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Transition charge and current densities of collective isoscalar quadrupole states in 208Pb
Volume
127B, number
TRANSITION
PHYSICS
6
LETTERS
11 August
1983
CHARGE AND CURRENT DENSITIES
OF COLLECTIVE
ISOSCALAR QUADRUPOLE
STATES IN 208Pb
H. SAGAWA National Institute for Nuclear Physics andHigh-Energy 1009 AJ Amsterdam, The Netherlands
Physics (NIKHEF,
section K), P.O.Box 4395,
and NGUYEN VAN GIAI Division de Physique Received
Thebrique
’ , Institut de Physique Nucliaire,
F-91405
Orsay-Ce’dex, France
1 April 1983
Transition charge densities and current distributions of isoscalar quadrupole states in ‘08Pb are calculated in the selfconsistent RPA model. We find clear differences between the low lying state and the giant resonance for transition charge densities as well as current distributions. The implications on the transverse form factors of electron scattering are discussed.
Transition charge densities and current densities are fundamental quantities for characterizing properties of nuclear excitations. The two types of transition densities are not independent since they are closely connected through the continuity equation.
In macroscopic models, the assumed shape of the nuclear velocity field is essential for describing the nuclear many-body dynamics. For instance, one takes often an irrotational and incompressible velocity flow in hydrodynamical models [l-3], which leads to the typical Bohr-Tassie shapes for charge and current densities. The Steinwedel-Jensen model [4], on the other hand, is characterized by a compressional velocity field. It is interesting to check to what extent these kinds of assumptions are valid by comparing with microscopic calculations and also with available experimental data. The transition charge densities have been analyzed extensively in medium and heavy nuclei by the Fourier-Bessel expansion method applied to electron scattering data [S ,6]. By comparing forward and backward differential cross sections, electron scattering experiments give also some indications about the ’ Laboratotie
Associe au CNRS.
0.031-9163/83/0000-OOOO/$
03.00 0 1983 North-Holland
current distributions. Until now, low lying states below 5 MeV have been analysed in this way and no corresponding information is available for giant resonances. Several theoretical studies of transition charge densities of heavy nuclei have been performed in the framework of microscopic models [7,8]. Recently, the current densities have been discussed also by using fluid-dynamical models [2,3,9] or microscopic models [ lo,11 1. In general, one should consider not only the convection current but also other types of currents, such as the magnetization current, if one aims at a comparison with the electron scattering data. However, so far, the convection current has been mainly considered in these theoretical calculations. The aim of this letter is to study the convection current and magnetization current distributions by using the self-consistent RPA theory. We also discuss the transition charge density because of its relationship with the divergence of the total current. The isoscalar quadrupole excitations in 208Pb are especially interesting because the low lying state has been observed as a collective one together with the giant resonance. We therefore concentrate our attention on this case. 393
Volume
127B, number
PHYSICS
6
Under the assumption that the nucleon is a nonrelativistic point particle, the charge density is given by p(r) =
Ce [i :
t,(i)] 6(r - ri) ,
i
(1)
and the current density is described by the sum of the convection current and the magnetization current, J(r)=Jc’
+JMC
=
c‘ f?[f-
+
fj$Ci qi(VX
i
t,(i)] &
(Vi - Vi)6(r -q>
Si)S(r - ri) ,
(2)
where the values of the spin g-factors are gp = 5 58 and gn = -3.82. The transition charge and current densities are just the matrix elements of fi and J between the ground and excited states. The conservation of electric charge is expressed by the continuity equation which, for a velocity independent hamiltonian, takes the simple form: -(i/i?) [H, b(r)] = V * J .
