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A new method for calculating rotational spectra for complex nuclei
Volume 58B, number 1
PHYSICS LETTERS
18 August 1975
A NEW METHOD FOR CALCULATING ROTATIONAL SPECTRA FOR COMPLEX NUCLEI S.K.M. WONG, W.Y. NG and L.E.H. TRAIN(JR Department of Physics, Universityof Toronto, Canada Received 5 May 1975 A new method has been devised for calculating the rotational spectra of complex nuclei which is much simpler than methods using projection techniques but which gives results comparable to them. The method is a modified version of the constrained Hartree-Fock-Skyrme technique.
In calculating rotational and quasi-rotational band structures associated with deformed nuclei, one runs into technical difficulties with the exact projection methods of Hill and Wheeler, even in Hartree-Fock approximation [1,2] ; these difficulties arise because of the large multi-shell calculations involved and the limitations associated with projection from a single intrinsic state. A partial solution to this problem has been to use an approximate projection method [3] in conjunction with the variation-after-projection (VAP) technique. The method still has limited practicability, however, since a great number of projections have to be carried out in VAP before numerical variations can be carried out. A trade-off has been made between the simplicity of projection and the frequency with which projection must be carried out. In this letter we propose a new method* which lends itself to large base calculations and gives results comparable to VAP. The method emphasizes the role of collective variables within the framework of Hartree-Fock-Skyrme (HFS) prescription previously outlined [4-6] which we call the WNT method. Specifically, if H has a well-defined ground state band for an E - E nucleus, so that the intrinsic Hamiltonian H ' = H - t~J 2 is degenerate with respect to all states in the band, the inertial parameter a is given precisely by the Skyrme formula a
(¢IHj21~) - ( ~/-/I ~>(~ lj21 ~> -
(1)
(~lJ41O> - (~b[J21~)2 * Apparently a somewhat similar approach has been suggested
by K. Kumar (Prec. Conference on Hartree-Fock and s~lfconsistent field theories on nuclei, Trieste, 1975).
where [¢0 is any linear combination of eigenstates in the band. The HFS method corresponds to choosing Iq~) to be the constrained HF solution o f H ' = H-¢~j'2 consistent with eq. (1). For spectra approximating true rotation, it has been shown [6] that eq. (1) gives a least squares estimate for fitting a J(J+ 1) spectrum to the exact spectrum Ej. In the method proposed here we write for the energy of an approximate rotational spectrum:
g j = (~IH - aJ21~) + etJ(Y+ 1)
(2)
where tv is now to be regarded as a function of J rather than being fixed by the self-consistent Skyrme formula. We regard the intrinsic state I~) as being parametrized by a set of, say N, independent, collective internal coordinates {Xi}, i = 1,2, ...,N, which can be determined, in principle, from the variational equations
aEj :xi )
a
[(¢ IHI ~b) - ,~(¢ I./'21~> + t v j ( j + 1)]
k
=0,
k = 1,2, ...,N,
(3)
where a depends explicitly on the X k through the Skyrme formula, eq. (1). The X k themselves may be regarded as expectation values of a set of collective operators Ck,
Xk = (~lCk[~). For rotational motion we assume, in fact, that a single collective variable is sufficient to give a good characterization of the dynamics. A natural choice [7] is X = (~lJ21~) - ( . / 2 ) so that eq. (3) becomes
(4)
Volume 58B, number 1
18August 1975
PHYSICS LETTERS Table 1 Nucleus = 2°Ne.
