Hypernuclei with A ⩾ 12

Volume 64B, n u m b e r 3

PHYSICS LETTERS

27 September 1976

H Y P E R N U C L E I W I T H A t> 12 A. BOUYSSY* and J. HUFNER Max-Planck-lnstitu t far Kernphysik and Institut far theoretische Physik tier Universitdt, Heidelberg, Germany Received 26 May 1976 We analyse recent e x p e r i m e n t s in which m e d i u m heavy hypernuclei 40 ;~ A ~ 12 are produced. The d e p t h o f the lambda-nucleus shell-model potential is f o u n d to be D A = 28 ± 3 MeV. We see no significant spin-orbit term. The magnitude o f the strangeness exchange cross section is not u n d e r s t o o d in all respects.

Strangeness exchange reactions are the modern way to produce and study hypernuclei: a K - meson hits a target nucleus AZ, changes a neutron into a lambda and a n - meson leaves the nucleus. The energy spectrum and the angular distribution of the n - carry the information about the hypernuclear state which has been produced. Very recently, new results have been published [ 1]. Spectra of medium heavy hypernuclei like 3 2 8 and 4AOCaare available for the first time. Like in similar experiments before, the kinematics is chosen to single out the events with low-momentum transfer: The momentum of the K - is PK = 900 MeV/c and the 7r- is observed in forward direction. Then the momentum transfer is IPK -P~ri ~ 80 MeV/c. No momentum transfer implies no transfer of angular momentum. Therefore, in targets AZ with j,r = 0 + only hypernuclear states (Az)* with j~r = 0 + can be excited with significant cross sections. While the complexity of the expected spectrum is drastically reduced in this way, the ground states of hypernuclei which carry important information, are also suppressed. The experimental spectra [1 ] for 1A2C, 160, 3A2Sand 4AOCalook rather similar. The ground state is invisible. A narrow structure (in the following called "peak") appears at B A ~ 0 to - 10 MeV (the binding energy B A of a lambda is negative for unbound states). A broad and structureless "bump" follows at higher excitation energies ( - 4 0 MeV ~
* On leave o f absence from Institut de Physique NucMaire, Orsay, France.

276

DA(MW) 35 25

++ ++

15 0

II

I

I

I

' ca

Fig. 1. Values for the d e p t h D A of the lambda nucleus shellmodel potential (r o = 1.1 fm). The value ,,oo,, is the extrapolated value from the literature [6].

We compare these data with a model calculation in order to extract information about the nature of the lambda-nucleus interaction. We use the shell-model to describe the structure aspect of hypernuclei and perform a distorted-wave-impulse-approximation calculation for the cross section (details in ref. [2]). The single particle states of the lambda are computed from an energy independent shell-model potential UAA(r) = --DAf(r ) + US~AI" s

1 d_ff rdr'

(1)

f(r) = (1 + exp ((r - R )la'))-I with R = roA 1/3 and a diffuseness a'= 0.6 fm. The two strength parameters D A for the central potential and U~A for the spin-orbit part, will be determined by comparing the calculation with the experimental spectra. For the first time it will be possible to obtain a value for U~A. The excited states of the hypernucleus are assumed to be of one-particle one-hole ( l p - l h ) struc-

Volume 64B, number 3

I

i

PHYSICS LETTERS

/ \ \ a = -0.05

\ \

I

I

I

I

40

o 10 u 5

0.50 0.25

•~ . 03 p-

,,

o

Z

o

IP3/2 *)

lyp)cal e r r o r bar

0.6 o oo

o 20

,

( $1/2 151/2

)

oo

o

"o

o

b o o

o

U

(1Pl/21P1/2

5

~

oo

o

~o

o

o o

o

°ooo

I

I

2

5

I

ture. Their exact structure is obtained by diagonalizing a residual lambda-nucleon interaction chosen o f the form VNA(r 1 -- t'2) = - V 0 8 ( r 1 - r2) (1 + a(I 1 " a 2 ) "

(2)

