Parity non-conservation at the peak of p-resonances in low-energy neutron-nucleus scattering

NUCLEAR PHYSICS A

Nuclear Physics A561 (1993) 189-200 North-Holland

Parity non-conservation at the peak of p-resonances in low-energy neutron-nucleus scattering B. Desplanques Universite'Joseph Fourier, CNRS-IN2P3, Institut des Sciences Nuclt!aires, 53 Avenue des Martyrs, 38026 Grenoble-Cgdex France S. Noguera Departamento de Fisica Teorica, Facultad de Ciencias Fisicas, Universitad de Valencia and IFIC, Centre M&e, Universitat de Valencia, CSIC, E-46100 Burjusot Valencia, Spain Received (Revised

4 February 1993 22 March 1993)

Abstract Parity-non-conserving effects at the top of p-wave resonances in low-energy neutron-nucleus scattering are revisited in view of recent measurements in 23xU and 23ZTh. This is done in the framework of the valence model. A quite simple expression in terms of the strength of the neutron-nucleus parity-non-conserving force is derived for the P(E,) asymmetry. The result, which is independent on the nucleus, can usefully be considered as a benchmark for those effects. Comparison of experiment to theory confirms earlier conclusions, namely the expected strength of the neutron-nucleus parity-non-conserving force is much too low to account for observations in this approach, even if the sign is well reproduced. Clues to explain discrepancies with other recent works are given.

The recent parity-non-conserving (pnc> effects observed in “sU and 232Th [1,2] have motivated several new theoretical developments [3-61. While the effects of the order of lo%, which is comparable to previous measurements in lX9La [7], are not surprising, the fact that most of them have the same sign is much more unexpected. At first sight, this result re-opens the debate as to know whether pnc effects observed in very low-energy neutron-nucleus scattering have their origin in a statistical mechanism [S-131, where signs are at random, or in the so-called valence mechanism [14-161, which is at present the only one to a priori provide a unique sign for effects observed in the same nucleus. The conclusion generally accepted before the new observations in 238U and 232Th was that the valence model could not work. The strength of the neutron-nucleus pnc force required to explain experimental results was too large with respect to usual expectations by one or two orders of magnitude and, perhaps worse, was not evidencing a unique sign [17]. This conclusion was based on a work incorporating as much as possible information about the physics accounted for by nucleon-nucleus optical potentials 0375.9474/93/$06.00

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[18]. In that work, specia1 attention was given to situations in between resonances, requiring to design particular models to overcome the difficulties raised by a naive use of optical models. It was also shown how to modify wave functions issued from standard optical models to recover most results obtained in the particular models. In the present work, we will use this last approach together with soluble optical models to describe the asymmetry, P(E,), measured at the peak of p-wave resonances, which is actually measured. Our aim is to provide for P(E,)expressions that can be simply related to quell-known measured quantities such as strength functions, or potential scattering radii. Using this approach, we are able to separate pnc contributions at the peak of p- and s-resonances, which are combined altogether in approaches using the full optical-model wave functions without any caution. We also examine the sensitivity of P(E > close to s-wave resonances, which in some cases may change its standard l/ f I?; dependence, as well as the possibility to get the same sign in different nuclei. Comments will be given as to the relationship of the present approach with that used by Bowman et al. 231and by Flambaum [4]. The single-particle hamiltonian relevant to discuss the valence contribution to pnc effects considered here may be written [39,20] as:

where XG has a direct relationship with the effective NN interaction in the nuclear medium [19] and, from it, to the free NN interaction. p(O) will be defined as _4/($rR3>, where R, for simplicity, will be chosen to be the same as the radius of the square potentials discussed later on, R = 1.3 Al/’ fm. This leads to a quite low density and a renormalization of some of the results by a factor 1.5 may be appropriate. The strength, Xg, can be easily compared to the corresponding strength of the pnc proton-nucleus force, Xs, whose value is rather we11 determined by pnc effects observed in odd proton systems and close to the value X,P = 3.5 X 10d6 [21]. This value is consistent with DDH expectations [22]. A value for XE may be obtained from the experimental limit on the y circular polarization in the transition 0’ (1.1 MeV) + l’(g.s.) in 18F_ This process is sensitive to the isovector part of the force and therefore to the difference \ Xi - XE /. On the other hand, due to the relationship with P-decay from ‘8Ne 1231,it is believed to be under control for the nuclear part of the estimate. One thus gets: 2.5 x lo+


< 4.5 x lo+.

