Parity mixing in 19F and the weak parity non-conserving nucleon-nucleon interaction

Ntrdear Physla A271 (1976) 412-428 ; © North-Nollatd Pubhahinp Co., Anvterdatn Not to be reprodaoed by photoprlnt or m}ao81m w}thout writtm P~lealoa flrom the pabliaher

PARITY 11~QIGNG IN "F AND THE WEAä PARITY NON-CONSERVING NUCLEON-NUCLEON INTERACTION M . A. BOX and A . J . GABRIC School of Physics, Universityy of Melbourne, Parkrille, Vic. 3032, Australia and BRUCE H . J . McKELLAR t Service de Physique Théorique, Centre d Etudes Nucléaires de Saday", BP no. 2, 91190 Gif-sur-Yoette, France Received 5 Jannary 1976 (Revised 6 April 1976) A6sfract : We report calculations of the asymmetry of the I10 keV photon emitted in the decay of the polarized =- first excited state of °F . The calculations employ short range oorreladons obtained by solving the Bethe-Goldstone equation in the .! = 20 system, and are carried out for a wide range of weak parity non-conserving potentials suggested in the literature. Because of the large experimental error of the observed asymmetry, many of these potentials give results compatible with the experimental value . A calculation of the parity allowed El matrix element suggests that the standard Cabibbo potential gives results which disagree in sign with the observed asymmetry, which has also been the case in other parity non~onserving transitions .

1 . Intro~cdon

The asymmetry A r (with respect to the spin direction of the i- initial state) of the 110 keV photon emitted in the transition between the ~- first excitod state and the i+ ground state of' 9F has recently ban measured by Adelburger et al. t). They find that Since the t9F system is much simpler than the heavier nuclei such as'e°Hf, tat Ta, etc., in which parity violations have ban observed Z) it is of greatinterest to calculate Py from the various weak parity non-conserving (PNC) potentials which are in current use. Results using the so called "standard PNC potential", which we define below, have ban reported previously by two of us s) and by Gari et al.'). In this paper we report on the extension of these calculations to include a more general form of the weak PNC potential. We also extend the previous calculations to compute the sign of A r. This requires a calculation ofthe El matrix element connecting the ~- and ~+ states . In our earlier f On leave from the School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia. 4}2

PARITY MIXING

41 3

work we were dissuaded from such a calculation by a statement of Benson and Flowers s) that this transition is forbidden in the simplest model. We now believe this statement follows from a misunderstanding of the role of the c.m. motion in the transition, and proceed to the calculation. Of course the hope in a calculation such as this is that the experimental data will permit a decision to be made between the various potentials. This is not possible with the present data . Firstly because of the large error which manages to embrace most of the calculated results, and secondly because this transition involves both dT = 0 and dT = 1 parts of the potential, which does not permit a distinction between enhancement in one part or the other of the potential. (It will be important to notice that this transition does not depend on the dT = 2 part of the potential.) In spite ofthis, many of the PNC potentials give results which are nearly eliminated by the present experiment, and we strongly urge that it be repeated to reduce the statistical errors . Our calculation of the sign of .! r is, with the standard potential, in disagreement with the observed sign. This confirms our previous experience, both in the reaction n+p -~ d+y, and in the transitions in heavier nuclei, where calculations with the standard potential have almost invariably given the wrong sign of the PNC observable . The difficulties which beset calculations of other parity violating observables that have been measured are too well known to repeat here Z). However there are a number of features which lead to the belief that one may be able to avoid many of these problems in this case. These are (i) The ~+ ground state and i - excited state are mixod with each other by the PNC interaction. Admixtures of other spini states, which aremuch further away in energy are estimated to be very small. (ü) The matrix element of the PNC interaction connecting the s+ and ~- states is largely determined by the dominant shell-model configurations for these states, and is therefore insensitive to the finer details of nuclear wave function . (iü) With the possible exception of the sign of the El matrix element, the parameters of the electromagnetic transitions which are required can be measured or reliably calculated. Following Blin-Stoyle e) and Manqueda'), or working from the general result given by Rose and Brink e) and Brink and Satchler ~ it is straightforward to show that the asymmetry, A r, defined by dQ/dQr ~ 1 +Ar cos Br,

