The formation of, and the γ-radiation from, the p-shell hypernuclei

ANNALS

OF PHYSICS

116, 167-243 (1978)

The Formation

of, and the y-Radiation

from,

the p-Shell Hypernuclei

R. H. DALITZ Department of Theoretical Physics, Oxford University, Oxford, England

A. GAL* Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel Received February 20, 1978

The physical factors relevant for the production of various low-lying A-hypernuclear states jZ* through the K- --f m- and K- -t@ reactions, in flight or from rest, on the corresponding target nuclei “Z and A(Z + 1) are discussed, on the basis of the shell model for these nuclei and hypernuclei, together with the characteristics of the dominant ytransitions resulting from the excited states thus produced. Detailed consideration is given for a number of hypernuclei of specific interest, including the cases of ,7He, ALi for A = 7, 9andlO,~BeforA=9andlO,~BforA= lO,ll,and12,~CforA = 12,13,14,and15, AN for A = 14 and 15, and ZO. The importance of (y, n-) correlation studies for the determination of hypernuciear spin values is stressed, with the discussion of several examples.

1. INTR~OUCTION Hypernuclear y-rays have already been reported from several expkriments on the interaction of K- mesons with target nuclei. In 1971, Bamberger et al. [I, 21 observed two nonnuclear y-rays, of energies 1.09 MeV and 1.42 MeV, following K- capture at rest on 6Li and ‘Li targets. Later observations [3] using a ‘Li target led to the conclusion that jH* decay was responsible for the 1.09 Mev line, with the suggestion that the 1.42 MeV line might come from jHe* decay. However, recent work [4] indicates that the y-rays from jH* and jHe* decay both lie in the range 1.04 to 1.10 MeV, whereas the existence of the 1.42 MeV line has not been confirmed. Bamberger et al. [2] also observed a non-nuclear y-ray of energy 0.31 MeV when a $Be target was used, and a similar observation is reported from the recent work [4]. Herrera et al. [5] have recently looked for hypernuclear y-rays in coincidence with the K- -+ rr- strangenessexchange reaction in fight and report the observation of an 0.79 MeV y-ray from a ‘Li target. Further information about such d-hypernuclear y-transitions will greatly * Work supported in part by the US-Israel Binational

Science Foundation.

167 0003-4916/78/1161-0167$05.00/0 All

Copyright Q 1978 rights of reproduction

by Academic Press, Inc. in any form reserved.

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AND GAL

advance our knowledge of /I-hypemuclear spectrosopy and our understanding of the A-nucleon interaction within nuclear matter. The emission of y-rays from /I-hypemuclei will arise predominantly from their particle-stable states. These have excitation energies E* not exceeding the groundstate binding energy BA for the (I-hypernucleus under consideration. Generally, the threshold for particle instability lies appreciably below B, , and corresponds to the emission of normal nuclear particles. For example, this threshold is at E,* = 3.50 f 0.04 MeV for IfBe, corresponding to 01emission, to be compared with its Bn value of 6.7.1 f 0.04 MeV. For other rl-hypernuclear species in the nuclear p-shell, these threshold excitation energies are listed in Table 1. Apart from a small number of exceptions (which will be discussed briefly in Sect 3), the particle-stable d-hypernuclear states which emit y-rays belong to the lowest shell-model conhguration an d our detailed remarks here will be confined to y-transitions W)&-J)~-~WA~ The Threshold

Excitation

TABLE I Energies for Particle-Instability Nuclei”

Hypemuclear Species

E* MeV (threshold)

Threshold Channel

jHe

0.24 f 0.10

N+jHe

4.07 + 0.23

;He

2.84 i 0.13”

N+jHe

2.00 f 0.13

;Li

3.94 f 0.04

2H + ;He

7.72 f 0.23

P+‘%eA

jBe

0.67 f 0.08

P + P + jHe

11.37 f 0.06

A + “B

jHe

1.56 i 0.70

N + ;Heb

w9.82”

P + llB A

jLi

6.i5 -+ 0.04

3H + :He

11.69 i 0.12

A +

‘*c

;Be

5.31 + 0.05

SHe + jHe

5.43 i 0.35

N

“C A

XLi

3.76 * 0.15

N+jLi

2.42 iz 0.35”

P + ‘SC A

;Be

3.50 -I 0.04

4He + ;He

8.97 * 0.35

P + wA

‘iB

1.18 f 0.16

P + ;Be

w8.9O

N

‘:Li

~4.8 MeVO

N + jLi

W7.58

P + A16N

for the d-Hypernuclei

Hypernuclear Species

Based on p-Shell

(thr;Told)

Threshold Channel

N-l-/p

+

+

14C A

0 The BA values used were taken from Juric et al. [22] and Cantwell et al. [36]. Not all of these hypemuclei are known, nor are these all the p-shell hypernuclei whose ground state may be particlestable. 0 BA value assumed equal to that known for the charge symmetric hypemucleus. c BA calculated with fit C.

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169

within this configuration. A further reason for this restriction is that all of the groundstate d-hypemuclei known to date, with A > 5, belong to this configuration and that detailed shell-model analyses have been given [6-81 for their binding-energies in terms of effective clN and llNN interactions; our remarks concerning these ytransitions will be based on predictions concerning the excited hypemuclear states for this configuration, based on these analyses. In Section 2 we shall obtain cross section expressions for two particular production processes: (i) the K- --+ 7~ (or roTTo) strangeness exchange reactions at 00. Our limitation to 8 = O” is only a simplifying assumption; the K-N-+ rr-A amplitudes are then spin-independent, and lead to simple spin-parity selection rules for spinless nuclear targets (see Sect. 2.1). For 8 # 00, the K- -+ 7~ amplitude has the general form MO + W a * II 1, w h ere n is the unit vector normal to the K-r plane; the spin-flip amplitude B is then effective, and more detailed calculations would be necessary than those given here. Also, the values of A(0) and B(B) separately are not as well known as are the values (I A( + / B(e)]“) from the differential cross section. (ii) K- absorption from an atomic orbit with quantum numbers (nKIk) about the nucleus. The absorption reaction processes K-N--+ n(A or Z) are very strong, which has the consequence that the capture process will generally take place before the K- meson reaches the (1s) atomic orbit. K- capture will generally occur from a number of orbits (nkZk) with Ik # 0 and will therefore occur predominantly on the periphery of the nucleus. We have little knowledge about the distribution of the capture rates as function of (nK , I,), and the expressions relevant for the data are necessarily averaged over the propability distribution P(n k , Ik) appropriate to the nucleus considered. We shall also discuss the selection rules appropriate for hypernuclear formation through the strangeness exchange reaction at O”, within the (one-step) approximation which we consider here. The treatment of y-transitions between hypernuclear levels is simplified by the fact that the (1 particle is neutral and in an s-state, since the dominant electromagnetic transitions are then essentially limited to y-transitions in the core nucleus and Ml spin-flip transitions for the cl-nucleus system. We shall give transitionrate formulae appropriate to these two situations in Section 2.4. In Section 3, we shall apply these expressions to the prediction of the characteristics to be expected for the production of excited hypernuclear states, for particular species of interest, on the basis of the shell-model configuration {(l~)~(lp)A,~(ls),}, in the following respects: (i) (ii) (iii) K- + n-

the relative production rates for the various excited states, the decay p attern of the y-transitions from these excited states, in some cases, the angular distribution for the y-radiation following the reaction at O”, relative to this direction,

We shall point out there the posibilities

for nuclear isomerism,

where excited cl-hy-

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AND GAL

pernuclei may decay dominantly through the decay interactions A + iVn (or the nonmesonic processes AN -+ NV), giving rise to the apparent occurrence of anomalous BA values. In several cases, we shall discuss the possibility of intruder states which stem from the configurations {(ls)~(lp)~-s’(2s/ld),(ls),} and may disturb the normal y-decay sequences. In Section 4, we turn our attention to estimating the absolute production cross sections, assuming that the strangeness exchange processes are due to a one-step transition (nl), -+ (1~)~ and taking into account the initial K- and final r interactions with the target nucleus in the eikonal approximation. These estimates will then be compared with what little empirical information exists at present, with some discussion. The mechanisms for, and the possible role of, the excitation of low-lying “wrongparity” hypernuclear states, arising mostly from excitations to the (2s, Id) shell, and the generation of hypernuclear y-spectra will also be discussed. We shall emphasize also the possible importance of small admixtures of the (1s)~~ configuration to the low-lying “wrong-parity” states of the core nuclei, and the possibility of obtaining significant nuclear information from observations connected with them. The purpose of experimental studies of these y-transitions may be viewed in two ways, as: (a) a test of the shell model of A-hypernuclear

states,

(b) a source of additional data which may be fed into the phenomenological fitting of A-hypernuclear data by the shell model theory, as function of the AN force parameters introduced, to provide new constraints for this fit. In some cases, the empirical study of y-y or r-y correlations in hypernuclear formation and/or decay will allow, at some time in the future, the determination of the spin-parity values for some of these states. Thus, we may envisage that a body of knowledge about hypemuclear spectroscopy will gradually be built up, and will teach us much about the character of AN and LtiVN forces.

2. DERIVATION

OF FORMULAS

In Section 2.1, we shall derive expressions for the cross sections for production of (1~)~ hypernuclear states in the in-flight K- + 7~ reaction at O”, on the basis of the shell-model. These expressions are appropriate for quite a wide range of target nuclei. Their extension to the O” reaction K- ---f 7~~is also indicated. In Section 2.2, we give expressions for (1~)~ hypernuclear formation rates for K- capture at rest. Their structure is very similar to that of the expressions given for the K- --+ n- forward cross sections in flight. In Section 2.3, we collect together all the selection rules effective in these hypernuclear production reactions, and in Section 2.4, we give the formulas appropriate to the calculation of the y-transition rates for transitions between Iowlying hypemuclear states.

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2.1. Formation Cross Sections of (1~)~ Hypernuclear States in the K- ---f w- Reaction in Flight at O” In the Distorted Wave Impulse Approximation for the strangeness exchange reaction

(DWIA),

we relate the cross section (2.la)

K- + “Z(i) --f n- + ;I”Z(f) to the basic K-n + T-A laboratory cross section by dU/d!Z(O

0’)

=

=

Vfi

dU/dQL(e

=

(2.2a)

0’)~

where vfi denotes the effective neutron number for the transition i --f f, defined by the expression

x

5

u-(j)

W’(r

-

rj)

/ ai ,

Ti~i, J&Ii)

j=l

12.

(2.3)

I

This approach has been used by Hufner et al. [9] for the study of ?C excitations and, recently, also for other closed shell nuclei [lo]. In this expression, the x(*) are distorted waves for the incoming kaon and outgoing pion, respectively, the product X(-)*X(+) reducing to exp(iq * r), with q = pi - pf , in the limit of no absorption. The operator u-(j) is the lowering component of the U-spin vector appropriate to baryonj, vanishing on a proton and transforming a neutron to rl in the same space-spin state; it is the upper component of an isospinor. Choosing the positive z-axis to lie along the beam direction, we expand the product of the distorted waves in expression (2.3) in partial waves:

y&.:*(r) &‘(r) = f

(47$X

+ l))““(i>“j,(qr)

Yko(E),

(2.4)

k=O

where the radial function j, reduces to j k , the spherical Bessel function of order k, as the initial state and final state interactions are turned off. We limit discussion to a single particle transition (nl), --f (Is),; in this case, only the term of (2.4) which has k = I contributes. The effective neutron number vfi , defined by Eq. (2.3), then assumes the form Q Ti2 vlcn~ - c2Ji1+ 1) ( * Ti )

after use of the Wigner Eckart theorem in both space-spin and isospin spaces. In this expression, 71 and Tf stand for the z projection of the initial and final isospin, respectively. The evaluation of the reduced matrix element in (2.5) is straightforward for a

172

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pure valent l-shell with Nl nucleons in the initial nuclear state, in which case the groundstate wavefunction may be characterized in terms of coefficients of fractional parentage (CFP), thus:

I ai , Ti , Ji> =

1

[Y, TN, JN;~,~~}O~~,T~,J~I{IY,TN,JN)OI~,~>>~,J,.

v.TNJNj

(2.6)

The coupling of the 1s A to the various nuclear states 1 y, TN , JN) is given by the following expansion:

We shall find that, in most cases, the sum (2.7) is dominated (2.6) and (2.7), expression (2.5) then reduces to

vani = N,W’i + 1) (-‘:,

X (-1)JM+f+J’(2j

:

by a single term. With

z)’ OJ, + 1)

+ l)l”

I

‘y/l2 / lrn

f.&,(r)jz(qr)

z&(r)

r2 dr

J2,

0

(2.8a) where u&) is the normalized nl radial wavefunction. For nuclei which have closed proton shells and whose excess neutrons belong to a pure Zj shell, expression (2.8a) may be simplified further. In this case, Ti = Nj/2 and Tf = (Nj - 1)/2, with the result:

It is instructive to consider expression (2.8a) for several limiting

cases:

(a) Nj = 1. The only nonzero CFP is that for Ji = j, JN = 0, T, = l/2, Tf = 0, for which its value is unity. Further, Jf = l/2 and Eq. (2.8a) takes the following form, appropriate for one neutron: Jl) 1s.nz

=

O" II 0

A dr).L(qr)

2 d(r)

r2 dr

.

(2.10)

The square of this matrix element therefore measures the strength of the strangeness transition (nZ), -+ (1~)~ in the extreme single particle picture. This strength clearly vanishes in the no-absorption limit with q + 0, unless I = 0. Furthermore, if the N

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HYPERNUCLEI:

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and A radial wavefunctions are generated in the same potential well, then only the (Is), + (1~)~ transition survives in this limit, with a strength of unity. (b) Summation of expression (2.8a) over the final quantum numbers, af, Tf , Jf , accessible for the hypernuclear (1~)~ configuration. The summation on af is first carried for tied values of TfJf , using the orthogonality relation: ; (&~~*(A;;~;~) then followed by summation

(2.11)

= &Y’~JNJ~N,

on Jf :

leading finally to the result Nt

1 BJNNiTt

=

(2T, + 1) ( Tf -7s

f 2

“)’ Ti, IF, Tf 3JN z 4sj I > ai 3 Ti 9Jill2 $nt

- 73 ~kfnl-

((NJ21

(2.13)

With our convention that T = +$ for a proton, the right-hand side of (2.13) is simply the number of (nl) neutrons in the target state 1 CQ, Ti , Ji), multiplied by expression (2. IO). (c) The same summations carried out on expression (2.9) yield the result: N5 1 l[P, JN ;j I > aiJil12$nz

=

(2.14)

N5Z.nl.

@Jo

Of course, this result may also be obtained from the result (2.13) by making the substitutions N& = Ni and 7i = -Nj/2. In these three cases, the summation over all possible final hypernuclear states leads to a summed strength given by the total number of neutrons in the valent shell, multiplied by expression (2.10), appropriate for one neutron. The expression (2.8a) is readily generalized for the charge-changing strangeness exchange reactions: (2.lb) K- + “(2 + l)(i) -+ 7~’ + 2(f) The final result is then the effective proton number appropriate to the transition i -+ f in the reaction (2.lb), namely:

X b%T, , JN ; i7.j I } cziTiJi](-1)JN+f+Jt(2j co

X II

0

+ I)*‘~ z

t

7

‘;I I2

2 &rlh(v)

d’dr>

r2

dr

.

(2.8b)

174

DALITZ

AND

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In applying expression (2.8b) one must recall that the forward nuclear cross section for K- -+ 7~~is related to this effective number as follows: do/dQ (6’ = 0’) = rfj duldi-2, (p + /Qoq = $rfi du/d!C?r. (n -+ A),0 ,

(2.2b)

where the factor l/2 comes from isospin considerations for the I = 1 transition KN -+ mt In some instances, as will become clear from the discussion of ‘Li, an LS formula may prove more transparent than the analogousJj expressions (2.8). If the LS decomposition of a nuclear state 1aTJ) is given by (2.15) .then the appropriate

neutron number takes the form:

=
(2TI+1)(2sN+1)LN[oIN].

a:~i;LN%ILi

x (-l)2S”+JN+t+J~[(2&+

is

22(4

1)(2Li+ 1)(2J,+

cc X

3

[

, ) (2r,+l)(2s,+l)Lj[a]]

1)]1’2 I?

z

2

“;I j$N

;

jN\ (

2 &Crhtqr)

d(r)

r2 dr

.

0

This expression simplifies

when the nuclear wavefunctions involved are given by levels of the core nucleus. Denoting the quantum numbers appropriate to the strangeness transition by

pure LS coupling and the A particle coupling is only with individual

Si+Li=Ji-tS,+L,=J,,

(2.17)

expression (2.16) becomes: vls.nz = WW ’ X

Lf Ji

Jf Li

I

m II

0

+ 1) (-‘:,

A uldrMqr)

:

SC 2 Jf Jiv 1 I! SN Si d(r)

2)” PJ, + 1)(2J, + WSi + 1)(2-h + 1) + 2 (21j+l)(2sN+l)L,[olf]; L, I [

r2 dr

26 w

’

22(4

, > (2r,+l)(2s,+l)Li[,i]]2

(2.18)

so that the spin SN of the nuclear core and the (1 spin must combine to give S, = S, . This last condition results from the spin independence of the B = 0” strangeness transition; it will hold also for nonforward directions only if the KN + 7rA amplitude remains spin-independent away from the forward direction.

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2.2. Hypernuclear

HYPERNUCLEI:

Formation

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175

& y-DECAY

Rates for K- Capture at Rest

We may treat K- capture at rest on a nuclear ground state ] aiT,Ji), leading to the formation of hypernuclear states 1c+T~J~) and the emission of a rr- meson with momentum q, by methods similar to those used in Section 2.1. We follow Hiifner et al. [9] in expressing the relationship of the hypernuclear formation rate to the reaction rate on a neutron in terms of an effective neutron number v appropriate to the transition i -f- Thus, we have

x

I

RnKLK(r) Y L~MJ?I1

d3r x:-‘*(r)

i

u-(j) @)(r - rJ I ai , Ti , Ji, Mi) I’, (2.19)

where the kaon initial state is described by atomic orbit nKL, . Expanding in partial waves x6-‘*(r)

= 477 LG i-LjXq.r)

CLLtB)

YLM~@)

(2.20)

L

we integrate over the direction of the outgoing pion to obtain

X 1

j

u-(j)

at3)(r

-

ri)

1 Eli ,

(2.21)

Ti , Ji , Mi) 1’.

From (2.21), by using the addition theorem for spherical harmonics (note that the resulting spherical harmonics all have order I since the transition is (nl), -+ (ls)J, summing over ML and MK , applying the Wigner-Eckart theorem, and summing over Mi and Mf , we obtain the expression

vfi = M2Ji + 1))(‘:,

; :)” c W + 1) (2

; “,,’

L

x Cc+, Tf , Jf il c u-(j)jL(qrd RnKL,(rJ(441’2 K(cJ 1)ai 9Ti 3Ji) 12- (2.22) j This expression has then the same structure as expression (2.5) for the in-flight case, provided that we make the identification

II

mA

udrMqr)

2 d(r)

r2 dr

0

+ ; (2L + 1) (2

595/x16/1-12

i

t)’

1 brn &r>jLtqr)

R,,L,(r)

d’dr) r 2dr 1’

(2.23)

176

DALITZ

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for the resulting radial matrix element squared. Hence the most general expression for the effective neutron number, given in Section 2.1 by Eq. (2.Q also covers the present case, after the substitution (2.23) is made. As mentioned in the introduction, the distribution function P(nK , LK) for the population of K--nuclear atomic states at the moment of nuclear capture is poorly known. This introduces further uncertainty into any quantitative estimate of the right-hand side of Eq. (2.23) since the latter must be weighted by P(n, , LK) for any comparison with the data. However, the relative formation rates predicted by Eq. (2.22) for the various hypernuclear states reached by the transition (nl), + (1s)” are independent of the capture distribution function P(n, , Lk). This conclusion stems from our tacit assumption, represented by the term 6(rj - r) in Eq. (2.19) and by the absence of any derivatives of it there, that the KN-+ TA reaction amplitude is independent of momentum and entirely S-wave. This assumption appears reasonable on the basis of our knowledge of the K-p interaction in the low-energy regime [l I], namely that the KN S-wave interaction is unusually strong, while the P-wave interactions are small and rather difficult to establish precisely by experiment at present [12, 131. 2.3. Selection Rules for (Is), Hypernuclear Formation The expression (2.8) for the effective neutron number in the nuclear shell nl, appropriate to transitions from the nuclear ground state to various hypernuclear states of the (1~)~ configuration, consists of two factors: (i) the square of the radial matrix element involved. This factor sets the scale for the strangeness exchange process, whether in-flight or from rest, and it is the same for all transitions discussed here. We shall discuss its magnitude for the p-shell nucleons in Section 4. (ii) the other factor depends on knowledge about nuclear and hypernuclear structure for the initial and final valent contigurations, through the CFPs and the coefficients denoted by A, and it depends in detail on the quantum numbers involved in the formation transitions. Within our approximations, the evaluation of formation branching ratios reflects completely the properties of this second factor. There are naturally selection rules which emerge from the expression (2.8). The simplest are as follows: (a) P,(-1)”

= Pf ,

(b) Ti + f =T,,

(4 Ji + 1 = J, ,

(2.24)

which just express the assumed one-step nature of the formation process and its spin-independence for 0 = 0”. With .Ti = 0 for a zero spin target, the selection rules (2.24) imply that the final hypernucleus can have only spin Jf = I and that its parity is then Pi(-l)Jf. These last selection rules hold more generally; for forward two-body processes, with conservation of intrinsic parity, only states with natural parity, i.e., JP = J,(-l)“‘, can be excited from a 0 + target.

