Numerical aspects of angular momentum projection for rotational nuclei

Nuclear Physics A321 (1979) 62-70; (~) North-Holland Publishing Co., Amsterdam

Not to be reproducedby photoprint or microfilmwithout written permissionfrom the publisher

NUMERICAL ASPECTS OF ANGULAR MOMENTUM FOR ROTATIONAL NUCLEI

PROJECTION

P. CHATTOPADHYAY and R. M. DRE1ZLER lnstitut fffr Theoretische Physik der Universitdt Frankfurt/Main, West German)"

Received 29 December 1978 Abstract: The question of complete and exact angular momentum projection from intrinsic Slater

determinant representations is studied for the example 162Er as a pilot case for the treatment of rotational nuclei. Adequate approximation schemes can be formulated.

1. Introduction

Recent criticism x) of the cranking approach to rotational states in nuclei has emphasised again the need for reasonably practical and accurate angular m o m e n t u m projection techniques 2- 4). In this note we investigate three aspects of this problem. (i) Supposing that an intrinsic state of a rotational nucleus is described as a Slater determinant composed of Nilsson orbitals, can exact projection of all angular m o m e n t u m components contained i n such a state be carried out with sufficient accuracy? (ii) H o w does the spectral decomposition of such a state vary with deformation ? (iii) Are there alternatives to exact numerical angular m o m e n t u m projection that are sufficiently economical while guaranteeing an acceptable standard of accuracy ? The vehicle for these investigations is the finite sum representation of the HillWheeler projector as given by Kelemen and Dreizler 5). Relevant details of the theory in juxta-position with other exact numerical schemes are given in sect. 2. As a test case we use the intrinsic ground state wave function of 162Er, which involves 24 neutrons and 28 protons outside an inert 140~170 lO7. core. Exact results are presented in sect. 3. In sect. 4 we address ourselves to the question of approximate projection, relying largely on previous work of Onishi and Sheline 6). The present work can be viewed as an extension, in spirit, of previous studies of L a m m e and Boeker 3) to a much m o r e involved situation. L a m m e and Boeker investigated the question of approximate angular m o m e n t u m projection for axially symmetric states of 8Be and 1'2C as well as asymmetric 4p-4h states in 160.

62

ANGULAR

MOMENTUM

PROJECTION

63

2. Projection technique For an axially symmetric, intrinsic state I~,,r), which is characterised by the z-component of the total angular momentum, the spectral weights of the angular momentum decomposition, Iff,,x) = ~ C~,KIu, JK),

(2.1)

J

can be calculated in terms of the Hill-Wheeler integral

IC~tKI2

J (~a, KIPKWI,, K)"

(2.2)

The basis of the spectral decomposition is characterised by Yz[0t, J K ) = J(J + 1)let, JK),

(2.3)

Jzl~, j K ) =/~1~, JK), and the projection operator is given as p s _ 2J +1 fdt2D~K(t2)./~(O), 8re2 d

(2.4)

/~(O) = exp ( - iaJz) exp ( - iflJy) exp ( - iyJz).

(2.5)

with

If we restrict our attention to the case K = 0 (for a discussion of the case K 4= 0 see ref. v)) the projector (2.4) simplifies to

f

P~ = (J + ½) +l d(cos fl)Pa(cos fl)/~(0, fl, 0).

(2.6)

In all practical cases the spectral decomposition terminates at a maximum value of the angular momentum, M

IqJ,,o)=

~

C~lct,J0).

(2.7)

J=O,2...

In this case one can evaluate each of the integrals (2.2) exactly using a suitable numerical integration formula 8) with n s = ~(M + J ) + 1 points, nJ

ICjI2 = Z

w.Ps(c°s/3.).

(2.8)

n=l

The quantities w, are the weights due to the particular quadrature scheme used. The maximum number of points is determined by the fact that the integrand is an even polynomial of degree (M + J) in x = cos 13. As a suitable integration scheme Gauss' formula a) offers itself. In this case (a similar statement can be made for any

64

P. C H A T T O P A D H Y A Y

AND

R. M. D R E I Z L E R

other, more sophisticated quadrature formula) the distribution of the points x, is given by the zeros of the Legendre polynomial P,j(x). In consequence, the expectation value of the rotation operator has to be recalculated for each value of J considered. For a complete and exact projection a total of (½M+ 1)(3M+ 1) points would be required. As the calculation of the expression <~,~.o[/~(fl)[~,o>, or in the more general case (~,,, ol0-g(/~)l~'=, o) (with 0 being a scalar operator), is the most time-consuming part of a projection programme, economy in this respect ought to be a major consideration. An alternative finite sum representation of the Hill-Wheeler integral was introduced by Kelemen and Dreizler 5) which avoids recalculation of the expectation value of the rotation operator for each J. The essence of the rather involved derivation can be summed up in the following: Given the representation (2.7) of the intrinsic state, a system of linear equations 4) for the spectral weights can be set up M

(g'.,01/~(/~.)lg'~,o) =

Z

Ifsl2Ps(cos~,,),

n = O, 1. . . . ½M.

