Electron excitation of isoscalar nuclear transitions

2.L

[ I

Nuclear Physics A248 (1975) 465476; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

ELECTRON

EXCITATION OF ISOSCALAR NUCLEAR TRANSITIONS L. J. TASSIE

Department of Theoretical Physics, Research School of Physical Sciences, The Australian National University, Canberra, ACT. 2600, Australia

Received 3 October 1974 (Revised 17 April 1975) Abstract: An infinite set of energy-weighted sum rules is given for longitudinal electric excitation of isoscalar nuclear transitions by inelastic electron scattering. The form factor and transition density for excitation of an arbitrary eigenstate of the nucleus are given as sums over the sum rules, and, for a particular choice of the operators in the sum rules, are given by series of which the t'irst terms are the same as the results of the hydrodynamical model.

1. Introduction

F o r m factors derived f r o m a h y d r o d y n a m i c a l model 1,2) have p r o v e d very useful in the analysis o f electron scattering 3) and also o f high energy p r o t o n scattering 4, 5). Deal and Fallieros 6) have derived a transition density o f the same form as that o f the h y d r o d y n a m i c a l model f r o m the assumption that a single state, called the d o o r w a y , dominates the sum rule for isoscalar electric transitions. Corrections to single doorway d o m i n a n c e have been considered by Fallieros 7) and by Ui and T s u k a m o t o 8). Using the sum rule o f Deal and Fallieros 5) and related sum rules, a series is obtained for the f o r m factor or transition density for a longitudinal electric isoscalar transition from the g r o u n d state to an arbitrary eigenstate o f the nucleus. The transition density obtained can be used to treat inelastic scattering o f other particles such as p r o t o n s 4, 5) and a-particles 9). A l t h o u g h first Born a p p r o x i m a t i o n is used t h r o u g h o u t this paper, it is easily shown that the same conclusions hold for D W B A , since the D W B A amplitude can be written as a folding over the first Born amplitude ?

fDw.A(ki, I,f) =J d3k, d3k2 ~(l,2)¢i(kl)f.or,,(kl, !,2),

(1)

where ~)i and ~bf are the initial and final distorted wave functions in the m o m e n t u m representation o f the electron. The f o r m factor for longitudinal electric excitation is the matrix element o f the operator z F(q) = ~ exp (iq. Vl), i=l

465

(2)

466

L.J. TASSIE

which, as has been emphasized by Fallieros 7.10), is a sum of single-particle operators ; and so, defining the no-particle-no-hole state as the ground state, electron scattering excites only one-particle-one-hole states of the nucleus. The I p-lh states are in general not eigenstates of the nucleus. The electroexcitation of any particular eigenstate of the nucleus will depend only on the lp-lh components of that eigenstate. This conclusion also applies to magnetic and transverse electric excitations, as again the appropriate operators are a sum of single-particle operators 3). The treatment in this paper is confined to the longitudinal electric excitation by inelastic electron scattering of isoscalar transitions of a nucleus with a ground state spin of zero. 2. Sum rules

For isoscalar longitudinal electric transitions

F(q) = ½ ~ iSFJ(q),

(3)

J

where A

FJ(q) = [4n(2J + 1)] ~ ~ js(qri) YJo((2i).

(4)

i=1

We define A

(2,, = 2 0~,,,,

(5)

i=1

~-~J,, i

= rJi + 2 , Yjo(~i).

(6)

Since

jj(Z) = [-(2J + 1)!!] -1 Z J[1 _ (1 !(2J + 3))- 1½Z2 + (2 !(2J + 3)(2J + 5))- 1(½)2Z4- ...],

(7)

the set of states

QjolO>

(8)

gives all the states excited by electron scattering. It should be noted that the set of states (8) does not necessarily include all lp-lh states. The final eigenstate of the nucleus can be written as I f ) = ~ b~Qj~lO)+ Irem),

(9)

where Irem) (remainder) does not contribute to electron scattering. The states Q j,10) are not eigenstates of any Hamiltonian and are not orthogonal. H is the nuclear Hamiltonian, which is invariant under time reversal, and has a

ELECTRON

EXCITATION

467

complete set of eigenstates

Hln) = E.{n).

(10)

Then for the operators A

X

=

EXi, i 1

(11)

A

r = yr

,

i=1

where X;, Y; are real functions of the position coordinates of the ith nucleon, there is a sum rule 11)

E, = (OIXHY[O> = ½(0I[X, [H, Y]][0>.