(2)

is given by Ar

_ Jn

~

Mp(1n_

}

NÜ"r)-~~~) E+-Er

.~ V~~~yPNCI~~

Fl(11li

El

414

M . A . BOX et al .

where 4~ U~~II k

[H, r2 C~7111~i,

(3')

with the conventions of Rose and Brink s) and Brink and Satchler regarding the C,,~, the reduced matrix elements, etc. Here, ~(.1) is the magnetic moment, and F(lC,,~) is the angular distribution coefFcient of Ferentz and Rosenzweig 8 9). In deriving (3) use has been made of the time reversal invariance of Vrr,o but no assumptions have been made about the time reversal properties of the states ~. In fact, as it is written (3) is independent of any assumptions about the phases of these states . With the cannonical phases (i.e. using z~Y~ for angular momentum wave functions) both El and <.!~'IV~cL~"i are separately real. With this definition ofE 1, the B(El ) ofBohr and Mottleson '~ isjust IE 1 IZ. Since this is the normalisation used by Blin-Stoyle and hence most of the literature on parity mixing in nuclei, we use it here rather then the slightly different multipole operators of ref. e). We refer the reader to that reference for a discussion of the pitfalls to be avoided if he wishes to rewrite (3) so that the matrix elements appear in the order (J~`*I . . . fix') . In the neat section we discuss our choice of wave functions and the electromagnetic parameters derived from these wave functions and from experiment . Then in sect. 3 we discuss the possible choices of V~c and present our results in sect . 4. Since the sign ofA~ is ofconsiderable importance and interest, we present aneffective potential calculation in appendix A which has the advantage of clearly demonstrating the origin of the sign . Indeed in this simple model the uncertainty regarding the sign of the El matrix element is completely eliminated - it cancels out. However in the complete calculation the sign of the El matrix element does not automatically cancel . For this reason we consider in appendix B a model for the effective charge of low energy electric dipole transitions. 9)

~

2. Wave f~ctlooe and electromagnetic parameters 2 .1 . INTRODUCTION

From eq. (3) it will be seen that one needs the initial and final states ~-~ and ~*) and the parameters El, N(i ) and p(~ * ). Experimentally ii) we find IElI

= 0.022 e ~ fm,

N(i * )

= 2.6287 n.m.

(4)

No experimental value of N(i -) is known, but it is expected to not be very different from the magnetic moment of the ~- ground state of 1sN, which is As a guide, let us see what information we obtain from a very simple model of the lowest two states in 19F. We consider the ground state as a 2sß hole in ~°Ne and the

PARITY MIXING

41 5

i_ excited state as a lp.~ hole in ~°Ne . This gives El 2.7923 n.m., = -0.24 fm. (6) N(~ +) = u(~ -) _ -0.2641 n.m., In calculating El we have discarded the argument of Benson and Flowers s) and Harvey 12) that El = 0 in this model. Their argument is that in the transition between these states only a proton moves. The other protons are then constrained to recoil to leave the proton c.m . fixed so that no dipole radiation results. While we accepted this argument in our earlier publication a) we now believe it to be in error, because of a misunderstanding of the role of recoil in the transition. Although our wave functions imply only proton movement, the entire core takes part in the recoil. This then introduces an effective charge in thê usual way ' 3). This effective charge is used in the calculations of this paper. The matrix element El is quoted for states with cannonical time-reversal properties . These simple wave functions reproduce the magnetic properties of the states of interest very well, but do not reproduce the value of El . This failure is very common for low-lying transitions, and is a consequence of the transfer of E1 strength to the giant dipole resonance (GDR). Any wave function which does not include the correlations which generate the GDR should be expected to fail to predict low-lying E1 strengths. Following our simple model further, we see that it predicts that the 2+ and ?states will be strongly mixed by the PNC potential. One can readily see this using the effective potential approximation ") e ~

e(NlA)

VPivc

^ -FQ . P~

(7)

We therefore expect the value of the matrix element <2+ ~ V~~-~ to be relatively

little affected by configuration mixing, in contrast to the situation in some heavy nuclei, and have confidence in the use of relatively simple wave functions to evaluate it. However it is desirable to go beyond this initial very naive model. 2.2 . THE ~+ GROUND STATE

There are now many Hartree-Fork calculations of the wave function of this state. We have usod the wave function of Ripka and Zamick i'), in which the three s-d nucleons occupy the deformed level, which for K = i is given by +,~; ~9Fi _ -0.7241~1d~) +0.5615~2s~)+0.4005~1d~~.