h
(H)
(j2)
o~
Eo
E2
E4
E6
Eg
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-34.8340 -35.5700 -35.8332 -35.8994 -35.8459 -35.7109 -35.5209 -35.2936 -35.0410 -34.7846 -34.5224 -34.2632 -34.0118 -33.7722
10.3992 13.0319 15.0469 16.3718 17.4303 18.3297 19.0883 19.7384 20.2998 20.7663 21.1707 21.5175 21.8140 22.0667
0.1508 0.1652 0.1702 0.1718 0.1719 0.1707 0.1687 0.1663 0.1633 0.1604 0.1573 0.1541 0.1510 0.1480
-36.4022 -37.7229 -38.3942 -38.7121 -38.8422 -38.8398 -38.7411 -38.5761 -38.3560 -38.1155 -37.8525 -37.5790 -37.3057 -37.0381
-35.4974 -36.7317 -37.3730 -37.6813 -37.8108 -37.8156 -37.7289 -37.5783 -37.3761 -37.1531 -36.9087 -36.6544 -36.3997 -36.1501
-33.3862 -34.4189 -34.9902 -35.2761 -35.4042 -35.4258 -35.3671 -35.2501 -35.0900 -34.9075 -34.7065 -34.4970 -34.2857 -34.0781
-30.0686 -30.7845 -31.2458 -31.4965 -31.6224 -31.6704 -31.6557 -31.5915 -31.4973 -31.3787 -31.2459 -31.1068 -30.9637 -30.8221
-25.5446 -25.8285 -26.1398 -26.3425 -26.4654 -26.5494 -26.5947 -26.6025 -26.5984 -26.5667 -26.5269 -26.4838 -26.4337 -26.3821
Table 2 Nucleus = 28Si. h
(H)
(j2)
oe
E0
E2
E4
E6
E8
-0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60
-122.6920 -122.9566 -123.1598 -123.2955 -123.3441 -123.2866 -123.0914 -122.7216 -122.1499 -121.3712 -120.4420
20.0203 20.7783 21.5960 22.5110 23.5017 24.6677 25.9767 27.4601 29.0954 30.8351 32.5268
0.1002 0.1030 0.1056 0.1077 0.1092 0.1098 0.1091 0.1067 0.1024 0.0964 0.0892
-124.6980 -125.0968 -125.4403 -125.7199 -125.9105 -125.9951 -125.9254 -125.6516 -125.1293 -124.3437 -123.3434
-124.0968 -124.4788 -124.8067 -125.0737 -125.2553 -125.3363 -125.2708 -125.0114 -124.5149 -123.7653 -122.8082
-122.6940 -123.0368 -123.3283 -123.5659 -123.7265 -123.7991 -123.7435 -123.5176 -123.0813 -122.4157 -121.5594
-120.4896 -120.7708 -121.0051 -121.1965 -121.3241 -121.3835 -121.3432 -121.1702 -120.8285 -120.2949 -119.5970
-117.4836 -117.6808 -117.8371 -117.9655 -118.0481 -118.0895 -118.0703 -117.9692 -117.7565 -117.4029 -116.9210
aE/fu2>) _ a a ( J 2)
a ( j 2)
[(qbIHI ¢) - ~(¢ IJ21 qb) + ¢xJ(J+1)]
=0.
(5)
In order to generate an intrinsic state for use in eqs. (2) and (5) we use the constrained Hartree-Fock method: 6(¢IH-)j21¢) = 0
(6)
where X is the Lagrange multiplier determined by the constraint condition <~(X)Ij21~(X)>=X. 10
(7)
This procedure generates a set of intrinsic states as functions of the collective variable X: =
= ¢(u2>).
(8)
The dependence of a on (,/2) is then determined from eq. (I), while the appropriate value o f ( J 2) and hence of ~ is then determined separately for each J value from eq. (5). While the method may seem involved, each step is in fact simple and the total work involved is almost trivial compared to standard projection techniques. In order to make comparison with exact projection we have carried out the calculations for 20Ne and 28Si using the Rosenfeld interaction described by Ripka
Volume 58B, number 1
PHYSICS LETTERS
-22.oc
18 A u g u s t 1 9 7 5
-I10.0C
-24.0C
-I12.0C
2ONe
-26.0C
28SI
-114.00
J*8
-28.0C
-I 16.0C
-30.oc
-118.00
/ z_
/
,.I-6
~-32.0C
z
~
~20.00
-34.0C ~
J'4
d=8
-122.00
~ -
-124.00 _
3
8
-40001 10.~
.
I 12.~
0
0
t
~
J'O
I I I I 14.00 16.~ 18.00 20.00 22.00 24.00
-126.00
-128.00 20.00
J,6 J-4 d*2 J'O
I I I I 1 I I 22.00 24.00 26.00 28.00 30.00 32.00 34.00 (d 2)
Fig. 1. Rotational energy versus collective variable (j2), the mean square angular momentum in the intrinsic ~tate; (a) 2UNe;
(b) 28Si. [2]. The results are shown in tables 1 and 2. Figs. l(a) and l(b) show graphically how the minimization of Ej versus (j2) was carried out. Finally, fig. 2(a) and fig. 2(b) show a comparison of our results with both
I IC ,>,, :E z
uJ 6 5 4
F
uJ 3 2
I 0
O+CHFS
H~ 2oNe
EIP
S O+CHF
HFP 28S1
EIP
Fig. 2. Comparison of the constrained Hartree-Fock-Skyrme results (present method) with exact projection and experiment; (a) 2°Ne; (b) 2SSi.