(12)

n

The Hamiltonian can he written H = T + V, A

T = ~ (2m)-

lp/2.

(13) (14)

i=1

Consider [Qse, [H, Qj~]]. For velocity independent V,

[ V,, Q,,~] = O. For a single-particle spin-orbit coupling or a two-particle spin-orbit coupling

[v, q.,3

o,

but has no momentum dependence so that [Q.,~, [v, Q~J] = 0.

(15)

Neglecting more general velocity dependent interactions A

[Qj~, [H, Qj,]] = (2m) 1 Z

[QJ~,i[P 2, QJ~,i]],

(16)

i=1

yielding the sum rule

(OIQjaHQj~[O> =

½
Qj~]][0>

= (h2/2m)(47r)-1[j(2J+2~+2fl+ 1)+4~[3]A,

(17)

where (,

(r"} = A- 1 j Poo(r)r,,d3r,

(18)

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L.J. TASSIE

and Poo(r) is the isoscalar density distribution of the ground state,

Poo(r) = (0[p(r)]0),

(19)

A

p(r)=

~b(r--r,).

(20)

i-1

If the distribution of neutrons in the ground state is the same as that of protons, Poo is proportional to the ground state charge distribution which can be determined from elastic electron scattering. The sum rule (17) has also been given by Ui and Tsukamoto 8). Similarly, sum rules are obtained for the form factor,



=

½ (" = _ (h2/2m)(4n)-~(2j + 1)~Jd3rjj(qr)rJ+ 2~- 2 x [2~(2~ + 2J + 1)Poo + (J + 2ot)rdpoo/dr ].

(21)

Using

Fa(q) = [4n(2J+ 1)]½f d3rjj(qr)Yjo(f~)p(r),

(22)

then <0[Qs~ Hp(r)10> = - (h2/2m)r J + 2,- 2[2~(2~ + 2J + 1)P0o + (J + 2~)r dOoo/dr ] Yso((2).

(23)

For J 4: 0, ~ = 0 eq. (23) reduces to the sum rule given by Deal and Fallieros 6) and corresponds to the transition density of the hydrodynamical model 1), and eq. (21 ) yields the sum rule of Deal and Fallieros ~2) which has the same form factor as the hydrodynamical model 2). For J = 0, ~ = 1 eqs. (21) and (23) yield the monopole sum rules given by Kao and Fallieros 13) and by Deal and Fallieros 6) which correspond to the transition density given for monopole transitions by Werntz and Oberal114). Eq. (23) has also been given by Ui and Tsukamoto 8). 3. Excitation of an eigenstate

Eq. (21) gives an infinite set of energy-weighted sum rules. In this section, this process is inverted, giving the form factor for an individual transition as a sum over all the energy-weighted sum rules. In order to obtain this result, we first introduce linear combinations of the Q~,,

mj~ = ~ A~Qs~, B

(24)

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469

with the A~e real, such that

(ormj, HmjplO) = 0

for ~ 4: /~.

(25)

Note that the states Mj, I0) are not necessarily orthogonal. In general

(OIMj~M~tj[O) 4: O.

(26)

Using eqs. (24) and (17), the condition (25) can be satisfied, although it should be noted that the condition (25) does not specify the M j, uniquely. A particular set of M j, is constructed in sect. 5. The transformation (24) can be chosen to be nonsingular and then Qs~ = ~ C ~ M ~ . (27)

p

We consider the electron excitation of a nucleus from the ground state of spin 0 to a final state of spin J. From eqs. (4), (7) and (27), we can write FJ(q)10) = ~ f~(q)mj~10).

(28)

Then using eq. (25), (flMj~.10) (0IMj~ HFJ(q)IO)/(OIMj~ HM t~IO) 7

= ~ (fIMj~IO)(OIMj~ HMj~[O)fa(q)/(OIMj~ HMj~[O) ylJ = ~ (flMj,lO)fe(q) = (flFS(q)10),

(29)

Y

the isoscalar longitudinal electric form factor for electron excitation from the ground state 10) to the final state If). From eqs. (29) and (27), (f]FJ(q)[0) = ~ B~,(O]Qj.~HFJ(q)JO),

(30)

Y

which with eq. (21) could be used to analyse electron scattering data. However the coefficients B~ are not simply related to the properties of the state If). The expansion in the Mj~ is more useful as the M~, can be constructed so that

(OIMj,;HFJ(q)IO) ~ qJ+2~,

as q ~ 0,

(31)

so that the term with 7 = 0 predominates at small q whereas

(OIQj,,HFJ(q)IO) ~ qJ,

as q ~ 0,

J 4: 0.