For internal consistency o f the shell-model, the overall strength V0 must be related to D A by V0 = DAfd3rf(r)/A, since VNA folded with the nuclear density should reproduce the average potential eq. (1), at least its central part. The spin dependence o f VNA, characterized by the parameter a seems small. Calculations of binding energies in light nuclei [3] suggest a = - 0 . 0 5 , while the possible 0 + - 1+ splitting observed [4] in the A = 4 system favours a = - 0 . 1 3 . We analyse the position o f the " p e a k " in the experimental spectra to deduce a value for D A. The shellmodel calculation suggests the " p e a k " to consist o f l p - l h states *l where a neutron from the least bound shell (p-shell in C and O, s-d shell for S and Ca) is converted into a lambda without changing the orbit. The absolute position o f the " p e a k " is related to the difference in single-particle energies for lambda and neutron. The neutron energies are taken from experiment [5]. The energies for the lambda are calculated from the potential eq. (1), where D A is adjusted so :14 Here and in the following we restrict ourselves to 0 + states, since we find states with J > 0 to contribute less than 20% to the cross section.

Joo° ooo

o

-30

-20

-10

0

10

o

20

BA(MeV)

10 LXE(MeV)

Fig. 2. A determination of the A-nucleus spin-orbit potential U~°A. The ratio of cross sections Xand the separation energy AE of the doublet in the "peak" of 160 are given as a function of usA~. Two different values for the spin dependence a of the residual interaction are used.

° o

~oo

oo

-40

0.3 ~

1)

ocoO

oe

0;

I

0.9 (IP3/2

,,,

\ \

I

o

UAA(MeV)UAA/~NA

\ \

I

160(K -, ~ - )lAB0

160(K-, ~ - ) 1 6 0 • ; "SO'-. ~SO /HS0

\~ \

10

L

I

t I /

27 September 1976

Fig. 3. Comparison of experimental (circles) and calculated (bars) spectra for the reaction 160(K-, ~r-)160*. The dominant configurations are indicated for each peak. Note the different vertical scales for experiment and calculation.

that the theoretical and experimental peaks coincide. The resulting values for D A are shown in fig. 1 as a function o f A . The error bars mainly reflect uncertainties in the identification o f structures in the theoretical and experimental spectra. Mean values for D A derived from this analysis are D A = 28 + 3 MeV,

for

r 0 = 1.1 f m ,

D A = 25 + 3 MeV,

for

r 0 = 1.2 f m .

(3)

These values obtained directly from an analysis of medium heavy nuclei can be compared with previous values for DA, obtained by extrapolating B A values from ground states of light hypernuclei [6]. Our values and the extrapolated one coincide, c.f. fig. 1, and thus support the concept of an average shell-model potential. Taking the values from eq. (3) we calculate the position of the hypernuclear ground state in 4h0Ca at B A = 17 -+4 MeV. An experimental value for this state could give information about a possible energy dependence (or effective mass) of the A-nucleus shell-model potential. The structure of the experimental " p e a k " contains information about the lambda-nucleus spin-orbit potential. We choose 160 to explain the physics of this dependence: Without residual interaction, the " p e a k " contains two states of the pure configuration (AP3/2, -1 p~-/12)0+ and ( A P l / 2 , Pl/2)0 ÷" Their energy separation A E is directly proportional to the difference in spinorbit strength for the lambda and the neutron ( A E = 0, 277

Volume 64B, number 3

PHYSICS LETTERS

if the strenghts are the same). Without residual interaction the ratio X of the cross sections in the two peaks is X -~ (2j> + 1)/(2]< + 1) = 2 f o r / > = 3/2 and ]< = 1/2. When the residual interaction VNA is switched on, the two configurations mix. AE and Xdepend in a complicated way on VNA and on the strength of the spin-orbit interaction. Keeping VNA fixed as described above, and taking the spin-orbit splitting of the nucleons from experiment, we determine U ~ . We use experimental values of X and AE, and thereby check the internal consistency of our model. Fig. 2 shows the dependence of X and A E on Us°A. We find from 1A60 (together with a less precise value'from 4AOCa)

U~)t/U~I°A = 0.0

-+0.3.

