Constant width approximations for large spectroscopic calculations

1.D.1

I I

Nuclear Physics A246 (1975) 29--42; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

CONSTANT WIDTH APPROXIMATIONS FOR LARGE SPECTROSCOPIC CALCULATIONS (3. D. L O U G H E E D and S. S. M. W O N G

Department of Physics, University of Toronto, Toronto, Ontario, Canada t Received 28 January 1975 Abstract: The widths which along with the centroids define the strength distributions for the subspaces of a given (mJT) space are found to be clustered within a narrow range of values. Approximations to the fixed-JT averaging spectral distribution method based on this factor are suggested and their accuracies discussed. Using these approximations, one can now perform nuclear structure calculations in a large model space with good accuracy and at a relatively low cost.

1. Introduction

It has been observed for some time that the level densities obtained in shell model calculations are well approximated by a Gaussian distribution for a sufficiently complicated model space. Subsequently, studies made with ensembles of random matrices 1, 2) established that the distribution of spectral strength converges rapidly to a Gaussian shape as the number of active nucleons m increases beyond two. The random matrix level densities are not only normal on the average (i.e., the average for each of the moments is the value expected for a Gaussian distribituon), but also these moments have small variances about these expected values 1, 3). This implies that it is, in general, difficult to find a spectroscopic strength distribution that deviates to any extent from a Gaussian form. Indeed, those distributions determined using realistic model interactions are all found to be nearly Gaussian. Since departures from normality are small, any corrections can be included using, for example, a Gram-Charlier series 4) involving up to the fourth moment. The fact that only a few moments are sufficient to describe the strength distribution for a given model space implies that accurate approximations for level densities and other spectroscopic quantities can be obtained in enormous vector spaces with relatively simple calculations. For example, the first four moments which define the scalar strength of m active nucleons can be expressed as combinations of the single particle energies and the two-body matrix elements and their products 5). These low order moments are usually represented by the centroid, width, skewness and excess, and since they are given by simple expressions involving the Hamiltonian and the labels of the particular model space, it is not surprising that their values vary slowly and smoothly as functions of the labels. This behavior of the strength moments has been Work supported in part by the National Research Council of Canada. 29

30

G.D. LOUGHEED AND S. S. M. WONG

confirmed by exact shell model calculations, and in fact, the centroids and widths for a set of model spaces with the same m- and T-values but differing in J-values can be fitted to within a few percent error using a polynomial expression which is quadratic in j 2 t. Ginocchio et al. 6) have used this property to deduce approximate Gaussian level densities for given (mJT) spaces. In this paper, we are mainly concerned with the simple behavior of the widths which along with the centroids describe the strength distributions for the subspaces of a given (mJT) space. Since these subspaces are chosen according to the irreducible representations of groups whose Casimir operators do not, in general, commute with the Hamiltonian, they are therefore not necessarily labeled by good quantum numbers. However, in the eigenvalue representation, these subspace distributions supply the detailed structure of the (mJT) space and can therefore be used to deduce the observables which describe the system. The particular subspaces we are interested in, as well as the behavior of their distribution moments, are described in sect. 2. Two methods for approximating the values of the subspace widths based on the particular behavior of this quantity are suggested. These methods, which we refer to as the root-mean-square and the random sampling constant width approximations, along with the accuracies with which they are able to reproduce the exact fixed-JT averaging level densities and level positions, are outlined in sects. 3 and 4 iespectively. In sect. 5, we conclude with some remarks concerning the implications of these approximations with respect to calculations involving large spectroscopic spaces.

2. Moments of the fixed-JT subspaees In spectroscopy, the most useful strength distributions are those for spaces with a definite number of active nucleons m and spin-isospin JT. Such a (mJT) space can be subdivided into subspaces according to the partition {rn~} of the m nucleons into the active orbits, the spin-isospin coupling ~ of the rn~ nucleons in the ith orbit and the spin-isospin coupling (Fi} between the orbits. We shall refer to such a subspace as a (m~,r) subspace 7, s). The dimensionality of each (mvF) subspace D s is usually veIy small since each of them contains states which, in the jj coupling language, differ only in the seniority, reduced isospin and ~edundancy labels for each orbit. If the strength distribution p.f(E) for t h e f t h subspace is normalised to Dy, the strength distribution p,,,sr(E) for the (mJT) space is given by

p=j,(e) = E$ pAE). Taking the zero of the energy scale at the

(mJT) centroid

(t) C, the moments Mu which

t The skewnesses and excesses have a more complicated behavior and are small in value.

