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Linear chain states with zero-range and other Skyrme interactions
Physics Letters B 293 (1992) 1-6 North-Holland
PHYSICS LETTERS B
Linear chain states with zero-range and other Skyrme interactions L, Z a m i c k
Department of Physicsand Astronomy, Rutgers University,Piscataway,NJ 08855, USA D.C. Z h e n g a n d D , W . L . S p r u n g
Department of Physicsand Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 Received 15 June 1992
With the usual Skyrme interactions, e.g., Skyrme III, the energies of the bandheads of the linear-alpha-chainstates in ~2Cand t60 come about a factor of two too high in energy. On the other hand, the Nilsson model gets the energies about right. In an analytic model using a zero-range Skyrmeinteraction with asymptotic deformed harmonic oscillator wave functions, the energies of the linear chain states come down considerably and are in better agreement with experiment. In more sophisticated HartreeFock calculationswith the above zero-rangeinteraction, these states come down too low in energy,but the overall analysissuggests lowering the magnitude Of the finite-range terms in the Skyrme interactions in order to get the linear chain states at the right energies. The zero-range Skyrme interaction gives a close realization of the Nilsson model when a constraint of volume conservation is imposed.
1. Introduction In a previous work [ 1 ] we noted that with the Skyrme III interaction, the energy of the J ~ = 0 + b a n d h e a d for the linear-alpha-chain states in ~2C a n d ~60 was approximately a factor of two too high compared with experiment. With the Nilsson model calculation [2 ], however, the states come at a reasonable energy a n d give support to the ideas of Brink [ 3 ] on alpha-clustering states. In this work we will investigate this problem more deeply by considering an analytic model invoking a zero-range Skyrme interaction with densities obtained from h a r m o n i c oscillator wave functions. We will also perform self-consistent Hartree-Fock ( H F ) calculations with the zerorange interaction. Our h a m i l t o n i a n is
/-/= E +t3
+to lE< J ~
i
8(ri-rj)t~(rj-rk).
the finite-range tl and 12 terms, the spin-orbit W t e r m a n d the C o u l o m b interaction are set equal to zero. The normalized one-dimensional h a r m o n i c oscillator wave functions are
1
q/.(x/b) = b-q)5 ~
1
exp( - x 2 / 2 b 2)
×H.(x/b),
(2)
where Hn are hermite polynomials which, for n = 0, 1, 2, a n d 3, are given by no(x) = 1 ,
Hl(x)=2x,
H2(x)=4x2-2,
H3(x)=8x 3-12x.
The density for a Slater d e t e r m i n a n t using harm o n i c oscillator wave functions is
p(x,y,z) (l)
=
E
[Vnx(x/bx)~ny(y/by)~n,(z/bz)] 2.
occupied
(3) This is equivalent to a Skyrme interaction in which
The energy is given by
0370-2693/92/$ 05.00 © 1992 Elsevier SciencePublishers B.V. All fights reserved.
1
Volume293, number 1,2
PHYSICSLETTERSB
1 (hco~G +hcorG +hco~Sz) + ]to f p : d 3 r + 3 ( ' 3 ~ ~p3d3 r 8\67
(4)
where ho~=h2/rnb~ and i=x, y, z. The first term on the right-hand-side of the above equation is the kinetic energy. The quantities X~ are given by X~= sum(N~+ ½) with N~ equal to the number of quanta in the/-direction. The factor ( A - 1 )/A is inserted to remove approximately spurious contributions from the center-of mass motion. This is a convention of Skyrme interactions which can be improved upon
22 October 1992
We next consider the linear-alpha-chain state for which we assume axial prolate symmetry (bx= by-b± ~ bz) and put nucleons in states in which all the quanta are in the z-direction. The occupied states are then (nx, ny, nz) = (0, 0, 0), (0, 0, 1 ), (0, 0, 2), and (0, 0, 3). To compute the kinetic energy we have
[11 2Z,± -X~+Xy=A,
Z==~A 2,
(7)
where A is the mass number. The density is 2
p(x,y, z ) _ ~r3/Eb3 (3+692-494+]g 6) ×exp( _ r E ) ,
[4].
(8)
where b 3 - b~bybzand we have defined
X=x/bx, y=y/by,
2o 160
For 260, the ground state is spherically symmetric so we take bx= by= bz = b. With four nucleons in the 0s shell and twelve in the 0p shell, X=Zx+27y+Z~= 36 and the density is 4
(
2r2~
:
2
p(r)= u3-57~b3 _1+ -~-) e x p ( - r /b ) (r2=x2 + y2 +z 2) .
699.859
(5)
/2.952454~
b-------v~+tok
-~
)
t3 (1.919973) + ~\. ~-g -].