(8)

ïn the calculation of Ripka ' e) the four s-d nucleons in the ground state of ~°Ne occupy the similar orbital + ;~;2°Ne~= -0.7184~1d~~+0.5673~2s~)+0.4025~1d~), (9) so to a sufficient approximation we may regard the 19F gmund state as a i+ hole in

M. A. BOX et a!.

416

zoNe, where a~, creates a particle in the state (9). The magnetic moment calculated from this wave function is's) which is intriguingly further from the observed value than our previous simple estimate, but is still within 10 ~ of it. 2.3 . THE }- EXCITED STATE

Many calculations exist of wave functions of negative parity states in t9F. We choose to use the Hartree-Fuck wave functions of Ripka 16), in order to be consistent with our ~+ wave function . Once again we can regard the i- state as a hole in z° Ne, where a~_ creates a particle in the state whose M = + i representative is ~~ = 0.9626~1p~~-0 .2710~1p~~.

For such a state the magnetic moment is simply l')

(13)

where gt, g, and go are the g-factors of the orbital motion, the spin, and the core (9, = 5.585, 8e = 1, g~ Z/.!) and a is the deooupling parameter t This gives which is very close to the "experimental" value. 2.4. THE El MATRIX ELEMENT

Using the wave fimctions described and the adiabatic method of obtaining wave functions of good angular momentum t6), we obtain which is four times the experimental value. We have yet to introduce the effects of coupling to the giant dipole state. In the simplest approximation, this introduces a further multiplicative correction effective charge of ' e) f

When the intrinsic stau

~~ ~ ~cmMl~, then a = ~(-1~-'~=U+})~C~ol' .

PARITY MIXING

41 7

where irmo is the oscillator energy and ED is the excitation energy of the dipole state. A modification ofthe Brown-Bolsterli model of the dipole state ' ~, which is outlined in appendix B replaces this by

> (18) é~i _ (~ -~/En = 0.33, where E is the average energy of the low-lying dipole states, and E is the energy of the state whose decay we are considering. Other estimates of the effect of coupling to the giant dipole resonance have been given by Bohr and Mottleson z~ and bY Johnstone and Caste1 21), for Z = Nt 1 they. give éc4>

-

1-

6Vi A~r2)

ra(E~E fic _ 1 es 4n2

E2-es,

where Yl x 100 MeV is the nuclear symmetry energy coefficient, and a{E) is the El y-absorption cross section on the neighbouring N = Z core . For the case of 19F this gives é~41 x 0.40.

(21)

These give Elh~ _ -0.049 e ~ fm,

E1~2 ~

_

-0.029 e ~

fm,

E1~3 ~ _

E1«~ _

-0.035 e ~

fm,

-0 .027 e ~

fm,

which are somewhat larger than, but in fair agreement with the experimental value of ~E1 ~. Our result for El in the case of Johnstone and Castel differs by a factor of two from their result. Thismay arise from a difference in intrinsic wave function, or from our use of the adiabatic rather than the projection method to obtain wave functions of good angular momentum . Given the fair agrcement we have obtained with the magnitude of ~El ~, and that the four models we have oonsider+ed for the effective charge all give the same sign for the effective charge, it is reasonable to assume that the calculated sign of El is also correct. The sign of the effective charge is determined in the models by the fact that the ~- state in question is at low excitation energy . If the excitation energy were to exceed the mean energy of the low-lying states (for the appendix H model) or the energy ofthe giant dipole state (for the other models) then the effective charge could be negative and the calculated and observed matrix elements dit%r in sign. In this case the excitation energy of the i_ state is low and we would seem to be on safe ground in assuming that the El matrix element has the calculated sign . Support . for this view comes from the calculations of Desplanques 2s) on 1 "Lu. Desplanques

41 8

M. A. BOX et al .

finds that his.calculated El agrces in sign with that obtained through the penetration effect and the El-M2 mixing ratio. Unfortunately we do not have the equivalent experimental information for ' 9F, and the more complicated nature of the penetration effect in this case zs), and the small calculated value ofthe El-M2 mixing do not give much hope of experimental verification of the- sign of El . Our procedure has been to give El the experimental value of IEl h with the calculated sign. 2.5 . SHORT RANGE CORRELATIONS

