A “potential” for low energy α-N interaction

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~WVFOR LOW ENERGY a-IN INUERACTIO-N A, N. MITRA, V. S. BHASIN and B. S. 1 L~fparrm,e,nj of P_4ysics, Unisersi~v of Def4is DrA" Ynd-~~2) Received 6 June 1962 on Hr A separiable mtential, mode] of alpha-nucleon interaCtion is propoud intaracuons m ~W s-and p-wava ouT aarhw aipproacb to p-p scattering. Only ~~ "uce the s- and p-phase shil qWw well up to about 20 ?*IeV. Mh sm marra=ons we ara-active and as expected the spin-orbit potential is much smau r,aj~ pmmtmL Tbe attrutive swavle interaction predicts a boun He --qale as W BVTem=t with the findings of other authors.

to Wpam. P-wave centem,

1. Introductim

!n st by a large Tb-i- interaaion of o:-particles with nuclear systems has I remning '. The n, umber of workers, both experimentally and by phenomenolo &W ~~= of the oparticle helps it to interact with many systems while retaining its, identity, especially when the energies involved are not high. In particx~lar, thetc is, a linnited range of energies (z. 20 MeV) for which an oc-N interaction m be usefully regm ded as a 2-body pro', Aem. This concept has Wved arnple expe"M con"ahm. It is of wurse possi We in principle to deduce an 3t-N force in terms of the more -fundamental two-nucleon forces. However, such attempts 2.3) have not usually gone beyond the generz] prediction that an m-N force is made up of: (i) a local farce which does not take account of the internal structure of the alpha particle, and (ii) an integral term, (non-local) which represents the effects of exchange of the outside nucleon with those residing inside the alpha particle. In particular, the relative strengths of the local and non local ternis ate not specified in any quantitafive way tt . Similar ambiguities have remained in the interpretation of the spin-orbit part of the 7.-N force in tenns of fundamental nucleon-nucleon forces. Thus Sugieet al. 4), employing two-nucleon potentials, derived from meson theory, found that the large doublet splittings of the p-phase shifts in scattering could be ascribed to the strong long-range (one-pion exchange) tensor force between two nucleons in triplet even states . However, it is not quaraitatively clear if tensor forces alone are adequate for the purpose or whether one needs in addition direct spin orbit FoTas between two nucleons, as other data like spin-orbit splittings of energy levels in certain nuclei suggest 5)fft. ' For a recent survey, see ref. 1 ).

tt

ttt

See, e.g., the general discussion on p. 467 of ref. 1). Gther references are given in ref. 5). 316

LOW

RGY 12'N INMACTION

317

Such inadequac;ies u above ofthe so-called "fundamental approaches" in providing a quantitative understanding of the mechalism of a-N interaction, perhaps give some practical Value to phenomenological attempts to represent the U-N force by ordinary two-body potentials, at least in so far as the alpha particle can be regarded as a distinct entity in a nuclear system. Representing the a-N interaction by an effective two-body potential (local or nonlocal) has some distinct advantages. In principle, it gives a concrete (almost selfthe evident) Wization to scheme of resonating group structures, with the alpha particle as the natural resonating group . In practice, an effective a-N potential is extremely useful for calculation of: (A) energy levels in nuclear systems where alpha particles can 4 regarded as distinct entities, eq., He', Lil, Be,, or (B) scattering parameters (cross-section, polarization) in systems where an alpha particle plays the role of a projectile . The simplest system of type (B) is of course cx-N scattering, b ut th!s is generally used to determine the parameters of the o(-N potential. However, once such a "potential" is obtained, its validity can and should be checked through its effect on systems like (A). Fon as is generally recognized, a potential always plays the role of a "'vertex" so that its effects in regions "'off the energy shell" are a matter for detailed scrutiny and cannot just 4 taken for granted on the basis of fits to scattering data 6). One of the earliest and most comprehensive phenomenological attempts to represent an a-N force by a two-body potential is that due to Gammel and Thaler 7), who were able to give almost precision hts to >N scattering data up to about 40 MeV. While such a fit is very impressive, there still temains a case for ensuring the validity of their model as a two-body potential, in the sense of the last paragraph . One could, e.g., test such a potential for correct off-energy shell contributions, with the help of nuclear systems involving 3 or more entities . Thus the calculation of a-d scattering as a 3-body problem by Gammel et al. 8 ) using the above potential 7 ) may be regarded as an attempt in that direction . However, in view of the mathematical complexities involved in such a problem the exigencies of approximation may largely obscure any specific conclusions that could otherwise be made about such a "potential" . In this paper we have taken a view similar to ref. 7) , but with a closer eye on the practical possibility of subjecting the Y-N "potential" to some simple 3-body systems . This necessarily implies a simple choice of the potential, sors to make the subsequent 3-body problem mathematically tractable. This is also in coniotmity with the philosophy we had laid down earlier in ref. 6) in connection with the 3-nucleun problem treated with 2-nucleon potentials. We have therefore explored the possibility of fitting a 46 separable" form of a-N potential to the x-N phase shifts . However, we consider an energy range of 40 MeV too large for neglecting, the structure effects of the a particle . In our estimation it would be quite satisfactory to fit a potential to the phase shifts up to about 20 MeV, but it is more important from our point of view to ensure that such a potential gives a reasonably quantitative description of ci--rtain low energy systems where an a-particle is involved as a distinct entity.