The energy is then
E(b,
b = ) = ( A _ ~ ) hrnF bz
A2 b 2
{2.933112' t3 (2.0b1235) +to k ~-3 ) + -~ •
We4hefi obtain the energy for the ground state of 16 0 as E(b)=
~,=z/b=,
(6)
To determine the parameters to and t3, we demand that
Since the potential energy depends only on b for a zero-range interaction, as is evident from the above equation, the condition OE(b, b~)/Ob~=O (with b fixed) is equivalent to minimizing the kinetic energy alone with respect to bz, i.e.,
NLA-b-+-i- g
E(b)=
These two equations are solved to give to --- -990.078 (MeV fm 3) , t3 = 17016.238(MeV fm 6) . The resulting ground state kinetic energy is 234.93 MeV and the potential energy is - 362.55 MeV.
xe=0.
(10)
This yields bffb=(A/4) ~/3. With this choice the expression for the linear chain state energy (9) becomes
E ( b = 1.72603)= - 127.624 MeV, db _Jb=1.72603= 0 "
(9)
740.638 {2.933112) b--V - - +to~ ~-3
t3 (2.021235~ + ~-\- ~-g .].
(11)
If we constrain b to be the same as that for the ground state, i.e., b= 1.72603 fro, the intrinsic state energy of the linear chain is -99.35 MeV. This can be broken up into a kinetic energy of EK=248.61 MeV and a potential energy of E p = - 347.96 MeV.
Volume 293, number 1,2
PHYSICS LETTERS B
The change relative to the ground state is as follows: A E = 28.27 M e V , AEK = 13.68 M e V ,
AEp = 14.59 M e V .
22 October 1992
netic a n d potential energies relative to the ground state in this volume n o n c o n s e r v a t i o n case are as follows: A E = 26.68 M e V ,
Note the near equality of AEK and AEp. In the Nilsson model with deformed h a r m o n i c oscillator wave functions, AEK is equal to AEp by virtue o f the virial theorem. Thus we see that the zero-range Skyrme interaction used here gives us a very close realization of the Nilsson model. The value of AE itself is also close to that obtained in the Nilsson model (25.89 MeV) [ 1 ]. Furthermore the value of AE is m u c h smaller than that obtained with the finite-range Skyrme interactions, e.g., the value of AE with the Skyrme III interaction is 37.23 MeV. Recalling from our previous work [ 1 ] that, after the angular m o m e n t u m projection, the J ~ = 0 + state comes about 10 MeV lower than the intrinsic state energy, we see that the zero-range Skyrme interaction gives us a linear chain state at an excitation energy of about 18 MeV, which is rather close to the experimental value of about 16 MeV [5 ]. If we allow the v o l u m e to change, we find that the linear chain state energy (11 ) has a m i n i m u m at b = 1.783 fm. The corresponding energy is - 100.94 MeV, which is 1.59 MeV lower than the energy when b=bg.s.= 1.72603 fm is used. The changes in the ki-
AEK = - 1.96 M e V ,
AEp = 2 8 . 6 5 M e V .
We see that the previous near equality of AEK a n d AEv in the volume conservation case is completely destroyed; indeed AE~¢ changes from positive to negative. A s u m m a r y of the results is given in table 1. The results for the ground state a n d the linear chain state it/'2C, which will be discussed next, are also shown / there.
3. 12C
We now consider ~2C. In ref. [ 1 ] it was noted that the excitation energy of the intrinsic linear-alphachain state in t2C in a Skyrme III H F calculation is much higher than that in the Nilsson model calculation. The respective n u m b e r s are 20.96 MeV for the Skyrme III H F and 9.64 MeV for the Nilsson model. The ground state in ~2C is of oblate shape. We again assume axial deformation and put the nucleons in the states (nx, nr, n~) = (0, 0, 0), ( 1, 0, 0), and (0, 1, 0).
Table 1 The ground state (ground) and the linear-alpha-chain state (chain) in ~60 and ~2Cwith a zero-range Skyrmeinteraction (see text) and densities obtained from asymptotic harmonic oscillator wave functions. Here E.r=EK+Ep where EK and Ep are kinetic and potential energiesrespectively.Changesin the various quantities listed are relative to the ground states. Valuesare consistent with the use of MeV as unit of energyand fm as unit of length. Nucleus
State
b
( bJb ) 3
Er~
160
ground chain a) change a)
1.726 1.726 0.000
1 4 3
234.93 248.61 13.68
chain b) change b~
1.783 0.057
4 3
ground chain a) change ")
1.670 1.670 0.000
chain b) change b~
1.699 0.029
~2C
Ep
Er
-362.55 347.96 14.59
- 127.62 - 99.35 28.27
232.96 -- 1.96
-- 333.90 28.65
-- 100.94 26.68
0.6 3 2.4
172.46 176.94 4.48
- 259.94 - 255.43 4.50
- 87.48 - 78.49 8.99
3 2.4
170.92 -- 1.54
-- 249.75 10.19
-- 78.83 8.65
a) b for the linear chain state is constrained to be the same as that for the ground state. b) b for the linear chain state is varied so as to obtain the lowest linear chain state.