The i?NC potential described in the next section contains important terms which have a range 1na 1, so that short range nucleon-nucleon oornlations are important in the calculation of '

The coefficients A;R' may be obtained from the wave functions described above. It must be emphasised thatthe sum in eq. (22) is over all the occupied lauds, and is not restricted to the valence levels only. However the sum over closed shells can be expressed in a useful more compact form which is described elsewhere s°) . The neat step is to replace ~ by d~a, the solution of the Hethe-Goldstone equation 26): Yaa = Xap +

~ V ~sA'

(23)

obtaining

In eq. (23) V is the strong parity conserving nucleon-nucleon interaction Q is the Pauli exclusion operator and e is the energy denominator. We .use the method of Harrett, Hewitt and McCarthy Z') to solve eq. (23) in the A = 20 systan. In this method 1~y is expanded in terms of wave functions ~~' for the relative motion of two

PARITY MIXING

41 9

particles in the potential Y+~2r2 , and XM.m., the harmonic oscillator wave function for the c.m . motion : ~Qe = ~ a(aß ; ~u)X~ .m.~x~. .. a~

(25)

This may be regarded as a generalisation of the Tahni-Moshinsky transformation

In our computation we found it most convenient to utilise the computer codes of Barrett, Hewitt and McCarthy 2') and Comins and Hewitt Z s) which calculate G-matrix elements for the Hamada-Johnson potential directly and to obtain a(aß ; J,~) from the equation

where

Q a(aß ; ~~) = b(aß; ~,~)+ ~ e b(Ya ; ~pKYaI GI~p>, re

(27)

The starting energy w enters through C = CO-Ex-Ed,

(29) and we have verified that our results vary by at most 2.5 ~ for variation of m in the range 40 < m < 65 MeV, which seems to be appropriate in this region of the s-d shell ~'). We should conclude this section by emphasing the logical inconsistency of this, and all other, calculations of the effects of short range correlations . The effective forces used to derive the basic wave functions ie the coefficients A;R~ in eq . (22) are not related to the G-matrix elements in eq. (2~. Providing such a relation is a well known problem in the theory of the shell model 2 ~, and until it is satisfactorily resolved we have no alternative but to proceed as we have done. 3. The weak PNC nucleon-nucleon potential Many forms of the weak PNC nucleon-nucleon potential, Y~,c, have been proposod on the basis of an assumed weak interaction Hamiltônian. These potentials may be written in the form y~ - Vu~+ PIO) +Vp l) +Vp 2) +V2~+ . . ., (~)

where the superscript labels the isotensor transformation property of the component of the potential, and the subscript indicates the meson exd~ange from which the component derived.

420

M. A. BOX et al .

In this work we have omitted Vz,~, the two pion exchange potential, since the dominant channel is well represented by p-exchange. Forthe other potentials we have

iô°~ _ pa> _

(A sl ~ sz+B)W_ +({1+ft}AT i ~ zz + {1+p,}B)Wx

(T«> + ~sz~xC t +c_)w_+(tc~~_ .~z~xC s s + _C_)Wt +(~'~+~2'}C : : x Wx~

(32a) (32b) (32c)

where Wt

_ _ Gm~~ (QCi~~QCZ~) . jPiz~ e r ~~, 4 N l

(33a) (33b)

In these equations g,~ is the strong NNa coupling constant (g~/4a = 14.4), fx is the weak NNn coupling constant (in Cabibbo theory, f,~ _ ,~ = 4.2 x 10' e), G is the Ferrai coupling constant (GmN = 1.01 x 10 -s), g,, is the ratio of the axial vector and vector nucleon form factors at zero momentum transfer (gw = 1.25), m,~, mo and mN are the n, p and nucleon masses, and Tc2 ~ _ ~C1~TC+~t~C+~zcz~~

with TN = ~(TN f i~).

(34b)

The potential parameters A, B, C+, C_, D .and E _ (fs/j,~)are calculated from the weak interaction Hamiltonian by a procedure we have reviewed in detail elsewhere 3~. There the potential was parameterisod in a slightly different way, and we provided for the readers' convenience a translation dictionary between the potential parameters H, I, l', K, L of ref. a~ and A, B, C+, C_, D of this paper : A = ~(2H+I~,

D = ~(H-I~,

It is important to emphasise that while the connection between J* and the weak I3amiltonian is generally regarded as having a sound basis sl), the values of A, B, Ct and D are much more tenuously linked to the weak I3amiltonian. They are best regarded as estimates.

