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Transverse sum rule for electrons scattered inelastically with large momentum transfers from light nuclei
ANNALS
OF
PHYSICS:
42,
119-143
(1967)
Transverse Sum Rule for with Large Momentum W. Insl,ylui
Czui, Fizyki
Electrons Transfers
L. LE~IAK, Jadwxuej,
Krakdw,
Scattered Inelastically from Light Nuclei A. ~~IA~EcK: Rronowice,
Poland
A isurn rule for the cross sections of electrons inelastically scattered from nuclei is analyzed. This sum rule contains contriblltions only from the transverse electron-nucleus interactions and is particularly suitable for analyzing the experiments of inelastic electron scattering at 0 = 180”. The sum rule is a fnnciion of the three-momentum transfer q. A discussion of this sum rule for momentum transfers 0.5f-1 5 q 5 2f-1 is presented. Strong dominance of the spiwcurrent contribution seems to offer a very atkactive possibility of measuring magnetic moments of nucleons bound in nuclei. The paper contains estimates of the short-range nucleon-nucleon correlation contributions to the sum rlile. I. INTRODUCTION
There are many ways of analyzing the inelastic electron scattering from nuclei. One of them is to measure experimentally the sums &a( 4, 8, U) and compare it with theoretical predictions. Here ~(4, 8, w) is the cross sect’ion for inelastically scattered eleckons with fixed three-momentum transfer 4, scattering angle 8, and energy loss w. In the sum over w we include the elastic scattering contribution (w = 0). We can, in the one-photon exchange approximation (t’he first Born approximation), always write the sum in the “canonical” form [see, e.g., (I)]: q
m 4% I‘ 0
dw
w, 0)
= s
[A(q)
I f(up”)12(e2/W2
+ B(q)
tan” ?$I
(1)
in which one separates the contributions from t,he longitudinal plus transverse elecbromagnet,ic interactions [the term with A ( q)] and the t’ransverse interact#ion alone [t,he term with B(q)]. Henceforth we shall call B(q) the transverse sum, and the various theoret’ical expressions obtained for it later the transverse sum rules (TSR). In Eq. (1)) f( q,,’ = q’ - w”) is t’he nucleon form factor related t’o the conventional nucleon form factors as followsf( q,“) = Fl,( qp2) = Fx,( qF2) = FPn(pC2), F1,( qp2) assumed to be zero; pi is the moment’um of the incident electron. A(q) and B(q) depend only on the absolute value of 1q 1 because one 119
120
CZTk,
LEhIAK,
.4iTIl
MALECKI
averages over the initial and sums over the final nuclear stnt,es compatible with the prescribed momentum transfer and energy loss. A (a) and B(q) can be cxpressed through the matrix elements of the nuclear charge and current operators [see, e.g., (I) and (g)] as folIows:
+
(9)
+
(@J:,,)(@*Jno) (“-“;,
“):’ 7 c2)
R where Qno= (n / Q 1O), Jno = (n 1J I 0) are the matrix elements between the ground and excited states, Q, J are the charge and current operators of the target nucleus and 4 = q/a; w,,o= E, - Eo is the excitation energy of the nucleus. The complete sum rule (1) is rather complicated and very many terms have to be evaluated in order to get numbers for comparison with the experimental data. It is therefore rather difficult to extract information about nuclear structure since the calculated numbers are produced by many effects of different physical origins which partly cancel each other. In other words, too many different effects contribute to (1) to let us decipher in a reasonably unique way the information contained in it. It is tempting therefore to analyze certain pieces of (1) which can be extracted experimentally and which contain more specific information about nuclear structure. First of all let’ us recall that the TSR can be measured directly. This can be done either by experimentally performing the Q = const, B = 180” sum for a(q, 8, w) by varying pi and the final-electron momentum, or by doing the complete sum for two different angles 0, but the same q. Then the slope of A(q) + B(q) tan” s0 vs tan” x0 gives us B(q). In fact, the first method has already been used for extracting B(q) both in the high- and the low-~ limit,.’ In this paper we limit ourselves to analysis of the B(q) term only. In Se&ion II we discussgeneral properties of the transverse sum rule. In Section III we evaluate TSR for the oscillator potential well shell model ground state of the 160 nucleus. Then we present a method of calculating corrections due to the nucleon-nucleon shortrange correlations which consists in modifying the relative two-nucleon wavefunctions at short internucleon distances. Section IV contains the numerical results and a discussion of their significance. Before concluding this Chapter let us point out that, although we do include 1 The measurements rateur Linkaire, Orsay. for informing us about in the measurements of paper.
of the high-q limit of R(p) are being done at Laboratoire de L’Accele(We are grateful to Professor Jo& Goldemberg and Dr. U. Isab6116 the measurements.) The low-p limit of B(q) was obtained essentially the giant dipole resonance and we discuss this problem in a separate
TRAKSVERSE
SUM
1“ 1
RULE
in principle the elastic scattering in our sum rule B(q), it turns out. to be immnteCal as the TSR seems to be most promising in cases of simplest nuclei with no magneti(a moment’s in t’he ground state, in which cast there will be no contrihution from t,he elastic scattering.’ The choice of spin-zero nuclei is again dictated by nur desire to extract few but well-defined pieces of information about ~luclenr strwt’ure, and avoid the embawas de richesse one gets into when dmling with c*oniplic:lt cd nuclei. II.
GENERAL
PROPERTIES
OF
TSR
In order to have a general idea as to how B(y) changes with y let us analyze it, assuming t,hat the final states onesums over in (3) form a complete set of states. Then, applying closure, one get.s WY)
(3)
= (0 I iJ+*J - (kJ+)(4.J)l 1%
where / 01 is the ground-state vector. We accept the nonrelativistic operator .J (without, exchange currents present) : Cpj eiq’ri + eiq”lpj)
+ 5
where IV is the nucleon mass, ej = >$[l + am], of t,he jth nucleon. From (3) and (4) we get
i(dj
form of the
x q)e’q”j
,
(,-I 1
and pi is t#he magnetic
moment
(6) +
;
lo
/ 2
pi~~(;J-“t)p.
2
1 O\\ I
\
ek
pj
problem
of elastic
scat,tering
I,h,
1-
at 0 = 150” is disrllssed
,’
9
/
where we chose q along the x axis. As B(y) depends only on the q, we can choose its direction as we please. In (6) we separated terms. The form (6) is more convenient for analyzing B(y) exhibits e:xplicitly the first’ leading term. Notice that (5) and the p-p and c-g correlations; the u-p correlations are absent. ? The
.c
3211
ill detail
absolute value of out the diagonal for large y, as it, (6) contain only in Ref.
(4).
in Eqs. 1.5) and (cij, that (0 1 cfk esp [1’~(x, - +zt11. e,e,,-p,,pl;, 10) = (0 j C$ esp - zkj]e,ekp,,pA, 1O), and similar rcl:ltions wit,h eJpl, ----) pygjr and -k .] The SL~II rule (5) or (6) contains contributions from some spurious cscitatiorw of the wntcr-of-mass (c.m.) degrees of freedom. Thew spurious contributions are, in general, small-A times smaller than the other term-but, for small q they become wry import.ant, (4). We can remove them employing the C;artenhnun-Sc.llw:trtz transformation (5) which in our case reduces t.o [We
:lSSLllll~,
[iy(Zj
ejpj,
/ljUjy
rl + rl - R,
pj + pj -
where R is the c.m. vector and P = c;” few simple niarlipulat,ions~
PK1,
(~71
pj . Instead of (6), we get now (after n
4 ej e,(P, A-’
-
pj.z
- /)Br)Pz 10’ I /’
[In obtaining expression (8) we employ the following symmetry of the ground state: (0 1piIPs / 0) = (0 1p&‘, / 0); ,i, Z-arbitrary. Hence il(O I PjzrPz IO) = (0 1CC: pjx)Px 10) = (0 I P,” I Oj.1 Before discussing more quantitatively the properties of the sum rule H(q), let us ment’ion some of its general properties. Let us (aall C,,(q) t,he spin-spin correlation fun&on [t,he third term of (S)] and CL,(y) = C,,(q) + AC,,(g) the c.m.motion-vorrectctl momentum-momentum correlation function [the fourth and the last term of (8) .“I They both do not) depend on the direction of q; hence
But for finite nuclei, the spatial correlation
functions
G<,(Y), G,,(V)
should
be
3 We shall see flIrther that there exist strong arguments that the correction for the cm. 1s negligible for the region of q considered in motion in C,‘,(p) [which we denote by K’,,(q)] this paper. So, in calculat,ions of the short-range correlation corrections we shall drop it and call Cr,,(q) the correla.tiun function [the fourth term of Eq. (S)].
TRBXSVEHSE
SUM
RULE
123
hounded for 3’ -+ * by an esponent.ial function e+’ \&eye 6 = ( ~:j[&Bjfi~‘)‘iu, EB being the binding energy of the most weakly bound I~UC~COII. For I:‘, e 8 ;\IeV (which is the (base for, e.g., 160), p corresponds to a niomc~ntum of :~bollt 1’20 AleV/c. Hence Co,(q) and (Y&,(q) are analytic in t,hc strip y = - $3, y = fib in the clomplex y-plane.4 Since they can depend only on y = ( q 1, U-P come to the c*onclusion t,hat CbO(~), C’,,(q), and hence R(y) cm be reprcscntcd by an absolutely convergent power series of y, only with even powers of y present, and the radius of convergence not smaller than 120 MeV/c, R(q)
= R(O)
+ Py”
+ ft4’y4 + . . . .
(IO)
It seems nit.hin t.he go with y, yg _ w:! =
that one can arrange esperiment,s which produce B:( (/I cwmfort:~blp radius of convergence. It is not clear however how far clonn OIW (x11 because for very small y the closure approximation brc&s down :ts 4pipf sin” !28; hence W’ 5 y’ , and for a fixed small Q, we can wvcr in the exper.iment only a small part of the excit,ation spectrum. There is a possibility of avoiding this trouble by splitting the series (10) into wntrihutions of differcntj multipoles. Then one may hope that specific multipolc cw~tributions arc concent,rat,ed at lowenough excitation energies and even for I~)x (1 the cffecstively coniplet,e set of states is contained in t,he comparatively lo\\-w cwitatiorr spectrum cwwecl in experiment,. This approach demands, from experimc~nt, ident,ificstion of t’hc levels excited--in order that the sum rule hc rcw~lvetl into the wntri butions of different multipoles. This is possible to obt:lin, for some multipolcs at least,. l’or instance, the measurement,s of the 31 ewitntions in the gi:~nt~-dipo’le-resorlan(.e energy region meet these recluiremcnts [FW, e.g., ( 7,~ and (S)]. A preliminary theoretical analysis in this spirit has also hccn tried and the result:; look encouraging (4 1. In fact, for low y, R(y) will bc titrongly clomnatetl by l;hc El transitions. Indeed, if we put y = 0 in (8), \YP get (II) whew f?j’ = N/A for protons; P,’ = -Z/A nuclei with Z = N = ,I z-4, WC have
for neutrons.
where rEI(w) is the El absorpt)ion cross serti(Jn (witjhout complete nwlear excitation spectrum is experimentally
(jr, if we consider the
wtnrdation). availabk
So, if the [hence y is
kept large in t,hf measurements of R(q)], H(O) is proportional to the cnergyweight,ed dipole sum rule-quite :m interesting quantit,y because of it,s ,~nsitivity to t’he short-range nuclear (especially neutron-proton) correlations. It seems, however, very doubtful whether one will be able to extrapolate R(O) from the measurements of R(q) with large 4. In t,his case, one is so far from the beginning of the R(p) curve that any reliable extrapolation looks rather impossible. It’ is perhaps worthwhile to point out, Ohat the measurements of the so-c*:dled in&&c form factors of the giant’ dipole resonance [see (7), (8), and also (,$)I, where one ran easily extrapolate B(p) to R(O), do not produce B(O) as given by Kq. (12). They contain only these excited stat’es (n ( which belong to the giant dipole resonance multiplet; so, if we extrapolate the inelastic form fxcators to 4 = 0 we get only a part of (13). For example, we miss completely the high-energy part, of the dipole absorption which rest& in production of correlated neutrorlPproton pairs (see, e.g., (9) 1. l~inally, one may also notice that only even powers of Q appear in (10) and this is a consequence of finiteness of nuclei [which is equivalent to t’he assumption that G(V) functions are bounded by an exponent#ial e-“‘I. This fact may help in cst,ablishing regions of validity of some nuclear models which produce odd powers of y in the expansion of R( 4). For instance the Fermi-gas model produces terms linear in 4; hence, ab least for small g, is rather inadequate and may introducee serious dist)ortions in the Q dependence of B(q). III.