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With an LS description, Eq. (2.16) is appropriate and we then have the further restriction Li + 1 = L, . Since the d particle is in the 1s shell, we have LN = L, and thus, (a> L + 1 = b ,

(2.25)

(b) Si = S, ,

where S, denotes the Pauli spin for the hypernuclear state (note that Si and S, are not necessarily unique). These LS selection rules (2.25) are useful for only a small number of cases where pure LS coupling holds to a sufficient approximation, and where the spins Si and S, are unique for both initial nuclear and final hypernuclear states. 2.4. Electromagnetic

Decay Modes for (Is), Hypernuclear

Levels

Excited hypernuclear states which are particle stable will generally decay predominantly through electrodynamic processes. Here we shall classify these decay modes. We shall also note those cases where the allowed electromagnetic rates are suppressed to the point where A-particle weak decay modes may successfully compete, the situation commonly referred to as “hypernuclear isomerism”. Hypernuclear y-transitions involving (Is), may be divided into the following two classes: (a) Doublet transitions. The ‘situation is depicted in Fig. 1, where both initial and final hypernuclear states are members of a doublet built on the nuclear state I CX,JX), with JN # 0. These two states have the same parity and their angular

lti.JN>

,//

IM,

la,J,,(l,),;Ji

>

(A-1$ 2

FIG. I. Figure illustrates Ml spin-tip transition Jj + J, between the two stateswith spins JN i g formed by the attachment of a A particle in the IsI/, orbital to the same core nucleus (spin JN).

momenta Ji and Jf differ by one unit, so that spin flip Ml radiation is allowed. For a doublet splitting AE less than about 5 MeV (which is always the case here), Ml radiation is expected to dominate over E2 radiation when the latter is also allowed. The deexcitation rate was first evaluated by de-Shalit [14] in his study of core excited multiplets in heavy nuclei and by Walecka [15] for the hypernuclear case, where the MI rate is given by:

where (Y~= 4.2 x 1012sec-1MeV-3, g, and gA are the effective nuclear (core) and A g-factors, respectively, both defined by p = gJ and expressed in terms of nuclear

178

DALITZ

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magnetons. In zero-order’approximation, g, and g, are taken from the measured magnetic moments for the nuclear core state 101,JN) and for the free A (thus g, = -1.34 f 0.13 [16]), respectively. We note that when g, = g,, the Ml transition operator lo = g,J, + g,j, reduces to gJ, where g is the common value of the gfactors. J, the total angular momentum, obviously cannot connect states with different J, which makes it clear why the right-hand side of Eq. (2.26) should vanish for g, = gA. We have tacitly assumed that the upper state Ji will not prefer to decay to some lower hypernuclear doublet. In fact, when such a lower doublet exists, the energy for transition to it often exceeds A,!?, and these interdoublet transitions are often the more favourable, as we shall discuss below. (b) Core transitions. The relevant situation is that depicted in Fig. 2. The hypernuclear y-transition Ji -+ Jf is induced by the (one-body) nuclear y-transition p,J,,(ls)A

; J, >

l%+q>(‘s),,;

J-)

Ek (or Mk) Ifi>Jl;Als)A

;J:>

FIG. 2. Figure illustrates the Ek (or Mk) transitions which are generally possible between the states of two doublets (J+ , JJ and (Jk , JI) of the hypernucleus jZ, which are built on two states (LX,JN) and @, &) of its core nucleus (“-l)Z, in consequence of an Ek (or Mk) transition between these two states in (A-l)Z. Of course, in order of increasing energy, these four states (J+ , .Ti , J- , JI) may appear in a variety of sequences, with a corresponding variety of transitions. If JN or Ji is zero, the corresponding doublet reduces to a singlet. If I JN - Jk I = k, then one of the four transitions conceivable will require 1Ji - .Tf / = (k + l), and will therefore be forbidden by the selection rules for Ek (or Mk) transitions.

The latter may be a parity-conserving Ml or E2 transition, or a paritychanging El transition. Denoting by k the multipolarity considered, the hypernuclear deexcitation rate rfi(k) is related to F,(k), that for the core nucleus transition, by [I41

J,-Ji,.

~,,(k)l~iv(k) = (25~ + 1X25, + 1) IF It is a simple matter to check that the summation

;

;I2 (~ti/&)2”+‘.

(2.27)

over both members of the final

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HYPERNUCLEI:

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hypernuclear doublet will yield, up to the appropriate of the core nucleus transition:

179

energy factor, the strength

(2.28)

3. HYPERNUCLEAR

Y-RAY SPECTRA

In this section, we shall apply the formulas derived and discussed&r Section 2 for the formation of hypernuclear states by the strangeness-exchange reactions (2.1) and the y-cascade from their subsequent decay. Where necessary, we shall speak in terms of the in-flight situation, but we remind the reader that similar remarks (involving branching ratios rather than absolute rates) hold for the K- -+ V- and K- --f no interactions from rest. We shall consider in turn the specific features of interest for Lhypernuclei with p-shell core nuclei, for increasing mass number A, but we shall not discuss every conceivable case, especially not those hypernuclei for which there exists no stable nucleus target from which the hypernucleus can be formed through reaction (2.1). The absolute formation rates are rather sensitive to the parameters of the K--nucleus and n--nucleus (or no-nucleus) interactions, which are not everywhere well known; we have used the K‘S and n% cross-sections tabulated by the Particle Data Group [16], the 01, values calculated from the n9I data by Hohler and Jakob [17] using dispersion relations, and the CLEvalues calculated from K-P and K-N scattering data by Jenni et al. [18] and by Baillon et al. [19] using dispersion relations. However, the relative formation rates for a given target nucleus and a given incident momentum PK are essentially independent of this uncertainty since the effect of these initial and final state interactions is concentrated in one matrix-element which is a common factor for all these formation rates. In discussing y-ray intensities, we shall therefore specify only relative formation rates for the K- + V- and K- -+ no reactions on ap-shell target nucleus AZ, normalized such that their sum over all hypernucIear states of the configuration (I~)~(lp)~~~(l,s), in jZ and i(Z - I), respectively, is one, The branching ratios for the various hypernuclear final states (2.1) are obtained from the expressions (2.8). These are made up of (a) the coefficients AZ&, f f ’ characterizing the wavefunctions for the states Z,A*, which we shall take from our shell-model analyses of the p-shell hypernuclei [7, 81. The fit (O+S+QL; 79) of Ref. [8] is the more conservative in spirit, being based on a simple central form for the (INN interaction. With a few exceptions, the wavefunctions #(;A*) for this fit are rather pure, consisting of one dominant term in the sum (2.7), and thus describing the coupling of the /1 to a unique nuclear core state, with spin JN and undistorted by the presence of the (1 particle. Admixtures with amplitude

180

DALITZ

AND

GAL

exceeding 0.1 in magnitude occur for only three ground states of the species considered, namely for jLi, jLi and :C; smaller admixtures will be ignored here. (b) the nuclear CFP appropriate for the ground state #(ZA). These have been calculated and tabulated by Cohen and Kurath [21], as part of a systematic shellmodel fitting to the nuclear data available for the lp, shell. The relative intensities for the y-rays emitted after the reaction process (2.1) are then obtained using the expressions (2.26) and (2.27), with the CFP of Cohen and Kurath when needed. As mentioned in Section 2.4, exceptional cases are possible and these will be discussed as they arise below. We note here that many nuclei in the (1~)~ shell have “intruder states” with parity opposite that of the ground state and lying at quite low excitation energies; for example, the first excited state of 9Be at 1.67 MeV has spin-parity (3 +). Obviously, there will be low-lying “wrong parity” states in the corresponding hypernuclei, and their effects will have to be considered in the discussion to follow concerning the y-rays emitted following the excitation of specific hypernuclear states. We shall also point out the angular distribution expected for each y-ray discussed, relative to the forward direction 0, = 0”, since its measurement depends only on the observability of the y-ray. Since the primary transition (Ip), - (1~)~ has dl = 1, this angular distribution can only be of the form W(4) - (1 + 4TJ)

cos2&,),

(3.1)

as is demonstrated explicitly in Appendix B. The coefficient A(ny) will be specified for each y-ray considered. In this discussion, we shall generally assume that the energy levels of these p-shell hypernuclei are approximately those calculated with intermediate coupling and with the dN and llNN force parameters of the fit (fl+S+&,,; 79), which we shall refer to hereafter as the fit C (= canonical). However, we shall also make use of the extreme limits of LS and jj coupling, where appropriate, for the purpose of illustration and comprehension. 3.1. The Hypernuclear

Species :Li and ;He

These systems are particularly important, since they are the lightest hypernuclei for which there will definitely exist particle-stable excited states and for which yemission is definitely expected. They are therefore species which should be studied in some detail, in that the y-rays characteristic of their decay should become wellestablished, and thus recognizable and removable in experiments using heavier target nuclei. However, these species and their y-rays have been discussed in considerable detail recently [20], so that we will be brief here. The I = 0 and I = 1 states of JLi and their relationship with the states of 6Li are shown on Fig. 3. The LS limit provides a good approximation for these states and we shall discuss their formation and decay on this basis, giving also numerical results obtained from the use of intermediate coupling wavefunctions. The discussion of the

P-SHELL

HYPERNUCLEI:

6Li

FORMATION

&

y-DECAY

181

7ALi

FIG. 3. The energy levels for BLi from experiment [23] and for JLi as calculated with the fit C. The numbers in the square brackets are the relative formation rates for the reaction K- + ‘Li -+ w- + jLi* at 0”.

spectroscopy of jLi is also simplified by the observation that each of these JLi* wavefunctions is dominated by a single 6Li parent. The properties of hypernuclear doublets are discussed in Ref. [8] and we summarize the essential points here. When there is a unique parent state, with spin JN , the spin-spin interaction A, the spin-orbit interaction S, , and the tensor interaction T do not contribute to the spin-weighted average E = ((JN + 1) E+ + J,E_)@J,

+ l),

(7.1)

where E* denotes the energy of the doublet state with spin (JN i ;t), for the two hypernuclear doublet states. The contributions of the spin-orbit interaction S, and the three-body interaction Qi, depend on JN , but do not contribute to the level separation (E+ - E-). The four lowest states of :Li form two hypernuclear doublets, based on: (i) 6Li(g.s.). This system has structure V. The addition of a 1s A particle generates the states 4S with J = 312 and 59 with J = l/2. The interactions S, , S, and T do not contribute to their energies because it has L = 0. The interaction A contributes energy &A to the J = 3/2 state, where all spins are parallel. The fact that the energy E has no contribution from d means that the spin-spin interaction contributes energy -A to the J = l/2 state. Thus their separation is given by d(JN = 1) - E+ - Em = +34/2.

(7.2)

The interaction Q& contributes the same energy -31/2Qi,, to all of the states discussed here.

182

DALITZ

AND

GAL

(ii) 6Li*(JN = 3) at 2.18 MeV. This system has structure 3D3 , and the addition of a 1s (1 particle generates the states *D with J = 7/2 and (71i2 2D + 2112*D)/3 with J = 5/2. Again, all spins are parallel for *D, and from Eq. (2.20) and Table III of Ref. [6], the interactions d, S, and T contribute 2 x (3/4) x (A/3 + 2S,/3 - 4T/5) to its energy E+ . Since these interactions do not contribute to (7.1), we conclude that d(J,

= 3) = (E+ - Em)=+;A+;&-;T

(7.3)

The interactions S, and Q&, contribute (S, - 3’/“Q&) to the mean energy E(JN = 3). The locations of these states as predicted for fit C is indicated on Fig. 3. All four are particle-stable, the lowest threshold being that for (iHe + 2H), which occurs at excitation energy E* = 3.94 ho.04 MeV. Two other states are to be mentioned: (iii) the lowest T = 1 level of lLi*, corresponding to iHe(g.s.). The fit C predicts this state to be particle-stable by 0.4 MeV, but this conclusion is related with its prediction of Bn = 5.54 MeV for JBe, which is 0.4 MeV greater than the empirical value Bn = 5.16 f 0.08 MeV [22]. Empirically, particle stability will be the case only if (B,* - B,(jHe)

- 2.088 & 0.005) MeV > 0,

(7.4)

where B1; denotes the separation energy for iLi*(T = 1) with respect to (/l + 6Li*(T = 1,0 +)). If this B1; is identified with Bn for JBe, as would be appropriate if charge-independence held exactly, then the expression (7.4) takes the value (-0.05 f 0.08) MeV, and this state is unstable, more likely than not. If unstable, JLi*(T = 1) will be a relatively narrow state, since its decay to (i;He + 2H) requires violation of isospin conservation and will take place only in consequence of electromagnetic effects. However, there are a number of uncertainties in this empirical argument, as are discussed in Ref. [20], and we must keep an open mind on this question, for the present. If jLi*(T = 1) is particle-stable, then it will be formed (see below) and there will be seen y-rays characteristic of its decay to the low-lying states of ILi. It is quite likely that these y-ray observations will be the means by which we learn whether or not the lowest T = 1 state of JLi is particle-stable. (iv) the J = 3/2 state of the hypernuclear doublet based on the JP = 2 +, T = 0 state 6Li* at 4.31 & 0.03 MeV. This sLi state has structure (2D + *D)/2112, in the LS limit. The separation of the doublet is given by d(JN = 2) = (5A + 25S/, + 42T)/12

(7.5)

and the interactions S, and Q& contribute (--+S, - 31i2Q&) to the mean energy E(JN = 2) here. This reduces the excitation energy by (-34 + SS, + 2S, + (42/5)2”)/4 = 0.50 MeV, with the parameters of fit C, placing this state at +3.8 MeV and so predicting it to be particle-stable.

P-SHELL

HYPERNUCLEI:

FORMATION

&

y-DECAY

183

With the limit of LS coupling, the target ‘Li has structure 2P and the selection rules (2.25) forbid transitions to the configurations 4S and 4D of interest here, while allowing transitions to 2S and 2D, the other two configurations of interest here. The relative formation probabilities calculated with fit C and intermediate coupling for all the states discussed are given on Fig. 3, in the square brackets against each level. The values calculated with fit C and LS coupling are as follows, given in the order of increasing excitation energy: 5/12 : zero : 7/30 : zero : 5136 : l/60, and show good qualitative agreement with the i.c. values. Thus about 28 % of the total transition rate goes to excitation of the (5/2) level, so that we can expect quite strong y-emission following the K- -+ r- reaction at 0”. Another 15 ‘A goes to excitation of the T = 1 (4) level, and 2 % to excitation of the T = 0 (3/2)* level, from which states there will be further y-emission if they are particle-stable. The strongly-excited (5/2+; 0) level will decay by E2 y-emission to the g.s. doublet states. These involve the E2 transition JN = 3 + JN, = 1 in the core nucleus 6Li, which is known [23] to have transition rate (6.7 f 0.5) x 101’ set-l for transition energy 2.185 Mev. For the energies given on Fig. 3, and expression (2.27), we deduce the transition rates 4.8 x IO9 set-l for 5/2+ + 3/2+, and 4.7 x lOlo set-l for 5/2+ + l/2+. Thus, emission of this last y-ray (energy 1.35 MeV) will occur for about 25 ok of the (lp), -+ (1~)~ hypernuclear formation strength in the O” reaction K- + ‘Li -+ rr- + JLi*,

(7.6)

so that it will be the dominant component of the y-ray spectrum following this reaction. We note, in passing, that the weak decay processes for the (1 are not negligible in this 5/2+ state of iLi*. Adopting the weak interaction rate 7;’ = 3.9 x lo9 set-l (the total (mesonic + nonmesonic) /1 decay rate in this state is probably 1.5 to 2.0 times this free decay rate [24]), we see that about 7 % of the (1 particles in this state decay before y-emission. The lesser y-emission, 5/2+ + 3/2+, is followed by the Ml spin-flip transition 3/2f -+ l/2+, which has rate 1.0 x IOr’ see-l for the calcuIated excitation energy 0.25 MeV. The angular distributions are given by A(7ry) = 3/4, 3/13 and 3/49 for the 1.35 MeV, 1.10 MeV and 0.25 Mev y-rays. The excitation of the (l/2 +*; 1) level, if it is particle-stable, will give rise to a second y-ray cascade. The Ml t ransitions to the g.s. doublet are fast, with rate of order lOI set-r (deduced from our knowledge of the (O+; 1) --P (If; 0) core transition [23]) and with ratio (&?(1/2* --f 1/2))3: 2(&(1/2* + 3/2))3. The transition to the 3/2+ level is then followed by the y-emission 3/2+ - l/2+, as discussed above. These y-emissions are all isotropic, of course. Transitions to the 5/2+ and 7/2+ levels, following (l/2*; 1) excitation, have negligible rate. We may also neglect the y-ray cascade from (3/2*; 0) excitation; if particle-stable, its leading transition is 3/2+ -+ 5/2+, with energy 2.5 MeV and intensity 1.5 % of the net y-ray emission following (7.6).

184

AND GAL

DALITZ

The structure of the hypernucleus following the related reaction:

jHe can be studied through the y-spectrum

K- + ‘Li + no + JHe*,

(7.7)

in coincidence with the detection of a no meson emitted at approximately o”, at least for K- incident momentum sufficiently low that this reaction can be clearly distinguished from similar multipion processes. From charge-independence, it follows that the absolute formation rate for lHe* in this reaction is the same as that for the isobaric state lLi*(T = 1). All the levels expected to be particle-stable are given on Fig. 4, together with their relative formation rates. The particle-stability threshold is at (2.84 & 0.13) MeV, for the channel (N + ,6He). The dominant JHe* excitation in reaction (7.7) is that for the 5/2+ state, although its cross section is in fact only about one-third of that for the excitation of the (5/2+; 0) state in reaction (7.6). It will decay by a rapid Ml spin-flip transition to the 3/2+ state, with rate 9.6(dQ3 x loll set-l MeV-3 = 1.7 x 1012 set-l for fit C, since the doublet splitting is given by

A(5/2+ --f 3/2+) = 0.464 + 2.03S, + O.OSS,- 2.6753f 0.22Qio,

(7.8)

which then takes the value 1.22 MeV, as given on Fig. 4. The subsequent decay of the 3/2+ level has been much discussed in the past [22, 25-261, and this discussion has recently been up-dated by Dalitz and Gal [20]. We shall not repeat this discussion here, except to say that: (2.8420.13)

-----

(N+iHe)

2.3 +I’ 2 \ \

1.80

go.351 Ml

\ 0.h

f

[ o.os]

1

l?

0’

0 (MeVI

-

-0

Ml ++[0.60]

(MeV)

6He

\He

FIG. 4. The energy levels for OHe, as known from experiment [23], and for XHe as calculated with the fit C. The numbers in square brackets are the relative rates for the reaction K- + ‘Li -+ Y#’+ ;He* at 0”. They are four times those for the analog T = 1 states in jLi, the latter being excited by (K-, n-) on ‘Li. The 3/2+ level of XHe may possibly be isomeric (see text).

P-SHELL

HYPERNUCLEI:

FORMATION

&

y-DECAY

185

(i) E2 decay to g.s. for either of the excited states has transition rate far below -r;l, the free A decay rate, (ii) Ml decay can occur for the 3/2+ state through small admixtures of the high-lying “PI configuration into the wavefunctions of the 3/2+ and l/2+ ,7He states. The up-dated discussion shows that the situation is rather sensitive to the tensor interaction T. The transition rate for fit C (where T = 0) is calculated to be 9.6 x log set-I, but this falls as T becomes nonzero and positive. The decay rate through weak interactions for the A particle in jHe is expected [24] to be quite comparable with that for free /1 decay, about 4 x log set-‘, although there now exists conflicting evidence concerning the relative strengths of the AN and AP nonmesonic weak decay interactions [27, 541. The conclusion is that the 3/2+ state is at least partially isomeric, and could well be strongly isomeric. The study of the y-rays following reaction (7.7) could settle this matter, by determining the intensity ratio for the 3/2+ + l/2+ y-rays, relative to the 5/2f --f 3/2+ y-rays. In this case, it will be necessary to take into account also the nonzero rate for the direct excitation of the 3/2+. It is useful to remark that the latter leads to an angular distribution given by A(rry) = +3/4, whereas the 3/2+ -+ l/2+ y-rays resulting from the 5/2+ excitation have A(ry) = -7/19, the same as for the 5/2+ + 3/2+ branch of this cascade. 3.2. The Hypernuclei

,8Be and jLi.

The particle-stable states predicted for :Be by fit C are indicated on Fig. 5, together with the relative formation rates from the intermediate coupling calculation. More than half the transition strength goes to particle-unstable jBe states, most of which will be states built on the I = 0 and I = 1 states of 8Be* with excitation around and above 17 MeV. $+[0.1131

3.56

/ 2.91 -2+

\

/

/

Ml \

\

3.5020.05 ---_ 5,He+‘He

E2

\ 2.&

f*[

0.1291

I-,

o-

o+

(Mei’)

0 ( MeVl

eOe

l!

E2

++[0.193]

gAOe

FIG. 5. The energy levels for *Be, as known up to 3 MeV from experiment [23], and for RBe as calculated with the fit C. The numbers in square brackets are the relative formation rates for the reaction K- + %e + ?T- + jBe* at 0”. Contrary to the figure, the y-rays from the 5/2+ level will be absent if the threshold for particle-instability lies below it.

186

DALITZ

AND

GAL

The fit C predicts only one particle-stable state for IBe*, that with spin-parity (3/2)+, built essentially on the 2+ broad level of *Be* at 2.94 MeV. This state will decay by E2 y-emission to jBe(g.s.), by deexcitation of the core state. Equation (2.27) for the transition rate requires I’, for *Be*(2+) as input. Since this is not known (r, = 1.45 MeV dominates the *Be*(2+) decay rate), we estimate it by calculating the transition rate for lD[4] * lS[4], adopting (&,) = 6.25 fm". Inserting this rate f,, in Eq. (2.27), the iBe* y-decay rate is r(3/2+ -+ l/2+; E2) = 1.72 x 101’ set-l,

(9.1)

its angular distribution being (3.1) with A(ny) = -l/2. Isoscalar E2 transition rates calculated for nuclei in this way are systematically low; the adoption of an effective charge q - 1.5 to 2.0 is conventional, so that the estimate (9.1) might rise as high as 7 x loll set-l. The (5/2)+ member of the hypernuclear doublet built on the 2.94 MeV state sBe*(2*) is only just particle-unstable for the fit C. For parameter values near those for fit C, the excitation energy for this state is well approximated by E*(5/2+) M (0.98OS, - 0.498T+

2.94) MeV.

(9.2)

A small reduction of S, and a moderate positive value for T (say +0.2) would bring this state below the stability limit, and it is for this reason that we have indicated on Fig. 5 the y-rays which would then result from the 5/2+ level. The dominant ytransition would then be Ml spin-flip to the 3/2+ level. With a calculated value g(2+) M 0.51, Eq. (2.26) gives the transition rate Q5/2+ --f 3/2+; Ml)

= 2.2 x 1013 set-l,

(9.3)

with angular distribution A(ry) = -7/19. For E2 y-emission to the l/2+ ground state, the estimate corresponding to (9.1) is I'(5/2+ + l/2+; E2) = 2.8 x 1012 set-l,

(9.4)

for effective charge q = 1, with angular distribution A(rry) = 3/4. For q N 2, this E2 rate would be competitive with the Ml .spin-flip rate. For parameter values close to those of fit C, the doublet separation is d = 2.45S, - 1.25T.

(9.5)

The rate (9.3) is proportional to d3, so that the branching ratio between these two y-rays from the 5/2 level will depend appreciably on its excitation energy, although generally dominated by the Ml transition. The observation of this y-ray will determine the quantity (9.5), which depends strongly on S, and T. The angular distribution of the E2 transition 3/2+ 3 l/2+ following this 5/2+ --t 3/2+ transition is given by A(xy) = 531197.

P-SHELL

HYPERNUCLEI:

FORMATION

&

y-DECAY

187

We now discuss briefly the y-rays emitted in the associated reactions of type (2.lb), K- + 9Be -+ r” + ,PLi*.

(9.6)

The particle-stable energy levels for fit C are given on Fig. 6, together with the relative probabilities of excitation. About half the transition strength goes to the excitation of jLi(g.s.); 20 % goes to excitation of the 3/2+ level at 1.88 MeV, 17 % to the 5/2+ level at 1.94 MeV, and the excitation of other particle-stable states can be ignored. The absence of excitation of the 5/2+ level at 0.99 MeV is readily understood from the selection rule (2.25b). BLi(g.s.) has structure dominantly 3P, , so that the hypernuclear doublet states based on it have structures (“P and 4P)3,2 and 4P,,2 . Since gBe has structure dominantly 2P,,2 , this selection rule allows the 3/2 excitation but not the 512 excitation. The absence of excitation of the l/2+ level at 1.14 MeV depends on more detailed dynamics; *Li(J, = 1) has both “PI and lP, components, whose amplitudes happen to lead to dominantly 4P,12 structure for this level of jLi, so that this selection rule (2.25b) is effective again here. The significant y-transitions following (9.6) are those indicated on Fig. 6. From I’J3+ + 2+) = (1 f 0.5) x 1014 set-l known for *Li*, the Ml rate I’,,(5/2+* -+ g.s.) for jLi* is (6 & 3) x 1013set-I; the Ml rate r(5/2+* -+ 5/2+) is two orders of magnitude slower. From I’,(l+ -+ 2+) = (8 & 3) x 1013 set-l known for *Li*, the Ml transition rates for 3/2* -+ 3/2 and 3/2* + 5/2 are both approximately (6 f 2) x 1013 set-l. The spin-flip transition 3/2* --f l/2 is two orders of magnitude slower (we used p = -2.36 n.m., the calculated value for *Li*(l)), and the g.s. spin-flip transition

(3.76‘-o.l5L!+!“--,+ 3.21

3.52, ,+*---3.45I I /’

2.26

3+ ----1,9L

j”[403,

:+* IO.17 I 3’” [ 0.20)

I.88

/’ Ml I I’ l.lL 1 + ‘-- - -0*9,9

0.98 -

, /’ 2+1---

* Li

“Ml

Ml

-0 li!B (MeV)

i+ ++

10.001 [O.Ol]

;+

IO.L9]

Ml

9

^Li

FIG. 6. The energy levels known up to 3.3 MeV for *Li from experiment [23] and as calculated for ALi from fit C. The numbers in the square brackets to the right are the relative formation rates for the reaction K- + OBe + no + jLi * at 0”. The main y-ray transitions which follow this jLi* excitation are indicated, as discussed in the text.

