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Spin saturation and the Skyrme interaction
Volume 56B. number 3
PHYSICS LETTERS
SPIN SATURATION
28 April 1975
AND THE SKYRME INTERACTION
~r
B.D. CHANG* Argonne National Laboratory, Argonne, ill. 60439, USA and The University o[ Chicago, Chicago, Ill. 60637, USA Received 18 March 1975 Of existing variants of the Skyrme interaction, those with strong three-body terms - in particular the variant SIII that is in best accord with experiment - overbind odd-mass and odd-odd nuclei and produce unstable spin-saturated Hartree-Foek ground states in nuclear matter and in even-even light nuclei. This difficulty can be removed either by imposition of an additional stability condition or by abandoning the three-body term in favor of the two-body density-dependent interaction equivalent to it in spin-saturated HF states. Nuclear Hartree-Fock (HF) calculations can be carried out within a finite basis of single-particle states provided that the free nucleon-nucleon interaction is replaced by a "soft" effective interaction. Such an effective interaction is necessarily density-dependent [1 ] and o f the variety of density-dependent interactions proposed the Skyrme interactions [2, 3] is particularly attractive because o f its simplicity. It contains a two-body and a three-body part, both o f zero range. In the notation of ref. [3] the matrix elements of the two-body term in momentum space are (kl V2 Ik') = / o ( 1 + x 0 P a) +-~ t 1 (k2+ k '2) + t 2 k ' k ' + iW0(. 1 + . 2 ) - ( k X k')
(1)
while the three-body term is V3 = t3~ (r 1 - r2)~ (r 1 - r3).
(2)
The Skyrme interaction is characterized by the six parameters t O, t 1, t2, Xo, W0 and t 3. The Skyrme interaction has been used with considerable success in HF studies of the ground-state properties of spherical and deformed nuclei and o f deformation- and fission-energy curves. A detailed analysis of the influence of the interaction parameters on the properties of spherical closed-shell nuclei, with references to earlier work, is given by Beiner et al. [4]. They list five of the six sets of interaction parameters given in table 1 ; the sixth (SI) is taken from ref. [3]. All six variants (SI to SVI) o f the Skyrme interaction give an adequate description o f the Table 1 The sets of parameters, defined as in eqs. (1) and (2), that specify the variants SI, SII, Sill, SIV, SV and SVI of the Skyrme interaction, to is given in MeV fm 3, tl, t2 and Wo in MeV fm s, t3 in MeV fm 6 while xo is dimensionless, as and a are the nuclear-matter parameters def'med by eq. (7) and (8); they are given in MeV. Interaction
to
tI
t2
I U fiI IV V VI
-1057.3 -1169.9 -1128.75 -1205.6 -1248.29 -1101.81
235.9 585.6 395 765 970.56 271.667
-100 -27.1 -95 35 107.22 -138.33
t3 14463.5 9331.1 14000 5000 0 '17000
WO
XO
as
a
120 105 120 150 150 115
0.56 0.34 0.45 0.05 -0.17 0.583
29.2 34.2 28.2 31.2 32.7 26.9
45.9 29.5 36.8 4.6 -16.1 46.6
¢,Work performed under the auspices of the U.S. Atomic Energy Commission. * Presented as a thesis to the Dept. of Physics, The University of Chicago, in partial fulfillment of the requirements for a Ph.D. degree. 205
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
ground-state properties of closed-shell spherical nuclei, but consideration of the density of single-particle states near the Fermi surface tends to favor those with stronger density dependence (Sill and SIV in particular). Most nuclear HF calculations have concerned even-even nuclei with HF ground states that are spin-saturated in the sense that each occupied state contains a nucleon with spin up and a nucleon with spin down. This property of spin saturation is usually enforced by the imposition on the HF ground state of time-reversal symmetry. To see the influence of this constraint we note that the three-body contribution to the HF energy is
H3:¼t 3 fd3r[Pn(r)p2p(r)+ pp(r)p2n(r) -S2n(r)pp(r) S2p(r)Pn(r)]
(3)
-
