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Electromagnetic properties of 7Li and 7Be in a cluster model
Nuclear Physics A413 (1984) 323-352 @ Noah-Holland abashing Company
ELEC’FROMAGNETFC
PROPERTIES
OF ‘Li AND ‘Be
IN A CLUSTER MODEL TOSHITAKA
KAJLNO, TAKEHIRO
MATSUSE and AKTl-0 ARIMA
&~~rtment of Physics, Fa&u~tyofScience, Universityof Tokyo, Bu~k~-k~ Tokyo 113, Japan Received 7 June 1983 (Revised 11 August 1983)
Abstrrctr The electromagnetic properties of 7Li (‘Be) are studied by using a microscopic a-t (‘He) cluster model. Among the several properties are the Coulomb and magnetic form factors for the ground spin-doublet states of ‘Li (‘Be), density distribution radii, static moments and radiative transition probabilities. Provided that the stability conditions for the binding energies are satisfied for both the constituent nuclei 1y and t (3He). the calculated results are in good agreement .with many availabfe data. Several interesting differences are predicted between ‘Li and ‘Be.
1. Introdmtion Many different kinds of experimental data have been accumulated for studying the electromagnetic structures of 7Li and ‘Be [refs. lmt4)], and a great many theoretical works 15-25) have also been devoted to study them. Especially, a very large electric quadrupole deformation of the ground spin-doublet states of 7Li has attracted several authors to theoretically investigate it 20-23*25).A deformed shell-model calculation by Kurath 23) showed that a 20% core excitation is needed to explain the large quadrupole deformation but which at the same time leads to an unacceptably large charge radius. Reasonable values for the both quantities have been obtained by a projected I&tree-Fock calculation ‘*), which could not, however, remove the large effect of spurious c.m. motion. Recently, an &t cluster-model calculation has been carried out by Kanada ef al. 25). They have succeeded in obtaining good agreement with the observations. Most of those theoretical works, however, have only been concentrated on a few properties (for example, the elastic Coulomb form factor or the electric quadrupole moment of 7Li) though many different kinds of data are available r-14). Quite recently, electron scattering experiments r3,14) have bean performed to measure the transverse form factors covering up to .XJ’= 9 fm-* with extremely high resolution and confirmed the accuracy of the data obtained in 1971 by the Amsterdam group ‘3. Though the observed inelastic data, not only of the transverse form factors but also of the lon~tudinal ones, are restricted to a momentum transfer region lower than q2 = 1.fm-*, several density distributjon radii, static moments and radiative transition probabilities could be extracted from the observed values by 323
T. Kajino et al. / Electromagnetic properties
324
the Amsterdam group since the resolution was quite good. Being stimulated by their works, we have investigated as many of the electromagnetic properties of ‘Li as possible in a microscopic a-t cluster model. This model has no difficulty in removing the spurious c.m. motion and can be very easily applied to cr-t scattering problems too. This is an advantage of the present model in comparison with shell-model and projected Hartree-Fock-model calculations. One purpose of the present paper is to show that the microscopic a-t (3He) cluster model provides a systematic understanding of the electromagnetic properties of ‘Li (‘Be). Special interest is placed on the point that the large electric quadrupole deformation can be produced by this model without changing many of the other electric and magnetic properties. The second purpose of this article is to clarify the size effect of the constituernt clusters on those quantities. In sect. 2, the resonating group method is briefly reviewed and several formulas are obtained for calculating the electromagnetic form factors and other basic observables. In sect. 3, various multipole contributions are separately calculated in the inclusive form factor. The size effect on each contribution is examined. Taking into account the size effect and the dependence of the result on an assumed force, we look into the consistency among the static moments, transition probabilities etc. in sect. 4. The Cy-3He (t) threshold effect on ‘Be (‘Li) and the difference between the isodoublet mirror nuclei are investigated in sect. 5. Finally, in sect. 6, the results are summarized.
2. Resonating group method The resonating group method is briefly discussed in this section. The method for calculating the RGM kernels for the electromagnetic multipole operators is shown. Notations and definitions are given in this section. 2.1. CLUSTER
WAVE
FUNCTION
AND
EFFECTIVE
INTERACTION
The
‘Li (‘Be) nucleus is assumed to consist of two clusters (Yand t ( 3He) throughout this paper. In the resonating group method (RGM) a total wave function is written as
in the U-coupling scheme, where J 12 is the antisymmetrizer between the two clusters, and & and Ni are the internal coordinates including spin-isospin variables and the mass number of the ith cluster. Here x&r) = xJL(r) YCL’(i) is the intercluster relative motion to be solved while the intrinsic wave function @(ti) of the ith cluster is approximated by a gaussian function with cluster size parameter pi.
T. Kajino et al. / Electromagneticproperties
325
The highest spatial symmetry is assumed in both the cluster wave functions. The total hamiltonian is written as H=i
k
tk-TG+
= lI,+ti,+
C Vij
(2)
icj
1
lr,,,+
is(cluster1)
c
V,
(2’)
9
je(cluster2)
where To and Trel are the kinetic energies of the c.m. motion and the inter-cluster relative motion, respectively, hi is the intrinsic hamiltonian of the ith cluster, and Vij a two-body effective interaction which consists of central, spin-orbit and Coulomb forces. As a central part, two typical phenomenological interations are used. One is the Volkov force No. 2 (V2) 28) which is a Serber-type gaussian potential having a long-range attractive part and a very weak short-range repulsive part only in the 3E and ‘E channels. The other is the modified-Hasegawa-Nagata force (MHN) 29) which is a three-range gaussian potential: a Rosenfeld-type longrange part simulating the tail of OPEP, an intermediate-range attractive part and a short-range repulsive part of moderate strength in the 3E, ‘E and 3O channels. They have the form
where PM, PB and PH are the Majorana, Bartlett and Heisenberg exchange operators, respectively; Wi, mi, bi and hi are the Wigner and their associated parameters of the exchange mixtures, Vi is the strength parameter, pi the range parameter, and rk! the relative coordinate between the kth and fth nucleon% The reason why we take these two interactions is because they have completely different exchange mixtures but give acceptable binding energies and rms charge radii to CX, t and 3He nuclei under the stability conditions
d(@(&)IhI@(6))=, dP
7
d2(@(6)IhI@(b)),o dP2 3
where /3 are their size parameters. As a spin-orbit interaction, a phenomenological Nagata force 43) is used. This force also has a gaussian form factor
V,,(rk,)=k=SC(Wi+miP~))ViexP[-CLiT2kll I
{ (Sk+%)
+klX~vkl))r
where Sk is the spin operator of the kth nucleon. Table 1 summarizes the parameters of the effective interactions adopted in this calculation.
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326
TABLE
1
Parameters of the two-body effective interactions i
IL~(fm-‘1
Vi (MeV)
wi
m,
hi
bi
cenrral interactions
v2 “)
1 f 2 3
0.3086 0.9803 0.16 1.127 3.4
-60.65 61.14 -6.0 -546.0 1655.0
1 2
3.0 5.0
-1519.0 1918.0
2
MHN b,
spin-whit
Nagata “f
“) Ref. 28),
“) Ref. 2g).
l-m l-m -0.2361 0.8297-m 0.4473
0.3785
0
0 0.5972 0.1405 0.1015
0
0 -0.5139 0.0302 0.0526
interaction
0.5 OS
-0.5 -0.5
0 0
0 0
‘) Ref. “).
In order to obtain the relative wave function differential SchrGdinger equation is solved:
= 0.
m
m 1.1528
,yJL(r), the following integro-
(4)
Three different sets of the cluster sizes are used. They are (i) /3(t) = P(a) = 0.32 fm-2; rather large size which reproduces the experimental rms charge radius of ‘Li approximately in the shell-model configuration of (OS)~(O~)~ [431. (ii) p(t) = Pfcr); very compact size which satisfies eq. (3) for the nucleus. (iii) P(t) and @(cr); they satisfy the stability condition (3) for both the nuclei t and a. These three cases are adopted to examine the size effect of the constituent nuclei. The V2 force is used for them. In addition, the MHN force is also used for case (iii) to see the force dependence. This additional case is called hereafter case’(iv). Only the two parameters are varied in the effective interaction, namely the Majorana mixture parameter m of the central force and the strength parameter kLs of the spin-orbit force. They are set to reproduce the separation energies of’ the two-cluster breakups 7Li+ a! + t and ‘Be-, a -t3He and the doublet-splitting energies for 7Li and 7Be, separately. With these conditions being satisfied, the cluster size effects on the electromagnetic structure of 7Li and 7Be can be discussed in connection with the stability conditions of the constituent nuclei. The values of these parameters are shown in table 2 with several properties of the gruund states of t, 3He, 4He, 7Li and ‘Be nuclei.
327
T. Kajino et al j Electromagnetic properties TABLE 2a
Parameter values varied in the calculations and several calculated results of the ground-state properties of the t, 4He and ‘Li nuclei Calculations Experiment
central force @{a) (frn+I WI (fm-? Majorana mixture m kLS LS strength E&1 (MeW E(!t-) WW BE (MeV)
-2.447 “) -1,989 “) 39.245 b, 28.296 b, 1.63(4) “) 8.482 b, 1.70(J) “f
a! t
BmeV) J{r&) @mm)
(ii)
(iii)
(iv)
V2
v2
v2
MHN
0.32 0.32 0.55 0.75
sire parameter
‘Li
W
-2.366 -1.850 31.942 22.820 2.028 6.756 1.896
0.5285 0.5285 0.585 0.54
0.5285 0.400 0.5735 0.60
-2.442 -1.983 36.600 27.973 1.651 6.186 1.537
-2.473 -1.963 37.638 27.973 1.651 7.192 1.725
0.574 0.460 0.39658 0.30 -2.538 -2.065 37.928 28.680 1.599 6.710 1.628
The four cases fi) to [iv) show the different sets of the chrster size parameters #frr) and #Z(t) for the alpha and triton nuclei, respectively. m is the Majorana mixture pantmeter of the centrat force and k= the strength parameter of the spin-orbit force. E(J*), BE and J(&> are the energy of the state having spin J and parity r from the a-t breakup threshold, binding energies and root mean square charge radii. “) F. Ajzenberg-Selove “). b, J.H.E. Mattauch et al. *‘). “) Ref. 2a). TANE
2b
The same as table 2a for the 3He, ‘$He and 7Be nuclei Calculations
Experiment central force size parameter
8(o) @n? /3f3He) (fm-‘) Majorana mixture m LS strength kLS ‘Be
a ‘He
E(f) WeVI E&I WeV) BE (MeV) BE_@feV) J(r$t UrnI BE MeVf sl ffm)
For footnotes see table 2a.
(iii)
deeply bound
V2
v2
0.5285 0.393 0.5775 0.55 -1.586 “) -1.157a) 37.601 ‘) 28.296 “) 1.63(4) “) 7.718 b) 1.8715) 3 1.88(S)
-1.548 -1.118 35.989 27.973 1.651 6,468 1.790
0.5285 0.393 0.5530 0.54 -2.476 -1.942 36.917 27.973 1.651 6.468 I.790
T Kajino et al. / Electromagnetic properties
328
Fig. 1. The reduced width amplitudes i’s of the a-t cluster wave functions of the ‘Li ground state. The dotted, dash-dotted, solid and dashed curves are obtained by using the different size parameter sets of (i), (ii), (iii) and (iv) d es&bed in the text, respectively. The short-dotted curve represents the harmonic oscillator function u,,(r; V) with Y= (3. 4/7)0.32 fm-*.
Fig. 1 shows the calculated reduced width amplitudes of the ground state of ‘Li:
0
y(l)((i)1(3/2)_
iqr-a) ~w/2~3,2(5152~) r2
65
d52
da.