IS\-ALGATION
OF
TSR
It is caomparatively easy to evaluate (8) with the shell-model [oscillatorpotential] ground-stat’e wavefunction. For instance, for the I60 nucleus, the completely nntisymmet,rized shell-model ground-state wavefunction gives ( for more det’uils see Appendix A)
where X is the parameter of the Gaussian factor exp ( -$zh?) of the oscillat,orpotent’ial wavefunctions. [I+kom elastic scattering of electrons from lfiO, one gets X z 0.36j’-‘.] The first two t,erms are the “diagonal” terms of (6)) the third and fourth are C,,( 4) and C,,(q) correlation functions of (6), the fifth and sixth terms are correctSions due to subtraction of the c.m. motion.5 They are small for B Notice that our results larger than the same term
for u-u correlat,ion in Eq. (22) of (2).
function
[the
third
term
of (13) ] is four
times
TIt.4XSVEltSE
SUM
1tULE
12.5
q 2 1 j-l, but for small y, they are very important. For instance, in the limit q + 0, thaw is :L 100 5 difference between B(q) with and wit.hout suhtr:wtion of the Cm. rll~JtiOll: from (6), B(O) = 8x/L122, from (8) H(O) = 4k/jr”. This f:lPt is essential in discussion of the lowy limit of B(q) [see (/t)]. Figure 1 ShOWS th(x plot of ( 12 1. 0nc should cwnsider the mnrly corrections to formula (13) before applying it wnfidentl,y to the analysis of the experimentally me>~suretl sum rules. I:irst of all, we shall not worry for the moment about the CXJrrertiOW to the nwlwr clwtromagnetic current (,4) ; this problem is tliscusscd in tho last Section. As wc ewluate sum rules which include summation Over ;1 host, Of very wmplicntcd cxcitat,ion;-;, the details of the shell model we ncwpt as 0111’ starting p(Jint [e.g., the spin-orbit8 coupling, the shape of the average potential] stem not to he important. A much more essential point> is to estimate t,hc effects of t.hc nuclco~lnucleon wrrclatjions; the wavelengt’h cSorrcsponding to q-values considered in t’his paper is small enough to show some det,ails of the fluctuations of nucleous around th(lir average shell-model orbits. This is a very complic~ntcd problem which deserves an exknded and thorough invkgntion. In the ws;t of c~alculttting B(q), however, we shall limit ourselves to some simple ost imntcs hccnuse the modclindependent term of (8) (the first one) is strongly domin:king for 1:wge y and there is no need for wry great’ accuracy. We shall ncglwt henceforth the NJiTCF tions of the (Am-motion subtraction, and so we should &&ate the c*orr&ion of short-range correlat’ions to C&q), C’,,(y) and (2,‘dl”)(O 1 c: e,p& / 0). To estimate C,,i y ) , one can USC the rluc~lcorl-nuclec,n IJair correlation function
The trouble is, however, that C,,i y ) and ( ?/‘A/“)(0 1 c: eipSI j 0) c!:tnnot b(h expressed through p( 1 , 2). So, RX’ prefer to use a method whirh c’xn be applied both for C,, and C,, . WC use the method of modifying the relative two-nucleon wavefunctions at short internucleon distances, employed, e.g., in ( 20). E’irst of all let us notice t,hat A(T,,(q
= 0) = A
where b.y A( . .) WC denote the correction
to ( . . . ) due to short-range
correla-
tions. Equation ( IA) says that we 1m~1 to know only C,,( r]l‘i :tntl clPP( 4) and the corrections to the c+onstant krm (2,‘;11’)(0 1 C;’ CjpS, 10) follow from the (‘orr&ions to C&q). J$‘c can see it, as follows:
where the summation in the right-hand side (r.h.s.) extends over t,he protons only. We USC [after Noshinky ill )] the following transformations of a pair of nucleons to their cm. and rclntivc coordinates: r = (1/2/:!)(r] p = !l/&)!Pl
- r2), - p2),
R = (l/g3)(r, P = ui’dwp1
+ r?J,\ + pd./
(18)
From (17 j and (18‘) WC see that if, in ( 17), we modify the relative motion of the j, 1; pair [in a unitary way: preserving normalizations and orthogonalities], the 1.11.~.of (17) does not, change; hence (16) follows. Let. us nom describe briefly t.he method of calculating short-range correlation chorrections. Consider t,he matrix clement (%J 1 (3 1q), where \Ir is a completeIy antisymmetrizcd shell-model wavefunction and 0 a two-part’icle operator, i.e., o = cj,, Oij, 1;). We introducac the correlations into the wavefunct’ion cmploying a unikvy operator U I +) = ‘u I w,
‘U+ zz ‘u-1
(19)
The unitary of U guarantees us t,he correct, normalization of the modified function. Taking into account only t,hc two nucleon correlations we can write the correlated matrix element as follows [see (10) and Appendix B] (\I,\ 01%) = (\lr(u-lmlI\k) = 5 [(cY(l)P(2) 1UPO!l, 2)U 1cu(l)P(2)) - (cr(l)P(‘L) / u-‘0(1,2)% = $ GCG@j
- b(lW)
/ /ql,Cx(“))]
I 0~1, 3 I a(1j~(2))
I (xl, 2) I P(Ik42)L
where CX,~arc complete sets of quantum numbers of single-particle states, and the summation rst,cnds over all occupied states. In (20) we have introduced the c~orrelatetl two-purticalc wavefunct’ions:
The incIic,es a, p . cont,ain, in :Idditiou to the spatiul cluarltum numhcrs, the spin and isospin quantum numbers. The summation owr the spin and isospiu quantum numbers can be easily performed as our simple oscillator potential model ( see Appendix B) does not introduw any spill- or isospitl-depcnclcnt couplings. The single-particle spatial c~u:tntum numbers IVY tlcnott 1)~ a, 0 . . , and in all the following formulas, the snmmntic~n ovw spin and isospin q~~antum numbers (see Appendix B) has been pcrformcd. The twwpartic*lc state iti’
Iiow > following Aloshinsky nucleon-nucleon pair:
(ii),
wc go ovc’r to t,he c.m. :tnd rclut,ivc mot,ions of a
where (n, I) are the quantum numbers of relative motion and (N, I, ) are quantum numbers of the cm. motion. The relative and the cm. coordinates defined in ( 18). Because of energy and orbital momentum conservation, 1\4oshinsky coefficients in (2X) arc different from zero if the following relations satisfied (11) : “?ll + 11+ 2?1?+ I, = 2% + 1 + 2N + I,,
124)
11 + 1, = 1 + L = a. These coefficients arc real and have the following
the arc the are
(2.5)
symmetry
properties
(11) :
(nl, NL, 1\ / ~11 , n& 1 A) = i - 1) ‘-‘( nl, NI,,, x / ~olz , ,&I , A] = ( - 1) “I-“{NL,
721,
X j nIlI , ,n:& , Xi
= i - 1j ‘I+“{ NJ+ nl, X 1&
(-“(ij
, nlll
, X).
12s I n1, NL, xp) = rz (Zdd
/ ILXp) I Edna)1NLJI),
(271
we have from (22), (23)) and ( 27) I ab) =
;< c (1,711,67nb j l,i*Xp)(lr~~L‘u ; ZLXp) hpinllII1Nl, . jnl, NL,
(28)
x / n,l,L , nblb , A} j nh)
/ NLM).
From (28) and the symmeky relations (26) WC:get 6 We we Edmonds (12) convenliwls for monics and the angular momentum algebra.
the
Clehsch-Gordon
cclcfliciellts,
spherical
har-
12s 1ba) =
CZTt,
C
C
Xp,~.tf
nlNL
LEhIAK,
ANI)
( - 1) 2(Z,~)~,,lh~7~6 1 l,lbX~j(ZmLIJ
1 I&L) i'28)
.j,/ll,
NL,
x
/ n,l,
We introduce the short-range corrclut,ions func%ions of the relat,ive motion I nh)
MhLECKI
= R.,liT) Yl”,(B, 41, N,,I = l-
, r&
, x)
1 dm)
1
NLU).
in (28) or (29) modifying
the wavei30)
GM1.) = [g(~)(Nn,)-li’l~~n2(l.), tzr~,~"~12,1(1*)g2(1.),
(31)
where the function g(l*), which modifies the standard radial oscillat.or function CR ,f,(Y) at short dist)ancesI‘, has the following properties: Cdl.1M 0 for
g(O) = 0,
7’ 5 (1/1/S) Y, ,
and Yi7.1 = 1 for
1’% (l,‘i/)~
,
g(y) = 1 for
1’i
00.
(32)
Here ).Pis the radius of the hard core, and l’h is the so called healing distance. We must, now cht4i whct.her t,he transformation (30), (31) is a unit,ary one, i.e., whether the correlated w:lvefunctions 1nl~) satisfy t’he same orthonormality condit,ions as / nbn). This we shall do for the particular case of t,he I60 nucleus for which WCare going to perform the numerical c*alculations.In this caseme ha,ve no = nb = 0 and 1, , lb = 0, 1. These states lead to the following ) nlm) states of the relative two-nuc~leonmotion: 1OOO),I 01)71),10”112),1100). After intJroduc4ngt,he modifications defined by (30) and (31), we get --four normalized states I nlrl~) which are orthogonal to each other except’ for (000 / 100) # 0. This situation can be remedied in the following way. Instead of ala(r) = 6If’-7r-“4ha;4[g(,-i.)( N10)-112](1 _ 3$ )+-(h/2)r”, me inb’Oducc
slightly different, radial dependence:
= -__;; il +; NE, 00
and E = Nol/N,,
-
1. After this modification of the state Ii?%), we recover
TRANSVERSE
SUM
l“9-I
RULE
orthonormality of all the states involved in calculations-hence the unitaritjy of the kansformation. Let us notice that E is very small compared to unity. For the values of parameters chosen in our calculations presented below, 6 is of t,he order of magnitude of few percent. Now lel: us calculate t,he corrections t’o the correlation functions C,,(p) and C,,(q) . Performing t’he summations over the spin and isospin quantum numbers we get, (for more details see Appendix B) C,,(y)
= - &
(pup2+ ~4,~‘) C
(a(l)bi2)j
exp (i2/‘2qz)(
b(l)a(2)),
(33)
n.b
(3-1)
where a, b run over the occupied orbital states. k’ormulas ( 33) and (84) arc valid for the nwlei with the same number of neutrons and protons in both spin states. Using (28) and (39) we get for IfiO, after some straight,forward algebra (for more details see Appendix B),
AC,c(q?