where Pn, Pp are the neutron and proton densities and S n, Sp are the corresponding spin densities S(r) = Y~a~*(r)~¢~(r). Since t 3 is positive, the spin-dependent terms give an attractive contribution to the energy that tends to cancel the repulsive influence of the spin-i.~dependent terms. Time-reversal symmetry forces S n, Sp to vanish since each spin-up state occurs with its spin-down partner and the contributions to S cancel in pairs. Thus there is a potential interplay between the three-body term that is suppressed by imposing time-reversal symmetry. To study the influence of this interplay, we studied the HF ground-state energy of the odd nucleus 14N 7 which has two unpaired nucleons and cannot be.forced to have a spin-saturated ground state. The calculations were carried out in a spherical oscillator basis truncated above the I f-2p shell, with Coulomb and center-of-mass contributions (including exchange terms) taken into account. The results of these deformed HF calculations, using the six interactions SI to SVI, are shown in fig. 1, where the ground-state energy is plotted as a function of the oscillator length parameter b. The fact that the results depend on b indicates that the basis has been too severely truncated; this is of no concern here since we wish to draw conclusions about overbinding for which purposes it suffices that the exact HF ground-state energy is bounded above by the minimum truncated-HF energy as a function of b. It is then clear from fig. 1 that SI, SIII and SIV (the variants of the Skyrme interaction with strong density dependence) seriously overbind 14N.
,',N -I00 ~ -IOC =--E(14;
-Vtl=O~
>
/" // /.
°°200 W //
,,4 -20f2
A/
I
/ //
-300
1.3 1.4 1.5 1.6 1.7 1.8 b (fro) Fig. 1. Plot o f Hartree-Fock binding energy against oscillator length b for 14N. The Roman numerals I, I1, etc. refer to the variant SI, SII, etc. o f the Skyrme interaction used in the HF calculations. The single-particle basis was truncated at the lf2p shell (N = Z = 20). The experimental ground-state energy of 14N is indicated.
206
/ I
1.3 114 b(fm) Fig. 2. Binding energy as a function of oscillator length for 160. The heavy line represents the results of Hartree-Fock calculations with time-reversal symmetry imposed; the dotted line was obtained by lifting this constraint. The interaction used was SIII.
Volume 56B, number 3
PHYSICS LEIffERS
28 April 1975
A similar phenomenon might be expected to occur in 160 if the customary constraint to a time-reversal symmetric HF ground state is lifted. Fig. 2 shows the HF ground-state energy of 160, with and without the constraint of time-reversal symmetry, computed and plotted as described for 14N; the interaction used was Sill but similar results are obtained with SI and SVI. The overbinding noted in connection with 14N is again manifest. The spinsaturated, time-reversal symmetric HF ground state of 160 is unstable against re-alignment of intrinsic spins. To study the dependence of spin-saturation on the interaction parameters consider a simple model of nuclear matter. Let us suppose that in the ground state single-particle (plane-wave) levels with tkl ~ k L are occupied by spin-up, spin-down pairs of nucleons while levels with k L ~< Ikl <~k F are occupied by one nucleon and one proton with spin up. The parameter D = kL/k F then provides a measure of the degree of spin saturation. The ground-state energy of nuclear matter at density P _ (I ÷ D3) ~.3 31r2
"F
and degree of spin saturation D with the Skyrme interaction is then E (kF, D) = h2 3 (1 +D 5) k 2 + _ _ m 10 (1 +D3) 4rt 2
-~(1+D 3) + -~(xo - -~) (1 (1 +D 3)
k5 / ( 1 - DS)(I---D3) l ' r t3k6. - - u " ..3, + 8Orr 2~F ~(3tl +'5t2)(1 +D5) + (t2 - tl) (1 +D 3) j 36rr4
(4)
where m is the nucleon mass. The equilibrium density and spin-saturation parameter are found by minimizing E/A; for a spin-saturated ground state (D = 1) the conditions for an extremum in k F and D are identical yielding a single equation for k F. All six sets of Skyrme parameters in table 1 yield k F ~ 1.3 fm -1 (smaller than the usual value of 1.36 fm-1). The condition that the spin-saturated ground state be stable can then be obtained from the requirement that at k F "" 1.3 fm -1 and D = 1,
a2(e/A!]
a2(E/A) ~2(E/A) ,~0.
