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The interaction of two alpha-particles
Nuclear Physics 7 (1958) 397--410; (~) North-Holland Publishing Co., Amsterdam Not
to be reproduced by photoprint or microfilm without written permission from the publisher
T H E I N T E R A G T I O N OF T W O A L P H A - P A R T I C L E S E. V A N D E R S P U Y and H. J. P I E N A A R
Department o] Physics, University, Stellenbosch Received 27 March 1958 A b s t r a c t : I t is investigated w h e t h e r the phase shift d a t a for a l p h a - a l p h a scattering up to b o m b a r d m e n t energies ~ 6 MeV can be represented b y a velocity-independent alphaa l p h a interaction potential, or a t w o - b o d y interaction. I t is found on the basis of square well potentials a n d of a general purely phenomenological analysis, which is here developed for interactions w i t h an infinite repulsive core, t h a t this is n o t possible even for this low energy domain. The results indicate, however, t h a t an outer interaction radius 4 × 10 -la cm w i t h a s t r o n g repulsive core of radius ~ 1.8 × 10 -13 cm is required. These dimensions are in line w i t h f u n d a m e n t a l g r o u p resonance calculations of the alphaa l p h a interaction s t a r t i n g from nucleon-nucleon forces, which also indicated t h a t the repulsive core of the interaction is velocity-dependent. This velocity-dependence may, for example, be phenomenologically indicated b y a lower effective repulsive core radius for higher I.
1. I n t r o d u c t i o n
The scattering of ~-particles on He 4 has recently been experimentally investigated with improved accuracy over an extended bombardment energy range, E ' = 0 to 22.9 MeV 1, 2, 3), and the results could be fitted with a unique set of phases for the S, D, F, G waves. It is therefore appropriate to investigate the phenomenological ~--~ interaction potential that will represent these results. The ~-particle is a saturated and tightly bound system. One might hence expect, in the collision of two such particles at sufficiently low energies, that the two interacting groups will act as two invariable, "rigid" units. For such a situation one would expect that polarization of the ~-particles, and exchange of its constituents would play no major role. At higher energies of collision the particles would interpenetrate more, and hence polarize each other, and call into play repulsive forces which will counteract the overlap of these two saturated groups, as is probably demanded b y the exclusion principle. One should, however, note that the exclusion principle does not necessarily demand repulsive forces in all cases of overlap of identical spin-½ particles, but rather that the overall wave-function for groups of such particles should be appropriately antisymmetrized. This antisymmetrization suffices in some cases 4) to secure the vanishing of the total wavefunction without the explicit intermediary of a repulsive force. 397
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At sufficiently low energies one would therefore e x p e c t t h a t two such e-particles would i n t e r a c t according to a v e l o c i t y - i n d e p e n d e n t t w o - b o d y i n t e r a c t i o n potential. The velocity-independence would include independence of l, the orbital angular m o m e n t u m q u a n t u m number. I t is the purpose of the investigation r e p o r t e d in this p a p e r to examine w h e t h e r the e x p e r i m e n t a l results p e r m i t of the l a t t e r i n t e r p r e t a t i o n at all, and w h a t features of the interaction are phenomenologically indicated. A sufficiently low e n e r g y of collision in the a b o v e sense is p r o b a b l y one which is, say, an order smaller t h a n the high e x c i t a t i o n e n e r g y of an ~-particle ( ~ 2 0 - - 3 0 MeV). Since a b o m b a r d m e n t e n e r g y of E ' = 6 MeV (centre of mass e n e r g y E = 3 MeV) would seem to be a m p l y allowed on this score, it seems n a t u r a l to limit this investigation to the results of H e y d e n b u r g and T e m m e r 1) and Russell et al. 2). These results go up to E ' = 5.63 MeV and only involve S and D phases. The n e x t results 3) start at 12.3 MeV, a n d are only used here to check the range of application of some of the potentials. Nevertheless, it should be m e n t i o n e d t h a t the phase results