55 0.14 0.14 -0.80 0.58 - 0.60 -0.04 0.21 0.18
-0.17 0.00 - 0.21 -0.11 -0.16 -0.11 -0.15 -0.21 - 0.02 -0.17 -0.24 0.25
1.30 1.35 1.47 1.53 1.90 1.42 2.22 1.50 1.67 1.13 2.18
0.89 0.99 0.66 0.58 0.00 0.12 0.41 1.66 0.67 0.94 1.04
-29.1 l -27.51 - 25.23 - 20.90 - 16.90 -22.01 - 19.89 - 17.57 - 18.59 - 17.02 - 16.60
0.03 0.01 0.06 0.03 0.10 0.08 0.09 -0.15 0.02 0.37 -0.12
-0.14 -0.02 -0.07 0.04 0.28 0.06 0.23 0.10 0.01 - 1.12 0.08
-0.26 -0.29 -0.31 - 0.35 -0.41 -0.19 0.16 -0.18 - 0.23 -0.20 -0.18
1.22 1.26 1.34 1.25 1.50 2.19
-
-0.12 -0.17 -0.24 - 0.15 -0.20 -0.04
0.08 0.16 0.19 -0.33 -0.45 -0.03
-0.18 -0.18 -0.19 -0.15 -0.14 -0.12
(/
S)
25
0
1.78
3 1
0.62 0.20 1.04 0.34
1.76 2.11 1.61 1.59
1z
0.75
1.10
17.35 16.76 16.32 16.21 16.50 16.89
The calculations for ~*F and 2°Ne are for a complete s-d basis; those for 2iF are in an SU(3) truncated basis. The expectation values of the two one-body operators must be multiplied by - 2 . 0 2 a n d - 1.16 to get the contribution to the energy.
The third technique to investigate the nature of the non-central components in a realistic interaction is to fit such 2BME with matrix elements obtained using the phenomenological interaction of eq. (1). However, since the energy eigenvalues of a many-particle system depend on the 2BME in a non-linear fashion, it is more reliable to fit the phenomenological interaction to eigenvalues obtained by diagonalizing the Kuo interaction in a several-particle system. Effects ofnon-linearity which would appear as an A-dependent interaction were investigated by fitting the parameters of the phenomenological interaction first to eigenvalues obtained by diagonalizing the Kuo interaction for all possible values of J and T in mass 18 and
210
N. A N Y A S - W E I S S
AND
D. S T R O T T M A N
19, and thirdly, only even values of J for the T = 0 states of mass 20. All eigenvalues of mass 18 and the six lowest eigenvalues for each J and T in mass 19 were included in the fit. The 2°Ne fit included the first 18 eigenvalues of angular momentum 0, 2 and 4 and all the angular momentum 6 and 8 eigenvalues. The resulting strengths for these fits to results of calculations using the Kuo interaction are given in table 3. TABLE 3 T h e strengths o f the p h e n o m e n o l o g i c a l interactions which provide the best fit to positions o f energy
eigenvalues as calculated using the K u o interaction Interaction
18
18,19
20
13 V~ 31 V~ ~ 1I.
37.85 - 29.16 84.38 -4.51 13.33 -- 71.65 -- 28.63 6.07 2.70 2.14 9.76 0.49
- 37.95 (0.01) - 29.42 (0.00) - 49.29 (82.83) 1.46 (0.22) - - 3 . 2 4 (11.10) --40.50 (2.05) -- 35.07 (0.62) 7.04 (0.06) 4.71 (0.03) 1.40 (0.00) -- 10.92 (2.08) 1.02 (0.04)
30.04 - 37.84 - 337.88 12.01 --21.48 -- 116.84 -- 70.96 19.07 15.66 -- 0.06 -- 58.25 3.26
33~ c
13VLS 331% s
l 3 Vl 33 VT
13I/QL 31 l/eL
11VQL 33
VQL
N
cr2/N
16 0.19
82 0,21
62 0.26
The n u m b e r s in parenthesis are the statistical uncertainties. The quantity N is the n u m b e r o f degrees 12) a n d a2/N is the error per degree o f freedom.