(2.9)

J=0,2...

For a numerical solution an arbitrary distribution of points fin could be used. It was demonstrated in ref. s), that there exists a specific choice, viz. n~

fin - M + 1'

n = 0, 1. . . . ½M,

(2.10)

for which the system of linear equations (2.9) can be solved analytically in the form ½M

ICjI 2 = ~ A,,(M, J)(g'~,01/~(/~,)lg'=,o).

(2.11)

n=O

The coefficients A.(M, J) are given by the relations J

Ao(M, J) -

2J+l M+I

1-[ ( M - - J + 2k) k=l s I-I(M-J+2k+l)

(2.12a)

k=O

A,,(M, J) -

]Oj, o + 2 Ib(J, m) cos .,=2 ., e v e n

mn M

1

n =~ 0

(2.12b)

'

where ½J- 1

[-I [m2-(2k) 2] -2

Ib(J, m) =

k=0 -14

for Iml ~ J

1-I [m2-( 2k+ 1)2] k=O

0

(2.12c) for Iml < J.

A N G U L A R M O M E N T U M PROJECTION

65

These coefficients can be calculated in a straightforward manner. For each value of J the same set of (½M+ 1) points fin is used, leading to a considerable reduction in the time required for the calculation of the matrix elements needed for each projection.

3. Exact projection for 162Er We consider, for the purpose of illustration of the projection technique, an intrinsic ground state wave function for 162Er in the form ofa Slater determinant of 24 neutrons 4o~L7o core. The single particle orbitals are Nilsson and 28 protons outside an inert 1~07. orbitals (from the oscillator shells with N = 4,5 for protons and N = 5,6 for neutrons) corresponding to the "experimental" deformation r / = 6. The maximum angular momentum contained in this state is 122. For a Slater determinant 1 ~ , o ) = det {l~ol),ltp 2) . . . . Iq~A)} one obtains 2) for the expectation value of the rotation operator (0~, oL/~(/~n)I0~,o) = det {Nzu(fl.) },

(3.1)

Nau(fln) = (~ozl e x p ( - ifl,~r)ko,).

(3.2)

with

In the case of Nilsson orbitals with the representation 2

IqOak) = ~ a iklX~k),

(3.3)

J

Nau(fln) has the explicit expression N ZR,.,R2(fln ) = ~, aj; a~kfl~,k2(fln ).

(3.4)

J

The determinant (3.1) decomposes into blocks according to isospin and parity. Table 1 shows the results for exact projection, ICjl z as a function of J, obtainable in single precision arithmetic. For comparison we have calculated the spectral decomposition of the intrinsic state at other deformations (q = 4, 2, 0). The Slater determinants at these deformations are not necessarily the groundstate configurations based on the picture of the Nilsson model, but are obtained by filling the same orbitals as in the case q = 6. For J-values above 42 the spectral weights are smaller than 10-5 and are thus not reliable in single precision. Repetition of the calculation in double precision reproduced the results obtained below J = 42 in single precision and yielded the remainder of the spectral decomposition as indicated in fig. 1 in a plot of loglo ICs[ z versus J for r / = 2. It is of interest to note that the peak value of ICjI 2 is reached for J = 8 independent of deformation. As q is reduced from 6 to 0 the components below J = 12 are

P. C H A T T O P A D H Y A Y A N D R. M. D R E I Z L E R

66

TABLE 1

ICa 12 for various deformations

The spectral weights t/=0

t/=2

~/=4

t/=6

ICs 12

ICj 12

[Cj 12

ICj 12

0.01613 0.07683 0.12353 0.14941 0.15335 0.13935 0.11435 0.08567 0.05897 0.03745 0.02201 0.01199 0.00606 0.00284 0.00124 0.00050 0.00019 0.00007 0.00002

0.01479 0.07075 0.11483 0.14096 0.14763 0.13762 0.11649 0.09051 0.06496 0.04326 0.02680 0.01547 0.00833 0.00419 0.00197 0.00087 0.00036 0.00014 0.00005 0.00002 0.00001

0.01350 0.06480 0.10613 0.13216 0.14114 0.13488 0.11766 0.09470 0.07080 0.04936 0.03219 0.01967 0.01128 0.00608 0.00307 0.00146 0.00065 0.00028 0.00011 0.00004 0.00001

0.01239 0.05970 0.09854 0.12419 0.13485 0.13160 0.11776 0.09768 0.07560 0.05483 0.03737 0.02398 0.01451 0.00829 0.00447 0.00228 0.00110 0.00050 0.00022 0.00009 0.00003 0.00001

J

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 j

10.58

11.09

11.49

12.17

For r/ = (0, 2, 4) the Slater determinants are obtained by filling the same single particle orbitals as in the case t/ = 6. J ( J + 1) = (¢~,olj21~,.o).