188

DALITZ

AND

GAL

5/2 - 3/2 is fast (- lOi set-l). Hence, for each (1~)~ -+ (1,~)~ transition, ing y-ray energies and intensities are predicted: Transition Energy (MeV) Rel. Intensity A(T)

the follow-

5*/2 - 312 3*/2 * 312 3*/2--f 512 512 -+ 312 1.94 1.88 0.89 0.99

=

17 % -7/19

Finally, we comment about to those from JLi*. The target and the P -+ rl transition rate 0” K- - rr reactions on target JLi(g.s.)/jBe(g.s.)

10 % -12129

10 % +I/8

10 % + 14137

the intensities for y-rays from lBe* formation relative 9Be has 3 p-shell neutrons, as against 2 p-shell protons, is 0.5 times that for N-+ (1, from isospin. Thus, in the 9Be, we expect the ratio = (0.489 x 2 x 0.5)/(0.193 x 3) = 0.84.

(9.7)

The total number of all the jLi* y-rays mentioned above bears ratio 1.2 to the number of y(3/2 + l/2)-rays from the jBe* excitations. However, if the 5/2 state of jBe* is particle-stable, there will then be more y-rays from jBe*; the number (0.129 x 3) for y(3/2 -+ l/2) will rise to (0.129 + 0.113) x 3 and that for y(5/2 + 3/2) becomes (0.113 x 3), assuming Ml dominance in the decay of jBe*(5+/2) as discussed above, so that the ratio (y’s from jLi*/y’s from jBe*) would fall to 0.46. 3.3 The Hypernuclear Doublet (:Be, YB) and the System ‘,oLi The particle-stable states predicted for :B and :Be for fit C are shown on Fig. 7. The level patterns are essentially identical, of course, but the thresholds for particleinstability are different, owing to Coulomb effects. We note that there is one “intruder” doublet (O+, 1’) with opposite parity, at excitation energy -2 MeV. We shall find below that this doublet will have little effect on the y-ray spectrum, so we ignore it for the present. Since the target l”B has J = 3, the 0” K- -+ 7~- reactions K- + l”B --+ rr- + 1,OB*

(10.1)

cannot lead to final states with J1 = 0 or 1, in consequence of the / Al j = 1 selection rule effective here: in particular, the formation of ‘jB(g.s.) is forbidden, since we expect it to be l-. The strongest transition is to its J1 = 2- partner. The transition to the J1 = 3- state is also quite strong, but this state lies well above the stability limit as long as S, is positive and substantial, in rough accord with fit C. There are transitions (~3 %) to the J1 = 2-* level predicted to lie at about 2.1 MeV, near the stability limit of 2.00 f 0.13 MeV; whether y-rays will or will not be emitted from this state will depend on whether it lies below or above this limit. At best, these latter y-rays will be weak. Thus, the dominant y-ray is expected to be that resulting from the spin-flip Ml transition from the first-excited state of :B*, formation of which is predicted to

p-SHELL

HYPERNUCLEI:

FORMATION

&

189

Y-DECAY (4.07iO.23) -----

($7 [0.1191

'.

2-j-2.06 ------(P*;Be)

/

3>L

3- [0.119]

;:

;36--.m4+---243 / /

210

\ \

1.6-(++)---A66

0‘ T-

\ '7 2.15

--Y

[0.032]

[0.200]

-*

228-l

(T‘7 -2.78.

3- -3.57

./

-106

(N+iBe)

)0+&l+ T'iO.0321

(2.00'0.13)

2-

0.92, _ /

1- 1

10 n0

0 ----($)

_ -0.92

-

2-

(MS!)

(M&i

\

\

0'

0

(MeV)

(Me'41 9El

[0.2001

O-

9Be

1-

'%e A

7. The energy levels known up to 3 MeV for the isobaric doublet (9B, 9Be) from experiment and as calculated for (SB, 2Be). The numbers in the square brackets are the relative formation rates for both the reactions K- + ‘OB + r- + 1,OB*and no + yBe* at 0”. The figure illustrates the difference between the y-ray spectra for the two mirror hypernuclei, resulting from their different thresholds for particle-instability. For YRe, we note that there occur two positive parity hypernuclear states in the midst of the spectrum of particle-stable states. FIG.

[23]

have 20 % of the N + (1 transition strength from initial state l”B. Using the calculated value g = 1.99 for gB, this Ml transition rate is I’,,(2- -+ l-) = 1.4 x lOI set-l, and A(~TY) = -l/7 for the y angular distribution. Measurement of this y-energy will determine the splitting for the g.s. doublet

d = 0.51d + 1.48S, - 1.27T,

(10.2)

and will confirm the spin-sequence expected for this doublet. If this doublet were inverted, the g.s. spin-flip y-ray would not be seen, since the N + A transition cannot excite the J1 = l- state. We note that, if we assumed the j-j coupling limit to hold here, so that the g.s. doublet had structure (ls,,,),(lp,&,, , this g.s. doublet splitting would be given by

d(JN = 3/2) = 2(A + 2S, - 122-‘/5)/3,

(10.3),

which is quite comparable with the i.c. result (10.2). This combination 6 = (A + 2S, 12T/5) occurs in a number of contexts; we have seen it already in the splitting of the fist-excited doublet in jLi (expression (7.3) is 78/6), and we shall see it again later in the splitting of the g.s. doublets for ?B and ?B. If the 2-* state is particle-stable, which requires 0.48S, + O.lZS, + 0.35T > (0.36 i 0.13) MeV

(10.4)

190

DALITZ

AND

GAL

for parameter values near those for fit C, it will be weakly excited, as noted above. Its dominant y-decay will be a fast Ml transition to g.s., with rate 1.4 x 1014 se&, the branch to the 2- first-excited state having rate only 2 % of this. This dominant y-ray has A(ry) = -2/23. We now consider the “intruder” hypernuclear states with J’P’ = Of and l+, which are built on the l/2+ state in (QBe, QB) with excitation energy about 1.7 MeV. To first approximation, this state may be described by coupling a 2s,,, proton to BBe(g.s.). The Id orbital is believed [28] to lie higher than the 2s orbital in this A-range, and its participation in the QB state under consideration would require coupling it to the 2+ 8Be* state at 2.94 MeV at least. Hence, we shall neglect here any contribution to the structure of the l/2+ state from the Id orbital. To test this simple 2s orbital model, we have evaluated the El radiative width of the l/2+ QBe state at 1.68 MeV and have obtained excellent agreement with the reported value. We shall make a rough estimate here as to where these “intruder” hypernuclear states may lie in (:Be, ?B). With J = l-, the g.s. binding energy for this isodoublet is given by Bn(g.s.) = b, + 5v + 0.284 + 0.965SA - 0.835T - 1.43SN + 9.85Q&, , (10.5a) where b, denotes B, for jHe, whereas the mean binding isodoublets based on the 2s orbital is B,(2s intruder)

energy for the (O+, l+)

= b,, - E*(2s) + 4it + V(2s) - O.SlS, + 9.82Q& . (10.5b)

The contributions from Q& are practically the same, because of the spin-isospin dependence assumed to hold for the (INN interaction and the fact that the (ls)“(l~)~ core remaining after removal of the 1s fl particle and the 2s nucleon has JN = 0, and a dominant component with the maximum supermultiplet symmetry allowed. The contribution of S, is different because there is no contribution from the 2s nucleon in the intruder state. As discussed in Appendix C, the mean potential r(2s) is expected to be approximately the same as the B for a lp nucleon, so that it contributes very little to the excitation energy of these intruder states. The mean excitation energy E(2.s) differs from the excitation energy E*(2s) in the parent QBe nucleus partly because it is raised by amount 0.81S, whereas the mean energy of the (l-, 2-) ‘jBe(g.s.) isodoublets is raised by 1.43S, , and partly because of the spin splitting within the latter, due to the interactions d, S, and T, which push ‘jBe(g.8.) down further. Neglecting Qi,, , the net effect is that E(2s) = E*(2s)

- 0.62S, + 0.284 + 0.965S, - 0.835T

m 1.68 + 0.13 + 0.04 + 0.55 = 2.40 MeV

(10.6)

using the parameters of fit C. The splitting d of the 2s hypernuclear doublet is expected to be small, since this can arise only from the spin-spin interaction d when IN = 0,

P-SHELL

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191

and d is small for fit C. For these (2~)~ intruder states, it is 42s) which is relevant, but this will be given by 423)/d = F(2S)/l7, (10.7) where d and rdenote the values for a lp nucleon, as in Refs. [6-81. From Appendix C we note that a reasonable choice of parameters gives the value 0.96 for the ratio (10.7). The value of d for these (Of, 1’) states is then d(2s) = 0.15 MeV. These intruder states are not expected to be particle-stable in ‘,OB, on the basis of our present knowledge. The situation for the intruder hypernuclear doublet based on the 5/2+ state in sBe* at 3.06 MeV can be discussed in a similar way. However, some further parameters now appear. First, we have v(ld) and d(ld); from Appendix C, these are given by V(ld)/B

= d(ld)/d

= 0.63,

(10.8)

this value being that resulting from a reasonable choice of parameters. Then we have new coefficients S,(ld) and S,(ld), also T(ld). The former are given by

where the functions JN and JA are defined in Appendix C, the suffix being necessary since these two spin-orbit interactions S, and S, will generally have different radial dependence. These four parameters introduce additional uncertainties but the most important effect is the reduction of the central spin-independent interaction by about 40 ‘A, since Vis large; with V = 1.23 MeV, for fit C, we have T(ld) M 0.77 MeV and this pushes the mean excitation energy for the intruder doublet (2+, 3’) up by about 0.46 MeV. On the other hand, their energy separation d receives a large contribution from S,(ld,,,), which may be estimated from Appendix C, for any particular model. On this basis, it appears likely that the 2f intruder state in :Be will be particle-stable, and perhaps even below the 3- state excited in the 0” K- -+ 7~’ reaction on l”B. Direct K- --j. W- and K- + 7r” excitation of such intruder states as these will be neglected, in general, in our discussions to follow below. Their excitation in this way would require either: (a) a two-baryon transition (lp)i(lp)E -+ (lp)~(lp)~(2s),/,(ls), in :B/?Be, or (b) appeal to the most likely components of (2 particle-2 hole) type in the lOB(g.s.) wavefunction which allow one-baryon transitions such as: (i)

[8Be(2f; 2.94 MeV) @ [(2s,,, , ~s~,J]~>‘=~]~+*~=~ dl=o- [gBe (T ; 3.06 MeV) @ (Is,~~),]~~’

(ii)

[*Wg.s.)

(iii)

[sBe(g.s.) 0 (2s,,, , ld,,,)]~;r==O 41=2

595/116/1-13

0 (14,

, ~Q,N~>~=~ xp

[sBe (s ; 3.06 MeV) @ (IS,,,)~]~+, FBe (g ; 1.68 MeV) @ (us,/,),]‘+,

192

DALITZ

AND

GAL

and we expect these multibaryon processes to be substantially suppressed relative to the one-baryon interaction terms. We note that the formation of :B*(O+) from target l”B(3+) is not possible through any such mechanism, since the transition 3+ -+ 0+ is forbidden by the selection rules (2.24) appropriate to the 0” K- -+ rr reaction processes. Thus, for the y-ray spectrum following the 0” K- -+ TIT-reaction on 1°B, we are led to a rather simple picture. Of the two states of the hypemuclear g.s. doublet, this reaction excites only the 2- state. With fit C, we expect this to be the first excited state. The y-ray spectrum will then be dominated by a - 1 MeV line, due to a fast Ml spin-flip transition 2- + I- between the g.s. doublet states. If such a line were definitely ruled out, we would have to conclude that the 2- state was the ground state of :B (or very low-lying in ‘:B). If such a line were definitely ruled out, we would have to conclude that S, was negative (or very small) in :B (contrary to our current belief, which depends inter alia on the evidence available from the study of ‘AB(g.s.) and ?B(g.s.)). If this were the case, then the upper doublet (3-, 2-*), built on “B(5/2-), would be inverted; if the 3- state were then at about 2 MeV excitation and particlestable, it would be excited strongly and would undergo a fast Ml transition (rate - 1014 set-l) to the ‘:B ground state (which would then have JP = 2-), emitting a y-ray of energy - 2 MeV and with A(sT~) = 9/17. If the - 1 MeV y-ray is seen, so that the situation is as expected for fit C, then there would still be the possibility of the emission of a much weaker - 2 MeV y-ray, which would then be an indication that the 2-* state lies below the stability limit. We now turn to the excitation of ‘:Be. This can be excited in two ways, and it is of interest to compare the information obtainable from each. The first process is K- + l”B -+ yr” + ‘>Be*,

(10.10)

which has cross sections just one-half those for the final state (7r- + ‘jB*), according to charge independence. The important difference is that the threshold for particleinstability is higher, the threshold for (N + JBe) being at excitation energy 4.07 & 0.2 MeV in YBe. The levels of ?Be are charge symmetric to those of :B, which are given in Fig. 7, but the hypernuclear doublets based on the 5/2- and l/2- levels of QBe are now particle-stable. Owing to the selection rule 1Al 1 = 1, reaction (10.10) at 0” can only produce states with spin 2-, 3- and 4-. The new features of reaction (10.10) are therefore the strong excitation of the 3- level and the lesser excitation of the 2-* level; it is worth giving some relative rates for 0” K- -+ 7r on 1°B target, -rr- + 1,OB(2-) = no + ljBe(2-) 0.200 0.100

= 7~~+ ‘;Be(3-) 0.060

= yr” + ‘jBe(2-*) 0.016

*

(10.11)

The y-decay of this 3- level is dominated by Ml decay to the 2- level (transition rate - 3 x 101’ set-l), although there is competition (- 3 %) from the spin-flip Ml transition 3- -+ 2-*, and (- 1 %) from the E2 transition 3- + l-. The first and third of these transition rates were obtained using expression (2.27) and the empirical

P-SHELL

HYPERNUCLEI:

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&

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193

data on the transition 5/2- --+ 3/2- in 9Be. The second rate obtained using (2.26) depends on the estimate g = -0.14 for the 5/2- state of 9Be; this estimate is based on the assignment of its lowest 3/2-, 5/2- and 7/2- levels to a K = 3/2 rotational band, a model which gives quite good accord with the data on the known 9Be y-transitions. Thus, the y-spectrum following reaction (10.10) is expected to be dominated by two strong lines and one weaker line, with energies - 1 MeV, - 2.7 MeV, and - 2.2 MeV and intensities in the ratio roughly 1 : 0.35 : 0.1. We note that the rate expected for the - 1 MeV line is 3.8 x loll set-l, considerably slower than for the analog ‘jB transition. The second process of :Be* excitation is the 0” K- -+ n- process on target loBe, K- + loBe -+ n- + 1,OBe*.

(10.12)

The important difference from process (10.10) is that the target has spin 0. The selection rule 1Al 1 = 1 tells us that this process (10.12) can excite only states with JP = l-, states which are not directly excited in reactions (10.1) and (10.10). There are only two l- states in ?Be expected to be particle-stable, :Be(g.s.) and a state I-* at about 3.8 MeV which is built on the l/2- state of 9Be* at 2.78 MeV. This is depicted on Fig. 8. (4.07

-+ 0.23) (N +,‘Bc) --e-w-

I O.lL61

3.06 2.70 2.L3

1.66

0

(MeV)

(MeV) ‘Be

‘iBe

FIG. 8. The energy levels known up to 3 MeV for @Befrom experiment [23] and as calculated for loBe for fit C. The reaction K- + loBe --f ?T- + 2Be* at 0” can excite only l- levels; the relative formation rates are given in square brackets. We note that the y-ray spectrum is now quite different from that given in Fig. 7; also that is quite possible that El transitions (indicated by dashed lines) to and from the (O+ & I+) states may now play a nonnegligible role in the r-cascade.

194

DALITZ

AND GAL

The y-decays from this l-* level of :Be to the ground hypernuclear doublet states arefast, the rates being0.40 I’g’ and 0.85 rpl for thefinalstates l- and 2-,respectively, where I”gl denotes here the rate for the Ml transition l/2- -+ 3/2- in the core nucleus sBe. Our calculated value for this rate is 9.0 x 1014 set-l. The rates for l-* -+ 3- and 2-* involve E2 core transitions and may be neglected, especially as their y energies (0.1 and 1.6 MeV, resp.) are not high. The rate for the I-* -j 0- spin-flip transition using the calculated estimate g = 1.7 for sBe( l/2-), is only 3.4 x 1012set-l. However, the existence of the intruder doublet (O+, l+) lying below the l-* state must also be considered, since an El transition l/2- -+ l/2+ is now possible for the core nucleus. Denoting its transition rate by rcE1, the l-* transition rates are given by 2I’g’/3 and rg1/3, for the final states l+ and O+, respectively, apart from their dependence (AE/1.10)3 on the y-ray energies. For a rough estimate, we calculate assuming the l/2+ state of QBe to have structure l?S,[4] @ (2s,,,), and the l/2- state to have the structure zzPl,z [4, 11. This leads to the value I’:: = 7 x 1013 set-l. Since BA for the l-* state is given by b, - E*(1/2-)

+ 5r + 0.084 - 0.32S, - 0.35SN - 1.57T + 9.91&,

, (10.13)

its comparison with (10.5b) for fit C indicates that it is reasonable to adopt a value AE M 1.4 MeV; the transition rates for l-* + l+ and l-* -+ 0+ are then estimated by 0.96 x 1014 set-l and 4.8 x 1013 set-l, respectively. We conclude that the y spectrum following reaction (10.12) will be dominated by the 2.9 MeV (- 60 %)l and 3.8 MeV ( N 28 %) Ml transitions to the ?Be g.s. doublet. However, there can well be also quite strong El transitions, with energies of order 1.4 MeV to the lowlying intruder states, possibly amounting to as much as 10 % of the y-transitions from the only ‘jBe* state appreciably excited.2 These (O+, l+) intruder states will then decay by El emission at a fast rate (- 5 x 1014 set-l), dominantly to ‘$Be(g.s.), generating two further y-ray lines in the spectrum. This picture would be modified if there were a significant rate of direct formation of these intruder states in the reaction (10.12). However, we note first that the excitation of ‘jBe*(l+) from target loBe is still not possible, since the transition 0+ + I+ is forbidden by the selection rules (2.24) appropriate to the 0” K- -+ n reaction processes. The excitation of IjlBe*(O+) requires appeal to (2 particle-2 hole) components in the lOBe(g.s.) wavefunction, since these allow the one-baryon transition [%e(g.s.) 0 (2s,,, , ~Q,)I~+*~=~ zp

[vBe (T ; 1.68 MeV) @ (ls,,,)A~ot.

This direct formation of ‘jBe*(O+) would then be followed by the El y-emission to :Be(g.s.) which has been discussed just above. The addition of such y-rays to those 1 Followed by the 0.9 MeV line discussed above. p The rate expected for I-* -+ 2+, in the event that the latter lies below the former level, is quite negligible, since it is an M2 core transition.

P-SHELL

HYPERNUCLEI:

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&

Y-DECAY

195

emitted in the El-El y-cascade following the direct excitation of ljBe(l-*) would cause the observed intensity ratios to differ from those expected for this cascade. The observation of the effects of such direct formation processes would have much intrinsic interest, although we do not expect them; we shall return to this topic in Section 4. Finally, we consider briefly the excitation and decay of yLi* states, in the 0” reaction with target loBe, K- + loBe -+ no + yLi*.

(10.14)

Again, this reaction will excite only J = 1 states of ?Li. The energy level pattern expected is shown in Fig. 9: the particle-stability threshold (N + ,BLi) is at -5 MeV. It consists of two hypernuclear doublets, based on the 3/2- and l/2- states of gLi, providing a further example of a general situation discussed in Section 3 (c) of Ref. 181. There may also exist one higher particle-stable doublet, based on the gLi state at 4.3 MeV, whose nature is not yet known. The reaction (10.14) thus excites only the upper I-* state, and that with N 13 % of the reaction strength, as calculated in the LS limit (the other 87 % then goes to ‘jLi(g.s.) formation); this state will then decay by fast Ml dominantly to the 2- state, which then undergoes a spin-flip transition to 1,oLi(g.s.).

1iT---

2.69

1-•[0.13] \

\

\

\

0-

f 0

2----2‘-

0.5

2-

-0

l- IO.871

LMeV)

( MeV) gLi

‘\Li

FIG. 9. The energy levels known up to 4.5 MeV from experiment [23] are given for gLi. The energy levels shown for ‘,OLi are a rough sketch only, with the relative formation rates at 0” as calculated for the nuclear configuration *P[3, 21 shown in square brackets on the right.

3.4. The Hypernucleus ‘,?,B

This is the most complicated case we shall consider, and we shall discuss it quite briefly, partly for lack of knowledge. In Fig. 10, we show the l”B core states in detail up to the (2+; 1) level at 5.17 MeV. Higher levels are known but mostly do not have unique spin assignments; a number of levels with (-) parity appear in this region,

196

DALITZ

6.56

AND

GAL

(7.72tO.23) ---me--

(T.1)

2+,

(~+'be)

-6*gj(T")

+? [O-026] 7+ 2

6.79 (1.1)

2.15 1 -74

1+ (T-1)

0'

0.72 0

[0.0601

3+

[0.274] 1 MeV)

IMeW ‘OB

11 AB

FIG. 10. The energy levels for l”B as known from experiment [23] (four T = 0 levels below 6.56 MeV are omitted or unlabelled: l+ at 5.18 MeV, 2+ at 5.92 MeV, 4+ at 6.02 MeV and 3- at 6.13 MeV) and as calculated for ZB with the parameters of fit C. The relative formation rates for IjB are given by the numbers in the square brackets to the right. All the y-transitions are shown whose intensity is more than 5 % of that for the dominant r-line. This is the most complex spectrum considered in this paper.

beyond the well established (22; 0) level at 5.11 MeV. The highest level we show explicitly is the (2+; 1) level at 6.56 MeV well known in loBe. The complexity of the hypemuclear spectrum shown on Fig. 10 is due to the high threshold energy for particle-instability, at 7.72 & 0.23 MeV for (P + 2Be). With fit C, the (5/2+; 1) level of ‘:B based on the l”B(2+; 1) level at 6.56 MeV is probably particle-unstable, whereas its doublet partner yB(3/2+; 1) is particle-stable. The pattern of excitations in the 0” reaction K- + llB + 7~- + yB*

(11.1)

is very simple, for only five states have transition probabilities exceeding 0.04, and these states account for more than half of the transition strength. These states are shown on Fig. 10; besides zB(g.s.), which accounts for 27 %, they are the (3/2f; 0) state at 3.58 MeV (- 11 “/,>, the (3/2+; 1) state at 5.19 MeV (- 13 %), the (l/2+; 0) state at 0.95 MeV (- 6 %) and the (l/2+; 1) state at 2.57 MeV (- 5 %).