42 1

PARITY MIXING T~ai.e 1 Parameters of weak interaction potentials Potential

Ref.

as) Cabibbo standard 34) Lee d'Espagnat Weinberg-Salsa as) +factorisation Cabibbo, SU(6), ") vanishingp-coupling abibbo, SU(6), t ") Goldbergen-Treiman

Islf~ °) 1

.l~C/S)2

A

B

C+

C_

D

3c~

0 0

-,it l

0

0 O

k,l

-2Sw

0

0

i(c~-2-4Sw)

5.7

0

0

0

-0 .9

0

0

0

3.~.3

(1+85,~,./3s=)

3(c=+1-25~,)

1

0

1

-5 .4

The notation used is c = coa B, s = sin B, S~ = sin B~, where B and B~, are Cabibbo and Weinberg angles respectively . In the Iast two cases the relative sign offR and .!, . . . D is feed, but the overall sign is not.In the othercasesthesign offianotknown, but the othersi~a arefixed.Forall potentials considered here C, _ (1 +p)C_+(1 +~C+.

To illustrate the range of potentials which have been considered in the literature, wequote a selection in table 1 . Morecomplete tabulations may be found inrefs . so, sz~ Potential no. 1 will be refered to as the standard potential . It is derived from the weak Hamiltonian of Cabibbo sa) using current algebra to eomputef,~ and the factorisation approximation for A, B, Ct and D, and has bcen the most widely used potential in calculations of parity mixtures in nuclei. In particular it was employed in the earlier calculations of A y reported in refs. '' 4). 4. Resalta and diealaeion

The most convenient way to present our results is to write (for the case of perfect spin-up polarisation [p(+i) = .1, p( -~) = 0]

a, = aA+ß6+y + C+ +y_C_ +y x Cx +8n+~(j*l.f`) Teats 2

Expansion parametero for the asymmetry Ay Parameter

Value x 10'

a

11 .47 3.40 4 .66 0.27 2 .06 0.0 -2 .61

Y+ Y_ y, b e It is asauaned that j, > 0. For j. < 0 reveree the sign of e.

(36)

eeThe form choose to exchange (32) function isospin values 4, Itvalues a,increase list difference is ß,and which we making SU(61 SU(6~, Salam the clear ytthe ofp-exchange sinhave used arc i, are (33), values Bforce value lists that binbetween the +factorisation ~Goldberger-Treiman given vanishing to = the this 0but used abe contribution the 0zz) ne= ofthe softer case band comparison and with This compared coefficients the == In coetFcients p~oupling 0The rz-exchange 4values sin coefficients and particular short the permits Owefor fm' coefficients replacement =reported is range of with typical of 0calculated unchanged we potentials t,the the rthe Awhich we should a, have the repulsion, asymmetry BOX weak p-exchange here reconstruction obtain experimental ~e-° been 43and from gives are The et interaction econsider for a!those in used experimental the the for enhanced table or a(65 areasonable 8947value the values the potential dipole of The potentials+f47) 2dipole uncertainties ref inclusion in2one, of Note two table value potential Ar of by potential reflect terms Ar approximation that for This 2isabout xofa listed AYlOs any the arc since xin is of finite lOs slightly respectively illustrated potential our in the i,(103 both =7table p-width, -(1819) between calculaform potential different states to ft +-f40) 47) 322the the of in r

422

M. .

.

TABLE

Asymmetry

A~

Potential Cabibbo l d'Espagnat Weinberg Cabibbo, Cabibbo, The contributions

+( .2t .6) +( .6f23) +( .5 +( .7t40) t(19 .4- .ti) .6) t

 ., .23

.53

.

a ß Y+ Y_ Y.

and have the These

+(11.5 .6) +( .58f23) +(15.7 +(13.7 .6) f(31 .6 .6) t

.

.

T~BLe

Expansion

Only

dipole

p-potential

18.19 5.55 7.942 0.458 3.158 .

;

.

. . .,

. .

.

(37) Before tions. would table eqs.

.

mzv We 2n t wave

--.

"~ .

.25

. .