o f f r e e d o m ( N = n u m b e r o f levels
For the combined mass 18 and 19 fit, an estimate of the uncertainty in the parameters is also given. The uncertainty is here defined as the diagonal element of the covariance matrix (see appendix) multiplied by cr2/N where
a2
= ~ (E~i' - - Ecale}2i " i
and N is the number of degrees of freedom. In contrast to fits which include only the first one or two levels of each angular momentum, the fits described here should adequately represent the odd state interaction. When comparing the mass 18 and the mass 18 and 19 fit, there is little change in the values of the parameters, save for those which have a large uncertainty. Amongst the latter are the singlet odd interactions, a 1 V (the value of which changes by 134 MeV) and 1 t VQL,and the two-body spin-orbit interaction. Otherwise the interaction has the expected properties: the singlet and triplet even central and quadratic spin-orbit interactions are well determined. The even tensor interaction is attractive.
N O N - C E N T R A L INTERACTIONS
211
However, the triplet odd central interaction in the fit is attractive, whereas in the investigations of Arima and collaborators 33V is repulsive, but since 33VQL is repulsive, the net effect on the triplet odd channel may still be one of repulsion. When restricting the fit to the levels of Z°Ne, there are changes in the strengths of the fitted interaction, even for those parameters which were seemingly well determined in the mass 18 and 19 fit. Of particular note are the values of the even central interaction; for mass 20 the singlet even is stronger than the triplet even, whereas in the mass 18 and 19 fit the opposite was true. However, the average central interaction strength (to which ~-particle nuclei are most sensitive), Vym = !~3t 2, V+13V~ c,,
(12)
is unchanged. A comparison between the two fits is instructive and sobering in one other respect. Even when the interaction is greatly overdetermined (92 levels were fit in the mass 18 and 19 fit), changing one's choice of fitted levels changes the nature of the interaction. One must therefore be extremely cautious in the interpretation of the results of calculations in which the number of degrees of freedom is much less. 4. The fit to A = 21 and 22 nuclei
From the results of the previous section one has an idea of the nature of a phenomenological interaction which simulates the realistic matrix elements of Kuo as well as the effects of non-central interactions on the spectra. As already mentioned in the introduction, the positions of several levels of nuclei near the beginning of the s-d shell are not reproduced by the realistic matrix elements; in this section we investigate what changes in the interaction need be made to reproduce the experimental level ordering. As many of the most notable disagreements occur for the mass 21 and 22 nuclei, it was decided to attempt a fit to the levels of 21Ne, 2ZNe and 22Na; the structure of the latter nucleus should be particularly sensitive to the tensor interaction. Difficulties inherent in the selection of a proper set of levels to include in a fit have been extensively discussed by MacFarlane 16); the two of most relevance here are firstly, that the spins of many levels are unknown (or perhaps incorrectly assigned) and secondly, the possibility that a level is an intruder state and should be described by a configuration lying outside the model space. The former difficulty is obviated by a circumspect selection of levels and restricts in most cases the number of levels fitted per angular momentum to one. Systematics based on the weak coupling model suggest there are no positive-parity intruder states in the energy region of interest in mass 21 and 22; the 7.20 MeV 0 + state in 2°Ne must, however, be excluded. A set of 19 levels from 21Ne, 22Ne and 22Na was selected to be included in the fit, and these levels are listed in table4. Additionally, some fits also included the members of the ground-state band and the excited 0 + level at 6.70 MeV of 2°Ne.
212
N. ANYAS-WEISS A N D D. S T R O T T M A N TABI,~i 4
Experimental energies of levels of mass 21 and 22 included in the fit Nucleus 21Ne
22Ne
22N[t
Spin ~ ",~
Ex(exp) 3.735
11 ~z
2.796 0 0.350 1.747 2.867 4.433 6.450
0 "~ 3 4 6
0.00 1.28 5.63 3.35 6.30
6.24 4.46
1 2 3 4 5 6 7
0.583 3.059 0.000 0.891 1.528 3.708 4.522
1.937
3.944
1.984
2.969
5.36
5.52
The bases for masses 21 and 22 were truncated to a maximum of 60 states by including only certain SU(3) and SU(4) irreducible representations. In order to facilitate comparison with earlier work, the truncation chosen was identical to that of Akiyama et al. 3). A comparison between calculations with the Kuo interaction in this basis and calculations in a complete s-d basis by the Darmstadt group 15) shows that for 2~Ne and 22Ne the omitted components account for less than 1 ')J,i of the complete wave function. The low-lying states of 22Na are dominantly S = 1 and the truncation is somewhat worse, although in most cases less than 2 oj/,, of the wave function is omitted. The many-particle matrix elements were calculated and stored on tape; the strengths of the interaction were then fitted using linear regression. Because of the success of the SU(3) model, the calculation usually converged after only three iterations unless a particularly "bad" set of initial choice of parameters was made in order to investigate the possibility of secondary minima. The low-lying levels of ZZNa are dominated by the SU(4) representation [42] and should therefore be particularly sensitive to the triplet-even tensor interaction. As a preliminary attempt to understand the poor agreement in 22Na between calculation and experiment, the triplet-even tensor interaction was varied whilst aH other parameters were fixed at the values given in table 3 as determined by the fit to the Kuo interaction for mass 18 and 19. The best fit was obtained by using a repulsive
213
NON-CENTRAL INTERACTIONS 21 He
5
3 ....