- &
( pp2 + p,b’> [ 7A(OOO / exp ( i&qz)
1 000)
+ ; A(100 ( exp (i2/?q.z)!
AC’,,l’y)
ii2
6.5A(OOO / exp (i2/2qx)
/ 000) + ;A(100
( exp (i&qx)
- 6-“‘la (100 1 cxp ( &‘“qz)
100)
1
(3.5)
,
1100) / 000)
1 C.36)
In proving (35) and (36) and evaluating numerical results, we employ the following :simplifications. We neglect t,he differences A(. . . ) for all the terms
130
CZTA,
LEhAK,
.4SD
JIALECKI
whose integrands behave like Y~(LY 2 4) for small Y. In the caorrelakion function this amount’s to keeping the short-range relative s-states only. In such a way we extract t’he leading range correlation corrections. This procedure simplifies enormously. Xotice that, according t’o (IB), the correction to the term
case of the Cc,(q) correlations in the terms of the shortt,he computations
becomes
+ ; A(100 / p,’ 1 100) We
give the formulas
for A(nO0 1esp (id2y~) c,,, A(nZm 1exp (idsy!qx)p?
in Appendix
1 .
1~~‘00) and 1ndj?l)
B. 11’.
NIJMERICAL
RESULTS
AND
DISCUSSIOK
To compute the curves shown in Figs l-3 and t#he numbers of Tables I and II, we used the formula (13) corrected for short-range correlat#ions as described in the previous section. In calculating these short-range correlation corrections we did not take into account the corrections for the c.m. motion (the GartenhausSchwartz transformation). As we have pointed out already, these corrections are unimportant in the region of momentum kansfer y discussed in this paper. Besides, the most important correction comes from t#he C,,(y) correlation function which is not influenced by t’he Gartenhaus-Schwartz transformation. We use the following form of the g(r) function:
g(y)
= 1 - e--‘x+
(38)
where X = 0.36f-* [the Gaussian factor of Eq. (13)J. We performed complete calculations for y = 8. This choice of y makes g( (I/&) re)/g( ~4 ) = e? for f., = O..i6j, hence one may interpret rc as being a hard-core radius. [We should emphasize t#hat (38) does not define precisely a hard-core radius. For example, we might use equally well the relation g( 1/42)r,)/ g( a ) = 10-l to define the hard core. In this case, y = 8 would correspond
6-
to
Ye =
relation however
0.27~“. So, (38) introdwcs certain short range wrwlutions but t,hc of (38) to the hard-~YIIY intfxwtion is :mbiguc~us. Wz’c che~lied the convention g( (l/d: 1,~~‘)/g( % ) = P-I assuming
got (’ = 0.3;‘#$, which wrresponds npprminmtely, according to the above convention, to y = 14, the sanw corrcct~ions, as is shown in the last two columns of Table II.] In order to check the senwitivit,y of the results to the c*hoic*e of y :md cover a fairly wide range of mrrelations, we calculated also the Icading I erms of shortrange correlation corrections for y = 5 ir, = 0.711’) :tnd y = 14 (Y, = O.ay). The form (38) of I enables us to perform all the integrations malyticnlly. The resuks are presented in Figs. l-3 ant1 Tables I and II show the nurnrricxl results. md
132
FIG. 2. The C,,(q) correlation function. The curve marked C,,(p) correlation corrections [it is equal to the third term of (13)]. The three culated for y = 5, 8, 14, as indicated.
has no short-range c,, curves are cal-
FIG. 3. The C,,(q) correlation fllnction. The curve marked C,,(q) contains no short-range correlation and no c.m.-motion corrections [it is equal to the fourth term of (13)]. K’,,(p) the c.m.-motion correction [the sixth term of (13)]. $,,(y; -, = 8) is the C,,,,(q) correlation function corrected for short-range correlations for y = 8.
is
From Fig. 1 we can see that the hopes expressed in (1’) are basically confirmed. That is to say the dominance of the leading term (q”/‘>M”) (2~~~ + Npn2) is SO strong that even drastic changes of the correlation functions caused by nucleonnucleon correlat,ions do not matter very much. For instance, at q M l.$-’ the leading term contributes almost 100 70 and at q z 2f-’ about’ 95 % to the sum rule B(q).’ The significance of this dominance was discussed in (1) and here we 7 Let
IIS emphasize
that
there
is no point
of going
to a higher
q than,
say,
qz
2f-‘.
If
TRANSVERSE
SUM
TABLE ‘TSR
0.0
0.000
0.5 1.0 1.5 2.0
0.506 2.055 4 ,556 8.099
‘1 See Eq.
WITHOUT
0.191 0.191 0.191 0.191 0.191
L‘JHORT-RANGE
0.000 -0.359 -0.751 -0.089 -0.2i3
133
RULE
I
CORREL.\TION
-0.oti3 -0.061 -0.038 -0.011 -0.002
CORREWIONS
0.128 0.277 l.-l27
FOR
- 0 ,008 -0.008 -0.008 -0.00s -0.008
4.047
8.015
WL
-0.056 -0.032 -0.004 -0.001 0.000
0 ,064 0. .,I ‘W 1.415 4 ,038 8.007
(13). TABLE: SHORT
RANGE
CORRELATION
II TO TSR
CORRECTIONS
FOR
YP
g(r) = 1 - cup (-yXr?1
q in j 1 y=8
y=5
y = 14 ilB(q)/Bt:.s.(q)
AC..
0.0 0.5 1.0 1.5 2.0
1 the
(4)
o.ooil 0.007 0.138 0.355 0.397
I";)
~C,,(n)
-0.013 -0.015 -0.020 -0.021 -0.019
0.013 0.013 0.013 0.013 0.013
0.0 2.1 9.3 8.6 6.1
0.0 5.5 16.5 13.0 8.3
1~See Eqs. (35)-(37). Iu calculatiug the short range correlation e--y@ with y = 5 and y = 14, and for g(r) = 1 - esp [-&r-c)], signifkant~ corrections come only from C,,(q).
0.0 1.-l 4.8 4.5 3.7
~
,
correction we have
0.0 1.2 4.2 4.1 3 .Y for ~(1,) = assttmrd that
repeat some points for the sake of completeness.* The sum rule R(q) is, for large q, very sensitive to the effective magnetic moment parameters in the current, (:4), and if one could measure B(q) with good accuracy one would have an important piece of information about the magnetic properties of nucleons bound one performed measurements of H(g) at p > 2f-‘, one would have ito clear distinction bet,ween purely nuclear contributions to B(q), i.e., contributions from nuclear escit ations, and the contributions from meson production processes whirh start with a cohrrrttt cxlcctr~,production at w = WI, ]see (I)]. 8 The estimates of the corrections to the leading term iu Table I of (1) contain 1111. merical mistakes. This does not alter however the content, suggestions and rottclttsiotls of this paper.
in nuclei. ITor esainple, an effect of 5 “Y’ or 7 ‘; “quenching” of 1ll~gIwt i (’ nlonlen t s would prod1we :Ibout 10 “; or 14 “; ch:mgw of the slml rule. In order to (~hwl< the cwwnt ( 4) one should c*hoose the sirnplwt nuc*lci whew the c~orrclation funcations :w as sirnplc tts possible and contribuk as little :IS pOsSible ihence it, is enough to t&c into account only short.-range cw’rclation~ ) ; but once the current (1) is well c:hec~l~ed,it w-oultl be interesting to measure BC q) for nuclei whrre the so-culled roar pol:~rization effeds ( 13) :we imporbnnt~ and to look for t.heir influence 011 t,he C’,,( yj cwwlnt.icJn Eunct.ion. We w-odd like to look upon the future Inensurements of the sum rule B(q) as :t c~licc~l; (Jf the nwlear elecStrcmagneticcurrent (4). It, is therefore inlportmt to how how wnsitive are the results to the details of t)he model of the ground state used in our cd~d~tions. First of all we do not, know the extent of the sensitivity of t>heresults to the pnrticular choice of the shell rnodrl which forms :L basisof our c~alculnt~ions (e.g., variations of the shape of t)he nuclear average pot)ential, introdu&m Of spinorbit force etch).WC expect this to have :I very sndl influence but any rcason:~bly :wcur:tte numerid malysis uf these effects is beyond our computing facilities. Howcvcr, from the numerid results we do present.in this paper in Figs. l-3 and in Tables I and II, we mn draw the following ~onchwions. ( nl The most important cmwc~tion of the shorty-range correlntions t,o the leading tcrrn (‘onles from the C,,,,(q) correlation function. The wrredions to CPP(q) nud (2 ‘J/*1 (0 1c;” ejpfI 10) arc srnnll, as Table II and Fig. 3 show. l\Ioreuvw these two corrcct~ionshave opposite signs and cancel each other to n large extent. In &cd, they are negligible. This is the reason we calculated them only for the y = 8 cdaseand neglected them in the caseswhere y = 5 and y = 14. (b) There is :\ region of q, l.-lf-’ 2 q 5 3j-‘, where the B( q; y = 5), B(q; y = S), and B( q; y = 14) curves intcrscrt the st,rnight’ line of the lending term (q’,;L’II/“) (Zp,,’ + Npn’). It would seemtherefore t,hwt somewhere in this region there is virtually nothing but the lending-term contribution. This region is thus most.suitnblc for fut,ure prcrise measurementsof t.he TSR clevixed to obtain data (Jn the nl:ig[iet~ic inonients of bound nucleon% ( (2) Sinw C,,( q) can bc tspressed through nuc,leon-Ilu~leoIt correlation function (c*ompare Eq. ( 15) ), an at#tractivc possibilit’y opens of employing rlut,leoIl-rluc.leolJ c*orrelat,ion furl&ions ohtaincd from other experiments (or cnlculatiorls). Then there would be no need for calculating correlation corrections of spec3ic short-range correlat~ionsand the analysis would have virtually no free parameters eswpt the parnmetcrs of the nuclear electromagnetic current (4,. (~1) As CP.‘,,( qj = CPP(q; y = m ) [compare formula (38)], Fig. 3 suggests that / AC&; Y) 1increases as y decreases.AS A[(Z,lllr”) (0 / cf ejp;= 10)] = - AC&( (I = 0), t.he same conclusion is true for the y dependence of
TRAR‘SVERSE
SUM
RULE
1 3.;
1(2/5Z”l (0 1 C,” ejpi, 10)] 1. In fact, one should expect such an dependence on y from t,.he uncertaintjy principle. If we let 7 -:- 0 we make t,he single-nucleon orbit, bett’er defined spatially, and hence we introduce higher momentum components into its Fourier transform. I’inally let us stress a few more points which are import~nnt t,o keep in mind when comparing the results of the present calculations wit’h experimental findings. Thr current given by (4) is the simplest possible single-particle currcnt. One (*x1 add to (4) a lot of different corrections coming from the presence of the ot.her nuc~lcons in the nucleus, for instance the two- , three- , etc. body currems (which depend on t,he coordinates of two, t.hrec, et.c. nucleons), or 01ir ca11 consider the changes of the single-pnrt,icle current introduced by t,he surface cffcrts, etc. All these effect’s are very difficult t)o calculate. It, is, however, comparatively easy to estimat,e the relativistic corrections which one obtains from the free nucleon relatjivistic current by applying the Foldy~\~‘outhuyscrl t ransformatictn. Then one can reprcsent~ the electron-nucleon interact,ion as a power scrics of ,K’ (J/ being the nucleon mass). This was done through order JI-” in, c.g., (2), and the result is that only the charge-density operator has a correction (the DarwinIJoldy term) ; the current operator, however, has no relativistic corrections through order il-‘. The trouble is t,hat we do not know to what cstcnt t,his result is rclcvant to the nucleons immersed in nuclei, and how it mixes with all the other effects described above. In conclusion, it. seems reasonable t,o accept that, only the agreement with experiment can have a clear interpretation, namely, t,hut all the described above modifications of the current, (4 j are negligible. If thcrr is a definite disngrecmentj with experiment the interpretation mill prob:lhly not be unicme. RECEIVEI):
August
17, 1966 REFERENCES
1. W. CZY~, Bull. ;icad. Polon. Sci., Ser. Sci. Malh. Astr. Phys. 12, 649 (19&l). 2. K. W. MCCOY AND L. VAN HOVE, Phys. Rev. 125, 1034 (1962). S. R. H. PR,IT.L., J. 11. W~LECILL, END T. A. (;RIFFY, ~\~ucl. Phys. 64, 677 (19ti5). 4. W. Czxt, Bull. Acad. Polon. Sci., Ser. Sci. Maih. Asir. Phys. 13, 893 (lYli5i, W. Cj:~t, I’hys. 42, 07-l 18 i lM7) 1. L. LIG~XIAK, AND A. MALECKI [accompanying paper, ilnn. 5. S. G.IR'~ENI~.\US.IND C. SCHW.\RTB,P~~.S. Rev. 106,482 (1957). 6. L. L. FOLEY .%ND J. D. W.\LECI+ Sztovo Cimenlo 36, 125i (19M). 7. F. H. LEWIS, J. D. WALECKA, J. GOLDEMISERO, .1x1) W. C. B.\RI~ISK, I’h,t/s. Nrc. /.cllr’~ 10, 4!13 (1963). 8. J. GOLDEMBERC: AND W. C. BARBER, Phys. Rev. 134, 13963 (l!)(S). 9. J. H. LICVINGER, Phgs. Rev. 84, 43 (1951), K. C~OTTFRIED, AVrrcl. Phys. 5, 555 (1958). 10. J. DA PROVIUENCIA AND C.M.SH.IKIK,~~~. Phys. (N.Y.) 30,95 (lM4); .Yrtc/. Php.65, 54 (1965). See also F. I'ILI,.\Rs. Proc. Zn/e~n. School. I’hp. “ICurie Fertui”, C:cjr~rw 23 (1961). Il. XI. MOHHINSKY, ,Vucl. Ph?ys. 13, 101 (1959).
