Inelastic electron scattering in the generalized collective model

Nuclear Physics .4,211 (1973) 393-404; ( ~ North-Holland Publishino Co., AmJterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

I N E L A S T I C E L E C T R O N S C A T I ' E R I N G IN T H E G E N E R A L I Z E D COLLECTIVE MODEL E. BORIE t D. DRECH.SEL and K. LEZUO lnstttut fiir Kernphysik, Universitiit Malnz, Mainz, Germany

Received 16 May 1973 Abstract: We have applied the generalized collective model to the description of electron scattering data for several nuclei. Three examples, namely I1"Cd, 64Zn and 24M8 are discussed explicitly. The theory is based on a quadrupole-type vibration of the nuclear surface which is governed by a collective Hamiltonian. The radial dependence of the nuclear surface has been assumed to be a two-parameter Fermi distribution. The model provides a reasonable description of both spectra and inelastic electron scattering. 1. Introduction The study of collective potential energy surfaces (CPES) of nuclei has been the subject of considerable interest recently especially with regard to heavy-ion scattering, fission and the possible existence of superheavy nuclei. Recent improvements in experimental techniques, particularly in the area of resolution, have made it possible to use electron scattering to shed considerable light on the properties of the CPES of medium and heavy nuclei.Unlike the situation in experiments with heavier particles ~-4), the interaction of the electron with the nucleus is weak and well known, and the contribution of two-step processes (dispersive e f f e c t s ) s - 7 ) is small. Experiments with heavy (usually a) particles have been performed with good energy resolution, but their interpretation is more model-dependent due to the large contribution of higher-order processes and the fact that the interaction between the target and projectile is not well known. By providing a direct measurement of the strength and spatial distribution of transition densities between the ground state and low-lying collective excited states of medium and heavy nuclei, electron scattering is sensitive to the properties of the CPES. Experiments of this nature have only recently been possible s - t t) due to the difficulty of obtaining sufficiently high energy resolution in experiments at large m o m e n t u m transfer. We point out that different spatial distributions of the multipole moments of nuclear deformation can lead to differences of orders of magnitude in the predicted lifetimes of superheavy nuclei ~2). A good understanding of these distributions in existing nuclei is essential before one can extrapolate to the superheavy ease, and electron scattering at high momentum transfer allows one to determine these distributions fairly unambiguously. t Fellow of the A.-v.-Humboldt-Stiftung 1970/72. Present address: Physics Department, Michigan State University, East Lansing, Michigan. 393

394

E. BORIE et al.

Though without doubt the most satisfactory procedure for obtaining the CPES is to derive it from a microscopic model, the Hartree-Fock approximation suffers from severe numerical problems if applied to heavy deformed nuclei. In light nuclei, however, such calculations have met with reasonable success in describing the lowlying states and defining some properties of the CPES t3-15). Neglecting the selfconsistency condition, calculations based on the shell model 16-18) allow for the inclusion of more complicated configurations, and also provide considerable information about nuclear deformations. The collective model used in this paper t 9), although rather phenomenological, has the advantage of being fairly easy to handle and allows an extension to the case of heavy nuclei. It is therefore useful as a first approximation in order to investigate the physical effects to be expected, and to study the sensitivity of electron scattering to the form of the CPES. Earlier collective theories 20- 22) were able to describe the CPES in certain limiting cases (deviating only slightly from the idealized cases of the spherical harmonic vibrator or rigid rotator). More recent developments 19.23), however, generalize the earlier work to provide a unified description of all types of collective surface motion, and are able to account for an astonishing variety of nuclear spectra 24.25). Although the present version of the model neglects higher multipoles (e.g. octupole and hecadecapole deformations) and single particle aspects, it is able to fit the spectra of the low-lying even parity states and the E2 transition rates reasonably well. Electron scattering, however, is sensitive to more detailed electromagnetic properties of the nucleus, e.g. monopole and hexadecapole transitions can be excited directly and measurements may be obtained over a wide range of momentum transfer. It is the purpose of the present paper to investigate the predictions of the model for electron scattering, and, where possible, to compare with experiment.

2. General formalism

The cross section for the electro-excitation of a transition from a state of spin Ji to one of spin Jt is given in the Born approximation as in ref. 26) by do"

dfl

4xtrM*tt /tA4 x~~ I
+ t a n 2 ½0 L=l 2 J i + l

(ll 2)

with o.Mott

~ cos 2 ½0 4e~ sin 4 ½0

(l)

where ,4 and q are, respectively, the four-momentum and three-momentum transfer

INELASTIC ELECTRON SCATTERING

395

to the nucleus, and the transition operators are given, e.g., by

M~M(q) = f p(e)jL(qr)gLM(~)d3r.