7
-
5,11
-
3
9
- - - 3 - - 1 1
3 -
-
5
3
3
3
- - . 5
~ ~
3 5
-
-
5
- - ' 5 - - 9 3
_
7
1
1
7
- - 5 7
~
7
- - 1
~
5
5
- - 1
- - 5
-----E---3
5
c------7--3c,--ET~-,r3 ~ 3 °OL
3
~ 3
5
E-E~--.3
*OI.°SP
Fig. 3. The spectrum of 2iNe calculated with the phenomenological interaction indicated which provides the best fit to levels of mass 21 and 22. The experimental levels shown are only those included in the fit.
t e n s o r i n t e r a c t i o n r a t h e r t h a n an a t t r a c t i v e i n t e r a c t i o n as r e q u i r e d by the K u o interaction. As this i n d i c a t e d t h a t a small a l t e r a t i o n in the K u o i n t e r a c t i o n was i n a d e q u a t e to fit the e x p e r i m e n t a l spectra, a general v a r i a t i o n of the H a m i l t o n i a n was c o n d u c t e d . T h e initial fits were restricted to the 19 levels of t a b l e 4. T h e singlet o d d i n t e r a c t i o n t e r m s of H were set to zero a n d n o t varied since it was a n t i c i p a t e d t h a t the levels w o u l d be insensitive to them. This was certainly true in the m a s s 18, 19 a n d 20 fit as well as the w o r k of A k i y a m a et al. 3). Several fits were p e r f o r m e d by successively
214
N. ANYAS-WEISS AND D. STROTTMAN Na -5
- - 2 7 - - ' 3
- - 5 27 ~
3
2 -
-
I
2 -
-
3
7 14
- - 5 2
~
-
2
. ~
7
1
4 3,1 2
- - 5
1
1 - - 6
6
-
-
"4
6
5
3
3
7
36
1,4
~
-
2
6
~
2
3
3
~
2 3
3
- - 3 - - 4 - - 1 5
- - 1 ~
1
5
5
-
-
5
2
3
3 ~
2
~
3
- - 1 - - 3
1 3
-
4
5
- - 2 ~
4
5
1
~
1
-
4
4
1 4
14
- - 1
- - 1
- - 1
- - 3
3
C
3
C *T
3
C * V*T
3
C *V*T • Or
3
C -V*T ~(~..*SP
-
-
3
0
EXPI".
Fig. 4. T h e s p e c t r u m o f 2 2 N a , o t h e r w i s e a s f o r fig. 3.
adding various terms of the interaction Hamiltonian. The calculated results obtained with the optimum strengths are shown in figs. 3-5 and the resulting strengths are given in table 5. It may be seen that when only the central components of the interaction are used, the results are very similar to the results ofAkiyama et al. 3) even though the strengths are not; in particular the lowest state of 22Na is still 1 + rather than the observed 3 +. Also, the ½+ level is calculated to be much to low. The error per degree of freedom, a2/N, is correspondingly large and hence, allowing only a central interaction, it is difficult to improve upon the results of ref. 3).
NON-CENTRAL 'INTERACTIONS
215
22 Ne
-
6 2
-
4
-
~
'6
4
0
2
6
~
2 -6
0 ~
2
~
0
O4 4
~ N
m
3
~ 4 - - 2
~ 3
~ ~
3
~
4
~
2
~
2
2
~
3 4 •
2
2
-5 M
2
4
4 4 ~
~ 2
4
-3
-2 ~
2
2 ~
~
0
0 c.T
~
0 C.V.T
2
0 C,V.T. QL
2
0 C.V..T *QL*SP
-0 EXP.T
Fig. 5. The spectrum of 2ZNe, otherwise as for fig. 3.
