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Mass parameters for quadrupole vibrations in the adiabatic approximation
4 August 1975
PHYSICS LETTERS
Volume 57B, number 5
MASS PARAMETERS FOR QUADRUPOLE VIBRATIONS IN THE ADIABATIC APPROXIMATION D. VAUTHERJN Institut de Physique Nuclkaire, Division de Physique Thkorique *, 91406Orsay-France Received 27 January 1975 (Revised version received 16 June 1975) Nuclear quadrupole vibrations are investigated in the framework of the adiabatic time-dependent Hartree-Fock equations. Using a continuity condition satisfied by these equations, we show that a simple analytic expression can be derived for the mass parameter in the case where the collective variable Q is a scaling variable i.e. changes the cartesian coordinates x, y, z into x/a, y/a, c?z. This result depends only on the existence of the continuity equation and thus holds for any nucleon-nucleon interaction. Applications are considered in the case of small amplitude motion in oxygen-16 and calcium40.
In this letter we apply the adiabatic time-dependent Hartree-Fock method [ 1,2] (as formulated by Baranger and VBn6roni [3]) to describe nuclear quadrupole oscillations. Let us consider the case of a single collective variable (Y,associated with a given set ao(cu) of time-even Slater determinants. Denoting by pa(o) the one-body densitymatrix corresponding to +O(CY)(and satisfying pg = po), the above authors give the mass parameter M(o) as M(ol)=ifiTrace where pl(o) +0
iA a(y=
[$
[Po,PI]]
,
is a time-odd hermitian
one-body
[p~WOPO+(l-~pg)~o(l-Po)IP~l
operator such that +
WIPPOI7
(2)
and Pl ‘POP1 +PlPo
*
(3)
In eq. (1) W. is a time-even operator whose definition FVo(l) =$-(1)
+ Trace2 V(1,2)po(2),
and the time-odd operator
is similar to that of the Hartree-Fock
field (4)
W, is given by
WI(l) = Trace2 Y(1,2)~1(2),
(5)
where Y(l, 2) denotes the antisymmetrized nucleon-nucleon interaction. Eq. (3) implies that the particle-particle and hole-hole matrix elements of pI with respect to p. are zero. As a consequence eq. (2) (which is linear in pl) has particle-hole matrix elements only and determines the particle-hole matrix elements of pl. Notice that for a rotating nucleus the previous formulae reduce to the result of Thouless and Valatin [4]. The time-dependent Hartree-Fock equations are known to lead to a continuity equation for the total particle density [S, 61. A similar relation holds in the limiting case of adiabatic motion namely
* Laboratoire associd au C.N.R.S.
425
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975 (6)
PO(r, or) + m div/(r, oe) = 0 , where
PO(r, ¢t) = (rlP0(t~) Ir),
(7)
f(r, c0 = ~-~(V 1 - V ')
.
r'=r Eq. (6) is obtained by taking the expectation value of equation (2) in a state Ir) and by noticing that since adiabaticity requires [W0, P0 ] to be small, the first commutator in eq. (2) may be replaced by [W0, p 1]" For a velocity independent force the proof follows immediately by using the observation of Thouless and Valatin [4] that only the exchange term contributes to equation (5). For Skyrme's interaction [7] the proof is slightly more complicated and may be found in ref. [6]. In order to exploit the continuity equation it is convenient to introduce the single particle wave functions ~ol(r, or) defining the Slater determinant ~0(~t) as well as a set of states It//) defined by [6] Ir//) = -iPll~Pj),
]= 1,2,...,a.
(8)
According to this definition the set It//) is invariant under time reversal and eq. (3) implies that (¢klr/i) = 0,
j , k = 1,2 ..... A .
(9)
Another consequence of eq. (3) is that Pl can be written as A Pl = i/=~l { Ir//) (~P]I- I~P/)(r//I} •
(10)
Inserting this expression into eqs. (1) and (7), and using the time reversal invariance of the sets ¢] and r//we obtain A M(a)= 2~ ~ f d 3 r 77~(,, a ) ~ ~o/(r, a) , ]=l
(11)
and A
7(., .) --
.) v r/;(,,
- r/;(., .) v
(12)
In ref. [6] the similarity between the two previous equations was used to derive an analytic expression for the mass parameter in the case of monopole vibrations. We will now show that the case of quadrupole vibrations can also be worked out analytically. Let us first denote by ~j(r) the single-particle wave functions of the Hartree-Fock ground state. In order to describe quadrupole vibrations we consider the following one-parameter set
~ol(r, ct) = ~ /(x/q, y/ot, tx2z) ,
(13)
where x , y and z are the cartesian coordinates ofr. From this definition it is easy to show that
a
ba ~Pl(r' a) = ~ V (2z 2 - x 2 - y2). V~p/(r, oe).