P-SHELL

HYPERNUCLEI:

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197

& y-DECAY

These states decay by a large number of y-cascade paths. We have estimated the rates for all these transitions and will just summarize the conclusions here. The numbers given are percentages of the 0” K- -+ 7~- total transition strength. &(MeV) Intensity

=

5.19 4.24

2.00

1.61

2.2

4.4

18.1 9.0

5.8

1.01

0.95

0.62

(Isomeric ?) 3.9

The three most energetic y-rays result from excitation of the (3/2+; 1) ‘iB* level at 5.19 MeV. Excitation of the (3/2+*, 0) level at 3.58 MeV leads to the 1.01 MeV y-ray emission to reach the (l/2+; 1) level at 2.57 MeV, together with a weaker 0.39 MeV y-ray emission to reach the (I /2+, 0) level at 3.19 MeV. This last level is also fed by the transition following the 2.00 MeV y-ray emission from the (3/2+; 1) level at 5.19 MeV, and the dominant y-transition from it is to the (l/2+; 0) level at 0.95 MeV. This 0.95 MeV level is also fed directly following the excitation of the (l/2+; 1) and (3/2+; 1) levels, following the emission of their 1.62 and 4.24 MeV y-rays, respectively. The result of all these y-cascades is that the intensity of the population of the 0.95 MeV level is 31.7, i.e. 91 % of the available excitation intensity. The decay of the lowest (l/2+; 0) level of YB has excited much comment in the past [29]. The reason is that its decay to the (5/2+; 0) ‘iB(g.s.) is due to an E2 transition in the core nucleus. This l”B transition is I+ + 3+, from the 0.72 MeV first excited state of 1°B, with transition rate (0.99 i 0.015) x log set-l. Using expression (2.27), the transition rate for the E2 transition l/2+ -+ 5/2+ from the first excited state of ‘;B is 4.0 x log set-l, using the calculated excitation energy of 0.95 MeV. We recall that the free A decay rate is .F’, = (3.88 & 0.05) x log set-‘. The weak interaction decay rate for the A in this l/2+ YB state is not known. Pionic decay is suppressed by the restrictions of the Pauli principle for the product nucleon, but there are nonmesonic decay channels which become available for a A particle in contact with nuclear matter. Estimates of the net weak decay rate have been made in the literature [24], which suggest that this rate is increased somewhat, by perhaps as much as a factor of two. If we adopt the estimate F,Wk(yB(1/2+, 0)) - 6 x log set-‘, then - 60 % of the A particles which reach this level will undergo weak decay, so that this is the situation which we describe as “partial isomerism.” These figures would still leave this 0.95 MeV y-ray resulting from this isomeric state with intensity 12, as one of the strongest three y-rays in the spectrum. Obviously the details of the above discussion depend crucially on the precise features of the spectrum; for example, if the excitation energy of this level were increased to 1.2 MeV (say), y-decay would have branching fraction about 70 %, whereas if it were as low as 0.8 MeV (say), its branching fraction would fall to 20 %. If the y-spectrum from the reaction (11.1) is ever understood, the intensity of the y-ray from the l/2+ + 5/2+ g.s. transition could provide a determination of the lifetime of a /1 particle in this l/2+ state, since the calculation of this y-transition rate is on a relatively firm footing. We comment briefly on the interpretation of the transition strengths for the various excitations. These depend primarily on the CFP for N + log* e, llB(g.s.), and we compare the i.c. values of Cohen and Kurath [21] with those for jj coupling for the

198

DALITZ

AND

GAL

states (JN ; T) of greatest strength (cf. Appendix A). Where two entries are given, they are for the hypernuclear spins (JN - l/2) and (JN + l/2), respectively. Core state (JN ; T)

(3+; 0) (O+; 1)

Cohen-Kurath

0.274

Jj

7116

cl+; 0)

v+; 1)

0.054

0.028,0.105

0.134,0.008

l/16

5/4&l/12

l/4,1/16

We note that the JJ values, obtained using the formula (cf. Appendix A): (11.2) show at least good qualitative accord with the i.c. values. The lowest intruder state in i”B is (2-; 0) at 5.11 MeV. Since the corresponding hypernuclear doublet (5/2-, 3/2-) lies close, or perhaps above, the strongly excited state ‘jB* at 5.19 MeV, these intruder states cannot be reached by y-decay following the dominant 0” K- + 7~-- excitations on llB target. However, there is a (3/2+; 1) state at 6.98 MeV, based on the (2+; 1) state of l”B at 6.56 MeV, which is weakly excited, with transition strength 0.026. Transitions from this 6.98 MeV state to the intruder states are possible, but can only play little role in the y-spectrum following reaction (11 .l), since the 6.98 MeV state will decay dominantly to low-lying states, by energetic and fast Ml y-emission from the nuclear core. The hypernuclear doublet splittings calculated with i.c. wavefunctions (but with coupling to a unique nuclear core state) are as follows: d(3) = 0.98d + 2.525, - 3.07T

(11.3a)

d(1) = -0.214

(11.3b)

d(l*)

+ 1.71S, - 1.35T

= 1.024 + 0.48S, - 3.281:

(11.3c)

The g.s. splitting (11.3a) is almost the same expression as that for the d(3) in JLi, cf. Eq. (7.3); it is almost equal to the value appropriate to the configuration to the g.s. [(P2/2)N1(P2/2);11’=3(1s)II 9 not surprisingly. It is also almost proportional doublet splitting in ?B. However, this simple picture does not hold for (11.3b, c), for which the jj coupling limit gives (A/2 + S, - 6T/5). In fact, it happens that neither of these adjacent core states is close to the pure jj coupling limit. We note that the g.s. doublet splitting d(3) cannot be readily determined from the study of this y-spectrum, because the 7/2+ state plays very little role in these y-cascades. It appears in a minor branch of the cascade leading from the (3/2*; 0) yB* state at 3.58 MeV, the 1.98 MeV y-ray from the transition (3/2*; 0) + (7/2+; 0) having branching fraction 4 %, and also in a minor branch leading from the (5/2*; 0) yB* state at 5.40 MeV (formation rate 0.025), the 3.80 MeV y-ray from the transition (5/2*; 0) + (7/2+; 0) having branching fraction 20 %. When formed thus, the (7/2+; 0)

P-SHELL HYPERNUCLEI:

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& y-DECAY

199

state emits a y-ray by a fast Ml spin-flip transition to the (5/2+; 0) g.s., the y-energy being 1.60 MeV according to our calculations. Thus, this y-ray is expected to have N 1.0 ‘A of the total K- + n- transition strength for (lp), -+ (1.~)~ . 3.5. The Hypernuclei :C and YB

These states form a hypernuclear isospin doublet. We shall discuss the energy level structure and the y-rays specifically for ‘jC*, formed in the 0” reaction K- + 12C -+ r- + ‘;C*,

(12.1)

since 12C is a particularly convenient target. However, more is known about llB* than about llC*, so we have to base some remarks on the former rather than the latter data, and it will then be natural to mention ?B* as well as ?C*. The energy level patterns for llC and ;C are shown on Fig. 11. The threshold for particleinstability lies high, at 9.82 MeV for the channel (P + ‘:B). For ‘jB, the lowest threshold is even higher, at 11.37 MeV for the channel (A + llB). Since J = 0 holds for 12C(g.s.), the 0” reaction (12.1) can lead only to :C* states with J,P = I-, owing to the selection rule I dl 1 = 1. The three excitations noted on Fig. 11 account for 99.4 % of the transition strength for states (ls)“(lp)‘(Is), . The dominant excitation is ‘jC(g.s.), as jj coupling would suggest, since 12C(g.s.) has a closed (l~,,,)~ shell, while llC is then represented by (lpSi2)‘, or equivalently (lp,,,)-l. The y-rays entered on Fig. 11 are all Ml llC* core transitions except for the Ml L. 80

32---

52--i

L. 32

0

5.11-

1-“”

L.59-

2-•

3; - 0.95 T-, '. 0

2l-

[O.O9L]

[.0.712

]

( MeV)

( MeV) “C

‘*AC

FIG. 11. The energy levels known up to 5 MeV for W from experiment [38] and as calculated for ZC using the parameters of fit C. The relative formation rates appropriate to the reaction K- + ‘T - Tr- + ‘jC* at 0” are given in the square brackets on the right. All the y-transitions aie shown whose intensity corresponds to 2 % of the (1~)~ + (1s)~ transition strength for this reaction.

200

DALITZ

AND

GAL

spin-flip transition 2- -+ l- for the ‘;C g.s. doublet. following the two :C* excitations.

We consider the y-cascades

(i) Following the l-** excitation, 80 % of the y-transitions are to ‘jC(g.s.), with two weaker transitions, each with branching fraction 10 %, to the lowest 2- and O- states. For the first two of these, the input was the transition rate for the 3/2-* ---f 312 4.80 MeV y-ray in llC; of course, the latter is not known, so the value has been taken from the llB data, the justification being that these Ml transition amplitudes are dominantly isovector. Both the 2- and 0- states decay to the g.s. by Ml y-emissions, the first by spin-flip (rate N 6.6 x loll set-l) and the second by the llC core-transition l/2- -+ 3/2- (estimated rate 3.6 x 1014 set-l, based on the data for this core transition in llB and the assumption of isovector dominance for the Ml amplitude). The angular distributions of the primary y-rays are given by A(~TY) = - 1, + 1, and -l/7 for the final states 0-, l-, and 2-, respectively. The y-ray 0- + g.s. will be isotropic, while A(rry) = -27/53 will hold for the 2- + g.s. y-ray which has been formed in this 1- * * decay. (ii) The l-* decay rates are given by C(J,)(dE(l-*

-+ 5,-)/2.00)3 r,M’(1/2-

-+ g.s.)

(12.2)

where C = l/6 for J, = l-, 5/6 for J, = 2-. The core transition rate is estimated from the llB data, as mentioned above. These rates are 1.3 x 1014set-r for Jz = l-, and 2.3 x 1014 set-l for Jz = 2-, both much larger than expected for the spin-flip transition l-* -+ O-. The angular distributions are again given by A(ny) = +l and -l/7 for Jz = l- and 2-, respectively. To summarize, the four strongest y-rays are 5.11 MeV (- 8 % of initial transition strength), 3.29 MeV (- 7 %), 2.34 MeV (- 12 %) and 0.95 MeV (- 14.5 %, coming mostly from the cascade l-* ---f 2----f l-), the latter being the g.s. hypernuclear doublet transition. From such y-ray observations, the excitation energies E*(J,P) can be determined for J,P = 2-, l-*, and l-**. For parameters near those of fit C, these energies are given by the following expressions, including all of the ANN forces defined in Refs. 16, 71 for the purpose of illustration:

E*(JP) ~5, d

S/I

2-

0.03

0.45

1.55

l-* l-**

2.00

0.12

1.37

4.80 0.05 -0.05

SN

T

Q:,

-1.89

0.15

-1.34

0.13

-1.47

-1.55

-0.02

-0.15

Q;,

Q;z

Q:,

Qk

-0.01 -0.04 (12.3a) 1.00 0.61 1.33 (12.3b) 1.38 0.37 1.51 (12.3~)

-0.02 -0.12

0.01

We note that the various parameters enter these expressions with widely differing coefficients. The g.s. doublet splitting E*(2-) has little dependence on the ANN parameters, since all the ANN interactions adopted (motivated by specific properties of the double-OPE mechanism [6]) are all independent of A spin. We note that this splitting agrees qualitatively with the expectation d(3/2) given by Eq. (10.3). We may

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note also that the doublet splitting calculated for the two states based on the l/2- state "C*(2.00 MeV) has the value d(J, = l/2) = -0.354

+ 1.35S, + 6.34T,

(12.4)

which is quite close to the value expected with jj coupling, djj(.JN = l/2) = (-A

+ 4S, + 24T)/3.

(12.5)

Returning to expressions (12.3) we note that the ANN parameters Qt can shift the mean energies for different doublets by quite different amounts, for E*(l-*) depends strongly on both Qz,, and Q& , and E*(l-**) depends strongly on Q& . Similarly, the dependence of level positions on the tensor interaction T varies widely, for the E* values for the l-* and l-** states are practically independent of T, whereas E*(2-) has a rather strong dependence on T. Between the uppermost level shown on Fig. 11, at 5.11 MeV, and the threshold for particle-instability at 9.8 rfrr0.1 MeV, there are certainly very many levels of ?C*, with both parity values, most of which have negligible formation strength for excitation in the 0” K- -+ r- reaction. However, the evidence now is [30, 311 that a large fraction of the total 0” K- -+ rr- hypernuclear formation rate (all configurations) in lzC is clustered around excitation energy E* = 11 MeV, which happens to be close to the threshold for (1 emission as well as above threshold for P emission. It is generally believed [32] that this is associated with a O+ state of ?C* belonging to the configuration (P~,&‘(P~~~)~ . This configuration also generates states I+, 2+, and 3+, but the general selection rules for the 0” K- -+ 7~- reaction require parity change (- 1)“’ for (vector) orbital angular momentum change AI. With J = 0 for target 12C, this means that only the Of and 2+ states of this configuration can be formed in the 0” reaction. For small momentum transfer q 5 120 MeV/c, the 2f formation cross section is expected to be considerably smaller than that for formation of the Of (partial analog) state [32-341 just mentioned, although still quite comparable with the cross section for formation of the states of the configuration (p,,,&‘(ls), , of the type with which this paper is primarily concerned. It is our expectation that this 2f state of ‘jC* will lie lower than the 0+ (partial analog) state, although the spacing between them depends on the details of the ilN interaction, such as its exchange character. in simple models, at least, the 2+ state also lies below the I+ and 3+ states. The latter can be formed in the K- + 7r- reaction for 8, # 0”. The 2+ state could well lie as low as 1.5 to 2.0 MeV below the Of state, in which case it would be particlestable. A simplified picture of the situation is given in Fig. 12, where it has been assumed that the (1 couples always with the same Y(g.s.) core. Coupling of the (1~)~ orbital generates the g-s. ?C doublet; coupling of a (p3!2)11orbital generates the states O+, If, 2+ and 3+ mentioned above and shown on Fig. 12. A more realistic description would require couplings with all the other llC core states, as well as the inclusion of the orbital (pl12)n . Nevertheless, no further states from the configuration (lp)$(lp), are

202

DALITZ

AND

GAL

-

2-

-1AJ’C

(MeV)

(A-orbitals)

12 AC

FIG. 12. The energy levels for l,zC are shown, corresponding to the structure (AW(g.s.)) in the j limit, where the excitation is provided by the A orbital alone. In other words, these are the levels for the two configurations (lp,&$(ls)~ and (lps&‘(lp3/,)~. The 0+ state is that generally known as the “Strangeness Analog State” in zC*, based on the lpslz shell. It lies above the particle-instability threshold, which lies at 9.8 MeV.

expected to lie below the particle-stability threshold, and this is uncertain even for the 2+ state. With this picture, and with particle-stability for the 2+ state alone, two El ytransitions are expected, 2+ -+ l- and 2-, corresponding to the transition (lp,,,), -+ (Is), . This is possible despite the fact that the A-particle is neutral, because the charge of the rlC nuclear core to which it is attached causes the A-particle to have a nonzero “effective charge” in the coupling of the system with the electromagnetic field (see below). We can calculate the transition rates using Eq. (2.27) if we interchange the roles of the A particle and the nuclear core there, i.e. if we treat the nuclear core as the spectator for a y-transition involving a change of orbital for the A-particle. We then have:

~i,wMEi,)3 =

@Ll + 1)(25, + 1) 1:

;;

?I2 (r,(El)/E,3),

(12.6)

where the hypernuclear El transition Ji -+ Jf is induced by the El transition j, -jL , E,, is an arbitrary energy inserted for dimensional reasons, and the last factor is given by

rAE1)/EA3 =

@,I + 1)-l I(& /I D(A)l/jJ12

x

3.80

x

1014fm-2 se+ MeV3, (12.7)

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where D(A) is the El dipole operator effective for a II particle in this cl-(core nucleus) state. The dipole operator D for the system is given by the translation-invariant expression (A-1)

D =

c

(12.8)

ei(r, - R),

i=l

where e, denotes the charge of the ith nucleon and R is the c.m. coordinate for the system, given by

where Mi denotes the mass of the ith nucleon and J4, = C:“;” Mi is the mass of the nuclear core. For the transition lPA ---t 1~~ of interest here, only the term proportional to r, in the dipole operator (12.8) is effective. After substituting expression (12.9) for R, we thus deduce from (12.8) the dipole operator appropriate to the “C* A situation , as D = -eQ(MAiWA

+ M&m

,

(12.10)

where Q = 6 denotes the total charge of the system. Expression (12.7) is then evaluated as r,(EJE43

= $[(Ls 11r /I lp3,2)12(QM~/(M~

+ Mc))2 x 3.80 x 1014fm-2 set-l MeV-3. (12.11)

Using un = 0.33 fm-2 for the harmonic oscillator wavefunction in the p-shell, we deduce the value 1.97 x 1014 set-l MeV-3. Assuming the 2+ state of ?C* to be at excitation 9 MeV, we then have the decay rates F(2++2-)

M 5.1 x 1016set-l, F(2++

I-) M 7.2 x 1Ol6 set-?

(12.12)

The angular distributions of these two y-rays are given by A(z-y) = +l and -3/5, respectively. The Of state in ?B, which is isobaric with the Of enhancement observed in 2C* at E* m 11 MeV, may well be particle-stable, since the stability threshold for ‘:B is at E* = 11.37 MeV. If this is the case, then it will be expected to decay rapidly by El transition to ‘jB(g.s.), with rate F(O+

-+ l-) w (11)3 x 1.97 x 1014 = 2.6 x 1017set-l.

(12.13)

Since the excitation of the 0+ state, involving coherence for the p-shell nucleons of the target, is expected to be much stronger than the excitation of the 2f state (which is certainly particle-stable in ‘:B), this energetic y-line is expected to have a really high intensity, considerably stronger than the intensities of the y-rays from the 2+ excitation, and very much stronger than the y-rays following the Ip, --+ Is, excitations

204

DALITZ

AND

GAL

which are the main topic of this paper. These interesting y-rays could conveniently be studied following the reaction K- + l*C -+ no + ljB*

(12.14)

for no formed near 19, = 0”. 3.6. The Hypernucleus YC The energy levels up to 8 MeV, as calculated for YC with fit C, are shown on Fig. 13 and these are the only levels excited with >I % of the total transition strength for lp, ---f lsn . The threshold for particle-stability lies at 11.69 & 0.12 MeV, where the channel (/I + rzC) opens. We note that the dominant transition (w 22 % of the Ip, -, Isn transition strength) is to the first excited state of ‘jC, with spin 3/2-. Its decay gives rise to a single y-ray of energy about 4.2 MeV. Although the selection rules allow an Ml transition, such a transition can only occur through admixtures of (I+; 0) states of 12C* into the initial and final states of YC, but the admixtures calculated in Refs. [7, 81 are too small to make this Ml amplitude significant. The core-transition is 2+ -, Of, and therefore E2, with a known rate. Using expression (2.27) and E*(3/2+) = 4.15 MeV, the 3/2+ b l/2+ transition rate is calculated to be 1.2 x 1013 set-l, and A(ny) = + 1 characterizes its angular distribution. The energy of this y-ray is given by E*(3/2’)

= 4.44 MeV - 0.144 - 1.36S, - 1.42SN - 1.72T - 2.22Qio + 0.115Qz2 + l.53Qt2 + 0.98Qi2 + 1.60Q& ,

(13.1)

where we have included all the Qtj of Ref. [7] because their coefficients are becoming large, with the increase of A. This expression (13.1) is only the diagonal term of the energy matrix; for ‘iC(g.s.) there is also a substantial off-diagonal term, which leads to a 10 % amplitude for admixture of the 0 + 12C* state at 7.66 MeV into the yC(g.s.) wavefunction, but this shifts ‘jC(g.s.) upwards in energy by less than 0.1 MeV. There is appreciable uncertainty in any calculation for the l/2* state of r,sJJ at 8.0 MeV, since the structure of the core nucleus 12C*(O+) at 7.66 MeV is not yet established, although it is well known that it does not correspond to any single shellmodel configuration. Probably its wavefunction includes a number of terms of the type (n particle-n hole) with appreciable amplitudes. Our shell-model estimate of transition strength gives about 5 % for the excitation of this state of yC*, but this result is sensitive to details of the wavefunctions; for example, the use of the i.c. nuclear wavefunctions of Soper [35] gives only 2 % for this transition strength, while giving 16 % for the transition strength to yC(g.s.) in fair accord with the value of Cohen and Kurath [21] entered on Fig. 13. It is quite possible that the transition strength for the l/2+* ‘jC* state may be much smaller than 5 %. If it is formed appreciably, the dominant y-transition will be l/2+* + 3/2+, an E2 transition with rate (5.0 & 1.6) x 1012 set-l for assumed energy 3.85 MeV. The other branches are 1/2+* h 5/2+, E2 with branching fraction m 15 %, and l/2+* 3 l/2+, an Ml

P-SHELL

HYPERNUCLEI:

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&

y-DECAY

205

----(11.7~0.1)

7.66

(A+‘%)

0:. elbow++*

L.44

[0.046]

5+ z

*‘T-5.50 \

f’

Ll5

[0.224

71 0 ( MeV)

0+ \

I 0 -$+

[0.123]

( MeV)

FIG. 13. The energy levels as known up to 9 MeV for r2C from experiment [38] (five further levels between 9.6 and 12 MeV, with both parity signs, have been omitted and are not invoked here) and as calculated (see Table II) for ;C using the parameters of fit C. The relative formation rates, given in square brackets for the reaction K- + 13C --f n- + qC at 0”, show that the 4.15 MeV r-line is expected to be by far the strongest r-line.

transition occurring through 1+ Y* admixtures in the initial and final ‘;C states, with branching fraction calculated to be M 5 %. The calculated value for the 5/2+-3/2+ doublet splitting energy is d(JN = 2) = 0.184 + 2.32S, + 2.73T,

(13.2)

but it is clearly not easy to measure, since the 5/2+ state plays relatively little role in the y-cascade just discussed, for it is excited only in a weak branch following a weak K- -+ W- excitation. Its decay has two branches, E2 decay to ‘jC(g.s.) with rate 5.0 x 1013 set-l and Ml spin-flip to its 3/2+ partner with rate 1.4 x 1013 set-l, assuming the energies shown on Fig. 13. If the two levels of the doublet were interchanged, the y spectrum would be markedly different. The upper level would be strongly excited and its decay would give rise to three strong y-lines, 70 % to the direct E2 transition (- 5.5 MeV) to g.s. and 30 % to the Ml spin-flip transition (- 1.5 MeV) to the lower doublet level, the latter followed by a fast E2 transition (- 4 MeV) to g.s. There is an important qualitative difference between the case of ?C and thep,,,-shell hypernuclei to be discussed below, and the case of the p,,,-shell hypernuclei discussed above. For the latter, the dominant formation rate was always that for the g.s. multiplet; for the former, as we shall see below, the dominant formation strength is always to excited multiplets, some of which may lie quite high in excitation. To

206

DALITZ

AND GAL

understand this for ?C, we consider the limit ofj coupling. This predicts that the (+) parity states of 12C*, (2+; 0) at 4.44 MeV, (l+; 0) at 12.71 MeV, (l+; 1) at 15.11 MeV and (2+; 1) at 16.11 MeV should all be assigned to the configuration (Ip,,2)-1(lp,,,) in 13C. yC(g.s.) is then formed by the transition (lp,,,), -+ (1~~)~ which involves only l/5 of the p-shell neutrons. The distribution of the other 4/5-th of the transition strength (all being of the type (lp,,,), -+ (1~)~) can be calculated using a simple model, given in Appendix A, which leads to branching ratios (13.3) irrespective of the final isospin Tf , where Ji = l/2 holds for the present case of ‘jC. This result has the same Jf dependence as that given by Eq. (2.8) when j is confined to the value 3/2. Thisjj coupling model leads to excellent agreement with calculation using (2.8) and the CFP’s given by Cohen-Kurath [21], as we see on Table II. TABLE

II

Relative Formation Rates for the J$’ = l/2+ and 3/2+ States of ?C Excited in the K- -+ n- Reaction on 13C at 0”, as Given by the jj Coupling Limit and as Calculated by Cohen and Kurath [21] (Of; oy

(2f; 0)

(I+; O)d

(l+; l)d

(2+; 1)c

Sum

ii

0.200

0.250

0.150

0.150

0.250

1.000

CK

0.123

0.224

0.133

0.120

0.202

0.802

(Jcfl; T) =

@The (Jc”; T) values refer to the W* states on which the hypernuclear states are built and which have the largest pllz and pal2strengths in 13C; their excitation energies are g.s., 4.44 MeV, 12.71 MeV, 15.11 MeV, and 16.11 MeV, respectively. b Jf = l/2 only, c The J, = l/2 hypernuclear state is not formed, d Shared in ratio 8:l between the states (J, = l/2):(5, = 312). TABLE Relative Formation

Jcn =

III

Rates for the 2N States Excited in the 0” Reaction K- + v- on IaN, as Given for jj Coupling and as Calculated by Cohen and Kurath [21] l/2-*”

3/2-* c

5/2-

ii

2115 = 0.133

4/15 = 0.267

215 = 0.400

CK

0.134

0.231

0.372

“The Jp values are those for the l3N* states on which the hypernuclear states are built and which have the largest pal2strengths in 14N; the excitation energies of these states are 8.92 MeV, 9.48 MeV, and 7.38 MeV, respectively. b Shared in ratio 2:l between the states (J, = O):(J, = 1). c Shared in ratio 5:l between the states (J, = l):(J, = 2).