When the tensor interaction is included, the error, 62/N, drops to 0.37 and the level ordering in 22Na is much improved. The even state central interaction, eq. (12) remains almost unchanged, but a strongly repulsive triplet even tensor interaction is required to produce a good fit. A d ~ ~g the two-body spin-orbit interaction causes the error per degree of freedom to increase slightly indicating the vector interaction is statistically irrelevant. Finally, the addition of the quadratic spin-ortrit term and variation of the singleparticle energies reduced a2/N to only 0.09. Again, the triplet even tensor interaction
216
N. A N Y A S - W E I S S A N D D. S T R O T T M A N 2ZNo !
i
w
I
E(MeV)
BE(MeV)
2.0
-35.5
1.0
-37.0 s
~
3
0.0
~
7---I.0
.
-38.5
~ I 5
'
l
I
-15
5
I
25
laVr Fig. 6. The spectrum of 2ZNa as a function of the triplet even tensor interaction ~3 Vf. The strengths of other interactions are those of table 5, column five.
TABLE 5 Values of the interaction parameters which give a best fit to experimental energy levels
C 13V~ 31V~
-12.7 -36.4
33V c
0.2
C+ T -20.4 -41.7 22.5
'3VLs 33VLs 13V r 33VT 13VQL 31VQL 33VQ1~ e Vsy m
96.3 --1.6
-2.03 -1.16 -28.1
-2.03 -1.16 -31.1
C+ T+ V+QL+SP -28.8 -36.9 -110.0 232.9 --242.9 65.3 31.2 --3.2 --2.0 62.7 - 1.36 -0,81 -32.6
(0.1) (0.1) (45) (106.6) (13.3) (1.8) (6.4) (0.2) (0.0) (l.1) (0.39) (0.12)
C+ T+ V+QL -31.6 -35.0 -97.2 326.5 --303.6 53.0 11.2 --7.0 --6.6 53.7 -2.03 -1.16 -33.3
(0.2) (0.1) (14.8) (147.2) (72.0) (11.2) (7.8) (0.5) (0.1) (3.8)
C o l u m n s two through four are fits to mass 21 and 22 levels; the fifth column also includes levels m 2°Ne. For reasons explained in the text ~~V, and ~~VQL were set to zero.
is repulsive contrary to expectation. By diagonalizing the error matrix, one may determine the linear combination of parameters which are best determined; Vsy m is invariably determined best as expected. The triplet even tensor interaction is also well determined, indicating the repulsive nature of 13 VTis not a statistical aberration. The two-body spin-orbit interaction is always very poorly determined and, as mentioned earlier, does not improve the goodness of the fit.
217
NON-CENTRAL INTERACTIONS
T o investigate the origin of the a p p a r e n t l y a n o m a l o u s value of 13 VT' levels were selectively omitted from the fit. First, all levels of ZZNa were omitted and only the levels of the three isotopes of Ne were included in the fit. Although good agreement was obtained with the experimental level ordering, the strength of the tensor interaction was very poorly determined. Thus, the tensor interaction is essentially determined by 22Na, as m a y be seen from fig. 6. T h e small calculated splitting between the K = 0 and the K = 2 bands in 2/Ne has been attributed to a variety of reasons ~8), including the presence of excitations to the lp-0f shell 19). T o check that these levels did not unduly affect the nature of the tensor interaction, all Z2Ne levels were omitted from the fit. Although the values of the fitted strengths changed, ~3V.r remained positive, or repulsive. Hence, we conclude that fitting the levels of 22Na forces the triplet even tensor interaction to be repulsive.