CONSTANT WIDTH APPROXIMATIONS

31

describe p,~r(E) are related to the subspace central moments M~s by Df

os E

Mrs cs ,

v=0

(2)

where D = ~sDs, (~) is the binomial coetiicient and Cs is the centroid of Ps(E). Since the subspace distributions are expected to be Gaussian on the average, it is convenient to make the approximation that they are exactly Gaussian and thus completely specified by their centroids Cs and widths trs(M2s = tr}) in addition to their dimensionalities D s. This is the basic assumption made in the fixed-JT averaging method and this method is found to generate reasonably accurate predictions for level densities and level positions 7). The distribution Projr(E) generated by

TABLE 1

Quantities which give the properties of the distribution o f the ( m y 1") subspace widths for several sd shell cases a n d a psd shell example (~60, J = 0, T = 0, 4p-4h) m

6 6 6 6 6 6 6 6 7 8 8 9 11 12 12 12 12 12 12 12 12 12 12 12 160

J

0 1 2 3 4 5 6 7 ~ 0 2 ~ ~ 0 1 2 3 4 5 6 7 8 9 10 0

T

0 0 0 0 0 0 0 0 ~r 0 0 ½ ½ 0 0 0 0 0 0 0 0 0 0 0 0

D

71 243 307 366 311 259 169 107 300 325 1206 315 290 839 2135 3276 3711 3793 3278 2667 1848 •205 657 334 2337

Number of (roT/')

tr(m,J,T)

~.t

(MeV)

(MeV)

56 183 230 265 230 194 136 90 245 186 672 265 250 337 856 1278 1451 1517 1413 1266 1004 740 465 261 2125

11.32 11.68 10.92 10.91 10.04 9.63 8.70

7.86 8.23 7.77 7.74 7.16 6.91 6.20 5.81 6.20 10.15 9.63 6.18 6.00 11.39 10,89 10.87 10.56 10.33 9.86 9.50 8.87 8.32 7.54 6.86 15.26

8.06 8.59 14.26 13.27 8.91 9.27 15.65 14.88 14.92 14.55 14.34

13.80 13.48 12.76 12.21 I 1.32 10.59 15.73

tTrms (MeV)

Variance of a s (MeV)

trmt, (MeV)

8.04 8.32 7.84 7.81 7.23 6.99 6.27 5.87 6.24 10.22 9.67 6.22 6.05 11.43 10.92 10.90 10.59 10.36 9.88 9.52 8.89 8.36 7.57 6.90 15.32

•.68 1.17 1.09 1.07 1.03 1,04 0.93 0,86 0.74 1.26 0.94 0.70 0.75 0.98 0.74 0.75 0.70 0.71 0.70 0.70 0.71 0,74 0.70 0.80 1.36

4.75 5.30 4.96 4.78 4.58 4.41 4.23 4.03 4.55 6,48 5.90 4.38 3.99 8.32 7.88 8.16 8.05 7.21 5.49 6.30 4.40 4.23 4.80 4.53 11.41

trm,x (MeV) 10.72 13.59 11.59 11.34 10.19 10.19 8.93 7.79 7.97 15.73 13.04 7.98 7.42 16.33 13.31 13.61 13.56 14.16 13.67 12.24 10.70 10.24 9.88 9.15 20.85

All the sd shell examples are calculated using a renormalized version o f the Lee-Scott-Kahana interaction xl) with the 170 single particle energies exa½ = 0,00 MeV, elak = 5.10 MeV, and e2,½ = 0.87 MeV. T h e 160 e x a m p l e is calculated using a Rosenfeld interaction with zero single particle energies.

