Interpretation of collective hypernuclear states produced in strangeness-exchange reactions

Volume 48B, number 1

PHYSICS LETTERS

INTERPRETATION PRODUCED

OF COLLECTIVE

7 January 1974

HYPERNUCLEAR

IN S T R A N G E N E S S - E X C H A N G E

STATES

REACTIONS

N. AUERBACH* Department of Phystcs, Rutgers University, New Brunswtck, New Jersey 08903, USA

and A. G A L Raeah Institute o¢ Physzcs, The Hebrew UntverslO,, Jerusalem, Israel Received 21 October 1973 The concept of Strangeness Analog Resonance may not be useful for describing low m o m e n t u m transfer strangeness exchange reactions on nuclei A band of collectwe excitations is expected where the lowest and most enhanced one Is invariably around 10 MeV above hypernuclear ground state

Intense K beams provide a new tool for producing and identifying hypernuclear states hitherto unexplored It has been suggested [1] that strangeness exchange reactions on nuclear targets rr +AAZ

(la)

7r° + A ( z - - 1)

(lb)

K - +Az-'~

could be done with an appropriate beam energy so that m o m e n t u m transfer to the A IS considerably less than nuclear Fermi momentum. The instantaneously resultant hypernuclear state (whether stable or not) is then very similar to the initial nuclear state, since (for reaction ( l a ) ) one of the nuclear neutrons (with an equal amphtude for each one of the neutrons) is replaced by A in the same spin-space configuration In analog with asospln analog resonance this state was called by Kerman and Llpkm (KL) [2] strangeness analog resonance (SAR). Observation of SAR depends of course on the extent to which this is an approximate elgenstates of the total hypernuclear Hamdtonlan. Kerman and Llpkm argued that SAR may be an approximate e~genstate even though the basic AN force IS considerably different from the NN force in the same spin space configurations The relevant hypernuclear "'symmetry breaking" consists of the difference between A-nucleus Interaction and N-nucleus lnterac* On leave of absence from Department of Physxcs and Astronomy, Tel-Avw Umverslty, Ramat Avw, Israel.

22

tlon. Since m a single particle description both A and N witness potentials which follow the shape of the nuclear core, the above difference is almost constant in the nuclear interior and hence ineffective in making A states different than nucleon states We find this argument somewhat misleading the " s y m m e t r y breaking" interaction is o f the same order o f magnitude as the "symmetric" component [3] and should prove essential for determination o f the self-conststent single particle potential. The phenomenologlcal observation that a nuclear singleparticle potential closely follows the shape o f nuclear matter does not indicate which component of the NN force is used up in gwmg this shape and which component is left to be treated as a residual interaction. In our view there ts a difference between the shapes of self-consistent single particle potentials witnessed by a A and a nucleon. This difference stems mainly from the restriction to hypernuclei o f strangeness - 1, that is, there is only one A partlcle but m a n y nucleons. Let us clarify this statement by assuming that for both A and N a selfconsistent field can be found which resembles an harmonic-oscillator well, except outside the nucleus The nuclear spring-constant coN is fixed by requiring the lowest lying levels in this well to be occupied by A nucleons and that the mean square radius (r 2) is given, as determined by various size measurements, by-~(roA1/3)2, w h e r e r o ~ 1 2 fm. This leads to the well known A-dependence

Volume 48B, number 1

PHYSICS LETTERS

hco N ~ 40A -1/3 MeV

(2)

. . . . Ip . . . .

Simdarly, by the varlal theorem for an hypernuclear ground state,/ko A =3h2/ma(r 2) Since the effect of one A on nuclear sizes is probably negligible in heavy hypernuclel one may safely assume (r2a) = ~(roA l/3 )2 so that the A pamcle can benefit in a maximal way from interacting with all nucleons in the hypernuclear ground state. Hence,

~coA~~- mAr2~ ~-A

>3

7 January 1974

=60A 2,3MeV.

o

x

....

o x x x

15 . . . .

x n

E=~oJ

E=~¢o A

N

(a)

~

--

(3)