. .')

PARITY MIXING

423

Desplanques' results for 41K (20 ~ enhancement) and the 19F results (a factor two enhancement) obtained by Gari 3s) . To estimate from Desplanques' curves ss), the 2n potential would give results between those of the p-potential and the dipole potential. The dipole potential results for each weak interaction theory are also listod in table 3. It is also important to realise that the contribution to a A~ from the a~xchange potential, while it is not very sensitive to the variation of the short range behaviour of the wave function, is sensitive to the variation in the long range behaviour. Gari et al. 4), who used different wave functions, obtained a contribution from V* 1 ' which was about two-thirds ours. These uncertainties, together with the large statistical error of the experimental result, make it very difficult to exclude any weak interaction theory . The only definite exclusion is of theories such as those with enhanced p-exchange terms such as those of McKellar-Pick type . The inclusion of 2n exchange terms serves to reinforce this exclusion . The recent potential of Altarelli et al. a~ with value of A enhanced by 5 is also excluded on this basis. Potentials with enhanced n-exchange terms, such as those d'Espagnat and Weinberg-Salam, are not quite excluded by the numbers in table 2, being about two standard deviations from the observed value (with a judicious choice of signs for~. However, the possibility of a larger effective p-exchange contribution from 2~ exchange, and a smaller n-exchange contribution as found by Gari et al. `) make such potentials definite candidates . Also it would seem necessary to exclude the conventional potential because it gives the wrong sign . The sign is however computed using the factorisation approximation, and we have no guarantee that it is unchanged by weak interaction renormalisation effects. The possibility that the conventional potential has the correct strength but incorrect sign for the p-exchange term cannot be ignored. 5. Cooclosioo We conclude by placing the new information gained from this calculation in the wider context ofwhat is known about the PNC nucleon-nucleon potential. (i) The a-decay of the 2 - state of 160, observed by Waffler and collaborators 4~ limits the strength ofthe dT = 0part of the potential to be about that ofthe standard potenha1 41). (ü) With the present results, this implies either (a) the dT = 0 part ofthe potential has the conventional sign, and the dT = 1 part in enhanced as one expects for various neutral current theories, or (b) the dT = 0 part of the potential has the opposite sign to that of the standard potential, and the d T = 1 part of the potential is weak, as is the standard motel. (iü) We are then forced to explain the anomolously large value observed by

424

M . A . HOX et al.

Lobashov 42) for the circular polarisation of the photôn emitted in n+p -~ d+y 1n terms Of an enhaaoed dT = 2 potential 43) (iv) Such an enhanced dT = 2 potential can explain the discrepancy in sign and magnitude observod in heavy nuclei ZZ .'3), and thus eliminates the possibility (üa) of an enhanced d T = 1 interaction. Alternatively, one could decide to believe in the enhanced dT = 1 potential, but leave Lobashov's result unexplained. The question of whether the dT = 1 or dT = 2 part of the PNC potential is enhanced is one which must be decided by further experiments. However it seems clear that at least one of them must be enhanced . We wish to thank Prof. E . Adelburger for discussions concerning the experiment, Drs. J. McGrory and M. Gari for information about their calculation, Drs. R. G. L. Hewitt and H. C. Comma for their G-matrix code, and for helpful discussions, Dr. B. Desplanques for information regarding his similar calculation in "'Lu, and Dr . K. R. Lassey for useful discussions. This work commenood while two of us (M.A.B. and B.H .J.McK.) were at the School of Physics, University of Sydney . We are grateful to the Science Foundation for Physics of that University and to its Director, Prof. H. Messel, for support at that stage of the work. We also gratefully acknowledge the support of the Australian Research Grants Committce. Bruce H. J. McKellar wishes to thank his colleagues of the Service de Physique Théorique for helpful discussions, and for their hospitality. A~peadix A A MODEL CALCULATION OF Ar

In this appendix we calculate Ar in the simplest model available. The i+ and =states are regarded as 2s} and lp~ harmonic oscillator hole states in 2°Ne, and the two-body PNC potential is replaced by a single particle effective potential 14 .30) W = -Fa ~ p.