The expected sign for the long-range T-exchange contribution the lower limit.

(21 would rather favor

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191

The advantage of relying on the X-parameters is to directly use constraints available from various processes without entering into detailed discussion involving uncertainties on short-range correlations or pnc coupling constants, whose effect on first-principle estimates may reach an order of magnitude in the case of Xi, without considering the sign which, contrarily to XE, is undetermined. The relation of X{ to strengths entering other parameterisations of the pnc neutron-nucleus force by Bowman et al. [3], Koonin et al. [6] is:

2Tp;jxG (=X$/150)

= E = E, x lo-‘.

(3)

Quite generally, the pnc forward-scattering amplitude estimate pnc effects such as the asymmetry P is given by:

which is needed

f,,,(O”) = G’(a.p),

to

(4)

where G’ has the following expression in our notations:

In Eq. (51, Mr represents the reduced mass, while g, is the usual spin statistical weight factor. W, and Wi are the appropriately normalized wave functions describing neutrons in s- and p-states, respectively. In absence of strong interaction, G& takes the simple expression: G& = - $X;Ag,.

(6)

As to the asymmetry P at the peak of p-resonances paper, it is given by P(E,)

=

a+- o2. P

considered

throughout

this

4rr Im G’ =

9

(7)

UP

where Im G’ is calculated at the peak of the p-resonance under consideration while ap represents the corresponding p-wave cross section: u+ and (T- represent the total neutron-nucleus cross sections for the corresponding helicity. The wave function !FO and W,, which enter Eq. (51, will be taken as:

(8) where

(9)

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192

In this last expression, S, and S, represent the strengths functions relative to sij2 E!, Z’/> are the reduced width, the and pi/2 resonances and rino, Eo, rio (cl, energy and the total width of the s-wave (p-wave) resonances. Wave functions To and ?Ir, given by Eq. (8) may be considered as the minimal generalization of the optical model wave functions accounting for the energy dependence due to the presence of resonances. Indeed by averaging over energy, the factors y take the value -i. Wave functions PO and Fi then become identical to the optical mode1 wave functions, ?FoM(r) = ~OMR(~) + ~~OM1(~), supposed to be normalized as

(11) In the above expressions, R, is defined as R, = 1.35 Ali3 fm while p. represents the nucleon momentum corresponding to the standard nucleon energy of 1 eV referred to in defining the strength functions So and S,. The quantities Rb and R; (which has the dimension of a volume) represent the scattering radii for each wave. They replace the usual scattering lengths, u, which characterize the scattering properties on inert objects. In contrast to them, they have the particularity to remain finite when the potential is getting close to bind an extra nucleon, in agreement with observations. An other advantage of wave functions (8) is that they are consistent with the low absorption observed for thermal neutrons, while optical-model wave functions generally predict too large absorption there due to ignoring the resonance structure. By inserting Eq. (8) of Ir$ and !If’, into Eq. (51, it can be seen that the amplitude G& will have the general expression: G& = -

$X;Ag,Z,

(12)

where

-I,, Z,, Z, and Z, being real quantities. Inspection of (12) immediately shows which term will generally contribute to the asymmetry P at the peak of some resonance. This effect involving the imaginary part of G& and therefore that of I, we get: p-resonance:

P( EP) a Im( G’) a Im( I) a I, + Re( yo) Z,,

s-resonance:

P( E,) a Im( G’) a Im( I) a I,+ Re( yr) Z,.