(2)

For electric collective transitions at moderate momentum transfer and scattering angles not close to 180 ° the transverse terms may be neglected and we will not consider them further here. The nuclear charge operator p appearing in eq. (2) can be decomposed into spherical tensor components: p = ~' f.EprLl x yt~.]]tol

(3)

L

£ = x/2L+ I. We then have MILl(q) =

f;

p[L](r)jL(qr)r 2 dr.

(4)

The inelastic electron scattering form factor for the excitation of the transition (L) is given by 4re F2(q) - 2J i + 1 ~ I
H = P2[~E2]xnt21]t°I+C2L2+CaLa+C4L22+CsL2L3+C6L2+D6L~, with L2 =

(5a)

[~[2] x ~[2]][o1,

L3 = [~t21 x ~t2] × ~t,l]to~.

(Sb)

The potential energy can be written in terms of the deformation parameters fl and )' of Bohr and Mottelson 20) in the intrinsic system. Thus

V(fl, y) = x/~C2 f12 _ ,~/~--~Caf13 cos 3y + ~C4 if" ~6 • - x/172 s C5 fls cos 3y + ~-,¢C6f16 cos 2 3), + Ls /vi ~-D6t,

(6)

This assumption allows for the existence of rather general types of collective potential energy surface. The case of coordinate-dependent inertial parameters (giving rise to terms in the Hamiltonian like P3[ct x n x n] t°J) is under investigation. The parameters C2, Ca, Ca, C5, C6, D6, and P2 are obtained by fitting to the spectrum. For a given set, the Hamiltonian H is diagonalized in the basis of a large number of eigenstates

396

E. B O R I E et ai.

of the five-dimensional harmonic oscillator, determining the spectrum and functions of the excited states. The charge density operator is assumed to be of the form

wave

p(r) = p(r-R(O, q~)),

(7)

g(0, ~) = c(l + aoo + 4 5 [ a t21 x Yt~3(O, ~,)]tol).

(8)

with

The quantity %0 describes monopole vibrations, and is determined as a function of the ctt2] by the requirement of volume conservation. For our numerical calculations we take the charge density to be a two-parameter Fermi distribution, p(r) = Po(l + e x p ( { r - R(O, t#)}/z))-'

(9)

However, many of our expressions will be valid for any charge distribution of the form (8). Other parametrizations are also possible; for example one could take the charge density to be of the form g(r/R(O, q~)), as would be the case if one calculated the charge density from shell model wave functions in a deformed harmonic oscillator potential. Our assumption is a natural one, however, if one regards the surface thickness as being due at least partly to collective surface oscillations. To evaluate matrix elements of the charge density operator ,o between the initial and final states of the nucleus, we must expand it in the collective coordinates. The nuclear states are superpositions of many oscillator states, so one should carry the expansion to a fairly high order. Keeping terms up to fourth order, we find, after some recoupling, for the tensor components of the charge density operator ptLl(r) = x/~6zo p(r - c) - c(ot6L2 + x/-4nOtootSLo)p'(r -- C) c2 ~___~ ( 2 0 2 +-~ o

L ) [ # × ot]tL]+2~XOOOt6L2+,,,/4--~Ot~O6LO 1 p"(r--c)

' ~) [[~ ×,]t"x,]tL' -

0

" o0( 0 0

+ -J4n

+ 4i (4n) --~

)

[~ × ~]tL] p(~)(r-c)

0

0

0 0

0]\0

0

x [I-ctx 0c]t'] x [:t x Ot]U']]tL]p(')(r--c)

+ ....

0o)

We have dropped the tensor index 2 on the collective variables. A similar expression would be obtained if collective oscillations of other multipolarities were allowed.

INELASTIC ELECTRON SCATTERING

397

The operator Ctoo is fixed by the requirement that

~/~o°P~oi(,.),.2dr

= Z,

(1 1)

which gives a relation between %0 and the tensor products of the ~tt21 coupled to angular momentum zero. Specifically, we have to fourth order in ~t x./5 L2 R 2 %0 = ~ 2R1

5 R3 12~x/1'4~ L3 Rt-+ i~-~ L~ 5

R--I

~-~i/

4R 2 1 '

with R,-

c"

f ~ r 2 d"

(,-l)!Jo

(13)

The explicit value of aoo for the case of a two-parameter Fermi distribution is given in the appendix.