the exact projection and with the experimental values. It should be noted that the definition o f intrinsic state used in the present paper implies a linear combination of eigenstates in the rotational band which can be recovered by angular momentum projection. Such intrinsic states necessarily obey time reversal invariance. This is in contrast to intrinsic states which correspond to actual eigenstates viewed from a rotating (body) reference frame, and which necessarily have time-reversed components arising from matrix elements of the Corioles force o~Jx. The present viewpoint, which involves only matrix elements o f c & 2 is in consonance with our previous work [5] and that of Kumar [8]. While it can be argued that projection is a more exact mathematical method, it may well be that our method is not only simpler but more physical. For one thing, our procedure implies that projection is being carried out from more than one intrinsic state, an advantage shared with the VAP method, but in contrast to VAP the complication of numerous projections is replaced by the simplicity associated with a judicious use of the Skyrme formula. This related 11
Volume 58B, number I
PHYSICS LETI'ERS
simplicity is clearly demonstrated in ref. [4]. Finally, we would argue that our method has an intrinsic advantage of making explicit use o f the collective variable X = (j2) and lends itself readily to large base, multi-shell calculations. Further developments along these lines will be reported elsewhere. Financial assistance from the National Research Council o f Canada is gratefully acknowledged.
12
18 August 1975
References [1] D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1106. [2] G. Ripka, in Advances in nuclear physics, Voi. I, eds. M. Baranger and E. Vogt (Plenum Press). [3] D. Justin, M.V. Mihailovic and M. Rosina, Nucl. Phys. A182 (1972) 54. [4] S.K.M. Wong, J. Le Tourneux, N. Quang-Hoc and G. Saunier, Nu¢l. Phys. A137 (1969) 318. [5] W.Y. Ng, L.E.H. Trainor and S.K.M. Wong, Phys. Lett. 33B (1970) 545. [6] W.Y. Ng and L.E.H. Trainor, Can. J. Phys. 52 (1974) 541. [7] R.K. Bhaduri and S. Das Gupta, Nucl. Phys. A212 (1973) 18. [8] K. Kumar, physica S~ipta 6 (1972) 270.
PHYSICS LETTERS
18 August 1975
A NEW METHOD FOR CALCULATING ROTATIONAL SPECTRA FOR COMPLEX NUCLEI S.K.M. WONG, W.Y. NG and L.E.H. TRAIN(JR Department of Physics, Universityof Toronto, Canada Received 5 May 1975 A new method has been devised for calculating the rotational spectra of complex nuclei which is much simpler than methods using projection techniques but which gives results comparable to them. The method is a modified version of the constrained Hartree-Fock-Skyrme technique.
In calculating rotational and quasi-rotational band structures associated with deformed nuclei, one runs into technical difficulties with the exact projection methods of Hill and Wheeler, even in Hartree-Fock approximation [1,2] ; these difficulties arise because of the large multi-shell calculations involved and the limitations associated with projection from a single intrinsic state. A partial solution to this problem has been to use an approximate projection method [3] in conjunction with the variation-after-projection (VAP) technique. The method still has limited practicability, however, since a great number of projections have to be carried out in VAP before numerical variations can be carried out. A trade-off has been made between the simplicity of projection and the frequency with which projection must be carried out. In this letter we propose a new method* which lends itself to large base calculations and gives results comparable to VAP. The method emphasizes the role of collective variables within the framework of Hartree-Fock-Skyrme (HFS) prescription previously outlined [4-6] which we call the WNT method. Specifically, if H has a well-defined ground state band for an E - E nucleus, so that the intrinsic Hamiltonian H ' = H - t~J 2 is degenerate with respect to all states in the band, the inertial parameter a is given precisely by the Skyrme formula a
(¢IHj21~) - ( ~/-/I ~>(~ lj21 ~> -
(1)
(~lJ41O> - (~b[J21~)2 * Apparently a somewhat similar approach has been suggested
by K. Kumar (Prec. Conference on Hartree-Fock and s~lfconsistent field theories on nuclei, Trieste, 1975).