-
-
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We have 27± = 20 and Zz = 6. The density is
p(x,y,z)-
4 zt3/Zb3 (1 +2~2+2)~ 2) e x p ( - F 2) . (12)
The expression for the energy is then obtained and is minimized with respect to bz at (bJb) = (3/5)t/3. The final expression for the ground state energy is
E(b)-
524.697
-~
)
4. H a r t r e e - F o c k c a l c u l a t i o n s
t3 (1.1~.8_750) +~.
(13)
For the linear chain state, we put nucleons in the states (nx, ny, nD = (0, 0, 0), (0, 0, 1 ), and (0, 0, 2). Eq. (7) with A = 12 also applies to the ~2C case. The density is 4 nS/:b3 (3+292) e x p ( - V : ) .
The energy is minimized at given by
E(b)-
(bz/b) =
(14)
(A/4)~/3 and is
538.3315 + t (1.910762"~
b~
ol
t3 / 1.1530875.~
"[- 6 ~
b6
If we allow the volume to change, the linear chain energy is minimized at b--1.699 fm. The minimal energy is - 7 8 . 8 3 MeV which is only 0.34 MeV smaller than the value for b = bg.s.= 1.670 fm. But the values of AEI< and AEp relative to the ground state become - 1.54 MeV and 10.19 MeV respectively so the near equality of AEK and AEp gets destroyed.
[ 1.904809"~
b ~ +to~
p(x,y, z ) -
22 October 1992
7~
] (15)
/]"
With the previously determined to and t3 parameters, the intrinsic energy of the ground state has a minimum of - 8 7 . 4 8 MeV at b = 1.670 fm (determined by h2/mb 2= 45A - ~/3- 25A - 2/3). This is 4.69 MeV higher than the expedmental value of - 9 2 . 1 7 MeV. But as we noted in ref. [ 1 ], the ~ = 0 ÷ state is about 4 MeV lower than the intrinsic state, thus in better agreement with experiment. The intrinsic energy of the linear chain state is - 7 8 . 4 9 MeV, which is 8.99 MeV above the intrinsic ground state energy. The values of AEK and AEp are 4.48 MeV and 4.51 MeV respectively, which are again fairly close to each other. The intrinsic excitation energy of 8.99 MeV for the linear chain state is very close to that in the Nilsson model (9.64 MeV) [1]. This is only 1.33 MeV higher than the experimental value of 7.65 MeV for the 0 + state in ~2C. But after the angular momentum projection, the J ~ = 0 + member of the linear chain is only about 6 MeV above the the 0 ÷ ground state.
In this section we use a Hartree-Fock program to calculate the excitation energies of the linear chain states. The program expands the deformed singleparticle orbits in terms of cylindrically symmetric harmonic oscillator wave functions in which the total number of major shells can be controlled. We will first give results with the minimal number of major shells ( N = 3) so a comparison with the results in the previous sections can be made. We will then perform a better HF calculation in which more major shells ( N = 13) are taken into account, The effects of the spin-orbit interaction are examined. Finally the HF results for the Skyrme I!I interaction are also shown for the sake of comparison. In table 2, we list various contributions to the total energy of ~80: kinetic (EK), volume (Ev), surface (Es), spin-orbit (Eso), Coulomb (Ec) and total (ET), energies. In order to compare the results with those in the previous sections which do not include the Coulomb 'energy, we also list the quantity E~r-Ex-Ec. We first perform the calculation in a small space ( N = 3) with a simplified Skyrme interaction which includes the Coulomb interaction but which is otherwise the same as the simple zero-range Skyrme interaction as discussed in the previous sections ( t o = - 9 9 0 . 0 7 8 , / 3 = 17016.238, t l = t 2 = W = 0 ) . The intrinsic energy for the ground state is e T = - 114.60 MeV which is somewhat higher than the experimental value of - 127.624 MeV. But when the Coulomb energy is excluded, the result is E~ = - 128.57 MeV, this is very close to the result in section 2. The intrinsic excitation energy of the linear chain state is 22.59 MeV which is smaller than the value obtained in section 2 (26.68 MeV). The main reason is the inclusion of the Coulomb interaction in the HF calculation. If we exclude the Coulomb contribution, the change in energy (AE~) is 26.25 MeV
Volume 293, number 1,2
PHYSICS LETTERS B
22 October 1992
Table 2 Hartree-Fock calculations for 160. Here N is the number of major shells allowed in the calculation, W the spin-orbit force strength, EK the kinetic energy, Ev the potential energy, Es the surface energy, Eso the spin-orbit energy, Ec the Coulomb energy, Ex the total energy. Changes in the various quantities listed are relative to the ground states. Values are consistent with the use of MeV as unit of energy and fm as unit of length. N
W
State
3
0
ground chain change
13
0
ground chain change
13
120.0
ground chain change ground chain change
Skyrme III
EK 234.51 236.93 2.42
Ev
Es
E~o
Ec
ET
Er--Ec
-363.09 - 339.25 23.84
0.00 0.00 0.00
0.00 0.00 0.00
13.97 10.31 - 3.67
- 114.60 - 92.01 22.59
- 128.57 - 102.32 26.25
- 381.11 - 344.98 36.13
0.00 0.00 0.00
0.00 0.00 0.00
14.11 10.50 -3.60