I

i

I

I

i

I

-Z

-5

-6 -|0

I

O

20"

t

I

I

I

40

50

80

100

]

I

IZO

1~0

-

Fig. 1. loglolC J 12 as a function of J in the case t/ = 2 in a double precision calculation.

ANGULAR MOMENTUM

67

PROJECTION

increased at the cost of the components above that value. One observes a fairly small variation with deformation. The slow variation with deformation is also indicated by the average value J of the angular momentum for each state, which decreases from 12.17 to 10.58 as r/changes from 6 to 0. The total distribution of weights (which, in general, is not a point of major interest) shows a steady decrease from the peak value over seven orders of magnitude until J ,~ 50. F r o m this value on one observes a modest rise of about one order of magnitude to Jmax = 122.

4. Approximate projection For exact projection the determinant (3.1) has to be evaluated at 62 points in the case considered here. The time requirement on a Univac 1108 computer for this calculation amounts to approximately 60 min. The relatively smooth behaviour of <@~,ol/~(~)lg'~,o> as a function of fl, as indicated in table 2 for r / = 6, has led quite early to various approximation schemes. The scheme suggested here involves truncation as well as interpolation. Scheme 1. The results quoted in table 2 show that the determinant (3.1) has decreased to a value below 10 - 7 for n > 25. As the coefficients An(M, J), viewed as a function of n, are bounded, one can truncate the sum in eq. (2.11) at a suitable upper value N N

ICjI 2 ~ ~

An(M,J).

(4.1)

n=0

TABLE 2 The determinants Det {N~v(fln) } for r/ = 6 for various n in an exact calculation and in approximations (schemes 2 and 3) using truncation and interpolation Det

{N~.(fl~)}

Det

/'/

exact 6 7 8 9 10 11 12 13 14 15

{N~(fl~)}

n

0.3897177 0.2771487 0.1869531 0.1196020 0.0725509 0.0417207 0.0227384 0.0117422 0.0057436 0.0026603

scheme 2

scheme 3

0.0417207* 0.0227267 0.0117422* 0.0057509 0.0026668

0.3897177* 0.2771320 0.1869391 0.1196020* 0.0725649 0.0417401 0.0227540 0.0117495 0.0057436* 0.0026563

16 17 18 19 20 21 22 23 24 25

exact

scheme 2

scheme 3

0.0011663 0.0004838 0.0001898 0.0000704 0.0000247 0.0000082 0.0000026 0.0000008 0.0000002 0.0000001

0.0011692 0.0004838* 0.0001888 0.0000695 0.0000242 0.0000081 0.0000026* 0.0000008 0.0000003 0.0000001"

0.0011618 0.0004805 0.0001881 0.0000698 0.0000247* 0.0000084 0.0000028 0.0000009 0.0000003 0.0000001"

Scheme 1 is a truncation at N = 25 (eq. (4.1)). Scheme 2 and 3 are combinations of truncation and interpolation. * Used in fitting.

68

P. C H A T T O P A D H Y A Y A N D R. M. D R E I Z L E R TABLE 3 The spectral weights for r/ = 6 in various approximations

ICjI2

IC~l 2

J

J scheme 1

0 2 4 6 8 10 12 14 16 18 20 22

0.01239 0.05970 0.09854 0.12419 0.13485 0.13160 0.11776 0.09768 0.07560 0.05483 0.03737 0.02398

scheme 2 0.01239 0.05970 0.09854 0.12419 0.13485 0.13160 0.11776 0.09768 0.07560 0.05483 0.03737 0.02398

scheme 3 0.01239 0.05970 0.09854 0.12419 0.13485 0.13160 0.11776 0.09768 0.07560 0.05483 0.03737 0.02399

scheme 1 24 26 28 30 32 34 36 38 40 42 44 46

0.01451 0.00829 0.00447 0.00228 0.00110 0.00050 0.00022 0.00009 0.00003 0.00001

scheme 2 0.01451 0.00829 0.00447 0.00228 0.00110 0.00050 0.00022 0.00009 0.00003 0.00001

scheme 3 0.01452 0.00829 0.00447 0.00228 0.00109 0.00049 0.00021 0.00008 0.00003 0.00001 0.00001 0.00001

Scheme 1 is a truncation at N = 25 (eq. (4.1)). Scheme 2 and 3 are combinations of truncation and interpolation.