32

G . D . L O U G H E E D AND S. $. M. WONG

t h e f i x e d - J T a v e r a g i n g m e t h o d is n o t e x a c t l y G a u s s i a n a n d its s k e w n e s s ?t a n d excess 72 c a n b e w r i t t e n explicitly u s i n g eq. (2) as

1 Z

(3)

Yx = Da---~ y 1

72

= -XZ-, ~V Ds(3a s4+ 6¢rsz Cs2 + C~) - 3, Da s

(4)

where

__

= 1 2

(5)

DI T h e v a l u e s o f 71 a n d 72 f o r t h e c a s e s l i s t e d in t a b l e 1 are t a b u l a t e d in t a b l e 2. F o r s p e c t r a l d i s t r i b u t i o n m e t h o d s , in g e n e r a l , t h e s u b d i v i s i o n o f a s p a c e i n t o a number

o f subspaces has several advantages. The

i n a c c u r a c i e s in t h e p r e d i c t e d

s t r e n g t h d i s t r i b u t i o n as a r e s u l t o f t h e G a u s s i a n a p p r o x i m a t i o n a r e u s u a l l y m o r e TABLE 2

The low order moments for various (mJT) cases; and the skewnesses and excesses produced by the fixed-JT averaging method and the rms constant width approximation m

6 6 6 6 6 6 6 6 7 8 8 9 11 12 12 12 12 12 12 12 12 12 12 12 x60

J

0 1 2 3 4 5 6 7 4} 0 2 4} 22_A 0 1 2 3 4 5 6 7 8 9 10 0

T

0 0 0 0 0 0 0 0 ½ 0 0 ½ ½ 0 0 0 0 0 0 0 0 0 0 0 0

D

71 243 307 366 311 259 169 107 300 325 1206 315 290 839 2135 3276 3711 3793 3278 2667 1848 1205 657 334 2337

Number of

(mTF)

C (MeV)

56 183 230 265 230 194 136 90 245 186 672 265 250 337 856 1278 1451 1517 1413 1266 1004 740 465 261 2125

--4.26 --5.14 --5.30 --5.86 --6.40 --7.26 -- 8.02 --9.06 --12.75 --14.91 --15.16 --25.34 --40.59 --46.12 --45.89 --46.15 --46.24 --46,56 --46.82 --47.21 --47.62 --48.19 --48.71 --49.44 --75.95

Fixed-JT averaging a 7~ (MeV) 11.32 11.68 10.92 10.91 10.04 9.63 8.70 8.06 8.59 14.26 13.27 8.91 9.27 15.65 14.88 14.92 14.55 14.34 13.80 13.48 12.76 12.21 11.32 10.59 15.73

0.34 0.21 0.17 0.19 0.15 0.12 0.15 0.09 0.13 0.26 0.21 0.13 0.16 0,21 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.17 0,17 0.15 0.02

y2

rms approximation

Yl 0.47 0.19 0.10 0.17 0.08 0.08 0.06 0,01 0.03 0.40 0.17 0.05 0.06 0.27 0.ll 0.13 0.13 0,12 0.09 0.12 0.08 0.07 0.05 0.04 0.10

0.32 0.26 0.22 0.25 0,21 0.16 0,19 0.14 0.15 0.29 0.24 0.14 0,15 0.21 0.20 0.19 0.19 0.19 0.18 0.18 0.17 0.16 0.15 0.15 0.00

Yz 0.21 0.13 0.02 0.09 0.00 --0.03 0.00 --0.09 --0.05 0,31 0.13 --0,05 --0,06 0.17 0.09 0.09 0.09 0.08 0.05 0.06 0.03 0.01 --0.02 --0.05 0.00

NUMBER

CF" W I D " ~ , S

NUMBER .

0

.

~_d__

I

o

i

-'I~"

I

~

p

1

p

"i

OF W I D T H S

.

!

i

8

1

]

--1

i

o

I

'l t'-~

-

'

t I

~

~E ~°

~

.!

"

-R

I

"

.

i

I

I

I

I 1

J

I

I

!

cL

¢._ o

¢.

o 0

I o

I

NUMBER o

;~1~

.o

I

I

I

I

I

OF" W I C T H S

NUMBER

o

p

9

9

oo

o

o

. . . .