- 119.31 - 98.96 20.35
- 133.41 - 109.46 23.95
252.18 245.96 - 6.22
-383.05 -350.58 32.46
0.00 0.00 0.00
-4.25 - 10.69 - 6.45
14.11 10.67 - 3.43
- 121.01 - 104.64 16.37
- 135.11 - 115.31 19.80
229.65 219.69 -9.96
-425.39 -380.00 45.39
54.91 68.23 13.32
-0.88 -8.03 -7.15
13.77 10.56 -3.21
- 127.94 -89.55 38.39
- 141.71 - 100.11 41.61
247.69 235.51 - 12.18
which is m u c h closer to the previous result. N o t e that the surface energy is zero with the simple zero-range interaction which does not have the finite-range t~ and t2 terms. This is mainly a m a t t e r o f definition: what the code calls surface energy is precisely the contribution from these terms in the interaction, which i n v o l v e gradients o f the density. In our zero-range interaction, we have changed the balance between the various terms, while preserving binding energy in the ground state. This is a key to understand why we are getting the linear chain state so low in this m o d el (as well as in the Nilsson m o d e l ) . Obviously the linear chain state will have a larger surface area than the ground state which is spherical (~60) or less d e f o r m e d (~2C). T h e surface energy is repulsive, as is well known from the semi-empirical mass formula. W h e n we use the same interaction but allow for 13 m a j o r shells, the excitation energy o f the linear chain state in ~60 changes from 22.59 M e V to 20.35 MeV. We are getting the linear chain state somewhat too low in energy as we recall that the J r = 0 + state energy will be about 10 M e V lower than the intrinsic state energy. This situation is exacerbated when we turn on the s p i n - o r b i t interaction. T h e linear chain state is lowered m o r e than the ground state by the s p i n - o r b i t interaction. I n d e e d for the ground state it is only sec-
o n d - o r d er effects o f the s p i n - o r b i t interaction that enter. The final intrinsic excitation energy o f the linear chain state is 16.37 MeV which is close to the experimental b a n d h e a d energy ( ~ 16 M e V ) . But as m e n t i o n e d before, the J r = 0 + m e m b e r is about 10 M e V lower than the intrinsic state energy. Finally with the Skyrme III interaction we find that the intrinsic state energy is very high at 38.39 MeV. the culprits are the changes in v o l u m e and surface energies which are 45.39 MeV and 13.32 M e V respectively. Fo r the zero-range interaction, the values were 32.47 and 0 MeV. The difference is 26.24 MeV.
5. Closing remarks We have shown that in an analytic model - the zerorange interaction with d e f o r m e d h a r m o n i c oscillator wave functions - the linear chain states can be brought to much lower excitation energies in ~2C and ~60. W h e n better H a r t r e e - F o c k calculations are perf o r m e d with the above zero-range interaction, there is an overshoot: The linear chain states c o m e to low in energy. This is in contrast to the usual Skyrme interactions, such as Skyrme III, which gives the linear chain states about a factor o f two too high in energy. Th e overshoot in the H F calculations is due to the fact that the C o u l o m b and the s p i n - o r b i t interac-
Volume 293, number 1,2
PHYSICS LETTERS B
tions as well as inclusion o f m o r e m a j o r shells, all conspire to give an a d d i t i o n a l lowering o f the linear chain states relative to the analytic m o d e l which uses asymptotic d e f o r m e d oscillator wave functions. O u r analysis clearly indicates that one can get the linear chain states to come down in energy using Skyrme interactions b y lowering the magnitude o f the finite-range potentials t, a n d t2. Obviously such a procedure will lower the surface energy (with a zerorange Skyrme interaction, the surface energy will be zero). Less obviously the change in v o l u m e energy in going from a ground state to a linear chain state becomes less when the finite-range terms are reduced. We should study this p o i n t further in the near future. We further note that we get a close realization o f the Nilsson m o d e l by using a zero-range Skyrme m o d e l a n d impose the condition o f the v o l u m e conservation.
22 October 1992
Acknowledgement We acknowledge the support o f the U S D O E u n d e r grant DE-FG05-86ER-40299 a n d o f N S E R C , Cana d a under research grant A-3198.
References [ 1] L. Zamick, D.C. Zheng, S.J. Lee, J.A. Caballero and E. Moya de Guerra, Ann. Phys. 212 ( 1991 ) 402. [2] S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955). [ 3 ] D.M. Brink, Many-body description of nuclear structure and reactions, Intern. School' of Physics Enrico Fermi, Course XXXVI, ed. C. Bloch (Academic Press, New York, 1966) p. 247. [4] M.N. Butler, D.W.L. Sprung and J. Martorell, Nucl. Phys. A 422 (1984) 157. [ 5 ] W.D.M. Rae, Proc. Fifth. Conf. on Clustering aspects in nuclear and subnuclear systems (Kyoto, 1988 ), J. Phys. Soc. Jpn. Suppl. 58 (1989) 77.