It should be noted that this truncation is not equivalent to a truncation of the original sum (2.7) at a value M' < M. Truncation of the sum (2.7) would amount to a loss of information on the maximum angular momentum state present and will introduce noticeable errors. The results of this scheme are shown in table 3 for N = 25. We have achieved a time saving of a factor of about 2.4 without any loss of accuracy in the single precision results. F o r further reduction we use an interpolation scheme first suggested by Onishi and Sheline 6). There one approximates the matrix element in question by the expression (~b~, o[ exp ( - iflJy)l~b~,, o) = e-~a),

(4.2)

with 5

e(fl) = ~ e k sin2k(fl).

(4.3)

k=l

The five parameters e k are to be determined by a suitable fitting procedure. The expression for the exponent is consistent with the requirements: (1) For fl = 0 we have e(0) = 0 and thus (~'~,01~'~,o)

= 1.

(2) As a consequence, of the hermiticity of the operator Jy we must have ~(#) = 8(-/~).

(3) For small values of fl and the choice of e 1 = ~1 ( J y2 ) one obtains the Gaussian overlap approximation.

ANGULAR MOMENTUM PROJECTION

69

The functional form of the determinant in connection with the relations (4.2) and (4.3) then suggests the following strategy. In addition to the truncation indicated above, (a) evaluate det {Na~(~n)} exactly at the first few points n --- 0, 1. . . . n o. (b) evaluate det {N~u(fln)} at five more points n = no+k 1, n = n 0 + k 2 etc., use these points to determine the parameters in eq. (4.3) and apply the interpolation scheme for the remainder of the points. We give explicit results for two possible options: Scheme 2. We use N = 25, n o = 10 and the five points indicated in table 2 for the fit of the interpolation function (4.2). The results for ICjI 2 (table 3) are accurate to the fifth digit. Scheme 3. We use N = 25, n o = 5 and the five points indicated in table 2. Here we find that values of ICjI 2 (table 3) are slightly affected in the fifth digit. The saving in computing time would amount to factors of 3.8 and 5.2 respectively. 5. Conclusion We have demonstrated that exact angular momentum projection is feasible for systems with large particle numbers. The economy of the finite sum representation used here as compared to exact integration using e.g. Gauss' quadrature formula is obvious in the case of exact projection and carries over to the case of approximate projection. The combination of truncation and interpolation reduces the computing time for the spectral weights of the angular momentum components of interest (J = 0 . . . . 42) to a few minutes, while maintaining a high level of accuracy. We have only considered the calculation of spectral weights. Calculation of energy values do not involve any novel aspects, except the fact that the calculation of the matrix elements (~,1/-?/~(/3)1~) is more time consuming. In this case the use of truncation interpolation schemes become even more essential. The method suggested can also be applied in the case of an intrinsic state of the Hartree-Fock-Bogoliubov type, where corresponding formulae for the evaluation of the matrix elements (~bl/~(fl)l~,) and (~kl/~/~(fl)l~') have been given. The question of an optimal access to simultaneous projection of both angular momentum and particle number is, however, still open. We thank the HRZ of the University of Frankfurt for the use of the computing facilities. One of the authors (R.M.D.) would like to thank Prof. K. Allaart for comments on the use of Gauss' integration technique in angular momentum projection calculations.

70

P. CHATTOPADHYAY AND R. M. DREIZLER

References 1) F. Grfimmer, K. W. Schmid and A. Faessler, Nucl. Phys. A306 (1978) 134; A305 (1978) 77; R. Bengtson, I. Hamamoto and B. Mottelson, Phys. Lett. 73B (1978) 259; E. R. Marshalek and A. L. Goodman, Nucl. Phys. A294 (1978) 92; I. Hamamoto, Nucl. Phys. A271 (1976) 15; R. A. Sorensen, Nucl. Phys. A369 (1976)301 2) N. MacDonald, Adv. in Phys. 19 (1970) 371 3) H. A. Lamme and E. Boeker, Nucl. Phys. A l l l (1968) 492 4) J. Raynal, J. Phys. Rad. 31 (1970) 3 5) A. Kelemen and R. MI Dreizler, Z. Phys. A278 (1976) 269 6) N. Onishi and R. K. Sheline, Phys. Rev. C2 (1970) 1304 7) P. Chattopadhyay, to be published 8) Handbook of mathematical functions, ed. M. Abramowitz and I. A. Stegun (Dover Publications, NY, 1972) sect. 25