I OF WIDTHS

g I

~,~

I

~a

I

3 ~

I

,

",,,

t~

, "

-t

;1

F t"

I --1

I

I

I I

I

1

-~I

I

I I

]

!

', I

~ .=[_ ~_.

I

e...

,t

;3 o

-4

-1

~,

z

I

,

I

P

I

f

I

.I

.

o

ff

S N O I I V / A / I X O ~ I d d V HZGIA~ J.NVJ~SNOS)

I

I

I

34

G . D . LOUGHEED AND S. S. M. WONG

severe the further away from the centroid one must go. There is, of course, no interest in the portion of the distribution where the integrated strength is much less than unity. Obviously, for large dimensionalities, such a region is much further away from the centroid than for small dimensionalities. The first advantage of subdividing the space, therefore, arises from the smaller dimensionality of each subspace. Furthermore, the summation over Ps(E) in eq. (1) tends to average out any non-systematic errors introduced in these distributions. There is also an increase in the information content of Proj r ( E ) as a result of the subdivision of the model space. At any energy position E, the composition of proj r ( E ) in terms of the subspace distributions can be determined and this composition allows one to deduce spectroscopic quantities other than level densities and level positions. The decomposition of a (mJT) space into (m~F)subspaces, in the fixed-JTaveraging method, also enables one to evaluate the angular momentum recouplings exactly. The subspace centroid C s is a simple quantity to calculate since it involves only the single particle energies and the diagonal two-body operator portion of the Hamiltonian. The subspace width a s, on the other hand, involves the square of the Hamiltonian operators, and therefore its evaluation is one of the most time consuming parts of the fixed-JTaveraging and other spectral distribution calculations. However, as is shown in fig. 1, the values o f a s for a given (mJT)space are clustered in a narrow region with a somewhat Gaussian-like distribution. This property seems to be common for all subspace widths and is not necessarily limited to fixed-JT spaces 9). The reasons for such clustering have not been fully explored yet: we believe it is related to the smooth behavior, mentioned earlier, of the moments of strength distributions for complex vector spaces consisting of several active orbits. If the range of the a s values is sufficiently narrow, they can be approximated by O'f ~ O'rms,

(6)

where arm, is the root-mean-square (rms) average of the subspace widths. This is the rms constant width approximation to be discussed in sect. 3. On the other hand, since there is no apparent systematic trend in the a s values, it would seem that the evaluation of a randomly selected sample of these widths will be sufficient, as the values for the remainder of the subspace widths can be approximated by that of a nearby sample member. This is the random sampling constant width approximation to be discussed in sect. 4. Note that for both approximation methods the subspace centroids C s are calculated exactly and that the subspace skewness ~ls and excess ~2s are taken to be zero as in the fixed-JT averaging method.

3. The rms constant width approximation

If the range of as values for a given (mJT) space is sufficiently small, they can be approximated by a constant width. We choose the rms average of the subspace

CONSTANT

WIDTH APPROXIMATIONS

35

widths o-rms as this quantity and since

2

O'rms =

a2 _ _1Z Dy C}, D

(7)

f

the width o- of the (mJT) space is preserved. Since, as mentioned previously, the subspace centroids Cf and hence the centroid C for the (mJT) space are evaluated exactly, the errors introduced by the rms approximation can be related to the spread of the ~ : values listed in table 1. From eqs. (3) and (4), it can be easily seen that these errors are of order h 0 " f ~- -o-f--O'rms • O'rms

(8)

O'rms

Table 2 compares the values of Yl and Y2 produced by the rms approximation with those generated by the fixed-tiT averaging method to give an idea of the effects of the constant width substitution on the low-lying distribution moments t. The density difference is given by 6Prms(E)

=

PmdT(E)-Prms(E)=

~ f

[Pf(E)-Pfrrns(E)],

py(E) and PSrms(E)are Gaussian functions defined by D:, C: and Dr, Cf and %ms, respectively. The density difference can be written as

where

6p,m~(E) = ~p:(E)

1-

f

e:-exp O'rms

(E-C:) 2 ~r;ms-~:ll

(~rm,O-:)~-I/'