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207

Some of these high-lying states based on the (p3,2)-1(p1,2) configurations for lzC* may have quite narrow widths. This is definitely the case for the T = 1 levels based on (I+; 1) and (2+; 1) and with spins l/2+, 3/2+ (twice) and 5/2+, since their decays to (A + 12C) violate isospin conservation; further, except for the case l/2+, this decay requires IA = 2. The next lowest threshold is at 12.35 MeV, for (4He + jBe), for which the same remark holds. The lowest T = 1 threshold is at 15.8 MeV, for (P + ?B), and the decay of these :C* states to this channel is possible with I, = 1, if allowed by energy conservation. No y-emission is expected from any of these levels above excitation energy I 1.7 MeV. 3.7. The Hypernuclei

‘$N and YC

The level diagrams for the charge-doublet states (?N, ?C) and their parent doublet CIN, 13C) are shown on Figs. 14 and 15. The hypernucleus ‘jC is known empirically [36] to have BA = 12.17 + 0.33 MeV. We shall assume charge symmetry, hence that this BA value holds also for ‘,4N. Only one particle-stable excited state is expected for :N, since the threshold energy for (P + :C) lies quite low, at 2.42 f 0.35 MeV. The particle-instability threshold lies much higher in ?C, at 5.4 & 0.35 MeV for the channel (N + YC). We consider ?N first, since it is the more convenient for study, using target 14N, through the reaction at O”, (14.1)

K- + 14N -+ ,r- + l;N.

The formation rate for particle-stable ?N states is only 14 % of the total lp, --f Is, transition strength, but in fact the maximum rate possible for (lp,,,), + (1~)~ is only 20 ‘A here, since 14N has only one pllz neutron in five p-shell neutrons. The 4.0_3 3.55

20 5+

3. 51 p3c--4.03

2.37

0

1 MeV)

‘3N

2’

[o.ooo]

,' 1-* [0.0331

-2 lf 2

-

i-2..

-

,?I

O’& 1’

(2.L2’0.35) ---_

'.

P?;C

0.70

l-

[0.082]

-0

o-

[0.056]

(MeVV) ‘LAN

FIG. 14. The energy levels as known up to 6 MeV for ‘$N from experiment [37] and as calculated for ?N using the parameters of fit C. The relative formation rates for the reaction K- + l*N --t X- + yN* at 0” are given in the square brackets, but only one excited state is particle-stable. 595/116/1-14

208

DALITZ

AND GAL E&3’5.00

5+

3.85 3.67

‘2 g-

3.09

2 1+ T---

---0.35)

1-21

,

1-‘[ 0.3391

/’ A.20

(MA’)

- 0.70 .

(N+‘3,C) 2-

--?L

0

-

O’&

ik

1’

l- [0.289]

.

-0 (MeV)

0-

FIG. 15. The energy levels as known up to 6 MeV for 18C from experiment [37] and as calculated for ?C using the parameters of fit C. The relative formation rates appropriate to the reaction K- + W - ST- + yC* at 0” are given in the square brackets. Three strong y-ray lines are expected.

ps12strength of 14N goes to high-lying states; the transition strengths according to the jj model, for the hypernuclear states based on 13N core states (JN) which saturate the paI2strength, are given by (4/3) times expression (13.3) (this factor comes from isospin; see Appendix A for details). They are listed in Table III and compared with the values calculated from the Cohen-Kurath tables, with remarkable agreement. The 13N core states specified in Table III are currently identified as the 8.92, 9.48 and 7.38 levels, for spins l/2-, 3/2-, and 5/2-, respectively. The y-spectrum is expected to consist of only one line, due to Ml spin-flip between the l- and O- levels. With the levels for fit C, as given on Fig. 14, the Ml transition rate is calculated to be 1.7 x loll set-l, and A(ny) = + 1. If the levels were interchanged, the Ml transition rate would be 5.1 x loll set-l and we would have A(~TY) = 0. With fit C, the y-ray energy is

E(l-)

- E(O-) = -0.304

+ 1.29S, + 7.307’,

(14.2)

which is in good accord with d(JN = l/2), as given by (12.5). As expected for the pllz shell [6, 71, this y-ray energy is especially sensitive to the tensor parameter T.

:C with with J1 =

is predicted to have at least four particle-stable levels with (-) parity, together several “intruder states” with parity (+) whose locations cannot be predicted any certainty. The l/2+ state lSC* at 3.09 MeV will generate states ‘jC* with 0+ and I+, whose mean B, value is given by &(2s) = bn - 3.09 + V(2s) + (SV - 3.24SN + 16.OOQL),

(14.3)

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HYPERNUCLEC

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&

where the quantity within the brackets is contributed W(g.s.) core, according to fit C. Since fit C gives

y-DECAY

209

by the A interactions with the

BA(g.s.) = bn + 9V - 0.294 + l.O4,S, - 2.75SN + 5.31T + lS.OlQ;,, ,

(14.4)

we see that the mean excitation energy for this lowest intruder doublet in ‘jC is then given by E*(2s) = 3.09 - (v(2s) - v) - 0.294 + IBIS,

+ 0.49SN + 5.31 T + 2.01Q;, ,

w 3.3 MeV.

(14.5)

The energies of these (If, 0+) states are then (+$, -&4(2s) relative to the mean value E*(2s), where d(2s) M d = 0.15 MeV. The locations of the 2+ and 3+ intruder states of the configuration (Id,,&,&), are much less certain, since there are additional AN interaction parameters involved, as we have discussed above for the case of :Be, so we have not entered these states on Fig. 15. However, it is to be expected that these 0+ and l+ states will both be particle-stable and that the lower state may even lie below the strongly-excited I-* state, shown on Fig. 15. There are at least two reactions possible for the formation and study of :C* and its y-rays: (a) through the reaction K- + 14N -+ yr” + yC*.

(14.6)

The relative formation rates for the final states ?C* are the same as those given on Fig. 14, and so are small. From charge independence, the absolute rates are one-half those for producing the corresponding states ?N* in reaction (14.1); (b) through the use of a 14C-enriched carbon target, to produce the 0” reaction K- +

14C

+ n- + 14C* A ’

(14.7)

whose y-rays can be identified by subtraction, after the similar experiments on 12C and enriched 13C targets have been carried out. The advantage of (14.7) is that the formation rates, given on Fig. 15, are much larger than for (14.6), first of all because the K- -+ n- interaction involved is with the p312 neutrons. The ratio of cross sections for the production of the particle-stable excited states (with (-) parity) of YC is given by (K-+14C+n-+ . ..) (0.628) x 6 = 13.1. (14.8) (K- + 14N -+ no + . ..) = 0.115 x 5 x (l/2) Since JP = Of holds for 14C, the I AI 1 = 1 rule implies that only ?C* states with JP = I- are excited in the 0” reaction (14.7) and there are two such states predicted to be particle-stable, as shown on Fig. 15. Three Ml y-transitions are expected, in these circumstances. Following formation of the l-* state, there will be y-transitions to yC(g.s.) and to the first excited state ?C*(l-), in consequence of the Ml transition 3/2- -+ l/2- in the

210

DALITZ

AND

GAL

core nucleus Y, a transition whose rate is known to be 6.1 x 1014 set-’ for AE = 3.68 MeV [37]. Using expression (2.27) and correcting to the calculated hypernuclear energies, we conclude that the l-* -+ O- and l-* + I- transitions will have rates 6.0 x 1014 set-l and 1.7 x 1014 set-l, respectively, with angular distributions given by A(ny) = -1 and +l. The Ml transition l- + O- has rate 2.7 x 1012 set-l, calculated for AE = 0.7 MeV. This y-ray arises both from the direct excitation of the l- state and from the y-cascade following direct excitation of the l-* state. The final intensities for these three y-rays, with energies 4.2 MeV, 3.5 MeV and 0.7 MeV according to fit C, are then in the ratio 1 : 0.29 : 1.39. The net angular distribution for the 0.7 MeV y-ray is given by (0.29 x (3/4)(1 + cos20) + 1.10 x (3/2)(1 - cos28)} (1 - 0.77cos28). The additional information provided by the ?C observations will be (i)

the excitation energy for the l-* level. According to fit C, this is given by E*(l-*)

= 3.68 - 0.814 + 0.31S, - 1.54S, + 5.07

- 1.33Q& - 0.27Qi2 + 2.10Qi2 + 0.70Qi2 + 1.32Q& .

(14.9)

We note that this expression has quite strong sensitivity to S, and Q,‘, , as well as to T (as is characteristic for nuclei in the pllz shell). Our fit C assumed that T 3 0 and that none of the (1NiV parameters beyond Q$, need be included, but this assumption might well turn out to be inadequate for the description of hypernuclei based on pl12 shell nuclei. (ii) the g.s. doublet splitting for YC may be compared with that for ‘,4N, as a test of charge symmetry. (iii) the hypernuclear states based on the l/2+ and 5/2+ intruder states of 13C are expected to be particle-stable, as discussed above, and they may conceivably play some role in the y-spectra resulting from (14.6) and (14.7). It is unlikely that any of these states will be reached with significant branching ratio by El transition from the I-* state, even if these transitions are energetically possible; although the El transition 3/2- -+ l/2+ in 13C has AE = 0.6 MeV, its rate is less than 1 o/0 of the Ml transition 3/2- 4 l/2- [37] responsible for the Ml transitions l-* + (0- and l-), and it is the core nucleus which must generally provide the major contribution to the El transition in the hypernucleus. The more interesting possibility is the direct excitation of these states in the reactions (14.6) and (14.7). This requires going beyond the single shellmodel configurations considered in Ref. [6] and including configurations which describe excitations to higher shells. The simplest possibility is the existence of (2 particle-2 hole) excitations in 14C(g.s.), involving two nucleons excited to the sd shell. The formation transition would then be (2~)~ ---f (1~)~ or (ld6,2)N --f (1~)~ , leading directly to the states of (+) parity required, with rate proportional to the intensity of these (2p-2h) components in the 14C(g.s.) wavefunction. These two transitions imply the selection rule Al = 0 or 1Al ] = 2, respectively. We note first that excitation of the Of state in reaction (14.6) and the I+ and 3f states in reaction (14.7) is already

P-SHELL

HYPERNUCLEI:

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y-DECAY

forbidden by the selection rules (2.24) for 0” K- -+ STreaction processes. For the reactions (14.1) and (14.6) on 14N, the Al = 0 mechanism just mentioned allows I+ excitation, while the / Al 1 = 2 mechanism allows further the excitation of 2f and 3+, but for ?N* these states are most unlikely to be particle-stable, from our discussion above for the charge-symmetric system ?C*. The states Of and 2+ in yC* can both be excited in reaction (14.7), the first by the Al = 0 mechanism and the second by the / Al 1 = 2 mechanism, and are both expected to be particle-stable, as discussed above. Any information which these y-ray studies may yield will be very informative, of course, concerning the intensity of (2p-2h) states in normal nuclei, but it is not our expectation that these states will contribute significantly to these excitation and decay processes. The jj coupling model again gives a good account of the relative formation rates for l- ?C states in the K- --+ n- reaction (14.7). The two initial pllz neutrons are coupled to give O+, with T = I. The substitution (lp,,,), -+ (1~)~ leads to a well-defined hypernuclear state where the final valent baryonic configuration is [(Is)~ @ (lp112)N]~1-;T=f) and for which the nuclear core state has perfect overlap with the 13C(g.s.) state. Since there are two pllz neutrons out of six p-shell neutrons in i4C, the relative formation rate for this (I-; T = 4) hypernuclear state is l/3, to be compared (cf. Table IV) with 0.289 from the Cohen-Kurath calculations. We note that the ji model does not allow the direct formation of the (I-; 3/2) hypernuclear state based on the (l/2-; 3/2) state of 13C, through the K- + 7r- reaction (14.7); the Cohen-Kurath tables give 0.018 as the largest formation strength for any state with these quantum numbers, and this value holds for the 13C*(18.8 MeV) level predicted by their energy level calculations. TABLE

IV

Relative Formation Rates for the ZC l- States Excited in the K- + W- Reaction on 14C at 0”, as Given by the jj Coupling Limit and as Calculated by Cohen and Kurath [21] (Jc*; T) =

u/2-; 112)

ii

l/3 = 0.333

CK

0.289

(l/Z-; 3/2)b 0.018

(3/2-i 112)

(312-i 3/2)

419 = 0.444

219 = 0.222

0.339

0.198

(1The (Jc”; T) values refer to the W* states which have the largest p,j2 strength (the first two) and the largest psla strength (the last two) in ‘“C. The energies of these states in 13C are 0 (g.s.), 18.8 MeV (estimated), 3.68 MeV and 15.11 MeV, respectively. b State not known empirically; calculated [21] to lie at 18.8 MeV in %.

The jj model for strangeness exchange excitations through the p312strength in these p,,,-shell nuclei now allows only a dependence on isospin, since Ji = Of for the target fixes Jf = I- and JN = 3/2- uniquely. The isospin factor is given by 1 [ -1

B Tf2 4 -*

1= :

for Tf = k, and i for T, = i ,

(14.10)

212

DALITZ

AND

GAL

corresponding to the coupling of a (T = 1, T, = - 1) (p&, pair with a psj2 neutron hole (for which T= 4, T, = +@ to yield the final nuclear core state (T = T,, T, = -4). Thus, the last two entries in Table IV correspond to sharing the net pslz strength of amount 2/3 among the two 3/2- nuclear states in the ratio 2 : 1, as given in Eq. (14.10). The Cohen-Kurath CFP values are in good accord with this simple model, the T = 3/2 levels calculated to lie in 13C at 18.8 MeV for J = l/2- and at 15.11 MeV for J = 3/2- (the latter being well-known empirically [37]) being identified with the two T = 3/2 levels listed on Table IV. 3.8. The Hypernuclei :N and ‘$2 The particle-stability threshold for :N lies quite high, at 8.97 f 0.35 MeV for the channel (P + ‘,“C), so that there can exist many y-ray transitions. The 14N levels of major importance for us are shown on Fig. 16. There is a doublet of intruder states known, based on the 2s,,, orbital, at energies 4.91 and 5.69 MeV, and another such doublet, based on the Id,,, orbital, at energies 5.11 and 5.83 MeV. We shall not discuss their possible role in the y-ray spectrum following the 0” K- -+ v- reaction on 14N, since they are expected to be only marginally populated via y-cascades from higher particle-stable ?N states with normal parity. The relative formation strengths are given on Fig. 16 and Table V, for the K- -+ nreaction on 15N at 0”. Since Ji = +, only hypernuclear levels with J, = l/2 or 3/2 can ++*

5.16 ,;L74

[O.OOl]

go.l15]

,/*--

3.95

1

1MeV)

(MeVI lLN

15 AN

FIG. 16. The energy levels as known up to 4 MeV for “N from experiment [37] and as calculated for yI’$ using the shell-model parameters of fit C. The relative formation rates for the reaction K- + lSN + T- + yN* at 0” are given in the square brackets. There are two strong y-rays expected, with energies 1.79 MeV and 1.07 MeV, according to these calculations. The particleinstability threshold is at about 9 MeV. Only the lower part of the energy level spectrum expected for ZN is shown, as explained in the text.

P-SHELL

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213

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V

Relative Formation Rates for the J,w = l/2+ and 3/2+ States of YN Excited in the K- -+ W- Reaction on lbN at 0”, as Given by the jj Coupling Limit and as Calculated by Cohen and Kurath [21] (Jc”; T) = ii CK

(l+; O)b

(Of; 1)

(l+; 0)

l/4=0.250

l/12=0.083

l/8=0.125

0.243

0.070

0.116

c2+;0)

e+; 1)

5/24=0.208

5/24=0.208

0.208

0.199

(l+; 1)”

Sum

l/8=0.125

1.000

0.125

0.961

0 The (Jc*; T) values refer to the lrN* states on which the hypernuclear states are built and which have the largest pliz and ~~1%strengths in r6N. The excitation energies for the first five of these I&N* states are gs., 2.31, 3.95, 7.03 and 9.17 MeV, respectively; the sixth state has not yet been identified empirically, but is predicted [21] to lie at 11.78 MeV. * Shared in ratio 1:s between the states (J, = l/2) : (J, = 312). c Shared in ratio 8:l between the states (Jf = l/2) : (Jf = 3/2).

be excited. The agreement of the formation namely

strengths given by thejj coupling model, (15.1)

with the net formation strengths obtained for each doublet with the Cohen-Kurath wavefunctions is excellent. However, although the sharing of this strength between the two hypernuclear doublet states shows some qualitative similarity, in the two calculations, the Cohen-Kurath result often deviates quite widely from the result (15.1). For example, for the g.s. doublet, the Cohen-Kurath wavefunctions give 24 : 1 for the formation ratio 3/2 + : l/2+, whereas (15.1) gives 8 : 1. Again, for the T = 0 doublet based on the state 1+*, the Cohen-Kurath result is that 3/2+ formation should amount to less than 1 % of the net formation strength for this doublet, whereas (15.1) gives the fraction l/9. Two higher states of 14N, (2+; 0) at 7.03 MeV and (2+; 1) at 9.17 MeV, are also likely to play an important role in the y-spectrum from ?N*, since their formation strengths represent a large fraction of the total (lp,,& + (1~)~ strength. The formation strength for the T = 0 3/2+** level of :N* at energy about 7.44 MeV is 21 %, while that for its T = 1 3/2+*** level is 20 %, although this latter level probably lies above the particle-stability threshold (as we shall henceforth assume to be the case). In IaN*, these two levels are observed to decay dominantly to 14N(g.s.), with branching ratio 98.6 % for the (2+; 0) level and 80 % for the (2+; 1) level. Hence, the formation of the (3/2+**; 0) level is expected to give rise to two strong y-rays, to the g.s. doublet states, in ratio 5.3 : 1 in favor of the g.s. transition and with energies approximately 7.44 and 6.37 MeV. We now consider the lower part of the y-ray spectrum. The (l+; 0) state in 14N at 3.95 MeV has branching fraction 96 % for an Ml transition to the (O+; 1) state at 2.31 MeV. The (l/2+**; 0) state in ?N which is based on it will similarly decay by a fast Ml transition to the (l/2 +*, 1) state, with transition rate 2.8 x 1014 set-l. There

214

DALITZ

AND

GAL

are also weak branches to the g.s. hypernuclear states, which we can neglect in the final spectrum. The (l/2+; 1) state, produced both directly and from the above cascade, decays to the g.s. doublet states by an Ml transition in the core nucleus, with transition rates of 8.0 x 1012 set-l and 4.1 x 1012 set-l to the g.s. and the first excited state, respectively, assuming the energies given on Fig. 16. The latter state (3/2+; 0), dominantly produced among the direct excitations as well as a consequence of the above cascades, will decay by Ml spin-flip to its doublet partner, with rate 5.2 x 1012 set-l for the energy separation 1.07 MeV given on Fig. 16. With these energy levels, the y-spectrum will consist of y-rays of energies 1.07 MeV, 7.44 MeV, 2.95 MeV, 1.79 MeV, 1.88 MeV and 6.37 MeV with relative intensities 1 : 0.57 : 0.40 : 0.36 : 0.20 : 0.11, neglecting all y-rays with relative intensity < 10 %. The third through fifth of these y-rays will be isotropic, while the others will have angular distributions given by A(ny) values between -0.50 and -0.40. We now make the following remarks, assuming that these y-rays can be determined empirically: (i) in principle, the energy measurements for the two y-rays emitted from the (l/2+; 1) level, or for one of them together with the g.s. spin-flip energy, give its excitation energy E* relative to E (defined by eq. (7.1)) for the g.s. doublet. With fit C, this energy E* has value 2.24 MeV, quite close to the excitation energy E* = 2.31 MeV of the parent (O+; 1) level in 14N. We recall that, whenever a doublet is based on a single core nucleus state, its mean energy E does not depend on A, S, or T. In the present case, both core states are approximately described by the same (P~,~)~ configuration, and the coefficients of S, and the Qfj are almost the same for the energies of them both. Explicitly, we have (neglecting core distortion by the A particle)

-(E*(1/2+;

1) - 2.31 MeV): -1.945

+21.28

+3.89

+2.75

+0.436 0.326

(15.2a)

-E (JN = l+; T = 0): -1.906

+21.88

+3.80

$2.81

+0.510

(15.2b)

0.513

The coefficient of S, is close to -2, in agreement with expectation (cf. [6, Eq. (2.16)]) for the cotiguration (l~)~(lp,,~)*(lp,,~~ appropriate to these two 14N states. The more general situation may be exemplified by comparison of the ‘jN doublet built on the (l+; 0) state of 14N at 3.95 MeV. If we neglect core distortion, then E for this doublet receives the following contributions, E(JN=l+*,

T=O): 0.324SN - 22.26&i,, - 4.01Q12 - 0.32&& - 1.39Q,2,+ 0.44Q;,. (15.3) The coefficient of S, is not far from the value -l/2 appropriate (cf. [6, Eq. (2.16)] for a (p$;12~~,~)excitation in 14N, and quite different from that for the lower ZN states. There are similar differences for the coefficients of Q:, , Q& and Qi, between (15.2) and (15.3). In the first instance, we might hope to account for all E locations observed, in relation to the levels of the core nucleus, in terms of S, alone, all Qi”, being zero except for the central ANN term Q&, .