from the different sources 1,2, 5) all fall on one s m o o t h continuous graph for a n y value of l. T h e analysis 3) of the S wave phases up to E ' = 5.63 MeV indicates t h a t these phases m a y be described b y 3 parameters. T h e S wave phases m a y therefore d e t e r m i n e 3 p a r a m e t e r s of the S wave i n t e r a c t i o n unambiguously. The D wave phases will d e t e r m i n e at least two f u r t h e r p a r a m e t e r s of a veloci t y - i n d e p e n d e n t interaction. If the l a t t e r is p e r m i t t e d , one m a y get not only the o u t e r i n t e r a c t i o n radius, core radius, d e p t h of the a t t r a c t i v e well, b u t also two shape-defining p a r a m e t e r s . T h e discussion of the e x p e r i m e n t a l results ~), as well as the theoretical discussion of H e r z e n b e r g 6), indicate an o u t e r radius of the ~ - - ~ nuclear i n t e r a c t i o n r N ~ 5/ (we express distance r in t e r m s of / = 10 -13 cm). T h e r e fore, in the phenomenological analysis, the logarithmic d e r i v a t i v e of the w a v e f u n c t i o n was calculated from the phases using Coulomb-functions 7), at assumed values of r N = 4, 4.5, 5[. R e a c t i o n data, b u t not clearly the scattering data, show an S wave resonance at 94.5 keV, at which e n e r g y it is permissible to t a k e the phase shift doN = 90 ° 1). Unless otherwise s t a t e d all potentials were selected to give 50N = 90 ° at E = 91.5 keV. It does not m a k e m u c h difference to the i n t e r a c t i o n w h e t h e r this point is c o u n t e d in or not, b u t t h a t actual i n t e r a c t i o n p o t e n t i a l m a y have a relatively large effect on the resonance energy. In section 2 it is investigated which square well p a r a m e t e r s give the best fit to the S and D wave results. This indicates the features of the interaction schematically. In section 3 we develop a general analysis for finding i n t e r a c t i o n param e t e r s from p a r a m e t e r s describing the calculated logarithmic derivatives at r N, for the case of a nuclear i n t e r a c t i o n with an infinite repulsive core.
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This analysis is a logical extension of a previously published 4) analysis relating phase parameters to interaction parameters. W i t h a repulsive core, the analysis can be put in a simple form, only needing sinusoidal radial wavefunctions, even for l > 0. The general analysis is applied in section 4 to the S wave d a t a to find a 3 or 4 parameter interaction giving a fit. These interactions are then tested for the D wave case. The general technique is finally applied to the x - - x interaction deduced b y Herzenberg s) from f u n d a m e n t a l considerations. In all cases for r < rN, the Coulomb-potential is understood as being added to the nuclear potentials.
2. Square Well Analysis If one starts from the SchrSdinger equation for two identical, spin-less, charged x-particles with velocity-independent interaction, one obtains in the usual way, after elimination of the angular dependence, the following equation for the radial part of the wave-function R~(r)/r:
d~Rz(r)
dr-----T - +
{
l(l+l) }
k s - V(r)
r2
R~(r) ---- 0,
(1)
where only even values of l need be considered; k S = 2#E/h 2, where = {m~ denotes the reduced mass of the two x-particles; and
U(r) : Ve(r)+UN(r), 2/~2r for r > r N Ue(r) : 0 forr
{S/ze 0
UN(r) :
for r > r N
2#VN(r)/?~~ for r < rN,
where VN(r) is the nuclear interaction potential. Now for the square well nuclear interaction of a sufficiently general t y p e for the present purposes one restricts V~(r) as follows:
VN(r) ---- + Go for r <_ rl, ---- - - V
for r 1 < r < rN, ~ 0
for r > r N,
(2)
which allows for an infinite repulsive core and has three parameters (rl, r N V) ; r 1 : 0 was always included as one possibility. The attractive outside well is d e m a n d e d b y the presence of a very low energy S wave resonance, the absence of b o u n d states, and the boN phase, which goes up to 180 °, at low energies, and then decreases from 180 ° and becomes negative at E ' ~ 19 MeV; as well as b y the positive D phases. Recognizing the possibility of velocitydependence of V Nat high energies, the fact t h a t SoNbecomes negative indicates a strong repulsive core at least at high energies. Such qualitative features have been noted b y Russell et al. ~).
