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Theory of overlap functions
I [
~
NuclearPhysics A213 (1973) 493 --509; (~) North-HollandPublishing Co., Amsterdam
I.E.3
[
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THEORY OF OVERLAP FUNCTIONS
0I). The overlap representation C. F. CLEMENT t TheoreticalPhysics Division, UKAEA Research Group, Atomic Energy Research Establishment, Harwell Received 8 January 1973
Abstract: Nuclear densities and the expectatiort values of electromagnetic operators are expressed explicitly in terms of the single particle overlap functions defined in the first paper I of this series. Using the sum rules of I some of the properties of this representation are discussed. In some cases values of nuclear electrical moments cart be obtained directly from experimental data on proton pick-up and stripping reactions. As an illustration the quadrupole and hexadecapole moments of slV are obtained with a discussion of the errors and uncertainties of the method. We find Q = -0.033±0.01 b, a possibly more accurate value than that found by other methods.
1. Introduction In the first paper o f this series 1), which we henceforth denote by I, sum rules were derived connecting the single particle overlap functions defined by projecting the states o f neighbouring nuclei onto one another. Here, we intend to show explicitly h o w the expectation value o f any single particle operator can be expressed in terms of these functions. Also we give an example o f h o w the representation can be used in practice to obtain electrical moments o f a nucleus. The potential uses o f experimental data on single-nucleon transfer reactions on odd mass nuclei are not realised at present. We h o p e that this paper will show experimentalists that accurate data will not only give values for the m o m e n t s but also, and possibly more importantly, set limits on their possible errors. The basic idea o f this paper is not new. French 2) has derived the shell-model equivalents o f the expressions given here, particularly for electrical moments. A b o u t the same time Berggren 3, a) realised that the overlap functions used in stripping and pick-up reactions could determine the expectation values o f one-body operators. However, awareness and application o f their ideas has been sadly lacking. We present here the exact equivalents o f the formulas derived by French 2) in the framework o f our general investigation o f overlap functions. Because it is exact, our representation contains contributions to expectation values f r o m deeply b o u n d nucleons which would not appear in his work. Using the sum rules o f I these contributions can generally be removed if the relevant shells are 'closed' in the sense discussed in I. This is where the t Attached to University of Minnesota, 1971-2, where part of this work was performed. 493
4.94
C.F. CLEMENT
work of Berggren a. 4) is extended. If the shells are not quite closed the best place to look for their contributions experimentally is in stripping reactions. This requires a particle-hole transformation in the representation as performed by French 2) in the shell model. The transformation is briefly discussed in sect. 3. With the exact relations of I any exact transformation would involve corrections to shell-model formulas. At the present time it is probably not worthwhile to include these as they are undoubtedly small though it is possible, at least with a perturbation approach, to go beyond the shell model in the present theory. In sect. 2 we give general formulas for the nucleon densities in terms of overlap functions and spectroscopic factors. It is already a general practice in nuclear physics to obtain the density distributions from single particle wave functions specified by their separation energies. We have shown that, with non-singular potentials, the best mean single particle overlap functions are specified by Hartree-Fock-like equations where the single particle energies are specified by the mean separation energy s). Thus the prescription is justified in this case. However, we need to extend this work to include c.m. corrections and two-body correlations arising from hard cores. There are undoubtedly corrections to the mean separation energy in the equations of ref. 5) and these will be discussed in the next paper of this series. For nuclei with non-zero spin the densities are in general no longer spherical and we give formulas for the appropriate L-multipoles, comparing them to commonly used deformation parameters in the case of L = 2. It is conceivable that one might be able to obtain a value for such deformation parameters from experimental pick-up data. The representations for electromagnetic moments and transition amplitudes are derived in sects. 3 and 4, the magnetic case being considered first. Whilst one could not hope to obtain more accurate values of magnetic moments from pick-up data than those obtained by conventional atomic means, it would be illuminating from the point of view of nuclear structure to use the known values to gain information about the overlap functions. In sect. 5 we present our main application of the representation to obtain the quadrupole and hexadecapole moments of s IV. Nuclear quadrupole moments given in the Nuclear Data Tables 6) are all the results of measurements by atomic means. It is not generally appreciated just how uncertain the values are. The trouble is one that experimentalists can blame on the theory needed to interpret their observations. The interpretation requires the gradient of the atomic electrical field at the nucleus and corrections from the polarization of core electrons, both of which require in turn extensive theoretical calculations. An example of the widely different results in some cases is shown in table 2. Whilst we cannot claim that the overlap representation does not require extensive theoretical calculations, these are mainly in the extraction of spectroscopic factors from experimental angular distributions using the distorted wave Born approximation. The rest of the calculation as shown here is relatively trivial. There are major uncertainties.connected with the relative phases of spectro-
O V E R L A P F U N C T I O N S (II)
495
scopic amplitudes but in certain special cases, such as s IV, these are not too important. It is important to put errors on the results obtained and it is certainly possible to do this in the overlap representation. The major sources of error in our case are discussed in sect. 5. The result for Q obtained for 51V is compared to the value from a fairly large shellmodel calculation 7). Since we find a major discrepancy between the shell-model wave function found there and one of the pick-up experiments we consider, we regard the theoretical value as being no more reliable than the value we obtain. Finally in sect. 6 we summarise the work of this paper and give the conclusions we have drawn from our application.
2. Nuclear density distributions In this section we give the densities in terms of the overlap functions defined in I. The nuclear densities and transition densities between two states of a nucleus enter into the description of all nuclear physics experiments involving one nucleon such as electron scattering and direct nucleon inelastic scattering. Thus the general formula giving the transition density between two states is first considered. We then specialize to the densities of a particular state. The wave functions of two states of an .4-particle nucleus may be written as X , ( 1 . . . A; J,M, TrT3,) and Z~(1...-4; J~M~TsT3~). The general single nucleon transition density operator between these states is defined as A
p(r; r M, sM~) = (X,, E b(r--ri), Z~) i=1
= "4(Z,, cS(r-ra), Z~),
(1)
since all the terms give the same contribution with antisymmetric wave functions. To obtain the neutron and proton densities, p, and pp, we must insert the operators ½ ( l + % a ) , respectively. For charge transfer and spin-flip reactions the operators z ] , z ; and aa could be inserted and it would be trivial to extend the following derivation to include these cases. We generally wish to refer the density to the c.m. of the nucleus considered so that we define A r' = r - - R A , x = r A - RA_I = - (r~-- R~). (2) .4-1 The J-function then becomes = J (r'\
A
.4-1)x --
A A-1
D
A-- 1
r '2
E rLM( )rMr'). LM
(3)
496
C.F. CLEMENT
The general method of this paper for any matrix element such as that in eq. (1) is to insert a complete set of states of the (A - 1)-particle Hamiltonian on each side of the single particle operator, and then use the definitions (3) and (8) of I for the overlap functions and modified spectroscopic factors• It is convenient to extend the complete set to cover the angular, spin and isospin co-ordinates of particle A so that we can use coupled states on the left and right in the forms [~'~(1 . . . A - l ; J~T~T3~) dp(f2~, ¢a, xa; jtltta)]~t'~ and [ ~ , ( 1 . . . A - l ; J~T~T3~)dp(f2x, ¢A, ZA; j212ta)]JMs, respectively. We have anticipated the diagonality of ~ and ta in the matrix element. From (3) and (8) of I and eq. (3) we obtain
p(r'; rMr, sM,) =
~.,
O',(jlllta)O'~(jzl2t3)
ctjlltj212t3
×<1~'~(1 A A- 1
• " "
A-1 r '2
.4-1;J~T~T3~)~,~(,,,,,A,~A;j,I~t
~'~
3]_lMr,
LM~YLu(x)Y~u(r'),
[~,~(1... A - 1; J~ ~ T~)~,(x, ~ , ~ ; J2 l, t3)]~). It is now straightforward to use the Wigner-Eckart theorem a) to evaluate the reduced matrix elements and we obtain 1
E rt,4.-M.ff') p(r'; rMr, sM~) - (4n) ½.j,z,j2,2t3z. Jr
L J2)
x [(2jt + 1)(2j2 + 1)(2Jr-l-1)(2L + 1)]'½11 +(-1) ''+'' +L] (L~ 0 x O'~,(j~ I 1 ta)O'~s(j2 12 ta)
~b~,
(Lr
; j~
½
11 t3)
(4) The neutron and proton densities may be obtained immediately by restricting the sum over ta to ta = +½, respectively. We note that with the c.m. factors given, the radial density is properly normalized: t ,2
drr
~r
A
r';jlllt3
q~,,
A
r';jlllt3
)
P = / d x x 2q~,(x, 2 . Jl It ta) = 1,
J
where the last line is the normalization condition (6) of I.
(5)
OVERLAP FUNCTIONS (II)
497
In the general expression (4) the values of It and 12 and therefore L are restricted by parity conservation ~,rc s = ( - 1 ) h+`= = ( - 1 ) L. Thus only odd or even values of L occur depending on whether r and s have the opposite or the same parity, respectively. One special case which often occurs in inelastic scattering is J~ = 0 when L = Jr and J~ = Jz and we have
p(r' ; rM~, sO) -
I ~°tT~) jlllJ212tz~
X
[(2J,+ 1)(2jl + 1)] ~r ( J ~
x O~,(jl ll
J'0 ~2) ½1"1+ ( - 1)'~+12+~']
t3)O's(j 2 l 2 t3)
~,
r'; Jl Itt 3 t~ s
r , J2 12 ta •
(6)
If, in the sense of paper I, the shell (Jl Ii ta) is closed in state s, or the shell (J2/2 ta) is closed in state r, then [eq. (L 32)]
Z
0;,(j212 t3)O',(j~ It t3) = O.
