Generalized nuclear polarizabilities in (e, e'γ) coincidence experiments

Nuclear Physics A233 (1974) 153-163;@

North-Holland

Publishing

Co., Amsterdum

Not to be reproduced by photoprint or microfilm without written permission from the publisher

GENERALIZED NUCLEAR POLARIZABILITIES IN (e, e’y) COINCIDENCE EXPERIMENTS H. ARENHdVEL Institut fiir Kernphysik,

and D. DRECHSEL

Unicersitiit

Mainz,

Mainz,

West Germany

Received 24 June 1974 Abstract: The (e. e’y) coincidence cross section is expressed in terms of the nuclear polarizabilities as a generalization of the standard (r, y’) formalism for polarized photons. This formalism is well suited to determine nuclear spins in a model-independent way. Due to the presence of virtual photons of longitudinal and transverse polarization and variable momentum transfer, electroexcitation gives more complete information than scattering of real photons and allows one to study the polarizabilities as function of excitation energy and momentum transfer.

1. Introduction The dynamical properties of nuclei which determine elastic and inelastic photon scattering are most conveniently described in terms of nuclear polarizabilities. This concept is analogous to the classical description of the scattering of light and has been applied to photon scattering on atoms and molecules ‘). Fano “) demonstrated the usefulness of the polarizabilities to describe nuclear photon scattering in the dipole approximation. Subsequently, this concept has been generalized to higher multipole radiation 3B“) and applied to analyse experimental data both in the region of isolated nuclear levels and in the dipole resonance region ‘). In the giant dipole region where dipole radiation dominates the scattering process, all information on the nuclear dynamics is contained in the well-known scalar, vector and tensor polarizabilities of the nucleus. The basic idea of the formalism is to perform a multipole expansion of the scattering amplitude by a coupling of the multipoles describing the excitation and deexcitation of the intermediate nuclear levels (fig. 1). This leads to the coupling scheme

Fig. 1. Contributions to the cross section arising from two-step transitions i, -+ J. or J, + J, proceeding via the electromagnetic field of multipolarity L, L’, K and K’. 153

154

H. ARENH&‘EL

AND D. DRECHSEL

of fig. 2a of the multipoles L and L’ (or K and K’) connecting the intermediate state J, (or J,) with initial and final states Ji and JI to total angular momentum transferj. For photon scattering this is a useful concept, because in this way one can separate the dynamical and kinematical properties of the scattering process.

(a)

(b)

Fig. 2. Coupling schemes for coincidence cross sections. Scheme (a) is used in this work and for photon scattering, scheme (b) has been used in refs. 6**).

Replacing now the real photons in the excitation process by virtual ones, we may generalize this concept to inelastic electron scattering in coincidence with photon decay. Such experiments have been proposed in the past 6), but it has been realized that the experiment is an extremely difficult one. However, with the advent of electron accelerators of high duty cycle such experiments seem to be possible in the next few years, even though the estimated counting rates are low and large angle photon counters seem to be necessary to improve on the statistics. As has been pointed out by Acker and Rose 6), the process under investigation competes - and in fact interferes - with the ordinary bremsstrahlung process and the geometry should be carefully selected to suppress this competition. According to these authors, however, the cross section for excitation of collective dipole levels rises to about two orders of magnitudes higher than the bremsstrahlung process - except at forward angles - and one should even be able to observe a collective quadrupote level as a distinct interference with the smooth bremsstrahlung background. Coincidence experiments in electron scattering have concentrated in the past on the “quasi-elastic” emission of protons ‘). The aim of these investigations has been primarily to determine the momentum distribution of the nucleons in the initial nuclear state. Final state interactions have been considered as perturbations and the kinematics have been chosen to keep them as small as possible. The present formalism, which directly connects the scattering of real and virtual photons, gives an opposite view particularly useful for a kinematical situation in which final state interactions are the dominant feature and which build up a (collective) resonance state. Earlier studies of such a situation “) have used a description both appropriate to photon and heavy particle emission following electroexcitation. Therefore, they have used the scheme of fig. 2b, which couples separately the multipoles L and K of the excitation (proceeding via electromagnetic forces) and the multipoles L’ and K’ of the de-excitation (proceeding via electromagnetic or nuclear forces). The aim of the present paper is to describe inelastic electron scattering in coincidence with photoemission in terms of nuclear polarizabilities, thus providing a

NUCLEAR

formalism which directly connects inelastic photon scattering.