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The d e r i v a t i o n of Rz(r ) for the inside domain, r < r N, follows in the wellk n o w n w a y from (1), (2), a n d one i m m e d i a t e l y deduces for the inside solution the logarithmic d e r i v a t i v e d R , (r) / = I r dr / R~(r) } ~= rN
/o = KrN cot K(rN--rl) , /2 =
xN~
-2+
(3)
3_1)
XN 2
with 1
cot ( K r l + ¢ ) - -
Krl
Krl/3,
= KrN,
= ks+
Now f o r b o t h S a n d D one can select a set (rN, r1, V) to give the best fit to the values for the outside logarithmic derivative as calculated in the usual w a y from the observed phases at the selected values of r N = 4, 4.5, 5/. The results were as follows: (a) F o r the S wave, the best c o m p r o m i s e fit for r N -= 4 / i s near r 1 = 1.7/, V = 7.2 MeV, if one includes the 94.5 keV resonance point; a n d near r 1 = 1.8/, V = 7.9 MeV, if one ignores the 94.5 keV resonance point. The corresponding p a r a m e t e r s for r N = 5/, are rl = 0.7/, V = 2.4 MeV (including resonance point); r I =: 0.8/, V ~ 2.4 MeV (not including resonance point). There is, if a n y t h i n g , a s o m e w h a t b e t t e r fit w h e n the curve is n o t m a d e to pass t h r o u g h the exact resonance point, a n d for rx = 4/. (b) F o r the D wave, the best c o m p r o m i s e fit for rN = 4/, is p r o b a b l y reached for r 1 = 1.S/, V = 10.5 MeV. L o w e r radii rl for lower V values are possible if one i m p r o v e s the fit at low energies at the expense of t h a t in middle e n e r g y range• F o r r N = 5 / t h e fit is generally m u c h worse; r~ = 0 to / a n d V--- 3.9 MeV give a good fit in the middle e n e r g y range (here E' = 4 to 5 MeV) ; b u t the well is too shallow at the higher e n e r g y end (here E' = 5.63 Me\ T) b y m 0.1 MeV, and m u c h too deep at low energies (here E' ~ 3 MeV, where 62x first deviates from zero). (c) Since we f o u n d a b e t t e r fit for r N = 4 / f o r b o t h S a n d D waves, the value r N ~-- 6 / w a s not tried• The conclusion was checked b y t a k i n g r N = 4.5/. Actually, the fit at r N = 4 / a l s o e x t r a p o l a t e s more satisfactorily to energies as high as E ' = 12.3 MeV. Hence one deduces, for b o t h the S and D wave, t h a t the best fit requires r N as low as 4/, a n d d e m a n d s a repulsive core of considerable radius: rl ---- 1.8/, b u t indicates V s = 7.9 MeV, V D ~ 10.5 MeV. The d i s p a r i t y between the latter two will increase for lower r N on a c c o u n t of the centrifugal potential. The investigation thus indicates for this square well t y p e t h a t even for
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E ' < 6 MeV one has a velocity-dependent interaction with a large radius core (r 1 = 1.8/). F r o m the f u n d a m e n t a l discussion of Herzenberg e) it appears t h a t the two aspects: repulsive core and velocity-dependence, can be m a d e one, because his calculations show t h a t the velocity-dependence resides in the repulsive core, while the outside a t t r a c t i o n is m a i n l y velocityindependent. If one m a y schematize the velocity-dependence of the core as a p u r e / - d e p e n d e n c e of rl, the outside a t t r a c t i o n V being common to all l, then, for example, interpolation indicates t h a t at r N ---- 4/, for rls = 2.0[, rXD = 0.8/; V ---- 9.5 MeV a reasonable fit is obtained. It is outside the scope of the present investigation to examine velocity-dependence, which a n y w a y can only be done most arbitrarily w i t h o u t more precise f u n d a m e n t a l indications, b u t if the correct t y p e of velocity-dependence is t h a t with V common to all l, a n d r~ velocity-dependent, then the above potential will probably give v e r y nearly the best compromise fit. Continuing the search for a velocity-independent potential, one notices t h a t the best solution given above has the D attractive well deeper t h a n the S well. On account of the centrifugal potential the D wave wants considerable attractive potentials at large radii. One would therefore t h i n k t h a t a compromise could be reached b y means of a stepwise variation of the potential: Vs(r ) - - o o
for r _ < r l ,
=--V1
for r l < r < r
2
-~--V~ for r 2 < r < r
N
--
0
for r > r N,
with (--V1) > (--V2). The domain r < r~ would then have relatively less effect on the D wave t h a n on the S wave; V2 can perhaps be selected large enough to give a sufficiently large 62N, while (--V1) would still be shallow enough to give the right b0N. This potential is m u c h more a r b i t r a r y (for one thing it has 4 free parameters) and was hence only investigated in one example. Take r N ---- 4/, r I = 1.8/ and V~ ---- 10.5 MeV, to agree with the best D well above; further, V1 ---- 5 MeV (note Vs = 7.9 m ½(10.5+5)), and select r 2 to give an S wave fit: one finds r~ ---- 3.3/. However, this well does not fit the D wave d a t a at all. It is too shallow, or the filling up for rl < r < r 2 still has a considerable D wave effect. One notes, however, t h a t this device brings the best fitting S well somewhat nearer fitting the D wave d a t a t h a n for the square well w i t h o u t step, b u t not so m u c h as to make this a promising t y p e of compromise. W i t h reference to the types of interaction potentials discussed b y Haefner s) and b y Herzenberg ~), and which we shall consider in section 4, one should note here t h a t the t y p e of change from the pure square well to the above step well works in the same sense as the change to the Haefner well, b u t in the opposite direction to the change to Herzenberg's interaction.