P-SHELL

HYPERNUCLEI:

FORMATION

82

y-DECAY

215

(ii) In this y-spectrum, the Ml spin-flip y-ray for the g.s. doublet is expected to dominate in intensity. According to fit C, its energy measures the quantity d = -0.4884

+ 1.985S, + 11.73T.

(15.4)

We note the very large coefficient for T in this expression. Even with a small tensor interaction, say with [ T / m 0.05 MeV, its contribution will far exceed the contribution expected for any reasonable d. To explore this at a more elementary level, we consider (15.4) in the jj coupling, describing 14N(g.s.) in terms of the valent configuration (p&+ . Consider the J, = J = 3/2 hypernuclear state based on this configuration. All three baryons have their orbitals aligned, so that both .LIN, pairs are in a pure Jk = 1 state. The (1 spindependent contribution to B, is then given by b = 2 x (d - 4S, - 24T)/12, according to eq. (2.20a) and Table III of Ref. [6], the factor 2 being required since there are two LIN, pairs. Since E = 0 holds for the J = l/2 and 312 states of this configuration, for the interactions A, S, and T, we can at once deduce their contribution to BA for the J = l/2 state to be (-2b), so that the doublet splitting is, forj = l/2, dij(JN = 1) = -$4

+ 2S, + 12T.

(15.5)

We note that this is (3/2) times the doublet splitting dij(JN = l/2) found for the (l-, O-) g.s. doublet in ?N, and for the first-excited doublet in !jC. The hypernucleus :C is not yet known. Fit C predicts its Bn value to be 12.9 MeV, and its threshold for particle-instability is then quite high, at 8.9 MeV for the channel (N + ‘,“C). The absolute formation rates for :C* states in the 0” K- + T# reaction on 15N are the same as those for the isobaric states in ‘jN. To obtain relative rates for :C* states from the entries for the T = 1 states on Fig. 16, we multiply those entries by 1215. The lowest (+) parity levels for 14C* are at 6.59 MeV (0+) and 7.01 MeV (2+). The relative formation rates are then 0.168 for the l/2+ g.s., 0.478 for the 3/2+ state based on 14C(2+), and 0.300 for the l/2+* and 3/2+* states based on the (unknown) state 14C(1+), with most of the formation strength going to l/2+*. Three (-) parity levels are known for 14C* in the range 6.0 to 7.0 MeV (at 6.09 MeV(l-), 6.73 MeV(3-), 6.90 MeV(O-), and 7.34 MeV(2-)), so there will be many intruder hypernuclear states here. The 3/2+ state of ?C* at 6.46 MeV (calculated energy) will decay mainly to g.s., by E2 y-emission, since the parent 14C*(2+) decays to 14C(g.s.) with branching fraction (98.6 i: 0.7)x. However, the El emission 14C*(2+) ---f14C*(1-) is known as the competing transition [37], so that there will exist also a weak El-El hypernuclear cascade 3/2+ ---f (l/2- and 3/2-) + l/2+ in competition with the direct E2 transition to g.s. 3.9. The Hypernucleus

:O

This species is not yet known empirically, so we shall adopt the value B&O) = 13.8 MeV, as given by the fit C. The particle-instability limit then lies at 7.5 MeV, for the channel (P + ?N). The energy level pattern for 20, shown in Fig. 17, takes a

216

DALITZ

AND

GAL

(-7.5) ----A5.24 6.18

$y -- _-- - 7.35 6.49

5.18

T1+

(P+15N) :-*[

l-r

&;,a_.70

0

2

2/3]

1’ [l/3] 0-

-0 li (MeV)

(MeVI 150

16

A0

FIG. 17. The energy levels as known up to 6.5 MeV for IsO from experiment [37] and as calculated for 20 using the shell-model parameters of fit C. The relative formation rates for the reaction K- + I60 -+ T- + ;O* at 0” are given in the square brackets. Three strong y-lines are predicted.

familiar form, for the states with (-) parity, consisting of an upper doublet (2-, l-*) and a lower doublet (l-, 0-). In the K- + x- 0” reaction on 160, only the I- states are excited. The total (lp), -+ (1~)~ strength is shared between these two states in ratio 1 : 2, corresponding to substitutions from the (pIlz)$ state (i.e., lsO(g.s.)) and from the (p3,&? state (IsO* at 6.18 MeV), respectively. Three Ml y-transitions result from these excitations. For the transitions l-* -+ 0- and l-* + l-, the transition rates are given by 0.77 r, and 0.27 I’, , respectively, for the fit C, where I’, denotes the rate for the Ml transition 3/2- -+ l/2- in lsO. The latter is not known, but we estimate r, N 5 x IO14 set-1 from the corresponding transition rate known for the mirror nucleus lSN, assuming isovector dominance. Their angular distributions are given by A(7ry) = - 1 and + 1, respectively. The third y-ray is the Ml spin-flip transition for the g.s. doublet, which is fed both from the y-cascade following l-* excitation (for which A(rry) = + 1) and from l- excitation (for which A(ny) = -1). On the basis of the above numbers, we would predict the 6.49 MeV, 5.79 MeV and 0.70 MeV y-rays to have intensity ratios 1 : 0.35 : 1.02 and the net angular distribution for the 0.7 MeV y-ray to be given by A(vy) = -0.59. The identification of the 0.7 MeV y-ray, with expected rate 2.8 x 10la set-‘, will determine the doublet splitting d(JN = l/2) given by Eq. (12.5). The intruder (+ parity) hypernuclear states based on the l/2+ and 5/2+ states of I60 certainly lie within the particle-stable range for ‘:O. The 0+ and l+ states of ‘jO* based on the l/2+ state IsO* (5.18 MeV) have mean A binding energy &(2s) = b,, - 5.18 + lOF+

v(2s) - 1.97SN + 21.2Q:, ,

the separate BA values being obtained by adding (+3/4,

-1/4)d(2s)

(16.1)

to this mean,

jhS.HELL HYPERNUCLEI:

FORMATION

217

& y-DECAY

respectively. The 2+ and 3+ states based on the 5/2+ state 160* (5.24 MeV) have binding energies ; z ;I; Bn(ld) = bA - 5.24 + lOv+

r(ld)

- 1.97SN - S,(ld)

(7/20) 414 + (7/5) S,(ld), + i-(1,4) Jld) - S/&d),

+ 21.2& (16.2a) (16.2b)

in the approximation that the intruder core state is represented as (P + ‘“C(g.s.)), as indicated by the (2s, Id& shell model calculations of Reehal and Wildenthal [47], for example. In order to estimate the excitation energies of these states, we note that, for “O* A 7 Bn(g.s.) = b, + 11 v - 4/4 + S, - SN + 25,98Q;,, , (16.3) neglecting (as above) the tensor term T. On this basis, we would conclude that the Of and l+ levels may lie at about 4.9 MeV, with separation about 0.15 MeV; adopting the estimates (10.8) and (10.9) for the Id-shell parameters, the 2f and 3+ levels may lie at about 4.4 and 5.9 MeV, respectively, as a first rough estimate. For 15N, it is known [37] that y-transitions from the 3/2- state to the l/2+ and 5/2+ states have branching ratio less than 1.5 %, even though the latter lie about 1 MeV below the 3/2- state, and it is reasonable to assume the same will hold true for the isobaric levels in the charge symmetric nucleus 150. Since El transitions in hypernuclei are primarily core nucleus transitions, we can reasonably conclude that these intruder states will play little role in the y-cascade following l-* excitation. The contribution of these intruder states to the y-spectrum could be of importance if they were excited directly in the K- + n- reaction, in consequence of (2p-2h) components in 160, for example. From the selection rules (2.24) for the 0” reaction, we know that it can excite only the 0+ and 2+ states of ‘iO*; the former can decay by El emission to the l- state at 0.7 MeV, and the latter can decay by M2 emission to both the I- and 0- states (note that the El transition 2f + l- is not forbidden by selection rules, but by the fact that the one-particle transition dsj2+pllz is forbidden -it may still occur through small admixtures of higher hypernuclear configurations, such as (Id,,,)&),). If y-rays beyond those in Fig. 17 are observed following ‘jO* formation, they would have to be attributed to these intruder states. Observation of the angular distributions for these y-rays would help to sort out the possible spin assignments. Following direct excitation of the 2+ state, the y-rays 2+ -+ 0- and 2+ -+ l- would have angular distributions sin28 cos28 and (1 - 3cos2 0 + 4cos48), respectively. If the 2+ state were reached indirectly, through l-* excitation, these y-rays would have angular distributions given by A(ry) = + 1 and +3/7, respectively. 4. DISCUSSION AND CONCLUSION

In the text above, the emphasis has been placed on the relative formation rates for a specific hypernuclear species from a related target nucleus. Now we turn to consider the absolute rates. These involve two further factors:

218

DALITZ

AND

GAL

(i) the magnitude of the 0” differential cross section for the T = 1 reaction i(iV-+ (lr. This has been plotted in Fig. 18 for incident K- meson momentum up to 2400 MeV/c, from the analysis of the available data in this range by Gopal et al. [39] and from 1850 MeV/c up to 2400 MeV/c from the data of Bellefon ef al. [40]. Its value is quite large (- 5 mb/sr) for pKL between 750 and 900 MeV/c, and it rises above 3 mb/sr again from 1700 MeV/c to 1900 MeV/c, after which it falls abruptly to values in the range 0.5 to 1.1 mblsr from 2000 MeV/c to 2500 MeV/c. Above this momentum region, the data available are quite sparse.

6t

\

RUf.[39] --.-.-.Ref.[LO] \

-

\ B-

-

3 dRL,b *(RN-AIX,T=l, -

-

Ip-cPKJ

- 250

e,= 00) mb/sr

MeVlc

\

-200 z

q-(m)

\ \ L3-

’

\

\

\ .2-

--

i

\

i i

\

5 E -100

i

\

,./---

i

l-

0

-150

.-. -A----

\\

cr’ -50

...R. I 1000

0

PKL I MoVlc)

I 1500

I 2000

0

FIG. 18. Displays the data available on du/dsz(RN ---t da, T = 1, 0, = OO)Labas function of incident laboratory R momentum PKL. Also displays the laboratory momentum transfer q given to the final d particle in the reaction RN -+ LLT, when this is projected backward (relative to the incident i? meson) in the c.m. frame.

(ii) the nuclear lp, -+ 1sd transition amplitude, which appears as the last factor in expressions such as (2.8). This depends first of all on the momentum transfer q, which is a function of the incident momentum pKL ; q also depends a little on the mass A of the nuclear target, since the kinematics of the reaction depend on A, but this is a minor effect, so we have neglected it in this work. The dependence of q on pKL is shown on Fig. 18, and a detailed discussion is given in Appendix D. When the nuclear interactions of the initial K- meson and the final r meson are omitted, the modulus squared of the lp, -+ IS, transition amplitude is given by I ~&)I2

= (q2/W(WV’2/(v

+ AN5 ew(--q2/(v + JW,

(4.1)

using harmonic shell-model wavefunctions, the appropriate parameter values being v = 0.41 fm-2 and h = 0.33 fm-2 [6]. We note that (4.1) vanishes for q = 0; this is a consequence of the orthogonality of the Ip and 1s wavefunctions. Expression (4.1)

P-SHELL

HYPERNUCLEI:

FORMATION

&

Y-DECAY

219

has its peak value of 0.40 for q = (v + h)li2, which is N 170 MeV/c for the above parameter values. Its value is still 0.23 for q = 90 MeV/c, which is about the same as the value 0.26 for the momentum q = 253 MeV/c which corresponds to capture from rest; however the value has fallen to 0.13 for q = 63 MeV/c, which corresponds to pKL = 900 MeV/c. The effect on (4.1) of the nuclear absorption of the incident K- and outgoing n mesons can be estimated by using the eikonal approximation, since our interest is in the incident momentum range 800-2000 MeV/c. This replaces F::(q) by

F,,(q)

=

j

#(lsJ

[exp [ iqz - q

(1 - iolg) L

p(b, z’) dz’

- + (1 - iE,) f mp(b, z’) dz’]l #(lpN) 2nb db dz,

(4.2)

B

where p(b, z) = p(v’(b2 + z2)), p(r) being the nuclear density at radius r in the target nucleus, (TV and C,, denote the RN and TN total cross sections averaged over the nucleons of the target nucleus, and & and Z,, denote the ratios Re(~~)/Im(f~) and Re(‘,,)/Im(TJ for the forward RN and TN elastic scattering amplitudes after they have been averaged over the nucleons, in the same way. I Fl,(q)12 is quite strongly damped by the total cross sections c?Rand 0, , so that their magnitudes will be an important factor in the choice of the best pKL for eXperhent. The cross SeCtiOn CiR is large, exceeding 40 mb, over the range 750-l 150 MeV/c for pKL , but falls quickly to N 30 mb as pKL increases to 1300 MeV/c, after which it falls more gradually, with value N 28 mb by pKL = 2000 MeV/c [16]. The cross section 6, is comparable with 30 mb all through this momentum range, despite some peaks and valleys in the n-+p and n-p cross sections separately [16]. The parameters CL~and 01, have lesser importance, although not negligible, and tabulations are available [17-191 from which their values can be estimated. The choice of an optimum pKL for these hypernuclear y-ray experiments must balance these various factors. The two studies to date on in-flight hypernuclear excitation without y-detection have usedp,, = 390 MeV/c [3 1] and pKL = 900 MeVlc 1411. Both situations have disadvantages. The former involves a small value for q (- 40 MeVjc) and therefore a small value for j F,,(q)/2. Although the 0” cross section for the K- + W- reaction is quite large there (- 3 mb/sr), it is difficult in practice to obtain K- beams with high intensity at such low momentum. The latter involves higher q(63 MeV/c) and a convenient beam momentum. As remarked above (see Fig. 18), the EN + (In cross section at 0” is then quite large (almost 5 mb/sr); on the other hand, the nuclear absorption is particularly strong in this momentum region, a fact not unrelated with this large value for the 0” reaction cross section, since the large value for the mean cross section f?R at 900 MeV/c is due to the ,Z* (1770) resonance at 950 MeV/c. The other possibility open is to work in the momentum range pKL = 1700-1900 MeV/c, where the momentum transfer q is 115-122 MeV/c, the 0” KN -+ /lm reaction cross section exceeds 3 mblsr, and the absorption cross

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sections are relatively small (- 30 mb), so that all the factors are as favorable as could be hoped. In Tables VI and VII, we have tabulated the quantity 1Fl,(q)(2 for the Ip, + Is, transition, for a typical target, lzC. The shell-model form p(r) - (1 +(A - 4) olr2/6) exp(--(ur2) has been adopted here for the nuclear density p(r), the parameter 01being chosen to fit the empirically observed charge radius for the nucleus, for the calculation of expression (4.2); the nuclear parameters adopted for each energy pKL are specified in the table caption. The sensitivity of the value calculated to the input parameters used, can be assessed from the entries on these tables and on Table VIII. Also, Table IX shows the effect of nuclear absorption, depressing the value of 1F,, 1% to an increasing degree as A increases through the p-shell. To obtain from these calculations the cross section for the excitation of a particular hypernuclear level by the K- -+ ?I- reaction at O”, 1Fl,(q)12 must be multiplied by n,(N)/3, where n,(N) is the number of p-shell neutrons in the target nucleus, and the factor I /3 is l/(21 + 1) for TABLE

NuclearTransition Amplitudes1F,, I* = I F(&

VI

+ ls~)l’ Calculatedfor l*C Target as Function

of L?Ffor pKL = 900 MeV/c, at which Momentum LYK =

We Have q = 63 MeV/c”

-0.1

+0.1

$0.4

+0.8

No Intns.

0.1606 0.0191 0.0139 0.0148

0.1552 0.0237 0.0134 0.0160

0.1444 0.0308 0.0139 0.0187

0.1262 0.0395 0.0162 0.0229

0.871 0.120 0.0028 0.0055

-__ ISN --f 1PN 2sN -+ I& +

ls‘q Is,, h/j Is/j

a The other nuclear parameters adopted were 57 = 40 mb, &, = 24.5mb, and &, = $0.1. The parameter or(K-N) is rather uncertain, but our best estimate [18, 191 is %x = +0.4. Here and in the following Tables, we have adopted equal values, X = Y = 0.37 fm-8, for the A and N harmonic oscillator constants. TABLE

VII

The Nuclear Transition Amplitudes 1& I2 = 1F(nlN -+ ls~)l~ Calculated for the Case of ‘*C as Function of d, for pKL = 1700 MeV/c, at which Momentum We Have q = 115 MeV/c” a, =

-0.6

-0.3

-0.1

+0.1

No Intns.

1SN 4 ISA 1PN ---t IsA &,J + Is,! lc& + lS‘.j

0.0959 0.0661 0.00255 0.0398

0.1071 0.0624 0.00207 0.0355

0.1141 0.0591 0.00212 0.0326

0.1207 0.0551 0.00257 0.0299

0.632 0.290 0.022 0.044

oThe othernuclearparameters adopted were 5~ = 28.9 mb, GF = +O.l, and en = 32.5 mb. The best estimate available for G,, is -0.3. Again, the values X = Y = 0.37 fm-% were adopted.

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Illustrates the Sensitivity of 1F,, J* with Respect to Changes in the Mean Nuclear Absorption Parameter d and Scattering Parameter 2 (i.e., with the Assumptions a, = o,, = B and 4 = OL,,= L?), for the Case of Incident Momentum pK = 1700 MeV/c on the Target ‘*CO $mb) =

27.8

30.8

33.8

,s = -0.1 = -0.3 = -0.5

0.0641 0.0634 0.0621

0.0552 0.0545 0.0531

0.0477 0.0470 0.0455

o Again, the values h = Y = 0.37 fm-8 were adopted. TABLE

IX

1Flp la Is Given as Function of Nuclear Target for the Two “Best” Nuclear Parameter Sets Specified in the Captions to Tables VI and VII for Incident Momenta Typical of the Two Most Useful Regimes for ExperimentO

PdMeVlc) 900

1700

dMeV/c)

‘Li

sBe

‘T

ISO

63 115

0.0511 0.1088

0.0416 0.0869

0.0308 0.0624

0.0207 0.0406

n Again, the values X = Y = 0.37 fm-a were adopted. the initial lp nucleons, by the relative formation rate for this level, and by the elementary cross section ~u/&~,~(RN -+ AT, T = I, oO).3 For excitation by the K- + TP reaction at O”, the final cross section has an additional factor l/2 due to isospin, and the number nl(P) of p-shell protons is to be used in place of n,(N). The cross sections predicted are not large, of course. To take a typical case, let us consider the 2.34 MeV y-ray resulting from the excitation of the l-* level of ‘jC* by the K- --+ n- reaction on 12C at O”, using incident K- momentum 1700 MeV/c. From Fig. 11, the relative formation rate for this level is 0.188, and from (12.2), we deduce that the resultant y-decay l-* -+ 2- has probability about 2/3. We have n,(N) = 4 for 12C, and Table VII gives us 0.062 as the appropriate value for 1F,,(q)12. Taking the elementary cross section to be 3.1 mb/sr, from Fig. 18, and A,, = -l/7 as given following Eq. (12.2), we finally predict:

dQd;QLa, (K-12C + y( I-* -+ 2-), 0, = 90”, 0, = 0”) Y = 3.1 x (0.062 x 4/3} x 0.188 x (2/3) x (21/20)/4~, = 34/47r ,ub/(srj2.

(4.3a) (4.3b)

8 We should remark here that the form-factors introduced in this section are related with those introduced in Eq. (2.10) for the most general case of distorted waves by the equation: I F&)1*

= (21-t I)$;,1

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The number 0.083 in the curly bracket in (4.3a) is the “effective neutron number J&ii) for all transitions lp, 3 IS, at pKL = 1700 MeV/c. This appears relatively large in view of the experimental measurement at 900 MeV/c by Bruckner et al. [41], giving an N$ of about l/3 for the transitions “(1~~ + lp,) -+ all,” in target 12C; for 900 MeV/c, Table VI gives 0.0308 in place of 0.062, thus an Nefr of 0.041 for all transitions IP, + Is, 3 which is a much larger fraction of the measured N$ than the observed spectrum would suggest. The possibility of a discrepancy between the measured and theoretical values for N$‘i has been discussed in some detail recently by Epstein et al. [42].

The only observation of a y-ray in coincidence with n- mesons emitted at 0” in the K- + n- reaction is that reported recently by Herrera et al. [5], who used a ‘Li target and K- beam momentum 1700 MeV/c. Possible interpretations for this y-ray (energy (energy 0.79 MeV) have been discussed quantitatively in Ref. [20], and we shall not repeat our discussion here. We note only that they report a total cross section of 48 f 12 pb/sr for this y-ray, assuming it to be emitted isotropically, and that this is quite comparable in magnitude with the calculated cross section for the interpretation of this y-ray as due to the 5/2+ + l/2+ transition in 2Li. Its interpretation as the 3/2+ -+ l/2+ y-ray is also conceivable, but its calculated cross section is less than 20 pb/sr.