155

POLARIZABILITIES

this experiment

to present

studies

of elastic and

2. Generalized pdarizabilities The “standard” experimental set-up for electron scattering in coincidence with photoemission without observation of spins in the initial and final states, correspond to the scattering of a virtual photon with a dejinite polarization in the initial state, which becomes real and unpolarized in the final state. Due to the nature of a virtual photon, its polarization may be both transverse and longitudinal. The cross section for photoabsorption integrated over a resonance near q, = w, is given by ‘) a(o,)dw,

s

= ;2Z;

n

II,’

(1)

&JiJ:d%~ *

where the density matrix cyi, describes the photon polarization in the initial state. We use the conventions h = c = 1 and a = ez = &. Due to the transverse character for real photons c$j, vanishes for i. = 0 and/or 1’ = 0. The spherical components of the nuclear current are ‘) Jj, = (-)“[2n(l+slo)]*

C (-i)“e(-)“-“’ L, M

( _J&f

h

:,)

(J/II~;:~‘IIJ,)D~,(R). (2)

The rotation matrices DhA describe a rotation R from a coordinate system whose z-axis points in the direction of the photon beam to an arbitrary coordinate system. of We use the definition given by Rose lo); note that in ref. ‘) the convention Edmonds I’) is used. The transition operators are @I i and may according

=

be calculated to ref. ‘):

@I ( T,:t1+ 1 T[“ mae from

for for

the nuclear

i. = 0 (Coulomb) L = + 1 (transverse), charge

and

current

(3) operators

p and j

s s

ML&f(q)= d7h.(v) YLAdQMrh

G&d = f d7V x C_f,(w)Y,,G-413% T;;*(q)

=

I

dt

jL(v)YdQ)

*i(r).

In the definitions above we have used the Coulomb gauge and formally the spherical component Jo of the transition current with the instantaneous

(4) identified Coulomb

H. ARENHdVEL

156

AND

D. DRECHSEL

interaction, which is actually the time-like component of the current four-vector. For real photons only the transverse terms (II.1 = [%‘I= 1) contribute, and eqs. (4) have to be evaluated at q = w (on sheh). To facilitate the notation we introduce two numbers %, r to describe both the helicity and parity of the operators (4):

Explicitely, we use LOO -= CL for Coulomb, L10 s EL for electric transverse and Lfl E ML for magnetic transitions of order L. The differential cross section for electroexcitation ‘) may be expressed similarly to eq. (1): do

(6)

dI2,dw where pg, = 2(-)“Q-,Qx++q;6,,.

l-i&,--

4r” 6,@ , ) q2

(7)

and F corresponds to the total final state of the target. The usual kinematical variables are defined as in fig. 3. Note that q,” = q2 -w2 = q2 -w2, where w = -q. is the excitation energy of the nucleus. Comparison of eqs. (I) and (6) show that electroexcitation is formally equivalent to photoexcitation with polarized real photons with a density matrix #,

= ’

z.?t_kt

2n2qf: k,

p::!6(0

- W,,)dW,

(8)

where

(i)

Pl-1

=

-K2/q2.

04

The remaining density matrices may be obtained from the symmetry relations pg, = p:‘;; = (-ypi’;_,,.