(3)
The rhs of eq. (3) may contain in general not only the divergence of the one-body current (2) but also contributions from two-body currents. We have performed self-consistent RPA calculations of 2+ excitations in 208Pb using the response function method. The self-consistent RPA approach is well founded and it can describe successfully nuclear collective excitations [7]. The response function calculation is performed in coordinate space. The effect of the single particle continuum is exactly taken into account, and there is no truncation of the particle-hole space. For the effective interaction we use the Skyrme type force SC11 [ 121. In general, the isoscalar natural parity states are not very sensitive to the choice of force parameters as long as the force is reasonable with respect to the effective mass value and the nuclear incompressibility value. Therefore, our results do not depend much on that choice. In the quadrupole case, the force SC11 yields better results than a more commonly used force like SIII. For instance, the low lyong 2+ state in 208Pb is found at 6.1 MeV with SIII and 5.1 MeV with SGII, while the observed excitation energy is 4.1 MeV. In our calculations, the full velocity dependence of the interac394
LETTERS
11 August
1983
tion is included, but we have dropped in the particlehole interaction the contributions from the two-body spin-orbit part and u * Q part since they are small for natural parity states. To calculate the various transition densities of eqs. (l)-(2), the isoscalar states are constructed as linear combinations of the one-particle - one-hole proton and neutron configurations. The effect of the excess neutrons is estimated by approximating their transition densities by the isoscalar ones. We obtain two collective 2+ states at 5 .l MeV and 11.4 MeV with the transition strengths B(E2) = 2.82 X lo3 e2 fm4 and 5.16 X lo3 e2 fm4, respectively. The low lying discrete 2+ state exhaust 14% of the energy-weighted sum rule value while the high lying state appears as a resonance having an escape width of 300 keV and contributing 70% to the sum rule value. This agrees well with experiment: the observed 2+ state at E, = 4 .I MeV has a B(E2) value of (3 .18 f 0.15) X lo3 e2 fm4 [6] and the giant quadrupole resonance around 10.9 MeV contains 213 of the sum rule value [13]. In the upper half of fig. 1 are shown the transition densities 6p,, -= CGlp 10) of the two calculated states along with the empirical one for the state at E, = 4.1 MeV. The transition density of the giant resonance at 11.4 MeV has a Tassie-type surface peak with additional small bumps around 2 and 4 fm. On the other hand, the 6p of the low lying state shows large negative values inside, in addition to the dominant peak at the surface. For this low lying state, the calculated transition density reproduces the main features of the experimental one. There is no corresponding data for the giant resonance. We now turn to a brief discussion of the continuity equation. The matrix element of the current operator (2), can be expanded in a complete set of vector spherical harmonics [lo].
NUlO~ = (-i) C JAr(r)Yi,,(f)
,
I where JAI(r) = i s Y,,,(r^)(h~lJlO)
dR .
(5)
The current components JL hk 1 contribute to electric X-pole transitions, while Jkh contributes to magnetic h-pole transitions. In the former case, the continuity equation (3) can be rewritten in terms of the
Volume 127B, number 6
6
PHYSICS LETTERS
5~ [~d~m~]
2O8
X~=
Pb 2÷
4
2
i
i 4
~ - - "2-: <
I / z
.
4:...
I
I 8
/ \
2
~
, 10
R (fm )
./
4 ~\
I ~-
/ "\
2
i 6
,1 6 /
8
10 R (frn)
[_~ Fig. 1. Transition densities 6p and divergence of transition currents'V-J of 2+ states in 2°sPb. The solid and dashed curves correspond to the calculated states at 11.4 MeV and 5.1 MeV, respectively. The dotted curve is the empirical transition density [6 ] of the observed state at E x = 4.1 MeV.