(5)
>
One of them assumes a harmonic oscillator wave function ul,(r; V) with V= (N1N2/A)0.32 fmm2 as the relative wave function x. This wave function is identical to the U-coupling shell-model wave function provided that the hw of the two clusters are taken to be common and hw = (h/p)v with /* = (NlN2/A)MN. They all show a remarkable clustering or core polarization effect in comparison with the shell-model wave function. This feature is in particular clearly seen in the nuclear surface region. The occupation probability of harmonic oscillator states with radial node Nra 2 is about 30% for case (i) and amounts to 40% for case (iii). Finally the calculational procedure of eq. (4) is briefly discussed before closing this section. When the spatial parts of the constituent nuclei are described by the harmonic oscillator wave functions with unequal size parameters, the GCM kernels using Brink’s GCM wave function 30) include spurious components of the c.m. excitation. However, since the procedure to obtain the GCM kernels is very simple, they are used to construct the RGM kernels which are completely free from spuriousness. In order to solve eq. (4) by means of the variational method 31), the essential points are how to calculate the matrix elements of the hamiltonian and norm kernels and what kind of basis vectors should be chosen to describe the inter-cluster relative wave function. We use locally peaked gaussian functions as
T. Kujino et al. / Electromagneticproperties
329
basis vectors and calculate the matrix elements by means of the new generator coordinate method (NGCM) 32) introduced by Tohsaki-Suzuki. The equivalence of this method to the RGM has already been proved by him. The advantages of this choice lie in the following two points: first, the c.m. excitation is completely removed from the NGCM kernels, and second, integrations of obtaining the matrix elements of very complicated operators such as electromagnetic multipole operators are very easily performed in the new generator coordinate space (NGC space). Details are given in appendix A. 2.2. BASIC REPRESENTATION OF THE ELECTROMAGNETIC MULTIPOLE OPERATORS The
squared Coulomb and magnetic form factors can be expanded into multipole components 44) as
where Ji and Jf are the total spins of the initial and final states, respectively. the multipole operators are defined by
T%I?)=
~A~A(qr)Y~'(~)~(r)
I
Here
dr,
T:” (4) = jAtqr)tY?l ($1.J(r)) dr,
(8)
where jh (qr) and Yz’ ($( Yy’ (P)) are spherical Bessel functions and spherical (vector spherical) harmonics. As for the charge and current density operators (p, J), we take a simple approximation in which they are given by the sum of free nucleon currents: p(r) = e; a(~--pk) J(r)
= JN(d
+v
,
XELN(d
,
where JN is the convection current density and j.&Nis the magnetization density. The g-factors are gzl,= $(l -I-T,( k>)gjp’ with gjp’ = 1.0 for the orbital part and gsL = i( 1 + r,( k))gLp) +i( 1 - r,( k))g$” with g$‘) = 5.584 and g$” = -3.826 for the spin part in units of nuclear magnetons j.&N = e@%&c. Here pk is the coordinate of the kth nucleon from the c.m. and ‘Vkthe derivative as to pk. The nucleon finite
T. Kajirw et al. / E~e~i~o~~g~e~~~properties
33Q
size effect is taken into account by multiplying the multipole operators by f&Y’)(4) = Gus(q) f G,“(q)
3
A!? (4)/~p=f~)(4)J~ELn=f~)(q) where the electric form factors and isovector parts 45),
of a
(9) 9
tw
single nucleon are expressed in terms of isoscalar
,+;;:“,, 7-1+;;6/026 7+oqlo , GES(q)E; 1 1.15
1+q2/8.1Q-O*16
=
In eq. (Q), the -I-(-) sign is for protons (neutrons). In this approximation with a nucleon finite size correction ((9), (lo)), the multipole operators are given as
x(f(l~T,(k))f~)(q)+l(l-T,(k))f~)(q)) TF” (q)=
z
2-Z:
+
r,
TZ$@,
(14)
5=**i
o=h*l
(13)
1 d4 (A)
0 G(expEiq- eJw%‘~(q)P~’ k
Y
iL
where the convection current part of eq. (14) TEL: and the spin current part Tg$” are written thus hereafter. %‘Wand yb are introduced for simplicity; they are as f5115ws:
I: Kajino et al. / Electromagneticproperties
331
In eqs. (13), (14), the relation h(qt) Yl”‘(i) =L4?ri”
Yt”’ (4) exp [iq r] dq^ l
(16)
is used to project the h-pole component out of the plane wave. It should be emphasized that the laborious task in obtaining the GCM kernels of the electromagnetic operators is extremely simplified by projecting the A-pole component after ~tegration with respect to the intrinsic coordinates and summation of the spin and isospin quantum numbers. Using the IS-coupling cluster wave function (l), the reduced matrix elements in eq. (6) can be rewritten in the following forms:
(17)
where 7’2;“’ (q) is defined by dropping all the spin observables out of Tskrr (4). In order to evaluate the reduced matrix elements in the right-hand sides of eqs. (17), (18), essentially the same technique as the last subsection and appendix A is applicable. The general form of the integrals in the new generator coordinate representation is given by
1 X
iXq iX’q : XX’
I
exp [-r,X’-
x~,,(X)X’~~X’X~~X,
T*X’~- tYq’]Ff$k
t
(19)
where F?$ is a geometrical factor depending on Li, Lf, I,, 12, l3 and A, and @j(q) presents symbolically the operators of p”(q), TzT (q) and T$‘f (4). Detailed forms of (19) are summarized in appendix B. It is relevant here to comment on the convergence of the summation concerning II, I2 and l3 in eq. (19). If the inter-cluster relative wave function is represented by
332
T. Kajino et al. / Electromagnetic
properties
a harmonic oscillator function such as +,(r), the summation stops at a finite integer. However, since the cluster wave function includes a great amount of core excited components, it runs in principle from zero to infinity. In the practical calculation, of course, we are limited to scan only one index among 1,, l2 and l3 from zero to a moderately large integer, because the variable range of them is mutually restricted through the relation in F+tk. The convergence is faster with a small q than a large q. Then the convergence of the calculation is checked in every multiple form factor at q2 = 9 fm-* which is the largest value in the present calculation. For the Coulomb form factor a good convergence is obtained before max [1i, 12,&] reaches 10. On the other hand, the magnetic form factor needs at least 15 to obtain an excellent convergency. Thus in practice we adopt 20 as upper limits of 1i, 1, and l3 in the summation. The computation is devised to check the convergence automatically with an accuracy of lo-” for the Coulomb form factors and lOPi4 for the magnetic form factors. Finally, the definitions of several important observables are given. The operators of the electromagnetic A-pole moments are related to the electromagnetic operators (7) and (8) in the limit of q = 0 as
where (in* = (2A + l)!!/i* and LYE*= ((2h + l)!!/i)m. Therefore, the electromagnetic moments and radiative transition probabilities are obtained by taking the limit of q = 0:
(22)
(23) Note that QE2 for the electric quadrupole mgment (47) and p( 0) for the magnetic dipole (octupole) moment (MM*> are used’heikjafte’“2t should be noticed that the conventional definition of the electric quadrupole moment QF$ventionalis given as 2QE2. Experimentally, the B(E(M)A; Ji+ Jr) strength is proportional to the inverse of the radiative lifetime from Ji to Jr. At the same time, however, the lifetime depends
T. Kajino et al. / E~e~t~o~g~t~~ properties
333
on the photon energy: r-‘(E(M)h;
Ji+Jf)=
*‘+’
B(E(M)A; Ji+JJ
Ey=IJ&-&,I.
9
(24)
Therefore the theoretical estimation of the lifetime has two sources of errors; one B(E(M)h ; Ji+ JJ and the calculated level spacing energy which depends on models. The mean square radii for charge distribution, neutron matter distribution and magnetization distribution are given by
(26) (27) respectively, where F g; is the monopole form factor namely the matter form factor of the neutron distribution.
3. Electromagnetic form factors The electromagnetic form factors of the 7Li nucleus are discussed in this section. Each magnetic and Coulomb contribution to the inclusive form factor is calculated separately. The size effect is also examined in connection with the stability of individual isolated clusters and the total binding energy. The inclusive form factor consists of the following four contributions: elastic longitudinal Coulomb (CO + C2), inelastic longitudinal Coulomb (C2’), elastic transverse magnetic (Ml + M3) and inelastic transverse (Ml’ + E2’) form factors. To see the nature of each contribution separately the calculated results for case (iii) are shown in fig. 2. Over a wide range of momentum transfer up to $= 5 fm-* the longitudinal Coulomb form factors are prominent, while at higher momentum transfer Ml + M 3 components compete with them. The E2’ component is neglected in the present calculation because it contributes little, i.e. only 1% of the inclusive data even at its maximum value near q2 = 2 fm-*. Fig. 3 shows the calculated inclusive yields of four cases compared with the experimental values 2). The calculated values for cases (i) to (iv) are explained in subsect. 2.1. They suggest clearly that only one case, in which the same large size is used for the two clusters, should be excluded from consideration because in this exceptional case the calculated inclusive yield decreases too rapidly. This behavior seems to be attributed to the broadening of the total wave function. Such a large size gives larger charge radii of the individual
334
T. Kajino et al. / Electromagnetic
properties
Fig. 2. The square of the form factors of ‘Li for the $-+ $- (elastic) and g-+ f- (inelastic) transitions, calculated with the a-t cluster wave function for case (iii) described in the text. The open circles are the observed inclusive date of ref. *). The solid, dashed, dotted and dash-dotted cures represent the individual contribution of the elastic Coulomb (CO+C2), inelastic Coulomb (C2’), elastic magnetic (Ml +M3) and inelastic magnetic (Ml’) form factors, respectively. The thick-solid curve represents the sum of them. The electric quadrupole (E2’) contribution to the inelastic transverse form factor is omitted in the present calculation.
nuclei, (Y and t, in comparison with the experimental values. In addition, it does not satisfy the energy stability condition: namely, the calculated binding energies are too small. On the other hand, we require that the threshold energy is equal to the empirical value. As a result, we encounter a serious problem: the total binding energy of ‘Li is too small. Thus the total system becomes very loosely bound and spreads too broadly. This is the reason why the calculated form factor decreases so rapidly with increasing momentum transfer. The calculated elastic magnetic form factors are shown in fig. 4. They are in a reasonable agreement with the experimental data 14). Near the photon point the dependence on the cluster size and the force assumed is negligibly weak, which reflects the fact that the Ml matrix element is almost independent of the wave function because the spin contribution dominates in the Ml operator and [43] symmetry is assumed in all wave functions. However, the four calculations produce different form factors from one another with increasing momentum transfer. This fact reflects the difference of the calculated magnetization radius and octupole moment. It is clear that the larger the cluster size, the broader the magnetization distribution and the larger the octupole deformation produced. Therefore all over
333
Fig. 3. The square of the inch&e frrrm factors of ‘Li for the SW-+$+ affd the g--f four curves are the same as those in fig. 1.
transitions. The
Fig, 4, The square of thg: magnetic form factors d ‘Li for the SW + STtransition. The four curves are the same as those in fig. 1, The magnetic form factors consist of the Mf and the M3 contributions. Each etlntribution is ilkstrated as the short-dotted curve for case {iii). The open circks reprewnt the data of ref. 12) and the others of ref. 14)_
336
T. Kajino et al. / Electromagnetic properties
the regions covering the low momentum transfer in which Ml dominates to the high momentum transfer in which M3 dominates, the calculation with the larger cluster size gives a smaller form factor. When the MHN force is used, the calculation overshoots the experimental values at the low momentum transfer but fits them very well at the high momentum transfer. There are two possible reasons. First, since the attractive part of this interaction in the intermediate range is shorter than that of the V2 force, the octupole deformation remains small, preventing a rapid decrease of the magnetic form factor at high momentum transfer. Second, the MHN interaction has a stronger short-range repulsive core than the V2 force. The nucleonnucleon short-range correlation enhances the high momentum components of the form factor. Thus a detailed comparison between the four calculated values and the observed ones shows some systematical discrepancy. There exists an incompatible result: in the Ml dominant region the V2 force is better than the MHN force, while the latter is better than the former in the M3 dominant region. In addition, at high momentum transfer the calculated yields decrease too rapidly compared with the experimental values. Such a discrepancy is similarly seen in a shell-model calculation of the 170 nucleus 33). In order to improve the present calculation one must find another effective interaction with at least the following two features: a longer attractive range and a strong repulsive core at the short range region. As another effect for improvement one should consider an admixture of lower symmetry supermultiplet 14,15)states which is caused mainly by the spin-orbit interaction.
‘Li
104
0.2
3/2--
111
o.4 Cleff(ffTi’
1 o.8
-
transition. The four curves are Fig. 5. The square of the magnetic form factors of ‘Li for the s--f the same as those in fig. 1. The E2’ contribution is omitted in the present calculation. The open circles, the black disks and the crosses represent the data of ref. 12), ref. 13) and ref. 14), respectively.
T. K~j~noet al. / Elec~romagne~cpro~~jes
337
F,:e,/Fei Transverse
._
0
0.2
q21fm2j
0.6
0.8
Fig. 6. The ratio of the squared magnetic form factors of 7Li for the 3-j i- transition to those for the s-+4- transition. The four curves are the same as those in fig. 1. The open circles represent the data of ref. 12).