(9) o-f, and by

(10)

to show the explicit dependence on the actual and the approximated widths o-: and a,m~. Thus, if Act:(=- o-:-¢rr,,~) is small for all subspaces, 6prm~(E) is also small. This illustrates the point that the density distribution is approximated accurately by the rms approximation if the variance of the individual a f widths is small. Fig. 2 shows proj r ( E ) and 6Prms(E ) for some typical cases. Note that in this figure the scale for 6Prms(E)is reduced by a factor of ten with respect to that of proj r ( E ) in order that the small level density differences become visible. The discrete level positions E~ can be deduced from a level density distribution by the following rule 1o)

fe, p,,~r(E)dE =

i-½,

for i = 1 to O.

(11)

--oO

We compare the rms approximation and fixed-JT averaging methods in terms of the low-lying level positions, since this is the region where they are of the most interest. The results are listed in table 3. * F o r this work, we used the exact centroids C a n d widths ~r obtained previously with the fixed-JT averaging m e t h o d : however, since the m a i n p u r p o s e o f the r m s a p p r o x i m a t i o n is to simplify this m e t h o d , a n o t h e r w a y to estimate a is required in realistic applications.

~

g

,

I

I

I

-]'°£~.~ --L

0.00 ~

1.00

0.00-55.00

iO.O(

20.00

30.0C

40,0C

~

-~9to.o

I '

-39.00

[

1

-23.00

1

/

I

t

I

I_

I

-7.00

I

-7.00 ENERGY IN MeV

l__i - 2 3 . 0 0

I "

I

I

I

I

t

9,00

I

9.00

I

I

I

I

25.00

25.00

Fig. 2a.

£

9

I

....

I

I

_4.O%o.oo..E~ -72.00 I

o.oc

4.0C

80.0¢

120.0C

160.OG

I.

1

t

I

I

I

I -36.00

I

ENERGY IN MeV

f--54,00

I

t

I

'1

(sd) 12

.36100

T----- t

54!00 I

I

-18.00

1

I

J=2

I

T=O

I

I

I

000

0.00

o

>

rn rn

t~ o

P

L

-8300

I

I

I

....

I

"

.J

{60,0C

I

, I ~103.00

1

-lO&O0

I

I ~'~--T"--

f I I -83.00 -6&O0 ENERGY IN MeV

I---

-63.00

I

I -4:5.00

I

-43.00

[

-23.00

-23.00

-4.0¢

O,OC

4.0C

~.00

L

O.OC ~89,00

40.0£

i

I

I

t

-7~.00

~----I

-71.00

J

I

I

I

I

I

[

I

L

-35.00

I ....

- 53.00 - 55.00 ENERGY IN MeV

I

-5.3.00

I

"I

-17.00

I

T

I

T=O

-17,00

tsd) Iz J=4

I

I

r

"

1,00

1.00

Fig.2b. Fig. 2. Partial level densities p,,~T(E) determined by the fixed-JT averaging method and the error, 6p,m~(E) = p,,jr(E)--p~mJE), caused by the rms constant width approximation. Note that the scale for 6prm~ is ~ of the level density scale.

-L6(3 l -i2&O0

0.0(3 ~

1.6C- -

-123.00

0.00

16.OO"

80.0(3

i

32.00

I-----T----F

t20.O0

I

48,00

64.00

Z

0

0 X

Z

>

,..]

O

G, D. LOUGHEED AND S. S. M. W O N G

38

15

15 -42.3 -42.9 --43.5 ~-44.4 ......

-45.1 45.6 -46.2

-10

51 -47.3 -47.9 -49.5

-49.6 hl

-5

-55.1 - s-W/~-56.8

FJT

RMS

S/~vIPLE5

S~E

10

0

Fig. 3. A comparison of the lowest 20 level positions for the 2¢Mg 2 + case calculated using the shell model (SM), fixed-//" averaging method (FJT), the rms constant width approximation (RMS), and the random sampling constant width approximation with sample sizes 135 (SAMPLE 5) and 68 (SAMPLE 10).