5. Discussion In sect. 4 it was d e m o n s t r a t e d that it is quite possible to obtain agreement with the k n o w n spectra of the mass 21 and 22 nuclei, although substantial modifications a p p e a r necessary in the interaction. The interaction fitted to experiment is s o m e w h a t closer to that fitted to the calculated levels of 2°Ne. T h e most substantial modification necessary was to m a k e the triplet even tensor force repulsive; slight changes were also necessary in the quadratic spin-orbit force. T h e role the quadratic spin-orbit force plays is interesting. F r o m figs. 3 and 5 it is seen that VQL acts to shift the K = 2 b a n d of 2:Ne u p w a r d and (since the I + of 2~Ne is d o m i n a n t l y (62)) it also separates different SU(3) representations. Further, it was seen that VQL tends to i m p r o v e the goodness of SU(3). Each of the latter two TABLE 6
Percentage of the leading SU(3) representations present in eigenstates of the phenomenological Hamiltonians with the parameters of table 5, columns two and four Nucleus
Rep.
Spin
C
C + NC
2tNe
(81)
~+ 2~+ 25+
73.2 25.2 68.4
89.2 9.5 83.4
22Ne
(82)
0+ 2+ 2~ 4+
60.0 65.4 60.6 66.5
73.3 74.4 58.7 68.4
22Na
(82)
3+ I1 1~2+
71.6 51.2 50.7 51.2
86.3 58.8 47.6 57.2
218
N. ANYAS-WEISS A N D D. S T R O T T M A N
characteristics is not unlike a quadrupole-quadrupole force; this perhaps should be anticipated from the definition of L12 of eq. (6): the first term is proportional to L(L + 1) as is the quadrupole-quadrupole interaction. Thus, the importance of the quadratic spin-orbit interaction may be suggestive that a longer range interaction should be included into the interaction. This possibility has already been investigated in 22Ne [ref. 18)]. A possible source of worry is that the truncation scheme which was based on SU(4) and SU(3) might break down in the presence of the strong non-central forces, with the result that the basis used in the work is inadequate. Although a definitive answer is impossible without diagonalizing the Hamiltonian within a complete basis, there are two items which suggest that the truncation scheme is still valid. An essential requirement for the validity of SU(3) in that the ls level lies below the 0d centroid. This is true for the single-particle energies of ~70 and if the singleparticle energies are allowed to vary, it remains true. Further, the variation of the one-body spin-orbit strength suggests a value smaller than that indicated by the l vO spectra which would help restore the SU(4) symmetry. As another check, one may simply examine the wave functions in the two cases of only central and central plus non-central interaction to note the degree of SU(3) and SU(4) breaking. In table 5 are listed the percentage of the leading SU(3) symmetry for the lowest few states. Surprisingly, when all the non-central interactions are included, SU(3) is a better truncation scheme contrary to common belief which would have SU(3) a less valid symmetry in the presence of non-central interactions. The simplest explanation is that the central interaction produces a too compact spectrum; the non-central interactions - particularly the quadratic spin-orbit interaction tend to expand the spectrum thus making large admixtures of lower SU(3) representations more difficult. Indeed, in the case of only central plus tensor, the amount of SU(3) breaking is somewhat greater than for the pure central interaction. This is opposite to the effect found by Manakos and collaborators 20) who investigated the effects of those two-body matrix elements which do not conserve S. One may note the anomalously small amount of (81) for the ½+ level of 21Ne. This is simply explained: the wave function is dominated by (62) L = 0 which is favoured by the attractive nucleon-nucleon interaction over the L = 1 state of the (81) representation. A similar situation occurs in 23Mg. With the possible exception of some states of 2°F, nuclei with fewer nucleons than six are not particularly sensitive to the tensor force, and therefore a repulsive triplet even tensor interaction may not cause undue difficulty. The interesting question remains, why should the effective triplet even tensor interaction be negative? Possible origins may lie in the convergence problems in calculating G as already mentioned, or the presence of three-body forces. The importance of three-body forces increases as (g), n = A - 16, and are thus more important in mass 22 than mass 19 and 20. Further, as much of the contribution to the three-body interaction arises from Pauli corrections, it may be anticipated that they are repulsive and further,
NON-CENTRAL INTERACTIONS
219
their effects must show up first in the even interaction. Another possibility is from the so-called Vary-Sauer-Wong effect: the contributions to G3p~h from intermediate states of up to 20h~o excitation are found to be repulsive and large 2 ~. 22). They arise primarily from the tensor interaction and could therefore explain some of the needed repulsion. Other investigations 15.2o) have been conducted to obtain an interaction which successfully reproduces the s-d spectra. One of the more successful is that of Preedom and Wildenthal 23) who varied several of the 2BME which, although l;roviding little physical insight into the interaction, did produce an interaction which fits levels remarkably well. Although an exhaustive investigation, such as that of sect. 3, into the nature of the effective interaction was not made, an examination of the transformed two-body matrix elements does provide some information. The matrix elements of the Preedom-Wildenthal interaction which violate S are larger than those of Kuo; this may be an effect of the fitting procedure used in ref. 23) in that matrix elements containing 0d~ particles were varied, but not those with 0d~, this producing essentially a non-local interaction. The effects of such matrix elements were investigated by the Darmstadt group 15, 2 0 ) and were not found to be essential. The authors wish to thank Sir Denys Wilkinson and Dr. Peter Hodgson for the hospitality of the Nuclear Physics Laboratory, Oxford where much of this work was done. Thanks also are extended to members of the experimental groups of Oxford, particularly Dave Rogers, for their ceaseless inquiring into the accuracy of the parameters of the levels and to Prof. Manakos and other members of the Darmstadt group for numerous discussions concerning their exhaustive calculations.