Comparing eqs. (11), (12) and (14), the expression for the mass parameter M(tx) becomes
426
(14)
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
M(ot)=_~_~f d3r,l~(2z2 x 2 - y 2-+ )ff(r,~).
(15)
Multiplying the continuity equation (6)by 2z 2 - x 2 - y 2 , integrating over d3r we obtain after integration by parts 1
a
M(o0 = - ~-~ m ~-~ a ( a ) ,
(16)
where Q(o0 is the expectation value of the quadrupole moment in the state qs0(a) i.e.
a(a) - - f d 3 r
(2z 2 - x 2 -
y2)po(r, a).
(17)
Using the scaling property (I 3) this quantity can be expressed in terms of the radius r 0 and the quadrupole moment Qo of the Hartree-Fock ground state, which corresponds to a = 1. Explicitly e(t~) =-~Ar 2 ~ - ~ - a 2 + ~ Q o
\or4
]
where A is the mass of the nucleus and r 0 and Qo are defined by
r2=l fonv(r)r2d3r,
Qo= fOrlF(r)(2z2-x2-y2)d3r.
(19)
Inserting eqs. (18) into (16) we obtain the following expression for the mass parameter
Notice that since this result depends only on the existence of the continuity equation, it is valid for any nucleonnulceon interaction. According to the authors of ref. [3] collective spectra are obtained by considering the collective Harniltonian h(a, or) =lM(a)~t2 + V(t~).
(21)
where the collective potential energy is V(a) = (qs0(a) IHIqS0(ot)).
(22)
As an application of eq. (20) let us now calculate the energy of the quadrupole vibration in oxygen-16 and calcium40, using Skyrme's interaction [7] to construct V(a). For simplicity we assume that the Hartree-Fock single particle wave functions can be approximated by pure harmonic oscillator wave functions with a parameter ~0 = ~ [ / f which minimizes the expectation value of the energy. In this case the expression for V(ct)is easily obtained from the formulae given in reference [8]. The result is V(~) = E 0 +
C-~mPO (27r)312(3Bltl
ol2
,
where t 1 and t 2 are the parameters of the Skyrme force and where E = V(~ -- 1) is the minimum of the energy calculated with pure oscillator functions. The coefficients B and C are [7, 8] C-- 36,
B 1 = 35•4,
B 2 ---4,
for oxygen-16,
C = 120,
B 1 = 2625•64,
B 2 --- 1760•64,
for calcium-40.
(24) In terms of/30, the mass parameter M(e) given by eq. (20) becomes 427
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
M(tv)=2mC1 (1+ ~ ) ,
(25)
t~0 where C is the same as in eq. (23). In the limit of small amplitude motion, i.e. ot ~, 1, the collective Hamiltonian (211 can be approximated by 1 h (a, &) = 1 M ( 1 ) &2 + E0 + ~ g(t~ - 1) 2 ,
(26)
with
I 1 r, h2 a2 + (27r)3/2 l (3B,t, + 5B2t2) t l
K = 24 [ ~ " - ~ - ~ v 0
35)
(27)
From equation (26) the energy EQ of the quadrupole mode i s - h ~
i.e.
h2 ~02 ~ K -. E Q = V -~m
(281
For interaction Skyrme-IIl, which gives a remarkable fit to a wide range o f nuclear ground state properties in Hartree-Fock calculations [9] (t 1 = 395.0 MeV fm 5 , t 2 = - 95.0 MeV fm 5) we obtain the following results 30 = 0.563 f m - 1 ,
K --- 2622 MeV,
EQ = 21.9 MeV
for oxygen-16,
30 = 0.507 f m - 1,
K = 6930 MeV,
Et2 = 17.5 MeV
for calcium-40.