Fig. 6 shows the ratio of the inelastic transverse form factor squared to the elastic one. The calculation gives reasonable agreement with the experimental values I*) except for the case with a large cluster size. Nevertheless, similarly to the elastic form factor, the absolute values are overestimated in the inelastic Ml’ form factor in the low momentum transfer region (fig. 5). The calculated elastic longitudinal form factors are illustrated in fig. 7 with the model-dependent data 2, compiled by Suelzle et al. They are cited only for reference, though they should be re-evaluated carefully because their values have been extracted from the inclusive form factor after subtracting the transverse contributions and the inelastic Coulomb quadrupole contribution. The subtracted transverse form factors are obtained by extrapolation into the high-q2 region from the observed data in the low-q2 region which have been measured by another group4). The subtracted C2’ form factor is calculated by using the odd proton model i7). Similarly to the magnetic form factors, the different size effects are remarkably seen in the Coulomb form factors too. In the low momentum transfer region the monopole strength dominates and the slope is essentially determined by the mean square charge radius. The form factor in the high momentum transfer region, on the other hand, is dominated by the quadrupole deformation. The calculated inelastic Coulomb form factors are shown in fig. 8 to which only the quadrupole component contribute. If the doublet-splitting energy is set to zero, the inelastic form factor gives completely the same value as the elastic quadrupole component. However, a finite splitting causes an unexpectedly large difference between the two. The calculated inelastic form factor squared has a 10% larger
338
T. Kajino et al. / Electromagnetic properties
0 Fig. 7. The square of the Coulomb form factors of ‘Li for the $- + $- transition. The four curves are the same as those in fig. 1. The form factors consist of the CO and the C2 contributions. Each contribution is illustrated as the short-dotted curve for case (iii). The black disks represent the model-dependent data of ref. 2, described in the text.
\ ’
’
’
4 q2t1m2
’
6
Y _
\ \
\\ >\
\ ’
*‘\I 8
1
Fig. 8. The square of the Coulomb form factors of ‘Li for the $-+ $- transition. The four curves are the same as those in fig. 1.
T: Kajino et aL/ Electromagneticproperties
339
value at the maximum point at q2 = 1.5 fm-’ and a 20% smaller value at q2 = 9 fme2 than the elastic quadrupole component. This shows clearly the fact that the clustering of the first excited state is more developed than the ground state. Since the threshold energy sensitively controls the spreading of the wave function, the inelastic form factor decreases more rapidly than the elastic quadrupole component. The observed ratio of the inelastic Coulomb form factor squared to the elastic one below q* = 0.8 fm-* was reported ‘*). The results of the present calculations are shown in fig. 9. They suggest that the dependence of the calculated ratio on the assumed force is unexpectedly large. However, further investigation on them is not relevant at the present stage, because there have been no reliable data of the longitudinal form factors covering up to q2 = 9 fme2 of an elastic contribution and an inelastic one, separately.
&I / Fe:
13 0
0.2
l.ongitudinrl
0.4 q2(fni2
0.6
0.6
1
Fig. 9. The same as fig. 6 for the squared Coulomb form factors.
One of the conclusions here is that the microscopic cluster model produces satisfacto~ly the el~troma~etic form factors of the ground spy-doublet states of ‘Li, although there is still a discrepancy between the observed data and the present calculations in the q2 dependence of the magnetic form factor. In order to improve the q2 dependence one must either find another effective interaction or consider another effect such as spatial symmetry breaking, i.e. a mixture of [421], [331] etc. The ele~troma~etic form factors also reflect the size effect very sensitively. A natural size or an alpha-like compact size produces a good fit. This size satisfies the stability condition and gives a binding energy closed to the observed value.
T. Kajino et al. f Eiectr~magnetic
340
p~opetiies
4. Systematic description of the electromagnetic moments and the transition probabilities We want to discuss in this section the electromagnetic moments, radiative transition probabilities and density distribution radii of ‘Li following the arguments of the last section. These observables are essentially determined by the derivative of the corresponding form factors in the limit of. 9 = 0. Therefore the static moments reflect the behavior of the form factors in the low momentum transfer region. The magnetic dipole moment and B(M1) strength are typical examples. Both quantities depend very weakly on the size and force. They are shown in table 3. This is because the spin-current contribution amounts to 90% of the total matrix element and only the remaining 10% comes from the convection current part which depends on the wave function. On the other hand, the root mean square radius J(&) of the magnetization density distribution and the octupole moment fl show a sizable dependence on the cluster size and the effective force. The G-coupling scheme in many shell-model calculations 17-19Z23) gives the best fit to the B(M1) strength or equivalently the radiative lifetime of the first excited state even at the [43] super-multiplet limit. Intermediate-coupling shell-model calculations 15*i6) also show that the doublets have a very similar structure to the pure U-coupling states. Indeed our calculation gives a similar agreement with the experimental 5,7,12)values as do the U-coupling shell-model calculations, though a symmetry breaking effect is not taken into account. It is partly a matter of course due to the spin dominance in the Ml operator. The experimental values of the magnetic dipole and octupole moments show 15% and 35 % quenching compared with the Schmidt values, namely ,_&h= 3.792~~ for the dipole moment and Lnsch=
TABLE The
3
magnetic properties of the ground spin-doublet
states of ‘Li
ex~r~rne~t “1
2.98 * 0.05
3.2*0.4 3.2564 b,
calculations (9
3.31
3.14
-12.6
2.34
0.88
(ii) (iii) (iv)
2.9s 3.04 2.81
3.16 3.16 3.15
-10.4 -10.5 -8.71
2.16 2.16 2.17
0.99 1.00 1.24
19.3*0.4/
2.48*0.12 1.06+0.14’)
LOS * 0.05 1.07*o.osd)
a, y and R represent the radius of the magnetization density distribution, the magnetic dipole and octupole moments of the ground state. B(M1; t) is the reduced transition probability for the $- -t fmagnetic dipole transition and 7(Ml; J.) the lifetime of the first excited state. “) Ref. I*). ‘) Ref. 9). “) Ref. ‘). d Ref. ‘O).
T. Kajino et al. / Electromagnetic properties TABLE The electric properties
m experiment :;
341
4
of the ground spin-doublet
states of ‘Li
JG5
BW; t)
(fm)
(fm)
( e2 *fm4)
2.55 * 0.07 (2.39)
(2.50)
13.8* 1.1) -4.l’), -4.44 d)
7*4 8.3kO.7’)
2.80 2.51 2.55 2.35
2.83 2.51 2.57 2.35
-4.58 -4.39 -4.41 -3.50
11.64 10.42 10.57 6.61
calculations if) (iii) (iv)
m, m and QE2 represent the rms radii of the charge density and neutron matter density distributions, respectively, and the electric quadrupole moment of the ground state. B(E2; t) is the reduced transition probability for the $- + f- electric quadrupole transition. “) Ref. I’). b, Ref. 35). ‘) Ref. 6). d, Ref. s). ‘) Ref. ‘).
-1.67!5&&) [ref. 34)] for the octupole moment, respectively. However, it is easily explained in the present calculation as the result of the LS-coupling scheme. The present calculation also succeeds in reproducing the large electric quadrupole moment and B(E2) strength between the ground spin-doublet states with eeff= e. This is an advantage of the present model in comparison with shell-model calculations where the effective charge is needed. The results are shown in table 4 with other electric observables compared with the experimental values 6-8*12P35). Since any specific condition such as a spin dominance in the Ml operator does not exist in the electric moments, they show a moderate size dependence as well as a force dependence. Except for case (i), in which a large cluster size used, their agreements with the empirical values are reasonable. Recently the rr* scattering on 7Li has been measured to determine the neutron matter radius of 7Li [ref. 35)]. Although its absolute value seems to be rather poorly determined because the extraction of the result includes many uncertainties, it should be pointed out that the observed neutron matter radius is larger than the charge radius. In the present calculations when an alpha-like compact size is used for the triton cluster (case (ii)), the neutron matter radius becomes as equally small as the charge radius. This is because two neutrons in the triton cluster are packed into too narrow a space. This fact is independent of the assumed interactions. Especially when the MHN force is used (case (iv)), the total wave function itself shrinks very much into the narrow space (see fig. l), since the attractive range is shorter than that of the V2 force. Therefore if a larger neutron matter radius suggested by the T* scattering experiment has indeed a crucial meaning, both the alpha and triton clusters of their natural sizes (case (iii)) are favorable in describing the nucleon matter distribution.
342
T. Kajino et al. / Electromagnetic properties
To summarize the results, the microscopic cu-t cluster wave function provides reasonable agreement with the observed data for all the electromagnetic moments, transition probabilities and radii of the ‘Li nucleus without any adjustable parameter such as an effective charge. In particular, a large electric quadrupole deformation is explained easily without serious change of the magnetic properties. As for the cluster size, the natural sizes of the two clusters are desirable in order to describe consistently many different properties. 5. Threshold effect on the mirror nuclei The purpose of this section is to predict several interesting differences between the isodoublet mirror nuclei ‘Li and ‘Be. Emphasis is placed on the difference of the degree of clustering in these nuclei which is caused by their different threshold energies and on the different role of the isovector part in the electromagnetic interaction. All the size parameters of t, 3He and cx nuclei satisfy the stability conditions (3). The V2 force is used as an effective central interaction. Those parameters are shown in table 2b. There have been few data on the electromagnetic properties of ‘Be. The magnetic dipole transition probability between the ground spin-doublet states 5-11) only has been measured. The present calculation agrees with the experiments in their errors (table 5). Other observables of ‘Be are calculated and compared with those of ‘Li. The o-t and cy-3He cluster configurations are bound below their thresholds by 2.467 MeV and 1.586 MeV, respectively. Since they are weakly bound and the difference between them 0.881 MeV is comparable to the threshold energies, the clustering of the cu-3He system is expected to exceed that of the a-t system. To see the threshold effect clearly we carried out the following two calculations on ‘Be. First, we artificially make the cy-3He system deeply bound just like the a-t system. This fictitious system is called hereafter the “deeply bound a-3He system or ‘Be” (fig. 10). In the second calculation, the calculated a-3He breakup threshold energy is adjusted to the experiment. This case is called the “natural a-3He system or ‘Be”. TABLE
5
The same as table 3 for 7Be
4%)
B(Ml; t)
(fm)
h44)
experiment
4Ml; i) (x10-‘3sec) 1.92kO.25 “), 2.7* 1.0 b,
calculations
(iii) deeply bound “) Ref. ‘).
3.50 3.38 b, Ref. II).
-1.28 -1.27
7.55 6.85
1.58 1.58
2.27 1.33
T Kajino et al. / Electromagnetic properties
2-cluster
343
threshold
:r/E
t
&I.,, ‘Li
deeply bound -,
‘Be
Fig. 10. The observed energy spectra of ‘Li and ‘Be from their two-cluster breakup thresholds.
Fig. 11 shows the reduced width amplitudes of the both wave functions together with that of the natural cr-t system. The reduced width amplitude of the deeply bound a-3He system resembles that of the cr-t system more than that of the natural n-3He system. Therefore, if there are some differences in the predictions of the physical quantities between the deeply bound 7Be and natural 7Li, they may be regarded as differences caused by isovector contributions. On the other hand, if some changes take place from the deeply bound cw-3Hesystem to the natural cw-3He system, they can be regarded as threshold effects. As for the electric properties, both the isovector effect and the threshold effect are equally imporant. As a typical example, the B(E2) strength changes as 10.57(7Li) A 22.7 (deeply bound 7Be) 5 29.2 (natural 7Be) , where “I” means the isovector effect and “T” the threshold effect mentioned above. Other moments and radii have already been shown in tables 4 and 6. The situation
-.-.-
deeply bound
Fig. 11. The same as fig. 1 for the ‘Be nucleus. The solid and dash-dotted curves are the reduced width amplitudes of the naturally bound and deeply bound a-3He systems, respectively. The reduced width amplitude of the a-t cluster wave function is also illustrated as the dotted curve.