G’b)

Relation (8) for the density matrix for virtual photons may be used to connect formally the cross sections of photoreactions with the corresponding electron scattering processes, in particular the cross sections for photon scattering and electron scattering in coincidence with photon emission. The cross section for the scattering of

NUCLEAR

POLARIZABILITIES

157

initially polarized photons with momentum k, to final momentum k,. without observation of polarization has been given in refs. 3*“). In a somewhat simplified notation of ref. “) one has da

a2 k I

dQ,

.Yf k,

_=--.-I

C LIA L’r’

C

C Pj(LlZ, L’lz’; k,, k,,)

~pp K!p’

j

K’lp’; k,, k7,)h~‘r’KPK’P’(di), R, II, /A),

x P;(Klp,

(9)

where the nuclear polarizabilities are

Pj(Lj.r,~‘-1’2’7’; k,, /q) = ( -)Jf+J/‘L”“‘jr’~‘2~ x II1+610)(1+~xo)l*f;~’ ; ($. ; 1

/

IJJK I. (10) ;) (J,II-ru’,f;‘l n

The small non-resonant contribution of the crossed diagram has been left out. For real photons only II.1 = Ii.‘1 = 1 contributes and we shall drop in the following the index 2.’in the final state but keep 1.in the initial state in order to be able to include longitudinal virtual photons (A = 0). In case of a simple resonance model R, = En-o+j+r,; for generalizations of the formalism see ref. “). The kinematical factor in eq. (9) is h~L’~‘GR’p’(a(O,R, I., cl) = T (_)L+K+j+p+ li--lr+pP-“+P’

x_? 2f2(1+(-) L+K+J+r+p) (Tl

‘: ;)(; -“, pfA)(KL’ ; 3)dzD;-,,,(R), (11)

where R is the rotation between the coordinate systems of the initial and final photon directions. With relation (8) for the density matrix of virtual photons we immediately obtain from eq. (I) da dQ, dQ,do

= -

x3

k2 k

-

-X C

2n2q,4 k, ./f

C C Pj(Llr, L’z’; q, k,)

LZA KP,, I L'r' K’p’

x PT(Kpp, K’p’; q,

ky)h$-KpK’p’(,ci),R, A, /A).

(12)

In this equation the initial photon momentum k, has been replaced by the momentum of the virtual photon y, and subsequently the prime on the final photon momentum has been dropped. It is important to note, however, that now the summation has to include the longitudinal photons (A = 0). Since the nuclear polarizabilities depend on the absolute values of 1, and /J only, the symmetry relation (7b) may be used to combine terms in (A, p) and (-A, -/.A) _._ da dC?,dQ,do x P~(LAT, L’t’)~f(~~p,

~‘p’)/if*~“‘~~~~‘(p(*), R, i, /I().

(13)

H. ARENHt)VEL

158

AND D. DRECHSEL

One obtains essentially four independent terms corresponding to the four independent functions py,! of eq. (7a), because the terms containing l;j (0, 1) and ~j (1,O) may be combined using eq. (15). The Rj contain only the geometry of the process and are defined as jiy’~‘(~(i),

R, I., P) 3 h~r’OK’f+(p(i),

R, 1, P) + h~r’Kpx’~‘(~(0,

El *--lr+pp--r’+p’ giLL’rKK’p(eY,I., /+::‘cos where 8, and 4, are the polar and azimuthal

R, _ 1, _ n)

((/I-+#+

04)

angles of the final photon

in the coor-

Fig. 3. Kinematics of electroexcitation: The initial and final electron momenta Jr,, k, define the scattering plane which contains the x- and z-axes; q = k2--k, and Q = &(k,+k,). The vector K = (k, xk,) is perpendicular to the scattering plane and parallel to the y-axis. Note that the z-axis of our coordinate system points opposite to momentum transfer q, and the x-axis is on the same side as the mean momentum Q.

dinate system as defined relation

in fig. 3. The geometrical

functions

gj obey the symmetry

g;L’rKK’P(61,) 2, p) = g:K’PLL’Z(O,, /f, A), and have been tabulated be obtained from gFK’p(e,

x(1+(-I

in the appendix

for some relevant

(15)

cases. Jn general they may

) A, p) =

L+K+J+r+p) (”

1

” 1

“)(” 0 I.