tional terms coming from the velocity dependent forces [ ! 4 ] . However, these corrections vanish if the commutator of the lhs is calculated with the isoscalar density. In fig. 1, one can see that eq. (3) is obeyed to a good accuracy in the present results. The small deviations could come from the spin-orbit interaction. We show the convection current patterns and the corresponding current components o f the 2 + states in fig. 2. For an irrotational and incompressible flow the jCC component would be identically zero. In our CC are indeed case, the irrotational components J21 dominant, but the jCC components are also appreciable. The values of J2C_C for the low lying 2 + state are large and negative at t~ae surface and this gives rise to a vortex in the current pattern near the surface region. On the other hand, the giant resonance has a relatively s m a l l 4 C component and its flow pattern is therefore close to that of the Bohr-Tassie model. Similar results have also been obtained in a cranking model calculation by Kunz et al. [11 ]. The total current jT / is defined as the sum of A the convection current J ~ + , and the magnetization current j M C I In fig 3 we show the components of • J M "~ and k~±±" j r for the low lying state and the giant resonance. In both cases the magnetization current patterns calculated from our jMC exhibit a number of vortices in the interior and at the surface. The magnetization contribution to the total current is relatively small compared to that of the convection current. In the plane wave Born approximation for electron scattering, the transverse form factor for a natural parity transition is expressed as [15]: lgT(q)l 2 = ~
transition density 5px u and the current components
11 August 1983
I(i/allkxu(q)10)12 ,
(7)
with
J x k-+l a s :
~x~ax.(r) = [V(2X+l)] 1/2 [(a-1)/r-d/~]J~, x - 1 (r) + [(X+I)/(2X+I)] 1/2 [(X+2)/r+d/dr]jx x+ l ( r ) ,
(6)
where con is the excitation energy. The lower half of fig. 1 shows the divergence of the convection current for the low lying state and the giant resonance. The magnetization current drops out of the continuity equation because it is divergenceless. If one works with a velocity-dependent Skyrme interaction, the rhs of eq. (3) or (6) should contain addi-
(XV 17~hu(q) 10)
= q1 f V X [fx(qr) Yxu(i)] (~a[JlO) d r .
(8)
In terms of the total current components, we have: (X~l fxu(q)10)
= f r 2 dr ([(X + 1)/(2X+ 1)11/2j T,x_ l(r)fx_ l(qr) - [X/(2X+ 1)] 1/2jT X+l(r)Jx+l(qr) ) .
(9) 395
PHYSICS LETTERS
Volume 127B, number 6
11 August 1983
8
6
Rtfmf
--2
_
- -4
--6
I
I *“Pb
8 ’
I
+
’
E,=51MeV
R (fm)
5, lu
.
\
* - -2 I
- -4
2 *
.
I
K-.-.+--CC
I
I.
-6.
2
-
2
’
4
1,.
6
I \
// ‘\_ /’ cc 37.3
--6
*. \
I 0
.
i
*
,~8~\
i
.I
- -8
8
Xtfm)
Fig. 2. Two-dimensional convection current distributions and the corresponding current components Jf(&l. Results for the states at E, = 11.4 MeV and Ex = 5.1 MeV are in the upper part and the lower part, respectively. The length of the arrow is proportional to the current strength.
396
Volume
127B, number
PHYSICS
6
[dfrc]
208
LETTERS
11 August
1983
Pb
- 10
208
Pb
-8
AK= 2’
o-
-10 L+ -.2
-4
R(fm)
-_4
T 21
-J [10m4frL3]
__~
16:
T J23
-10 J
MC 21
0
E,=
51
MeV
--2
\\ \
-.4
R(fm)
\_,’
I’
-.6
Fig. 3. Components
of the magnetization
and of the total current
Jl
current
Jy:*I
hkl.
The calculated transverse form factors are shown in fig. 4 where we have taken into account the single nucleon form factor as a factor (1 + 42/h2)-2 with X2 = 18.1 fmp2 [16]. Two main differences between the form factors can be noted. Firstly, the first minimum in the form factor of the giant resonance occurs at 4 = 0.8 fm- 1 whereas the corresponding minimum for the low lying state is at 4 = 1 .OS fm- l. Secondly, the ratio of the first to the second maximum is about 40 for the giant resonance and only 13 for the low lying state. Since the contributions of the magnetization current to the form factors are rather small in the present calculations, these differences must mainly be due to the convection currents of the two states. In this letter, we have discussed the transition
1
q ( fm-’
)
2
3
Fig. 4. Transverse form factors FT(q)12 of the quadrupole resonance at E, = 11.4 MeV (solid curve), and of the 2+ state at E, = 5.1 MeV (dashed curve).
densities and current distributions of collective quadrupole states in 208Pb. We find clear differences in both transition charge and current densities between the low lying and high lying excitations. While the giant quadrupole resonance, which exhausts a large fraction of the sum rule, corresponds well to the irrotational and incompressible flow, the low lying state -has a transition density quite different from the Bohr-Tassie one and has a current distribution with vortices at the surface. Differences are also clearly shown up in the transverse form factors which are predominantly determined by the convection current. The measurement of these transverse form factors in electron scattering would give direct information on the current distributions of these collective excitations . After we had completed the calculation reported here, we received a preprint by Serr et al. [17] in which the convection currents of collective excitations in 4oCa and 208Pb are studied in the selfconsistent RPA approach. The results are quite similar to ours.