T Kajino et al. / Electromagnetic properties
344
~~%il/Fz~7Bcl 312 -
3/2-
I
0
6
2
8
Fig. 12. The ratio of the squared form factors of ‘Li to those of ‘Be for the s---f $- transition. The upper part is for the magnetic form factors and the lower part for the Coulomb form factors. The solid and dotted curves represent the calculated results using the naturally bound and deeply bound cluster wave functions of the cx-3He system, respectively. For the a-t system, the naturally bound cluster wave function is used.
holds true for the Coulomb form factor too. In fig. 12 the form-factor ratios of 7Li/7Be and 7Li/(deeply,bound 7Be) are illustrated by the solid and dotted curves, respectively. The deviation of the dotted curve from 1 is considered to show the isovector effect and the change from the dotted to the solid curves is the threshold effect. On the magnetic properties, however, the effect of the isovector difference is prominent in the low momentum transfer region. This fact is clearly shown in the extremely large deviation of the ratios 7Li/7Be and 7Li/(deeply bound 7Be) of ,the magnetic form factors from 1. In this region, the threshold effect is relatively weak, which explains why the two ratios are almost identical. In the Ml operator, the spin current and the convection current work in phase for 7Li and out of phase for TABLE 6 The same as table 4 for ‘Be
J(4)
m
(fm)
(fm)
2.74 2.65
2.50 2.40
WE& t) (e’. fm4)
calculations
(iii) deeply bound
-5.51 -4.89
29.2 22.7
I: Kajino et al. / Electromagneticproperties
345
7Be. This interference in the 7Be nucleus decreases the Ml form factor. Thus the ratios become very large in the Ml-dominant region. On the other hand, in the high momentum transfer region the ratios are not so large. This is because the M3 contribution dominates in this region. No interference in the M3 operator occurs because only the spin current part survives in the M3 form factor. The threshold effect becomes important again in this region as for the Coulomb form factor. Next we shall discuss briefly the importance of the doublet-splitting energy. Compared with the threshold energy difference of 0.88 MeV between 7Li and ‘Be, the doublet-splitting energies are very similar: 0.477 MeV for 7Li and 0.429 MeV for ‘Be. Therefore the threshold energy difference between the two first excited states is almost the same as that of the ground states. It is thus expected that the clustering of the first excited state relative to the ground state may be more remarkable in ‘Be than ‘Li. This leads to the prediction that the inelastic Coulomb and magnetic dipole form factors of ‘Be have their maximum values at smaller momentum transfers than those of 7Li. Therefore, the ratios of those inelastic form factors to the elastic ones increase more steeply for ‘Be than 7Li at the lower momentum transfer than at their maxima and, on the contrary, decrease more
Fig. 13. The ratios of the squared Coulomb form factors for the $-+ k- transition to those for the $--P $transition of the ‘Li and ‘Be nuclei. The naturally bound cluster wave functions are used for both nudei.
rapidly at the higher momentum transfer than at the maxima. In figs. 13 and 14, the tendency is seen as a different q2 dependence. The summa~ of this section is that clustering affects s~gni~cantly the electromagnetic structure difference between the isodoublet mirror nuclei through the isovector part of the elecromagnetic interaction. Experiments on 7Be are highly desired.
346
Fig 14. The same 2s fig, $3 for the squared magnetic form factors.
6. Conclusions
ft was shownthat the microscopic CY-tcluster model explains the large electric quadrupole deformation af the ground spin-doublet states of 7Li without any adjustable parameter such as an effective charge. It should be emphasized that the calculated results are obtained by using the resonating group wave function which has no spurious center-of-mass motion, The calculated results for other electromagnetic properties, namely the density distribution radii, static moments and radiative transition probabilities are in reasonable agreement with the available data, too. Both the elastic and inelastic lon~t~din~ Coulomb and transverse magnetic form factors are reproduced well in the present calculation. It was pointed out, however, that in a detailed comparison with the observed data the 4’ dependence of the ealeulated magnetic form factors is not very satisfactory and the force dependence is unexpectedly large. Furthermore, the effects of symmetry breaking or mesonexchange currents 24) should be examined. It is an open problem to find better effective interactions. The cluster-size effect on the electromagnetic properties was also studied from the viewpoint of the stability of the constituent nuclei, One learned that the natural sizes for both clusters are desirable in order to have a consistent description of many different properties. It was predicted that the difference of the average distance between the two clusters produces a sizable difference in the physical quantities between the isodoublet mirror nuclei ‘Li and 7Be. This is attributed to the threshold effect in conjunction with the isovector part of the electromagnetic interaction.
T. K@w
et at. j Electromagnetic properties
347
This article shows clearly the advantage of the microscopic cr-t c3He) cluster wave function in systematically describing the electromagnetic structure of ‘Li (‘Be). Being stimulated by the present successful results, we are planning to investigate the 4He(3He, y)%e and ‘Be(p, rj8B radiative capture reactions. Both reactions are closely related to the solar neutrino problem that the observed flux on the earth is smaller than the theoretical value predicted by a standard solarmodel 42).The products of these reactions, ‘Be and a& are considered to be the main sources of the solar neutrino. Recent ex~rimental development on the both reactions 3&-41)requires accurate theoretical calcu’lations to extract the low-energy capture cross sections. All possible electromagnetic multipole contributions to the capture process should be evaluated. It seems that the resonating group wave function is most desirable for this purpose, because both the bound and scattering states can be described in a consistent way and especially the effect of antisymmetrization is fully taken into account. The authors would like to thank members of the Nuclear Theory Group in University of Tokyo for valuable discussions. The numerical calcuiations of this work have been performed by using M-280 H at the Computer Center, University of Tokyo and ~nan~ially supported by the Research Center for Nuclear Physics, Osaka University.
Appendix A
A space called NGC space (the new generator coordinate space) is defined as a parameter space spanned by basis functions {ff”’ (X)), which are related to the RGM basis functions (x!“(t)) as
l-(&x;
S)=
0 T
3/4
exp [-S(r-X)']
,
fA.2)
where S is the size parameter for the gaussian folding transform. Reduced matrix elements of the operator 15,which represents symbolically the total hamiltonian H, 1 and the electromagnetic operator, in the cluster states (1) are expressed in the RGM and the new generator coordinate representations by CCWf>JfPilCbWi~
348
T Kajino et al. / Electromagnetic properties
with K RGM(r’,r) = j ([~‘s1’(51)O~‘s2’(52)]‘sf’ 6(r’-a)(OI X .d12{[@(‘l)(.&) 0
K NGCM(X’,X) =
I X
@(‘z)( tJ2)]‘$’S(r - a)}) d& d& da
,
(A.5)
([cPp(s~‘(&)O@‘s2’(52)]‘Sf’r(r, X’; S)lOl d12{[ Ocsl)( tl) 0
@csz)( [2)]‘si’r(r, X; 6))) de, d&
dr
,
(A.61
where the relations (A.l) and (A.2) are used. Using Tohsaki’s formula 32), we obtain the NGCM kernels KNCCMfrom the Brink GCM kernels KGCM, 1 1 K NGCMW’, X) = -(2r)6 &
(
-yy’o &
3/2 >
x
-iK’*
~Ko,,(R’,R)exp X
[
1[
(R’-r’)+$
iK*(R-r)+E
exp
(UK-~/K’)* 4w
1
I
T(r, X; 6) dR’ dR dK’ dK dr’ dr.
(A.7)
Here R’ and R are the original generator coordinates in the Brink GCM wave functions. This formula (A.7) for the inseparable double Fourier transform is one of important results derived by Tohsaki-Suzuki. The kernels KNGCMare free from the c.m. motion. Applying eqs. (A.3)-(A.7) to eq. (4), we can solve the Schrodinger equation in the NGC space. In eq. (A.7), u, y etc. are given by u = (~W%V2/
T)~‘* ,
w = WW’IP~~Y
+NNP:P:IW
~‘=a%+P;,P*+P;)
7
64.8)
3
in a general case with four size parameters differing from one another, of which two, /3; and p;, belong to a bra state and the others, pi and p2, to a ket state, respectively. In this paper locally peaked gaussian wave functions 31) are used as the basis vectors of the inter-cluster relative wave function: 3/4
exp [-8(r2+X?)]iL(2
SXir) YcL)(E) .
(A.9)
349
T. Kajitw et al. / Electromagnetic propeties
These wave functions correspond p’
(X)
to =
S(Xx;x,)ywyl&
,
(A.lO)
in the NGC space. Here the function iL is a modified spherical Bessel function and Xi’s are mesh points. We use 6 =~(N,N,/A)j3(a) and 11 mesh points, namely {X$=1-11}={0.6, 1.2, 1.8, 2.4, 4.15, 5.9, 7.65, 9.4, 11.15, 12.9, 14.65). Appendix B The kernels of el~tromagnetic multipole operators are here given in terms of the new generator coordinate representation. Since the projection in the operators (13) and (14) can be performed in the NGC space, the Brink GCM kernels of the unprojected operators are first derived, which are essentially classified into three types:
~expEig+pkl,
for Ch
g
exp
[b
l
Pk)&,!k7
for MA (convection)
T
exp
14
l
Pkh#k~
for Mh (spin) .
,
,
(B.1)
In order to hold the translational invariance, any operator should be written only in terms of the internal coordinates. Using the relations
c,6
pk=rk-rc=rk---
exp
[i4
l
Pkllk
=+
exp
[@
03.2)
A ’ ’ pklpk
xvpk
=~fvrX{expiiq’pkl(v,~-~)},
03.3)
one can easity obtain the GCM kernels. The general form is &,,(R’,
R)
[
tl/WXq
J
- Qq’ldq^ for the two (Os~-cluster system. The outer products of R
03.4) X q,
R’ x q and R’ x R
?I Kajino et aL / EIectromagnerie prqwties
350
come from the convection current part, while only the scalar term comes from the spin-current and Coulomb parts. Inserting eq. (B.4) into eq. (A.7), one gets the NGCM kernels free from the c.m. excitation. For the A-pole operators, they are as follows:
Xexp
CfW’, X
u3.6)
411 d4,
where u = h Z+Z 1 and the function f(X’,
x, q) = -T,x2- ?2x%r3xt
l
x+itpx*
q+i(bb’X’
l
fB.8)
q-eq2
is used for simplicity. In eq. (BS)-(B.7), the coefficients C, 4, TV etc. are very simple functions of a, a’, b, B, B’, Q and quantities in eq. (A.@, which depend on the number of nucleons exchanged due to the Pauli principle and the size parameters pi, /3&,PI and &. The summation on the right-hand side is taken over those types s~boiically. In the same manner as eq. (I??), the scalar function G^(X’, X; q) is defined by
Connecting the NGCM kernels (ES)-(B.7) projected NGCM scalar kernels as GCh(X’, X; q) -
with eq. (B.91, one finally gets the
-J-(l)“(4~)“/” exp[-~,X2-*
Xf2-
2
8q2]
T. Kajino et al. 1 Electromagneticproperties
G%i
351
W’, X; 4)
+G G$;“‘(X’,
3
Ax’xF3 i )H,
(B.ll)
X; q)
(B.12)
(B.13)
References 1) J. Goldemberg and Y. Torizuka, Phys. Rev, 129 (1963) 312 2) L.R. Suelzle, M.R. Yearian and H. Crannell, Phys. Rev. 162 (1967) 992 3) 4) 5) 6) 7)
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T. Kajino et al. / Electromagnetic properties
8) S.L. Kahalas and R.K. Neset, J. Chem. Phys. 39 (1963) 529 9) A. Beckmann, K.D. Boklen and D. Elke, Z. Phys. 270 (1974) 173 10) T. Lauritsen and F. Ajzenberg-Selove, Nucl. Phys. 78 (1966) 41 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36)
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D. St. P. Bunbury, S. Devons, G. Manning and T.H. Towle, Proc. Phys. Sot. A69 (1956) 165 G.J.C. van Niftrik, L. Lapikes, H. de Vries and G. Box, Nucl. Phys. Al74 (1971) 173 P.E. Burt, L.W. Fagg and H. Crannell, Phys. Rev. C26 (1982) 713 J. Lichtenstadt, J. Alster, M.A. Moinester, J. Dubach, R.S. Hicks, G.A. Peterson and S. Kowalski, Phys. Lett. 121B (1983) 377 S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1; Al01 (1967) 1 F.C. Barkar, Nucl. Phys. 83 (1966) 418 R.S. Willey, Nucl. Phys. 40 (1963) 529 A.M. Lane and L.A. Radicati, Proc. Phys. Sot. 67 (1954) 167 A.M. Lane, Proc. Phys. Sot. 68 (1955) 189 M. Bouten, M.C. Bouten and P. van Leuven, Nucl. Phys. A102 (1067) 322; All1 (1968) 385 J. Kruger and P. van Leuven, Nucl. Phys. Al39 (1969) 418 CM. Chesterfield and B.M. Spicer, Nucl. Phys. 41 (1963) 675 D. Kurath, Nucl. Phys. 14 (1960) 398; Phys. Rev. 101 (1956) 216; 140 (1965) B1190 J. Dubach, J.H. Koch and T.W. Donnelly, Nucl. Phys. A271 (1976) 279 H. Kanada, Q.K.K. Liu and Y.C. Tang, Phys. Rev. C22 (1980) 813 Atom. Nucl. Data Tables, 14 (1974) 485 J.H.E. Mattauch, W. Thiele and A.H. Wapstra, Nucl. Phys. 67 (1965) 1; F. Ajzenberg-Selove, Nucl. Phys. A320 (1979) 1 A.B. Volkov, Nucl. Phys. 74 (1965) 33 F. Tanabe, A. Tohsaki and R. Tamagaki, Prog. Theor. Phys. 53 (1975) 677 D. Brink, Proc. Int. School of Physics, Enrico Fermi 36 (Academic Press, New York, 1966), p. 247 M. Kamimura, Prog. Theor. Phys. Suppl. No. 62 (1977) 236 A. Tohsaki-Suzuki, Prog. Theor. Phys. 59 (1978) 1261; Suppl. No. 62 (1977) 191 A. Arima, Y. Horikawa, H. Hyuga and T. Suzuki, Phys. Rev. Lett. 40 (1978) 1001 C. Schwartz, Phys. Rev. 97 (1955) 380 E.F. Gibson, J.J. Kraushaar, T.G. Masterson, R.J. Peterson, R.S. Raymond and R.A. Ristinen, Nucl. Pys. A377 (1982) 389 H. Krawinkel, H.W. Backer, L. Buchmann, J. Giirres, K.U. Kettner, W.E. Kieser, R. Santo, P. Schmalbrock, H.P. Trautvetter, A. Vlieks, C. Rolfs, J.W. Hammer, R.E. Azuma and W.S. Rodney, Z. Phys. A304 (1982) 307 J.L. Osborne, C.A. Barnes, R.W. Kavanagh, R.M. Kremer, G.J. Mathews, J.L. Zyskind, P.D. Parker and A.J. Howard, Phys. Rev. Lett. 48 (1982) 1664 R.G.H. Robertson, P. Dyer, T.J. Bowles, R.E. Brown, N. Jarmie, C.J. Maggiore and SM. Austin, Phys. Rev. C27 (1983) 11 B.W. Filippone, A.J. Elwyn, W. Ray, Jr. and D.D. Koetke, Phys. Rev. C25 (1982) 2174 A.J. Elwyn, R.E. Holland, C.N. Davids and W. Ray, Jr., Phys. Rev. C25 (1982) 2168 B.W. Filippone, A.J. Elwyn, C.N. Davids and D.D. Koetke, Phys. Rev. Lett. 50 (1983) 412 J.N. Bahcall, N.A. Bahcall and R.K. Ulrich, Astrophys. J. 156 (1969) 559; J.N. Bahcall and R.K. Ulrich, Astrophys. J. 160 (1970) L57 H. Furutani, H. Kanada, T. Kaneko, S. Nagata, H. Nishioka, S. Okabe, S. Saito, T. Sakuda and M. Seya, Prog. Theor. Phys., Suppl. No. 68 (1980) 215 T. de Forest and J.D. Walecka, Adv. in Phys. 15 (1966) 1 T. Janssens, R. Hofstadter, E.B. Hughes and M.R. Yearian, Phys. Rev. 142 (1966) 922
ELEC’FROMAGNETFC
PROPERTIES
OF ‘Li AND ‘Be
IN A CLUSTER MODEL TOSHITAKA
KAJLNO, TAKEHIRO
MATSUSE and AKTl-0 ARIMA
&~~rtment of Physics, Fa&u~tyofScience, Universityof Tokyo, Bu~k~-k~ Tokyo 113, Japan Received 7 June 1983 (Revised 11 August 1983)
Abstrrctr The electromagnetic properties of 7Li (‘Be) are studied by using a microscopic a-t (‘He) cluster model. Among the several properties are the Coulomb and magnetic form factors for the ground spin-doublet states of ‘Li (‘Be), density distribution radii, static moments and radiative transition probabilities. Provided that the stability conditions for the binding energies are satisfied for both the constituent nuclei 1y and t (3He). the calculated results are in good agreement .with many availabfe data. Several interesting differences are predicted between ‘Li and ‘Be.