(A.1)

The determination of the constant F is discussed in detail in ref. 3~. For the standard Cabibbo potential and ' 9F it is F = ~~ Gp {0.12 cos2 6~f 0.09~~ = 1 x 10 -6(1 f 0.3) MeV ~ fin,

(A.2)

which has been adjusted to approximately reproduce the matrix elements of Yom. Since such amodel gives a bad value for E l , we do notexpect this calculation to give reliable values for the magnitude of d~ . However since cancellations occur ia. the ratio <~-~Vrnici~+~/El, this model calculation is less sensitive to possible sign

PARITY MIXING

42 5

errors through inconsistent sign conventions for operators and states, and so provides a useful check that our calculation does not in fact suffer from such problems . We begin with eq . (3), for the case of perfect spin-up polarisation [p(+z) = 1, p( -i) = 0], which we write as (for j = i). 3 ~i+ ) - ~~ ) (A .3) where Using the fact that the i+ and i- states are single hole states, we obtain [ct: refs. e . ~l
where the matrix elements on the right are single particle matrix elements. So whence a straightforward calculation gives S = +~~

m ücoF, e

(A .~

where the relationship p = (mli~)[r, Ho], with Ho the single particle harmonic oscillator Hamiltonian has bcen used to reduce the matrix elements ofp to those of r, which then cancel in S. In this sense the El matrix element cancels out in this simple calculation, and hence the sign is determined independently of the sign of El. So we see that S is positive, and thus Ay is positive, in agreement with the more detailed calculation, but disagreeing with experiment . Numerically, in this model As we know E1 is overestimated by about 5 in this model, we would expect Ar to be underestimated by this same factor, and it is. Appeadlx H THE GIANT DIPOLE RESONANCE AND THE EFFECTIVE CHARGE FOR EI TRANSITION

In this appendix we describe a simple model for discussing the effect of the giaat dipole resonance on the effective age for El transitions. Such a model is necessary

426

M . A . BOX et al.

if we are to obtain a reasonable value for low-lying El matrix elements, and indeed a number of them exist in the literature is .2o,~t) . The simple model proposed here is a natural extension of the Brown-Bolsterli schematic model for the giant dipole resonance. In its simplest form the Brown-Bolsterli model for the giant dipole resonance has the Hamiltonian H = H o + VOID)
(B.1)

where ID) is the dipole state. Ifthe eigenstetes of Ho are degenerate, with the eigenvalue ~, then all but one of the eigenstetes of H have the eigenvalue s, and the other, which is just the state ID), has the eigenvalue ~+ Vo. In this way the giant dipole state is shifted above the other states and acquires all the dipole strength. The dipole strength of a state ICY) is just proportional to I
The dipole amplitude of the low-lying state is now
Since
if we take «,lye,) x 1 . As s, measures the shift of the state from the mean particlahole state energy ~, and Vo mesures the shift of the dipole state energy from the same mean, we get

as quotod in the text . A better approximation to «,I~,) is CY'ilWi) -

1-~I
1+'.6)

PARITY MIXING

427

since this enforces the orthogonality of ~~,) and ~D~ to lowest order in (D~d~,i. This additional correction is similar to the correction introduced by Johnstone and Castel s t), but because of the large number of approximations inherent in our model we do not feel justified in using it for anything more than an simple estimate. Refereaces