(29) Similar results are obtained for interactions Skyrme-I and II [10], namely Et2(I ) = 21.8 MeV, EO(II ) = 24.2 MeV for oxygen-16 and Et2(I) = 17.2 MeV, Et2(II) = 19.7 MeV for calcium-40. The above values of Et2 are in good agreement with the theoretical results of references [8, 1 1 - 1 3 ] , with the analysts of reference [ 14] and are also consistent with available experimental data in the case of calcium-40 [ 15, 16] In oxygen-16 the experimental situation is still an unsettled question, but seems to indicate that the isoscalar quadrupole strength is spread over a wide region [17]. I would like to thank D.M. Brink, H. Flocard, N. Rowley and M. V6n~roni for stimulating discussions.
References [1] S.T. Belyaev, Nucl. Phys. 64 (1965) 17. [2] F.M.H. Villars, Proc. 1971 Mont Tremblant Intern. Summer School, ed. D.J. Rowe, p. 3. [31 M. Baranger, Journal de Physique, Supplement 33 (1972) 61; M. Baranger and M. Veneroni, to be published. [4] D.J. Thouless and J.G. Valatin, Nucl. Phys. 31 (1962) 211. [5] L.P. Kadanoff and G. Baym, Quantum statistical mechanics(Benjamin, New York, 1962) p. 56. [6] D.M. Brink, Y.M. Engel, K. Goeke, S.J. Krieger and D. Vantherin, University of Oxford preprint 76/74. [71 T.H.R. Skyrme, Nucl. Phys. 9 (1959) 615. [8] H. Flocard and D. Vautherin, Phys. Lett. 55B (1975) 259. [9] M. Beiner, H. Flocard, N.V. Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [10] D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. [11] G.F. Bertsch and S.F. Tsai, preprint, 1974. [121 S. Krewald and J. Speth, Phys. Lett. 52B (19741 295. [ 13 ] K.F. Liu, Thesis, University of Stony-Brook, 1974, unpublished. [14] H.V. Geramb, private communication and to be published. [15] M.B. Lewis, Phys. Rev. Lett. 29 (1972) 1257. [16] N. Marty, M. Morlet, A. Willis, V. Comparat and R. Frascaria, Nucl. Phys. A238 (1975) 93. [17] S.S. Hanna, Proc. Intern. Conf. on Nuclear Structure and Spectroscopy, Amsterdam, 1974, Vol. II, p. 270. 428
PHYSICS LETTERS
Volume 57B, number 5
MASS PARAMETERS FOR QUADRUPOLE VIBRATIONS IN THE ADIABATIC APPROXIMATION D. VAUTHERJN Institut de Physique Nuclkaire, Division de Physique Thkorique *, 91406Orsay-France Received 27 January 1975 (Revised version received 16 June 1975) Nuclear quadrupole vibrations are investigated in the framework of the adiabatic time-dependent Hartree-Fock equations. Using a continuity condition satisfied by these equations, we show that a simple analytic expression can be derived for the mass parameter in the case where the collective variable Q is a scaling variable i.e. changes the cartesian coordinates x, y, z into x/a, y/a, c?z. This result depends only on the existence of the continuity equation and thus holds for any nucleon-nucleon interaction. Applications are considered in the case of small amplitude motion in oxygen-16 and calcium40.
In this letter we apply the adiabatic time-dependent Hartree-Fock method [ 1,2] (as formulated by Baranger and VBn6roni [3]) to describe nuclear quadrupole oscillations. Let us consider the case of a single collective variable (Y,associated with a given set ao(cu) of time-even Slater determinants. Denoting by pa(o) the one-body densitymatrix corresponding to +O(CY)(and satisfying pg = po), the above authors give the mass parameter M(o) as M(ol)=ifiTrace where pl(o) +0
iA a(y=
[$
[Po,PI]]
,
is a time-odd hermitian
one-body
[p~WOPO+(l-~pg)~o(l-Po)IP~l
operator such that +
WIPPOI7
(2)
and Pl ‘POP1 +PlPo
*
(3)
In eq. (1) W. is a time-even operator whose definition FVo(l) =$-(1)
+ Trace2 V(1,2)po(2),
and the time-odd operator
is similar to that of the Hartree-Fock
field (4)
W, is given by
WI(l) = Trace2 Y(1,2)~1(2),
(5)
where Y(l, 2) denotes the antisymmetrized nucleon-nucleon interaction. Eq. (3) implies that the particle-particle and hole-hole matrix elements of pI with respect to p. are zero. As a consequence eq. (2) (which is linear in pl) has particle-hole matrix elements only and determines the particle-hole matrix elements of pl. Notice that for a rotating nucleus the previous formulae reduce to the result of Thouless and Valatin [4]. The time-dependent Hartree-Fock equations are known to lead to a continuity equation for the total particle density [S, 61. A similar relation holds in the limiting case of adiabatic motion namely
* Laboratoire associd au C.N.R.S.