(13)

B. Desplanques, S. Noguera / Parity-non-conseruation

Fig, 1. I,, 12, I3 and I, defined in Eq. (121, as a function of the mass absorption potential (full lines) and a surface absorption potential

number (dotted

193

A, for a volume lines).

All factors Ii present in (12) have a quite smooth energy dependence (neglected here). Nevertheless, contributions involving Z4 may introduce some energy dependence through factors Re(y). We shall come back to this point later on. Two soluble optical models have been used to calculate the quantities I,, Z2, I, and I,. The first one assumes that absorption takes place at the surface while the real part is described by a square well. The corresponding hamiltonian is:

fJ”=

p2

(14)

--VVB(R-r)-ig26(r-R).

2Mr

The second nian

model

H’2L 2W

i

assumes

.+ig

a volume

i

The strengths of the potentials have been chosen respectively term, W/R, in (15) is constant.

B(R-r).

absorption.

It is described

by the hamilto-

(15)

for s 1,2(! = 0) and pr,J! = 1) single-particle states as V0 = 54 MeV and VI = 51 MeV. The imaginary A factor l/R, where R is the nuclear radius, has

194

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been introduced so that the integral of W/R over the nuclear volume behaves like R*, W being independent on the nucleon number, A. This allows us to account in some part for the fact that absorption is mainly a surface effect, while providing a soluble model significantly different from (14). The strengths of the imaginary parts in (14) and (15) have been determined to reproduce the strength functions S, and S, at their maxima. They approximately obey the relation 2g2 = W. For the first model (Eq. (14)), it can be shown that the introduction of the energy dependence into the wave function, as assumed in eq. (81, is correct for an infinite set of equally spaced resonances. For the second model (eq. (15)), the introduction of the energy dependence as done in Eq. (8) is correct outside the interaction region and only approximate within. Possible corrections have their origin in the internal part of the nucleus, whose contribution is suppressed, and one does not expect this could be a serious difficulty. The absence of sensitivity to the internal part of the nucleus may be checked by comparing results obtained with volume and surface absorptions. This can be done by comparing the corresponding quantities I,, Z, and Z,, drawn in Fig. 1 as a function of the nucleon number A. Differences can hardly be seen in those cases. Some differences may be seen for the quantity Z4, but its contribution is generally expected to be smaller. While curves given in Fig. 1 evidence sometimes complicated structure, (whose size varies like A - ‘13), they can be approximated by quite simple expressions, given in terms of quantities such as potential scattering radii or strength functions: I,( surface or volume) = 3 [($-$-($-I)],

Z,( surface) = - 3

($+:)’

Z,(volume) = - 3

(16) The quantity Z/X is related to the ratio of imaginary and real part of the optical potential (z/x = g2/(VoR>>. It represents a minor correction, except in cases where S, or S, become quite small (where the surface absorption model fails in any case>. The accuracy of the above approximation can be appreciated by looking

195

B. Desplanques, S. Noguera / Parity-non-conservation

60

100

160

200

Fig. 2. I,, f,, and I, defined in Eq. (12), as a function of the mass number A, for a volume absorption potential. The full lines correspond to the exact solution and the dotted lines correspond to the approximated expressions given in Eq. (16).

at Fig. 2 for the case of a volume absorption. Slight deviations occur around A = 50 and 150 for I,, A = 100 and 230 for 1,. Quite similar results are obtained for a surface absorption. Using the approximate

expressions

(16) for Z,, remembering

that the a,, cross

section entering the expression (7) of PC,!?,) is given by aP = -4rr(p/p,) S,$-Rig, and neglecting terms proportional to Z4 in the calculation (Eq. (13)), one gets a very simple expression for P(E,):

Im(r,) of Im G’,

(17a) or also for R = 1.3 Ail3 P(E,)

-274X;

An even simpler qq

fm:

=

expression

( 17b) is obtained

2ME, x 10-T= P

z, P

in terms of the parameters

l7 or c(Eq. (3)):

( 18a)

2f

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196

or in terms of the ratio (pa/p):

P(E,)

= 0.436, x 1o-2 ( 2 P

) = 0.436 x lo5

5 ( P 1

.