K -p

’ .)(“’ P--A L

L’ K

“) d;+@,). j

(16)

By straightforward but tedious recoupling techniques our eqs. (3)-(S) may be transformed to eqs. (39)-(43a) of ref. ‘), while earlier references “) on the subject contain some misprints. 3. Results and discussion The formalism in the preceeding section has been applied to two special cases of current interest in order to give an estimate of the expected cross sections and angular distributions. The first example describes electroexcitation of two overlapping giant (l-) and quadrupole (2+) resonances in an even nucleus, followed by subsequent photo-

NUCLEAR

emission leading to the nuclear dominates, the cross section is da dn, dQ, dw, =

~Mott 4$$ -2,/3

ground

+5&(C2,

state.

sinZ0,(31P,,(C1,

Re (P,(Cl,

159

POLARIZABILITIES

Assuming

that

the Coulomb

term

El; q, o,)\~

El; q, w,)Pz(C2,

E2; q, a./)) cos 8,

E2; q, w,)12 cos’ O,),

(17)

where cMolc is the Mott cross section for scattering on a point charge Ze. The scalar polarizabilities contributing have been calculated in the Goldhaber-Teller model 9*12) for a uniform charge distribution with radius R,: ,~31W jl(qR,) P,,(CI, El ; q, w,) = !J&!!?! 2AM co, -my-&T, -qR, ’ P,,(C2, E2; q, co,) = -

43 x 5NZ 2AM

R,0;/0, o,-o,-+iT2

j2(qRo). qR,

(18)

The total mass of the nucleus is AM, the resonance energies and widths are ol, rl and 02, r2 for the dipole and quadrupole state, respectively. Fig. 4 shows the angular distributions calculated for a model nucleus with R, = 3.5 fm, momentum transfer q = 0.6 fm-’ and two sets of resonance energies: o1 = 22 MeV, o2 = 26 MeV, I-, = T2 = 4 MeV and o1 = 22 MeV, 0.1~= 24 MeV, rI = r2 = 2 MeV. At

20

LO

60

100

120

1LO

160

1

Fig. 4. Angular distributions for photon emission from the giant dipole and quadrupole resonances leading to the ground state, calculated for two sets of parameters and excitation energies o = UJ, and o=u~, corresponding to the dipole and quddrupole energies. Full curves: o, = 22 MeV, 02 = 26 MeV, I’, = r, = 4 MeV (cross sections multiplied by 5). Dashed curves: w, = 22 MeV, 02 = 24 MeV, rl = r2 = 2 MeV. The momentum transfer is 4 = 0.6 fm-‘.

160

H. ARENHOVEL

AND

D. DRECHSEL

oy = 02, the dipole interference produces a distinct asymmetry about 6, = 90”, which determines the quadrupole strength in a model-independent way. By varying the momentum transfer q, the interference term may be enhanced even further, while for real photon scattering the quadrupole mode is strongly suppressed in both decay and excitation 13). As compared to particle emission, the total cross section for photon emission is reduced by the ratio of radiation width to total width by about two orders of magnitude. Other problems of interest are the scalar and tensor polarizabilities of deformed rotational and soft vibrational nuclei, which lead to photon decay through the giant dipole resonance to both the ground state and the low lying rotational and vibrational 2+ states. For an even nucleus, and keeping only the Coulomb term in the excitation, we obtain

I/

c

=0+,2+

da

crk,. ffMuloCC 4n2Z2

-=

dQ,dQ,do, x ii

sin’

B,IP,(Cl, E1)12 + (1 +i sin28,)(P2(C1, El)I’).

(19)

This cross section has a form similar to the cross section for the scattering of photons which are polarized parallel to the scattering plane 3), except that 8, has to be replaced by +r-8, because here we consider only longitudinally polarized photons. In particular, the scalar polarizability does not contribute at 8, = 0” and 180”. Furthermore, the tensor polarizability gives a rather isotropic angular distribution as is well known from photon scattering. For a simple rotator nucleus with intrinsic dipole modes according to the Goldhaber-Teller model, the scalar and tensor polarizabilities are P,(Cl, El) = $(AL+2AT),

(20)

P,(Cl, El) = ,/&-AT), where A L/T

NZ =

2AM

Of/WL,T i (dh,T). 1

iLiT

-WY--fir‘L/T

The transverse and longitudinal parameter /IO:

‘#L/T

radii R, and R, are related by the deformation

RL = (1+0.945j?,)RT.