PHYSICS LETTERS B
Linear chain states with zero-range and other Skyrme interactions L, Z a m i c k
Department of Physicsand Astronomy, Rutgers University,Piscataway,NJ 08855, USA D.C. Z h e n g a n d D , W . L . S p r u n g
Department of Physicsand Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 Received 15 June 1992
With the usual Skyrme interactions, e.g., Skyrme III, the energies of the bandheads of the linear-alpha-chainstates in ~2Cand t60 come about a factor of two too high in energy. On the other hand, the Nilsson model gets the energies about right. In an analytic model using a zero-range Skyrmeinteraction with asymptotic deformed harmonic oscillator wave functions, the energies of the linear chain states come down considerably and are in better agreement with experiment. In more sophisticated HartreeFock calculationswith the above zero-rangeinteraction, these states come down too low in energy,but the overall analysissuggests lowering the magnitude Of the finite-range terms in the Skyrme interactions in order to get the linear chain states at the right energies. The zero-range Skyrme interaction gives a close realization of the Nilsson model when a constraint of volume conservation is imposed.
1. Introduction In a previous work [ 1 ] we noted that with the Skyrme III interaction, the energy of the J ~ = 0 + b a n d h e a d for the linear-alpha-chain states in ~2C a n d ~60 was approximately a factor of two too high compared with experiment. With the Nilsson model calculation [2 ], however, the states come at a reasonable energy a n d give support to the ideas of Brink [ 3 ] on alpha-clustering states. In this work we will investigate this problem more deeply by considering an analytic model invoking a zero-range Skyrme interaction with densities obtained from h a r m o n i c oscillator wave functions. We will also perform self-consistent Hartree-Fock ( H F ) calculations with the zerorange interaction. Our h a m i l t o n i a n is
/-/= E +t3
+to lE< J ~
i
8(ri-rj)t~(rj-rk).
the finite-range tl and 12 terms, the spin-orbit W t e r m a n d the C o u l o m b interaction are set equal to zero. The normalized one-dimensional h a r m o n i c oscillator wave functions are
1
q/.(x/b) = b-q)5 ~
1
exp( - x 2 / 2 b 2)
×H.(x/b),
(2)
where Hn are hermite polynomials which, for n = 0, 1, 2, a n d 3, are given by no(x) = 1 ,
Hl(x)=2x,
H2(x)=4x2-2,
H3(x)=8x 3-12x.
The density for a Slater d e t e r m i n a n t using harm o n i c oscillator wave functions is
p(x,y,z) (l)
=
E
[Vnx(x/bx)~ny(y/by)~n,(z/bz)] 2.
occupied
(3) This is equivalent to a Skyrme interaction in which
The energy is given by
0370-2693/92/$ 05.00 © 1992 Elsevier SciencePublishers B.V. All fights reserved.
1
Volume293, number 1,2
PHYSICSLETTERSB
1 (hco~G +hcorG +hco~Sz) + ]to f p : d 3 r + 3 ( ' 3 ~ ~p3d3 r 8\67
(4)
where ho~=h2/rnb~ and i=x, y, z. The first term on the right-hand-side of the above equation is the kinetic energy. The quantities X~ are given by X~= sum(N~+ ½) with N~ equal to the number of quanta in the/-direction. The factor ( A - 1 )/A is inserted to remove approximately spurious contributions from the center-of mass motion. This is a convention of Skyrme interactions which can be improved upon
22 October 1992
We next consider the linear-alpha-chain state for which we assume axial prolate symmetry (bx= by-b± ~ bz) and put nucleons in states in which all the quanta are in the z-direction. The occupied states are then (nx, ny, nz) = (0, 0, 0), (0, 0, 1 ), (0, 0, 2), and (0, 0, 3). To compute the kinetic energy we have
[11 2Z,± -X~+Xy=A,
Z==~A 2,
(7)
where A is the mass number. The density is 2
p(x,y, z ) _ ~r3/Eb3 (3+692-494+]g 6) ×exp( _ r E ) ,
[4].
(8)
where b 3 - b~bybzand we have defined
X=x/bx, y=y/by,
2o 160
For 260, the ground state is spherically symmetric so we take bx= by= bz = b. With four nucleons in the 0s shell and twelve in the 0p shell, X=Zx+27y+Z~= 36 and the density is 4
(
2r2~
:
2
p(r)= u3-57~b3 _1+ -~-) e x p ( - r /b ) (r2=x2 + y2 +z 2) .
699.859
(5)
/2.952454~
b-------v~+tok
-~
)
t3 (1.919973) + ~\. ~-g -].