397
PHYSICS LETTERS
Volume 127B. number 6
We wish to thank J.H. Koch and Toshio Suzuki for helpful
discussiorrs.
This work is part of the research National
Institute
for Nuclear
program
Physics
Physics (NIKHEF, section K), made possible financial support from the Foundation for Fundamental Netherlands Pure Research
of the
and High-energy by
Research on Matter (FOM) and the Organization for the Advancement of (ZWO).
References [l] A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26 (1952) 1;
[2] [3] [4] [5] [6]
398
A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 2 (Benjamin, New York, 1975). G.F. Bertsch, Ann. Phys. 86 (1974) 138; Nuci. Phys. A249 (1975) 253;Phys. Rev. Cl3 (1976) 1312. J.J. Griffin and K.K. Kan, Mod. Phys. 48 (1976) 467. H. Steinwedel and J.D. Jensen, Z. Naturforsch. 59 (1950) 413. J. Heisenberg, Adv. Nucl. Phys. 12 (1981) 61, and references therein. J. Heisenberg et al., Phys. Rev. C25 (1982) 2292.
11 August 1983
[7] G.F. Bertsch and S.F. Tsai, Phys. Rep. 18C (1975) 125; Nguyen Van Giai and H. Sagawa, Nucl. Phys. A371 (1981) 1. [8] I. Hamamoto, Phys. Rep. 1OC (1974) 2; Phys. Lett. 66B (1977) 410; J. Speth, Proc. Sendai Conf. on Electra- and photoexcitations (Sendai, Japan, 1977) p. 65; J. Decharge’ and D. Gogny, Phys. Rev. C21 (1980) 1568. ]91 H. Koch et al, Nucl. Phys. A373 (1982) 173. 1101 T. Suzuki and D.J. Rowe, Nucl. Phys. A286 (1977) 307. [Ill J. Kunz et al., Proc. 5th Kyoto Summer Institute (Kyoto, Japan, 1982). [la Nguyen Van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379. [I31 Y. Torizuka, Proc. of Sendai Conf. on Electra- and photoexcitations (Sendai, Japan, 1977) p. 7; C. DjaIaIi et al., Nucl. Phys. A380 (1980) 42. [I41 S. FaIReros, Lecture Notes (Orsay, France, 1975); Y.M. Engel et al., Nucl. Phys. A249 (1975) 215. [I51 T. de Forest and J.D. Walecka, Adv. Phys. 15 (1966) 1. 1161 A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 1 (Benjamin, New York, 1969) p. 386. 1171 F.E. Serr et aI.,preprint (Nordita, 1983).
127B, number
TRANSITION
PHYSICS
6
LETTERS
11 August
1983
CHARGE AND CURRENT DENSITIES
OF COLLECTIVE
ISOSCALAR QUADRUPOLE
STATES IN 208Pb
H. SAGAWA National Institute for Nuclear Physics andHigh-Energy 1009 AJ Amsterdam, The Netherlands
Physics (NIKHEF,
section K), P.O.Box 4395,
and NGUYEN VAN GIAI Division de Physique Received
Thebrique
’ , Institut de Physique Nucliaire,
F-91405
Orsay-Ce’dex, France
1 April 1983
Transition charge densities and current distributions of isoscalar quadrupole states in ‘08Pb are calculated in the selfconsistent RPA model. We find clear differences between the low lying state and the giant resonance for transition charge densities as well as current distributions. The implications on the transverse form factors of electron scattering are discussed.