A . N . MITRA

318

et at.

model of oc-n interaction is prOPOsed on lines In sect. 2, a separable potential 1). The Parameters of the potential which similar to our earlier work on p-p scattering phase shifts, are found to reproduce the latter are determined from the observed this potential are also found to agree broadly rather well. Certain general features of with the conclusions of other authors . 2. The %-N Potenti we choose an ot-n potential in Analogously to our earlier approach to p-p scattering 9) momentum space of the form *.(P)(L- &+t)yl .(p,) . (1) 2M,(pjVjP') 2,-, -1OVO(P)VO(P')-4nA,vj(p)vj(p') M Y1M

E l

Here M, is the reduced mass given by Mr

Ma

Mn

while cr is the Pauli spin vector for the neutron and L is the relative orbital angular momentum . The first term in (1) represents pure s-wave interaction . The second tem stands for p-wave interaction, both central and spin-orbit; for simplicity we hue taken the shape factors - to be the same for both these varieties . The reaction matrix for a given c.m. momentum k which satisfies the inte al equation , (pl Vjp')(p'jTjk) (plTik) = (pjVjk)+2M, d 3p (3) 9

f

k'+ic_p#2

is related to the scattering amplitude F(O, 0) by the normalization F(O, 0)

where ' 0)

= - 4,n

2

M,,(pjTjk)(P=k)9

F(O, 0) = f(0) + (er - n)g(0).

(4)

(5)

Here the non-spin-flip amplitude f(O), which is given by

j?1 (cos 0)[(I+ I)q +

(6)

± == eiô "sin ô,,, il

(7)

kf(O) = '

00

is sufficient to provide complete identification of our parameters in terms of the phase shifts . Thus after straightforward calculations we find (8)

10 = lro(I-P,)-"

-

+ = -r,

1

T I (t-2)[1 -iii(t-2)1-',

(9) (10)

LOW EMGY 49-N INTERACTION

319

AI J dpv,(p)(p2 -k 2 _ic)- 1

(11)

= cl+ir,

and T I (k)

W kAIV12(k) ,

(12) I = 0, 1 .

(13)

It is easy to see that the Unitarity condition is explicitly satisfied by (8)-(10). Eqs . (8~(10) lead finally to the effective range formul ae 2P - 1 k cot 6 0 = (2,r 2A O )- I (p2 +k 2)2 + jk

0

(14)

0

k 3 Cot 6+1 = [27t 2A I (I +t)]-1(fl2I+k2)2 _ 1# 1 (#21+3k 2), k 3 Cot 6-1 = [2n2AI(t -2)]- 1 (fl2 +k 2)2 _ 1# 1 (fl2 + 3k2),

where we

,ve chosen Vo (p ) = V ( p) I

(15) (16)

(p2 , + .0 2) - 1 , 0

(17)

= p(fl2 +p 2 ) - 1 . I

(18)

As eqs. (14-16) show, the fbims (17) and (18) are obviously compatible with the requirements of correct threshold behaviour of the various phase shifts. To fix the nwnerical values of the various parameters A 0 , A,, fl o , # I we proceed as follows. Since the Gammel-Thaler 7) fits to the observed phase shifts are extremely good, we could directly make use of the curves and tables of ref. 7) . Thus we find that the s-wave phase shift given in fig. (2) of ref. '7 ) is well-represented by the formula 6 0 ;k - 20(Elab)l _" - 0.46 k,

1

(19)

where EIA is expressed in MeV and k, the c.m. momentum is expressed in units of the deuteron binding parameter 2(,22/M = 2 .25 MeV). Eq. (19) leads to the developMent (20) k cot 6 0 ~z: - 2.174 + 0.153 V + 0.00214 0. A comparison of the coefficients in eq. (14) with those in (20) gives the "best values" of 0 0 , ).,0 a s 3 (21) 2 7r 2 0 8#01 fi 0

= 5.52.