( 18b)

Expressions (171 and (18) for P(E,) are essentially independent on the shape of the absorption potential as well as on the nucleon number. Departures are possible in the A-region where I, (surface) f I3 (volume), around A = 50 and 150, where the surface model badly reproduces the S, strength. The above result can be compared to that obtained by Koonin et al. [6]. It is slightly larger than theirs in regions where the S, strength dominates (A 1: 30, 100 and 250). It strongly disagrees, in sign and in magnitude, in regions where the S, strength dominates (A 2: 50 and 150). We checked the sensitivity to the approximation we made: I, (volume) = 3LS,rrRi/2p,R3). While it removes some structure from P(E,) in regions where the strength 5, is small, it cannot explain the above discrepancy. Instead, if the full optical-model wave function is used as did Koonin et al., therefore adding together the contributions to (a’a-) at the s- as well as the p-resonance peaks, it is possible to reproduce results close to theirs. Typically, always using the approximate expressions given above, (Eq. (1611, one gets:

(19) and

P (p+s)

(20)

independently of the use of a volume or surface absorption. This expression allows us to recover several features of the variation of P with A presented in Fig. 1. of Koonin et al., such as the values of A where P = 0 or the ratio of maxima and minima (using their values for S, and S,). Obviously, eq. (20) is inappropriate for a comparison with present experiments which measure the P-asymmetry only at the peak of p-resonances. It is based on an incorrect use of the optical-model wave functions for the present purpose. This difficulty was already perceived few years ago 1171, being at the origin of tedious developments for its overcoming. In the previous discussion, we retained the contribution to P(E,) arising from the term I, and neglected that one involving Re(y,)Z, (see Eq. (13)). In between resonances, the real part of y0 is expected to be of the order (T”/(iAErrS,)) = 2/rr. As I4 is smaller than I, by an order of magnitude on the average, this assumption may be safe. There are a few cases however where it may not be so.

B. Desplanques, S. Noguera / Parity-non-conserr;ation

i-.16

197

-l

”

“1

60

1

““‘I

loo

“““I

160

“1 200

25:

Fig. 3. I, as a function of A for a volume absorption potential. The full line corresponds solution

with V, = 54 MeV (s-wave) and V, = 51 MeV (p-wave). The dotted exact solution with VO= P’, = 54 MeV.

to the exact line corresponds to the

They involve nuclei where the ratio Z4/Z3 is relatively enhanced, around A = 45-60 or 120-160 (see Fig. 2), and particular situations where Re(y,) is larger than the above value of 2/7r. This could occur when a s-resonance is very close to the p-resonance or/and when its width is exceptionally large. An example is ‘39La which combines two of these effects (Rely,) = 5-10 and Z4/Z3 - -(0.6-l.O)), leading to a change in the sign of the effect, which does occur in refs. [14,171. For 232Th, two effects may be sensitive to a nearby resonance. They concern the levels at 64.50 and 167.17 eV where close s-resonances at 69.23 and 170.39 eV could give rise to factors Re(y,) as large as -4, -5, respectively. The size of the correction due to the proximity of a s-wave resonance is somewhat uncertain however. As shown in Fig. 3, the quantity Z4 which enters into this correction is very sensitive to the inclusion of the spin-orbit interaction for which we used a volume one for convenience. While this may not be quite appropriate, one cannot discard an energy dependence of the asymmetry P(E,) more complicated than the one given by the standard factor l/a, emphasized once more in recent works [3,6]. At this point, it is possible to compare experimental results in 232Th and 238U with the contribution of the valence mechanism. This shows an important discrepancy since the strength of the neutron-nucleus force XG should be of the order of (300 _t 225) X 10e6 [2,3], (e7 x lo-’ = E = (2 k 1.5) X 10P6), while a value of about 3 X 10-6(~7 X lo-’ = E = 2 X 10-s) is expected from theory complemented by some phenomenology, as pointed out at the beginning of this paper. A theoretical value of Xi (or E) larger by a factor 3, which is consistent with the largest DDH range and quoted by Bowman et al. [3] or Koonin et al. [6] seems excluded. The possibility of an enhancement of the weak NN interaction in heavy nuclei has also been mentioned in the literature (see comments made in ref. [6]). The confirmation in light odd-proton systems (r9F, p(~ scattering) of the strength of the pnc proton-nucleus force, X8, first determined in heavy nuclei (18’Ta, 175Lu, 41K) [19,201 makes this conjecture unlikely.