Bamberger et al. [2] and Piekarz [4] have also mentioned the observation of a y-ray with energy 0.31 MeV, which may be attributed to the p-shell hypernuclei, emitted in coincidence with a z-- meson following K- capture from rest in sBe. In principle, this could be due to the 5/2+ -+ 3/2+ transition in Fig. 5, if the doublet splitting were five times smaller than that predicted on the basis of fit C, in which case the 5/2+ state would indeed be particle-stable, with excitation energy about 3 MeV. However, E2 decay to lBe(g.s.) would then be expected to dominate, with rate about an order of magnitude greater than that for the Ml spin-flip transition. A more likely possibility is that the K- -+ n- reaction generates the T = 1 states of ,BBe* which have spin values l/2+, 3/2+ and 5/2+ and excitation energies of order 16-18 MeV. Although these states lie far above the (4He + jHe) threshold, their decay into this channel will be inhibited by the requirements of isospin conservation; there are also dynamical reasons for this inhibition-the nuclear cores for these jBe* states have structure 33P[4, 3, I], from which it is difficult for the nucleons to reach the S = 0, symmetry [4, 41, states appropriate for any 24He system, whether existing as two separate a-particles (e.g., as the system A,,) or within other structures (as for the system (4He + jHe)). Disregarding /l-emission threshold, for this reason, the next lowest threshold is that for (P + jLi), at 17.17 5 0.05 MeV; a T = 1 state jBe* not more than 6 MeV above this threshold will prefer to decay through this channel or through the consecutively opening thresholds (I’ + jLi*), (N + iBe) (threshold at 18.77 & 0.07 MeV), and (N + jBe*). The decays to :Li* or iBe* will then lead to y-emissions characteristic of the A = 8 hypernuclei. It is reasonable to suppose that some of the >Be* states based on (1 +; 1) EBe* (17.64 MeV) and (3+; 1) 8Be* (19.22 MeV) will lie above the (P + jLi) threshold, and perhaps sufficiently far above ,for decay to (P + jLi*). These jBe* states include the (3/2+; 1) and (S/2+; 1) states (cf. Fig. 6 for

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their isobaric counterparts in :Li; n‘Be*@ = 1) formation rates for K- + n- on %e target are one-third of those given for iLi* states on Fig. 6), whose formation rates are 0.067 and 0.058, respectively, dominant among the JBe* excitations in this energy interval. With fit C, their excitation energies are 17.15 and 17.51 MeV, respectively, the latter hardly exceeding (by to.03 -& 0.05 MeV) the threshold (P + jLi* (0.31 MeV)) based on the assumption that there is a state jLi* giving rise to this 0.31 MeV y-ray through its decay jLi* -+ y + ,SLi(g.s.). The characteristics of the level spectrum for :Li are even less certain, because the two low-lying configurations A - 7Li(3/2-, g.s.) and A - 7Li*(1/2-) both contribute to iLi(J = l-). This situation has been reviewed recently 181; with lit C, the excited states :Li* resulting from these configurations are predicted to be at 0.55 MeV (J” = O-), 0.88 MeV (2-) and 1.28 MeV (I-*). The most logical hypernuclear interpretation of the 0.31 MeV y-ray observed would then be its assignment to either the Ml transition O- -+ l-, following decay from the (3/2+; 1) jBe* state mentioned above (the decay 5/2+ --, 0- is expected to be slowed down becausef-wave proton emission is involved), or the Ml transition 2- + l-, following decay from either of the two jBe* states mentioned above. Other transitions in the 0” K- --+ z- reaction on 9Be could lead to jBe* states which decay predominantly by nucleon emission to low-lying jLi* and jBe* states whose radiative decay would yield y rays in coincidence with the fast 7~- meson at 0”. For example, it is now generally believed that the two peaks observed at 16 MeV and 27 MeV for lBe* excitation in this reaction [41]-the “strangeness analog states”are due to coherent lp, - lp, transitions, as envisaged by Lipkin [33] and others [34, 491, and some of the states in this configuration (ls)~lp)~(lp), might decay preferentially by nucleon emission to low-lying states of jLi* and jBe*, as Pniewski [43] has conjectured.4 We may note that, according to Ref. [44], the lower peak consists of T = 0 states, whose symmetry does not forbid, nor even inhibit, a-decay. In the region of the 27 MeV peak, both T = 0 and T = 1 states are expected; the former may decay rapidly through ol-emission, A-emission, and nucleon-emission, while allowed A-emission from the latter will begin to compete strongly with nucleon emission only above the threshold energy of 23.34 -& 0.04 MeV for (A + 8Be*(T = 1)). Whether or not hypernuclear states reached by the transition lp, + lp, from 9Be exist between 17.5 MeV and 23.3 MeV, which have the properties of (i) reasonably large formation rates for the 0” K- - rr- reaction, and (ii) inhibited e-decay, remains to be seen. Close above 23.3 MeV, many additional channels open up, and the formation of low-lying states of jLi* and iBe* by nucleon-emission will become correspondingly rare as the energy rises to and above the second peak. It is worth mentioning that the T = 0 channel (2H + :Li) has threshold at 23.41 & 0.05 MeV and 4 The 2- state *Be* (IS.91 MeV) to which Pniewski particularly drew attention does not actually lead to this hypernuckar configuration but most probably to the configuration (I~)~(lp)~(2~)~(ls)~ , which is of course coupled with it. In addition, the decay 8Be*(2-) -+ 2~xis forbidden by parity, so that nucleon emission is its only nuclear decay process, but the decay iBe* -+ (a + AHe) is not forbidden for the hypernuclear doublet states built on *Be*(2-), although this or-emission is no doubt dynamically inhibited to a considerable degree because of the difference between their internal (permutation) symmetry and that for the final state (Al + ,5He).

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may sometimes compete with a-emission in the region of the upper peak, since formation of the first excited state JLi*(3/2+) will give rise to a low-energy y-ray by an Ml spin-flip transition to iLi(g.s.), the y-energy being predicted by fit C to be 0.25 MeV. However, there is no reason to expect JLi* production to be in any way prominent. The above remarks illustrate how important it is for experiments to be capable of either measuring accurately the final z- momentum relative to the incident K- momentum, or of being able to set a precise upper limit on the energy transfer in the K- + TITreaction, in coincidence with the low energy y-detection, since this energy transfer will be a major clue for identifying which hypernuclear system gave rise to the observed ‘y-ray. For example, if only r- mesons which correspond to jBe* excitation of less than 10 MeV (say), are recorded then we can be sure that the y-rays observed in coincidence with them can come only from iBe* and not from lighter species, since the only other hypernucleus which could be formed then is :He, which cannot contribute further y-rays. If y-ray measurement could be cotined to K- -+ r- events such that ,DBe* has excitation energy between 17 MeV and 23 MeV, then we would know that any additional y-rays could come only from jLi* or jBe*. And so on. Our discussion in Section 3 has led us to the conclusion that the existence of intruder states, by which we mean “wrong parity” states most probably due to excitations into the sd shell, plays little role in the y-cascades resulting from the formation, in the 0” K- -+ n- reaction, of iZ* states with parity opposite that for “Z(g.s.), their contribution being below the level of 1 % per K- + r- process. This is even the case for YBe following reaction (10.12), where we found that about 10 % of the y-rays emitted could well be from El transitions to or from these levels, since the l-* level which gives rise to them, and which is the only level directly excited, itself has a formation strength of only 15 %. On the other hand, the interpretation of the y-rays emitted from hypernuclei in the upper half of the p-shell could be much complicated if the direct excitation of lowlying “wrong-parity” states were strong relative to that for the normal-parity states. Excluding from the present discussion consideration of two-baryon transitions, due to total ignorance, we have pointed out the possibility that wrong-parity hypernuclear states may be excited directly from (np-nh) components in the g.s. target nucleus, for example by transitions such as ( lp);;2(2s)i -+ ( lp)i2(2s),( 1~)~ , or ( lp)G2( 1ds12)k -+ through the W~2WddW~ 9 which are calculated in the impulse approximation one-baryon transitions (2~)~ + (1~)~ and (ld5,2)N + (1.~)~ , respectively. The intensity of such (2p-2h) admixtures from the nuclear sd shell in the g.s. nuclear wavefunctions has not yet been well studied over the whole range of p-shell nuclei, except for A 3 14. The theoretical indications are that the summed sd intensity grows rapidly as one approaches W, from about 4 % in 14N(g.s.) and r4C(g.s.) [45] to about 15 % in r5N(g.s.) [46] and a similar magnitude for W(g.s.) [47, 481. In order to provide some rough estimates, we have made calculations of the basic transition matrixelements involved, namely (2~)~ + (1~)~ and (14, + (Is), , and these are given in Tables VI and VII, where we have denoted the squared transition amplitudes for these two transitions by ] F,,(q)12 and 1Fl,(q)j2, respectively.

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The (2~)~ + (IS), amplitude

FORMATION

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225

takes the following form:

when the initial and final nuclear interactions of the K and n mesons are neglected. This is rather small in the circumstances of interest here, because q is relatively small and the states are orthogonal (for h = v) in the limit q = 0. In this case, 1Z$~‘j2 is more than one order-of-magnitude less than 1Z$12, over the range of q considered. When the nuclear absorptions are included, the additional factor in the integral of (4.2) removes the orthogonality to some variable degree. To illustrate this explicitly, we consider the expression (4.2) for the special case where GK = 0, = C?and C& = Cu,= 0, taking also h = v, since the integrals required are then one-dimensional; the result is F2s

= LO

- Wo + q2/%2/3>1’2,

(4.5)

where the integral Z, is given by Z, = (v/n) 1 (vb2)” Exp(-A2

- gOT(b)) d2b,

(4.6)

where T(b) is

T(b) = Ia p(b2+ z”) dz. --m

(4.7)

We note that the integrals Z, are independent of q. It is useful to note here also that, for this special case, we have

F,, = Ciq/(2v)‘/2)Fls,

(4.8a)

Fl;s = Z,Exp(-q2/4v).

(4.8b)

Returning to expression (4.5), we note that I0 = Z1 for 6 = 0, but that I, < Z1 for 6 # 0, so that F,,(q) approaches the finite limit F,,(O)(l - Z,/Z,,)(2/3)1/2 as q -+ 0. For q f o, the term q2/4v in (4.5) has sign opposite to (I - ZJZ,). Over the range of q considered here, 0 < q < 130 MeV/c, / F,, I2 is therefore expected to fall drastically with increasing q, its minimum value (zero, with expression (4.5)) occurring for q N 150 MeV/c. This expectation is indeed confirmed for the more general case by the values given in Tables VI and VII. For q = 63 MeV/c, 1F2s I2 is greater than / F:z’ I2 by a factor 5, whereas for q = 115 MeV/c, 1F,, I2 is smaller than 1Fif’ j2 by a factor of 10. The value of / F,, I2 itself falls by a factor of 6 over this momentum range, and the ratio I F2,(q)j2/j F,,(q)j2 ranges from values of order l/2 to values of order l/20 as q increases from 63 MeV/c to q = 115 MeV/c. Hence, the (2~)~ + (Is), transition could only be of relevance if the corresponding (2~2h) components of “Z(g.s.) were unexpectedly large. For pK- = 1700 MeV/c this (AZ)” = 0+ transition may safely be neglected relative to the normal (AZ)” = I- transitions.

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The (14, + (1~)~ amplitude is nonzero only for m, = 0, since the eikonal factor in (4.2) has no azimuthal dependence. In the absence of the R and rr nuclear interactions, the amplitude Fii) is given by the expression (4.9) This is very small in the region of interest here because q is small and the expression vanishes for q = 0 owing to the orthogonality of the IS and ldangular functions. This orthogonality is removed either by the exp(iqz) factor or by the eikonal factor. To illustrate this explicitly, we consider again the special case 0, = 5, = 5 and ‘Ye = ol, = 0, for which expression (4.9) becomes replaced by Fld = F,,(l

- Z,/& - q2/2v)/31i2,

(4.10)

which approaches the finite limit F,,(O)(l - Z,/Z,)/31/2 as q + 0. For q # 0, the terms (1 - ZJZ,) and -q2/2u in (4.10) have the same sign, and 1Fld I2 is expected to increase gradually as q increases over the range of interest here. This general expectation is confirmed by the values given for 1Fl,, I2 in Tables VI and VII. For q = 63 MeV/c, / Fld I2 and j F2F,,I2 are found to be comparable in magnitude, whereas for q = 115 MeV/c, I Fld I2 exceeds I F,, I2 by more than an order of magnitude. Indeed, over this momentum range, j Fld I2 is comparable with ] F19 12, the calculated ratios 1Fld I”/ ( F,, I2 ranging between about 0.55 and 0.77, for various parameter choices. Although the summed intensity for (2p-2h) excitation in 16N and lsO into the sd shell may be as much as 15 %, this is generally spread over a number of states, only a few of which are likely to be of any importance to the mechanism discussed just above. Consider briefly the case of 160 and the ‘jO*(O+) state based on the l/2+ state 150*(5.18 MeV). The calculations by Reehal and Wildenthal [47] give about 0.01 for the ratio of the spectroscopic factor for this l/2 + 150* level to that for l/2- 160(g.s.), so that the formation rate for the ?O*(O+) state relative to that for the I-* state (based on 3/2- 150*(6.18 MeV)) is expected to be considerably less than I %, except perhaps at the lower part of the q-range of interest, namely for pK = 900 MeV/c, where it may be as large as 1 %. The situation is somewhat more favorable for the excitation of ‘:0*(2+) based on the 5/2+ state 150* (5.24 MeV), since these calculations give about 0.2 for the ratio of the spectroscopic factor for this state to that for 150(g.s.). The formation rate for 20*(2+) is then predicted to bear a ratio of roughly 0.2(1 Fld(q)~2/5)/(~Fl,(q)~2/3)

N 0.12 x 0.66 = 0.08

(4.11)

to that for $O*(l-), where we have used a mean value of 0.66 for the calculated ratio 1Fl,(q)/Fl,(q)12. The y-yield of 20*(2+) is therefore expected to be about 3 % of the y-yield of the negative parity levels of ?O* shown on Fig. 17. For p-shell nuclei, where theoretical analyses of intruder states are lacking, the use of measured spectroscopic factors may yield some meaningful estimates. As one example of interest here, we may mention the values (0.007 f 0.003) and (0.18 & 0.07)

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reported by Auton [50] for the l/2 + 9Be* (1.68 MeV) and 5/2+ 9Be*(3.06 MeV) states, respectively, from the reaction ‘OBe(d, t)9Be, which are to be compared with the value 2.19 & 0.48 mesured for 9Be(g.s.). The latter is in good agreement with the value of 2.36 calculated by Cohen and Kurath [21]. From experiment this time, we find again that the l/2+ spectroscopic factor is small relative to that for the 5/2+ state, so that the formation rate for ‘jBe*(O+) based on the l/2+ state is well below 1 % per K- ---f w reaction. The ratio of the formation rate for l”Be*(2+), based on the 5/2+ state, relative to that for the ground state ‘$Be*(l-), is given by an expression similar to (4.11), except that the factor 0.2 is now replaced by (0.18/2.19). Hence, this ratio is expected to be about 0.03, from which we would predict that the absolute rate for ?Be*(2+) formation will be less than 2 % per K- -+ T- reaction. There is one further point concerning the excitation of intruder hypernuclear states whose JP coincides with that for the target nucleus, which deserves comment here. The low-lying intruder states of the core nucleus are predominantly of the type (lP)i2@, wiv , relative to the ground state of the target nucleus, but their wavefunctions may include also a component of the form (Is);‘. Let us characterize its amplitude by the coefficient p. The amplitude for the excitation of the intruder hypernuclear state will then consist of a sum of terms proportional to F,, , Fld and fiFls, the latter corresponding to Is, -+ Is, , the transition leading to the final hypernuclear configuration (I~)~~(ls)~ . This amplitude F,, has its maximum value for q = 0, and we see from Tables VI and VII that it is much larger than F2’zsor Fld in the regime of interest. For simplicity, therefore, we shall focus attention here on the F,, contribution, disregarding the F,, and Fld contributions which will be coherent with it. For definiteness, we consider the target 12C. For the intruder hypernuclear state ?C*(O+) based on W*(1/2+) at 6.34 MeV, the formation rate relative to that for ?C(g.s.) is given by 2 I B I2 x I f-1, I2 --Y(Of *) m 3.6 I ,d 12, (4.12) 4 x (0.712/3) x 1F,, I2 Yks.) for pK = 1700 MeV/c, using the g.s. formation strength given on Fig. 11 and the amplitudes 1Fnl I2 given in Table VII. Thus, we see that the ‘jC*(O+) formation rate is rather sensitive to the (Is)-, component in the intruder nuclear core state W*(1/2+). There are two relevant experiments on the K- --f n- excitation of ?C*, those of Bonazzola et al. [31] at 390 MeV/c and of Brueckner et al. [41] at 900 MeV/c. The observation of ‘jC(g.s.) formation is claimed in the first experiment but the data would not exclude ‘jC*(6 MeV) formation at a comparable rate; in the second experiment, it is not clear whether either state is formed. However, what is clear is that more accurate experiments would allow at least an upper limit to be placed on the formation ratio (4.12), and therefore on j fl /, which would give valuable information5 j The only shell model calculation we have found in the literature, which includes a quantitative treatment of the contribution of these (Is)-’ excitations, is that of Teeters and Kurath [51]. Their work is for llC and lSC, using two possible sets of one-particle levels. The admixture 1 j3 1%is not stated for IV, whose properties are calculated only for their set 1. This set 1 leads to 1fi I* = 0.166 for “C(1/2+), but their set 2, considered more reasonable for the case of V, leads to ( ,8 IS = 0.089, which, although smaller, is still a rather large value. Further, the intensity of the configuration

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concerning the parent intruder state W*(l/2+). Similar discussions can be given for other intruder states of llC, and for other nuclear species. These (ls);‘(lp), components in the core nucleus may be expected to be more pronounced in the lower half of the lp-shell, since there are then more vacant Q-states available6. If we now focus attention on the intruder hypemuclear state rather than on the nuclear core state, we may characterize these components of its wavefunction as having the structures (1~);~(2s, l&(ls), and (I~);~(ls), , relative to the ground state of the target nucleus. Let us consider again the specific case of 12C target, for which Hufner et al. [32] have discussed the coherent excitation of the configurations (ls)$(ls), and (lp);‘(lp), . These authors place the (lp),l(lp)d excitations at excitation energy about 10 MeV in ?C*, the (l~);~(ls)~ excitations being placed at about 20 MeV. The (ls);‘(l~)~ admixture to the intruder hypemuclear state, discussed above, is now seen as the result of mixing between the (l~);‘(ls)~ strangeness analog state @AS) and the (lp)~2(2s),(ls), intruder state, possible only when the latter has the same f” as the former. This mixing is generated by NN interaction amplitudes such as (ls;;‘lp, I V,, / l&2sN), which we estimate to be about 0.5 MeV, so that the admixture would be expected to be rather small (I /3 1 less than 0.03 in amplitude) in view of the large energy separation ( w 15 MeV) between these two cotigurations. The configuration (lp);‘(lp), lies closer in energy (dE w 4-5 MeV) to the intruder hypernuclear state and also couples directly with it, through AN interaction amplitudes such as (Ip$lpd / V,, 1 lp~22s,lsn), which we estimate to be about (or less than) 0.3 MeV, so that it might have a larger mixing amplitude. Its excitation through the K- -+ W- process involves the baryonic transition (lp), --f (lp), , and we note that 1F(O)&, + lpA)12 does not vanish, but is large, for q = 0. Since the 0” differential cross section for the excitation of this SAS is known to be large (= 500 pb/sr at 900 MeV/c [41]), such SAS admixtures to the low-lying intruder hypernuclear states with J” the same as that for the g.s. target nucleus could increase significantly the excitation cross sections for these “wrong-parity” hypernuclear states, making them perhaps even as large as those for the low-lying normal-parity hypernuclear states. These (lplh) configurations can contribute also to states with J” different from that for the g.s. target nucleus, although not coherently, so that their excitation will have substantially lower cross sections. Any observations of, or upper limits for, (Is)-’ ‘“C(g.s.) in l’C(1/2+) is 0.082, which is more than 90 % of the calculated value for I fl Ia. It is clear that K- + n- hypernuclear excitation studies now being planned should fmd no difficulty in measuring such a large intensity as this, or at least in obtaining an upper limit which could provide a severe teat for such calculations. * We note that, quite generally, the leading (lp)$(2~)~ excitation of the core nucleus contains a spurious center of mass component, the elimination of which requires the presence of (ls)$(lp)~ components, as well as of others. These additional components usually enter with amplitudes < l/A in magnitude, so that they give rise to nonnegligible modifications for light nuclear species. For example, the l/2+ state OBe(1.68 MeV) contains the component l”Be(g.s.)(ls)&l with intensity about 3.3 %, due to this c.m. correction alone. Taken at face value, these considerations would imply significant excitation rate for the yBe(O+) state in the (K-, Z-) reaction on loBe, comparable with that for yBe(l-*) excitation.

P-SHELL

HYPERNUCLEI:

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&

Y-DECAY

229

these low-lying hypernuclear excitations would be of much interest for nuclear and hypemuclear physics. The excitation of wrong-parity (or SAS or other) hypernuclear states at higher excitation energies could have a great effect on the y-ray spectra observed, if these excitations lie below the lowest threshold for particle stability. This will not be the case for most of the hypernuclear species discussed here, as we see from Table I and Figs. 3-17, for the particle-stability thresholds mostly lie about 5 MeV or lower. The (‘;B, ‘$Z) doublet is one exception to this remark, since the SAS excitation energy for ‘;C* is known to be comparable with its particle-stability threshold. lt is quite possible for izC that at least a component of the SAS excitation may be particlestable, and it is quite likely for ‘;B that most components of the SAS excitation may be particle-stable. YC is another case where some SAS components might be particlestable. ‘:B, ‘:N, 20, and 20 are cases where the SAS will lie well above the particlestability limit, but where there are still quite a few particle-stable hypernuclear states, with both parity values, lying above the hypernuclear states which are discussed in this paper, displayed in Figs. 3-17, and correspond to the transition lp, + lsn from the target nucleus. Some of these higher particle-stable states may have an appreciable component of the configuration (lp)$(lp), and may be strongly excited by the transition lp, - lp, . The excitation of these higher states would give rise to many y-cascades additional to those discussed in this paper, some of which may pass through the normal-parity levels we have considered, and so lead to a rather complicated y-spectrum to be interpreted. In these cases, the capability of accurate measurement of the primary pion energy in coincidence with the y-ray detection and measurement becomes essential for any detailed understanding of the excitation and decay processes taking place; this is a requirement which we have already emphasized above. The main objective of the possible hypernuclear experiments envisaged and discussed here would be to establish the systematics of a hypernuclear spectroscopy, from which we would hope to learn more about the clN nuclear interaction. Such experiments appear at present the most accessible way for us to learn about the detailed properties of this interaction. Low energy (lp scattering studies (pllr. 2 300 MeV/c) have taught us a good deal about the s-state clN interaction, but it appears unlikely that we shall be able to learn anything appreciable about the noncentral /IN interactions through /l,a elastic scattering experiments in the foreseeable future. Let us close by reviewing briefly several of the outstanding questions: (i) the nature of the clN spin-orbit force. There are two parameters involved, S, and S, , and the theoretical estimates of them [6] are S, = -0.1 MeV and S, = -0.2 MeV (note that the ANN interaction does not contribute to these matrixelements if it is assumed purely central, as in Ref. [S]), to be compared with the values S, = +0.57 MeV and S, = -0.21 MeV for fit C. The parameter S, = +(S, + S,) = +0.18 MeV is that most closely connected with the NN spin-orbit force S(NN); its magnitude appears reasonable but its sign is opposite that known for S(NN). This question will be closely connected with the determination of g-s. spin values for the

230

DALITZ

AND GAL

p,,,-shell hypernuclei. A negative S,, and/or a positive spin-spin interaction d will place lowest in energy the hypernuclear state with spin (JN + l/2), whereas a positive S, will place lowest the state with spin I(JN - l/2)], where JN is the spin of the core nucleus. (ii) the existence of a /1iV tensor force. We have at present no evidence on this question. Theoretical estimates [6] suggest that the tensor parameter T should have a nonnegligible value (of order 0.05 MeV), which would play an appreciable role in the p,,,-shell hypernuclei, since the coefficient with which it appears in various doublet splittings can be remarkably large; for example, see Eq. (15.5) for the g.s. doublet splitting of :N. (iii) the existence and nature of a llNN three-body force. It appears generally recognized now that there must exist some effective llNN interaction at least, and the question is whether or not this should have strong noncentral components. The effects of these latter components are strongest for hypernuclei in the upper part of the nuclear p-shell, and their character could be inferred from the systematics of hypernuclear spectroscopy in this mass region.

5. ACKNOWLEDGMENTS Both authors wish to acknowledge the hospitality of the Physics Department of Brookhaven National Laboratory at one stage (1976) during this work, and of the Center for Theoretical Physics of MIT. at a later stage (1977) of the work. The tist author (R.H.D.) also wishes to acknowledge the hospitality of the Racah Institute of Physics of the Hebrew University of Jerusalem in 1974, during a visit when this program of work was begun. We wish to acknowledge helpful discussions with Drs C. L. Wang, J. J. Kolata and A. Kanofsky, and with Dr H. Piekarz, concerning possible experiments of this type, and to thank Dr G. P. Gopal for providing us with a table of the 0” reaction amplitude RN -+ (lo as function of incident momentum. We have benefitted much from the experience of Dr S. Y. Lee with the computation of hypernuclear formation rates in K- --+ rr- reactions and we acknowledge great help from Miss C. Hemming who calculated for us the numerical values of the matrixelements used in Tables VI-IX, using a program developed in another connection. After the completion of this work, but before the final manuscript was complete, we received from Drs A. Banasiuk and D. Zieminska a manuscript [55] concerning the calculation of y-ray spectra from hypernuclear excited states following K- capture from atomic orbits about nuclei,’ which has some overlap with our paper. ’ In this work, Banasiuk and Zieminska conclude that, assuming the Cohen-Kurath wavefunctions 1211, the total rate of formation for the T = 1 jBe* states 3/2+ and S/2+ (cf. Sect. 3 above) in Kcapture from rest on *Be is less than 0.5 x lo-* per stopped K- meson. The reported rate [2] for the 0.31 MeV y-ray, namely 2 x 1O-S per K- stop, is so much greater than this estimate that any mechanism for its production which involves the production and decay of these particular (+) parity ,jBe* states appears excluded on this ground alone.