Transition charge densities and current densities are fundamental quantities for characterizing properties of nuclear excitations. The two types of transition densities are not independent since they are closely connected through the continuity equation.
In macroscopic models, the assumed shape of the nuclear velocity field is essential for describing the nuclear many-body dynamics. For instance, one takes often an irrotational and incompressible velocity flow in hydrodynamical models [l-3], which leads to the typical Bohr-Tassie shapes for charge and current densities. The Steinwedel-Jensen model [4], on the other hand, is characterized by a compressional velocity field. It is interesting to check to what extent these kinds of assumptions are valid by comparing with microscopic calculations and also with available experimental data. The transition charge densities have been analyzed extensively in medium and heavy nuclei by the Fourier-Bessel expansion method applied to electron scattering data [S ,6]. By comparing forward and backward differential cross sections, electron scattering experiments give also some indications about the ’ Laboratotie
Associe au CNRS.
0.031-9163/83/0000-OOOO/$
03.00 0 1983 North-Holland
current distributions. Until now, low lying states below 5 MeV have been analysed in this way and no corresponding information is available for giant resonances. Several theoretical studies of transition charge densities of heavy nuclei have been performed in the framework of microscopic models [7,8]. Recently, the current densities have been discussed also by using fluid-dynamical models [2,3,9] or microscopic models [ lo,11 1. In general, one should consider not only the convection current but also other types of currents, such as the magnetization current, if one aims at a comparison with the electron scattering data. However, so far, the convection current has been mainly considered in these theoretical calculations. The aim of this letter is to study the convection current and magnetization current distributions by using the self-consistent RPA theory. We also discuss the transition charge density because of its relationship with the divergence of the total current. The isoscalar quadrupole excitations in 208Pb are especially interesting because the low lying state has been observed as a collective one together with the giant resonance. We therefore concentrate our attention on this case. 393
Volume
127B, number
PHYSICS
6
Under the assumption that the nucleon is a nonrelativistic point particle, the charge density is given by p(r) =
Ce [i :
t,(i)] 6(r - ri) ,
i
(1)
and the current density is described by the sum of the convection current and the magnetization current, J(r)=Jc’
+JMC
=
c‘ f?[f-
+
fj$Ci qi(VX
i
t,(i)] &
(Vi - Vi)6(r -q>
Si)S(r - ri) ,
(2)
where the values of the spin g-factors are gp = 5 58 and gn = -3.82. The transition charge and current densities are just the matrix elements of fi and J between the ground and excited states. The conservation of electric charge is expressed by the continuity equation which, for a velocity independent hamiltonian, takes the simple form: -(i/i?) [H, b(r)] = V * J .
(3)
The rhs of eq. (3) may contain in general not only the divergence of the one-body current (2) but also contributions from two-body currents. We have performed self-consistent RPA calculations of 2+ excitations in 208Pb using the response function method. The self-consistent RPA approach is well founded and it can describe successfully nuclear collective excitations [7]. The response function calculation is performed in coordinate space. The effect of the single particle continuum is exactly taken into account, and there is no truncation of the particle-hole space. For the effective interaction we use the Skyrme type force SC11 [ 121. In general, the isoscalar natural parity states are not very sensitive to the choice of force parameters as long as the force is reasonable with respect to the effective mass value and the nuclear incompressibility value. Therefore, our results do not depend much on that choice. In the quadrupole case, the force SC11 yields better results than a more commonly used force like SIII. For instance, the low lyong 2+ state in 208Pb is found at 6.1 MeV with SIII and 5.1 MeV with SGII, while the observed excitation energy is 4.1 MeV. In our calculations, the full velocity dependence of the interac394
LETTERS
11 August
1983