Similarly one obtains for the frequencies w, and oL, corresponding dipole modes transverse and longitudinal to the symmetry axis, or = (1+0.872/3,)oL.

(21) to the giant (22)

Fig. 5 shows angular distributions obtained for scattering on a nucleus with a static deformation PO = 0.3. Similar to real photon scattering 14~‘ “) the tensor polarizability can be determined from angular distributions even if the resolution is not sufficient to separate transitions to various members of the ground state band.

0

20

NUCLEAR

POLARlZABILlTlES

LO

60

60

100

120

161

140

160 Qr

100

ps]

Fig. 5. Angular distributions for photon emission from the giant resonance of a rotational nucleus leading to the O+ and 2+ states of the ground state rotational band, calculated for the two excitation energies wx = 15.1 MeV and oL = 12 MeV corresponding to the transverse and longitudinal dipole modes. The nuclear deformation is /? = 0.3, the momentum transfer q = 0.35 fm-r. The full curves are calculated for a common width of 3 MeV, the dashed curves for 1.5 MeV.

The simple model calculation above gives, of course, only a crude estimate of the effects to be encountered in (e, e’ y) experiments. The potential power of this process to determine spins of unresolved levels, particularly of overlapping resonances, in a model-independent way and to experimentally separate the contributions of various multipolarities, should instigate experimental investigations even though the experiment will certainly be a very tough one compared to (e, e’ p) and related processes. In particular, present discussions indicate different conclusions on the distribution of the sum rule strength of higher giant resonances (e.g. EO and E2) as derived from inelastic electron ’ 6), heavy particle scattering ’ ‘) and radiative capture reactions I*). A model-independent separation of the various multipoles would clearly settle this question and also make it possible to measure sum rules as function of momentum transfer for each multipolarity separately. Such experiments should therefore be invaluable for a more direct approach to fundamental questions like exchange forces and mesonic currents in nuclei. The formulation proposed in this paper is particularly amenable for an evaluation of experimental data in terms of generalized nuclear polarizabilities and known geometrical factors (if only the experimentalists will get sufficient counting rates!). Further, the formalism directly connects the (e, e’ y) process to elastic and inelastic photon scattering, another difficult experiment of current interest, which only recently

162

H. ARENHOVEL

AND

D. DRECHSEL

has become possible with sufficient energy resolution 19*‘O). In the limit of q = to, at the “photon point”, the momentum dependent polarizabilities derived from (e, e’ 7) have to be identical with the polarizabilities obtained from photon scattering. Due to the longitudinal virtual photons and the variable momentum transfer, however, the process (e, e’ y) could give more detailed information on nuclear dynamics, particularly also on resonances different from the giant dipole state. Appendix (0,) I., p) as defined in eq. (16) have been tabulated The angular functions gjLL’rKK’p below for electric dipole and quadrupole transitions. We use the abbreviations x = cos 0, and y = sin O,, and denote the multipolarity of the transition by CL and EL, respectively. Except for the interference term, the angular functions are symmetric against the exchange LL’r ++ KK’p. TABLE Al Coulomb

i Multipolarity

Cl El

Cl El,

C2E2

(1 = p = 0)

0

\

Cl El,

term

1 +‘y*

+(I +x2)

3Y’

-&x

-&

-&(1-3x2+4x4)

A(5

- Jfxy2

C2 E2, C2 E2

2

1

*x’y’

.I-Y2) -t- 9x2 - 12x4)

TABLE A2 Interference

Multipolarity Cl El, Cl El,

1’

term

(I. = 0, p = 1)