(7)
~, d~ fixed
In other words the off-diagonal contribution of these shells to the density sums vanish. The actual diagonal density distribution in state J,M, is obtained by putting s = r in eq. (4);
;(~'; ,M,) = -
1
E e~(cos o) art gjlllj212taL × (s, M, L01S, M,)(- 0'o+~,+~[(2j~ + 1)(2j~ + 1)(2L + 1)~(2J, + 1)]~
{J~ J2 L t [ j , × J,
J,
J2(-½
L
J2}
o
½
×½[l+(--1)h+t2+L]o'r(jlllt3)O'r(j212t3)
c~,
r, jtl, t a
x ¢~, (A-~I r'; jz12t3).
(8)
Only even values of L occur. The number of nucleons is given by
A = fP(r') d3r' = E S',(jlt3), J
(9)
~jlt3
where we have used eq. (5) and, since L = O,j l = Jz and only 11 = lz can contribute. Clearly we also have
N = Z S~,(jlta = +½), ~tjl
These results agree with eq. (14) of I.
Z = ~. S;~(jlt a = - ~ ) . ajl
(10)
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C.F. CLEMENT
In fact for L = 0 no off-diagonal terms where (j111) density. For spin zero nuclei where J, = 0 the density is p(r')
l~s,r(jlt3 = ~- ~1u3
)
[A '~3d~2( A ~-~-~1 ~-~, ~ - ~
For a general spin nucleus where the shell [eq. (I. 34)] E
# (j212) contribute to the )
r' ; jlt3 .
(11)
(flt3) is closed we can use the sum rule
S;, - 2 J ~ + l . 2Jr + 1
(12)
aJotfixed The sum over J~ becomes
j Jrj J~LI}= F(2Jr+l)(2j+l).]~'6z °" L ~
J.E(-1)s'+J'+J(2J~+l)tJr
(13)
Thus, as expected, the contribution of any closed shell to the density is
P r ( r ' ; j l t 3 ) - 2 j +41rl / A ~-~-~)'~3"2 9~, ( ~ A
r'; jIG ) •
(14)
For nuclei with finite Jr, L goes up to the maximum even value < 2./,. For L = 2 deformations, which are mostly considered, we can compare the density deformation in the state M r = Jr to two model formulae: (A)
p(r) = po(r)[1 + flA r2o(0~b)], x2 + y 2
(B)
p = const,
a2
(15)
g2
+ ~ < 1,
(16)
where
To compare to our result we demand the equivalence of the quantity
f p(r t ; rJ,)Y2o(P t)d 3r .I Then the result is, to first order in fiB,
flA
=
3fls
=
1 (4~ ~ , (J,J,2OIJ, J,) A \ 5 ] ajill j2123
t •
t .
X O~r(J , l, t3)Oar(J 2 12 t3)
~- x 2dx~b~,(x,.
Jl l, t3)~)~,(x ; 22 12t3)"
From our general result (7) no closed shells can contribute to this sum.
(18)
OVERLAP FUNCTIONS
(II)
4.99
No attempt has been made yet to evaluate the above density expressions. However, one possible application is to the difference between neutron and proton densities in nuclei. From observed spectroscopic factors, it would be possible to calculate the overlap functions with the aid of the Hartree-Fock equations of ref. 5) (or extensions of them) and hence the density distribution for even Jr = 0 nuclei. For odd nuclei the relative phases of the spectroscopic amplitudes are needed too. Theoretical investigations, such as the effect of adding more neutrons to the lf~ shell in the Ca isotopes, might be worthwhile in order to compare the results to those from experimental investigations of changes of radii with mass number. However, we intend to investigate c.m. and correlation corrections to the Hartree-Fock equations before proceeding with such applications. 3. Magnetic moments and M1 transition amplitudes
Here, we apply the same techniques as in the previous section to evaluate magnetic moments and M1 transition operators in terms of spectroscopic factors and overlap functions. We include the case of the transition operators because, although the spectroscopic factors for excited states may not be known, it is a trivial extension of the magnetic moment operator case and could be a useful parametrization. Higher magnetic multipoles can be treated in the same way. In units of Bohr magnetons (eh/2Mpc) the magnetic moment operator may be written as /~ = E [¼(1 +'r3,)g n a, + ¼(1 - z3,)gp a, + ½(1 - z3,)l,]. (19) i The magnetic moment of a state X,(1 • • • A; p, = ( Z , ( 1 . . . A ; J , J , T ,
J,M, T, T3r) is conventionally defined as
T3~),p~,Z,(1...A;J,J~T~T3,)).
(20)
The M1 transition matrix element between two states s and r is
( 3 ~ ~ eh
[4-~n] 2-~pc (Zr(1 " " " A; S~ M~ T, T3~), p ~ , Zs(1 • • • A ; Ss Ms T~Tas)), where the operator is written in its tensor form. To evaluate this general matrix element we insert complete sets for the ( A - 1)particle system as in the previous section• Then
= ~,
<[~P~(1...A-1;J~T~T3~)(o~,(x, ea,zA,
~jtllt3J212
¼(1 + ~ ) g ° ~ , + ¼(1 - ~ ) g ~ ~ , + ½0 - ~)(/~)~,, I-~0
• • • a-
•
Js
!
•
a; s~ T~ T~)O~s(x, ~ , ~,; s2 l~ t ~ ) ] ~ ) o = 0 , 1 ~
=(-1)s"-u"
( J~
--M r
t3o~sO~ l~ t~)
1 Y~) ~ O,(j llG)O,'s(jzlt3) m Ms ,S,S~u~
t
•
t3)]t~r,s~
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C.F. CLEMENT
x(-l)S'+'"+t+Y'+Y2+'[(2J,+l)(2J,+1)(2j, +l)(2j2+l)]½{j I J,
+ ( - 1)i2-J~(½-t3)[l(l + 1)(2/+ 1)] ~ {jl2
l
Jl
× fax x2+,,(x; J, lt3)dP,~(x ;J2 lt3).
(21)
To a good approximation, especially when r = s, the radial integral will be unity, which it is of course defined to be when Jt = J2 and r = s. The magnetic moment of a state r then becomes
]It
= FJr(2Jr+l)-] * 20'-r(j, lt3)O;r(j21t3) L ~
I ~tjlj2lt3
× (I(½+t3) gf +½(1-t3)~--] ~/(){)2
+(_l)i2_j,(½_t3)[l(l+l)(21+l)] ~ {jl
J,
Jl
"
It can easily be shown, using eq. (7), that no closed shells can contribute to the sum. Furthermore, i f a shell is nearly closed, we can use sum rule (25) of I to represent the operators in terms of the spectroscopic factors and overlap functions for stripping reactions. From a practical point of view this might be advisable because of the large number of pick-up states which would be involved and large cancellations in the sums over J~ in (21) and (22). The transformation from the ( A - 1)-nucleus to the (A+ 1)nucleus representation is the exact analogue of the particle-hole transformation in the second quantisation representation of the shell model. We restrict the discussion to the case when all the q~ are approximately equal and continuum contributions and I/A corrections are ignored. It is possible, but probably not worthwhile at present, to include these corrections. To transform generalized expressions such as (21) and (22) with an operator rank L (:/: 0) we require the recoupling identity
s.
[J~ J. J~I(Jl J~ J.
(23) =
J2
Jr
"
OVERLAP
FUNCTIONS
(II)
501
From eq. (13) the contribution of any diagonal sum rule term vanishes so that, for example, the contribution of a given (JlJ2 l) to eq. (22) for/1 r is transformed to [J,(2J,+l)]~2Jn+l
~r(Jl J2 1) = . j - - ~
, .
2J---~ Or~(ll lt3)O;.(j2 It3)
{j, 1}
×(--1)z"+z'+t-½[(2j'+l)(2j2+l)]& Jr J, J,, x {[(½+t3)9"+½(1--t3)~lx/'6{~j2
½ ~}
+(-1)J~-J'(½-t3)[l(l+l)(21+l)J~{J2
Jl
'
where 0~= is the spectroscopic amplitude for stripping to a final state n, as defined by (4) and (8) of I. For a general L a representation such as (21) assumes a similar form with the suffix g replaced by n, a factor of ( 2 J , + l ) / [ ( 2 J , + 1)(2Js+ 1)] ~r included and an additional phase factor of ( - 1 ) J'-~'+il+j2+L. For Jl = J 2 and Jr = Js this phase factor is ( - 1 ) L+z, identical to that of the particle-hole transformation in the shell model as discussed by Rowe 9), for example. Since accurate values for magnetic moments of nuclei are known from atomic methods, it is unreasonable to expect the overlap representation to provide better values. However, one might use the known values to compare with data on pick-up reactions and, in particular, to help to discover the phases of the products 0~, (Jl = l+½ lta) 0'~, (J2 = l-½ lt3) which enter into expression (22). As is further discussed in the next section, these products are important physical quantities which enter into the expressions for all electric and magnetic moments and also play a role in any dynamical theory of overlap functions. The latter point will be shown in a future publication and we hope that eventually known magnetic moments will be compared to the values given by (22) and (24). Furthermore, if there are any considerable exchange current corrections to the expression for the magnetic moment operator, they will form a discrepancy between a value from (22) and a measured atomic value. It would be interesting if one could get sufficient accuracy on the overlap representation value of the magnetic moment to detect exchange current corrections. 4. Electric moments and EL transition amplitudes
The electric moments and EL transition rates for an A-particle nucleus are obtained by taking matrix elements of the operators A
Q(ELM)
= e ~ ½(1 - T3,)lri i
-
-
RAILYLM(r,-- RA).