(5)
an In other words, the extremum at k F ~ 1.3 fm -1 , D = 1 is not a saddle point. Assuming that the parameters of the Skyrme interactions have been chosen such that the nuclear compressibility parameter is positive (true of all six interactions in table 1), substitution of the expression (4) for E/A in the equality (5) casts the spin-stability condition into the explicit form a s >~ c~;
(6)
where a s , the symmetry energy, is given by
as =
6m - 6,2
91r-"2
36rr4
(7)
and
~=
-
-3n- X2 o t O.
(8) 207
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
Values o f a s and a for interactions SI to SVI are listed in table 1 ; the values o f a s are close to the 30 MeV obtained from the semi-empirical mass formula since this was used as a constraint in determining the interaction parameters. It is seen from table 1 that the interactions SI, Sill and SV1 which were found to yield an unstable spin-saturated HF ground state of 160 and to permit overbinding due to spin realignment in 14N violate the condition (6) for the stability of the spin-saturated ground state of nuclear matter. These interactions are the ones with the strong three-body term that seems to be required to reproduce the observed density of single-particle levels near the Fermi surface of spherical nuclei [4]. In particular the spin-stability condition is violated by the interaction Sill that is in best overall accord with experiment. The problem of spin-instability is a direct cons.-quence of the fact that the Skyrme interaction contains a zerorange three-body term rather than a density-dependent two-body term. This is clear on inspection of the expression (3) for the three-body contribution to the HF energy. The first two terms give the desired repulsive effect (t 3 > 0) of a density-dependent interaction. The additional terms involving the spin averages then yield attractive contributions that lead to overbinding or spin collapse. This can be prevented by constraining the spin averages to be zero, by forcing the spin-dependent two-body part of the interaction to compensate (hence the stability condition) or by keeping the entire three-body term small. If the three-body term V3 in the Skyrme interaction is simply replaced by the density-dependent two-body term
t3(l + Po)p(R)6 (r I - r2)
(9)
that is equivalent to V3 in time-reversal symmetric HF states, then the problems with spin-instability and overbinding do not arise. There are then two possible responses to the problem of spin instability. 1) If we wish to use a three-body rather than a density-dependent two-body interaction, the search must continue for a Skyrme parametrization that satisfies the stability condition (6) and accounts satisfactorily for the experimental data. 2) The three-body term V3 in the Skyrme interaction should be replaced by the density-dependent two-body term given by eq. (9). This is what must be done if the interaction Sill is to be used - certainly desirable in view of the success with which Sill reproduces a wider variety of nuclear properties [4]. In other words, the contact three-body term in the Skyrme interaction is to be viewed merely as a device that introduces a suitable density-dependence; all physical consequences are to be based on this equivalent density-dependent interaction. Note finally that problems with spin-stability can also be avoided by lifting the restriction that the three-body interaction be of zero range. However, introduction of a finite-range three-body force destroys the simplicity and numerical convenience of the Skyrme interaction; it is better then to introduce a density-dependent two-body interaction directly. The author is deeply grateful to his thesis advisor, Professor Malcolm H. Macfarlane, for his valuable guidance and constant encouragement throughout the course of this work. Thanks are also due to Dr. B. Day for stimulating discussions concerning spin saturation in nuclear matter.
References [1] [2] [3] [4]
208
J.W. Negele, Phys. Rev. C1 (1970) 1260. T.H.R. Skyrme, Phil. Mag. 1 (1956) 1043. D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. M. Beiner, H. Flocard, Nguyen van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29.