136
CZYi,
LEhIAK,
AiYI)
MALECKI
Princeton 12. A. It. EDMONDS, “Angular Momentum in Qllatltllm Mechanics,” Princeton, New Jersey, 1957. 13. E. BODENSTEUT ;\NI) J. D. ROGERS, Magnetic Momellts of Nuclear Excited tlu-bed Angular Correlations, North-Holland, Amsterdam, 19ti-I. APPENDIX
Univ. States,
Press, Fer-
A
In t,his appendix we give some details of calculat’ions leading from Eq. (8) to Eq. (13). We use t.he completely antisymmetrized oscillator-well shell-model ground-state wavefunction corkructed from the following single-particle wavefunctions: I a)) = I uua7a) = I a) I ua) I 7a), (AlI where / a,) and 17a) are the spin and isospin functions respectively, and I a) = I dma)
(Al”)
= I aL,l,(~)YZ,?n,(~, +>>,
where Yl, are spherical harmonics defined as in (22). 112
%a! R,t(?*)
=
[
r(n
+
2 +
Fi)
1
x(~+3'2)'21'lLfi+li2(xrz)
exp(
_ g$),
(A3)
I,,,k being the Laguerre polynomials. The energy corresponding to this wavefunkon is E,l = (2n + 1 + ~$)fiw,
x = ivlw.
(A4)
Firstly, let us rewrite the matrix elements of the last two terms of (8) :
and
+ $ (0‘\
2 2 2 C?i’(zJ-zk)ej ekplz flnz O‘\ . j#k,Z.mk#Z,mZ#m /
In order to work out B,.s.(q) we have to evaluate eight different matrix elements. One can express them through one-particle matrix elements as follows
TRAljSVERSE
SUM
RULE
137
n-here Ly, /3 ... run over all (neutron and proton) occupied states, and (oc), i/3) . . . run over the occupied states of protons. In the case of 160 WC get the following results:
p2i2Xe
JI, = 2 From
>
(A7) we immedint’ely
get, (13). BPPENDIS
B
In this appendix we give some details of calculations of Sect,ion III, connected with int,roducing short-range correlations int,o t,he ground-state expectat’ion values. A two-particle operator fl = ,z w,
(Bl)
k)
can be expressedthrough creat’ion and annihilation operators as follows: B=
c (a(l>P(2) abY8
1O(1, 2) / y(l)&(2))akaa+as a,,
iB2)
where CY,p . . . specify the quant’um numbers of the single-particle states. Since in the ground state 10) all single-particle states are filled up to certain state (corresponding to t’he Fermi momentum of t’he degenerate Fermi gas), we get (0 I C o(j, k)l 0) = l&l8 MM2)l j#k = s
(apt
O(1, ?)I a(l)f?(2!)
0(1,2)1 -dl)Wj(o -2
(a(l)P(2)[
I G+Q+U~ay I 0) 0(1,2)~p(lM:!,),
(H3)
where the summation extends over all occupied one-parkle stat’es. If we want, to introduce only the two-particle correlations, the unitary operator u acts ouly on a nucleon-nucleon pair, hence it. modifies the two-nucleon states j ty@) [see (IO)]. Thus we have
(0 I J#k c Nj, k)l 0) -+ (0 I 6 z w, km I 0) = 5 (41M2)l UT,‘NL ~ha I 4lM2)) - 2 (a(l)P(2)l u~;o(l, = 5 (a(ljP(2jl O(l, 2)l n(lW~)
2h2
I /3(lht4)
- 5 b(~)PW O(1, YJl P(lML’)),
(B-L)
mherc 1 a7) denotes the two-particle correlated st’ate. The summation over spin and isospin quantum numbers for C,,(q) correlnt ion fuwtions is performed a:’ follows:
and C,,( qj
where a. b run over the awupied states (AZ). The formulas (B5) and (B6 j are v:tlid for nuclei xvith the same number of neutrons and protons in bot,h spin stntrs. X\TUIVxc give wmc details of calculations leading t’o (35) and (36). Let us start with (‘,,(q) (*orrelation function. Using (38) and (29), we obtain
140 In proving
CZYi,
LEh-IAK,
ANIl
M.4LEChI
(B7) we employed t’ho relations
where 6(1,&X) = 1 for l,lJ, satisfying the triangle relation, and zero otherwise. For the I60 nucleus, where n, = ?&b = 0, I, , 16= 0, 1, WCcan put, beenuseof (24) and (25), I = I’ and n = n’. So, for ‘“0 we have
Then we use t,he simplifying approximation that, A{nlm
1 exp (id5q.z)
1nlm) # 0
only
for
1 = 0,
@lo)
where A( . . . ) denot,esthe difference between correlated and uncorrelated magnt.udes. Thus, we assumethe presenceof short-range correlations in the relat,ive s-&a& only. Employing (B9) and (BlO) we get for I60 [the squares of the RIoshinsky coefficients are either 1 or f’2 in our case]
. A(nO0 / exp ( id2
qx) j nO0) Wl)
= - L9
hi
•k ~1,‘) [7A(OOO 1 exp ( i& + !,,A(100
qx) / 000)
j exp (ifi
qz) / loo)].
The formula (36) for C&p) correlation function is obtained as follows. First of all, the calculations proceed differentIy for the pa&s of C,,(q) containing P,’ and those containing p: [see (BS)]. Let us denote these parts C:;(q) and C:;(q), respectively. Applying (28) and (29) we obtain
(hL’6f
1tL’$)
{d, NL, k [ da, n& , A) (n’i, N’L’, x 1?1&,, n& ,A)
(NLAI I P: 1N’L’M)
(nh j exp ( id$qz)
( Bl:! 1
I n’l’m).
In proving (B12) we employed t’he propert’ies of Clebsch-Gordan and Moshinsky coeffkients and the relations
TRAKSVERSE
(nlm / exp(+&qz)
SUM
1 ~‘Z’UZ’) = (nlm 1exp(i2/2qx
(NLM I P,2 I N’L’M’)
141
RULE
# 0 only for
/ n’l’m’)
S,,,\
1 L - L’ even. J
(B13)
We use now the simplifying approximat,ion that A(nh
1exp(i-\/“qx)
I ~z’l’m) # 0 only for
(1314)
I = 1’ = 0.
As 1 - 1’ in (Bl2) can be an cvcn number only this simplification amounts to neglecting the same kind of terms as in (BlO). Icrom (B12) and (B14) we get for ‘“0:
(NhO I P” / N’XO) A(nO0 I exp (idsqx)
/ n’OO),
where P’* = P,” + Pyz + P,*. The only matrix elements needed in (Bl.5) are (010 I P” 1010) = 3; x,
(000 / P2 I 000) = 3i x,
(020 1P” I 020) = 7,; x,
(100 / PZ 1100) = TJ:!h,
CB16)
(000 / P” / 100) = (>$P)X. Finally we get AC$‘i(q:l = g2
[
6.5A(OOOI exp i i2/?qr) 1000)
+ i A(100 1exp (i2/qx)
1100) - s(i A(100 I exp (idsq(x)
1
(B17)
1000) .
In a very similar fashion as in proving (B9), we get for 160 cb2;(q j = - &
C
l,.Z,,=O,l
C
C
hZL?LN
pv:M
[2 - ( - ijz] (hm
/ zLxpj2
. 1711, NL, X 101,) Oh, X}“(nlm 1exp ( id2qz)pz2 I nh) 7 (000 1exp (i~?qz)&
/ 000)
i Bls)
+ ;1,(100 I exp (i&qdp:
j 100).
142
CZYi,
LEkNlAK,
ASD
.\IALECKl
According to (Bll), (Bl7),:mcl (B18), in order to evalu:~tcAC,,(rl)nrld AC,,(q) lye must work out A(nO0 1exp ( idL)qz) / ~‘00) and c.? A(nl~ 1 cxp ( i2/!~lz)p,~’ ;~ll~r\. One can carry out t,he angle int’egrals employing the relations
-0
+ 1(1 + 1) j’- flT iF12,1(,,) j”(4).) [
- ;.j2&.)