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3. G e n e r a l A n a l y s i s
A former analysis 4) relating phase parameters to potential parameters is extended bere to the case of an interaction with an infinitely repulsive core. F o r convenience one writes (1) in a different form: d2
dr---2 Rz(r)+{k2--U,(r)--Uec(r)}Rz(r)
-= O,
(4)
where
u.o(r) = l
Ul(r) =
t| l ( l + l ) ~ / ;
8/~e~ + ~---;
for r >
rN,
for r <
rN;
0 for r > r s, 2/~VN (r) l(l+ 1) 2#Vs, (r) h2 + ~ r -?~2 = UN~(r) for r 1 < r < rN, Vc:
+oo
for r < r l ;
here r 1 is the radius of the repulsive core, and UN,(r) ~ 0 outside the domain
rl < r < r~. In the usual w a y the b o u n d a r y conditions for Rz(r ) are 1
R~(r) , - . - ~ sin (kr--ll~+~zc+b~N),
R~(r <_ rl) : O,
where ~, is expressed as the sum of 6, s, a n d the usual Coulomb phase shift ~ze ---- --~7 log 2kr + arg P(l+l+i~?), with
~7 ~ 4#e2/hZk. In the above case the outside solution (r > r~) is 1 R~ (r) = - - {cos ~ Nl
F , (r) + sin ~lNG~(r)},
where F~ (r), G~(r) are the regular a n d irregular Coulomb-functions as defined b y Breit et al. 10); the normalization c o n s t a n t Nz is now chosen so t h a t Rz(rs) ---- 1: N z ---- cos ~NF~(rN) + sin ~G~(rN). Introduce now the auxiliary function R',(r) defined as follows: d2
~r 2 R'z(r)+{k2--U'ce(r)}R',(r)
= O,
where
U'ec(r) =
Ve = + o o for r < r 1 0 for r 1 < r < rN, l ( l + 1) 8/ze~ for r > r N.
r~
+ ~Wr
(5)
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The normalization of R'~(r) is effected in the same w a y to that for R,(r), except for the substitution of 60 for ~N above. It follows in the usual way that the Wronskian R'z(r)
dR,(r) dr
dR',(r) Rz(r)
dr
--const.
for r > r
N.
(6)
In general from (4) and (5),
{R'z (r) dRz(r) -- dr
Rz(r)
dR',(r)t~° = dr ]0
Therefore 1
(1 _1o) • -rN- =
f
o
Um(r)Rz(r)R'~(r)dr.
,, UNz(r)Rz(r)R',(r) dr,
(7)
using the definition of Um(r ) and the boundary conditions for Rz(r) and R'z(r), and eq. (6); [z,/0 are the logarithmic derivatives of R~(r), R'z(r ) at
r~rN. Now R', (r) clearly is only required for the domain r 1 < r < rN, when, with the boundary conditions,
R',(r) = sin k(r--rl) cosec k(rN--rl).
(8)
Similarly p m a y be found b y continuity at r = r N from (8):
]o = kr N cot k(rN--rl).