The energy is then
E(b,
b = ) = ( A _ ~ ) hrnF bz
A2 b 2
{2.933112' t3 (2.0b1235) +to k ~-3 ) + -~ •
We4hefi obtain the energy for the ground state of 16 0 as E(b)=
~,=z/b=,
(6)
To determine the parameters to and t3, we demand that
Since the potential energy depends only on b for a zero-range interaction, as is evident from the above equation, the condition OE(b, b~)/Ob~=O (with b fixed) is equivalent to minimizing the kinetic energy alone with respect to bz, i.e.,
NLA-b-+-i- g
E(b)=
These two equations are solved to give to --- -990.078 (MeV fm 3) , t3 = 17016.238(MeV fm 6) . The resulting ground state kinetic energy is 234.93 MeV and the potential energy is - 362.55 MeV.
xe=0.
(10)
This yields bffb=(A/4) ~/3. With this choice the expression for the linear chain state energy (9) becomes
E ( b = 1.72603)= - 127.624 MeV, db _Jb=1.72603= 0 "
(9)
740.638 {2.933112) b--V - - +to~ ~-3
t3 (2.021235~ + ~-\- ~-g .].
(11)
If we constrain b to be the same as that for the ground state, i.e., b= 1.72603 fro, the intrinsic state energy of the linear chain is -99.35 MeV. This can be broken up into a kinetic energy of EK=248.61 MeV and a potential energy of E p = - 347.96 MeV.
Volume 293, number 1,2
PHYSICS LETTERS B
The change relative to the ground state is as follows: A E = 28.27 M e V , AEK = 13.68 M e V ,
AEp = 14.59 M e V .
22 October 1992
netic a n d potential energies relative to the ground state in this volume n o n c o n s e r v a t i o n case are as follows: A E = 26.68 M e V ,
Note the near equality of AEK and AEp. In the Nilsson model with deformed h a r m o n i c oscillator wave functions, AEK is equal to AEp by virtue o f the virial theorem. Thus we see that the zero-range Skyrme interaction used here gives us a very close realization of the Nilsson model. The value of AE itself is also close to that obtained in the Nilsson model (25.89 MeV) [ 1 ]. Furthermore the value of AE is m u c h smaller than that obtained with the finite-range Skyrme interactions, e.g., the value of AE with the Skyrme III interaction is 37.23 MeV. Recalling from our previous work [ 1 ] that, after the angular m o m e n t u m projection, the J ~ = 0 + state comes about 10 MeV lower than the intrinsic state energy, we see that the zero-range Skyrme interaction gives us a linear chain state at an excitation energy of about 18 MeV, which is rather close to the experimental value of about 16 MeV [5 ]. If we allow the v o l u m e to change, we find that the linear chain state energy (11 ) has a m i n i m u m at b = 1.783 fm. The corresponding energy is - 100.94 MeV, which is 1.59 MeV lower than the energy when b=bg.s.= 1.72603 fm is used. The changes in the ki-
AEK = - 1.96 M e V ,
AEp = 2 8 . 6 5 M e V .
We see that the previous near equality of AEK a n d AEv in the volume conservation case is completely destroyed; indeed AE~¢ changes from positive to negative. A s u m m a r y of the results is given in table 1. The results for the ground state a n d the linear chain state it/'2C, which will be discussed next, are also shown / there.
3. 12C
We now consider ~2C. In ref. [ 1 ] it was noted that the excitation energy of the intrinsic linear-alphachain state in t2C in a Skyrme III H F calculation is much higher than that in the Nilsson model calculation. The respective n u m b e r s are 20.96 MeV for the Skyrme III H F and 9.64 MeV for the Nilsson model. The ground state in ~2C is of oblate shape. We again assume axial deformation and put the nucleons in the states (nx, nr, n~) = (0, 0, 0), ( 1, 0, 0), and (0, 1, 0).
Table 1 The ground state (ground) and the linear-alpha-chain state (chain) in ~60 and ~2Cwith a zero-range Skyrmeinteraction (see text) and densities obtained from asymptotic harmonic oscillator wave functions. Here E.r=EK+Ep where EK and Ep are kinetic and potential energiesrespectively.Changesin the various quantities listed are relative to the ground states. Valuesare consistent with the use of MeV as unit of energyand fm as unit of length. Nucleus
State
b
( bJb ) 3
Er~
160
ground chain a) change a)
1.726 1.726 0.000
1 4 3
234.93 248.61 13.68
chain b) change b~
1.783 0.057
4 3
ground chain a) change ")
1.670 1.670 0.000
chain b) change b~
1.699 0.029
~2C
Ep
Er
-362.55 347.96 14.59
- 127.62 - 99.35 28.27
232.96 -- 1.96
-- 333.90 28.65
-- 100.94 26.68
0.6 3 2.4
172.46 176.94 4.48
- 259.94 - 255.43 4.50
- 87.48 - 78.49 8.99
3 2.4
170.92 -- 1.54
-- 249.75 10.19
-- 78.83 8.65
a) b for the linear chain state is constrained to be the same as that for the ground state. b) b for the linear chain state is varied so as to obtain the lowest linear chain state.