PHYSICS LETTERS
SPIN SATURATION
28 April 1975
AND THE SKYRME INTERACTION
~r
B.D. CHANG* Argonne National Laboratory, Argonne, ill. 60439, USA and The University o[ Chicago, Chicago, Ill. 60637, USA Received 18 March 1975 Of existing variants of the Skyrme interaction, those with strong three-body terms - in particular the variant SIII that is in best accord with experiment - overbind odd-mass and odd-odd nuclei and produce unstable spin-saturated Hartree-Foek ground states in nuclear matter and in even-even light nuclei. This difficulty can be removed either by imposition of an additional stability condition or by abandoning the three-body term in favor of the two-body density-dependent interaction equivalent to it in spin-saturated HF states. Nuclear Hartree-Fock (HF) calculations can be carried out within a finite basis of single-particle states provided that the free nucleon-nucleon interaction is replaced by a "soft" effective interaction. Such an effective interaction is necessarily density-dependent [1 ] and o f the variety of density-dependent interactions proposed the Skyrme interactions [2, 3] is particularly attractive because o f its simplicity. It contains a two-body and a three-body part, both o f zero range. In the notation of ref. [3] the matrix elements of the two-body term in momentum space are (kl V2 Ik') = / o ( 1 + x 0 P a) +-~ t 1 (k2+ k '2) + t 2 k ' k ' + iW0(. 1 + . 2 ) - ( k X k')
(1)
while the three-body term is V3 = t3~ (r 1 - r2)~ (r 1 - r3).
(2)
The Skyrme interaction is characterized by the six parameters t O, t 1, t2, Xo, W0 and t 3. The Skyrme interaction has been used with considerable success in HF studies of the ground-state properties of spherical and deformed nuclei and o f deformation- and fission-energy curves. A detailed analysis of the influence of the interaction parameters on the properties of spherical closed-shell nuclei, with references to earlier work, is given by Beiner et al. [4]. They list five of the six sets of interaction parameters given in table 1 ; the sixth (SI) is taken from ref. [3]. All six variants (SI to SVI) o f the Skyrme interaction give an adequate description o f the Table 1 The sets of parameters, defined as in eqs. (1) and (2), that specify the variants SI, SII, Sill, SIV, SV and SVI of the Skyrme interaction, to is given in MeV fm 3, tl, t2 and Wo in MeV fm s, t3 in MeV fm 6 while xo is dimensionless, as and a are the nuclear-matter parameters def'med by eq. (7) and (8); they are given in MeV. Interaction
to
tI
t2
I U fiI IV V VI
-1057.3 -1169.9 -1128.75 -1205.6 -1248.29 -1101.81
235.9 585.6 395 765 970.56 271.667
-100 -27.1 -95 35 107.22 -138.33
t3 14463.5 9331.1 14000 5000 0 '17000
WO
XO
as
a
120 105 120 150 150 115
0.56 0.34 0.45 0.05 -0.17 0.583
29.2 34.2 28.2 31.2 32.7 26.9
45.9 29.5 36.8 4.6 -16.1 46.6
¢,Work performed under the auspices of the U.S. Atomic Energy Commission. * Presented as a thesis to the Dept. of Physics, The University of Chicago, in partial fulfillment of the requirements for a Ph.D. degree. 205
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
ground-state properties of closed-shell spherical nuclei, but consideration of the density of single-particle states near the Fermi surface tends to favor those with stronger density dependence (Sill and SIV in particular). Most nuclear HF calculations have concerned even-even nuclei with HF ground states that are spin-saturated in the sense that each occupied state contains a nucleon with spin up and a nucleon with spin down. This property of spin saturation is usually enforced by the imposition on the HF ground state of time-reversal symmetry. To see the influence of this constraint we note that the three-body contribution to the HF energy is
H3:¼t 3 fd3r[Pn(r)p2p(r)+ pp(r)p2n(r) -S2n(r)pp(r) S2p(r)Pn(r)]
(3)
-