The lowest twenty levels for the 2 4 M g 2 + case determined using the rms approximation are plotted in fig. 3 and are compared with the results given by the shell model, the fixed-JT averaging method and the random sampling approximation which is described in the next section. In general, we find that, for the measures used here, the errors introduced by the rms approximation are small. In fact, they are usually less than the inherent errors of the fixed-JT averaging method 7). Furthermore, we find that the approximation improves withincreasing dimensionality and with increasing values of spin and isospin, although the evidence for the latter is somewhat incomplete. The main reason for this can be attributed to the sharper distribution of the values of a I with increasing dimensionality and spin-isospin as can be seen from table 1. 4. Random sampling constant width approximation If N, the number of (re?F) subspaces in a given (rnJT) space, is large, a reasonable representation of the closely bunched distribution of subspace widths shown in fig. 1 may be given by a sample of k randomly selected widths. Let the (m~,F) subspaces be ordered 1, 2 , . . . , N, then the k members of the sample can be chosen by selecting those subspaces at N/k intervals and the exact widths for these subspaces

CONSTANT

WIDTH APPROXIMATIONS

39

TABLE 3 C o m p a r i s o n o f the lowest three level positions p r o d u c e d by the fixed-JT averaging m e t h o d a n d the r m s c o n s t a n t width a p p r o x i m a t i o n with ZIE~ = E~(JT)--El(rms) m

J

T

D

Number of

(myr') 6 6 6 6 6 6 6 6 7 8 8 9 11 12 12 12 12 12 12 12 12 12 12 12 160

0 1 2 3 4 5 6 7 ~ 0 2 ~ 2.2_1 0 1 2 3 4 5 6 7 8 9 10 0

0 0 0 0 0 0 0 0 ½ 0 0 ½ ½ 0 0 0 0 0 0 0 0 0 0 0 0

71 243 307 366 311 259 169 107 300 325 1206 315 290 839 2135 3276 3711 3793 3278 2667 1848 1205 657 334 2378

56 183 230 265 230 194 136 90 245 186 672 265 250 337 856 1278 1451 1517 1413 1266 1004 740 465 261 2166

Spectrum span (MeV) 58.04 68.74 65.13 61.66 59.85 56.21 48.25 40.70 50.47 90.62 92.17 52.93 54.40 90.04 96.08 100.66 98.10 97.70 92.24 89.38 81.88 76.46 67.08 59.14 113.7

G r o u n d state fixed-JT rms (MeV) (MeV) --30.03 --36.46 --35.56 --36.69 --34.42 --34.04 --30.84 --29.41 --36.79 --55.11 --56.54 --50.15 --65.85 --95.04 --93.93 --96.48 --95.29 --95.41 --92.94 --91.90 --88.56 --86.42 --82.25 --79.01 --132.8

--29.43 --35.57 --34.40 --35.43 --33.35 --33.16 --30.16 --28.64 --36.15 --53.41 --55.40 --49.79 --65.65 --93.06 --93.48 --95.47 --94.55 --94.41 --92.37 --91.04 --88.00 --85.92 --81.77 --78.47 --131.3

AEx (MeV) 0.60 0.89 1.16 1.26 1.07 0.88 0.68 0.75 0.64 1.70 1.14 0.36 0.20 1.98 0.45 1.01 0.84 1.00 0.57 0.86 0.56 0.50 0.48 0.44 1.5

ZIE2 (MeV)

,dEn (MeV)

0.21 0.67 0.86 0.98 0.80 0.72 0.44 0.45 0.43 1.13 0.88 0.27 0.20 1.26 0.37 0.80 0.55 0.72 0.37 0.63 0.35 0.39 0.30 0.39 0.9

0.05 0.60 0.73 0.82 0.69 0.63 0.34 0.31 0.36 0.91 0.80 0.23 0.20 0.97 0.29 0.66 0.49 0.65 0.39 0.57 0.34 0.34 0.23 0.32 0.8

evaluated using the fixed-JT averaging program. For the other (N-k) subspaces, only the centroids C/are calculated exactly and the values for the widths are taken to be that of the nearest previous sample member. Since, for a space with several active orbits, the number of (m~F)subspaces with a given partition of active nucleons m (or m~,) varies in a sufficiently complicated way, a reasonable randomness is achieved in this sampling method. The approximation with several different sample sizes k and the fixed-JT averaging method are compared in table 4 for a few of the larger examples using the standard measures introduced in the previous sections. The rms approximation results for these examples have also been listed to give a comparison between the two approximation methods. The 2 + states for 24Mg considered as (sd) s and using a KSL interaction ~l) are plotted in fig. 3 for two different sample sizes (N/k = 5 and N/k = 10). The differences in the results obtained using the different sample sizes are small and the agreement of these results with the exact fixed-JT averaging levels is excellent.