198

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S. Noguera / Parity-non-conservation

In their work, Bowman et al. concluded that results in 238U and 232Th might be consistent with DDH expectations. As mentioned above, they referred to the largest range of DDH predictions, which are probably excluded now. On the other hand, their prediction for P(E,) relies on an approach based on a neutron-nucleus potential, which is in principle a particular contribution of those considered in works developed for instance in ref. [171 and here. They ignore absorption, whose effect however should not prevent one from making a meaningful comparison in the A-region where the strength S, is small as in Th and U. This can be done by introducing in the calculation of the pnc matrix element (U- ] VP,, 1T’), a complete set of states corresponding to an infinitely deep potential. One thus gets: cos( KR)

P&J

(21)

= 2 PR

where the states I n) and I p) are normalized to 1 within the interaction volume, K2 = 2M(L’+ E), Ki = 2M(I/ + E,) = ([(n + $h-r]/Rj2. From Eqs. (17) and (181, it is clear that phases entering the states 1n) and 1p) are not arbitrary. The above result (Eq. (21)) is quite similar to that obtained by Bowman et al., especially if one notices that their condition to have a p single-particle state just bound at threshold implies (cos(KR) I = 1. There are a few differences however. Our expression does not contain the factor 6 present in their definition of yPy,/yS. This factor may be related to a question of normalization and should be cancelled by another one in the calculation of the pnc matrix element. While the factor cos (KR) in eq. (21) is unimportant for a formal comparison, it does matter for numerical estimates. Its presence cancels possible singularities coming from a zero-energy denominator in the second member of eq. (21) and ensures the internal consistency of the result. By taking information from different sources, this internal consistency may be lost. Together with the above factor a, we believe that it is the main origin of the discrepancy between Bowman et al.? result, P(E,) = 2.9 X 10'dp,,/p>, and ours, P(JZJ = 0.43 X 10’~ (p,/p) (Eq. (18)). We cannot however exclude that their inputs have some relevance in more elaborate models. For instance, possible effects may originate from the deformation of the nucleus which is important for both 238U and 232Th. While some part could be incorporated in a mean field, for which the simple result of Eq. (18) is likely to

Fig. 4. Diagrams

describing

the ‘quasielastic

scattering’ interaction.

contribution.

The cross represents

the weak

3. Desplanques,

S. Noguera / Parity-non-conservation

199

Fig. 5. Some diagrams included in the valence mechanism which can be related with the ‘quasielastic contribution’.

hold, another part, of a less statistical nature, could be incorporated in the term Re(y,)L,. At this point, it is necessary to consider the compound-nucIeus mechanism. At present, this is the only one able to explain the magnitude of the observed effects [ll-131. Nevertheless in our opinion, within this model there is no explanation of the systematics in the sign of the asymmetry observed in the experiments. En a recent work 141, a mechanism has been suggested. It assumes that there is some coherent contribution due to the “quasielastic scattering” which is represented in diagrams of Fig. 4. The proposed asymmetry (Eq. (35) of ref. [4]) is of the form: P=P,N,,