(25)
502
C.F. CLEMENT
Thus the quadrupole moment of a state Z , ( 1 . . . A; J, MrT, Tar) is defined as
eQ =('~106.
(26)
In our representation the general matrix element is
= eA ( ~ - ~ ) L,
(27)
where we have used the relation (2) to obtain the correct c.m. correction. We note that, in this representation, as opposed to the more usual shell-model representation, the effective operator still refers to protons only even in the electric dipole case. The matrix element is evaluated as before to obtain <)~r(dr Mr) , Q(ELM), Z,(J, Ms)> = e((A - l ) / A f
~tJlllj212
--M,
M
M~
4x
/. X
O~r(jl lx - ½)0~,(j212 - ½)3~b=r(x; Jt 11 --½)~)~ts(X;J2 12--½)XL+ 2dx.
(28)
The quadrupole moment, given by eq. (26), becomes
x O;r(j, I, -½)0"r(j2 I2-½)Q~(j, 1, ; J212),
(29)
where we define the radial integrals as
II, J2 12) = f 6~tr(X; Jl 11--½)6~r( X; J2 12 -
½)x4 d x .
(30)
Alternatively, the quadrupole moment can be obtained from the density distribution given by eq. (8).
eQ = (-l~)½e J r'Ey2o(V)pp(r'; rMr = Jr) dad, where pp is the proton density obtained by including only t3 = - ½ in eq. (8).
(31)
OVERLAP FUNCTIONS (II)
503
This result is equivalent to eq. (27) since we have
(~)
2Q~(jl I1; j 2 1 2 ) =
A 3 (a-~--l) f
dr'r'4
x q~, ( A - ~ r'; Jt l1-½)~b~, (A-~I r';
J212--½)"
(32)
The higher static electric moments may be obtained in the same way. For example, the hexadecapole moment is (other definitions may differ by a constant)
Z.(1 • • • A;
eH = J, Jr T, T3r)>
(-~670½(Z,(1... A; J, J, T~ Tar), Q(E40),
= (J~rc)½ej -r '6 r,o(P')pp(r';
=e
rM.
Jr -Jr
= dr)d3r '
2 0 " , ( j l lt-½)
jlllJ212ct,
×O',(j2l: -½)(- 1)~+s-++[(2j~ + 1)(2j2 + 1)]½[1+ ( - 1)t'÷t~] x (j~
4~ 2 ) { J j 0
JrJ2 j4}H,(j,l~;j212) '
(33)
where
t" H~(ji ll ; J2 12) = J dx x6$.r(x; Jl lt)~b~r(x;
J2 •2).
(34)
To calculate the moments from these expressions we need to know the form factors, ~b,r, and the spectroscopic amplitudes, 0",. The former may be obtained fairly reliably from the shell model, but from pick-up reactions we generally only know S~,, which are the squares of the latter. Thus the phase of O'~,(jtll) 0",(j212) for (j~ lx) # (J2/2) is undetermined. In some special cases, such as the one discussed in the next section, this may not be an important drawback. We also have some hope of determining the important phases within the framework of this theory by constructing dynamical equations which contain them. They are also given by complete shell-model wave functions for the states in question. Of course the phase is independent of the phases chosen for the states ~ and r, and is thus a physical quantity which plays a very important role in the overlap representation. To transform any of eqs. (28), (29) or (33) to a representation in terms of stripping amplitudes we can use the general result of the last section to replace ct by n and multiply by the factor (-- 1)s'-s'+i' +Yz+L(2J,, + l)/[(2Sr + 1)(2Ss + 1)] ½. For this transformation we assumed that all the ~b were equal so that the states n included must only contain dpr.(x;jl-½) corresponding to the same major shell.
504
C.F. CLEMENT 5. The electrical moments of StV
In this section we describe an application of the formulae of sect. 4 to experimental results for pick-up reactions on 51V. This will illustrate some of the points involved in using the overlap representation and also gives a respectable value of the quadrupole moments and a prediction for the hexadecapole moment. The former is usually measured by atomic methods which are subject to some uncertainties. Our application shows that, for some nuclei at least, a careful analysis of the results of proton pick-up reactions is a very competitive method. Nuclei in the lf~_ shell are a particularly favourable case since only a few (fl) values are involved, the dominant contributions being expected to be from the lf~ and 2p~ shells. In order to evaluate the electrical moments we need estimates for the single particle integrals Q~ and H~ given by eqs. (30) and (34) respectively. To obtain these we have adopted the simplest possible approach and have used harmonic oscillator wave functions. The integrals are relatively insensitive once the size of the nucleus is fixed and the use of more realistic form factors, such as Woods-Saxon shapes, is hardly worthwhile in view of larger uncertainties in experimental spectroscopic factors. Taking into account the fact that the usual oscillator wave functions are defined in in the variable r' the results are Q((lf)2) = Q((2p)2) =
Q(lf, 2p) = -
(A)2 ~
4v' (14)~ 2v '
H ( ( l f ) 2 ) = (A__~) 4 99 16v z ' n ( l f , 2p) = -
( A ) ' 2 7 x ( 1 4 ) ~~ 16v2
(36)
A good estimate for the oscillator parameter v is obtained from ho~ ,,~ 40 A-* MeV giving v- 1 = 2.065 A ¢ fm 2, (37) As an illustration of the type of numerical form of Q obtained we give explicitly the contributions to eq. (29) for a J, = ½ nucleus from lf~, lf¢ and 2p t protons. (any 2p~ contribution is expected to be negligible for our example of 5tV):
Q = Q(lf~)+a(lf~)+Q(2p~)+Q(lf~,
2p~)+Q(lf~, lfl), II
Q(lf~) =
-
9"S-i+VS2-2S3ta~-T-s-7v -
- ~ S1,4 -
- r ~aS75-
--
+~s6),
-
'7
--
OVERLAP
Q(2p~) = Q(lf~, 2p~)
=
9 lOv
-- _ _
FUNCTIONS
(II)
505
_ ]-xS4
_ ~(})~ 1 [02 + (~x-)~O~ + (~)~0~ -
9~ ½
-
],
V
Q(lf~, lf~)
-
-
+
~O£-,-s-/
T~', ---~v6 j ,
(38)
where
S;,(jl)=
E
S'r(jl-½),
e, d~ fixed
Of~(jt 11; j212) =
E
O'er(jr l~-½)O'r(j212-½)
~¢,J= fixed
=
E
+__[S'r(jt l ~ - } ) S ' , ( j 2 1 2 - ½ ) ] ~.
(39)
~, J~ fi xed
In the above formulae it has been assumed that the overlap functions, and hence the single particle quadrupole moments, given by eq. (34), are the same for f~ and ft" The use of more realistic functions would, relative to Q(1 f~), increase Q(1 f~) because of less binding energy for lf~ particles and decrease Q(lf~, lf~) because of less than perfect overlap. An obvious feature of the expressions (38) is that the signs of the various terms are such that cancellations are expected to occur. Indeed we know that if a shell is full there must be complete cancellation of the appropriate contributions to Q. For an odd proton nucleus such as S~V little or no contribution is expected from what would be the unnatural parity states of spin 1 +, 3 +, 5 + and 7 +. Such states in S°Ti could have n o (fqr) 2 components to connect to a dominant (f~)a configuration in SlV. Similar formulae giving the contributions of ld and 2s holes to the quadrupole moment can be written down using the rules of sect. 4 to relate them to the results of proton stripping reactions on 51V. In (aHe, d) reactions performed on 5tV [ref. t o)] there is no reported ld or 2s strength up to an excitation energy of 8.7 MeV in 52Cr. This makes it very unlikely that there is any significant contribution to the quadrupole moment from ld or 2s holes. One would like, however, to have an experimental upper limit on the strength in order to put errors from this source into our expression for the quadrupole moment. Theoretically there can be no contribution from pairing force mixing across a closed shell since the pairs are always coupled to J = 0 giving no contribution to Q. However, there could be a contribution from 'polarization" of the core in which, for example, a ld particle is excited into a 2d state. Any such contributions from d- or s-shell protons cannot interfere with the dominant lf~ contribution. Because of the different parities there are no terms like 0(ld~)0(lf~). The potential error from neglecting the s-d shell is thus reduced and is independent of the f-p shell contribution apart from the requirement that the total number of protons is fixed.
506
C . F . CLEMENT
Turning to the I f-2p contribution several experiments 11- ~3) have been performed which have found l = 3 proton pick-up leading to J = 0 ÷, 2 +, 4 ÷ and 6 + states with the possibility of small l = 1 contributions 13). We first assume that all the l = 3 strength arises from lf~ transfer and the l = 1 strength from 2p~ transfer. Then the quadrupole moments calculated from eqs. (37)-(39) and the hexadecapole moments calculated from similar expressions obtained from eqs. (33), (36) and (37) are shown in table 1. The variation from Q = - 0 . 0 3 3 b to -0.041 b gives an indication of the error expected from the f~ contribution. We see, however, that even a small l = 1 contribution can significantly change the quadrupole moment according to the sign
of 0~ (~, ~). TABLE 1 Electrical moments of s ~V from proton pick-up reactions
~So- (~)
$2- ({)
Spectroscopic factors S , - ({) $ 6 - (~)
0.73 a) 0.74 b) 0.72 c)
0.39 0.37 0.34
0.64 0.75 0.81 a)
1.05 1.14 1.08
0.75 f)
0.42
0.75
1.08
a) Ref. 11).
b) Ref. 12).