The different A dependencies m eqs. (2) and (3), resultmg from having only one A versus many (A) nucleons, gwe rise to different from each other shapes of the self consistent A-nucleus and N-nucleus potentials. The same holds for the corresponding single particle wavefunctionsH Therefore, already m first approximation, namely that of single particle model, the symmetry between N-excitations and A-exmtations is broken Consider the strangeness exchange process ( 1a) at zero m o m e n t u m transfer. Applying the KL picture the resultant state ~s a linear c o m b l n a n o n of various particle (A) - hole (n) exmtatlons, sketched m fig 1 for a two major shell hypernucleus such as 12 xC. Shown m fig la are two components of the formed 1A2C* state and thmr unperturbed exmtatmn energy relative to I~C ground state (fig lb). In the KL picture coN = coA and the coherent SAR is indeed produced with an enhanced cross-section, the enhancement factor given by the total number of neutrons. In the alternatwe p~cture the two particle-hole components are not degenerate to start with, the energy difference being roughly gwen by/}(co N coA) ~ 5.5 MeV (forA = 12) Residual mteractmn has to be extremely strong to overcome this energy difference and cause the produced state resemble that pamcular linear c o m b i n a n o n characterized by the coherent SAR. Most of the strangeness exchange strenght will probably be distributed over these two separate excitations. Moreover, additional components, not present m the SAR, will be excited m the above reaction. Such a component ia shown m fig lc for 1A2C* consisting o f A exmtatmn +1 Eq (3) may be checked ar present only m the p-shell Ihts of binding energies around A = 10 gwe [41 ~eoA ~ 11.5 MeV, to be compared with a value of 12 8 MeV gwen by eq (3)

x

----2s

° x ~ A

n

E=O(g (b)

s)

A

. . . . ]s lp

n E=2hco^ (c)

Fig 1 Excitations relevant for Ii~c* (a) Particle-hole excitations which SAR consists of forA = 12 together with their unperturbed exmtatlon energms (b) A shell model description of l~C (ground state) (c) An exc)tanon, not present in SAR description, of A into its 2s orbit following strangeness ex change reaction (lal rote its 2s-shell Since the 1s nucleon-wavefunctlon is not orthogonal to the ls A-wavefunctlon it becomes possible to mix m this component. Hence m IA2C* one would expect a lowest exmtatlon at hcoA = 11 5 MeV The next exmtatlon hes around 17 MeV, but could well be located considerably higher (see also footnote 3). The approximate ratio between enhancement factors is 2 (gwen by the ratio of number of neutrons m the p-shell) m favor of the lowest exmtatlon Taking account of residual lnterachon may shghtly change these numbers Higher excitations are expected to be weakly produced relative to the above mentioned two. Experimentally [5] a state at 10 11 MeV has been observed and tentatively assigned as a collectwe one Kerman and Llpkm used this experimental number to evaluate m their model the difference between A welldepth and a neutron well-depth and thus to predict SAR excitation energms m other hypernuclm) 2 For 1A4N* they would have arrived at 20 MeV and for 1,60* ~2 By deahng with the difference between well-depths they Justly avmd, m a single pamcle approach, the use of quantities hke hc~N and need not consider the effect of resldua| interaction, and extremely lmportnat effect m view of the large degeneracy m theLr unperturbed particlehole exmtataons 23