(32)

It is necessary to compare the relative magnitudes of the terms for different ? in the expansion (29) for the form factor. If MjylO) is an eigenstate of H,

If) = kMj,;lO),

(33)

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L.J. TASSIE

the state If) exhausts the energy-weighted sum rule

(OIMj~In)E,,(nIMj~LO).

(34)

n

Then 1 = (flf) = Ik[ZE(I(OLMjvHMj,,.[O),

(35)

(flMjel0) : E(~(0[Mj, HMj~IO)L

(36)

and

For an arbitrary state If), we can write (f[Mj~10) = c,I E~-~(OIMj7 HMj~I0) ½.

(37)

Then the state If) satisfies the fraction Lc~[2 of the sum rule (34). The c~ could be obtained by calculating (flMs~10) from nuclear wave functions in some model, and then using eqs. (37), (24) and (17). For fitting electron scattering data, the c 7 are adjustable parameters, with the restriction Ic,I _-< 1,

(38)

and the values of the c~ determined from experiment could then be compared with results of nuclear structure calculations. The form factor is (flFJ(q)10) = E ~ - ~ %(O[Mj>.HFJ(q)IO)/(OIMjTHMjT[O)~.

(39)

Y

The condition (25) does not uniquely specify the M j , , so that there is more than one way to proceed. Before the M j, satisfying eq. (31) are constructed in sect. 5, another possibility is briefly discussed in sect. 4. From (37) we have

c~ = E? ~(flHM j~LO)/(OIMj~ HMj~I0) -~. Then, if Mj~ [0) is an eigenstate of H, i.e. for an eigenstate of the form (33), c~ = E f ~k(0IMs, HMj~I0)/(0IMj~HMj~[0) ~ =0,

for e : P 7 . 4. Use of orthogonal states

It is possible to construct a basis of orthonormal states Nj~[0) by constructing operators Nj~ = ~

D~Qj~,

(40)

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471

(OINj~NjaIO) = 6~,

(41)

such that

and then diagonalize the Hamiltonian H using this basis. This procedure will yield orthogonal states Mj,(orth)10) satisfying the condition (25), and for which the form factors for electron excitation can be calculated. The form factor for excitation of an arbitrary state is then given by eq. (39). If velocity dependent forces, other than a single-particle spin-orbit coupling, are neglected, [H, Qj~] is a sum of single-particle operators, so that HQj~I0> = [H, Qj=]I0> is also a lp-lh state. The nuclear Hamiltonian does not directly connect the states Qj~[0> with 2p-2h or higher states, so that the states Mj~(orth)10> should be approximate eigenstates of the nuclear Hamiltonian and should be giant resonances. To obtain the Ms~(orth), it is necessary to have not only the ground state matrix elements of single-particle operators (OIQj~HQj ~10> given by eq. (17) but also the (O[Qj~Qjt~IO> which are ground state matrix elements of two-particle operators which must be obtained from some detailed nuclear model which yields a good ground state wave function. Such a procedure involves a separate calculation for each nucleus. It is more convenient at this stage to have one procedure which can be used for all nuclei and this is given in the next section. 5. Use of non-orthogonal states

A particular set of

Mj~,with the property (25) can be constructed as Mj~ = ~ A~pQj~,

(42)

/~=0

with A~, = 1. Then

Qjp =

~

Ct~Ms,"

(43)

co=0

The condition (25) is satisfied by

(OIQjBHMj~IO} = 0,

for fl < ~,

(44)

which for given c~ provides e simultaneous linear equations

L A~,~(OIQj~HQjTIO)= 0, y=0

to determine the coefficients A,,.

for fl < ~,

(45)

472

L.J. TASSIE

F r o m eqs. (4), (5), (6) and (7), we can write

FS(q) = ~ b~qS+ ZOQjo.

(46)

/~=0

Then, using eq. (44)

(OIMjTHFJ(q)[O) = ~ baqS+2a(olms~HQap[O) oc qJ+2~,

as q --* 0.

(47)

Writing only the first two terms, the form factor is given by (f[FJ(q)[0) = [(8mn/h2)Ef A]~(2J + 1)- ~{c o J - ~(r 2s- 2) - ½(01QJo HFJIO) + c 1[(2J + 1)(J(2J + 5) + 4)(r 2J + 2) _ J(2J + 3) 2(r2J) 2(r 2s- 2 ) - 1] -

x((2J+I)(OIQj1HFJIO)-(2J+3)(rZ')(r2J-2)-I(O[QjoHFJ]O))+

...},

(48)

where

(r") = A - 1j d 3 r r , Poo(r).