where Pn, Pp are the neutron and proton densities and S n, Sp are the corresponding spin densities S(r) = Y~a~*(r)~¢~(r). Since t 3 is positive, the spin-dependent terms give an attractive contribution to the energy that tends to cancel the repulsive influence of the spin-i.~dependent terms. Time-reversal symmetry forces S n, Sp to vanish since each spin-up state occurs with its spin-down partner and the contributions to S cancel in pairs. Thus there is a potential interplay between the three-body term that is suppressed by imposing time-reversal symmetry. To study the influence of this interplay, we studied the HF ground-state energy of the odd nucleus 14N 7 which has two unpaired nucleons and cannot be.forced to have a spin-saturated ground state. The calculations were carried out in a spherical oscillator basis truncated above the I f-2p shell, with Coulomb and center-of-mass contributions (including exchange terms) taken into account. The results of these deformed HF calculations, using the six interactions SI to SVI, are shown in fig. 1, where the ground-state energy is plotted as a function of the oscillator length parameter b. The fact that the results depend on b indicates that the basis has been too severely truncated; this is of no concern here since we wish to draw conclusions about overbinding for which purposes it suffices that the exact HF ground-state energy is bounded above by the minimum truncated-HF energy as a function of b. It is then clear from fig. 1 that SI, SIII and SIV (the variants of the Skyrme interaction with strong density dependence) seriously overbind 14N.
,',N -I00 ~ -IOC =--E(14;
-Vtl=O~
>
/" // /.
°°200 W //
,,4 -20f2
A/
I
/ //
-300
1.3 1.4 1.5 1.6 1.7 1.8 b (fro) Fig. 1. Plot o f Hartree-Fock binding energy against oscillator length b for 14N. The Roman numerals I, I1, etc. refer to the variant SI, SII, etc. o f the Skyrme interaction used in the HF calculations. The single-particle basis was truncated at the lf2p shell (N = Z = 20). The experimental ground-state energy of 14N is indicated.
206
/ I
1.3 114 b(fm) Fig. 2. Binding energy as a function of oscillator length for 160. The heavy line represents the results of Hartree-Fock calculations with time-reversal symmetry imposed; the dotted line was obtained by lifting this constraint. The interaction used was SIII.
Volume 56B, number 3
PHYSICS LEIffERS
28 April 1975
A similar phenomenon might be expected to occur in 160 if the customary constraint to a time-reversal symmetric HF ground state is lifted. Fig. 2 shows the HF ground-state energy of 160, with and without the constraint of time-reversal symmetry, computed and plotted as described for 14N; the interaction used was Sill but similar results are obtained with SI and SVI. The overbinding noted in connection with 14N is again manifest. The spinsaturated, time-reversal symmetric HF ground state of 160 is unstable against re-alignment of intrinsic spins. To study the dependence of spin-saturation on the interaction parameters consider a simple model of nuclear matter. Let us suppose that in the ground state single-particle (plane-wave) levels with tkl ~ k L are occupied by spin-up, spin-down pairs of nucleons while levels with k L ~< Ikl <~k F are occupied by one nucleon and one proton with spin up. The parameter D = kL/k F then provides a measure of the degree of spin saturation. The ground-state energy of nuclear matter at density P _ (I ÷ D3) ~.3 31r2
"F
and degree of spin saturation D with the Skyrme interaction is then E (kF, D) = h2 3 (1 +D 5) k 2 + _ _ m 10 (1 +D3) 4rt 2
-~(1+D 3) + -~(xo - -~) (1 (1 +D 3)
k5 / ( 1 - DS)(I---D3) l ' r t3k6. - - u " ..3, + 8Orr 2~F ~(3tl +'5t2)(1 +D5) + (t2 - tl) (1 +D 3) j 36rr4
(4)
where m is the nucleon mass. The equilibrium density and spin-saturation parameter are found by minimizing E/A; for a spin-saturated ground state (D = 1) the conditions for an extremum in k F and D are identical yielding a single equation for k F. All six sets of Skyrme parameters in table 1 yield k F ~ 1.3 fm -1 (smaller than the usual value of 1.36 fm-1). The condition that the spin-saturated ground state be stable can then be obtained from the requirement that at k F "" 1.3 fm -1 and D = 1,
a2(e/A!]
a2(E/A) ~2(E/A) ,~0.