$2- ({t)
Q (b)
02-(5, {)
0.023
+0.088 --0.088
--0.041 --0.033 --0.036 e) --0.024 --0.038
H (10- 5* cm 4) 0.21 0.21 0.30 =) 0.15 0.22
o) Ref. aa).
d) Includes two states. e) Values correspond to 4- signs in the previous column. t) Theory based on (fk)3 configuration. In order to discuss this 2p contribution further and to consider a possible lf~ contribution we refer to a large shell-model calculation for s ~V performed by Rustgi et aL 7). The ground state proton wave function they find is 7J = 0.94(f~)3-0.13[(p~)2]°f~-0.25[(f~)2]°f~. The coefficients differ very slightly in the three approximations they make and we have taken the median values. The spectroscopic factors for pick-up of p~ and f~ protons leading to states [p~f~]s and [f~f~]s respectively are Sj(2p~) = (0.13) 2 2 J + 1 , 16
Sj(lf~:) = (0.25) 2 2 J + l 24
The first thing we notice is that the value for S;(2p~) is completely inconsistent with the observed value of 0.023 to the lowest 2 + state found by Zeidman et aL ~3). Even if this 2 + state is 100 Y/oo[p~f~]2, which of course is extremely unlikely, the value of S;(2p~) would only be 0.005 with the above wave function. This indicates that either the DWBA analysis of the experiment is incorrect or there should be very much larger 2p~ components in the wave function. The factor 2 J + 1 in Sj indicates
507
OVERLAP FUNCTIONS (II)
that, if the major component is [(2p~)2] °, one should observe experimentally l = 1 transfer strength to 4 + states which is nearly a factor 2 bigger than the 2 + strength. Secondly, a wave function which has only admixtures of [(p~)2]o and [(f~)2]o gives no p~ or fl contribution to Q. This is of course an immediate result because of the zero spin but can also be seen in our formalism from the factor 2 J + 1 contained in Sj and eq. (13). In other words the contributions to all the terms in eq. (38) except Q(lf~) must cancel completely. In the spirit of our approach working directly from experiment we only have S~ (2p~) and 0 2 (2p~, lf~) non-zero in the 2p~ terms. Thus cancellation does not occur with the result that Q takes on the two quite different values of - 0 . 0 2 4 b or - 0 . 0 3 6 b as shown in table 1. It is possibly more realistic to go some way towards theory and say that admixtures of states such as [[(lf~)2] 2 2p~] ~ or [[(lf~)2] 2 lf~] ~ which would produce 2p~ or lf~. contributions to Q are relatively small. Then an error of 0.01 in Q from taking the value of Q(lf_~) alone would be an overestimation. TABLE 2 Quadrupole moment of 51V Method
(bQ)
atomic and molecular beams nuclear magnetic resonance
--0.052 a) ~0.04 a) +0.28 c) +0.26 r) --0.033-4-0.01 --0.042 :k0.002 h)
optical spectroscopy proton pick-up reactions shell-model theory z) a) Ref. 14).
b) Ref. 15).
c) Ref. 16).
~:0.0073 b) :~0.26 c)
d) Ref. xT).
e) Ref. is).
r) Ref. 19).
s) Ref. 7).
h) Error from three different calculations. Values of the quadrupole moment of 5 t V obtained from various sources are shown in table 2. It is apparent that some values obtained by atomic methods are badly inaccurate; though they may be old experiments they are still quoted in Nuclear Data Tables 6). The value and error we give from proton pickup reactions cover the values shown in table 1. From the above discussion we are fairly confident that the error covers all missing f-p shell contributions. There is no experimental evidence at present for any s-d contributions though we would like the (3I-Ie, d) spectrum on 51V to be closely examined for l = 0 and 2 transfer. We would not regard the shellmodel theory value as being the final answer, particularly in view of the discrepancy commented on above with experimental l = I transfer. The actual value may be quite close to the pure (f~)3 value. Without invoking interactions with the s-d core, the above examination of the various terms in the overlap representation has indicated that one needs large and unlikely constituents of lf-2p wave functions in 51V to obtain a much different value for Q.
508
C.F. CLEMENT 6. Conclusions
We have expressed the principal one-body operators in nuclear physics in terms o f spectroscopic factors and overlap functions. Collectively the representations, including those for the densities, apply directly to almost all nuclear physics experiments involving the electromagnetic field and to many other experiments such as inelastic scattering. Since the single particle wave functions are relatively well known, the important unknown parameters which specify the representations are quantities like E~O'~s(j212 ta)O'ar(j 1 ll ta), wherejl ll, J2/2 and possibly J~ are fixed. For ground states these quantities are, apart from phases, accessible to experiment. For excited states this is not so, apart from a few special cases where one can, for example, perform inelastic scattering via an analogue state. There are, nevertheless, reasons why the representations may be useful. As discussed in the introduction to I, general microscopic comparisons between theory and experiment in nuclear physics involve calculating theoretical many-body wave functions for the states involved. This procedure has two drawbacks, the first being that the model space is necessarily limited. Matrices to be diagonalized grow enormously with the addition of extra configurations. The second drawback is that the many-body correlations built into the wave functions are generally not accessible to experimental test. One therefore in effect has to calculate the single particle quantities expressed here before comparison to experiment can take place anyway. The main exceptions are the actual energies of the states, determined by two-particle correlations, which are in many cases notoriously insensitive to small one-body properties which can drastically change electromagnetic transition rates, for example. Thus our first main point is that the parametrization in terms of overlap functions can be a very useful intermediate stage in the comparison between microscopic theory and experiment. This brings us on to the discussion of errors. With the usual many-body wave functions it is difficult if not impossible to put errors on the single particle quantities obtained. For example the inclusion of the next major shell may be beyond the bounds of calculational possibility. Working directly with the overlap representation the calculation of possible errors is relatively easy. We have to specify a certain small contribution from the spectroscopic factors of the next shell when the calculation of possible errors is straightforward. Thus our second point is that when single particle quantities are looked at via the overlap representation any discussion of errors in a theory becomes immeasurably simpler. The example we have discussed in sect. 5 exemplifies both points. A fairly large shell-model calculation was performed for 5iV [ref. 7)] but does not appear to give a wave function which has configurations consistent with single particle transfer data [ref. la)]. Working directly from experimental data we have obtained Q = -0.033 + +0.01b with an estimated error from the lf-2p shell. To add an error from the 2s-ld shell we need to put an upper limit on l = 0 and l = 2 stripping transfers on 5iV.
OVERLAP FUNCTIONS (II)
509
I n the existing e x p e r i m e n t 10) there is no evidence for such transfers b u t we urge t h a t there be further e x a m i n a t i o n o f the data. The o v e r l a p r e p r e s e n t a t i o n also gives the o t h e r m o m e n t s a n d we find H = (0.22-t-0.8)x 1 0 - 5 4 c m 4. I n conclusion we t h i n k there m u s t be m a n y m o r e examples in n u c l e a r physics where e x p e r i m e n t a l d a t a c o u l d be p r o f i t a b l y l o o k e d a t f r o m the p o i n t o f view o f the o v e r l a p representation. I t also m a k e s w o r t h w h i l e the careful e x a m i n a t i o n b y experimentalists o f single n u c l e o n transfer d a t a for w e a k transitions f r o m o t h e r m a j o r shells t h a n the m a i n one for the nucleus concerned. T h e a u t h o r w o u l d like to t h a n k once m o r e the staff o f the University o f M i n n e s o t a for their hospitality and, in p a r t i c u l a r , Dr. N. H i n t z for his help in finding the d a t a on 5iV.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
C. F. Clement, Nucl. Phys. A213 (1973) 469 J. B. French, Phys. Lett. 15 (1965) 327 T. Berggren, Nucl. Phys. 72 (1965) 337 H. Tyr6n, S. Kullander, O. Sundberg, R. Ramachandran, P. Isaacsort and T. Berggren, Nucl. Phys. 79 (1966) 321 C. F. Clement, Phys. Lett. 2~B (1969) 398 Nucl. Data Tables A5 (1969) 433, ed. G. H. Fuller and V. W. Cohert (Academic Press, New York and London) M. L. Rustgi, R. P. Singh, B. Barman Roy, R. Raj and C. C. Fu, Phys. Rev. C3 (1971) 2238 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, New Jersey, 1957) D. J. Rowe, Nuclear collective motion, models and theory (Methuen, London, 1970) D. D. Armstrong and A. G. Blair, Phys. Rev. 140 (1965) B1226 F. Hintenberger, G. Mairle, U. Schmidt-Rohr, P. Turek and G. J. Wagner, Z. Phys. 202 (1967) 236 E. Newman and J. C. Hiebert, Nucl. Phys. A l l 0 (1968) 366 B. Zeidman, T. H. Braid and J. A. Nolen, Jr., private communication (1968) W. Childs, Phys. Rev. 156 (1967) 71 H. Nagasawa, K. Takeshita and Y. Tonomo, J. Phys. Soc. Jap. 19 (1964) 764 J. O. Artman, Phys. Rev. 143 (1966) 541 V. Saraswati, J. Phys. Soc. Jap. 23 (1967) 761 K. Murakawa, J. Phys. Soc. Jap. 11 (1956) 422 V. S. Korolkov and A. G. Makhanek, Opt. i Spektroskopiya 12 (1963) 163; Optical Spectroscopy (USSR) 12 (1962) 87
~
NuclearPhysics A213 (1973) 493 --509; (~) North-HollandPublishing Co., Amsterdam
I.E.3
[
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THEORY OF OVERLAP FUNCTIONS
0I). The overlap representation C. F. CLEMENT t TheoreticalPhysics Division, UKAEA Research Group, Atomic Energy Research Establishment, Harwell Received 8 January 1973
Abstract: Nuclear densities and the expectatiort values of electromagnetic operators are expressed explicitly in terms of the single particle overlap functions defined in the first paper I of this series. Using the sum rules of I some of the properties of this representation are discussed. In some cases values of nuclear electrical moments cart be obtained directly from experimental data on proton pick-up and stripping reactions. As an illustration the quadrupole and hexadecapole moments of slV are obtained with a discussion of the errors and uncertainties of the method. We find Q = -0.033±0.01 b, a possibly more accurate value than that found by other methods.