1. Introdmtion Many different kinds of experimental data have been accumulated for studying the electromagnetic structures of 7Li and ‘Be [refs. lmt4)], and a great many theoretical works 15-25) have also been devoted to study them. Especially, a very large electric quadrupole deformation of the ground spin-doublet states of 7Li has attracted several authors to theoretically investigate it 20-23*25).A deformed shell-model calculation by Kurath 23) showed that a 20% core excitation is needed to explain the large quadrupole deformation but which at the same time leads to an unacceptably large charge radius. Reasonable values for the both quantities have been obtained by a projected I&tree-Fock calculation ‘*), which could not, however, remove the large effect of spurious c.m. motion. Recently, an &t cluster-model calculation has been carried out by Kanada ef al. 25). They have succeeded in obtaining good agreement with the observations. Most of those theoretical works, however, have only been concentrated on a few properties (for example, the elastic Coulomb form factor or the electric quadrupole moment of 7Li) though many different kinds of data are available r-14). Quite recently, electron scattering experiments r3,14) have bean performed to measure the transverse form factors covering up to .XJ’= 9 fm-* with extremely high resolution and confirmed the accuracy of the data obtained in 1971 by the Amsterdam group ‘3. Though the observed inelastic data, not only of the transverse form factors but also of the lon~tudinal ones, are restricted to a momentum transfer region lower than q2 = 1.fm-*, several density distributjon radii, static moments and radiative transition probabilities could be extracted from the observed values by 323
T. Kajino et al. / Electromagnetic properties
324
the Amsterdam group since the resolution was quite good. Being stimulated by their works, we have investigated as many of the electromagnetic properties of ‘Li as possible in a microscopic a-t cluster model. This model has no difficulty in removing the spurious c.m. motion and can be very easily applied to cr-t scattering problems too. This is an advantage of the present model in comparison with shell-model and projected Hartree-Fock-model calculations. One purpose of the present paper is to show that the microscopic a-t (3He) cluster model provides a systematic understanding of the electromagnetic properties of ‘Li (‘Be). Special interest is placed on the point that the large electric quadrupole deformation can be produced by this model without changing many of the other electric and magnetic properties. The second purpose of this article is to clarify the size effect of the constituernt clusters on those quantities. In sect. 2, the resonating group method is briefly reviewed and several formulas are obtained for calculating the electromagnetic form factors and other basic observables. In sect. 3, various multipole contributions are separately calculated in the inclusive form factor. The size effect on each contribution is examined. Taking into account the size effect and the dependence of the result on an assumed force, we look into the consistency among the static moments, transition probabilities etc. in sect. 4. The Cy-3He (t) threshold effect on ‘Be (‘Li) and the difference between the isodoublet mirror nuclei are investigated in sect. 5. Finally, in sect. 6, the results are summarized.
2. Resonating group method The resonating group method is briefly discussed in this section. The method for calculating the RGM kernels for the electromagnetic multipole operators is shown. Notations and definitions are given in this section. 2.1. CLUSTER
WAVE
FUNCTION
AND
EFFECTIVE
INTERACTION
The
‘Li (‘Be) nucleus is assumed to consist of two clusters (Yand t ( 3He) throughout this paper. In the resonating group method (RGM) a total wave function is written as
in the U-coupling scheme, where J 12 is the antisymmetrizer between the two clusters, and & and Ni are the internal coordinates including spin-isospin variables and the mass number of the ith cluster. Here x&r) = xJL(r) YCL’(i) is the intercluster relative motion to be solved while the intrinsic wave function @(ti) of the ith cluster is approximated by a gaussian function with cluster size parameter pi.
T. Kajino et al. / Electromagneticproperties
325
The highest spatial symmetry is assumed in both the cluster wave functions. The total hamiltonian is written as H=i
k
tk-TG+
= lI,+ti,+
C Vij
(2)
icj
1
lr,,,+
is(cluster1)
c
V,
(2’)
9
je(cluster2)
where To and Trel are the kinetic energies of the c.m. motion and the inter-cluster relative motion, respectively, hi is the intrinsic hamiltonian of the ith cluster, and Vij a two-body effective interaction which consists of central, spin-orbit and Coulomb forces. As a central part, two typical phenomenological interations are used. One is the Volkov force No. 2 (V2) 28) which is a Serber-type gaussian potential having a long-range attractive part and a very weak short-range repulsive part only in the 3E and ‘E channels. The other is the modified-Hasegawa-Nagata force (MHN) 29) which is a three-range gaussian potential: a Rosenfeld-type longrange part simulating the tail of OPEP, an intermediate-range attractive part and a short-range repulsive part of moderate strength in the 3E, ‘E and 3O channels. They have the form
where PM, PB and PH are the Majorana, Bartlett and Heisenberg exchange operators, respectively; Wi, mi, bi and hi are the Wigner and their associated parameters of the exchange mixtures, Vi is the strength parameter, pi the range parameter, and rk! the relative coordinate between the kth and fth nucleon% The reason why we take these two interactions is because they have completely different exchange mixtures but give acceptable binding energies and rms charge radii to CX, t and 3He nuclei under the stability conditions
d(@(&)IhI@(6))=, dP
7
d2(@(6)IhI@(b)),o dP2 3
where /3 are their size parameters. As a spin-orbit interaction, a phenomenological Nagata force 43) is used. This force also has a gaussian form factor
V,,(rk,)=k=SC(Wi+miP~))ViexP[-CLiT2kll I
{ (Sk+%)
+klX~vkl))r
where Sk is the spin operator of the kth nucleon. Table 1 summarizes the parameters of the effective interactions adopted in this calculation.
T. Kajino et al / Ekctromagnetic prop&es
326
TABLE
1
Parameters of the two-body effective interactions i
IL~(fm-‘1
Vi (MeV)
wi
m,
hi
bi
cenrral interactions
v2 “)
1 f 2 3
0.3086 0.9803 0.16 1.127 3.4
-60.65 61.14 -6.0 -546.0 1655.0
1 2
3.0 5.0
-1519.0 1918.0
2
MHN b,
spin-whit
Nagata “f
“) Ref. 28),
“) Ref. 2g).
l-m l-m -0.2361 0.8297-m 0.4473
0.3785
0
0 0.5972 0.1405 0.1015
0
0 -0.5139 0.0302 0.0526
interaction
0.5 OS
-0.5 -0.5
0 0
0 0
‘) Ref. “).
In order to obtain the relative wave function differential SchrGdinger equation is solved:
= 0.
m
m 1.1528
,yJL(r), the following integro-
(4)
Three different sets of the cluster sizes are used. They are (i) /3(t) = P(a) = 0.32 fm-2; rather large size which reproduces the experimental rms charge radius of ‘Li approximately in the shell-model configuration of (OS)~(O~)~ [431. (ii) p(t) = Pfcr); very compact size which satisfies eq. (3) for the nucleus. (iii) P(t) and @(cr); they satisfy the stability condition (3) for both the nuclei t and a. These three cases are adopted to examine the size effect of the constituent nuclei. The V2 force is used for them. In addition, the MHN force is also used for case (iii) to see the force dependence. This additional case is called hereafter case’(iv). Only the two parameters are varied in the effective interaction, namely the Majorana mixture parameter m of the central force and the strength parameter kLs of the spin-orbit force. They are set to reproduce the separation energies of’ the two-cluster breakups 7Li+ a! + t and ‘Be-, a -t3He and the doublet-splitting energies for 7Li and 7Be, separately. With these conditions being satisfied, the cluster size effects on the electromagnetic structure of 7Li and 7Be can be discussed in connection with the stability conditions of the constituent nuclei. The values of these parameters are shown in table 2 with several properties of the gruund states of t, 3He, 4He, 7Li and ‘Be nuclei.
327
T. Kajino et al j Electromagnetic properties TABLE 2a
Parameter values varied in the calculations and several calculated results of the ground-state properties of the t, 4He and ‘Li nuclei Calculations Experiment
central force @{a) (frn+I WI (fm-? Majorana mixture m kLS LS strength E&1 (MeW E(!t-) WW BE (MeV)
-2.447 “) -1,989 “) 39.245 b, 28.296 b, 1.63(4) “) 8.482 b, 1.70(J) “f
a! t
BmeV) J{r&) @mm)
(ii)
(iii)
(iv)
V2
v2
v2
MHN
0.32 0.32 0.55 0.75
sire parameter
‘Li
W
-2.366 -1.850 31.942 22.820 2.028 6.756 1.896
0.5285 0.5285 0.585 0.54
0.5285 0.400 0.5735 0.60
-2.442 -1.983 36.600 27.973 1.651 6.186 1.537
-2.473 -1.963 37.638 27.973 1.651 7.192 1.725
0.574 0.460 0.39658 0.30 -2.538 -2.065 37.928 28.680 1.599 6.710 1.628
The four cases fi) to [iv) show the different sets of the chrster size parameters #frr) and #Z(t) for the alpha and triton nuclei, respectively. m is the Majorana mixture pantmeter of the centrat force and k= the strength parameter of the spin-orbit force. E(J*), BE and J(&> are the energy of the state having spin J and parity r from the a-t breakup threshold, binding energies and root mean square charge radii. “) F. Ajzenberg-Selove “). b, J.H.E. Mattauch et al. *‘). “) Ref. 2a). TANE
2b
The same as table 2a for the 3He, ‘$He and 7Be nuclei Calculations
Experiment central force size parameter
8(o) @n? /3f3He) (fm-‘) Majorana mixture m LS strength kLS ‘Be
a ‘He
E(f) WeVI E&I WeV) BE (MeV) BE_@feV) J(r$t UrnI BE MeVf sl
For footnotes see table 2a.