(9)
Eq. (7) is clearly an integral equation for
G'z(r) = Um(r)R,(r),
(10)
in terms of the experimental function [z (for assumed values of the parameters rl, rN) and the analysis can now proceed substantially as detailed in a previous paper 4); it will therefore only be briefly sketched. The only differences are that now [,, rather than the phase function k 2z+1 cot ~zN of the previous analysis, is the experimental function; that one now has a repulsive core; and that the analysis need now only use the simple l = 0 type radial functions for all l: this effects a great simplification in practice. The analysis, starting from ]z but without core, follow analogous lines and will not be given. The final form of the analysis below apply also if one has no Coulomb forces for r > r N. Since G',(r) = 0 for r ---- r 1 and r = rN, the solution of the integral equation in (7) m a y be obtained b y expanding in terms of orthogonal functions:
G'~(r) = ~ A~ sin kn(r--rl) ,
(11)
n=l
where k~ = nk 1 is defined by kl(rN--rl) = ~. The integer n denotes in fact the nth harmonic of the representation. A~ depends of course on l, but the
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dependence has no formal significance. Hence eq. (7) reduces to { / z _ k r N c o t k(rN--rl)}'---1 = ~ (--1) n+~ k n A s rN , =1 (ks~--k ~) Now in the domain r l < r < r
(12)
N eq. (4) reads
d2 dr 2 R t ( r ) + k ~ R~(r) = U m ( r ) R z ( r ) = G',(r) =
A n sin kn(r--rl). n=l
Thus, on account of the boundary conditions, _
R~(r)
sin k ( r - - r l ) sin k(rN
~ 2_
+ s
=
1
As (k~--kn 2) for
sin kn(r--rl), r~
N.
With (10) and (11) this leads to UN~ (r)
sin k (r--r1) ~ An I + 2, sin ks(r--r1) ) sin k ( r N - - r l ) ~=1 (k2--k~ 2) =
~ A s sin ks(r--r1).
(13)
•=1
Eq. (13) clearly shows t h a t the A n, functions of k 2 or the energy, are "interaction parameters" defining U m (r). If one k n o w s / , as a function of energy, the parameters defining the latter are related to the interaction parameters by (12), and hence to the interaction by (13). Again, (12) and (13) m a y be used to calculate/, if Urn(r) is known. The practical way of effecting these connections is to expand both sides of equations (12) and (13), as one may ~), in power series in k 2 oc E, and to equate coefficients of the same powers. The equations thus obtained do not contain k 2 and hence probably apply over a larger energy range than t h a t determined by the radius of convergence of the relevant expansions. To illustrate the procedure: As --= ]~12
C~o+C,np2+ C~2p~+ . . . .
with p = k/k s.
(14)
Experimentally one has cot 6tN and using Coulomb functions here one may calculate /l for any chosen r N. This m a y be fitted with a polynomial in powers of E' /~ = c¢+flE'-t-TE'2+ . . . . (15) Since k 2 = 2#E/'h 2 = / , E ' / h 2, p2 = #E,/f~2 k2 = / ~ (r N_r~)2 E,/h2 ~2; and /~ = a + ~ 2 bp2+~ a cp4+ . . . . where b = h2fl/tz(rN--rl) 2,
c
=
]i4y/#2(rN--rl)4
.....
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The a, b, c are thus experimental parameters determining the logarithmic derivatives for a given set (rl, rN), from the parameters ~,/3, ~ . . . . which are generally applicable for a given r N. Eq. (13) now yields a number of equations, of which the following comes from the lowest order term in the expansion in powers of k 2 (it m a y be directly deduced from (13) by letting k - + 0):
UNl(r) l ( r _ r l I
oo C
~ ~n°sin
- -
kl 2
t\rN~rl!
n~l
} k~(r--rl) :
n 2
~ Cno sin k n(r--rl).