P-SHELL

APPENDIX

HYPERNUCLEI:

FORMATION

& y-DECAY

231

A: THEEVALUATIONOFRELATIVEFORMATIONRATESFORTHEFORWARD K---+w REACTION, DUE TO A CLOSED NEUTRON SHELL

Consider the forward K- -+ rr- reaction on target nucleus (Ji, TJ. Suppose that a A-orbital (lZ4j,) is formed from a closedneutron shell (Injn) in this nucleus. The final hypernuclear states (Jr, Tf) reached may be characterized by the nuclear core states Jx to which ,jA is coupled to give J, , the isospin being limited to the values Tf = / T, 5 l/2 I. The evaluation of the absolute formation rates for these hypernuclear states would require the evaluation of a “sticking probability” integral depending on the single particle radial wavefunctions for the orbits iA and Z, and on the momentum transfer and distortion factors appropriate to a transition with multipolarity k. We note that k must obey triangular conditions with Ji , Jf , with 1, , I,, and with j, , j,, ; for IA = 0, the case of particular interest in this work, k is limited to the value I, . Here, we shall consider only the relative formation rates with the shell-model, since these depend only on geometrical considerations. These may be evaluated in three steps: (i) For K- -+ x- at 8, = 0”, the component M of the total angular momentum along this direction is conserved. For multipolarity k, the averaging over initial m-states then leads necessarily to the following statistical factor,

(ii) The space-spin decomposition of the hypernuclear state formed is determined by the transition (0+) + (k(-1)‘“) f rom the closed neutron shell to a final II particleneutron hole excitation with multipolarity k, the latter being coupled with Ji (which is the total spin of the valent configuration, for the particular initial nuclei being considered here) to give the total spin Jf, thus giving the final state,

l[j, OK1l’k’, Ji ; Jf).

(A.21

This state is then to be projected on the possible hypernuclear final states given by,

IJo 3K’ 0 Jf”‘;

Jf>.

(4.3)

This is accomplished by a unitary recoupling transformation, the expression,

(2k + lUJ,

+ 1) (k

2

;j’.

with intensity given by 64)

In these remarks, we have, of course, suppressed the isospin degrees of freedom, which we now consider. (iii) The isospin manipulations depend on whether or not the proton shell ([J,) which has the same 1 and j as the neutron shell (!,j,) under consideration is also closed.

232

DALITZ

AND

GAL

(a) If it is closed, as holds for the psj2 proton shell for all the pllz shell targets discussed in this work, then the isospin of the closed proton and neutron shells, both with the same (rj), has value zero. Since the II particle has zero isospin, the final hypernuclear isospin T, and its projection 7j are obtained by coupling the (t, T) = (4,:) for the neutron hole with (Ti , TJ for the initial valent configuration; no recoupling is necessary. This leads to intensity factor,

(A.9 Taking this factor together with (A.]) and (A.4) leads us to the desired relative formation rate: (2JN

+ 1)(2Jj + 1) Jo .in k 24 Ti T, 2 (2Ji+ 1) IJi Jf JN I[ 4 ri rf ’

1

(A.6)

We note that summation of (A.6) over J N, Jf and Tj gives unity. Summation over Jr alone leads to an expression whose complete JN dependence is given by the factor (2J, + 1). This formula (A.6) has been used in the text for the casej, = 3/2, j, = l/2 and k = I, which holds for the species YC, :N, and $N. It also applies trivially for the species YC, where j,, = 312, and 70, where both j, = 312 and j, = l/2 are relevant. (b) If it is empty, and all other shells are either closed or empty for both neutrons and protons, as is the case for 14C with j, = l/2, then the final hypernuclear isospin Tj and its projection 7j are obtained by coupling (t, T) = (Ti - l/2, - (7’i - l/2)) for the neutron hole with (0, 0) for the closed (or empty) nucleon shells. This leads to the factor (27 - l/2) -(Ti _ ]/‘4

0

0 -(TiT’ ]/2) I2= ‘7t.@-1/2).

64.7)

This result holds for N -+ (1 substitution for any shell occupied only by excessneutrons, when all proton and other neutron shells are either closed or empty. (c) two cases discussed in this paper but not covered by the remarks above are ?N, where the core nucleus is described by the valent configuration (p1,2)P(p112)N , and ‘;B, where the core nucleus is described by the valent configuration(p,&?(p,,,)~‘. In these cases, for strangeness exchange on the relevant shell, it happens that the space-spin coupling leading to JN determines uniquely the final nuclear isospin Tj , so that the isospin factor is unity. This will happen generally when the valent neutrons and protons (or the corresponding valent holes) are in the same shell (lj). We note that the J, dependence of expression (A.6) is the same as that given in Eq. (2.8) for the one-term case j = j, , for fixed JN and /3 and for k = I, (necessarily, because the final (1 is in the state Is,,,). In this appendix, we have effectively derived by a direct means the closed form expressions appearing in Eq. (2.8).

P-SHELL

APPENDIX

HYPERNUCLEI:

B: DERIVATION FOLLOWING

FORMATION

& Y-DECAY

OF THE (w, y) ANGULAR CORRELATION THE FORWARD K--+PREACTION

233 EXPRESSIONS

Consider the simplest case occurring, as illustrated in Fig. Bl, where the K- --+ Wreaction on the g.s. AZ(spin J) leads to the hypernuclear state A,Z*(spin 1,) which decays directly to some lower state $Z(spin J,) by y-emission with multipolarity k # 0 and parity (- 1)k+l-A, where X can take the values 0 and 1. The vector potential A,kf” for the photon can be written as follows: (a) A,k*Oz y’(k*l),

(b) A;.’ E -$,

x +*l),

P.1)

where py denotes the photon momentum, with direction given by (0,) $,,), and yf(~=ba=l)(&, d) d enotes the vector spherical harmonic, as defined and used in Chapter 1 of Ref. [52], for example. If we denote the multipolarity of the K- --+ w transition by (Al), then the W- wavefunction for this reaction is Yz”*(0, , d,,). Angular momentum conservation require that (a) Al + J = J, ,

(b) J, = k + J, .

03.2)

For fixed M and M, , the projections of spins J and J, on the quantization axis, the amplitude for the two-step process is obtained by summing the product of these two amplitudes over the intermediate m, Ml and K, with result (B.3)

The angular correlation intensity between fi,, = (6,) A) and fi,, = (13,) &) is then obtained by squaring (B.3), averaging over the initial M values and summing over the final M, values and the photon polarisation directions & (if the quantization axis is chosen along p,, , there will be no TV= 0 contribution since the vector spherical harmonic Ykfk*l) is chosen so as to satisfy the transversality condition p,, 3 Y,“‘“*“($,,) = 0 appropriate to a photon field). The latter sum gives A;?+

. A;.A

=

y$@l)t

. y;(kl)

It is then convenient to choose p,, as the polar axis, so that 0, = 0. Then Yhk(&, , 4) = 6,,((2k + 1)/47~)l/~, and (B.4) simplifies to

234

DALITZ

AND

GAL

Aif

3

Bl. Shows the strangeness exchange transition %5(J) + jZ(.l,) and the subsequent hypernuclear y-decay of multipolarity k. FIG.

Finally, after the M average and the M2 sum, we have apart from common factors: Al c [ M,M~.m.Mpt.M; m '

J M

J1 MI

([flk

J1 44;

I[+1k

x

J, M2

I[Alm Jz M,

J J1 A4 Ml

I

J1 M1 +

1 [-1k

J, M,

J1 k h4; I[ -1

J, M,

J1 Ml

I)

which reduces to the expression:

We note that, in the present case, the transition is (lp), -+ (1~)~) so that AZ = 1. Hence, for the K- + z-- transitions considered in this paper, the angular correlation (B.7) is limited to the (normalized) form (1 + A cos28)/(1 + 43).

tB.8)

An alternative possibility would have been to choose p,, as the polar axis, so that 0, = 0, and 8 is then equal to the angle 0, . Indeed, since we are considering forward K- + T- excitation, this might well appear a more appropriate choice. With this choice, however, the result obtained would have been a sum of powers of (cos2@” up to n = k, a much more complicated expression. Of course, since we know that the (7r, 7) angular correlation cannot depend on the choice of polar axis, the coefficients of (~0~~8)~ must turn out to be zero for n > 1, as a result of cancellations. Therefore, expression (B.7) is much more convenient to use. The result (B.7) can readily be generalized to a chain of y-transitions following the forward K- -+ w- reaction. Let us consider the successive y-transitions 1 ---f 2 + 3, for example, the first y-ray ~(12) being of multipole order k,, , and the second y-ray

P-SHELL HYPERNUCLEI:

FORMATION

~(23) being of multipole order k,, . The angular distribution relative to the forward n- direction then has the form:

the summation

APPENDIX

235

& Y-DECAY

for the second y-radiation

being over all magnetic quantum numbers.

C: EVALUATION

OF &V

PARAMETERS

FOR SPECIFIC POTENTIAL

FORMS

Here we shall give the expressions for the clN potential parameters V, d, S* and TV appropriate for a Is n particle in interaction with Id and 2s nucleons. We consider only nonexchange interactions and follow closely the procedures used in Appendix B of Ref. [6]. (a) Central Interactions The normalized

wavefunction for a nucleon in the 2s state is given by #(2s) = N,(2s)(l

where N,(2s) = (~/7r)“/“(3/2)‘/“;

- 2vrN2/3) exp(--vr,“/2),

the (is), wavefunction is given by

#(Is) = N,(k) where N,(k)

(C.1)

exp(--hrA2/2),

= (X/Z-)~/~. For a (1N potential

(C.2)

of the form

V(r) = u exp(-ar2),

the mean interaction energy V(2s) is to be obtained by integrating #A( 1~)~. This leads to the expression

(C.3) it over &(2~)~

V(2s) = V(ls)(v2(h + a)” + 3h2a2/2)/(Av+ a(h + v))~, where V(ls), the mean interaction

in the IS shell, is given by

V(k) = u(XV)~I”/{~V + a(h + v)}~/“. The normalized

(ld), #I)

where iV,(ld) by

(C.4)

(C.5)

wavefunction for m = f2 has the following form, = NN(ld)(xN

+ i.~~)~ exp(---vrN2/2),

VW

= (u/ ~Z)(V/~T)~/~. The mean clN interaction energy in this state is given V(M) = F(ls)v2(a + h)2/(hv + a(h + v))“.

(C.7) With the parameter values v = 0.41 fm-2 and h = 0.33 fm-2 for the wavefunctions, 595/116/r-16

236

DALITZ

and a = 0.936 fm-a for the potential numerical values are

AND

GAL

range parameter, as discussed in Ref. [6], the

V(2s)/ V( 1s) = 0.602,

V(ld)/V(ls)

= 0.393.

(C.8)

We note from Ref. [6], that these parameters lead to the ratio V(lp)/V(ls) = 0.627. These ratios (C-8) hold separately for both the spin-average and the spin-spin terms of the AN central interaction. (b) Tensor Interaction We need consider the tensor interaction only for the (I&&), configurations, since it necessarily gives zero for the (2s),(ls), configuration. We adopt the following form for it, VT = u(r)(on * laN * f -

aA * 6#/3),

W.9)

where r denotes the AN separation vector, (C.10)

r = r, - rN , and we choose to consider the particular * = N,(k)

AN state with J = 3, m = +3, for which

NN(lcI)(xN + &J2 exp(--hrA2/2 - vrJ2)

u+U+ ,

(C.11)

where U, U denote the A, N spin states, respectively. The expectation value ( Vr) is then calcuIated by integrating over all coordinates the product of (z2 - r2/3)u(r)/rz with the modulus square for this wavefunction (C.ll). To carry out this integration, the integral over rA and rN is transformed to an integration over r and R, where

R=

rN

+ [Uh + 4lr.

(C.12)

In the variables (r, R) the integral separates, and the R-integration once for all. The result is as follows:

(Fy) = NA2(lS) NN2(14

(xy;;7,2

’ 1+ A2fv (x2+ Y3+ SI

x (z2-$-)(F)exp(-&r”)@r.

2(A;

can be carried out

42

(x2+ Y2121 (C.13)

On averaging over angles, this reduces to (W

312 8h2v2 AU = - 45(A + v)” ( no + 4 )

x l/v@)r’(l + 7cf~vj

rz)ew(-&ra)d%j.

(C.14)

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HYPERNUCLEI:

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237

& Y-DECAY

With the canonical form CT(pnS/4it4,,MN)h(mr) for v(r) (cf. eq. (3.4) of ref. [6] for h(x)), this integral inside the curly brackets takes the form (nCT/m2MhMN)1Y(ld), where K(ld)

= Jm;rd.r(3

2h2 ) (-x + 3x + x2) (1 + 7m2(h + v) x2 exp

-

(h ;:)m2

x2),

(C.15) where d denotes the cut off radius. For m = 500 MeV, numerical integration of (C.15) leads to the values8 K = 11.52 for d = 0, and K = 10.15 for d = 0.4 fm. (c) Spin-Orbit Interactions

These interactions are zero in the (2s),(ls), configuration, so again we need consider only the (ld),(.ls)n configuration. It is most convenient to consider the I1N state (C.11) for which J = 3, m = +3. The interaction we wish to consider is

As explained in Ref. [6], the partial derivatives within LAN act in the variable r, the (1N center of mass coordinate B being held constant, where

it = of/F, + MNrNMMA + MN).

(C.17)

We now consider the effect of L,, on the wavefunction $I given by (C.l l), after it has been transformed to the variables (r, R). The angle-dependent terms in t/~are then (i) those in the argument of the exponential factor. As explained in Ref. [6], the result of differentiating this factor always becomes zero after integration over the variable, R, simply because this leaves only one vector quantity r, from which it is not possible to construct an axial vector. (ii) the spherical harmonic factor (xN + iyN)2. The operator L . S = L . (un -t- a#)/2 acting on (xN + iyN)2u+U+ gives rise to u+U+L,(x, + iyN)2, since the spin wavefunctions are the same for both initial and final states and it is only the component S, which yields a nonvanishing expectation value, namely fl. Since L, is a differential operator of first-order, we have: L,(xN + iyNj2 = 2(xN + iyN) Lz(xN + iyN).

(C. 18)

We note that (C.17) may be rewritten in the convenient form w = rN + Try where 77 = M&M,, (C. 18) becomes: 2(xN

(C.19)

+ MN). Since L, gives zero acting on i?, the right-hand

+ iyN) ’ (-> &(x + b) =

B The value K(1 p) = 11.I2 given for d = 0 and by 10.87.

m

-2qh

+ iyN)(x + iy).

side of (C.20)

= 500 Mev in Ref. [6] should be replaced

238

DALITZ

AND

GAL

We now transform back to the original variables rN and r, , and consider the surviving terms of the integral over (#V”o$), with the result: (YSO) = -2$V,2(lS) x

il

iv&ld)

(xN - iyN)(xN2 + yN2)(x + iy) u(r) exp(--hrd2 - vrN2) d3rN d3rA . I (C.21)

Next we transform to the variables r and R (see Eq. (C.12)), with respect to which the exponential factor separates, giving the two factors

(exp(-(A + 4 R2N@xp(-W/O + 41 r2)I.

(C.22)

After integrating out the variable R, the expression within the curly brackets in (C.21) takes the form

&$-$

(*)3’2

J (x2 + ~3 (1 + &

(x2 + YB))4-1 exp(- &

r2) d+/ (C.23)

Averaging over the angles leads then to the final result:

r2) u(r) exp (- & With the canonical form Cso(ms/2~2)g(mr) for v(r) (cf. Eq. (3.10) of Ref. [6] for the form of g(x)), the integral within the curly brackets of (C.24) takes the form (4SSo/ 3m2m2) J(ld), where we have J(l+J~~xdu(l+x)(l+

2h2

)

.5m2(X + v) x2 exp

(-x-

hv

m2(X + v)

x2).

(C.25) This integral has been evaluated for the following cases. For m = 750 MeV, J(ld) has the value 2.77 for d = 0 and 1.91 for d = 0.4 fm; for m = 500 MeV, J(ld) has the value 2.52 for d = 0 and the value 2.09 for d = 0.4 fm. Finally, we consider the reduced matrix-elements or,(A) for dN interactions linear in the spin s,, , acting in a /I-hypernuclear state with spin J built on a nuclear state with spin JN and within the nuclear lj shell, which is defined by the relationship A(i) . sA) = a.j(A)(J(J + 1) - JN(JN + 1) - 3/4)/2.

(i

(C.26)

i=l

As discussed in Ref. [6], in Eq. (2.18) et seq., these a,(A) depend both on the nature of A and on the particular shell lj considered; there is in general no simple relationship

&SHELL

HYPERNUCLEI:

FORMATION

239

& y-DECAY

between the CL~(A)for different Z-values. For 1 = 0, there is only one matrix-element, that appropriate to the spin-spin interaction. However, for I = 2, as for the 1 = 1 shell [6], there are two matrix-elements for each A, corresponding to the (l&l, and (l&l, shells, and the relationship between them is given on Table Cl, for the three operators A of interest in this work. TABLE

Cl

Values of the Coefficient a,(A), Defined by Eq. (C.26), for the Nuclear d-Shell interaction A = -s A = --I A = 6(8~)‘9s

APPENDIX

operator

j = 512 -l/5 -415 24135

@ Y&1”’

D: KINEMATICS

We consider here the kinematics

OF THE REACTION RN+/lr

j = 3i2 +1/5 -615 -1215

AT 00

of an exothermic reaction a+A+b+B

CD.11

for particle a incident on stationary target A, transforming to the particles (b, B), for the case 8, = 0”. We shall have in mind the case where (a, A) stand for incident R meson and nucleon target N, and (b, B) for the final 7~and L’I particle. In the formulas to be derived, these symbols will be used to denote the rest masses of these particles; otherwise they will be used as suffices, for example pnc will denote the momentum of particle a in the c.m. system. The momentum transfer q given to B in the laboratory frame is a quantity of special interest for us. For incident laboratory momentum paL , it satisfies the equation (a2 + piL)1/2 + A = (b2 + (paL - q)31j2 + (B2 + q2)l12.

03.2)

We first give the relationship explicitly for three cases of special interest. For an exothermic reaction, we have (a + A) > (b + B), so that there is a physical threshold at c.m. energy (a + A). The cases of interest are: (i)

at the threshold (a + A), B (and b) recoil with the momentum

q. , where

q. = {J&4 + a rt B -f b)Y/2/2(A + a),

(D.3)

where we use the notation 17(a f P * I’) = (a + P + Y)(‘JJ+ p - y)(a - p + y)(‘y - fi - y).

(D.4)

For RN -+ AT, the value of q. is 253.4 MeV/c, using the masses for K-, N and r-.

240

DALITZ

(ii)

the momentum

AND

GAL

par. at which q = 0 holds must satisfy the equation (a” + &Y2

+ A = (b2 + P:~F’~ + B,

(D-5)

obtained from (D.2) for q = 0. To discuss its solution, let us consider the function F(p)

= A + (a’ + p2)lj2 - B - (b2 + p2)l12,

(D.6)

so that the equation F(p) = 0 gives Eq. (D.5). Its derivative F’(p) is one-signed for p > 0, positive if b > a and negative if b < a, so that F(p) is monotonic. For an exothermic reaction, we have (a + A) > (b + B), so that F(0) > 0. As p + co, F(p) --+ (A - B). Hence, for B > A, the case of interest here, we have F(co) < 0. It then follows that F(p) crosses zero once and onZy once, for some positive value p,, . The explicit solution of (D.5) is par. = p0 = {lT(B - A f a & b)}1’2/2(B - A).

(D-7)

For RN -+ AT, the value of p0 is 531 MeV/c. Attention was ‘first drawn to this special configuration by Podgoretsky [53], and to its special interest for the strangeness exchange reaction by Feshbach and Kerman [49] and by Kerman and Lipkin [34]. (iii) the momentum transfer q, in the limit as par. -+ co, is obtained from the limit of Eq. (D.2), A = -q(“o)

+ (B2 + q(m)2)1’2,

03.8)

with the solution q(c0) = (B2 - A3/2A.

(D.9)

We note that this limit q(a) does not depend on the masses of the particles (a, b), which are the mesons in our case. However, the approach to this limit does depend on the masses (a, b), the leading terms being as follows

q + q(m) + (b2 - u2)/2p,r. + *** .

(D. 10)

For RN -+ rlrr, we have q(a) = 192.5 MeV/c; also the limit is approached from below, since (b2 - u2) = (mw2 - mK2) < 0. As is well known, the c.m. velocity is given in terms of the initial uA system by the expression (D.ll) B = Pa,/@ + (a” + P:J1’2). If we now consider the reaction uA ---fbB for an arbitrary c.m. reaction angle &e, the momentum transfer q, which is the laboratory momentum for the recoil particle B, is given by q2 = y2(-pBc

~0s bc + b-h2

+ (psl sin Rd2,

(D.12)

P-SHELL

HYPERNUCLEI:

FORMATION

&

Y-DECAY

241

FIG. Dl. Shows the curves for q+and q- for the reaction a + A ---t b + B on a stationary target A as function of incident laboratory momentum par, where q* denote the laboratory momentum of the recoil system B along the direction paLfor the cases (+) of c.m. production angle &c = 180”, and (-) of c.m. production angle S,c = 0”. The curve 1q- 1 is the reflection of the curve q- in the axis pan. As the angle &,c varies from 0” to 180”, the laboratory recoil momentum of B varies along the line pan= constant, from the curve labelled I q- I to the curve labelled q+ , according to the formula (D.13).

where 7 = l/(1 - p)lj2 and (pBC , wBc) denote the momentum and total energy of B in the c.m. frame. This expression may be rewritten in the form q2 = T2(wBc - ,&,

cos &,c)a - B2,

(D.13)

whence we see that the magnitude q lies between the two limits reached for Oat = 0” and 180“. The momentum transfer q2 can vanish only if both squared terms in expression (D.12) vanish at the same time; this clearly requires sin &,o = 0, and a particular value for pet . Specializing to e,, = 0” and 180”, the two limiting values [ q- 1 and q+ , respectively, shown on Fig. Dl, correspond to the two functions 4;t = 7@%c f Pm).

(D.14)

These functions q+ are both smooth function of paL or E. The ebC = 0” value qfollows the lower curve shown on Fig. Dl ; it becomes negative for par. below the value (D.7), which means that B recoils backwards relative to the incident particle a in the laboratory frame. However, Eq. (D.12) is concerned only with the magnitude of the momentum transfer, not its direction, and Eq. (D.13) makes it clear that q varies only between q+ and j q- 1as B,o varies from 180” to 0”. The curve j q- j therefore has a discontinuous tangent at the incident momentum par. = p. given by (D.7); the slope to the right is (B - .4)/(a2 + po2)l12,whereas the slope to the left is -(B - A)/ (a” + p$-y.

The general situation is summarized on Fig. Dl. The curve for 1 q- 1is plotted on Fig. 18 for the case RN --f LIT of physical interest here.

242

DALITZ

AND GAL

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