where [¢0 is any linear combination of eigenstates in the band. The HFS method corresponds to choosing Iq~) to be the constrained HF solution o f H ' = H-¢~j'2 consistent with eq. (1). For spectra approximating true rotation, it has been shown [6] that eq. (1) gives a least squares estimate for fitting a J(J+ 1) spectrum to the exact spectrum Ej. In the method proposed here we write for the energy of an approximate rotational spectrum:
g j = (~IH - aJ21~) + etJ(Y+ 1)
(2)
where tv is now to be regarded as a function of J rather than being fixed by the self-consistent Skyrme formula. We regard the intrinsic state I~) as being parametrized by a set of, say N, independent, collective internal coordinates {Xi}, i = 1,2, ...,N, which can be determined, in principle, from the variational equations
aEj :xi )
a
[(¢ IHI ~b) - ,~(¢ I./'21~> + t v j ( j + 1)]
k
=0,
k = 1,2, ...,N,
(3)
where a depends explicitly on the X k through the Skyrme formula, eq. (1). The X k themselves may be regarded as expectation values of a set of collective operators Ck,
Xk = (~lCk[~). For rotational motion we assume, in fact, that a single collective variable is sufficient to give a good characterization of the dynamics. A natural choice [7] is X = (~lJ21~) - ( . / 2 ) so that eq. (3) becomes
(4)
Volume 58B, number 1
18August 1975
PHYSICS LETTERS Table 1 Nucleus = 2°Ne.
h
(H)
(j2)
o~
Eo
E2
E4
E6
Eg
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
-34.8340 -35.5700 -35.8332 -35.8994 -35.8459 -35.7109 -35.5209 -35.2936 -35.0410 -34.7846 -34.5224 -34.2632 -34.0118 -33.7722
10.3992 13.0319 15.0469 16.3718 17.4303 18.3297 19.0883 19.7384 20.2998 20.7663 21.1707 21.5175 21.8140 22.0667
0.1508 0.1652 0.1702 0.1718 0.1719 0.1707 0.1687 0.1663 0.1633 0.1604 0.1573 0.1541 0.1510 0.1480
-36.4022 -37.7229 -38.3942 -38.7121 -38.8422 -38.8398 -38.7411 -38.5761 -38.3560 -38.1155 -37.8525 -37.5790 -37.3057 -37.0381
-35.4974 -36.7317 -37.3730 -37.6813 -37.8108 -37.8156 -37.7289 -37.5783 -37.3761 -37.1531 -36.9087 -36.6544 -36.3997 -36.1501
-33.3862 -34.4189 -34.9902 -35.2761 -35.4042 -35.4258 -35.3671 -35.2501 -35.0900 -34.9075 -34.7065 -34.4970 -34.2857 -34.0781
-30.0686 -30.7845 -31.2458 -31.4965 -31.6224 -31.6704 -31.6557 -31.5915 -31.4973 -31.3787 -31.2459 -31.1068 -30.9637 -30.8221
-25.5446 -25.8285 -26.1398 -26.3425 -26.4654 -26.5494 -26.5947 -26.6025 -26.5984 -26.5667 -26.5269 -26.4838 -26.4337 -26.3821
Table 2 Nucleus = 28Si. h
(H)
(j2)
oe
E0
E2
E4
E6
E8
-0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60
-122.6920 -122.9566 -123.1598 -123.2955 -123.3441 -123.2866 -123.0914 -122.7216 -122.1499 -121.3712 -120.4420
20.0203 20.7783 21.5960 22.5110 23.5017 24.6677 25.9767 27.4601 29.0954 30.8351 32.5268
0.1002 0.1030 0.1056 0.1077 0.1092 0.1098 0.1091 0.1067 0.1024 0.0964 0.0892
-124.6980 -125.0968 -125.4403 -125.7199 -125.9105 -125.9951 -125.9254 -125.6516 -125.1293 -124.3437 -123.3434
-124.0968 -124.4788 -124.8067 -125.0737 -125.2553 -125.3363 -125.2708 -125.0114 -124.5149 -123.7653 -122.8082
-122.6940 -123.0368 -123.3283 -123.5659 -123.7265 -123.7991 -123.7435 -123.5176 -123.0813 -122.4157 -121.5594
-120.4896 -120.7708 -121.0051 -121.1965 -121.3241 -121.3835 -121.3432 -121.1702 -120.8285 -120.2949 -119.5970
-117.4836 -117.6808 -117.8371 -117.9655 -118.0481 -118.0895 -118.0703 -117.9692 -117.7565 -117.4029 -116.9210
aE/fu2>) _ a a ( J 2)
a ( j 2)
[(qbIHI ¢) - ~(¢ IJ21 qb) + ¢xJ(J+1)]
=0.