3.1. ELASTIC SCATTERING

3. Applications

AS has been pointed out many times 2~-33) oscillations and deformations of the nuclear surface lead to an effective smearing which results in an increase of the skin thickness (it is sometimes assumed that the entire observed skin thickness is due to surface oscillations). Therefore the experimentally determined parameters of the charge distribution will be different from the intrinsic parameters entering into the charge density (eq. (9)). We have required that the moments ~r2) and (r 4) determined from the experimental values c* and z* of the Fermi distribution be the same as would be calculated using the intrinsic parameters and including surface oscillations. This enables us to determine the intrinsic parameters e and z. The exact connection for the case of the Fermi distribution is given in the appendix. The expansion of the charge density in the description of electron scattering must be treated extremely carefully because the truncation of the expansion (10) introduces small, spurious oscillations in the density operator. At sufficiently high momentum transfer q the Fourier components of the neglected terms can make an important contribution (the expansion for the form factor is effectively a series in qcfl, with fl a measure of the deformation). The ground state charge density for example, is a perfectly smooth function of the radius when terms of all orders are kept. The oscillations found in ref. 31) are therefore unphysical and should not be interpreted as the fluctuations suggested in ref. 34). 3.2. INELASTIC SCA'VI'ERING We have investigated a variety of CPES ranging from the pure harmonic oscillator through the transition nuclei to strongly deformed rotators. The shape of the form

398

E. BORIE et al.

factor for excitation of the first 2 + state was extremely insensitive to this variation, explaining the success of the Helm model. This, however, is not the case for the socalled forbidden states, higher 2 + states or excited 0 ÷ states. A typical example is shown in fig. I. Both transition densities and form factors for the E2 transition to the second 2 + state differ appreciably in spite of the fact that the transitions to the first 2 + state were practically identical in all three cases: spherical, oblate, and prolate nuclei. In the following we show specific applications with comparisons to the experimental data for lt4Cd, 6"Zn, and 24Mg. For 114Cd the CPES has been calculated by the Frankfurt group [ref. 24)] and reproduces the energy level scheme and E2 transition rates quite well. This CPES which is shown in fig. 2 has a prolate minimum and a second spherical minimum of about the same depth. The heavy line indicates the energy Eo of the ground state above the minimum; the spacing between the lines is -~Eo. The spectrum may be interpreted as that of an anharmonic vibrator plus a rotational band situated in the second 10-3_,1F] 2

"',,

10-!

10-q

/

\

QSO

1.00

1

/

2.0

/,'\

,

/

,

/t

2.[~0 q (fnq l)

150

2.00

\;

/

,0 0

1.50

;'~ \ '

t r * 10-"

0150

\\ 1

rio

- 1.0 Fig. 1. Transition densities and form factors for an E2 transition to the second 2* level. Full line: harmonic oscillator; dashed line: prolate deformed; dotted line: oblate deformed. The transition was adjusted to give the same B(E2) value for excitation o f the first excited state.

399

INELASTIC ELECTRON SCATTERING

I -a~/ d~/~M 11~ Cd 10-

I

°'~ 1

/'----,. [

.

2

o.o

-/_

lO

o. •

~22.

2"

2"

.

10

Q561vk, V

1(3

05 1.0 15 q (f~=~) exp O" I-F,~ O" 0 Fig. 2. Potential energy surface, spectrum and form factors for ss4Cd. The potential energy surface (upper right) has been taken from ref. 2,,) (theory I), see text for explanation. The square of the "form factor", (da/d.Q)/aM as function o f momentum transfer q has been calculated in DWBA and compared to experimental data [ref. to)] for the first (triangles: Yale; full circles: NBS) and second (open circles: NBS) 2 + states. For the latter we also show the result if an oblate minimum is assumed, which yields exactly the same spectrum (dashed curve). A ~/~;

/~/

:6 3_

6&Zn

~=~o

M

~/

Eo, 2,o

°%

II

h

.

/

*,

+:f

:

3O.