425
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975 (6)
PO(r, or) + m div/(r, oe) = 0 , where
PO(r, ¢t) = (rlP0(t~) Ir),
(7)
f(r, c0 = ~-~(V 1 - V ')
.
r'=r Eq. (6) is obtained by taking the expectation value of equation (2) in a state Ir) and by noticing that since adiabaticity requires [W0, P0 ] to be small, the first commutator in eq. (2) may be replaced by [W0, p 1]" For a velocity independent force the proof follows immediately by using the observation of Thouless and Valatin [4] that only the exchange term contributes to equation (5). For Skyrme's interaction [7] the proof is slightly more complicated and may be found in ref. [6]. In order to exploit the continuity equation it is convenient to introduce the single particle wave functions ~ol(r, or) defining the Slater determinant ~0(~t) as well as a set of states It//) defined by [6] Ir//) = -iPll~Pj),
]= 1,2,...,a.
(8)
According to this definition the set It//) is invariant under time reversal and eq. (3) implies that (¢klr/i) = 0,
j , k = 1,2 ..... A .
(9)
Another consequence of eq. (3) is that Pl can be written as A Pl = i/=~l { Ir//) (~P]I- I~P/)(r//I} •
(10)
Inserting this expression into eqs. (1) and (7), and using the time reversal invariance of the sets ¢] and r//we obtain A M(a)= 2~ ~ f d 3 r 77~(,, a ) ~ ~o/(r, a) , ]=l
(11)
and A
7(., .) --
.) v r/;(,,
- r/;(., .) v
(12)
In ref. [6] the similarity between the two previous equations was used to derive an analytic expression for the mass parameter in the case of monopole vibrations. We will now show that the case of quadrupole vibrations can also be worked out analytically. Let us first denote by ~j(r) the single-particle wave functions of the Hartree-Fock ground state. In order to describe quadrupole vibrations we consider the following one-parameter set
~ol(r, ct) = ~ /(x/q, y/ot, tx2z) ,
(13)
where x , y and z are the cartesian coordinates ofr. From this definition it is easy to show that
a
ba ~Pl(r' a) = ~ V (2z 2 - x 2 - y2). V~p/(r, oe).
Comparing eqs. (11), (12) and (14), the expression for the mass parameter M(tx) becomes
426
(14)
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
M(ot)=_~_~f d3r,l~(2z2 x 2 - y 2-+ )ff(r,~).
(15)
Multiplying the continuity equation (6)by 2z 2 - x 2 - y 2 , integrating over d3r we obtain after integration by parts 1
a
M(o0 = - ~-~ m ~-~ a ( a ) ,
(16)
where Q(o0 is the expectation value of the quadrupole moment in the state qs0(a) i.e.
a(a) - - f d 3 r
(2z 2 - x 2 -
y2)po(r, a).
(17)
Using the scaling property (I 3) this quantity can be expressed in terms of the radius r 0 and the quadrupole moment Qo of the Hartree-Fock ground state, which corresponds to a = 1. Explicitly e(t~) =-~Ar 2 ~ - ~ - a 2 + ~ Q o
\or4
]
where A is the mass of the nucleus and r 0 and Qo are defined by
r2=l fonv(r)r2d3r,
Qo= fOrlF(r)(2z2-x2-y2)d3r.
(19)
Inserting eqs. (18) into (16) we obtain the following expression for the mass parameter
Notice that since this result depends only on the existence of the continuity equation, it is valid for any nucleonnulceon interaction. According to the authors of ref. [3] collective spectra are obtained by considering the collective Harniltonian h(a, or) =lM(a)~t2 + V(t~).
(21)
where the collective potential energy is V(a) = (qs0(a) IHIqS0(ot)).