G. D. L O U G H E E D A N D S. S. M. W O N G

40

TABLE 4 Comparison of low order fixed-JT averaging and rms approximation central moments, as well as their ground state energies with those given by the random sampling approximation using several different sample sizes k N/k

(sd) 12, J = 0, T = 0

(sd) 12, J = 2, T = 0

(sd) ~2, J := 3, T = 0

~60, J = 0, T = 0 [(psd) ~2, 4p-4h]

k

Width (MeV)

~'t

~'2

Ground state (MeV) error (MeV)

fixed JT 3 5 12 20 rms

337 112 67 28 16 1

15.65 15.58 15.71 15.67 15.40 15.65

0.21 0.22 0.19 0.24 0.24 0.21

0.27 0.22 0.33 0.22 0.20 0.17

--95.04 --93.33 --97.06 --93.12 --92.33 --93.06

1.71 --2.02 1.92 2.71 1.98

fixed JT 3 5 8 12 20 rms

1278 426 255 159 106 63 1

14.92 14.90 14.93 14.59 14.93 15.02 14.92

0.19 0.20 0.21 0.21 0.20 0.20 0.19

0.13 0.12 0.14 0.10 0.12 0.05 0.09

--96.48 --95.83 --95.58 --95.57 --95.91 --95.44 --95.47

0.35 0.90 0.91 0.57 1.04 1.01

fixed JT 3 5 8 12 16 20 rms

1451 483 290 181 120 90 72 1

14.55 14.57 14.58 14.57 14.55 14.60 14.58 14.55

0.19 0.20 0.21 0.23 0.21 0.24 0.22 0.19

0.13 0.12 0.13 0.14 0.15 0.14 0.13 0.09

--95.29 --94.96 --95.23 --94.85 --95.19 --94.68 --94.88 --94.55

0.33 0.06 0.44 0.10 0.61 0.41 0.84

fixed JT 3 5 l0 12 16 20 rms

2125 708 425 213 177 133 106 1

15.73 15.75 15.71 15.74 15.72 15.66 15.81 15.73

0.02 0.02 0.01 0.00 0.00 --0.01 --0.01 0.00

0.10 0.10 0.09 0.09 0.11 0.08 0.10 0.00

--132.8 --132.9 --132.8 --133.3 --133.5 --132.8 --133.9 --131.3

0.1 0.0 0.5 --0.7 0.0 --1.1 1.5

Since the r a n d o m s a m p l i n g a p p r o x i m a t i o n d o e s i n t r o d u c e s o m e s p r e a d i n t h e a p p r o x i m a t i o n o f try, w e w o u l d e x p e c t this m e t h o d t o p r o d u c e m o r e a c c u r a t e results t h a n t h o s e g i v e n b y t h e r m s a p p r o x i m a t i o n . O n t h e o t h e r h a n d , t h e e x a c t t o t a l w i d t h is n o t n e c e s s a r i l y g i v e n c o r r e c t l y b y this a p p r o x i m a t i o n . S i n c e t h e w i d t h f o r m s t h e scale o f a n y G a u s s i a n - l i k e d i s t r i b u t i o n , a n y i n a c c u r a c y in t h e e s t i m a t i o n o f its v a l u e r e p r e sents a s e r i o u s p r o b l e m . U n l e s s t h e s a m p l e o f a s is a " t r u e " r e p r e s e n t a t i o n o f all t h e a s , t h e e r r o r s will b e large. W e find t h a t in g e n e r a l , i f k is o f t h e o r d e r o f 100 o r m o r e , t h e a c c u r a c y is sufficient. H e n c e f o r a l a r g e case w i t h N i n t h e o r d e r o f t h o u s a n d s , t h e r a n d o m s a m p l i n g m e t h o d b e c o m e s v e r y a t t r a c t i v e as t h e r e w o u l d b e a c o m p u t a tional saving of order kiN.