(17)
n=l
Note that the function in the curly brackets is the zero-energy form of Rz (r). This equation read at a sufficient number of r values in the domain r 1 < r < rN, yields the C~0 in terms of Um(r)/kl ~ or vice versa. Having Cno, the next order in the expansion of (13) yields an equation for the C,1 and so on. All such equations shall be collectively indicated by (17). Expanding (12) and using certain identities between C,o, C,~, etc. derived from eqs. (17), one readily arrives at
__1{l_a(rN_rl)/rN ) = ~ 7/7
(_1) C~0
n=l
=
n
__X n
(-
+ -"
1 oo
2
.2C,,o
~a(--1/ 45--c (rN--rl) /rN} : ~= l ( (--1) --~-
--
,
Os)
n 4
,,C,,1
~ C~o
- ~ - - I J"
One notices that (18) relates the experimental parameters a, b, c to the interaction parameters C,,~. If the experiments can be described by only 2 parameters, the first two equations m a y suffice, in which case we only need calculate the C~0. If the next order coefficient is important, or has to be checked, the Cni must be calculated from the next order equation of (17) above. Note that (17) only uses sinusoidal and power functions of (r--r1). It is clear from (17) that on the left hand side the effect of the higher harmonics is reduced in the term in the curly brackets which means that this term is usually a smooth function of (r--r~). Hence C~0 will only be appreciable for higher n, if U m (r) has sharp or rapid changes with (r--r1). Since in all the equations (18), the higher harmonics are relatively weighed down, one need, except for very rapidly changing potentials, only determine the lower harmonics to any accuracy. One further has a natural way of selecting a number of potential parameters: namely to use just the lower harmonics or C,o. The only arbitrariness involved herein is the perhaps natural one of the consequent limitation of the investigation to smooth potentials which do not show rapid fluctuations. The advantage of introducing r 1 explicitly is that all l m a y be dealt with by one set of simple functions and that the
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infinite repulsive core need not be represented by the C,0--which would have called for m a n y higher harmonics.
4. Application of the General Analysis The q u a n t i t y / ~ is determined from the phases for r N = 4 and 5/ (found useful in the square well analysis) for both S and D waves. These functions are then fitted by least squares with polynomials as in (15), using the least number of parameters demanded to fit the results within the experimental errors, and letting the S curve pass through the 94.5 keV resonance. This yields 3 parameters for the S wave, and at least 2 more for the D wave. The former number agrees with that calculated by Russell et al. 2), using another phase function. One therefore has 4 to 5 parameters altogether if a velocityindependent potential is possible; because one must recognize that one of the parameters for the S wave, y, has relatively little effect and m a y perhaps not be insisted on. The general analysis is first of all applied to the S wave to find, for a given (rN, rx) combination, the Ci0, C20 from (18) which will give the correct ~, fl of (15), (16). This gives 4 parameters (rN, r 1, Clo, C2o), or 3 when one takes the case r I = 0, which fixes a potential by the first of eqs. (17). The C,1 follow for the latter potential from the second of the eqs. (17). The y corresponding to this potential m a y then be calculated by (16), (18), and hence the corresponding [0 compared with the experimental/0. One m a y then t r y other sets of r N, r 1 (and C10 and C2o) till the ~ is equal to that required by the experimental fit, or if this is not accurately possible, till the [o calculated from the potential, found in this way, agrees as nearly as can be with the experimental data, over the chosen energy range. The above procedure always ensures a fit at low energies for a start. Potentials which are found in this way to fit the S wave data well, are then tested for their fit of the D wave data. In this way, the 4 to 5 parameters from the S and D phases m a y suffice to determine the 4 parameters of the potential uniquely, if the latter can be velocity-independent at all. There is little reason to start with 5 interaction parameters (adding C30, say), since one of the experimental parameters (y from/0) gives a small contribution in the chosen energy range, and it is doubtful whether it m a y be insisted on. In any case, C3o for the class of smooth, non-rapidly varying potentials will contribute little to/o, but would tend to make the analysis prohibitively cumbersome for a desk calculating machine. Since the infinite repulsive core is explicitly introduced (and need hence not be represented by the harmonics), and since one knows from the phases that R o(r), R 2(r) have only part of a loop over the chosen low energy range, two harmonics can permit of a U m (r) with one sign reversal in the domain r 1 < r < r N (see eq. (17)), which should suffice to cover as general a class