(49)

F r o m eqs. (48), (22) and (23) the transition density is (flp(r)[0) = [(2rch2/m)Ef A]~(2J + 1)- ~ Yjo(~){ - c o J½(r 2J - 2 ) - ~r s - Xdpoo/d r - c I [(2J + 1)(J(2J + 5) + 4)( r 2J + 2 )

_

J(2J + 3)2(r 2s) 2(r2S- 2) - 1] -.~

x ((2J + 1)rJ[2(2J + 3)po o + (J + 2)r dpoo/dr ] - (2J + 3)(rZJ)(r 2s-

2) -

1jr J- 1 dpoo/dr)

+ ...}.

(50)

It should be noted that, even in the higher terms, the transition density does not involve any derivatives higher than the first derivative of the ground state density distribution. By repeated application of

fdrp(r)rL+2"jL(qr)=

--(2L+2n+l)-lfdrrL+2"{(dp/dr)r(jL(qr)+

JL+2(qr))

+2(n-- l)pjL+z(qr)},

(51)

eq. (21) can be written as (01Qs~ HFS(q)[O) = - (h2/2m)[4n( 2J + 1)]3 f dr(dpoo/dr)r J + 25 +1 {jjs(qr)

+ ~ ( - 1)~2i(2j + 4i + 1)[~ [/(~ - i)!][(2J + 2ct + 1) ! !/(2J + 2~ + 2i + 1)[ !]jj + 2i(qr)}. (52) i=l

ELECTRON EXCITATION

473

For a uniform density distribution

(53)

dpoo/dr = - [ (3 A/4n)R313(r- R ), (0[ Qs, HFS(q)[ O) --- 3( hZ/2m)(2J ~- 1)~(4n) - ~AR J +2~- 2 {jjs(qR)

+ ~ ( - 1)iU(2J+4i+ 1)[c~!/(~-i)!][(2J + 2~ + 1)! !/(2J+2ot+2i+ 1)! !]js+2i(qr)}. (54) i= 1

From eqs. (42) and (47) it follows that (OIMs~HFS[O) is given by the last term with i = ~ in eq. (54) so that (01Mso HFJ(q)IO) = 3(h2/2m)(2J + 1)½(4n)- ~AR s- 2jjs(qR), (01Ms, H FJ( q)IO) = ( - 1)'3( h2 /2m)(2J + 1)-~(4n)- I A R J +2, - 2 xU~![(2J+2~+l)!!/(2J+4e-1)!!]js+2,(qR),

for c~ :~ 0,

(55)

x 2~eoHFS(q)lO).

(56)

(OIMj~HFS(q)[O) -- ( - 1)~(J + 2 a ) - ' [(2J + 1)/(2J + 4a + 1)] 3

The result ofeq. (55) for ~ = 1 and 2 has been given by Ui and Tsukamoto 8). Eq. (56) shows that higher order terms in the form factor for excitation of a state of spin J have the same q-dependence as the first term of excitation of states of higher spin, J + 2, J + 4, etc., as also noted by Ui and Tsukamoto 8) for c~= 1 and 2. However, from eq. (52), it is seen that this relation does not hold for an arbitrary density distribution, as the power o f t multiplying js + 2i(qr) in the integrand depends on ~. Even so, eq. (56) shows that caution is needed in allocating spins to nuclear states according to the shape of the angular distribution of inelastic scattering. A well known special case is the similarity for electric monopole and quadrupole transitions (0[M0a HFS(q)[O) = - 5- ~(01M2o HFS(q)[O),

(57)

which is valid for an arbitrary density distribution. From eqs. (42) and (44) (OIMs~HMs~[O) =- (OlMs~HQs~[O) = ( - 1y2"[4n(2J+ 1 ) ] - ~ ! ( 2 J + 2 ~ + 1)! !lira q-(S+2")(OlMs~HFS(q)[O).

(58)

q-~O

For the uniform distribution (OIMso HMso[O ) = (h2/8mn)3AJRZS - 2, (O]Ms~ HMs~[O ) -= (hZ/8mn)3AR~J +4~- 2 4~(2j + 4~ + 1)- 1 x[c~I(2J+l+2et)!I/(2J+4c~-l)!!] 2,

for ~ :p 0,

(59)

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L.J. TASSIE

and the form factor (39) is

(flFJ(q)[O) = [3El- 1(hZ/2m)(2J+ 1)AJAR-1 {c o J½jj(qR) + ~ ( - 1)%~(2J+4c~+

1)½Js+2a(qR)}.