(4)

The lambda-nucleus spin-orbit potential is much smaller than the one for the nucleon. The reduction seems even larger than for the central part of the shell-model potential (where the lambda potential is about one half of the nucleon one). However, there is no reason for the two quantities to be coupled, since there is no simple relation between A-N and N-N forces.(e.g, one pion exchange is impossible for A-N). Using the average potential eq. (1) (of which now all parameters are determined) and the residual interaction eq. (2), we calculate the full spectrum of l p - l h hypernuclear states and the cross section for each state. Fig. 3 shows a typical example. While the positions of tile calculated states essentially follow the experimental pattern, there is some disagreement in a quantitative comparison of the intensities: (i) The integrated experimental cross section is 1.2 mb/sr practically independent of the target nucleus. Our calculated numbers reproduce the independence on A. The absolute value of the cross section depends sensitively on the elementary K - n --> A n - forward cross section (we use 2.5 mb/sr) and on the total K - - N and lr--N cross sections (we take 50 mb for both). With these values we achieve agreement with experiment, but the uncertainties in the inputs may change the calculated value by a factor of two, roughly. (ii) The experimental ratio of cross section in the "peak" and in the " b u m p " is between 1 : 1 (1A60) and 1 : 2 (12C and 4AOCa). We calculate ratios of about 2:1 ( I ~ c and 4AOCa)and 3:1 (1A60). This discrepancy is independent of the uncertainties in the parameters

278

27 September 1976

discussed above. We investigated the possibility whether the " b u m p " does not only contain l p - l h states but also 2p-2h states created by the incoming or outgoing mesons. The correction turns out to be less than 10%. States with 1- can change the ratio by 20% at most. Is the "peak" the strangeness analogue state as proposed in ref. [7]? Then in 160 it should be described by the wave function [SAS) = ½ [x/~ lAP3/2 , p~-/1) + [APl/2' p~-/1) + [ASl/2 ' S-{[12)]. The immediate consequence is that all intensity of the spectrum must be concentrated in a "peak" and a " b u m p " should not exist. Experimentally in G O , the " b u m p " carries about as much intensity as the "peak". If one assumes the "peak" to be a coherent superposition of l p - l h states from the P3/2 and Pl/2 shells, "the peak" should consist of one line only. The shoulder in the experiment indicates the presence of another state. Our conclusion: a traditional shell-model calculation is sufficient to understand most features of the present data, and there is no experimental indication which supports the interpretation of some state as a strangeness analogue state. Dalitz and Gal [8] reach essentially the same conclusion. We have profited a lot from discussions with Dr. Kilian, Prof. Povh and Prof. Weidenmfiller. One of us (A.B.) thanks the Max-Planck-Institute for financial support and the hospitality. References

.[1] W. Brueckner, B. Granz, D. Ingham, K. Kilian, U. Lynen, J. Niewisch, B. Pietrzyk, B. Povh, H.G. Ritter and H. Schroeder, CERN Preprint, submitted to Phys. Lett. [2] J. H~ifner, S.Y. Lee and H.A. Weidenm~lller, Phys. Lett. 49B (1974) 409; Nucl. Phys. A234 (1974) 429. [3] A. Gal, J.M. Soper and R.H. Dalitz, Ann. Phys. 63 (1971) 53, 72 (1972) 445. [41 A. Bamberger et al., Nucl. Phys. B60 (1973) 1. [5] G.J. Wagner, Lecture Notes in Physics, Vol. 23 (Springer, Berlin 1973) p. 16. [6] J.D. Walecka, Nuovo Cim. 16 (1960) 342. [7] J.P. Schiffer and H.J. Lipkin, Phys. Rev. Lett. 35 (1975) 708. [8] R.H. Dalitz and A. Gal, Phys. Rev. Lett. 36 (1976) 362.