1i
(I3201
, )
where I, are spherical Bessel functions. Applying the simplifications (BlO) and (B14) we extract the leading terms of the short-range correlation corrections. In order to proceed consistently, in the radial integrals calculation, we neglect the differences A(. . . ) for all the terms whose integrands behave like ra (a 2 4) for small I’. In such a way we exkact the leading terms. One can expect that t’he contributions of terms wit)h a~= 4 and (Y = 6 (there are only even powers of T) are equal to about (NoI - 1) ! (No0 - 1) and ( No2 - l)/(N, - 1) times the leading terms, respectively. For q(r) = 1 - exp (- 8Xr*), we get irV,, - l)/(N,,c, - 1) = 0.124and (N, - I)/ (No,, - 1 ) = 0.015. IJsing this procedure we get A(000 j exp ( 1’2/2qxj ( 000j = 4 -l)l’X:%% x A(100 / cxp (iz/?qx)l
100)
A(100 j exp (i&qz)(
000)
1
g”c - 1 ?j”(~~I~), Noo
(B21)
TRASSVERSE
SUM
RULE
143
OF
PHYSICS:
42,
119-143
(1967)
Transverse Sum Rule for with Large Momentum W. Insl,ylui
Czui, Fizyki
Electrons Transfers
L. LE~IAK, Jadwxuej,
Krakdw,
Scattered Inelastically from Light Nuclei A. ~~IA~EcK: Rronowice,
Poland
A isurn rule for the cross sections of electrons inelastically scattered from nuclei is analyzed. This sum rule contains contriblltions only from the transverse electron-nucleus interactions and is particularly suitable for analyzing the experiments of inelastic electron scattering at 0 = 180”. The sum rule is a fnnciion of the three-momentum transfer q. A discussion of this sum rule for momentum transfers 0.5f-1 5 q 5 2f-1 is presented. Strong dominance of the spiwcurrent contribution seems to offer a very atkactive possibility of measuring magnetic moments of nucleons bound in nuclei. The paper contains estimates of the short-range nucleon-nucleon correlation contributions to the sum rlile. I. INTRODUCTION
There are many ways of analyzing the inelastic electron scattering from nuclei. One of them is to measure experimentally the sums &a( 4, 8, U) and compare it with theoretical predictions. Here ~(4, 8, w) is the cross sect’ion for inelastically scattered eleckons with fixed three-momentum transfer 4, scattering angle 8, and energy loss w. In the sum over w we include the elastic scattering contribution (w = 0). We can, in the one-photon exchange approximation (t’he first Born approximation), always write the sum in the “canonical” form [see, e.g., (I)]: q
m 4% I‘ 0
dw
w, 0)
= s
[A(q)
I f(up”)12(e2/W2
+ B(q)
tan” ?$I
(1)
in which one separates the contributions from t,he longitudinal plus transverse elecbromagnet,ic interactions [the term with A ( q)] and the t’ransverse interact#ion alone [t,he term with B(q)]. Henceforth we shall call B(q) the transverse sum, and the various theoret’ical expressions obtained for it later the transverse sum rules (TSR). In Eq. (1)) f( q,,’ = q’ - w”) is t’he nucleon form factor related t’o the conventional nucleon form factors as followsf( q,“) = Fl,( qp2) = Fx,( qF2) = FPn(pC2), F1,( qp2) assumed to be zero; pi is the moment’um of the incident electron. A(q) and B(q) depend only on the absolute value of 1q 1 because one 119
120
CZTk,
LEhIAK,
.4iTIl
MALECKI
averages over the initial and sums over the final nuclear stnt,es compatible with the prescribed momentum transfer and energy loss. A (a) and B(q) can be cxpressed through the matrix elements of the nuclear charge and current operators [see, e.g., (I) and (g)] as folIows:
+
(9)
+
(@J:,,)(@*Jno) (“-“;,
“):’ 7 c2)
R where Qno= (n / Q 1O), Jno = (n 1J I 0) are the matrix elements between the ground and excited states, Q, J are the charge and current operators of the target nucleus and 4 = q/a; w,,o= E, - Eo is the excitation energy of the nucleus. The complete sum rule (1) is rather complicated and very many terms have to be evaluated in order to get numbers for comparison with the experimental data. It is therefore rather difficult to extract information about nuclear structure since the calculated numbers are produced by many effects of different physical origins which partly cancel each other. In other words, too many different effects contribute to (1) to let us decipher in a reasonably unique way the information contained in it. It is tempting therefore to analyze certain pieces of (1) which can be extracted experimentally and which contain more specific information about nuclear structure. First of all let’ us recall that the TSR can be measured directly. This can be done either by experimentally performing the Q = const, B = 180” sum for a(q, 8, w) by varying pi and the final-electron momentum, or by doing the complete sum for two different angles 0, but the same q. Then the slope of A(q) + B(q) tan” s0 vs tan” x0 gives us B(q). In fact, the first method has already been used for extracting B(q) both in the high- and the low-~ limit,.’ In this paper we limit ourselves to analysis of the B(q) term only. In Se&ion II we discussgeneral properties of the transverse sum rule. In Section III we evaluate TSR for the oscillator potential well shell model ground state of the 160 nucleus. Then we present a method of calculating corrections due to the nucleon-nucleon shortrange correlations which consists in modifying the relative two-nucleon wavefunctions at short internucleon distances. Section IV contains the numerical results and a discussion of their significance. Before concluding this Chapter let us point out that, although we do include 1 The measurements rateur Linkaire, Orsay. for informing us about in the measurements of paper.
of the high-q limit of R(p) are being done at Laboratoire de L’Accele(We are grateful to Professor Jo& Goldemberg and Dr. U. Isab6116 the measurements.) The low-p limit of B(q) was obtained essentially the giant dipole resonance and we discuss this problem in a separate
TRAKSVERSE
SUM
1“ 1
RULE
in principle the elastic scattering in our sum rule B(q), it turns out. to be immnteCal as the TSR seems to be most promising in cases of simplest nuclei with no magneti(a moment’s in t’he ground state, in which cast there will be no contrihution from t,he elastic scattering.’ The choice of spin-zero nuclei is again dictated by nur desire to extract few but well-defined pieces of information about ~luclenr strwt’ure, and avoid the embawas de richesse one gets into when dmling with c*oniplic:lt cd nuclei. II.
GENERAL
PROPERTIES
OF
TSR
In order to have a general idea as to how B(y) changes with y let us analyze it, assuming t,hat the final states onesums over in (3) form a complete set of states. Then, applying closure, one get.s WY)
(3)
= (0 I iJ+*J - (kJ+)(4.J)l 1%
where / 01 is the ground-state vector. We accept the nonrelativistic operator .J (without, exchange currents present) : Cpj eiq’ri + eiq”lpj)
+ 5
where IV is the nucleon mass, ej = >$[l + am], of t,he jth nucleon. From (3) and (4) we get
i(dj
form of the
x q)e’q”j
,
(,-I 1
and pi is t#he magnetic
moment
(6) +
;
lo
/ 2
pi~~(;J-“t)p.
2
1 O\\ I
\
ek
pj
problem
of elastic
scat,tering
I,h,
1-
at 0 = 150” is disrllssed
,’
9
/
where we chose q along the x axis. As B(y) depends only on the q, we can choose its direction as we please. In (6) we separated terms. The form (6) is more convenient for analyzing B(y) exhibits e:xplicitly the first’ leading term. Notice that (5) and the p-p and c-g correlations; the u-p correlations are absent. ? The
.c
3211
ill detail
absolute value of out the diagonal for large y, as it, (6) contain only in Ref.
(4).
in Eqs. 1.5) and (cij, that (0 1 cfk esp [1’~(x, - +zt11. e,e,,-p,,pl;, 10) = (0 j C$ esp - zkj]e,ekp,,pA, 1O), and similar rcl:ltions wit,h eJpl, ----) pygjr and -k .] The SL~II rule (5) or (6) contains contributions from some spurious cscitatiorw of the wntcr-of-mass (c.m.) degrees of freedom. Thew spurious contributions are, in general, small-A times smaller than the other term-but, for small q they become wry import.ant, (4). We can remove them employing the C;artenhnun-Sc.llw:trtz transformation (5) which in our case reduces t.o [We
:lSSLllll~,
[iy(Zj
ejpj,
/ljUjy
rl + rl - R,
pj + pj -
where R is the c.m. vector and P = c;” few simple niarlipulat,ions~
PK1,
(~71
pj . Instead of (6), we get now (after n
4 ej e,(P, A-’
-
pj.z
- /)Br)Pz 10’ I /’
[In obtaining expression (8) we employ the following symmetry of the ground state: (0 1piIPs / 0) = (0 1p&‘, / 0); ,i, Z-arbitrary. Hence il(O I PjzrPz IO) = (0 1CC: pjx)Px 10) = (0 I P,” I Oj.1 Before discussing more quantitatively the properties of the sum rule H(q), let us ment’ion some of its general properties. Let us (aall C,,(q) t,he spin-spin correlation fun&on [t,he third term of (S)] and CL,(y) = C,,(q) + AC,,(g) the c.m.motion-vorrectctl momentum-momentum correlation function [the fourth and the last term of (8) .“I They both do not) depend on the direction of q; hence
But for finite nuclei, the spatial correlation
functions
G<,(Y), G,,(V)
should
be
3 We shall see flIrther that there exist strong arguments that the correction for the cm. 1s negligible for the region of q considered in motion in C,‘,(p) [which we denote by K’,,(q)] this paper. So, in calculat,ions of the short-range correlation corrections we shall drop it and call Cr,,(q) the correla.tiun function [the fourth term of Eq. (S)].
TRBXSVEHSE
SUM
RULE
123
hounded for 3’ -+ * by an esponent.ial function e+’ \&eye 6 = ( ~:j[&Bjfi~‘)‘iu, EB being the binding energy of the most weakly bound I~UC~COII. For I:‘, e 8 ;\IeV (which is the (base for, e.g., 160), p corresponds to a niomc~ntum of :~bollt 1’20 AleV/c. Hence Co,(q) and (Y&,(q) are analytic in t,hc strip y = - $3, y = fib in the clomplex y-plane.4 Since they can depend only on y = ( q 1, U-P come to the c*onclusion t,hat CbO(~), C’,,(q), and hence R(y) cm be reprcscntcd by an absolutely convergent power series of y, only with even powers of y present, and the radius of convergence not smaller than 120 MeV/c, R(q)
= R(O)
+ Py”
+ ft4’y4 + . . . .
(IO)
It seems nit.hin t.he go with y, yg _ w:! =
that one can arrange esperiment,s which produce B:( (/I cwmfort:~blp radius of convergence. It is not clear however how far clonn OIW (x11 because for very small y the closure approximation brc&s down :ts 4pipf sin” !28; hence W’ 5 y’ , and for a fixed small Q, we can wvcr in the exper.iment only a small part of the excit,ation spectrum. There is a possibility of avoiding this trouble by splitting the series (10) into wntrihutions of differcntj multipoles. Then one may hope that specific multipolc cw~tributions arc concent,rat,ed at lowenough excitation energies and even for I~)x (1 the cffecstively coniplet,e set of states is contained in t,he comparatively lo\\-w cwitatiorr spectrum cwwecl in experiment,. This approach demands, from experimc~nt, ident,ificstion of t’hc levels excited--in order that the sum rule hc rcw~lvetl into the wntri butions of different multipoles. This is possible to obt:lin, for some multipolcs at least,. l’or instance, the measurement,s of the 31 ewitntions in the gi:~nt~-dipo’le-resorlan(.e energy region meet these recluiremcnts [FW, e.g., ( 7,~ and (S)]. A preliminary theoretical analysis in this spirit has also hccn tried and the result:; look encouraging (4 1. In fact, for low y, R(y) will bc titrongly clomnatetl by l;hc El transitions. Indeed, if we put y = 0 in (8), \YP get (II) whew f?j’ = N/A for protons; P,’ = -Z/A nuclei with Z = N = ,I z-4, WC have
for neutrons.
where rEI(w) is the El absorpt)ion cross serti(Jn (witjhout complete nwlear excitation spectrum is experimentally
(jr, if we consider the
wtnrdation). availabk
So, if the [hence y is
kept large in t,hf measurements of R(q)], H(O) is proportional to the cnergyweight,ed dipole sum rule-quite :m interesting quantit,y because of it,s ,~nsitivity to t’he short-range nuclear (especially neutron-proton) correlations. It seems, however, very doubtful whether one will be able to extrapolate R(O) from the measurements of R(q) with large 4. In t,his case, one is so far from the beginning of the R(p) curve that any reliable extrapolation looks rather impossible. It’ is perhaps worthwhile to point out, Ohat the measurements of the so-c*:dled in&&c form factors of the giant’ dipole resonance [see (7), (8), and also (,$)I, where one ran easily extrapolate B(p) to R(O), do not produce B(O) as given by Kq. (12). They contain only these excited stat’es (n ( which belong to the giant dipole resonance multiplet; so, if we extrapolate the inelastic form fxcators to 4 = 0 we get only a part of (13). For example, we miss completely the high-energy part, of the dipole absorption which rest& in production of correlated neutrorlPproton pairs (see, e.g., (9) 1. l~inally, one may also notice that only even powers of Q appear in (10) and this is a consequence of finiteness of nuclei [which is equivalent to t’he assumption that G(V) functions are bounded by an exponent#ial e-“‘I. This fact may help in cst,ablishing regions of validity of some nuclear models which produce odd powers of y in the expansion of R( 4). For instance the Fermi-gas model produces terms linear in 4; hence, ab least for small g, is rather inadequate and may introducee serious dist)ortions in the Q dependence of B(q). III.