.o.,~

o / %

~o/

:/"----"q*-

4

"o

,,

~

,;~

'

70

'

?~rl

10-5..1]I

1~

/

/

,'

,,

"', ' . V

""

,,

',, t L{ ~

,,

"..~'~

'~,~,,

,, J . , , , t . A ~ w \ , ,

'

10-"

,' ~**o °~

,,~\ '~ , , / C~r

0'5

10

1'5

\

q

Hm_lT

~o" --~" __~: 099 MeV - - 2 "

p"~'~-=x O"

~ d - -

o'. 2"

--theory O+

Fig. 3. Potential energy surface, spectrum and form factors for 64Zn. The square of the "formfactor", (da/dD)/aM as function of momentum transfer q has been calculated in DWBA and compared to experimental data [ref. as)] from Mainz and NBS. Open circles: first 2 + state; full circles: second 2 + state; squares: first 4 + state.

400

E. BOR1E et al.

minimum. The form factors which result from this CPES are also shown in fig. 2 and are compared with the electron scattering data from ref. to). Of particular interest is the E2 transition to the second excited 2 + level which has considerably more strength than would be expected from a harmonic oscillator model, and at the same time has its maximum somewhat shifted to lower values o f q - resembling the form factor of the first excited 2 + level. A similar situation is known for some other nuclei and has been discussed by Lightbody 11) by introducing adhoc mixing of one- and two-phonon states. The present theory which accounts simultaneously for the spectrum and several B(E;t) values 2+) also gives the right behavior for the form factors. We observe, however, that a consistent treatment of the anharmonicity not only gives rise to a mixing of one- and two-phonon components in the 2 ÷ states but also introduces higher phonon components into all states, including the ground state. In the case of ~+Cd the probability of finding such components is of the order of 30 ~o and this has an appreciable effect on the transition probabilities. We should remark that an even better fit to the experimental spectrum has been obtained by including higher-order terms in the effective mass [theory II of ref. 2+)]. We turn now to the case of 6+Zn. The structure of the CPES has been determined from the energy spectrum using the computer code of Habs 25). The result is shown in fig. 3; the CPES is that of an anharmonic vibrator with a shallow second minimum at an oblate deformation. Again the order of the triplet states is not exactly reproduced, which might be improved by introducing higher-order terms in the kinetic energy. The cross section is compared to experimental data from ref. 35). The form factor for thefirst 2 + level is described very well up to the second maximum (note that

. ~ M c V

24Mg

_; -a-E/ m Io

./

*'%

/

;o-

.40 •

/

/

.20

;o

I

.40

T

L

I

.60

,

,

80

--2" °

°°];"

.

__o: - -

00.. 3"

10

3" _ _ 2 " -

1.3"/MeV d5

If0

1.5

2(3 q(fm 'l)

--0 exp.

-

L

2* °

,

+

2" ~ 0

°

Fig. 4. Potential energy surface, spectrum and form factors for Z+Mg. The experimental data are from refs. 3~.39.+o). For the first 2 + state, experiment and theory have been divided by 10.

INELASTIC ELECTRON SCA'I-FERING

401

only the "photon point", q = to, has been fitted by the value of 8o). The shape of the second 2 + cross section is well reproduced, although the amplitude is too low by about 40 ~o. Similarly the first 4 + level is fairly well described by quadrupole oscillations only, without the introduction of hexadecapole deformations. Though from a theoretical point of view one should not expect the collective model to be applicable to light nuclei, we have also treated the case of 24Mg and found astonishing agreement with experiment. The potential energy surface and the spectrum obtained are shown in fig. 4. We find a prolate deformation, but the nucleus is extremely soft against ),-vibrations. A triaxial deformation would slightly improve the spectrum at the cost of reducing the monopole transition strength. The present fit gives good agreement with the experimental form factors 36) with the exception of the transition to the second 4 + level which can only be fitted by assuming additional static hexadecapole deformation. Our results on the CPES, the spectrum and electro-excitation are compatible with microscopic calculations ~5) existing for this nucleus. This gives us some confidence that interpretation of the data with the collective model does indeed give some insight into the nature of the CPES. This model should be of particular utility, however, in the analysis of new highresolution electron scattering data, which will be available in the near future. It will then be possible to study the low-lying states of rare-earth and actinide nuclei as well as transitional nuclei, for which the model is expected to work well. The authors are grateful to many members of the Institut f/Jr Theoretische Physik in Frankfurt for fruitful discussions on the generalized collective model, particularly to L. v. Bernus. We are also thankful to Dr. Habs, Heidelberg, for making available to us his computer code. We wish to thank Dr. Neuhausen, Mainz, for making available to us his experimental results prior to publication. Appeadix We list here the specific expressions for %o, the renormalized rms radius and (r4} for the case of a two-parameter Fermi distribution. Defining