1. Introduction In the first paper o f this series 1), which we henceforth denote by I, sum rules were derived connecting the single particle overlap functions defined by projecting the states o f neighbouring nuclei onto one another. Here, we intend to show explicitly h o w the expectation value o f any single particle operator can be expressed in terms of these functions. Also we give an example o f h o w the representation can be used in practice to obtain electrical moments o f a nucleus. The potential uses o f experimental data on single-nucleon transfer reactions on odd mass nuclei are not realised at present. We h o p e that this paper will show experimentalists that accurate data will not only give values for the m o m e n t s but also, and possibly more importantly, set limits on their possible errors. The basic idea o f this paper is not new. French 2) has derived the shell-model equivalents o f the expressions given here, particularly for electrical moments. A b o u t the same time Berggren 3, a) realised that the overlap functions used in stripping and pick-up reactions could determine the expectation values o f one-body operators. However, awareness and application o f their ideas has been sadly lacking. We present here the exact equivalents o f the formulas derived by French 2) in the framework o f our general investigation o f overlap functions. Because it is exact, our representation contains contributions to expectation values f r o m deeply b o u n d nucleons which would not appear in his work. Using the sum rules o f I these contributions can generally be removed if the relevant shells are 'closed' in the sense discussed in I. This is where the t Attached to University of Minnesota, 1971-2, where part of this work was performed. 493
4.94
C.F. CLEMENT
work of Berggren a. 4) is extended. If the shells are not quite closed the best place to look for their contributions experimentally is in stripping reactions. This requires a particle-hole transformation in the representation as performed by French 2) in the shell model. The transformation is briefly discussed in sect. 3. With the exact relations of I any exact transformation would involve corrections to shell-model formulas. At the present time it is probably not worthwhile to include these as they are undoubtedly small though it is possible, at least with a perturbation approach, to go beyond the shell model in the present theory. In sect. 2 we give general formulas for the nucleon densities in terms of overlap functions and spectroscopic factors. It is already a general practice in nuclear physics to obtain the density distributions from single particle wave functions specified by their separation energies. We have shown that, with non-singular potentials, the best mean single particle overlap functions are specified by Hartree-Fock-like equations where the single particle energies are specified by the mean separation energy s). Thus the prescription is justified in this case. However, we need to extend this work to include c.m. corrections and two-body correlations arising from hard cores. There are undoubtedly corrections to the mean separation energy in the equations of ref. 5) and these will be discussed in the next paper of this series. For nuclei with non-zero spin the densities are in general no longer spherical and we give formulas for the appropriate L-multipoles, comparing them to commonly used deformation parameters in the case of L = 2. It is conceivable that one might be able to obtain a value for such deformation parameters from experimental pick-up data. The representations for electromagnetic moments and transition amplitudes are derived in sects. 3 and 4, the magnetic case being considered first. Whilst one could not hope to obtain more accurate values of magnetic moments from pick-up data than those obtained by conventional atomic means, it would be illuminating from the point of view of nuclear structure to use the known values to gain information about the overlap functions. In sect. 5 we present our main application of the representation to obtain the quadrupole and hexadecapole moments of s IV. Nuclear quadrupole moments given in the Nuclear Data Tables 6) are all the results of measurements by atomic means. It is not generally appreciated just how uncertain the values are. The trouble is one that experimentalists can blame on the theory needed to interpret their observations. The interpretation requires the gradient of the atomic electrical field at the nucleus and corrections from the polarization of core electrons, both of which require in turn extensive theoretical calculations. An example of the widely different results in some cases is shown in table 2. Whilst we cannot claim that the overlap representation does not require extensive theoretical calculations, these are mainly in the extraction of spectroscopic factors from experimental angular distributions using the distorted wave Born approximation. The rest of the calculation as shown here is relatively trivial. There are major uncertainties.connected with the relative phases of spectro-
O V E R L A P F U N C T I O N S (II)
495
scopic amplitudes but in certain special cases, such as s IV, these are not too important. It is important to put errors on the results obtained and it is certainly possible to do this in the overlap representation. The major sources of error in our case are discussed in sect. 5. The result for Q obtained for 51V is compared to the value from a fairly large shellmodel calculation 7). Since we find a major discrepancy between the shell-model wave function found there and one of the pick-up experiments we consider, we regard the theoretical value as being no more reliable than the value we obtain. Finally in sect. 6 we summarise the work of this paper and give the conclusions we have drawn from our application.
2. Nuclear density distributions In this section we give the densities in terms of the overlap functions defined in I. The nuclear densities and transition densities between two states of a nucleus enter into the description of all nuclear physics experiments involving one nucleon such as electron scattering and direct nucleon inelastic scattering. Thus the general formula giving the transition density between two states is first considered. We then specialize to the densities of a particular state. The wave functions of two states of an .4-particle nucleus may be written as X , ( 1 . . . A; J,M, TrT3,) and Z~(1...-4; J~M~TsT3~). The general single nucleon transition density operator between these states is defined as A
p(r; r M, sM~) = (X,, E b(r--ri), Z~) i=1
= "4(Z,, cS(r-ra), Z~),
(1)
since all the terms give the same contribution with antisymmetric wave functions. To obtain the neutron and proton densities, p, and pp, we must insert the operators ½ ( l + % a ) , respectively. For charge transfer and spin-flip reactions the operators z ] , z ; and aa could be inserted and it would be trivial to extend the following derivation to include these cases. We generally wish to refer the density to the c.m. of the nucleus considered so that we define A r' = r - - R A , x = r A - RA_I = - (r~-- R~). (2) .4-1 The J-function then becomes = J (r'\
A
.4-1)x --
A A-1
D
A-- 1
r '2
E rLM( )rMr'). LM
(3)
496
C.F. CLEMENT
The general method of this paper for any matrix element such as that in eq. (1) is to insert a complete set of states of the (A - 1)-particle Hamiltonian on each side of the single particle operator, and then use the definitions (3) and (8) of I for the overlap functions and modified spectroscopic factors• It is convenient to extend the complete set to cover the angular, spin and isospin co-ordinates of particle A so that we can use coupled states on the left and right in the forms [~'~(1 . . . A - l ; J~T~T3~) dp(f2~, ¢a, xa; jtltta)]~t'~ and [ ~ , ( 1 . . . A - l ; J~T~T3~)dp(f2x, ¢A, ZA; j212ta)]JMs, respectively. We have anticipated the diagonality of ~ and ta in the matrix element. From (3) and (8) of I and eq. (3) we obtain
p(r'; rMr, sM,) =
~.,
O',(jlllta)O'~(jzl2t3)
ctjlltj212t3
×<1~'~(1 A A- 1
• " "
A-1 r '2
.4-1;J~T~T3~)~,~(,,,,,A,~A;j,I~t
~'~
3]_lMr,
LM~YLu(x)Y~u(r'),
[~,~(1... A - 1; J~ ~ T~)~,(x, ~ , ~ ; J2 l, t3)]~). It is now straightforward to use the Wigner-Eckart theorem a) to evaluate the reduced matrix elements and we obtain 1
E rt,4.-M.ff') p(r'; rMr, sM~) - (4n) ½.j,z,j2,2t3z. Jr
L J2)
x [(2jt + 1)(2j2 + 1)(2Jr-l-1)(2L + 1)]'½11 +(-1) ''+'' +L] (L~ 0 x O'~,(j~ I 1 ta)O'~s(j2 12 ta)
~b~,
(Lr
; j~
½
11 t3)
(4) The neutron and proton densities may be obtained immediately by restricting the sum over ta to ta = +½, respectively. We note that with the c.m. factors given, the radial density is properly normalized: t ,2
drr
~r
A
r';jlllt3
q~,,
A
r';jlllt3
)
P = / d x x 2q~,(x, 2 . Jl It ta) = 1,
J
where the last line is the normalization condition (6) of I.
(5)
OVERLAP FUNCTIONS (II)
497
In the general expression (4) the values of It and 12 and therefore L are restricted by parity conservation ~,rc s = ( - 1 ) h+`= = ( - 1 ) L. Thus only odd or even values of L occur depending on whether r and s have the opposite or the same parity, respectively. One special case which often occurs in inelastic scattering is J~ = 0 when L = Jr and J~ = Jz and we have
p(r' ; rM~, sO) -
I ~°tT~) jlllJ212tz~
X
[(2J,+ 1)(2jl + 1)] ~r ( J ~
x O~,(jl ll
J'0 ~2) ½1"1+ ( - 1)'~+12+~']
t3)O's(j 2 l 2 t3)
~,
r'; Jl Itt 3 t~ s
r , J2 12 ta •
(6)
If, in the sense of paper I, the shell (Jl Ii ta) is closed in state s, or the shell (J2/2 ta) is closed in state r, then [eq. (L 32)]
Z
0;,(j212 t3)O',(j~ It t3) = O.