Appendix We briefly discuss here the fitting procedure and certain results which are very useful in the interpretation of the parameters resulting from the fit. The Hamiltonian is defined here as M
H = ~hH
+Ho,
(A.1)
~=1
where the ha are the parameters to be determined and H o is that part of the Hamiltonian left unchanged. The parameters {hi} are found by varying the quantity m
c~~ = ~ ( E I - - ( H ) , ) z i=1 m
M
= ~ ( E , -
a=l
(A.2)
220
N. ANYAS-WEISS AND D. STROTTMAN
with respect to the h,. In eq. (A.2) the E i are the experimental energies, m is the number of levels and N = m - M is the number of degrees of freedom. The hessian matrix is the M x M matrix consisting of second derivatives of 0.2. ('320.2
m
(A.3) Let K be the inverse of G; then an unbiased estimate of the variance ofg~ is 0-2
var,q~=~K
.
(A.4)
In table 5 the values of var.G are given in parentheses. References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12)
J. P. Elliott and B. H. Flowers, Proc. Roy. Soc. A229 (1955) 536 T. Inoue, T. Sebe, H. Hagiware and A. Arima, Nucl. Phys. 59 (1964) 1; 85 (1966) 184 Y. Akiyama, A. Arima and T. Sebe, Nucl. Phys. A138 (1969) 273 T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 T. T. S. Kuo, Nucl. Phys. AI03 (1967) 71 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 D. Strottman, Particles and Nuclei 5 (1973) A. Arima, S. Cohen, R. Lawson and M. Macfarlane, Nucl. Phys. AI08 (1968) 94 J. P. Elliott, H. A. Mavromatis and E. A. Sanderson, Nucl. Phys. AI21 (1968) 241 M. E. Fosada and K. B. Wolf, Rev. Mex. Fis. 14 (1965) 29 T. lnoue, T. Sebe, K. K. Huang and A. Arima, Nucl. Phys. 99 (1967) 365 E. C. Halbert, J. B. McGrory, B. H. Wildenthal and S. P. Pandya, in Advances in nuclear physics, vol. 4, ed, M. Baranger and E, Vogt (Plenum, NY, 1971) 13) B. R. Barrett and M. W. Kirson, Phys. Lett. 30B (1969) 8 14) R. R. Whitehead, A. Watt, B. J. Cole and 1. Morrison, Advances in nuclear physcis, vol. 9 (Plenum, NY, 1977) p. 123, and references therein 15) R. Manakos, M. Conze and H. Feldmeier, private communication 16) M. MacFarlane, in Proc. Int. School of Physics Enrico Fermi, course 40, Varenna 1969, ed. M. Jean (Academic Press, NY, 1969) p. 457 17) D. Kurath, Phys. Rev. 130 (1963) 1525 18) C. Abulaffio, Nucl. Phys. A144 (1970) 225 19) A. Arima and D. Strottman, Nucl. Phys. A162 (1971) 605 20) M. Conze, H. Feldmeier and P. Manakos, Phys. Lett. 43B (1973) 101; H. Feldmeier, Dissertation, Technische Hochschule Darmstadt, 1974 21) J. P. Vary, P. U. Sauer and C. W. Wong, Phys. Rev. C7 (1973) 1776 22) C. L. Kung, private communication 23) B. M. Preedom and B. H. Wildenthal, Phys. Rev. C6 (1972) 1632