IS\-ALGATION
OF
TSR
It is caomparatively easy to evaluate (8) with the shell-model [oscillatorpotential] ground-stat’e wavefunction. For instance, for the I60 nucleus, the completely nntisymmet,rized shell-model ground-state wavefunction gives ( for more det’uils see Appendix A)
where X is the parameter of the Gaussian factor exp ( -$zh?) of the oscillat,orpotent’ial wavefunctions. [I+kom elastic scattering of electrons from lfiO, one gets X z 0.36j’-‘.] The first two t,erms are the “diagonal” terms of (6)) the third and fourth are C,,( 4) and C,,(q) correlation functions of (6), the fifth and sixth terms are correctSions due to subtraction of the c.m. motion.5 They are small for B Notice that our results larger than the same term
for u-u correlat,ion in Eq. (22) of (2).
function
[the
third
term
of (13) ] is four
times
TIt.4XSVEltSE
SUM
1tULE
12.5
q 2 1 j-l, but for small y, they are very important. For instance, in the limit q + 0, thaw is :L 100 5 difference between B(q) with and wit.hout suhtr:wtion of the Cm. rll~JtiOll: from (6), B(O) = 8x/L122, from (8) H(O) = 4k/jr”. This f:lPt is essential in discussion of the lowy limit of B(q) [see (/t)]. Figure 1 ShOWS th(x plot of ( 12 1. 0nc should cwnsider the mnrly corrections to formula (13) before applying it wnfidentl,y to the analysis of the experimentally me>~suretl sum rules. I:irst of all, we shall not worry for the moment about the CXJrrertiOW to the nwlwr clwtromagnetic current (,4) ; this problem is tliscusscd in tho last Section. As wc ewluate sum rules which include summation Over ;1 host, Of very wmplicntcd cxcitat,ion;-;, the details of the shell model we ncwpt as 0111’ starting p(Jint [e.g., the spin-orbit8 coupling, the shape of the average potential] stem not to he important. A much more essential point> is to estimate t,hc effects of t.hc nuclco~lnucleon wrrclatjions; the wavelengt’h cSorrcsponding to q-values considered in t’his paper is small enough to show some det,ails of the fluctuations of nucleous around th(lir average shell-model orbits. This is a very complic~ntcd problem which deserves an exknded and thorough invkgntion. In the ws;t of c~alculttting B(q), however, we shall limit ourselves to some simple ost imntcs hccnuse the modclindependent term of (8) (the first one) is strongly domin:king for 1:wge y and there is no need for wry great’ accuracy. We shall ncglwt henceforth the NJiTCF tions of the (Am-motion subtraction, and so we should &&ate the c*orr&ion of short-range correlat’ions to C&q), C’,,(y) and (2,‘dl”)(O 1 c: e,p& / 0). To estimate C,,i y ) , one can USC the rluc~lcorl-nuclec,n IJair correlation function
The trouble is, however, that C,,i y ) and ( ?/‘A/“)(0 1 c: eipSI j 0) c!:tnnot b(h expressed through p( 1 , 2). So, RX’ prefer to use a method whirh c’xn be applied both for C,, and C,, . WC use the method of modifying the relative two-nucleon wavefunctions at short internucleon distances, employed, e.g., in ( 20). E’irst of all let us notice t,hat A(T,,(q
= 0) = A
where b.y A( . .) WC denote the correction
to ( . . . ) due to short-range
correla-
tions. Equation ( IA) says that we 1m~1 to know only C,,( r]l‘i :tntl clPP( 4) and the corrections to the c+onstant krm (2,‘;11’)(0 1 C;’ CjpS, 10) follow from the (‘orr&ions to C&q). J$‘c can see it, as follows:
where the summation in the right-hand side (r.h.s.) extends over t,he protons only. We USC [after Noshinky ill )] the following transformations of a pair of nucleons to their cm. and rclntivc coordinates: r = (1/2/:!)(r] p = !l/&)!Pl
- r2), - p2),
R = (l/g3)(r, P = ui’dwp1
+ r?J,\ + pd./
(18)
From (17 j and (18‘) WC see that if, in ( 17), we modify the relative motion of the j, 1; pair [in a unitary way: preserving normalizations and orthogonalities], the 1.11.~.of (17) does not, change; hence (16) follows. Let. us nom describe briefly t.he method of calculating short-range correlation chorrections. Consider t,he matrix clement (%J 1 (3 1q), where \Ir is a completeIy antisymmetrizcd shell-model wavefunction and 0 a two-part’icle operator, i.e., o = cj,, Oij, 1;). We introducac the correlations into the wavefunct’ion cmploying a unikvy operator U I +) = ‘u I w,
‘U+ zz ‘u-1
(19)
The unitary of U guarantees us t,he correct, normalization of the modified function. Taking into account only t,hc two nucleon correlations we can write the correlated matrix element as follows [see (10) and Appendix B] (\I,\ 01%) = (\lr(u-lmlI\k) = 5 [(cY(l)P(2) 1UPO!l, 2)U 1cu(l)P(2)) - (cr(l)P(‘L) / u-‘0(1,2)% = $ GCG@j
- b(lW)
/ /ql,Cx(“))]
I 0~1, 3 I a(1j~(2))
I (xl, 2) I P(Ik42)L
where CX,~arc complete sets of quantum numbers of single-particle states, and the summation rst,cnds over all occupied states. In (20) we have introduced the c~orrelatetl two-purticalc wavefunct’ions:
The incIic,es a, p . cont,ain, in :Idditiou to the spatiul cluarltum numhcrs, the spin and isospin quantum numbers. The summation owr the spin and isospiu quantum numbers can be easily performed as our simple oscillator potential model ( see Appendix B) does not introduw any spill- or isospitl-depcnclcnt couplings. The single-particle spatial c~u:tntum numbers IVY tlcnott 1)~ a, 0 . . , and in all the following formulas, the snmmntic~n ovw spin and isospin q~~antum numbers (see Appendix B) has been pcrformcd. The twwpartic*lc state iti’
Iiow > following Aloshinsky nucleon-nucleon pair:
(ii),
wc go ovc’r to t,he c.m. :tnd rclut,ivc mot,ions of a
where (n, I) are the quantum numbers of relative motion and (N, I, ) are quantum numbers of the cm. motion. The relative and the cm. coordinates defined in ( 18). Because of energy and orbital momentum conservation, 1\4oshinsky coefficients in (2X) arc different from zero if the following relations satisfied (11) : “?ll + 11+ 2?1?+ I, = 2% + 1 + 2N + I,,
124)
11 + 1, = 1 + L = a. These coefficients arc real and have the following
the arc the are
(2.5)
symmetry
properties
(11) :
(nl, NL, 1\ / ~11 , n& 1 A) = i - 1) ‘-‘( nl, NI,,, x / ~olz , ,&I , A] = ( - 1) “I-“{NL,
721,
X j nIlI , ,n:& , Xi
= i - 1j ‘I+“{ NJ+ nl, X 1&
(-“(ij
, nlll
, X).
12s I n1, NL, xp) = rz (Zdd
/ ILXp) I Edna)1NLJI),
(271
we have from (22), (23)) and ( 27) I ab) =
;< c (1,711,67nb j l,i*Xp)(lr~~L‘u ; ZLXp) hpinllII1Nl, . jnl, NL,
(28)
x / n,l,L , nblb , A} j nh)
/ NLM).
From (28) and the symmeky relations (26) WC:get 6 We we Edmonds (12) convenliwls for monics and the angular momentum algebra.
the
Clehsch-Gordon
cclcfliciellts,
spherical
har-
12s 1ba) =
CZTt,
C
C
Xp,~.tf
nlNL
LEhIAK,
ANI)
( - 1) 2(Z,~)~,,lh~7~6 1 l,lbX~j(ZmLIJ
1 I&L) i'28)
.j,/ll,
NL,
x
/ n,l,
We introduce the short-range corrclut,ions func%ions of the relat,ive motion I nh)
MhLECKI
= R.,liT) Yl”,(B, 41, N,,I = l-
, r&
, x)
1 dm)
1
NLU).
in (28) or (29) modifying
the wavei30)
GM1.) = [g(~)(Nn,)-li’l~~n2(l.), tzr~,~"~12,1(1*)g2(1.),
(31)
where the function g(l*), which modifies the standard radial oscillat.or function CR ,f,(Y) at short dist)ancesI‘, has the following properties: Cdl.1M 0 for
g(O) = 0,
7’ 5 (1/1/S) Y, ,
and Yi7.1 = 1 for
1’% (l,‘i/)~
,
g(y) = 1 for
1’i
00.
(32)
Here ).Pis the radius of the hard core, and l’h is the so called healing distance. We must, now cht4i whct.her t,he transformation (30), (31) is a unit,ary one, i.e., whether the correlated w:lvefunctions 1nl~) satisfy t’he same orthonormality condit,ions as / nbn). This we shall do for the particular case of t,he I60 nucleus for which WCare going to perform the numerical c*alculations.In this caseme ha,ve no = nb = 0 and 1, , lb = 0, 1. These states lead to the following ) nlm) states of the relative two-nuc~leonmotion: 1OOO),I 01)71),10”112),1100). After intJroduc4ngt,he modifications defined by (30) and (31), we get --four normalized states I nlrl~) which are orthogonal to each other except’ for (000 / 100) # 0. This situation can be remedied in the following way. Instead of ala(r) = 6If’-7r-“4ha;4[g(,-i.)( N10)-112](1 _ 3$ )+-(h/2)r”, me inb’Oducc
slightly different, radial dependence:
= -__;; il +; NE, 00
and E = Nol/N,,
-
1. After this modification of the state Ii?%), we recover
TRANSVERSE
SUM
l“9-I
RULE
orthonormality of all the states involved in calculations-hence the unitaritjy of the kansformation. Let us notice that E is very small compared to unity. For the values of parameters chosen in our calculations presented below, 6 is of t,he order of magnitude of few percent. Now lel: us calculate t,he corrections t’o the correlation functions C,,(p) and C,,(q) . Performing t’he summations over the spin and isospin quantum numbers we get, (for more details see Appendix B) C,,(y)
= - &
(pup2+ ~4,~‘) C
(a(l)bi2)j
exp (i2/‘2qz)(
b(l)a(2)),
(33)
n.b
(3-1)
where a, b run over the occupied orbital states. k’ormulas ( 33) and (84) arc valid for the nwlei with the same number of neutrons and protons in both spin states. Using (28) and (39) we get for IfiO, after some straight,forward algebra (for more details see Appendix B),
AC,c(q?