(7)
~, d~ fixed
In other words the off-diagonal contribution of these shells to the density sums vanish. The actual diagonal density distribution in state J,M, is obtained by putting s = r in eq. (4);
;(~'; ,M,) = -
1
E e~(cos o) art gjlllj212taL × (s, M, L01S, M,)(- 0'o+~,+~[(2j~ + 1)(2j~ + 1)(2L + 1)~(2J, + 1)]~
{J~ J2 L t [ j , × J,
J,
J2(-½
L
J2}
o
½
×½[l+(--1)h+t2+L]o'r(jlllt3)O'r(j212t3)
c~,
r, jtl, t a
x ¢~, (A-~I r'; jz12t3).
(8)
Only even values of L occur. The number of nucleons is given by
A = fP(r') d3r' = E S',(jlt3), J
(9)
~jlt3
where we have used eq. (5) and, since L = O,j l = Jz and only 11 = lz can contribute. Clearly we also have
N = Z S~,(jlta = +½), ~tjl
These results agree with eq. (14) of I.
Z = ~. S;~(jlt a = - ~ ) . ajl
(10)
498
C.F. CLEMENT
In fact for L = 0 no off-diagonal terms where (j111) density. For spin zero nuclei where J, = 0 the density is p(r')
l~s,r(jlt3 = ~- ~1u3
)
[A '~3d~2( A ~-~-~1 ~-~, ~ - ~
For a general spin nucleus where the shell [eq. (I. 34)] E
# (j212) contribute to the )
r' ; jlt3 .
(11)
(flt3) is closed we can use the sum rule
S;, - 2 J ~ + l . 2Jr + 1
(12)
aJotfixed The sum over J~ becomes
j Jrj J~LI}= F(2Jr+l)(2j+l).]~'6z °" L ~
J.E(-1)s'+J'+J(2J~+l)tJr
(13)
Thus, as expected, the contribution of any closed shell to the density is
P r ( r ' ; j l t 3 ) - 2 j +41rl / A ~-~-~)'~3"2 9~, ( ~ A
r'; jIG ) •
(14)
For nuclei with finite Jr, L goes up to the maximum even value < 2./,. For L = 2 deformations, which are mostly considered, we can compare the density deformation in the state M r = Jr to two model formulae: (A)
p(r) = po(r)[1 + flA r2o(0~b)], x2 + y 2
(B)
p = const,
a2
(15)
g2
+ ~ < 1,
(16)
where
To compare to our result we demand the equivalence of the quantity
f p(r t ; rJ,)Y2o(P t)d 3r .I Then the result is, to first order in fiB,
flA
=
3fls
=
1 (4~ ~ , (J,J,2OIJ, J,) A \ 5 ] ajill j2123
t •
t .
X O~r(J , l, t3)Oar(J 2 12 t3)
~- x 2dx~b~,(x,.
Jl l, t3)~)~,(x ; 22 12t3)"
From our general result (7) no closed shells can contribute to this sum.
(18)
OVERLAP FUNCTIONS
(II)
4.99
No attempt has been made yet to evaluate the above density expressions. However, one possible application is to the difference between neutron and proton densities in nuclei. From observed spectroscopic factors, it would be possible to calculate the overlap functions with the aid of the Hartree-Fock equations of ref. 5) (or extensions of them) and hence the density distribution for even Jr = 0 nuclei. For odd nuclei the relative phases of the spectroscopic amplitudes are needed too. Theoretical investigations, such as the effect of adding more neutrons to the lf~ shell in the Ca isotopes, might be worthwhile in order to compare the results to those from experimental investigations of changes of radii with mass number. However, we intend to investigate c.m. and correlation corrections to the Hartree-Fock equations before proceeding with such applications. 3. Magnetic moments and M1 transition amplitudes
Here, we apply the same techniques as in the previous section to evaluate magnetic moments and M1 transition operators in terms of spectroscopic factors and overlap functions. We include the case of the transition operators because, although the spectroscopic factors for excited states may not be known, it is a trivial extension of the magnetic moment operator case and could be a useful parametrization. Higher magnetic multipoles can be treated in the same way. In units of Bohr magnetons (eh/2Mpc) the magnetic moment operator may be written as /~ = E [¼(1 +'r3,)g n a, + ¼(1 - z3,)gp a, + ½(1 - z3,)l,]. (19) i The magnetic moment of a state X,(1 • • • A; p, = ( Z , ( 1 . . . A ; J , J , T ,
J,M, T, T3r) is conventionally defined as
T3~),p~,Z,(1...A;J,J~T~T3,)).
(20)
The M1 transition matrix element between two states s and r is
( 3 ~ ~ eh
[4-~n] 2-~pc (Zr(1 " " " A; S~ M~ T, T3~), p ~ , Zs(1 • • • A ; Ss Ms T~Tas)), where the operator is written in its tensor form. To evaluate this general matrix element we insert complete sets for the ( A - 1)particle system as in the previous section• Then
<[~P~(1...A-1;J~T~T3~)(o~,(x, ea,zA,
~jtllt3J212
¼(1 + ~ ) g ° ~ , + ¼(1 - ~ ) g ~ ~ , + ½0 - ~)(/~)~,, I-~0
• • • a-
•
Js
!
•
a; s~ T~ T~)O~s(x, ~ , ~,; s2 l~ t ~ ) ] ~ ) o = 0 , 1 ~
=(-1)s"-u"
( J~
--M r
t3o~sO~ l~ t~)
1 Y~) ~ O,(j llG)O,'s(jzlt3) m Ms ,S,S~u~
t
•
t3)]t~r,s~
500
C.F. CLEMENT
x(-l)S'+'"+t+Y'+Y2+'[(2J,+l)(2J,+1)(2j, +l)(2j2+l)]½{j I J,
+ ( - 1)i2-J~(½-t3)[l(l + 1)(2/+ 1)] ~ {jl2
l
Jl
× fax x2+,,(x; J, lt3)dP,~(x ;J2 lt3).
(21)
To a good approximation, especially when r = s, the radial integral will be unity, which it is of course defined to be when Jt = J2 and r = s. The magnetic moment of a state r then becomes
]It
= FJr(2Jr+l)-] * 20'-r(j, lt3)O;r(j21t3) L ~
I ~tjlj2lt3
× (I(½+t3) gf +½(1-t3)~--] ~/(){)2
+(_l)i2_j,(½_t3)[l(l+l)(21+l)] ~ {jl
J,
Jl
"
It can easily be shown, using eq. (7), that no closed shells can contribute to the sum. Furthermore, i f a shell is nearly closed, we can use sum rule (25) of I to represent the operators in terms of the spectroscopic factors and overlap functions for stripping reactions. From a practical point of view this might be advisable because of the large number of pick-up states which would be involved and large cancellations in the sums over J~ in (21) and (22). The transformation from the ( A - 1)-nucleus to the (A+ 1)nucleus representation is the exact analogue of the particle-hole transformation in the second quantisation representation of the shell model. We restrict the discussion to the case when all the q~ are approximately equal and continuum contributions and I/A corrections are ignored. It is possible, but probably not worthwhile at present, to include these corrections. To transform generalized expressions such as (21) and (22) with an operator rank L (:/: 0) we require the recoupling identity
s.
[J~ J. J~I(Jl J~ J.
(23) =
J2
Jr
"
OVERLAP
FUNCTIONS
(II)
501
From eq. (13) the contribution of any diagonal sum rule term vanishes so that, for example, the contribution of a given (JlJ2 l) to eq. (22) for/1 r is transformed to [J,(2J,+l)]~2Jn+l
~r(Jl J2 1) = . j - - ~
, .
2J---~ Or~(ll lt3)O;.(j2 It3)
{j, 1}
×(--1)z"+z'+t-½[(2j'+l)(2j2+l)]& Jr J, J,, x {[(½+t3)9"+½(1--t3)~lx/'6{~j2
½ ~}
+(-1)J~-J'(½-t3)[l(l+l)(21+l)J~{J2
Jl
'
where 0~= is the spectroscopic amplitude for stripping to a final state n, as defined by (4) and (8) of I. For a general L a representation such as (21) assumes a similar form with the suffix g replaced by n, a factor of ( 2 J , + l ) / [ ( 2 J , + 1)(2Js+ 1)] ~r included and an additional phase factor of ( - 1 ) J'-~'+il+j2+L. For Jl = J 2 and Jr = Js this phase factor is ( - 1 ) L+z, identical to that of the particle-hole transformation in the shell model as discussed by Rowe 9), for example. Since accurate values for magnetic moments of nuclei are known from atomic methods, it is unreasonable to expect the overlap representation to provide better values. However, one might use the known values to compare with data on pick-up reactions and, in particular, to help to discover the phases of the products 0~, (Jl = l+½ lta) 0'~, (J2 = l-½ lt3) which enter into expression (22). As is further discussed in the next section, these products are important physical quantities which enter into the expressions for all electric and magnetic moments and also play a role in any dynamical theory of overlap functions. The latter point will be shown in a future publication and we hope that eventually known magnetic moments will be compared to the values given by (22) and (24). Furthermore, if there are any considerable exchange current corrections to the expression for the magnetic moment operator, they will form a discrepancy between a value from (22) and a measured atomic value. It would be interesting if one could get sufficient accuracy on the overlap representation value of the magnetic moment to detect exchange current corrections. 4. Electric moments and EL transition amplitudes
The electric moments and EL transition rates for an A-particle nucleus are obtained by taking matrix elements of the operators A
Q(ELM)
= e ~ ½(1 - T3,)lri i
-
-
RAILYLM(r,-- RA).