(22)

Eq. (21) has been obtained from the condition that the constant term in eq. (14) should be a minimum - a requirement necessitated by the large and negative value Of the constant term. in eq . (20). The values given by (21) and (22), represent quite well the first two coelincients in (20), but give a rather small value for the ceefficient of k 4 . However, up to about 20 MeV the s-phase shifts are well represented by eqs, (2 1) and (22). I

j'~ .

320

N.

Mllr",et

dl-

ofcourse impli anattractive It maybe noted from eq. (21) ihat A 0 is positive. This s-wave interaction and therefore a bound state of He"', though of course such a a: state is not observed . This paradox is usually understood in tems of 5-b0dY problern where the Pauli exclusion principle rules out such a configuration . As a tWO-body problem, however, this difficulty is unavoidable since, as has also been noted by other . authors ', 1) it is only with attractive s-wave interaction that the observed s phase shifts can be properly fitted. To fix the p-wave parameters t, #I and A I of our representation, we note that acco rd. lyses or ing to the inverted doublet scheme (which is well substantiated by the ana various authors in terms of resonance levels and widths 11)) the initial rise with energy of 6 1+ is much faster than that of 6TI . This means that cot b,' is appreciably smaller than cot 6 In terms of eqs. (15) and (16), this implies that the "effective strength should be larger than the cor. parameter" A, (I+ t) for interaction in the state j responding parameter A, (t-2) in the case of j = 1 . Thus in our model, t must be. positive . A detailed numerical analysis of eqs. (15) and (16) with respect to the observed phase shifts gives the following "best values" : 2jr2Aj (l + t) = 7.87, 2n2AJ(l - 2) = 6.80, #I

(23) (24)

= 4.2oc.

(25)

A comparison of the calculated and observed phase shifts which is given in table 1 . shows that the agreement is quite good . TABLE I

Comparison of the calculated and observed p-phase shifts in m-n scattering Energy in MeV Calculated 6-1 Observed 61 Calculated 6.L2 Observed .3-1

1 .25 26' 21' 4.2' 4.0'

2.5

5.0

86' 900

109 .51, 112.0'

13' 13'

35.5" 35.0*

7.5 109 0 111 0 53.0' 52.0'

10

15

106' 108 0

98.5 100"

62.5" 64.0*

The value of t is found from (23) and (24) to be t z, 22.0.

(26)

Such a large and positive t means that the central part of the (p-wave) interaction is attractive and enormously larger than the spin-orbit part, since the eigenvalues of L - tr are confined to - 2 and 41 only. This result, obtained with separable potent "als, is quite in harmony with the zorresponding findings by other authors 2,7 ) who had used ordinary static potentials .

LOW ENERGY M-N INTERACTION

321

3. Conclusion

We have found a very simple parametrization of a-n interaction regarded as a

two.body problem. The parameters fit both s- and p- phase shifts quite well . The main draw.back of this parametrization is of course that it fails to rule out a bound He-5 . a feature it shares with similar approaches made by other authors. Perhaps it may not be difficult il, practice to identify the effects of such spurious levels in a nuclear s ctroscopic calculation and subsequently to drop suchlevels out offurther consideration. it should be emphasized however, that a representation as above, lends itself rather easily to physical calculations in more complicated systems in which an alpha particle is involved as an entity. Thus m-d scattering, or energy levels in He' which can be regarded as approximately 3-body systems for low excitation energies, are ideally suited for calculations with such potentials 1). Calculation of the low-lying energy levels of He' on the lines of ref. 6) using the above model of a-n interaction, is now in progress . We are indebted to Professor R. C. Majumdar for his interest in this work. References 1) P. G. Burke, Nuclear Forces and the Few Nucleon Problem (Pergamon Press, London, 1960) Vol. 11; p. 413. 2) E. Van der Spuy, Nuclear Physics 1 (1956) 381 3) A. Herzenberg and E. J. Squires, Nuclear Physics 19 (1960) 280 4) A. Sugic el aL, Proc. Phys . Soc. A 70 (1957) 1 5) See, e.g., A. N. Mitra. and V. L. Narasimham, Nuclear Physics 14 (1960) 407 6) See, e.g. A. N. Mitra, Nuclear Physics 32 (1962) 529 7) J. Gammet and R. Thaler, Phys. Rev. 109 (1958) 2041

8) 9) 10) 11)

J. Gammel et al., Phys. Rev. 119 (1960) 267 A. N.forMitra and J. H. Naqvi, Nuclear Physics 25 (1961) 307 See, notation, J. Hamilton, Theory of Elementary Particles (Oxford Univ. Press, 1960) p. 385 See, e.g. P. D. Miller and G. C. Phillips, Phys. Rev. 112 (1958) 2043