- &
( pp2 + p,b’> [ 7A(OOO / exp ( i&qz)
1 000)
+ ; A(100 ( exp (i2/?q.z)!
AC’,,l’y)
ii2
6.5A(OOO / exp (i2/2qx)
/ 000) + ;A(100
( exp (i&qx)
- 6-“‘la (100 1 cxp ( &‘“qz)
100)
1
(3.5)
,
1100) / 000)
1 C.36)
In proving (35) and (36) and evaluating numerical results, we employ the following :simplifications. We neglect t,he differences A(. . . ) for all the terms
130
CZTA,
LEhAK,
.4SD
JIALECKI
whose integrands behave like Y~(LY 2 4) for small Y. In the caorrelakion function this amount’s to keeping the short-range relative s-states only. In such a way we extract t’he leading range correlation corrections. This procedure simplifies enormously. Xotice that, according t’o (IB), the correction to the term
case of the Cc,(q) correlations in the terms of the shortt,he computations
becomes
+ ; A(100 / p,’ 1 100) We
give the formulas
for A(nO0 1esp (id2y~) c,,, A(nZm 1exp (idsy!qx)p?
in Appendix
1 .
1~~‘00) and 1ndj?l)
B. 11’.
NIJMERICAL
RESULTS
AND
DISCUSSIOK
To compute the curves shown in Figs l-3 and t#he numbers of Tables I and II, we used the formula (13) corrected for short-range correlat#ions as described in the previous section. In calculating these short-range correlation corrections we did not take into account the corrections for the c.m. motion (the GartenhausSchwartz transformation). As we have pointed out already, these corrections are unimportant in the region of momentum kansfer y discussed in this paper. Besides, the most important correction comes from t#he C,,(y) correlation function which is not influenced by t’he Gartenhaus-Schwartz transformation. We use the following form of the g(r) function:
g(y)
= 1 - e--‘x+
(38)
where X = 0.36f-* [the Gaussian factor of Eq. (13)J. We performed complete calculations for y = 8. This choice of y makes g( (I/&) re)/g( ~4 ) = e? for f., = O..i6j, hence one may interpret rc as being a hard-core radius. [We should emphasize t#hat (38) does not define precisely a hard-core radius. For example, we might use equally well the relation g( 1/42)r,)/ g( a ) = 10-l to define the hard core. In this case, y = 8 would correspond
6-
to
Ye =
relation however
0.27~“. So, (38) introdwcs certain short range wrwlutions but t,hc of (38) to the hard-~YIIY intfxwtion is :mbiguc~us. Wz’c che~lied the convention g( (l/d: 1,~~‘)/g( % ) = P-I assuming
got (’ = 0.3;‘#$, which wrresponds npprminmtely, according to the above convention, to y = 14, the sanw corrcct~ions, as is shown in the last two columns of Table II.] In order to check the senwitivit,y of the results to the c*hoic*e of y :md cover a fairly wide range of mrrelations, we calculated also the Icading I erms of shortrange correlation corrections for y = 5 ir, = 0.711’) :tnd y = 14 (Y, = O.ay). The form (38) of I enables us to perform all the integrations malyticnlly. The resuks are presented in Figs. l-3 ant1 Tables I and II show the nurnrricxl results. md
132
FIG. 2. The C,,(q) correlation function. The curve marked C,,(p) correlation corrections [it is equal to the third term of (13)]. The three culated for y = 5, 8, 14, as indicated.
has no short-range c,, curves are cal-
FIG. 3. The C,,(q) correlation fllnction. The curve marked C,,(q) contains no short-range correlation and no c.m.-motion corrections [it is equal to the fourth term of (13)]. K’,,(p) the c.m.-motion correction [the sixth term of (13)]. $,,(y; -, = 8) is the C,,,,(q) correlation function corrected for short-range correlations for y = 8.
is
From Fig. 1 we can see that the hopes expressed in (1’) are basically confirmed. That is to say the dominance of the leading term (q”/‘>M”) (2~~~ + Npn2) is SO strong that even drastic changes of the correlation functions caused by nucleonnucleon correlat,ions do not matter very much. For instance, at q M l.$-’ the leading term contributes almost 100 70 and at q z 2f-’ about’ 95 % to the sum rule B(q).’ The significance of this dominance was discussed in (1) and here we 7 Let
IIS emphasize
that
there
is no point
of going
to a higher
q than,
say,
qz
2f-‘.
If
TRANSVERSE
SUM
TABLE ‘TSR
0.0
0.000
0.5 1.0 1.5 2.0
0.506 2.055 4 ,556 8.099
‘1 See Eq.
WITHOUT
0.191 0.191 0.191 0.191 0.191
L‘JHORT-RANGE
0.000 -0.359 -0.751 -0.089 -0.2i3
133
RULE
I
CORREL.\TION
-0.oti3 -0.061 -0.038 -0.011 -0.002
CORREWIONS
0.128 0.277 l.-l27
FOR
- 0 ,008 -0.008 -0.008 -0.00s -0.008
4.047
8.015
WL
-0.056 -0.032 -0.004 -0.001 0.000
0 ,064 0. .,I ‘W 1.415 4 ,038 8.007
(13). TABLE: SHORT
RANGE
CORRELATION
II TO TSR
CORRECTIONS
FOR
YP
g(r) = 1 - cup (-yXr?1
q in j 1 y=8
y=5
y = 14 ilB(q)/Bt:.s.(q)
AC..
0.0 0.5 1.0 1.5 2.0
1 the
(4)
o.ooil 0.007 0.138 0.355 0.397
I";)
~C,,(n)
-0.013 -0.015 -0.020 -0.021 -0.019
0.013 0.013 0.013 0.013 0.013
0.0 2.1 9.3 8.6 6.1
0.0 5.5 16.5 13.0 8.3
1~See Eqs. (35)-(37). Iu calculatiug the short range correlation e--y@ with y = 5 and y = 14, and for g(r) = 1 - esp [-&r-c)], signifkant~ corrections come only from C,,(q).
0.0 1.-l 4.8 4.5 3.7
~
,
correction we have
0.0 1.2 4.2 4.1 3 .Y for ~(1,) = assttmrd that
repeat some points for the sake of completeness.* The sum rule R(q) is, for large q, very sensitive to the effective magnetic moment parameters in the current, (:4), and if one could measure B(q) with good accuracy one would have an important piece of information about the magnetic properties of nucleons bound one performed measurements of H(g) at p > 2f-‘, one would have ito clear distinction bet,ween purely nuclear contributions to B(q), i.e., contributions from nuclear escit ations, and the contributions from meson production processes whirh start with a cohrrrttt cxlcctr~,production at w = WI, ]see (I)]. 8 The estimates of the corrections to the leading term iu Table I of (1) contain 1111. merical mistakes. This does not alter however the content, suggestions and rottclttsiotls of this paper.
in nuclei. ITor esainple, an effect of 5 “Y’ or 7 ‘; “quenching” of 1ll~gIwt i (’ nlonlen t s would prod1we :Ibout 10 “; or 14 “; ch:mgw of the slml rule. In order to (~hwl< the cwwnt ( 4) one should c*hoose the sirnplwt nuc*lci whew the c~orrclation funcations :w as sirnplc tts possible and contribuk as little :IS pOsSible ihence it, is enough to t&c into account only short.-range cw’rclation~ ) ; but once the current (1) is well c:hec~l~ed,it w-oultl be interesting to measure BC q) for nuclei whrre the so-culled roar pol:~rization effeds ( 13) :we imporbnnt~ and to look for t.heir influence 011 t,he C’,,( yj cwwlnt.icJn Eunct.ion. We w-odd like to look upon the future Inensurements of the sum rule B(q) as :t c~licc~l; (Jf the nwlear elecStrcmagneticcurrent (4). It, is therefore inlportmt to how how wnsitive are the results to the details of t)he model of the ground state used in our cd~d~tions. First of all we do not, know the extent of the sensitivity of t>heresults to the pnrticular choice of the shell rnodrl which forms :L basisof our c~alculnt~ions (e.g., variations of the shape of t)he nuclear average pot)ential, introdu&m Of spinorbit force etch).WC expect this to have :I very sndl influence but any rcason:~bly :wcur:tte numerid malysis uf these effects is beyond our computing facilities. Howcvcr, from the numerid results we do present.in this paper in Figs. l-3 and in Tables I and II, we mn draw the following ~onchwions. ( nl The most important cmwc~tion of the shorty-range correlntions t,o the leading tcrrn (‘onles from the C,,,,(q) correlation function. The wrredions to CPP(q) nud (2 ‘J/*1 (0 1c;” ejpfI 10) arc srnnll, as Table II and Fig. 3 show. l\Ioreuvw these two corrcct~ionshave opposite signs and cancel each other to n large extent. In &cd, they are negligible. This is the reason we calculated them only for the y = 8 cdaseand neglected them in the caseswhere y = 5 and y = 14. (b) There is :\ region of q, l.-lf-’ 2 q 5 3j-‘, where the B( q; y = 5), B(q; y = S), and B( q; y = 14) curves intcrscrt the st,rnight’ line of the lending term (q’,;L’II/“) (Zp,,’ + Npn’). It would seemtherefore t,hwt somewhere in this region there is virtually nothing but the lending-term contribution. This region is thus most.suitnblc for fut,ure prcrise measurementsof t.he TSR clevixed to obtain data (Jn the nl:ig[iet~ic inonients of bound nucleon% ( (2) Sinw C,,( q) can bc tspressed through nuc,leon-Ilu~leoIt correlation function (c*ompare Eq. ( 15) ), an at#tractivc possibilit’y opens of employing rlut,leoIl-rluc.leolJ c*orrelat,ion furl&ions ohtaincd from other experiments (or cnlculatiorls). Then there would be no need for calculating correlation corrections of spec3ic short-range correlat~ionsand the analysis would have virtually no free parameters eswpt the parnmetcrs of the nuclear electromagnetic current (4,. (~1) As CP.‘,,( qj = CPP(q; y = m ) [compare formula (38)], Fig. 3 suggests that / AC&; Y) 1increases as y decreases.AS A[(Z,lllr”) (0 / cf ejp;= 10)] = - AC&( (I = 0), t.he same conclusion is true for the y dependence of
TRAR‘SVERSE
SUM
RULE
1 3.;
1(2/5Z”l (0 1 C,” ejpi, 10)] 1. In fact, one should expect such an dependence on y from t,.he uncertaintjy principle. If we let 7 -:- 0 we make t,he single-nucleon orbit, bett’er defined spatially, and hence we introduce higher momentum components into its Fourier transform. I’inally let us stress a few more points which are import~nnt t,o keep in mind when comparing the results of the present calculations wit’h experimental findings. Thr current given by (4) is the simplest possible single-particle currcnt. One (*x1 add to (4) a lot of different corrections coming from the presence of the ot.her nuc~lcons in the nucleus, for instance the two- , three- , etc. body currems (which depend on t,he coordinates of two, t.hrec, et.c. nucleons), or 01ir ca11 consider the changes of the single-pnrt,icle current introduced by t,he surface cffcrts, etc. All these effect’s are very difficult t)o calculate. It, is, however, comparatively easy to estimat,e the relativistic corrections which one obtains from the free nucleon relatjivistic current by applying the Foldy~\~‘outhuyscrl t ransformatictn. Then one can reprcsent~ the electron-nucleon interact,ion as a power scrics of ,K’ (J/ being the nucleon mass). This was done through order JI-” in, c.g., (2), and the result is that only the charge-density operator has a correction (the DarwinIJoldy term) ; the current operator, however, has no relativistic corrections through order il-‘. The trouble is t,hat we do not know to what cstcnt t,his result is rclcvant to the nucleons immersed in nuclei, and how it mixes with all the other effects described above. In conclusion, it. seems reasonable t,o accept that, only the agreement with experiment can have a clear interpretation, namely, t,hut all the described above modifications of the current, (4 j are negligible. If thcrr is a definite disngrecmentj with experiment the interpretation mill prob:lhly not be unicme. RECEIVEI):
August
17, 1966 REFERENCES
1. W. CZY~, Bull. ;icad. Polon. Sci., Ser. Sci. Malh. Astr. Phys. 12, 649 (19&l). 2. K. W. MCCOY AND L. VAN HOVE, Phys. Rev. 125, 1034 (1962). S. R. H. PR,IT.L., J. 11. W~LECILL, END T. A. (;RIFFY, ~\~ucl. Phys. 64, 677 (19ti5). 4. W. Czxt, Bull. Acad. Polon. Sci., Ser. Sci. Maih. Asir. Phys. 13, 893 (lYli5i, W. Cj:~t, I’hys. 42, 07-l 18 i lM7) 1. L. LIG~XIAK, AND A. MALECKI [accompanying paper, ilnn. 5. S. G.IR'~ENI~.\US.IND C. SCHW.\RTB,P~~.S. Rev. 106,482 (1957). 6. L. L. FOLEY .%ND J. D. W.\LECI+ Sztovo Cimenlo 36, 125i (19M). 7. F. H. LEWIS, J. D. WALECKA, J. GOLDEMISERO, .1x1) W. C. B.\RI~ISK, I’h,t/s. Nrc. /.cllr’~ 10, 4!13 (1963). 8. J. GOLDEMBERC: AND W. C. BARBER, Phys. Rev. 134, 13963 (l!)(S). 9. J. H. LICVINGER, Phgs. Rev. 84, 43 (1951), K. C~OTTFRIED, AVrrcl. Phys. 5, 555 (1958). 10. J. DA PROVIUENCIA AND C.M.SH.IKIK,~~~. Phys. (N.Y.) 30,95 (lM4); .Yrtc/. Php.65, 54 (1965). See also F. I'ILI,.\Rs. Proc. Zn/e~n. School. I’hp. “ICurie Fertui”, C:cjr~rw 23 (1961). Il. XI. MOHHINSKY, ,Vucl. Ph?ys. 13, 101 (1959).