(5)
In order to generate an intrinsic state for use in eqs. (2) and (5) we use the constrained Hartree-Fock method: 6(¢IH-)j21¢) = 0
(6)
where X is the Lagrange multiplier determined by the constraint condition <~(X)Ij21~(X)>=X. 10
(7)
This procedure generates a set of intrinsic states as functions of the collective variable X: =
= ¢(u2>).
(8)
The dependence of a on (,/2) is then determined from eq. (I), while the appropriate value o f ( J 2) and hence of ~ is then determined separately for each J value from eq. (5). While the method may seem involved, each step is in fact simple and the total work involved is almost trivial compared to standard projection techniques. In order to make comparison with exact projection we have carried out the calculations for 20Ne and 28Si using the Rosenfeld interaction described by Ripka
Volume 58B, number 1
PHYSICS LETTERS
-22.oc
18 A u g u s t 1 9 7 5
-I10.0C
-24.0C
-I12.0C
2ONe
-26.0C
28SI
-114.00
J*8
-28.0C
-I 16.0C
-30.oc
-118.00
/ z_
/
,.I-6
~-32.0C
z
~
~20.00
-34.0C ~
J'4
d=8
-122.00
~ -
-124.00 _
3
8
-40001 10.~
.
I 12.~
0
0
t
~
J'O
I I I I 14.00 16.~ 18.00 20.00 22.00 24.00
-126.00
-128.00 20.00
J,6 J-4 d*2 J'O
I I I I 1 I I 22.00 24.00 26.00 28.00 30.00 32.00 34.00 (d 2)
Fig. 1. Rotational energy versus collective variable (j2), the mean square angular momentum in the intrinsic ~tate; (a) 2UNe;
(b) 28Si. [2]. The results are shown in tables 1 and 2. Figs. l(a) and l(b) show graphically how the minimization of Ej versus (j2) was carried out. Finally, fig. 2(a) and fig. 2(b) show a comparison of our results with both
I IC ,>,, :E z
uJ 6 5 4
F
uJ 3 2
I 0
O+CHFS
H~ 2oNe
EIP
S O+CHF
HFP 28S1
EIP
Fig. 2. Comparison of the constrained Hartree-Fock-Skyrme results (present method) with exact projection and experiment; (a) 2°Ne; (b) 2SSi.
the exact projection and with the experimental values. It should be noted that the definition o f intrinsic state used in the present paper implies a linear combination of eigenstates in the rotational band which can be recovered by angular momentum projection. Such intrinsic states necessarily obey time reversal invariance. This is in contrast to intrinsic states which correspond to actual eigenstates viewed from a rotating (body) reference frame, and which necessarily have time-reversed components arising from matrix elements of the Corioles force o~Jx. The present viewpoint, which involves only matrix elements o f c & 2 is in consonance with our previous work [5] and that of Kumar [8]. While it can be argued that projection is a more exact mathematical method, it may well be that our method is not only simpler but more physical. For one thing, our procedure implies that projection is being carried out from more than one intrinsic state, an advantage shared with the VAP method, but in contrast to VAP the complication of numerous projections is replaced by the simplicity associated with a judicious use of the Skyrme formula. This related 11
Volume 58B, number I
PHYSICS LETI'ERS
simplicity is clearly demonstrated in ref. [4]. Finally, we would argue that our method has an intrinsic advantage of making explicit use o f the collective variable X = (j2) and lends itself readily to large base, multi-shell calculations. Further developments along these lines will be reported elsewhere. Financial assistance from the National Research Council o f Canada is gratefully acknowledged.
12
18 August 1975
References [1] D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1106. [2] G. Ripka, in Advances in nuclear physics, Voi. I, eds. M. Baranger and E. Vogt (Plenum Press). [3] D. Justin, M.V. Mihailovic and M. Rosina, Nucl. Phys. A182 (1972) 54. [4] S.K.M. Wong, J. Le Tourneux, N. Quang-Hoc and G. Saunier, Nu¢l. Phys. A137 (1969) 318. [5] W.Y. Ng, L.E.H. Trainor and S.K.M. Wong, Phys. Lett. 33B (1970) 545. [6] W.Y. Ng and L.E.H. Trainor, Can. J. Phys. 52 (1974) 541. [7] R.K. Bhaduri and S. Das Gupta, Nucl. Phys. A212 (1973) 18. [8] K. Kumar, physica S~ipta 6 (1972) 270.