1) E. G. Adelberger, H. E. Swanson, M. D. Cooper, J. W. Tape and T. A. Tremor, Phys. Rev. Lett . 34 (1975)402 2) M. Gari, Phys. Reports 6 (1973) 318 ; E . Fischbach and D. Tactic, Phys . Reports 6 (1973) 123 and the papers and discussion in Proc. Symp. on interaction studies in nuclei, Mainz 1975 3) M. A. Box and B. H. l. McKellar, Phys . Rev. Cll (1975) 1859 4) M. Gari, A. H. Huffman, J. B. McGrory and R. Offerman, Phys . Rev. Cll (1975) 1485 5) H. G. Benson and B. H. Flowers, Nucl. Phys . Alb (1969) 305 6) R. J. Blin-Stoyle, Phys. Rev. 1211(1960) 181 7) E. Maqueda, Phys . Lett . 23 (1966) 571 8) H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1%7) 306 9) D. M. Brink and G. R. Satchler, Angtil$r momentum, 2nd ed. (Oxford University Press, 1971) 10) A. Bohr and B. R. Mottleson, Nuclear structure, vol. 1 (Benjamin, NY, 1969) 11) L. M. Lederer, J. M. Hollander and I. Perlman, Table of isotopes, 6th ed . (Whey, NY, 1%7) ; F. Ajzenberg-Selove, Nucl . Phys. A190 (1972) 1 12) M. Harvey, Nucl . Phys . 52 (1964) 542 13) A. de Shalit and H. Fichbach, Theoretical nuclear physics, vol. 1 (Whey, NY, 1974) 14) F. C. Michel, Phys . Rev. 133 (1964) B329 15) G. Ripka and K. Zarnick, Phys . Lett . 23 (1966) 347 16) G. Ripka, Adv. in Nucl . Phys . 1 (1968) 183 ; Centre d'Etudea Nucléaires de Saclay, rapport CEA-R3404 (1968) 17) S. G. Nilsson, Mat. Ft's. Medd. Dan. Vid. Selsk. 29 (1955) no . 16 18) G. E. Brown, Nucl . Phys . 57 (1964) 339 19) G. E. Brown and M. Bolsterli, Phys . Rev. Lett. 3 (1959) 472 20) A. Rohr and B. R. Mottelson, Proc . Int. Sym. on neutron captive gamma-ray spectroscopy, Studsvik (IAEA, Vienna, 1969) p. 7 21) I. P. Johnstone and B. Castel, Nucl . Phys . A213 (1973) 341 ; Can. J. Phys. 52 (1974) 1998 22) B. Desplanquea, Thèse Université de Paris-Sud, Orsay 1975 23j E. L. Church and J. Weneser, Ann. Rev. Nucl . Sci . 10 (1960) 193 24) H. H. l. McKellar, Phys . Rev. Lett. 20 (1968) 1542 25) M. A. Box, Thesis, University of Sydney, 1975 26) H. A. Bethe and J. Goldstone, Proc . Roy. Soc. A238 (1957) 1531 27) B. R. Barrett, R. G. L. Hewitt and R. J. McCarthy, Phys . Rev. C3 (1970) 1137 28) H. N. Convins and R. G. L. Hewitt, Nucl . Phys . A228 (1974) 153 29) B. R. Banrett and M. W. Kirson, Adv. in Nucl . Phys. 6 (1973) 219 30) M. A. Box, B. H. J. McKellar, P. Pick and K. R. Lassey, J. of Phys. Gl (1975) 493 31) J. F. Donoghue, Phys . Rev., to be published 32) K. R. Lassey and B. H. J. McKellar, Saclay Preprint 75/74, 1975 ; Nucl . Phys ., A260 (1976) 413 33) N. Cabibbo, Phys. Rev. Lett . 10 (1%3) 531 34) T. D. Lee, Phys. Rev. 171 (1968) 1731 35) B. d'Espagnat, Phys. Lett. 7 (1%3) 209 36) M. Gari and J. H. Reid, Phys . Lett . 53B (1974) 237 37) B. H. J. McKellar and P. Pick, Phys. Rev . D6 (1972) 2184, ibid. (1973) 260 38) M. Gari, Proc . Symp . on interaction studies in nuclei, Mainz 1975 (North-Holland, Amsterdam . 1975) p. 307 39) G. Altarelli, R. K. Ellis, L. Maini and R. Petronzio, Nucl . Phys . B88 (1975) 215

428

M. A. HOX et al.

40) N. Neuheck, H. Schober and H. Waftler, Phys. Rev. C10 (1974) 3M 41) M. Gari, H. Kummel and J. G. Zabolitsky, Nucl . Phys. A161 (1971) 625 42) H. lochim and B. Ziegler, ed., Interaction studies in nuclei, Froc. Symp. on interadion studies in nuclei, Mainz 1975 (North-Holland, Amsterdam, 1975); and ref. ") 43) B. H. J. McKellar, M. A. Hox, K. R. Lassey and A. J. Gabric, Proc. Symp . on interaction studies in nuclei, Mainz 1975 (North-Holland, Amsterdam, 1975) p. 61 ; B. H. J. McKellar, Nucl . I?hys., AZi4 (1975) 349 M. Simonius, froc. Symp . on interaction studies in nuclei, Mainz 1975 (North-Holland, Amsterdam, 1975) p. 3; M. A. Box, A. J. Gabric, K. R. Lassey and B. H. J. McKellar, J. of Phys . G, to be published