,42 = ~

[~ × .,]I0J,

A~ =

5

[~ x ~, x ,,]to3,

12n~/l~

A4 =

5 [,a,x (x]t°J[~(x c(]t°j, Y]

:

~Z/C,

(A.I)

402

E. BORIE

et al.

we find from eq. (12), %0

-I(--A2+A3+~rl2A4). l +~n ~

(1 +4~,/2)2

In terms of the experimental charge density parameters c*, z* we have (r2)

=

3C.2Zt | V

j r ~-r/ "7 *2,,),

3,._,4~), < r4) = ~3c,4q+ ~ 6 ~, 2 + --S-q

(A.2)

while in terms of the surface oscillation parameters, we have, with ao, az, a3, a4 the ground state matrix elements of %0, A2, A 3, A4 respectively, 3c 2 (r2) --- 3c2(1 +It/2) + ~ 2 [ 212(a2+a'/12)+ao13-6II(a3+a'/l')+~aJo]'


c'(1 +

2+

+

3c 4 -- [aols+31,(a2+a,/l~)--15(a3+a,/l~)13+~S-a, 1+~, 2

I2],

(A.3)

with n + l f~o r'dr 1. = c-Tg-/j ° l +exp ((r-c)/z)"

(A.4)

The integrals I n have been evaluated by Schucan 3 7). For completeness they are also given here:

I,, = 1 - 2 [~r(,,+ E ,)](-1)* ( n + l ~ (22._ ' - l ) B 2 , r / 2 . ,=1

\2v/

+(n+ 1)r(-1)"(z/c) "÷ % + , ( - e - C / z ) ,

(A.5)

where the B2. are Bernoulli numbers. For Ixl ~_ 1 the generalized Spence functions are given by the series x*

(A.6)

L.+,(x) = V=, v.+, .

For the case of monopole transitions, the integrals over r appearing in eq. (4) can be performed analytically. The Fourier transform of the Fermi distribution was first studied by Petkov et al. 3s) and the main contribution is evaluated using their method. However, we include explicitly correction terms neglected in their work, which are important at high momentum transfer (q > c/nz2). Proceeding in the usual manner we find for the form factor of the Fermi distribution:

F(q) = 47zP°rr(e'q" 2--~qgo - e -'q')

(

l+exp

(V))'

dr.

(A.7)

INELASTIC ELECTRON SCATTERING The integral has simple poles at x ~ = zero for Re(r) ~ + co. As in ref. 3a) we find fo ~

re±'dr

1), s = 0, 1, 2 . . . . and approaches

±2ni~Res(x~,)-f °

=

t +exp ((,-c)/z)

c+iz(2s+

403

re±iq'dr

i +exp

,=o

c)/z)'

(A.8)

The second term in the right hand side can be transformed by use of the change of variable r = ±iy into f~ y exp ( - q y ) d y J o 1 + exp ((- c + iy)/z)

o

The first term is equal to oc

:]:2niz ~, (c+ i~z(2s +

1)) exp ( +

iqz -

nqz(2s + 1)).

s=0

The series can be summed. Setting a = 0.5 sech

F(q) = 4n2po z/q(~z sin qz coth

nzq - c cos

c/z we find

qc) csch xzq

+ 4nz2po a/q

~Ue-~" sin

u du 1 + 2a cos u

fo

(A.9)

Since the quantity a ~, exp(-c/z) is small, we may expand the integrand as a power series in a and integrate term by term, obtaining for the second term of (A.9):

{

,

8rc2p°z2a ~ +q2z2)2 (1--4a2) ½

(2n-f~F

a2,,_,"it .=, ,=o\

_ ~,

k

2(n-k)

2(n-k-l)

"]

]L(4(n-k--k-~-+q2zZ)2-(4(n'-k---O2-+--qq2z2)2j

+.~=a,.:~='o(2~) F

2(n-k+½)

_

L(4(n--~k--~~ :'~---qq2z2) 2

2(n-k-½)] (4(n-k-½12+q2z2)Zd}

= 8rcpo z2a/(l + q2z2)2.

(A.10)

The quantity Po is determined by the requirement that F(0) = Z. We find

Po =

47rc---~

\c/

(

--

'

in agreement with well-known results. For small values of q the first term of F(q) dominates, giving rise to sharp diffraction minima and an exponential decrease in the form factor with increasing q. However, for sufficiently large momentum transfer, the second term, which for large q is proportional to q-4, dominates and the minima become progressively less deep.

404

E. BOILIE et al.

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