tion is included, but we have dropped in the particlehole interaction the contributions from the two-body spin-orbit part and u * Q part since they are small for natural parity states. To calculate the various transition densities of eqs. (l)-(2), the isoscalar states are constructed as linear combinations of the one-particle - one-hole proton and neutron configurations. The effect of the excess neutrons is estimated by approximating their transition densities by the isoscalar ones. We obtain two collective 2+ states at 5 .l MeV and 11.4 MeV with the transition strengths B(E2) = 2.82 X lo3 e2 fm4 and 5.16 X lo3 e2 fm4, respectively. The low lying discrete 2+ state exhaust 14% of the energy-weighted sum rule value while the high lying state appears as a resonance having an escape width of 300 keV and contributing 70% to the sum rule value. This agrees well with experiment: the observed 2+ state at E, = 4 .I MeV has a B(E2) value of (3 .18 f 0.15) X lo3 e2 fm4 [6] and the giant quadrupole resonance around 10.9 MeV contains 213 of the sum rule value [13]. In the upper half of fig. 1 are shown the transition densities 6p,, -= CGlp 10) of the two calculated states along with the empirical one for the state at E, = 4.1 MeV. The transition density of the giant resonance at 11.4 MeV has a Tassie-type surface peak with additional small bumps around 2 and 4 fm. On the other hand, the 6p of the low lying state shows large negative values inside, in addition to the dominant peak at the surface. For this low lying state, the calculated transition density reproduces the main features of the experimental one. There is no corresponding data for the giant resonance. We now turn to a brief discussion of the continuity equation. The matrix element of the current operator (2), can be expanded in a complete set of vector spherical harmonics [lo].
NUlO~ = (-i) C JAr(r)Yi,,(f)
,
I where JAI(r) = i s Y,,,(r^)(h~lJlO)
dR .
(5)
The current components JL hk 1 contribute to electric X-pole transitions, while Jkh contributes to magnetic h-pole transitions. In the former case, the continuity equation (3) can be rewritten in terms of the
Volume 127B, number 6
6
PHYSICS LETTERS
5~ [~d~m~]
2O8
X~=
Pb 2÷
4
2
i
i 4
~ - - "2-: <
I / z
.
4:...
I
I 8
/ \
2
~
, 10
R (fm )
./
4 ~\
I ~-
/ "\
2
i 6
,1 6 /
8
10 R (frn)
[_~ Fig. 1. Transition densities 6p and divergence of transition currents'V-J of 2+ states in 2°sPb. The solid and dashed curves correspond to the calculated states at 11.4 MeV and 5.1 MeV, respectively. The dotted curve is the empirical transition density [6 ] of the observed state at E x = 4.1 MeV.
tional terms coming from the velocity dependent forces [ ! 4 ] . However, these corrections vanish if the commutator of the lhs is calculated with the isoscalar density. In fig. 1, one can see that eq. (3) is obeyed to a good accuracy in the present results. The small deviations could come from the spin-orbit interaction. We show the convection current patterns and the corresponding current components o f the 2 + states in fig. 2. For an irrotational and incompressible flow the jCC component would be identically zero. In our CC are indeed case, the irrotational components J21 dominant, but the jCC components are also appreciable. The values of J2C_C for the low lying 2 + state are large and negative at t~ae surface and this gives rise to a vortex in the current pattern near the surface region. On the other hand, the giant resonance has a relatively s m a l l 4 C component and its flow pattern is therefore close to that of the Bohr-Tassie model. Similar results have also been obtained in a cranking model calculation by Kunz et al. [11 ]. The total current jT / is defined as the sum of A the convection current J ~ + , and the magnetization current j M C I In fig 3 we show the components of • J M "~ and k~±±" j r for the low lying state and the giant resonance. In both cases the magnetization current patterns calculated from our jMC exhibit a number of vortices in the interior and at the surface. The magnetization contribution to the total current is relatively small compared to that of the convection current. In the plane wave Born approximation for electron scattering, the transverse form factor for a natural parity transition is expressed as [15]: lgT(q)l 2 = ~
transition density 5px u and the current components
11 August 1983
I(i/allkxu(q)10)12 ,
(7)
with
J x k-+l a s :
~x~ax.(r) = [V(2X+l)] 1/2 [(a-1)/r-d/~]J~, x - 1 (r) + [(X+I)/(2X+I)] 1/2 [(X+2)/r+d/dr]jx x+ l ( r ) ,
(6)
where con is the excitation energy. The lower half of fig. 1 shows the divergence of the convection current for the low lying state and the giant resonance. The magnetization current drops out of the continuity equation because it is divergenceless. If one works with a velocity-dependent Skyrme interaction, the rhs of eq. (3) or (6) should contain addi-
(XV 17~hu(q) 10)
= q1 f V X [fx(qr) Yxu(i)] (~a[JlO) d r .