136
CZYi,
LEhIAK,
AiYI)
MALECKI
Princeton 12. A. It. EDMONDS, “Angular Momentum in Qllatltllm Mechanics,” Princeton, New Jersey, 1957. 13. E. BODENSTEUT ;\NI) J. D. ROGERS, Magnetic Momellts of Nuclear Excited tlu-bed Angular Correlations, North-Holland, Amsterdam, 19ti-I. APPENDIX
Univ. States,
Press, Fer-
A
In t,his appendix we give some details of calculat’ions leading from Eq. (8) to Eq. (13). We use t.he completely antisymmetrized oscillator-well shell-model ground-state wavefunction corkructed from the following single-particle wavefunctions: I a)) = I uua7a) = I a) I ua) I 7a), (AlI where / a,) and 17a) are the spin and isospin functions respectively, and I a) = I dma)
(Al”)
= I aL,l,(~)YZ,?n,(~, +>>,
where Yl, are spherical harmonics defined as in (22). 112
%a! R,t(?*)
=
[
r(n
+
2 +
Fi)
1
x(~+3'2)'21'lLfi+li2(xrz)
exp(
_ g$),
(A3)
I,,,k being the Laguerre polynomials. The energy corresponding to this wavefunkon is E,l = (2n + 1 + ~$)fiw,
x = ivlw.
(A4)
Firstly, let us rewrite the matrix elements of the last two terms of (8) :
and
+ $ (0‘\
2 2 2 C?i’(zJ-zk)ej ekplz flnz O‘\ . j#k,Z.mk#Z,mZ#m /
In order to work out B,.s.(q) we have to evaluate eight different matrix elements. One can express them through one-particle matrix elements as follows
TRAljSVERSE
SUM
RULE
137
n-here Ly, /3 ... run over all (neutron and proton) occupied states, and (oc), i/3) . . . run over the occupied states of protons. In the case of 160 WC get the following results:
p2i2Xe
JI, = 2 From
>
(A7) we immedint’ely
get, (13). BPPENDIS
B
In this appendix we give some details of calculations of Sect,ion III, connected with int,roducing short-range correlations int,o t,he ground-state expectat’ion values. A two-particle operator fl = ,z w,
(Bl)
k)
can be expressedthrough creat’ion and annihilation operators as follows: B=
c (a(l>P(2) abY8
1O(1, 2) / y(l)&(2))akaa+as a,,
iB2)
where CY,p . . . specify the quant’um numbers of the single-particle states. Since in the ground state 10) all single-particle states are filled up to certain state (corresponding to t’he Fermi momentum of t’he degenerate Fermi gas), we get (0 I C o(j, k)l 0) = l&l8 MM2)l j#k = s
(apt
O(1, ?)I a(l)f?(2!)
0(1,2)1 -dl)Wj(o -2
(a(l)P(2)[
I G+Q+U~ay I 0) 0(1,2)~p(lM:!,),
(H3)
where the summation extends over all occupied one-parkle stat’es. If we want, to introduce only the two-particle correlations, the unitary operator u acts ouly on a nucleon-nucleon pair, hence it. modifies the two-nucleon states j ty@) [see (IO)]. Thus we have
(0 I J#k c Nj, k)l 0) -+ (0 I 6 z w, km I 0) = 5 (41M2)l UT,‘NL ~ha I 4lM2)) - 2 (a(l)P(2)l u~;o(l, = 5 (a(ljP(2jl O(l, 2)l n(lW~)
2h2
I /3(lht4)
- 5 b(~)PW O(1, YJl P(lML’)),
(B-L)
mherc 1 a7) denotes the two-particle correlated st’ate. The summation over spin and isospin quantum numbers for C,,(q) correlnt ion fuwtions is performed a:’ follows:
and C,,( qj
where a. b run over the awupied states (AZ). The formulas (B5) and (B6 j are v:tlid for nuclei xvith the same number of neutrons and protons in bot,h spin stntrs. X\TUIVxc give wmc details of calculations leading t’o (35) and (36). Let us start with (‘,,(q) (*orrelation function. Using (38) and (29), we obtain
140 In proving
CZYi,
LEh-IAK,
ANIl
M.4LEChI
(B7) we employed t’ho relations
where 6(1,&X) = 1 for l,lJ, satisfying the triangle relation, and zero otherwise. For the I60 nucleus, where n, = ?&b = 0, I, , 16= 0, 1, WCcan put, beenuseof (24) and (25), I = I’ and n = n’. So, for ‘“0 we have
Then we use t,he simplifying approximation that, A{nlm
1 exp (id5q.z)
1nlm) # 0
only
for
1 = 0,
@lo)
where A( . . . ) denot,esthe difference between correlated and uncorrelated magnt.udes. Thus, we assumethe presenceof short-range correlations in the relat,ive s-&a& only. Employing (B9) and (BlO) we get for I60 [the squares of the RIoshinsky coefficients are either 1 or f’2 in our case]
. A(nO0 / exp ( id2
qx) j nO0) Wl)
= - L9
hi
•k ~1,‘) [7A(OOO 1 exp ( i& + !,,A(100
qx) / 000)
j exp (ifi
qz) / loo)].
The formula (36) for C&p) correlation function is obtained as follows. First of all, the calculations proceed differentIy for the pa&s of C,,(q) containing P,’ and those containing p: [see (BS)]. Let us denote these parts C:;(q) and C:;(q), respectively. Applying (28) and (29) we obtain
(hL’6f
1tL’$)
{d, NL, k [ da, n& , A) (n’i, N’L’, x 1?1&,, n& ,A)
(NLAI I P: 1N’L’M)
(nh j exp ( id$qz)
( Bl:! 1
I n’l’m).
In proving (B12) we employed t’he propert’ies of Clebsch-Gordan and Moshinsky coeffkients and the relations
TRAKSVERSE
(nlm / exp(+&qz)
SUM
1 ~‘Z’UZ’) = (nlm 1exp(i2/2qx
(NLM I P,2 I N’L’M’)
141
RULE
# 0 only for
/ n’l’m’)
S,,,\
1 L - L’ even. J
(B13)
We use now the simplifying approximat,ion that A(nh
1exp(i-\/“qx)
I ~z’l’m) # 0 only for
(1314)
I = 1’ = 0.
As 1 - 1’ in (Bl2) can be an cvcn number only this simplification amounts to neglecting the same kind of terms as in (BlO). Icrom (B12) and (B14) we get for ‘“0:
(NhO I P” / N’XO) A(nO0 I exp (idsqx)
/ n’OO),
where P’* = P,” + Pyz + P,*. The only matrix elements needed in (Bl.5) are (010 I P” 1010) = 3; x,
(000 / P2 I 000) = 3i x,
(020 1P” I 020) = 7,; x,
(100 / PZ 1100) = TJ:!h,
CB16)
(000 / P” / 100) = (>$P)X. Finally we get AC$‘i(q:l = g2
[
6.5A(OOOI exp i i2/?qr) 1000)
+ i A(100 1exp (i2/qx)
1100) - s(i A(100 I exp (idsq(x)
1
(B17)
1000) .
In a very similar fashion as in proving (B9), we get for 160 cb2;(q j = - &
C
l,.Z,,=O,l
C
C
hZL?LN
pv:M
[2 - ( - ijz] (hm
/ zLxpj2
. 1711, NL, X 101,) Oh, X}“(nlm 1exp ( id2qz)pz2 I nh) 7 (000 1exp (i~?qz)&
/ 000)
i Bls)
+ ;1,(100 I exp (i&qdp:
j 100).
142
CZYi,
LEkNlAK,
ASD
.\IALECKl
According to (Bll), (Bl7),:mcl (B18), in order to evalu:~tcAC,,(rl)nrld AC,,(q) lye must work out A(nO0 1exp ( idL)qz) / ~‘00) and c.? A(nl~ 1 cxp ( i2/!~lz)p,~’ ;~ll~r\. One can carry out t,he angle int’egrals employing the relations
-0
+ 1(1 + 1) j’- flT iF12,1(,,) j”(4).) [
- ;.j2&.)
1i
(I3201
, )
where I, are spherical Bessel functions. Applying the simplifications (BlO) and (B14) we extract the leading terms of the short-range correlation corrections. In order to proceed consistently, in the radial integrals calculation, we neglect the differences A(. . . ) for all the terms whose integrands behave like ra (a 2 4) for small I’. In such a way we exkact the leading terms. One can expect that t’he contributions of terms wit)h a~= 4 and (Y = 6 (there are only even powers of T) are equal to about (NoI - 1) ! (No0 - 1) and ( No2 - l)/(N, - 1) times the leading terms, respectively. For q(r) = 1 - exp (- 8Xr*), we get irV,, - l)/(N,,c, - 1) = 0.124and (N, - I)/ (No,, - 1 ) = 0.015. IJsing this procedure we get A(000 j exp ( 1’2/2qxj ( 000j = 4 -l)l’X:%% x A(100 / cxp (iz/?qx)l
100)
A(100 j exp (i&qz)(
000)
1
g”c - 1 ?j”(~~I~), Noo
(B21)
TRASSVERSE
SUM
RULE
143