-
-
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PHYSICSLETTERSB
We have 27± = 20 and Zz = 6. The density is
p(x,y,z)-
4 zt3/Zb3 (1 +2~2+2)~ 2) e x p ( - F 2) . (12)
The expression for the energy is then obtained and is minimized with respect to bz at (bJb) = (3/5)t/3. The final expression for the ground state energy is
E(b)-
524.697
-~
)
4. H a r t r e e - F o c k c a l c u l a t i o n s
t3 (1.1~.8_750) +~.
(13)
For the linear chain state, we put nucleons in the states (nx, ny, nD = (0, 0, 0), (0, 0, 1 ), and (0, 0, 2). Eq. (7) with A = 12 also applies to the ~2C case. The density is 4 nS/:b3 (3+292) e x p ( - V : ) .
The energy is minimized at given by
E(b)-
(bz/b) =
(14)
(A/4)~/3 and is
538.3315 + t (1.910762"~
b~
ol
t3 / 1.1530875.~
"[- 6 ~
b6
If we allow the volume to change, the linear chain energy is minimized at b--1.699 fm. The minimal energy is - 7 8 . 8 3 MeV which is only 0.34 MeV smaller than the value for b = bg.s.= 1.670 fm. But the values of AEI< and AEp relative to the ground state become - 1.54 MeV and 10.19 MeV respectively so the near equality of AEK and AEp gets destroyed.
[ 1.904809"~
b ~ +to~
p(x,y, z ) -
22 October 1992
7~
] (15)
/]"
With the previously determined to and t3 parameters, the intrinsic energy of the ground state has a minimum of - 8 7 . 4 8 MeV at b = 1.670 fm (determined by h2/mb 2= 45A - ~/3- 25A - 2/3). This is 4.69 MeV higher than the expedmental value of - 9 2 . 1 7 MeV. But as we noted in ref. [ 1 ], the ~ = 0 ÷ state is about 4 MeV lower than the intrinsic state, thus in better agreement with experiment. The intrinsic energy of the linear chain state is - 7 8 . 4 9 MeV, which is 8.99 MeV above the intrinsic ground state energy. The values of AEK and AEp are 4.48 MeV and 4.51 MeV respectively, which are again fairly close to each other. The intrinsic excitation energy of 8.99 MeV for the linear chain state is very close to that in the Nilsson model (9.64 MeV) [1]. This is only 1.33 MeV higher than the experimental value of 7.65 MeV for the 0 + state in ~2C. But after the angular momentum projection, the J ~ = 0 + member of the linear chain is only about 6 MeV above the the 0 ÷ ground state.
In this section we use a Hartree-Fock program to calculate the excitation energies of the linear chain states. The program expands the deformed singleparticle orbits in terms of cylindrically symmetric harmonic oscillator wave functions in which the total number of major shells can be controlled. We will first give results with the minimal number of major shells ( N = 3) so a comparison with the results in the previous sections can be made. We will then perform a better HF calculation in which more major shells ( N = 13) are taken into account, The effects of the spin-orbit interaction are examined. Finally the HF results for the Skyrme I!I interaction are also shown for the sake of comparison. In table 2, we list various contributions to the total energy of ~80: kinetic (EK), volume (Ev), surface (Es), spin-orbit (Eso), Coulomb (Ec) and total (ET), energies. In order to compare the results with those in the previous sections which do not include the Coulomb 'energy, we also list the quantity E~r-Ex-Ec. We first perform the calculation in a small space ( N = 3) with a simplified Skyrme interaction which includes the Coulomb interaction but which is otherwise the same as the simple zero-range Skyrme interaction as discussed in the previous sections ( t o = - 9 9 0 . 0 7 8 , / 3 = 17016.238, t l = t 2 = W = 0 ) . The intrinsic energy for the ground state is e T = - 114.60 MeV which is somewhat higher than the experimental value of - 127.624 MeV. But when the Coulomb energy is excluded, the result is E~ = - 128.57 MeV, this is very close to the result in section 2. The intrinsic excitation energy of the linear chain state is 22.59 MeV which is smaller than the value obtained in section 2 (26.68 MeV). The main reason is the inclusion of the Coulomb interaction in the HF calculation. If we exclude the Coulomb contribution, the change in energy (AE~) is 26.25 MeV
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22 October 1992
Table 2 Hartree-Fock calculations for 160. Here N is the number of major shells allowed in the calculation, W the spin-orbit force strength, EK the kinetic energy, Ev the potential energy, Es the surface energy, Eso the spin-orbit energy, Ec the Coulomb energy, Ex the total energy. Changes in the various quantities listed are relative to the ground states. Values are consistent with the use of MeV as unit of energy and fm as unit of length. N
W
State
3
0
ground chain change
13
0
ground chain change
13
120.0
ground chain change ground chain change
Skyrme III
EK 234.51 236.93 2.42
Ev
Es
E~o
Ec
ET
Er--Ec
-363.09 - 339.25 23.84
0.00 0.00 0.00
0.00 0.00 0.00
13.97 10.31 - 3.67
- 114.60 - 92.01 22.59
- 128.57 - 102.32 26.25
- 381.11 - 344.98 36.13
0.00 0.00 0.00
0.00 0.00 0.00
14.11 10.50 -3.60