0

\

El El

1 - Jfxy

&/zyV JZ(

E2 E2

1-2x+

--Jgxzy

C2 E2, El El

-#(I

C2 E2, E2 E2

2 S&Y

J&(1

i-4x%

-Jg$ --2xZ)y -2xZ)xy

+&(3

J&(1

f4yZ)v

-&g1+2x21y

-8x2)xy

-+&3-8x2)xy

TABLE A3 Transverse term (1 = p = 1) MultipolaritY \ El El,

El El

El El,

E2 E2

E2 E2. E2 E2

j

1

0 $1

+x2)

$3

2

-x2)

-2&$x3

&2x*

‘(1-3x2+4x4) 5

&(3+15x2-

+(13+x2) -3)x

-Jg(zx2+3)x 16x“)

&(13-15x2+16x4)

NUCLEAR

Transverse

i

0

Multipolarity\ +Y*

El El,

- .2 J&-xy

E2 E2, E2 E2

-$(I

TABLE A4 term (1 = I, u =

163

-1 b

2

1

El El, El El E2 E2

POLARlZABILlTIES

=

-4xZ)yZ

-+yv’

*Y=

2Jfxy=

- .2&&v’

i1c(1-16x2)y2

+(I

- 16xz)y2

References I) G. Plazcek, Marx Handbuch der Radiologie VI.2 (1934) 205 2) U. Fano, NBS technical note no. 83 (1960) unpublished 3) H. Arenhovel and W. Greiner, Prog. Nucl. Phys. 10 (1969) 167 4) R. Silbar and H. Uberall, Nucl. Phys. A109 (1968) 146; R. Silbar. Nucl. Phys. A118 (1968) 389 5) H. Arenhbvel, Proc. Int. Conf. on photonuclear reactions and applications, ed. B. L. Berman (Asilomar, California, 1973) 6) D. F. Hubbard and M. E. Rose, Nucl. Phys. 84 (1966) 337; H. L. Acker and M. E. Rose, Ann. of Phys. 44 (1961) 336 7) G. Jakob and Th. A. J. Maris, Rev. Mod. Phys. 38 (1966) 121; T. de Forest, Ann. of Phys. 45 (1961) 365; U. Amaldi, Jr., G. Campos Venuti. G. Cortellessa, E. de Sanctis, S. Frullani, R. Lombard and P. Salvadori, Phys. Lett. 22 (1966) 593; A. Bussiere, J. Mougey, Phan Xuan Ho, M. Priou and I. Sick, Nuovo Cim. Lett. 2 (1971) 1149 8) D. Drechsel and H. Uberall, Phys. Rev. 181 (1969) 1383 9) T. de Forest and J. D. Walecka, Adv. Phys. 15 (1966) 1 IO) M. E. Rose, Elementary theory of angular momentum (Wiley, NY, 1957) 11) A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, NJ, 1960) 12) H. Uberall, Electron scattering from complex nudei, part B (Academic Press, NY, 1971) 13) H. Arenhovel, M. Danos and W. Greiner, Phys. Rev. 157 (1967) 1109 14) H. Arenhovel and J. M. Maison, Nucl. Phys. Al47 (1970) 305 15) E. Hayward, W. C. Barber and J. Sazama, Phys. Rev. C8 (1973) 1065 16) R. Pitthan and T. Walcher, Phys. Lett. 36B (1971) 563; S. Fukuda and Y. Torizuka, Phys. Rev. Lett. 29 (1972) 1109 17) G. R. Satchler, Nucl. Phys. Al% (1972) I; M. B. Lewis, Phys. Rev. Lett. 29 (1972) 1257; M. B. Lewis and F. E. Bertrand, Nucl. Phys. Al% (1972) 337 18) S. S. Hanna, Phys. Rev. Lett. 32 (1974) 114; and private communication 19) M. Hass, R. Moreh and D. Salzmann. Phys. Lett. 368 (1971) 68 20) H. E. Jackson and K. J. Wetzel, Phys. Rev. I.&t. 28 (1972) 513; H. E. Jackson, G. E. Thomas and K. J. Wetzel, Phys. Rev. C9 (1974) 1153