(22)

where P, is the valence contribution (the one calculated here in Eq. (17) and Ng (- 103-104) is the number of the “quasielastic” components in the full compound state. We believe that Eq. (22) has some kind of double counting of the compound nucieus effect. The first one in the resonant parts of G& in (12) which gives our result for P, expressed in (17) and the second one in the factor N,. If one takes Eq. (22) for granted, then our expression for the valence mechanism should also include the factor N4, corresponding to including diagrams of Fig. 5. In fact, these diagrams (and many other ones in which any particle of the compound-nucleus state is in the cpSor ‘pi, state) are included in the valence contribution, inside the resonance factors, and there is no place for any additional N, factor. In our opinion, Eq. (22) does not really correspond to what the author intended to estimate, namely the contribution of diagrams represented in Fig. 4. In this paper, we considered the helicity dependence in the total scattering cross section of neutrons on nuclei at the peak of p-resonances. We studied the vaience contribution and found a quite simple expression for it, which can be used in any discussion as a benchmark and leads to a unique sign, independent of the nucleus.

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200

We looked at possible deviations from this simple expression. They may mainly arise from close s-resonances and be sizeable in some cases. The discrepancy with results of other studies has been examined. The comparison of theory to new measurements in 238U and 232Th reinforces conclusions reached previously for other nuclei. The strength of the pnc neutronnucleus force required to account for these effects is about two orders of magnitude larger than what is reasonably expected from non-compound nuclear theory together with some phenomenology involving pnc effects in other systems. In view of that, the reproduction of the sign in most cases does not seem to be relevant. From our study, we do not believe there is much hope to explain these effects from a valence model. It probably requires more elaborate explanations than those presented until now.

One of us (B.D.) would like to thank the institute of Nuclear Theory (Seattle) where part of this work was started. The present work has also been supported by the grants CICYT AEN-90-0040 and DGICYT pb 91-01119-CO2-01.

References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] 1121 [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] 1231

X. Zhu et al., Phys. Rev. 46 (1992) 768 C.M. Frankle et al., Phys. Rev. 46 (1992) 778 J.D. Bowman et al., Phys. Rev. L&t. 68 (1992) 780 V.V. Flambaum, Phys. Rev. C45 (1992) 437 N. Auerbach, Phys. Rev. C45 (1992) R514 S.E. Koonin, C.W. Johnson and P. Vogel, Phys. Rev. Lett. 68 (1992) 1163 V.P. Alfimenkov et al., Nucl. Phys. A398 (1983) 93 R. Haas, L.B. Leipuner and R.K. Adair, Phys. Rev. 116 (1959) 1221 R.J. Blin-Stoyle, Phys. Rev. 120 (1960) 181 V.A. Karmanov and G.A. Lobov, Pi’sma Zn. Ebsp. Teor. Fiz. 10 (1969) 332 G.A. Lobov, Iz. Akad. Nauk USSR (Ser. Phys.) 34 (1970) 1141 [Sov. J. Nucl. Phys. 35 (1982) 8 221 O.P. Sushkov and V.V. Flambaum, Sov. Phys. Usp. 25 (1982) 1; Nucl. Phys. A412 (1984) 13 V.E. Bunakov and V.P. Gudkov, Z. Phys. A303 (1981) 285; Nucl. Phys. A401 (1983) 93 B. Desplanques and S. Noguera, Phys. Lett. B143 (1984) 303 D.F. Zaretskii and V.K. Sirotkin, Sov. J. Nucl. Phys. 37 (1983) 361; 45 (1987) 808 V.S. Olkhovsky and A.K. Zaichenko, Phys. Lett. B130 (1983) 127 S. Noguera and B. Desplanques, Nucl. Phys. A457 (1986) 189 B. Buck and F. Perey, Phys. Rev. Lett. 8 (1962) 444 B. Desplanques and J. Missimer, Nucl. Phys. A300 (1978) 286; Phys. Lett. B84 (1979) 363 B. Desplanques, Nucl. Phys. A316 (1979) 244 B. Desplanques, Nucl. Phys. A335 (1980) 147 B. Desplanques, J.F. Donoghue and B.R. Holstein, Ann. of Phys. 124 (1980) 449 E.G. Adelberger et al., Phys. Rev. C27 (1983) 2833