CONSTANT WIDTH APPROXIMATIONS

41

5. Conclusion

The two approximations for the fixed-JT averaging method presented here are complimentary in their usefulness. If the (mJT) width a is known with sufficient accuracy, the rms approximation provides a simple way to approximate the subspace widths and to incorporate the advantages of subdividing the space into the calculation. However, in our tests of this approximation method, we have used the exact width cr obtained previously by the fixed-JT averaging method, whereas, in any real application, o- must be obtained by some simpler procedure in order for the approximation method to be of any real value. For example, if the (mJT) width is given exactly by a polynomial in jz, its value could be obtained using scalar-Taverages 5) of the products of the powers of j2 with H z. However, any inaccuracy in the prediction of # implies additional errors in the level density, especially in the tail regions. Therefore, any application of the rms approximation is limited until the behavior of a for a given (mJT) space is better understood. On the other hand, since a fixed-JT averaging computer code is available, a sample consisting of the order of a hundred subspace widths can be determined at a relatively low cost. We also find that, at least for the cases studied and the particular measures used, the random sampling method yields better accuracies. For both approximations, the subspace centroids C: are calculated exactly. A method to estimate these values may also exist, but they are such simple and inexpensive quantities to calculate that it does not seem worthwhile to sacrifice the accuracy. A Gaussian approximation is used for the subspace distribution p:(E) and as discussed in an earlier work 7), this is a good approximation. Using the random sampling method (or the rms method if a can be determined), spectloscopic quantities can now be determined using model spaces orders of magnitude larger than those used by the present shell model techniques. This is true due to the fact that only the number of subspace centroids C: to be evaluated increases linearly with the dimension, while the number of subspace widths 0-: remains essentially unchanged (one for the rms approximation and k ~ 100 for the random sampling approximation). The sd .~hell, as made clear by the examples cited in this work, forms simple applications of the constant width approximations. The kinds of applications we are more interested in involve active orbits spanning more than one major shell. The level densities produced in such large vector spaces can be meaningfully used in nuclear reaction studies involving compound nucleus formation 12). It also becomes possible to study problems such as the "goodness" of shell closure. More ambitiously, by treating a nucleus with different truncations of the vector space, one can hope to learn more about certain aspects of the renormalization of the nuclear Hamiltonian. It is a pleasure to thank Dr. J. B. French for providing the encouragement and for suggesting many of the ideas in this work.

42

G . D . L O U G H E E D A N D S. S. M. W O N G

References 1) J. B. French and S. S. M. Wong, Phys. Lett. 33B (1970) 447; 35B (1971) 5; O. Bohigas and J. Flores, Phys. Lett. 34B (1971) 261; 35B (1971) 338; S. S. M. Wong and J. B. French, Nuel. Phys. A198 (1972) 188 2) A. Gervois, Nucl. Phys. A184 (1972) 507 3) J. K. M o n and J. B. French, to be published and private communication 4) H. Cramer, Mathematical methods of statistics (Princeton University Press, Princeton, N.[, 1946) 5) F. S. Chang, J. B. French and T. H. Thio, Ann. of Phys. 66 (1971) 137 6) J. N. Ginocchio, Phys. Rev. Lett. 31 (1973) 1260; J. N. Ginocchio and M. M. Yen, Nucl. Phys. A239 (1975) 365 7) G. D. Lougheed and S. S. M. Wong, Nucl. Phys. A243 (1975) 215 8) J. B. French, P. Mugambi, S. K. M. Wong and S. S. M. Wong, to be published 9) F. S. Chang and J. B. French, in Statistical Properties of Nuclei, ed. J. B. Garg (Plenum Press, NY, 1972) 10) K. F. Ratcliff, Phys. Rev. C3 (1971) 117 11) S. Kahana, H. C. Lee and C. K. Scott, Phys. Rev. 185 (1969) 1378 12) S. Ayik and J. N. Ginocchio, Nucl. Phys. A234 (1974) 13