(iii)
deeply bound
V2
v2
0.5285 0.393 0.5775 0.55 -1.586 “) -1.157a) 37.601 ‘) 28.296 “) 1.63(4) “) 7.718 b) 1.8715) 3 1.88(S)
-1.548 -1.118 35.989 27.973 1.651 6,468 1.790
0.5285 0.393 0.5530 0.54 -2.476 -1.942 36.917 27.973 1.651 6.468 I.790
T Kajino et al. / Electromagnetic properties
328
Fig. 1. The reduced width amplitudes i’s of the a-t cluster wave functions of the ‘Li ground state. The dotted, dash-dotted, solid and dashed curves are obtained by using the different size parameter sets of (i), (ii), (iii) and (iv) d es&bed in the text, respectively. The short-dotted curve represents the harmonic oscillator function u,,(r; V) with Y= (3. 4/7)0.32 fm-*.
Fig. 1 shows the calculated reduced width amplitudes of the ground state of ‘Li:
0
y(l)((i)1(3/2)_
iqr-a) ~w/2~3,2(5152~) r2
65
d52
da.
(5)
>
One of them assumes a harmonic oscillator wave function ul,(r; V) with V= (N1N2/A)0.32 fmm2 as the relative wave function x. This wave function is identical to the U-coupling shell-model wave function provided that the hw of the two clusters are taken to be common and hw = (h/p)v with /* = (NlN2/A)MN. They all show a remarkable clustering or core polarization effect in comparison with the shell-model wave function. This feature is in particular clearly seen in the nuclear surface region. The occupation probability of harmonic oscillator states with radial node Nra 2 is about 30% for case (i) and amounts to 40% for case (iii). Finally the calculational procedure of eq. (4) is briefly discussed before closing this section. When the spatial parts of the constituent nuclei are described by the harmonic oscillator wave functions with unequal size parameters, the GCM kernels using Brink’s GCM wave function 30) include spurious components of the c.m. excitation. However, since the procedure to obtain the GCM kernels is very simple, they are used to construct the RGM kernels which are completely free from spuriousness. In order to solve eq. (4) by means of the variational method 31), the essential points are how to calculate the matrix elements of the hamiltonian and norm kernels and what kind of basis vectors should be chosen to describe the inter-cluster relative wave function. We use locally peaked gaussian functions as
T. Kujino et al. / Electromagneticproperties
329
basis vectors and calculate the matrix elements by means of the new generator coordinate method (NGCM) 32) introduced by Tohsaki-Suzuki. The equivalence of this method to the RGM has already been proved by him. The advantages of this choice lie in the following two points: first, the c.m. excitation is completely removed from the NGCM kernels, and second, integrations of obtaining the matrix elements of very complicated operators such as electromagnetic multipole operators are very easily performed in the new generator coordinate space (NGC space). Details are given in appendix A. 2.2. BASIC REPRESENTATION OF THE ELECTROMAGNETIC MULTIPOLE OPERATORS The
squared Coulomb and magnetic form factors can be expanded into multipole components 44) as
where Ji and Jf are the total spins of the initial and final states, respectively. the multipole operators are defined by
T%I?)=
~A~A(qr)Y~'(~)~(r)
I
Here
dr,
T:” (4) = jAtqr)tY?l ($1.J(r)) dr,
(8)
where jh (qr) and Yz’ ($( Yy’ (P)) are spherical Bessel functions and spherical (vector spherical) harmonics. As for the charge and current density operators (p, J), we take a simple approximation in which they are given by the sum of free nucleon currents: p(r) = e; a(~--pk) J(r)
= JN(d
+v
,
XELN(d
,
where JN is the convection current density and j.&Nis the magnetization density. The g-factors are gzl,= $(l -I-T,( k>)gjp’ with gjp’ = 1.0 for the orbital part and gsL = i( 1 + r,( k))gLp) +i( 1 - r,( k))g$” with g$‘) = 5.584 and g$” = -3.826 for the spin part in units of nuclear magnetons j.&N = e@%&c. Here pk is the coordinate of the kth nucleon from the c.m. and ‘Vkthe derivative as to pk. The nucleon finite
T. Kajirw et al. / E~e~i~o~~g~e~~~properties
33Q
size effect is taken into account by multiplying the multipole operators by f&Y’)(4) = Gus(q) f G,“(q)
3
A!? (4)/~p=f~)(4)J~ELn=f~)(q) where the electric form factors and isovector parts 45),
of a
(9) 9
tw
single nucleon are expressed in terms of isoscalar
,+;;:“,, 7-1+;;6/026 7+oqlo , GES(q)E; 1 1.15
1+q2/8.1Q-O*16
=
In eq. (Q), the -I-(-) sign is for protons (neutrons). In this approximation with a nucleon finite size correction ((9), (lo)), the multipole operators are given as
x(f(l~T,(k))f~)(q)+l(l-T,(k))f~)(q)) TF” (q)=
z
2-Z:
+
r,
TZ$@,
(14)
5=**i
o=h*l
(13)
1 d4 (A)
0 G(expEiq- eJw%‘~(q)P~’ k
Y
iL
where the convection current part of eq. (14) TEL: and the spin current part Tg$” are written thus hereafter. %‘Wand yb are introduced for simplicity; they are as f5115ws:
I: Kajino et al. / Electromagneticproperties
331
In eqs. (13), (14), the relation h(qt) Yl”‘(i) =L4?ri”
Yt”’ (4) exp [iq r] dq^ l
(16)
is used to project the h-pole component out of the plane wave. It should be emphasized that the laborious task in obtaining the GCM kernels of the electromagnetic operators is extremely simplified by projecting the A-pole component after ~tegration with respect to the intrinsic coordinates and summation of the spin and isospin quantum numbers. Using the IS-coupling cluster wave function (l), the reduced matrix elements in eq. (6) can be rewritten in the following forms:
(17)
where 7’2;“’ (q) is defined by dropping all the spin observables out of Tskrr (4). In order to evaluate the reduced matrix elements in the right-hand sides of eqs. (17), (18), essentially the same technique as the last subsection and appendix A is applicable. The general form of the integrals in the new generator coordinate representation is given by
1 X
iXq iX’q : XX’
I
exp [-r,X’-
x~,,(X)X’~~X’X~~X,
T*X’~- tYq’]Ff$k
t
(19)
where F?$ is a geometrical factor depending on Li, Lf, I,, 12, l3 and A, and @j(q) presents symbolically the operators of p”(q), TzT (q) and T$‘f (4). Detailed forms of (19) are summarized in appendix B. It is relevant here to comment on the convergence of the summation concerning II, I2 and l3 in eq. (19). If the inter-cluster relative wave function is represented by
332
T. Kajino et al. / Electromagnetic
properties
a harmonic oscillator function such as +,(r), the summation stops at a finite integer. However, since the cluster wave function includes a great amount of core excited components, it runs in principle from zero to infinity. In the practical calculation, of course, we are limited to scan only one index among 1,, l2 and l3 from zero to a moderately large integer, because the variable range of them is mutually restricted through the relation in F+tk. The convergence is faster with a small q than a large q. Then the convergence of the calculation is checked in every multiple form factor at q2 = 9 fm-* which is the largest value in the present calculation. For the Coulomb form factor a good convergence is obtained before max [1i, 12,&] reaches 10. On the other hand, the magnetic form factor needs at least 15 to obtain an excellent convergency. Thus in practice we adopt 20 as upper limits of 1i, 1, and l3 in the summation. The computation is devised to check the convergence automatically with an accuracy of lo-” for the Coulomb form factors and lOPi4 for the magnetic form factors. Finally, the definitions of several important observables are given. The operators of the electromagnetic A-pole moments are related to the electromagnetic operators (7) and (8) in the limit of q = 0 as
where (in* = (2A + l)!!/i* and LYE*= ((2h + l)!!/i)m. Therefore, the electromagnetic moments and radiative transition probabilities are obtained by taking the limit of q = 0:
(22)
(23) Note that QE2 for the electric quadrupole mgment (47) and p( 0) for the magnetic dipole (octupole) moment (MM*> are used’heikjafte’“2t should be noticed that the conventional definition of the electric quadrupole moment QF$ventionalis given as 2QE2. Experimentally, the B(E(M)A; Ji+ Jr) strength is proportional to the inverse of the radiative lifetime from Ji to Jr. At the same time, however, the lifetime depends
T. Kajino et al. / E~e~t~o~g~t~~ properties
333
on the photon energy: r-‘(E(M)h;
Ji+Jf)=
*‘+’
B(E(M)A; Ji+JJ
Ey=IJ&-&,I.
9
(24)
Therefore the theoretical estimation of the lifetime has two sources of errors; one B(E(M)h ; Ji+ JJ and the calculated level spacing energy which depends on models. The mean square radii for charge distribution, neutron matter distribution and magnetization distribution are given by
(26) (27) respectively, where F g; is the monopole form factor namely the matter form factor of the neutron distribution.
3. Electromagnetic form factors The electromagnetic form factors of the 7Li nucleus are discussed in this section. Each magnetic and Coulomb contribution to the inclusive form factor is calculated separately. The size effect is also examined in connection with the stability of individual isolated clusters and the total binding energy. The inclusive form factor consists of the following four contributions: elastic longitudinal Coulomb (CO + C2), inelastic longitudinal Coulomb (C2’), elastic transverse magnetic (Ml + M3) and inelastic transverse (Ml’ + E2’) form factors. To see the nature of each contribution separately the calculated results for case (iii) are shown in fig. 2. Over a wide range of momentum transfer up to $= 5 fm-* the longitudinal Coulomb form factors are prominent, while at higher momentum transfer Ml + M 3 components compete with them. The E2’ component is neglected in the present calculation because it contributes little, i.e. only 1% of the inclusive data even at its maximum value near q2 = 2 fm-*. Fig. 3 shows the calculated inclusive yields of four cases compared with the experimental values 2). The calculated values for cases (i) to (iv) are explained in subsect. 2.1. They suggest clearly that only one case, in which the same large size is used for the two clusters, should be excluded from consideration because in this exceptional case the calculated inclusive yield decreases too rapidly. This behavior seems to be attributed to the broadening of the total wave function. Such a large size gives larger charge radii of the individual
334
T. Kajino et al. / Electromagnetic
properties
Fig. 2. The square of the form factors of ‘Li for the $-+ $- (elastic) and g-+ f- (inelastic) transitions, calculated with the a-t cluster wave function for case (iii) described in the text. The open circles are the observed inclusive date of ref. *). The solid, dashed, dotted and dash-dotted cures represent the individual contribution of the elastic Coulomb (CO+C2), inelastic Coulomb (C2’), elastic magnetic (Ml +M3) and inelastic magnetic (Ml’) form factors, respectively. The thick-solid curve represents the sum of them. The electric quadrupole (E2’) contribution to the inelastic transverse form factor is omitted in the present calculation.
nuclei, (Y and t, in comparison with the experimental values. In addition, it does not satisfy the energy stability condition: namely, the calculated binding energies are too small. On the other hand, we require that the threshold energy is equal to the empirical value. As a result, we encounter a serious problem: the total binding energy of ‘Li is too small. Thus the total system becomes very loosely bound and spreads too broadly. This is the reason why the calculated form factor decreases so rapidly with increasing momentum transfer. The calculated elastic magnetic form factors are shown in fig. 4. They are in a reasonable agreement with the experimental data 14). Near the photon point the dependence on the cluster size and the force assumed is negligibly weak, which reflects the fact that the Ml matrix element is almost independent of the wave function because the spin contribution dominates in the Ml operator and [43] symmetry is assumed in all wave functions. However, the four calculations produce different form factors from one another with increasing momentum transfer. This fact reflects the difference of the calculated magnetization radius and octupole moment. It is clear that the larger the cluster size, the broader the magnetization distribution and the larger the octupole deformation produced. Therefore all over
333
Fig. 3. The square of the inch&e frrrm factors of ‘Li for the SW-+$+ affd the g--f four curves are the same as those in fig. 1.
transitions. The
Fig, 4, The square of thg: magnetic form factors d ‘Li for the SW + STtransition. The four curves are the same as those in fig. 1, The magnetic form factors consist of the Mf and the M3 contributions. Each etlntribution is ilkstrated as the short-dotted curve for case {iii). The open circks reprewnt the data of ref. 12) and the others of ref. 14)_
336
T. Kajino et al. / Electromagnetic properties
the regions covering the low momentum transfer in which Ml dominates to the high momentum transfer in which M3 dominates, the calculation with the larger cluster size gives a smaller form factor. When the MHN force is used, the calculation overshoots the experimental values at the low momentum transfer but fits them very well at the high momentum transfer. There are two possible reasons. First, since the attractive part of this interaction in the intermediate range is shorter than that of the V2 force, the octupole deformation remains small, preventing a rapid decrease of the magnetic form factor at high momentum transfer. Second, the MHN interaction has a stronger short-range repulsive core than the V2 force. The nucleonnucleon short-range correlation enhances the high momentum components of the form factor. Thus a detailed comparison between the four calculated values and the observed ones shows some systematical discrepancy. There exists an incompatible result: in the Ml dominant region the V2 force is better than the MHN force, while the latter is better than the former in the M3 dominant region. In addition, at high momentum transfer the calculated yields decrease too rapidly compared with the experimental values. Such a discrepancy is similarly seen in a shell-model calculation of the 170 nucleus 33). In order to improve the present calculation one must find another effective interaction with at least the following two features: a longer attractive range and a strong repulsive core at the short range region. As another effect for improvement one should consider an admixture of lower symmetry supermultiplet 14,15)states which is caused mainly by the spin-orbit interaction.