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of potentials as one can perhaps expect from the n a t u r e of the problem in hand. One finds then: (a) Neither r N ----- 4/, nor r N = 5/, with r 1 ----- 0, allows of a n y real solutions for Cxo, C2o. This does not m e a n t h a t there are no solutions possible if one were to t a k e sufficiently strong higher harmonics to describe the sharply changing potentials, with large values for low r, which seem to be d e m a n d e d b y the results given below for rl > 0. In this sense this finding is in line with the requirement of a strong repulsive core. 0 (b) W i t h r N = 5 / n o value for r I > 0 permits of a real solution for C10 , C20, or a smooth non-steep potential. (c) W i t h r N = 4/, the values r I : 1.2, 1.4, 1.8/ all permit of solutions; 2 each, in fact. The one solution could be selected as being of the same order of general depth as for the square well potentials; the other solution corresponds to m u c h deeper wells a n d hence to a bound state. The two solutions come from the quadratic n a t u r e of (18) for 2 harmonics C~0. W i t h this selection the complete interaction potential curve can be drawn and the corresponding ]0 calculated. As higher values of r a are t a k e n in the above set the fit becomes slightly better. Together with the D wave findings below, r 1 ~ 1.8] is perhaps the best compromise for the S well. Comparing the corresponding wells in the order r a = 1.2, 1.4, 1.8/, the first two show, for r just greater t h a n rl, a repulsive part which is strongest for r 1 ---- 1..2/. The effect of this is to indicate clearly the need for a bigger repulsive core radius, which supports the finding t h a t r~ ~ 1.8[ is probably the best compromise. F u r t h e r m o r e for r~ = 1 . 8 / t h e potential (for r 1 < r < rN) is purely negative, and up to radii r ~ 3], resembles the square well of best fit in being nearly of c o n s t a n t depth, a n d of its order of depth. For larger r the potential rises gradually to 0 at rE = 4/. (d) Testing these three potentials for the D wave one finds t h a t none nearly represents the results, the larger r 1 being further from a fit. E v e n potentials of this kind with lower rl do not come close to a D wave fit. I t is for this reason t h a t in (c) above, the compromise S wave fit is t a k e n as corresponding to r~ ~ 1.8/. The lack of D wave fit is not unexpected from the square well results (especially in view of the potential for r = 1.8] being so similar to the square well); a n d as an estimate from the latter, one would say t h a t the wells required b y the S wave results are too shallow for the D wave phases b y ~ 2.7 MeV. These results indicate, more generally, t h a t a velocity-independent potential fitting b o t h S a n d D results is not possible, even for the low energy domain. For the S waves, the results confirm the demand, for an infinite repulsive core of the considerable radius r 1 ~ 1.8]. T h e y also indicate r N m 4]. It is outside the scope of the present analysis to determine the na-
408
E. V A N
DER
SPUY
AND
H.
J. P I E N A A R
ture of the velocity dependence, but it is nevertheless clear that if the outside attractive well is to have the same shape up to the core, the velocity dependence could consist in r: depending on l, being in fact smaller for the D wave than for the S wave (for which, say, r: = 1.8/). The order of the compromise core dimension is near the suggestion of Herzenberg 6), rl = 2 to 3/. The latter analysis in any case shows from fundamental arguments that the core interaction is repulsive and velocity-dependent. Since it is possible to test any interaction with a core and an effectively finite range rN, the analysis was used to test the potential given in the paper b y Herzenberg 6), with r 1 = 2.5/. Using this potential for the S wave, the/0 was nowhere near a fit, the attractive well being, effectively speaking, too shallow, or alternatively the core radius being too large, to a considerable degree. Although no calculations were made, this m a y also be said with great probability of /2. Since the Herzenberg potential s) is uncertain in actual detail and velocity dependence below r = 2.5/, this analysis cannot usefully be pursued further at this stage. The fact is that if one permits of velocity-dependent effects a purely phenomenological analysis becomes impossible or at least quite arbitrary, unless a fairly precise indication of the type of velocity-dependence is furnished from other considerations (for example, fundamental considerations). Even so the question m a y well arise of the possible use of such a phenomenological potential, apart from being a mere shorthand for scattering results (which m a y be done more simply with square wells). Indeed, the mere fact that, in spite of the considerations in the introduction, the interaction is velocity-dependent indicates that the problem is not precisely enough a two-body problem. It is hence natural