(60)

a

For J = 2, taking all c a to be of the same magnitude, taking the summation in (60) to ~ = 2 is adequate up to qR = 4, and the predominance of the first term up to qR = 3 shows why the hydrodynamical model has been so useful in analyzing electron scattering data. Electron scattering data has frequently been fitted 15- 21) using the hydrodynamical model with a density distribution whose radius parameter ctr and surface thickness ttr differ from the values for the ground state charge distribution and also vary from level to level. Such measurements of ctr and ttr contain some information about the relative magnitude of the coefficients c 7 and show that the contributions from the non-leading terms in eqs. (39), (48) and (50) are not negligible. We now prove that i f a final state If) saturates one of the sum rules, i.e. Ic,[ = 1 for some value of ~, then c~, = 0 for all 7 :~ ~. The phase of If) can always be chosen so that c a = 1. We then write If) =

E~(OIMj~HMj~,IO)- ~Mjal0) + [rem),

(61)

and show that ( r e m l r e m ) = 0. For, since If) saturates the sum rule, it follows that

( OlM ja M ja[O) = Ef 1( OIM j~ n M sa]O).

(62)

Since ca = 1,

Ef~(O[Mj~HMj,IO) ~ = (flMjal0) = E~f(OIMj, HMja[O)-~(O[MjaMja[O) + (rem[Mj,]0) = E f ~(0[Mja HMjaIO) ~ + (remIMj,[0), so that (remlMsa[0) = 0.

(63)

F r o m the normalization of (61), (flf) =

Ef(O[Mj~ HMja[O)- I(0IMj~ Mjal0)

+ E~(O[MjaHMjaIO)-I 2 Re (rem[Mj~[0) + ( r e m l r e m ) = 1 + ( r e m l r e m ) = 1, using eqs. (62) and (63), so that Irem) = 0. Then, from eqs. (61) and (25), (flMj~[0) = EfX(flHMj~[0) =0,

for 7 ~ ~.

F o r c 7 = 6,~ there is only one non-zero term in the series (39). The transition density is uniquely determined for a state which saturates the sum rule and is an eigenstate of the nuclear Hamiltonian. This contrasts with the conclusion of Ui and

E L E C T R O N EXCITATION

475

Tsukamoto 8) that the transition density of a giant multipole state saturating a sum rule is not uniquely determined. Of course, observed giant multipole states neither completely saturate a sum rule nor are eigenstates of the nuclear Hamiltonian. 6. Discussion

Inelastic electron scattering excites only lp-lh states whereas inelastic scattering by strongly interacting particles can excite higher excitations, 2p-2h states and so on, because of the effects of coupling between channels which are negligible for electron scattering because of the weakness of the electromagnetic interaction. In principle then it is possible to learn more about nuclear wave functions from inelastic scattering of strongly interacting particles, but in practice this is very difficult because of the complication due to the strength of the interaction and the sensitivity to so many nuclear parameters. However, if the l p-1 h components of nuclear wave functions were first determined by a systematic study of electron scattering, the scattering of strongly interacting particles could be used to study the other components of nuclear wave functions. The hydrodynamical model has been widely used to analyse electron scattering data because it is a reasonable first approximation to the data and because it can be incorporated in a DWBA program 22) more easily than complicated models such as the Hartree-Fock shell model. Although the transition density of eq. (50) is more complicated than that of the hydrodynamical model which it contains as its first term, it is still simple enough to be used in a DWBA program to analyse data. Because the recoil of the nucleus has been neglected, the motion of the c.m. of the nucleus has not been treated correctly in this paper. Deal 23) has shown that the sum rules for electron scattering are modified when the motion of the c.m. is treated correctly. For heavy nuclei, the c.m. corrections are small, except for A T = 0 electric dipole transitions where the main effect when the Mj~ are constructed according to eq. (42) is to ensure c o = 0. The finite size of the proton, which has been neglected here, can be taken into account in the same way as in the hydrodynamical model 24), namely by unfolding the charge density of the proton from the ground state charge density, forming a transition density, and then folding back the charge density of the proton.

It is a pleasure to acknowledge the kind hospitality of the Institut fiir Kernphysik, Technische Hochschule Darmstadt, where much of this work was done.

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476 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

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