(8)
In terms of the total current components, we have: (X~l fxu(q)10)
= f r 2 dr ([(X + 1)/(2X+ 1)11/2j T,x_ l(r)fx_ l(qr) - [X/(2X+ 1)] 1/2jT X+l(r)Jx+l(qr) ) .
(9) 395
PHYSICS LETTERS
Volume 127B, number 6
11 August 1983
8
6
Rtfmf
--2
_
- -4
--6
I
I *“Pb
8 ’
I
+
’
E,=51MeV
R (fm)
5, lu
.
\
* - -2 I
- -4
2 *
.
I
K-.-.+--CC
I
I.
-6.
2
-
2
’
4
1,.
6
I \
// ‘\_ /’ cc 37.3
--6
*. \
I 0
.
i
*
,~8~\
i
.I
- -8
8
Xtfm)
Fig. 2. Two-dimensional convection current distributions and the corresponding current components Jf(&l. Results for the states at E, = 11.4 MeV and Ex = 5.1 MeV are in the upper part and the lower part, respectively. The length of the arrow is proportional to the current strength.
396
Volume
127B, number
PHYSICS
6
[dfrc]
208
LETTERS
11 August
1983
Pb
- 10
208
Pb
-8
AK= 2’
o-
-10 L+ -.2
-4
R(fm)
-_4
T 21
-J [10m4frL3]
__~
16:
T J23
-10 J
MC 21
0
E,=
51
MeV
--2
\\ \
-.4
R(fm)
\_,’
I’
-.6
Fig. 3. Components
of the magnetization
and of the total current
Jl
current
Jy:*I
hkl.
The calculated transverse form factors are shown in fig. 4 where we have taken into account the single nucleon form factor as a factor (1 + 42/h2)-2 with X2 = 18.1 fmp2 [16]. Two main differences between the form factors can be noted. Firstly, the first minimum in the form factor of the giant resonance occurs at 4 = 0.8 fm- 1 whereas the corresponding minimum for the low lying state is at 4 = 1 .OS fm- l. Secondly, the ratio of the first to the second maximum is about 40 for the giant resonance and only 13 for the low lying state. Since the contributions of the magnetization current to the form factors are rather small in the present calculations, these differences must mainly be due to the convection currents of the two states. In this letter, we have discussed the transition
1
q ( fm-’
)
2
3
Fig. 4. Transverse form factors FT(q)12 of the quadrupole resonance at E, = 11.4 MeV (solid curve), and of the 2+ state at E, = 5.1 MeV (dashed curve).
densities and current distributions of collective quadrupole states in 208Pb. We find clear differences in both transition charge and current densities between the low lying and high lying excitations. While the giant quadrupole resonance, which exhausts a large fraction of the sum rule, corresponds well to the irrotational and incompressible flow, the low lying state -has a transition density quite different from the Bohr-Tassie one and has a current distribution with vortices at the surface. Differences are also clearly shown up in the transverse form factors which are predominantly determined by the convection current. The measurement of these transverse form factors in electron scattering would give direct information on the current distributions of these collective excitations . After we had completed the calculation reported here, we received a preprint by Serr et al. [17] in which the convection currents of collective excitations in 4oCa and 208Pb are studied in the selfconsistent RPA approach. The results are quite similar to ours.
397
PHYSICS LETTERS
Volume 127B. number 6
We wish to thank J.H. Koch and Toshio Suzuki for helpful
discussiorrs.
This work is part of the research National
Institute
for Nuclear
program
Physics
Physics (NIKHEF, section K), made possible financial support from the Foundation for Fundamental Netherlands Pure Research
of the
and High-energy by
Research on Matter (FOM) and the Organization for the Advancement of (ZWO).
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11 August 1983
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