(22)
As an application of eq. (20) let us now calculate the energy of the quadrupole vibration in oxygen-16 and calcium40, using Skyrme's interaction [7] to construct V(a). For simplicity we assume that the Hartree-Fock single particle wave functions can be approximated by pure harmonic oscillator wave functions with a parameter ~0 = ~ [ / f which minimizes the expectation value of the energy. In this case the expression for V(ct)is easily obtained from the formulae given in reference [8]. The result is V(~) = E 0 +
C-~mPO (27r)312(3Bltl
ol2
,
where t 1 and t 2 are the parameters of the Skyrme force and where E = V(~ -- 1) is the minimum of the energy calculated with pure oscillator functions. The coefficients B and C are [7, 8] C-- 36,
B 1 = 35•4,
B 2 ---4,
for oxygen-16,
C = 120,
B 1 = 2625•64,
B 2 --- 1760•64,
for calcium-40.
(24) In terms of/30, the mass parameter M(e) given by eq. (20) becomes 427
Volume 57B, number 5
PHYSICS LETTERS
4 August 1975
M(tv)=2mC1 (1+ ~ ) ,
(25)
t~0 where C is the same as in eq. (23). In the limit of small amplitude motion, i.e. ot ~, 1, the collective Hamiltonian (211 can be approximated by 1 h (a, &) = 1 M ( 1 ) &2 + E0 + ~ g(t~ - 1) 2 ,
(26)
with
I 1 r, h2 a2 + (27r)3/2 l (3B,t, + 5B2t2) t l
K = 24 [ ~ " - ~ - ~ v 0
35)
(27)
From equation (26) the energy EQ of the quadrupole mode i s - h ~
i.e.
h2 ~02 ~ K -. E Q = V -~m
(281
For interaction Skyrme-IIl, which gives a remarkable fit to a wide range o f nuclear ground state properties in Hartree-Fock calculations [9] (t 1 = 395.0 MeV fm 5 , t 2 = - 95.0 MeV fm 5) we obtain the following results 30 = 0.563 f m - 1 ,
K --- 2622 MeV,
EQ = 21.9 MeV
for oxygen-16,
30 = 0.507 f m - 1,
K = 6930 MeV,
Et2 = 17.5 MeV
for calcium-40.
(29) Similar results are obtained for interactions Skyrme-I and II [10], namely Et2(I ) = 21.8 MeV, EO(II ) = 24.2 MeV for oxygen-16 and Et2(I) = 17.2 MeV, Et2(II) = 19.7 MeV for calcium-40. The above values of Et2 are in good agreement with the theoretical results of references [8, 1 1 - 1 3 ] , with the analysts of reference [ 14] and are also consistent with available experimental data in the case of calcium-40 [ 15, 16] In oxygen-16 the experimental situation is still an unsettled question, but seems to indicate that the isoscalar quadrupole strength is spread over a wide region [17]. I would like to thank D.M. Brink, H. Flocard, N. Rowley and M. V6n~roni for stimulating discussions.
References [1] S.T. Belyaev, Nucl. Phys. 64 (1965) 17. [2] F.M.H. Villars, Proc. 1971 Mont Tremblant Intern. Summer School, ed. D.J. Rowe, p. 3. [31 M. Baranger, Journal de Physique, Supplement 33 (1972) 61; M. Baranger and M. Veneroni, to be published. [4] D.J. Thouless and J.G. Valatin, Nucl. Phys. 31 (1962) 211. [5] L.P. Kadanoff and G. Baym, Quantum statistical mechanics(Benjamin, New York, 1962) p. 56. [6] D.M. Brink, Y.M. Engel, K. Goeke, S.J. Krieger and D. Vantherin, University of Oxford preprint 76/74. [71 T.H.R. Skyrme, Nucl. Phys. 9 (1959) 615. [8] H. Flocard and D. Vautherin, Phys. Lett. 55B (1975) 259. [9] M. Beiner, H. Flocard, N.V. Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [10] D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626. [11] G.F. Bertsch and S.F. Tsai, preprint, 1974. [121 S. Krewald and J. Speth, Phys. Lett. 52B (19741 295. [ 13 ] K.F. Liu, Thesis, University of Stony-Brook, 1974, unpublished. [14] H.V. Geramb, private communication and to be published. [15] M.B. Lewis, Phys. Rev. Lett. 29 (1972) 1257. [16] N. Marty, M. Morlet, A. Willis, V. Comparat and R. Frascaria, Nucl. Phys. A238 (1975) 93. [17] S.S. Hanna, Proc. Intern. Conf. on Nuclear Structure and Spectroscopy, Amsterdam, 1974, Vol. II, p. 270. 428