(5)
an In other words, the extremum at k F ~ 1.3 fm -1 , D = 1 is not a saddle point. Assuming that the parameters of the Skyrme interactions have been chosen such that the nuclear compressibility parameter is positive (true of all six interactions in table 1), substitution of the expression (4) for E/A in the equality (5) casts the spin-stability condition into the explicit form a s >~ c~;
(6)
where a s , the symmetry energy, is given by
as =
6m - 6,2
91r-"2
36rr4
(7)
and
~=
-
-3n- X2 o t O.
(8) 207
Volume 56B, number 3
PHYSICS LETTERS
28 April 1975
Values o f a s and a for interactions SI to SVI are listed in table 1 ; the values o f a s are close to the 30 MeV obtained from the semi-empirical mass formula since this was used as a constraint in determining the interaction parameters. It is seen from table 1 that the interactions SI, Sill and SV1 which were found to yield an unstable spin-saturated HF ground state of 160 and to permit overbinding due to spin realignment in 14N violate the condition (6) for the stability of the spin-saturated ground state of nuclear matter. These interactions are the ones with the strong three-body term that seems to be required to reproduce the observed density of single-particle levels near the Fermi surface of spherical nuclei [4]. In particular the spin-stability condition is violated by the interaction Sill that is in best overall accord with experiment. The problem of spin-instability is a direct cons.-quence of the fact that the Skyrme interaction contains a zerorange three-body term rather than a density-dependent two-body term. This is clear on inspection of the expression (3) for the three-body contribution to the HF energy. The first two terms give the desired repulsive effect (t 3 > 0) of a density-dependent interaction. The additional terms involving the spin averages then yield attractive contributions that lead to overbinding or spin collapse. This can be prevented by constraining the spin averages to be zero, by forcing the spin-dependent two-body part of the interaction to compensate (hence the stability condition) or by keeping the entire three-body term small. If the three-body term V3 in the Skyrme interaction is simply replaced by the density-dependent two-body term
t3(l + Po)p(R)6 (r I - r2)
(9)
that is equivalent to V3 in time-reversal symmetric HF states, then the problems with spin-instability and overbinding do not arise. There are then two possible responses to the problem of spin instability. 1) If we wish to use a three-body rather than a density-dependent two-body interaction, the search must continue for a Skyrme parametrization that satisfies the stability condition (6) and accounts satisfactorily for the experimental data. 2) The three-body term V3 in the Skyrme interaction should be replaced by the density-dependent two-body term given by eq. (9). This is what must be done if the interaction Sill is to be used - certainly desirable in view of the success with which Sill reproduces a wider variety of nuclear properties [4]. In other words, the contact three-body term in the Skyrme interaction is to be viewed merely as a device that introduces a suitable density-dependence; all physical consequences are to be based on this equivalent density-dependent interaction. Note finally that problems with spin-stability can also be avoided by lifting the restriction that the three-body interaction be of zero range. However, introduction of a finite-range three-body force destroys the simplicity and numerical convenience of the Skyrme interaction; it is better then to introduce a density-dependent two-body interaction directly. The author is deeply grateful to his thesis advisor, Professor Malcolm H. Macfarlane, for his valuable guidance and constant encouragement throughout the course of this work. Thanks are also due to Dr. B. Day for stimulating discussions concerning spin saturation in nuclear matter.
References [1] [2] [3] [4]
208
J.W. Negele, Phys. Rev. C1 (1970) 1260. T.H.R. Skyrme, Phil. Mag. 1 (1956) 1043. D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. M. Beiner, H. Flocard, Nguyen van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29.