Volume 48B, number 1

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17 MeV assuming an extrapolation of measured hypernuclear binding energies [6] whereas we expect a lowest excitation around 10 MeV ( ~ C O A ) . The situation in heavier hypernuclel is more comphcated because of the neutron excess Ignore for a moment lsospin, since many major shells are occupied by neutrons one would expect to see in reaction (la) a band o f states, the corresponding excitation energies of the most favorable ones are v~co^, ( v - 1)~6o A +rico N , .. u~w N (for u+ 1 major shel'Is). In 20A8pb~ these excitations are located around 8.5 MeV, (8.5 +5)MeV,... 34 MeV'I"3 respectively. Each member of a band consists, in turn, of states formed by replacing one of the neutrons in each of the orbits of the appropriate major shell by a A in one of a corresponding set of states, by which are meant those A-orbits having large enough overlap with the neutronhole orbit. Each member of a band is produced with an enhancement factor proportional to the number of neutrons in that major shell times a generahzed overlap Integral. The lowest, and mostly enhanced, excitation lies at about a constant (over the periodic table) value of v h w A ~ (10 -+ 1.5) MeV while the next ones appear at intervals of roughly 5 - 6 MeV. In 20aSpb this lowest excitation is a pure T -Jl one because it is obtained by exciting a A into an excess-neutron hole. The next excitations are split into T ± ~- components, but for T N 1 most of the above considerations remain valid for the T--½ components (the T+½ components may be obtained in a similar way from the various A particle -- proton hole excitations). In the KL approach the T - ½ component of SAR is about 30 MeV above ground statet 4 . The two approaches are seen to give totally opposing pre&ctions for heavy elements and any experimental evidence would appear most interesting. Nevertheless, it is quite probable that the actual situation is somewhere in the middle, as the two approaches concern, for the following reasons: (i) The residual lnteractmn may admix members of the band thus leading to more coherent states. This may part 3 The last excitation energies should be taken wlth more than a gram of salt since for such deeply lying nuclear hole states the use of eq. (2) is highly questmned. Also, sphttmg within major shells should be taken into account, preferably on a phenomenologlcal basis Actually this number is obtained [2] for reaction (lb), but since symmetry energy largely components for Coulomb energy the above statement follows 24

7 January 1974

tlculalry turn unportant for heavy hypernuclel where, in addition, splitting within major shells could simulate a similar effect. However, the resultant widths of these produced states may become quite large, obstructing their direct experimental identification. The above discussion does not touch upon the more conventional type of admixing np-nh (n >1 2) components which obviously contribute to the widths (n) The A dependence, eq. (3), advocated here for the shape of the A single particle potential may strictly hold only for low-lying hypernuclear states. For higher A-excitations the A dependence is phenomenologically unknown and could in principle be different, reflecting an appreciable state-dependence of the A single particle potential. Nevertheless, since AN interaction is quite different from NN interaction it is inconceivable that the emergent A single particle potential for these high lying states would cope in shape with nucleon single particle potential. Future experimental work, especially In heavy nuclei, will clarify the underlying picture of strangeness exchange reactions. [1] H J Llpkm, Phys Rev Lett. 14 (1965) 18, H Feshbach and A K Kerman, m Preludes of Theoretical Physics, eds A de-Shaht, H Feshbach and L. Van Hove (North-Holland, Amsterdam, 1966) p 260; R H Dahtz, in Proc Intern Conf on Hypernuclear physics, Argonne (1969) eds. A.R Bodmer and L.G Hyman, p 728-729 [2] A K Kerman and H J Llpkm, Ann Phys. (N.Y) 66 (1971) 738 [3] J. Brown, B. Downs and C Iddmgs, Ann Phys (N Y ) 60 (1970) 148, J J. De-Swart, M.M. Nagels, T A. Rijken and P.A Verhoeven, Srpinger Tracts m Modern Physics 60 (1971) 138, A Gal, J.M Soper and R H. Dahtz, Ann Phys. (N.Y) 63 (1971) p. 56, 72-84, D O Rlska, Phys Lett 40B (1972) 177 [4] A. Gal, J M. Soper and R H Dahtz, ref [3] p. 63. [5] G. Bohm et al, Nucl Phys B24 (1970) 248, M. Junc et al, Nucl. Phys. B47 (1972) 36; T Bressanl, to appear m Proceedings of Meeting on Kbeams and hyp0rnuclear physics, Brookhaven, July 1973, G.C Bonazzola et al, contributed paper to the Fifth Intern. Conf on High energy physics and nuclear structure, Uppsala, June 1973 and a contributed paper to the Intern Conf on Nuclear physics, Mumch, August 1973, M.A Faessler et al, (CERN-Hmdelberg-Warsaw Collaboratio), Plays Lett. 46B (1973) 468 Letters [6] M Junc et al, Nucl. Phys B52 (1973) 1, A Gal, J M Soper and R H Dahtz, Ann. Phys (N Y.) (1972) 445