(25)
502
C.F. CLEMENT
Thus the quadrupole moment of a state Z , ( 1 . . . A; J, MrT, Tar) is defined as
eQ =('~106
(26)
In our representation the general matrix element is
= eA ( ~ - ~ ) L
(27)
where we have used the relation (2) to obtain the correct c.m. correction. We note that, in this representation, as opposed to the more usual shell-model representation, the effective operator still refers to protons only even in the electric dipole case. The matrix element is evaluated as before to obtain <)~r(dr Mr) , Q(ELM), Z,(J, Ms)> = e((A - l ) / A f
~tJlllj212
--M,
M
M~
4x
/. X
O~r(jl lx - ½)0~,(j212 - ½)3~b=r(x; Jt 11 --½)~)~ts(X;J2 12--½)XL+ 2dx.
(28)
The quadrupole moment, given by eq. (26), becomes
x O;r(j, I, -½)0"r(j2 I2-½)Q~(j, 1, ; J212),
(29)
where we define the radial integrals as
II, J2 12) = f 6~tr(X; Jl 11--½)6~r( X; J2 12 -
½)x4 d x .
(30)
Alternatively, the quadrupole moment can be obtained from the density distribution given by eq. (8).
eQ = (-l~)½e J r'Ey2o(V)pp(r'; rMr = Jr) dad, where pp is the proton density obtained by including only t3 = - ½ in eq. (8).
(31)
OVERLAP FUNCTIONS (II)
503
This result is equivalent to eq. (27) since we have
(~)
2Q~(jl I1; j 2 1 2 ) =
A 3 (a-~--l) f
dr'r'4
x q~, ( A - ~ r'; Jt l1-½)~b~, (A-~I r';
J212--½)"
(32)
The higher static electric moments may be obtained in the same way. For example, the hexadecapole moment is (other definitions may differ by a constant)
Z.(1 • • • A;
eH = J, Jr T, T3r)>
(-~670½(Z,(1... A; J, J, T~ Tar), Q(E40),
= (J~rc)½ej -r '6 r,o(P')pp(r';
=e
rM.
Jr -Jr
= dr)d3r '
2 0 " , ( j l lt-½)
jlllJ212ct,
×O',(j2l: -½)(- 1)~+s-++[(2j~ + 1)(2j2 + 1)]½[1+ ( - 1)t'÷t~] x (j~
4~ 2 ) { J j 0
JrJ2 j4}H,(j,l~;j212) '
(33)
where
t" H~(ji ll ; J2 12) = J dx x6$.r(x; Jl lt)~b~r(x;
J2 •2).
(34)
To calculate the moments from these expressions we need to know the form factors, ~b,r, and the spectroscopic amplitudes, 0",. The former may be obtained fairly reliably from the shell model, but from pick-up reactions we generally only know S~,, which are the squares of the latter. Thus the phase of O'~,(jtll) 0",(j212) for (j~ lx) # (J2/2) is undetermined. In some special cases, such as the one discussed in the next section, this may not be an important drawback. We also have some hope of determining the important phases within the framework of this theory by constructing dynamical equations which contain them. They are also given by complete shell-model wave functions for the states in question. Of course the phase is independent of the phases chosen for the states ~ and r, and is thus a physical quantity which plays a very important role in the overlap representation. To transform any of eqs. (28), (29) or (33) to a representation in terms of stripping amplitudes we can use the general result of the last section to replace ct by n and multiply by the factor (-- 1)s'-s'+i' +Yz+L(2J,, + l)/[(2Sr + 1)(2Ss + 1)] ½. For this transformation we assumed that all the ~b were equal so that the states n included must only contain dpr.(x;jl-½) corresponding to the same major shell.
504
C.F. CLEMENT 5. The electrical moments of StV
In this section we describe an application of the formulae of sect. 4 to experimental results for pick-up reactions on 51V. This will illustrate some of the points involved in using the overlap representation and also gives a respectable value of the quadrupole moments and a prediction for the hexadecapole moment. The former is usually measured by atomic methods which are subject to some uncertainties. Our application shows that, for some nuclei at least, a careful analysis of the results of proton pick-up reactions is a very competitive method. Nuclei in the lf~_ shell are a particularly favourable case since only a few (fl) values are involved, the dominant contributions being expected to be from the lf~ and 2p~ shells. In order to evaluate the electrical moments we need estimates for the single particle integrals Q~ and H~ given by eqs. (30) and (34) respectively. To obtain these we have adopted the simplest possible approach and have used harmonic oscillator wave functions. The integrals are relatively insensitive once the size of the nucleus is fixed and the use of more realistic form factors, such as Woods-Saxon shapes, is hardly worthwhile in view of larger uncertainties in experimental spectroscopic factors. Taking into account the fact that the usual oscillator wave functions are defined in in the variable r' the results are Q((lf)2) = Q((2p)2) =
Q(lf, 2p) = -
(A)2 ~
4v' (14)~ 2v '
H ( ( l f ) 2 ) = (A__~) 4 99 16v z ' n ( l f , 2p) = -
( A ) ' 2 7 x ( 1 4 ) ~~ 16v2
(36)
A good estimate for the oscillator parameter v is obtained from ho~ ,,~ 40 A-* MeV giving v- 1 = 2.065 A ¢ fm 2, (37) As an illustration of the type of numerical form of Q obtained we give explicitly the contributions to eq. (29) for a J, = ½ nucleus from lf~, lf¢ and 2p t protons. (any 2p~ contribution is expected to be negligible for our example of 5tV):
Q = Q(lf~)+a(lf~)+Q(2p~)+Q(lf~,
2p~)+Q(lf~, lfl), II
Q(lf~) =
-
9"S-i+VS2-2S3ta~-T-s-7v -
- ~ S1,4 -
- r ~aS75-
--
+~s6),
-
'7
--
OVERLAP
Q(2p~) = Q(lf~, 2p~)
=
9 lOv
-- _ _
FUNCTIONS
(II)
505
_ ]-xS4
_ ~(})~ 1 [02 + (~x-)~O~ + (~)~0~ -
9~ ½
-
],
V
Q(lf~, lf~)
-
-
+
~O£-,-s-/
T~', ---~v6 j ,
(38)
where
S;,(jl)=
E
S'r(jl-½),
e, d~ fixed
Of~(jt 11; j212) =
E
O'er(jr l~-½)O'r(j212-½)
~¢,J= fixed
=
E
+__[S'r(jt l ~ - } ) S ' , ( j 2 1 2 - ½ ) ] ~.
(39)
~, J~ fi xed
In the above formulae it has been assumed that the overlap functions, and hence the single particle quadrupole moments, given by eq. (34), are the same for f~ and ft" The use of more realistic functions would, relative to Q(1 f~), increase Q(1 f~) because of less binding energy for lf~ particles and decrease Q(lf~, lf~) because of less than perfect overlap. An obvious feature of the expressions (38) is that the signs of the various terms are such that cancellations are expected to occur. Indeed we know that if a shell is full there must be complete cancellation of the appropriate contributions to Q. For an odd proton nucleus such as S~V little or no contribution is expected from what would be the unnatural parity states of spin 1 +, 3 +, 5 + and 7 +. Such states in S°Ti could have n o (fqr) 2 components to connect to a dominant (f~)a configuration in SlV. Similar formulae giving the contributions of ld and 2s holes to the quadrupole moment can be written down using the rules of sect. 4 to relate them to the results of proton stripping reactions on 51V. In (aHe, d) reactions performed on 5tV [ref. t o)] there is no reported ld or 2s strength up to an excitation energy of 8.7 MeV in 52Cr. This makes it very unlikely that there is any significant contribution to the quadrupole moment from ld or 2s holes. One would like, however, to have an experimental upper limit on the strength in order to put errors from this source into our expression for the quadrupole moment. Theoretically there can be no contribution from pairing force mixing across a closed shell since the pairs are always coupled to J = 0 giving no contribution to Q. However, there could be a contribution from 'polarization" of the core in which, for example, a ld particle is excited into a 2d state. Any such contributions from d- or s-shell protons cannot interfere with the dominant lf~ contribution. Because of the different parities there are no terms like 0(ld~)0(lf~). The potential error from neglecting the s-d shell is thus reduced and is independent of the f-p shell contribution apart from the requirement that the total number of protons is fixed.
506
C . F . CLEMENT
Turning to the I f-2p contribution several experiments 11- ~3) have been performed which have found l = 3 proton pick-up leading to J = 0 ÷, 2 +, 4 ÷ and 6 + states with the possibility of small l = 1 contributions 13). We first assume that all the l = 3 strength arises from lf~ transfer and the l = 1 strength from 2p~ transfer. Then the quadrupole moments calculated from eqs. (37)-(39) and the hexadecapole moments calculated from similar expressions obtained from eqs. (33), (36) and (37) are shown in table 1. The variation from Q = - 0 . 0 3 3 b to -0.041 b gives an indication of the error expected from the f~ contribution. We see, however, that even a small l = 1 contribution can significantly change the quadrupole moment according to the sign
of 0~ (~, ~). TABLE 1 Electrical moments of s ~V from proton pick-up reactions
~So- (~)
$2- ({)
Spectroscopic factors S , - ({) $ 6 - (~)
0.73 a) 0.74 b) 0.72 c)
0.39 0.37 0.34
0.64 0.75 0.81 a)
1.05 1.14 1.08
0.75 f)
0.42
0.75
1.08
a) Ref. 11).
b) Ref. 12).