- 119.31 - 98.96 20.35
- 133.41 - 109.46 23.95
252.18 245.96 - 6.22
-383.05 -350.58 32.46
0.00 0.00 0.00
-4.25 - 10.69 - 6.45
14.11 10.67 - 3.43
- 121.01 - 104.64 16.37
- 135.11 - 115.31 19.80
229.65 219.69 -9.96
-425.39 -380.00 45.39
54.91 68.23 13.32
-0.88 -8.03 -7.15
13.77 10.56 -3.21
- 127.94 -89.55 38.39
- 141.71 - 100.11 41.61
247.69 235.51 - 12.18
which is m u c h closer to the previous result. N o t e that the surface energy is zero with the simple zero-range interaction which does not have the finite-range t~ and t2 terms. This is mainly a m a t t e r o f definition: what the code calls surface energy is precisely the contribution from these terms in the interaction, which i n v o l v e gradients o f the density. In our zero-range interaction, we have changed the balance between the various terms, while preserving binding energy in the ground state. This is a key to understand why we are getting the linear chain state so low in this m o d el (as well as in the Nilsson m o d e l ) . Obviously the linear chain state will have a larger surface area than the ground state which is spherical (~60) or less d e f o r m e d (~2C). T h e surface energy is repulsive, as is well known from the semi-empirical mass formula. W h e n we use the same interaction but allow for 13 m a j o r shells, the excitation energy o f the linear chain state in ~60 changes from 22.59 M e V to 20.35 MeV. We are getting the linear chain state somewhat too low in energy as we recall that the J r = 0 + state energy will be about 10 M e V lower than the intrinsic state energy. This situation is exacerbated when we turn on the s p i n - o r b i t interaction. T h e linear chain state is lowered m o r e than the ground state by the s p i n - o r b i t interaction. I n d e e d for the ground state it is only sec-
o n d - o r d er effects o f the s p i n - o r b i t interaction that enter. The final intrinsic excitation energy o f the linear chain state is 16.37 MeV which is close to the experimental b a n d h e a d energy ( ~ 16 M e V ) . But as m e n t i o n e d before, the J r = 0 + m e m b e r is about 10 M e V lower than the intrinsic state energy. Finally with the Skyrme III interaction we find that the intrinsic state energy is very high at 38.39 MeV. the culprits are the changes in v o l u m e and surface energies which are 45.39 MeV and 13.32 M e V respectively. Fo r the zero-range interaction, the values were 32.47 and 0 MeV. The difference is 26.24 MeV.
5. Closing remarks We have shown that in an analytic model - the zerorange interaction with d e f o r m e d h a r m o n i c oscillator wave functions - the linear chain states can be brought to much lower excitation energies in ~2C and ~60. W h e n better H a r t r e e - F o c k calculations are perf o r m e d with the above zero-range interaction, there is an overshoot: The linear chain states c o m e to low in energy. This is in contrast to the usual Skyrme interactions, such as Skyrme III, which gives the linear chain states about a factor o f two too high in energy. Th e overshoot in the H F calculations is due to the fact that the C o u l o m b and the s p i n - o r b i t interac-
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PHYSICS LETTERS B
tions as well as inclusion o f m o r e m a j o r shells, all conspire to give an a d d i t i o n a l lowering o f the linear chain states relative to the analytic m o d e l which uses asymptotic d e f o r m e d oscillator wave functions. O u r analysis clearly indicates that one can get the linear chain states to come down in energy using Skyrme interactions b y lowering the magnitude o f the finite-range potentials t, a n d t2. Obviously such a procedure will lower the surface energy (with a zerorange Skyrme interaction, the surface energy will be zero). Less obviously the change in v o l u m e energy in going from a ground state to a linear chain state becomes less when the finite-range terms are reduced. We should study this p o i n t further in the near future. We further note that we get a close realization o f the Nilsson m o d e l by using a zero-range Skyrme m o d e l a n d impose the condition o f the v o l u m e conservation.
22 October 1992
Acknowledgement We acknowledge the support o f the U S D O E u n d e r grant DE-FG05-86ER-40299 a n d o f N S E R C , Cana d a under research grant A-3198.
References [ 1] L. Zamick, D.C. Zheng, S.J. Lee, J.A. Caballero and E. Moya de Guerra, Ann. Phys. 212 ( 1991 ) 402. [2] S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955). [ 3 ] D.M. Brink, Many-body description of nuclear structure and reactions, Intern. School' of Physics Enrico Fermi, Course XXXVI, ed. C. Bloch (Academic Press, New York, 1966) p. 247. [4] M.N. Butler, D.W.L. Sprung and J. Martorell, Nucl. Phys. A 422 (1984) 157. [ 5 ] W.D.M. Rae, Proc. Fifth. Conf. on Clustering aspects in nuclear and subnuclear systems (Kyoto, 1988 ), J. Phys. Soc. Jpn. Suppl. 58 (1989) 77.