‘Li
104
0.2
3/2--
111
o.4 Cleff(ffTi’
1 o.8
-
transition. The four curves are Fig. 5. The square of the magnetic form factors of ‘Li for the s--f the same as those in fig. 1. The E2’ contribution is omitted in the present calculation. The open circles, the black disks and the crosses represent the data of ref. 12), ref. 13) and ref. 14), respectively.
T. K~j~noet al. / Elec~romagne~cpro~~jes
337
F,:e,/Fei Transverse
._
0
0.2
q21fm2j
0.6
0.8
Fig. 6. The ratio of the squared magnetic form factors of 7Li for the 3-j i- transition to those for the s-+4- transition. The four curves are the same as those in fig. 1. The open circles represent the data of ref. 12).
Fig. 6 shows the ratio of the inelastic transverse form factor squared to the elastic one. The calculation gives reasonable agreement with the experimental values I*) except for the case with a large cluster size. Nevertheless, similarly to the elastic form factor, the absolute values are overestimated in the inelastic Ml’ form factor in the low momentum transfer region (fig. 5). The calculated elastic longitudinal form factors are illustrated in fig. 7 with the model-dependent data 2, compiled by Suelzle et al. They are cited only for reference, though they should be re-evaluated carefully because their values have been extracted from the inclusive form factor after subtracting the transverse contributions and the inelastic Coulomb quadrupole contribution. The subtracted transverse form factors are obtained by extrapolation into the high-q2 region from the observed data in the low-q2 region which have been measured by another group4). The subtracted C2’ form factor is calculated by using the odd proton model i7). Similarly to the magnetic form factors, the different size effects are remarkably seen in the Coulomb form factors too. In the low momentum transfer region the monopole strength dominates and the slope is essentially determined by the mean square charge radius. The form factor in the high momentum transfer region, on the other hand, is dominated by the quadrupole deformation. The calculated inelastic Coulomb form factors are shown in fig. 8 to which only the quadrupole component contribute. If the doublet-splitting energy is set to zero, the inelastic form factor gives completely the same value as the elastic quadrupole component. However, a finite splitting causes an unexpectedly large difference between the two. The calculated inelastic form factor squared has a 10% larger
338
T. Kajino et al. / Electromagnetic properties
0 Fig. 7. The square of the Coulomb form factors of ‘Li for the $- + $- transition. The four curves are the same as those in fig. 1. The form factors consist of the CO and the C2 contributions. Each contribution is illustrated as the short-dotted curve for case (iii). The black disks represent the model-dependent data of ref. 2, described in the text.
\ ’
’
’
4 q2t1m2
’
6
Y _
\ \
\\ >\
\ ’
*‘\I 8
1
Fig. 8. The square of the Coulomb form factors of ‘Li for the $-+ $- transition. The four curves are the same as those in fig. 1.
T: Kajino et aL/ Electromagneticproperties
339
value at the maximum point at q2 = 1.5 fm-’ and a 20% smaller value at q2 = 9 fme2 than the elastic quadrupole component. This shows clearly the fact that the clustering of the first excited state is more developed than the ground state. Since the threshold energy sensitively controls the spreading of the wave function, the inelastic form factor decreases more rapidly than the elastic quadrupole component. The observed ratio of the inelastic Coulomb form factor squared to the elastic one below q* = 0.8 fm-* was reported ‘*). The results of the present calculations are shown in fig. 9. They suggest that the dependence of the calculated ratio on the assumed force is unexpectedly large. However, further investigation on them is not relevant at the present stage, because there have been no reliable data of the longitudinal form factors covering up to q2 = 9 fme2 of an elastic contribution and an inelastic one, separately.
&I / Fe:
13 0
0.2
l.ongitudinrl
0.4 q2(fni2
0.6
0.6
1
Fig. 9. The same as fig. 6 for the squared Coulomb form factors.
One of the conclusions here is that the microscopic cluster model produces satisfacto~ly the el~troma~etic form factors of the ground spy-doublet states of ‘Li, although there is still a discrepancy between the observed data and the present calculations in the q2 dependence of the magnetic form factor. In order to improve the q2 dependence one must either find another effective interaction or consider another effect such as spatial symmetry breaking, i.e. a mixture of [421], [331] etc. The ele~troma~etic form factors also reflect the size effect very sensitively. A natural size or an alpha-like compact size produces a good fit. This size satisfies the stability condition and gives a binding energy closed to the observed value.
T. Kajino et al. f Eiectr~magnetic
340
p~opetiies
4. Systematic description of the electromagnetic moments and the transition probabilities We want to discuss in this section the electromagnetic moments, radiative transition probabilities and density distribution radii of ‘Li following the arguments of the last section. These observables are essentially determined by the derivative of the corresponding form factors in the limit of. 9 = 0. Therefore the static moments reflect the behavior of the form factors in the low momentum transfer region. The magnetic dipole moment and B(M1) strength are typical examples. Both quantities depend very weakly on the size and force. They are shown in table 3. This is because the spin-current contribution amounts to 90% of the total matrix element and only the remaining 10% comes from the convection current part which depends on the wave function. On the other hand, the root mean square radius J(&) of the magnetization density distribution and the octupole moment fl show a sizable dependence on the cluster size and the effective force. The G-coupling scheme in many shell-model calculations 17-19Z23) gives the best fit to the B(M1) strength or equivalently the radiative lifetime of the first excited state even at the [43] super-multiplet limit. Intermediate-coupling shell-model calculations 15*i6) also show that the doublets have a very similar structure to the pure U-coupling states. Indeed our calculation gives a similar agreement with the experimental 5,7,12)values as do the U-coupling shell-model calculations, though a symmetry breaking effect is not taken into account. It is partly a matter of course due to the spin dominance in the Ml operator. The experimental values of the magnetic dipole and octupole moments show 15% and 35 % quenching compared with the Schmidt values, namely ,_&h= 3.792~~ for the dipole moment and Lnsch=
TABLE The
3
magnetic properties of the ground spin-doublet
states of ‘Li
ex~r~rne~t “1
2.98 * 0.05
3.2*0.4 3.2564 b,
calculations (9
3.31
3.14
-12.6
2.34
0.88
(ii) (iii) (iv)
2.9s 3.04 2.81
3.16 3.16 3.15
-10.4 -10.5 -8.71
2.16 2.16 2.17
0.99 1.00 1.24
19.3*0.4/
2.48*0.12 1.06+0.14’)
LOS * 0.05 1.07*o.osd)
a, y and R represent the radius of the magnetization density distribution, the magnetic dipole and octupole moments of the ground state. B(M1; t) is the reduced transition probability for the $- -t fmagnetic dipole transition and 7(Ml; J.) the lifetime of the first excited state. “) Ref. I*). ‘) Ref. 9). “) Ref. ‘). d Ref. ‘O).
T. Kajino et al. / Electromagnetic properties TABLE The electric properties
m experiment :;
341
4
of the ground spin-doublet
states of ‘Li
JG5
BW; t)
(fm)
(fm)
( e2 *fm4)
2.55 * 0.07 (2.39)
(2.50)
13.8* 1.1) -4.l’), -4.44 d)
7*4 8.3kO.7’)
2.80 2.51 2.55 2.35
2.83 2.51 2.57 2.35
-4.58 -4.39 -4.41 -3.50
11.64 10.42 10.57 6.61
calculations if) (iii) (iv)
m, m and QE2 represent the rms radii of the charge density and neutron matter density distributions, respectively, and the electric quadrupole moment of the ground state. B(E2; t) is the reduced transition probability for the $- + f- electric quadrupole transition. “) Ref. I’). b, Ref. 35). ‘) Ref. 6). d, Ref. s). ‘) Ref. ‘).
-1.67!5&&) [ref. 34)] for the octupole moment, respectively. However, it is easily explained in the present calculation as the result of the LS-coupling scheme. The present calculation also succeeds in reproducing the large electric quadrupole moment and B(E2) strength between the ground spin-doublet states with eeff= e. This is an advantage of the present model in comparison with shell-model calculations where the effective charge is needed. The results are shown in table 4 with other electric observables compared with the experimental values 6-8*12P35). Since any specific condition such as a spin dominance in the Ml operator does not exist in the electric moments, they show a moderate size dependence as well as a force dependence. Except for case (i), in which a large cluster size used, their agreements with the empirical values are reasonable. Recently the rr* scattering on 7Li has been measured to determine the neutron matter radius of 7Li [ref. 35)]. Although its absolute value seems to be rather poorly determined because the extraction of the result includes many uncertainties, it should be pointed out that the observed neutron matter radius is larger than the charge radius. In the present calculations when an alpha-like compact size is used for the triton cluster (case (ii)), the neutron matter radius becomes as equally small as the charge radius. This is because two neutrons in the triton cluster are packed into too narrow a space. This fact is independent of the assumed interactions. Especially when the MHN force is used (case (iv)), the total wave function itself shrinks very much into the narrow space (see fig. l), since the attractive range is shorter than that of the V2 force. Therefore if a larger neutron matter radius suggested by the T* scattering experiment has indeed a crucial meaning, both the alpha and triton clusters of their natural sizes (case (iii)) are favorable in describing the nucleon matter distribution.
342
T. Kajino et al. / Electromagnetic properties
To summarize the results, the microscopic cu-t cluster wave function provides reasonable agreement with the observed data for all the electromagnetic moments, transition probabilities and radii of the ‘Li nucleus without any adjustable parameter such as an effective charge. In particular, a large electric quadrupole deformation is explained easily without serious change of the magnetic properties. As for the cluster size, the natural sizes of the two clusters are desirable in order to describe consistently many different properties. 5. Threshold effect on the mirror nuclei The purpose of this section is to predict several interesting differences between the isodoublet mirror nuclei ‘Li and ‘Be. Emphasis is placed on the difference of the degree of clustering in these nuclei which is caused by their different threshold energies and on the different role of the isovector part in the electromagnetic interaction. All the size parameters of t, 3He and cx nuclei satisfy the stability conditions (3). The V2 force is used as an effective central interaction. Those parameters are shown in table 2b. There have been few data on the electromagnetic properties of ‘Be. The magnetic dipole transition probability between the ground spin-doublet states 5-11) only has been measured. The present calculation agrees with the experiments in their errors (table 5). Other observables of ‘Be are calculated and compared with those of ‘Li. The o-t and cy-3He cluster configurations are bound below their thresholds by 2.467 MeV and 1.586 MeV, respectively. Since they are weakly bound and the difference between them 0.881 MeV is comparable to the threshold energies, the clustering of the cu-3He system is expected to exceed that of the a-t system. To see the threshold effect clearly we carried out the following two calculations on ‘Be. First, we artificially make the cy-3He system deeply bound just like the a-t system. This fictitious system is called hereafter the “deeply bound a-3He system or ‘Be” (fig. 10). In the second calculation, the calculated a-3He breakup threshold energy is adjusted to the experiment. This case is called the “natural a-3He system or ‘Be”. TABLE
5
The same as table 3 for 7Be
4%)
B(Ml; t)
(fm)
h44)
experiment
4Ml; i) (x10-‘3sec) 1.92kO.25 “), 2.7* 1.0 b,
calculations
(iii) deeply bound “) Ref. ‘).
3.50 3.38 b, Ref. II).