$2- ({t)
Q (b)
02-(5, {)
0.023
+0.088 --0.088
--0.041 --0.033 --0.036 e) --0.024 --0.038
H (10- 5* cm 4) 0.21 0.21 0.30 =) 0.15 0.22
o) Ref. aa).
d) Includes two states. e) Values correspond to 4- signs in the previous column. t) Theory based on (fk)3 configuration. In order to discuss this 2p contribution further and to consider a possible lf~ contribution we refer to a large shell-model calculation for s ~V performed by Rustgi et aL 7). The ground state proton wave function they find is 7J = 0.94(f~)3-0.13[(p~)2]°f~-0.25[(f~)2]°f~. The coefficients differ very slightly in the three approximations they make and we have taken the median values. The spectroscopic factors for pick-up of p~ and f~ protons leading to states [p~f~]s and [f~f~]s respectively are Sj(2p~) = (0.13) 2 2 J + 1 , 16
Sj(lf~:) = (0.25) 2 2 J + l 24
The first thing we notice is that the value for S;(2p~) is completely inconsistent with the observed value of 0.023 to the lowest 2 + state found by Zeidman et aL ~3). Even if this 2 + state is 100 Y/oo[p~f~]2, which of course is extremely unlikely, the value of S;(2p~) would only be 0.005 with the above wave function. This indicates that either the DWBA analysis of the experiment is incorrect or there should be very much larger 2p~ components in the wave function. The factor 2 J + 1 in Sj indicates
507
OVERLAP FUNCTIONS (II)
that, if the major component is [(2p~)2] °, one should observe experimentally l = 1 transfer strength to 4 + states which is nearly a factor 2 bigger than the 2 + strength. Secondly, a wave function which has only admixtures of [(p~)2]o and [(f~)2]o gives no p~ or fl contribution to Q. This is of course an immediate result because of the zero spin but can also be seen in our formalism from the factor 2 J + 1 contained in Sj and eq. (13). In other words the contributions to all the terms in eq. (38) except Q(lf~) must cancel completely. In the spirit of our approach working directly from experiment we only have S~ (2p~) and 0 2 (2p~, lf~) non-zero in the 2p~ terms. Thus cancellation does not occur with the result that Q takes on the two quite different values of - 0 . 0 2 4 b or - 0 . 0 3 6 b as shown in table 1. It is possibly more realistic to go some way towards theory and say that admixtures of states such as [[(lf~)2] 2 2p~] ~ or [[(lf~)2] 2 lf~] ~ which would produce 2p~ or lf~. contributions to Q are relatively small. Then an error of 0.01 in Q from taking the value of Q(lf_~) alone would be an overestimation. TABLE 2 Quadrupole moment of 51V Method
(bQ)
atomic and molecular beams nuclear magnetic resonance
--0.052 a) ~0.04 a) +0.28 c) +0.26 r) --0.033-4-0.01 --0.042 :k0.002 h)
optical spectroscopy proton pick-up reactions shell-model theory z) a) Ref. 14).
b) Ref. 15).
c) Ref. 16).
~:0.0073 b) :~0.26 c)
d) Ref. xT).
e) Ref. is).
r) Ref. 19).
s) Ref. 7).
h) Error from three different calculations. Values of the quadrupole moment of 5 t V obtained from various sources are shown in table 2. It is apparent that some values obtained by atomic methods are badly inaccurate; though they may be old experiments they are still quoted in Nuclear Data Tables 6). The value and error we give from proton pickup reactions cover the values shown in table 1. From the above discussion we are fairly confident that the error covers all missing f-p shell contributions. There is no experimental evidence at present for any s-d contributions though we would like the (3I-Ie, d) spectrum on 51V to be closely examined for l = 0 and 2 transfer. We would not regard the shellmodel theory value as being the final answer, particularly in view of the discrepancy commented on above with experimental l = I transfer. The actual value may be quite close to the pure (f~)3 value. Without invoking interactions with the s-d core, the above examination of the various terms in the overlap representation has indicated that one needs large and unlikely constituents of lf-2p wave functions in 51V to obtain a much different value for Q.
508
C.F. CLEMENT 6. Conclusions
We have expressed the principal one-body operators in nuclear physics in terms o f spectroscopic factors and overlap functions. Collectively the representations, including those for the densities, apply directly to almost all nuclear physics experiments involving the electromagnetic field and to many other experiments such as inelastic scattering. Since the single particle wave functions are relatively well known, the important unknown parameters which specify the representations are quantities like E~O'~s(j212 ta)O'ar(j 1 ll ta), wherejl ll, J2/2 and possibly J~ are fixed. For ground states these quantities are, apart from phases, accessible to experiment. For excited states this is not so, apart from a few special cases where one can, for example, perform inelastic scattering via an analogue state. There are, nevertheless, reasons why the representations may be useful. As discussed in the introduction to I, general microscopic comparisons between theory and experiment in nuclear physics involve calculating theoretical many-body wave functions for the states involved. This procedure has two drawbacks, the first being that the model space is necessarily limited. Matrices to be diagonalized grow enormously with the addition of extra configurations. The second drawback is that the many-body correlations built into the wave functions are generally not accessible to experimental test. One therefore in effect has to calculate the single particle quantities expressed here before comparison to experiment can take place anyway. The main exceptions are the actual energies of the states, determined by two-particle correlations, which are in many cases notoriously insensitive to small one-body properties which can drastically change electromagnetic transition rates, for example. Thus our first main point is that the parametrization in terms of overlap functions can be a very useful intermediate stage in the comparison between microscopic theory and experiment. This brings us on to the discussion of errors. With the usual many-body wave functions it is difficult if not impossible to put errors on the single particle quantities obtained. For example the inclusion of the next major shell may be beyond the bounds of calculational possibility. Working directly with the overlap representation the calculation of possible errors is relatively easy. We have to specify a certain small contribution from the spectroscopic factors of the next shell when the calculation of possible errors is straightforward. Thus our second point is that when single particle quantities are looked at via the overlap representation any discussion of errors in a theory becomes immeasurably simpler. The example we have discussed in sect. 5 exemplifies both points. A fairly large shell-model calculation was performed for 5iV [ref. 7)] but does not appear to give a wave function which has configurations consistent with single particle transfer data [ref. la)]. Working directly from experimental data we have obtained Q = -0.033 + +0.01b with an estimated error from the lf-2p shell. To add an error from the 2s-ld shell we need to put an upper limit on l = 0 and l = 2 stripping transfers on 5iV.
OVERLAP FUNCTIONS (II)
509
I n the existing e x p e r i m e n t 10) there is no evidence for such transfers b u t we urge t h a t there be further e x a m i n a t i o n o f the data. The o v e r l a p r e p r e s e n t a t i o n also gives the o t h e r m o m e n t s a n d we find H = (0.22-t-0.8)x 1 0 - 5 4 c m 4. I n conclusion we t h i n k there m u s t be m a n y m o r e examples in n u c l e a r physics where e x p e r i m e n t a l d a t a c o u l d be p r o f i t a b l y l o o k e d a t f r o m the p o i n t o f view o f the o v e r l a p representation. I t also m a k e s w o r t h w h i l e the careful e x a m i n a t i o n b y experimentalists o f single n u c l e o n transfer d a t a for w e a k transitions f r o m o t h e r m a j o r shells t h a n the m a i n one for the nucleus concerned. T h e a u t h o r w o u l d like to t h a n k once m o r e the staff o f the University o f M i n n e s o t a for their hospitality and, in p a r t i c u l a r , Dr. N. H i n t z for his help in finding the d a t a on 5iV.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
C. F. Clement, Nucl. Phys. A213 (1973) 469 J. B. French, Phys. Lett. 15 (1965) 327 T. Berggren, Nucl. Phys. 72 (1965) 337 H. Tyr6n, S. Kullander, O. Sundberg, R. Ramachandran, P. Isaacsort and T. Berggren, Nucl. Phys. 79 (1966) 321 C. F. Clement, Phys. Lett. 2~B (1969) 398 Nucl. Data Tables A5 (1969) 433, ed. G. H. Fuller and V. W. Cohert (Academic Press, New York and London) M. L. Rustgi, R. P. Singh, B. Barman Roy, R. Raj and C. C. Fu, Phys. Rev. C3 (1971) 2238 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, New Jersey, 1957) D. J. Rowe, Nuclear collective motion, models and theory (Methuen, London, 1970) D. D. Armstrong and A. G. Blair, Phys. Rev. 140 (1965) B1226 F. Hintenberger, G. Mairle, U. Schmidt-Rohr, P. Turek and G. J. Wagner, Z. Phys. 202 (1967) 236 E. Newman and J. C. Hiebert, Nucl. Phys. A l l 0 (1968) 366 B. Zeidman, T. H. Braid and J. A. Nolen, Jr., private communication (1968) W. Childs, Phys. Rev. 156 (1967) 71 H. Nagasawa, K. Takeshita and Y. Tonomo, J. Phys. Soc. Jap. 19 (1964) 764 J. O. Artman, Phys. Rev. 143 (1966) 541 V. Saraswati, J. Phys. Soc. Jap. 23 (1967) 761 K. Murakawa, J. Phys. Soc. Jap. 11 (1956) 422 V. S. Korolkov and A. G. Makhanek, Opt. i Spektroskopiya 12 (1963) 163; Optical Spectroscopy (USSR) 12 (1962) 87