-1.28 -1.27
7.55 6.85
1.58 1.58
2.27 1.33
T Kajino et al. / Electromagnetic properties
2-cluster
343
threshold
:r/E
t
&I.,, ‘Li
deeply bound -,
‘Be
Fig. 10. The observed energy spectra of ‘Li and ‘Be from their two-cluster breakup thresholds.
Fig. 11 shows the reduced width amplitudes of the both wave functions together with that of the natural cr-t system. The reduced width amplitude of the deeply bound a-3He system resembles that of the cr-t system more than that of the natural n-3He system. Therefore, if there are some differences in the predictions of the physical quantities between the deeply bound 7Be and natural 7Li, they may be regarded as differences caused by isovector contributions. On the other hand, if some changes take place from the deeply bound cw-3Hesystem to the natural cw-3He system, they can be regarded as threshold effects. As for the electric properties, both the isovector effect and the threshold effect are equally imporant. As a typical example, the B(E2) strength changes as 10.57(7Li) A 22.7 (deeply bound 7Be) 5 29.2 (natural 7Be) , where “I” means the isovector effect and “T” the threshold effect mentioned above. Other moments and radii have already been shown in tables 4 and 6. The situation
-.-.-
deeply bound
Fig. 11. The same as fig. 1 for the ‘Be nucleus. The solid and dash-dotted curves are the reduced width amplitudes of the naturally bound and deeply bound a-3He systems, respectively. The reduced width amplitude of the a-t cluster wave function is also illustrated as the dotted curve.
T Kajino et al. / Electromagnetic properties
344
~~%il/Fz~7Bcl 312 -
3/2-
I
0
6
2
8
Fig. 12. The ratio of the squared form factors of ‘Li to those of ‘Be for the s---f $- transition. The upper part is for the magnetic form factors and the lower part for the Coulomb form factors. The solid and dotted curves represent the calculated results using the naturally bound and deeply bound cluster wave functions of the cx-3He system, respectively. For the a-t system, the naturally bound cluster wave function is used.
holds true for the Coulomb form factor too. In fig. 12 the form-factor ratios of 7Li/7Be and 7Li/(deeply,bound 7Be) are illustrated by the solid and dotted curves, respectively. The deviation of the dotted curve from 1 is considered to show the isovector effect and the change from the dotted to the solid curves is the threshold effect. On the magnetic properties, however, the effect of the isovector difference is prominent in the low momentum transfer region. This fact is clearly shown in the extremely large deviation of the ratios 7Li/7Be and 7Li/(deeply bound 7Be) of ,the magnetic form factors from 1. In this region, the threshold effect is relatively weak, which explains why the two ratios are almost identical. In the Ml operator, the spin current and the convection current work in phase for 7Li and out of phase for TABLE 6 The same as table 4 for ‘Be
J(4)
m
(fm)
(fm)
2.74 2.65
2.50 2.40
WE& t) (e’. fm4)
calculations
(iii) deeply bound
-5.51 -4.89
29.2 22.7
I: Kajino et al. / Electromagneticproperties
345
7Be. This interference in the 7Be nucleus decreases the Ml form factor. Thus the ratios become very large in the Ml-dominant region. On the other hand, in the high momentum transfer region the ratios are not so large. This is because the M3 contribution dominates in this region. No interference in the M3 operator occurs because only the spin current part survives in the M3 form factor. The threshold effect becomes important again in this region as for the Coulomb form factor. Next we shall discuss briefly the importance of the doublet-splitting energy. Compared with the threshold energy difference of 0.88 MeV between 7Li and ‘Be, the doublet-splitting energies are very similar: 0.477 MeV for 7Li and 0.429 MeV for ‘Be. Therefore the threshold energy difference between the two first excited states is almost the same as that of the ground states. It is thus expected that the clustering of the first excited state relative to the ground state may be more remarkable in ‘Be than ‘Li. This leads to the prediction that the inelastic Coulomb and magnetic dipole form factors of ‘Be have their maximum values at smaller momentum transfers than those of 7Li. Therefore, the ratios of those inelastic form factors to the elastic ones increase more steeply for ‘Be than 7Li at the lower momentum transfer than at their maxima and, on the contrary, decrease more
Fig. 13. The ratios of the squared Coulomb form factors for the $-+ k- transition to those for the $--P $transition of the ‘Li and ‘Be nuclei. The naturally bound cluster wave functions are used for both nudei.
rapidly at the higher momentum transfer than at the maxima. In figs. 13 and 14, the tendency is seen as a different q2 dependence. The summa~ of this section is that clustering affects s~gni~cantly the electromagnetic structure difference between the isodoublet mirror nuclei through the isovector part of the elecromagnetic interaction. Experiments on 7Be are highly desired.
346
Fig 14. The same 2s fig, $3 for the squared magnetic form factors.
6. Conclusions
ft was shownthat the microscopic CY-tcluster model explains the large electric quadrupole deformation af the ground spin-doublet states of 7Li without any adjustable parameter such as an effective charge. It should be emphasized that the calculated results are obtained by using the resonating group wave function which has no spurious center-of-mass motion, The calculated results for other electromagnetic properties, namely the density distribution radii, static moments and radiative transition probabilities are in reasonable agreement with the available data, too. Both the elastic and inelastic lon~t~din~ Coulomb and transverse magnetic form factors are reproduced well in the present calculation. It was pointed out, however, that in a detailed comparison with the observed data the 4’ dependence of the ealeulated magnetic form factors is not very satisfactory and the force dependence is unexpectedly large. Furthermore, the effects of symmetry breaking or mesonexchange currents 24) should be examined. It is an open problem to find better effective interactions. The cluster-size effect on the electromagnetic properties was also studied from the viewpoint of the stability of the constituent nuclei, One learned that the natural sizes for both clusters are desirable in order to have a consistent description of many different properties. It was predicted that the difference of the average distance between the two clusters produces a sizable difference in the physical quantities between the isodoublet mirror nuclei ‘Li and 7Be. This is attributed to the threshold effect in conjunction with the isovector part of the electromagnetic interaction.
T. K@w
et at. j Electromagnetic properties
347
This article shows clearly the advantage of the microscopic cr-t c3He) cluster wave function in systematically describing the electromagnetic structure of ‘Li (‘Be). Being stimulated by the present successful results, we are planning to investigate the 4He(3He, y)%e and ‘Be(p, rj8B radiative capture reactions. Both reactions are closely related to the solar neutrino problem that the observed flux on the earth is smaller than the theoretical value predicted by a standard solarmodel 42).The products of these reactions, ‘Be and a& are considered to be the main sources of the solar neutrino. Recent ex~rimental development on the both reactions 3&-41)requires accurate theoretical calcu’lations to extract the low-energy capture cross sections. All possible electromagnetic multipole contributions to the capture process should be evaluated. It seems that the resonating group wave function is most desirable for this purpose, because both the bound and scattering states can be described in a consistent way and especially the effect of antisymmetrization is fully taken into account. The authors would like to thank members of the Nuclear Theory Group in University of Tokyo for valuable discussions. The numerical calcuiations of this work have been performed by using M-280 H at the Computer Center, University of Tokyo and ~nan~ially supported by the Research Center for Nuclear Physics, Osaka University.
Appendix A
A space called NGC space (the new generator coordinate space) is defined as a parameter space spanned by basis functions {ff”’ (X)), which are related to the RGM basis functions (x!“(t)) as
l-(&x;
S)=
0 T
3/4
exp [-S(r-X)']
,
fA.2)
where S is the size parameter for the gaussian folding transform. Reduced matrix elements of the operator 15,which represents symbolically the total hamiltonian H, 1 and the electromagnetic operator, in the cluster states (1) are expressed in the RGM and the new generator coordinate representations by CCWf>JfPilCbWi~
348
T Kajino et al. / Electromagnetic properties
with K RGM(r’,r) = j ([~‘s1’(51)O~‘s2’(52)]‘sf’ 6(r’-a)(OI X .d12{[@(‘l)(.&) 0
K NGCM(X’,X) =
I X
@(‘z)( tJ2)]‘$’S(r - a)}) d& d& da
,
(A.5)
([cPp(s~‘(&)O@‘s2’(52)]‘Sf’r(r, X’; S)lOl d12{[ Ocsl)( tl) 0
@csz)( [2)]‘si’r(r, X; 6))) de, d&
dr
,
(A.61
where the relations (A.l) and (A.2) are used. Using Tohsaki’s formula 32), we obtain the NGCM kernels KNCCMfrom the Brink GCM kernels KGCM, 1 1 K NGCMW’, X) = -(2r)6 &
(
-yy’o &
3/2 >
x
-iK’*
~Ko,,(R’,R)exp X
[
1[
(R’-r’)+$
iK*(R-r)+E
exp
(UK-~/K’)* 4w
1
I
T(r, X; 6) dR’ dR dK’ dK dr’ dr.
(A.7)
Here R’ and R are the original generator coordinates in the Brink GCM wave functions. This formula (A.7) for the inseparable double Fourier transform is one of important results derived by Tohsaki-Suzuki. The kernels KNGCMare free from the c.m. motion. Applying eqs. (A.3)-(A.7) to eq. (4), we can solve the Schrodinger equation in the NGC space. In eq. (A.7), u, y etc. are given by u = (~W%V2/
T)~‘* ,
w = WW’IP~~Y
+NNP:P:IW
~‘=a%+P;,P*+P;)
7
64.8)
3
in a general case with four size parameters differing from one another, of which two, /3; and p;, belong to a bra state and the others, pi and p2, to a ket state, respectively. In this paper locally peaked gaussian wave functions 31) are used as the basis vectors of the inter-cluster relative wave function: 3/4
exp [-8(r2+X?)]iL(2
SXir) YcL)(E) .
(A.9)
349
T. Kajitw et al. / Electromagnetic propeties
These wave functions correspond p’
(X)
to =
S(Xx;x,)ywyl&
,
(A.lO)
in the NGC space. Here the function iL is a modified spherical Bessel function and Xi’s are mesh points. We use 6 =~(N,N,/A)j3(a) and 11 mesh points, namely {X$=1-11}={0.6, 1.2, 1.8, 2.4, 4.15, 5.9, 7.65, 9.4, 11.15, 12.9, 14.65). Appendix B The kernels of el~tromagnetic multipole operators are here given in terms of the new generator coordinate representation. Since the projection in the operators (13) and (14) can be performed in the NGC space, the Brink GCM kernels of the unprojected operators are first derived, which are essentially classified into three types:
~expEig+pkl,
for Ch
g
exp
[b
l
Pk)&,!k7
for MA (convection)
T
exp
14
l
Pkh#k~
for Mh (spin) .
,
,
(B.1)
In order to hold the translational invariance, any operator should be written only in terms of the internal coordinates. Using the relations
c,6
pk=rk-rc=rk---
exp
[i4
l
Pkllk
=+
exp
[@
03.2)
A ’ ’ pklpk
xvpk
=~fvrX{expiiq’pkl(v,~-~)},
03.3)
one can easity obtain the GCM kernels. The general form is &,,(R’,
R)
[
tl/WXq
J
- Qq’ldq^ for the two (Os~-cluster system. The outer products of R
03.4) X q,
R’ x q and R’ x R
?I Kajino et aL / EIectromagnerie prqwties
350
come from the convection current part, while only the scalar term comes from the spin-current and Coulomb parts. Inserting eq. (B.4) into eq. (A.7), one gets the NGCM kernels free from the c.m. excitation. For the A-pole operators, they are as follows:
Xexp
CfW’, X
u3.6)
411 d4,
where u = h Z+Z 1 and the function f(X’,
x, q) = -T,x2- ?2x%r3xt
l
x+itpx*
q+i(bb’X’
l
fB.8)
q-eq2
is used for simplicity. In eq. (BS)-(B.7), the coefficients C, 4, TV etc. are very simple functions of a, a’, b, B, B’, Q and quantities in eq. (A.@, which depend on the number of nucleons exchanged due to the Pauli principle and the size parameters pi, /3&,PI and &. The summation on the right-hand side is taken over those types s~boiically. In the same manner as eq. (I??), the scalar function G^(X’, X; q) is defined by
Connecting the NGCM kernels (ES)-(B.7) projected NGCM scalar kernels as GCh(X’, X; q) -
with eq. (B.91, one finally gets the
-J-(l)“(4~)“/” exp[-~,X2-*
Xf2-
2
8q2]
T. Kajino et al. 1 Electromagneticproperties
G%i
351
W’, X; 4)
+G G$;“‘(X’,
3
Ax’xF3 i )H,
(B.ll)
X; q)
(B.12)
(B.13)
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T. Kajino et al. / Electromagnetic properties
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