to expect that a velocitydependent phenomenological potential deduced for the ~--x interaction would only apply strictly to it, and not to x - - x cluster interaction in the Be 9 and Cn problems, for example. The need for such care is further indicated b y the consideration that whereas the a - - n interaction 4) involves no sensible velocity-dependence (apart from the spin-orbit coupling, of course), the ~ - - a interaction does. There is another difficulty in the w a y of an application of velocity-dependent forces such as is likely here: namely, that the strong repulsive forces would require due allowance for cluster correlation effects in problems involving ,t-clusters, in line with the work of Brueckner :1). Since one would then have to deal with repulsive velocity-dependent forces within a region of large radius, involving in general a few clusters which are themselves compound particles, the use of a phenomenological potential looks very unpromising. However, if it is possible to formulate the velocity-dependence in a reasonably simple form (/-dependence of core radius, say), connecting it with nucleon-nucleon forces b y means of a group-resonance calculation, then the
THE
INTERACTION OF T~,VO ALPHA-PARTICLES
409
phenomenological potential could provide an avenue for testing these forces and the ~--~ interaction derived from such a calculation. It m a y then furnish information on the properly velocity-independent part of the ~--~ interaction, and indicate features of the velocity-dependence which m a y qualitatively elucidate the application of the ~-cluster model. With electronic computers a direct calculation of ~--~ scattering from nucleon-nucleon forces m a y be possible, using the group-resonance formalism. The equivalent velocity-dependent potential m a y then be calculated from this solution. This would be another way of arriving at the x--~ potential, and would presumably have the restricted use indicated above. The trend of the above phenomenological conclusions, arrived at by using square wells and more general smooth potentials, is confirmed by Nilson etal.'s 5) t fitting of the experimental data to the Haefner potential 8) q2 h~ VN(r) ---- 0
for
r > rN,
= --D+
2~tr 2
for r < rN,
where q~ = 30 for convenience. To fit the 94.5 keV resonance, D may be calculated as a function of r~. The rest of the data then provide r N, and hence D. Since these fits cover a wider energy range, the experimental points naturally have a somewhat wider scatter round the theoretical curves than was found in the present investigation over a more limited energy range. Nilson et al. 5) found with the Haefner potential (the subscripts S, D, G refer to the partial wave): (a) r~s = 3.49/ for E' up to 20 MeV. (b) r•D :- 4.44! for E' up to 6 MeV. No fit here could represent D wave phases at E' > 6 MeV as well, the well with r~D = 4.44! being too shallow for higher energy data. (c) rNG just smaller than 4.44/would represent the G wave data up to E' = 20 MeV, but rapidly becomes too shallow at larger energies up to E ' : 22.9 MeV. These results thus also indicate velocity-dependence, inasmuch as r~ is not common to all the l values, and as a fit at lower energies deteriorates at higher energies. Similar results have been presented in a different form by Humphrey 9): he used a modified Haefner potential and found a general fit up to E' = 22 MeV with r~ = 3.75/ and D :
21 MeV for the S wave,
D = 25 MeV for the D wave, D = 32 MeV for the G wave. ? The result of this w o r k became available w h e n the calculations of the p r e s e n t p a p e r were nearly completed; the a u t h o r s are indebted to Prof. J e n t s c h k e for c o m m u n i c a t i n g these results to t h e m before publication.
410
E. VAN" DER SPUY AND H. J. PIENAAR
The value of r N and the trend of D agrees with the general findings of the present paper. These modified Haefner potentials go through zero at a radius between 1 and 2], and rise rapidly for smaller r. This indicates a repulsive core with an effective radius comparable to that, r 1 ~ 1.8/, found in this paper, and this effective radius also becomes smaller for higher I. It should be noted that the Haefner potential, like the square well potential, is merely a convenient form for expressing repulsion between overlapping systems, which to some extent can fit the results, but has no theoretical foundation as yet. One of the authors (H. J. P.) wishes to acknowledge the benefit of a bursary of the South African Council for Scientific and Industrial Research during the present investigations. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll)
N. P. Heydenburg and G. M. Temmer, Phys. Rev. 104 (1956) 123 J. L. Russell, G. C. Phillips and C. W. Reich, Phys. Rev. 104 (1956) 135 Nilson, Kerman, Briggs and Jentschke, Phys. Rev. 104 (1956) 1673 E. v a n der Spuy, Nuclear Physics 1 (1956) 381 Nilson, Jentschke, Briggs, Kerman and Snyder, to be published A. Herzenberg, Nuclear Physics 3 (1957) 1 Bloch, Hull, Broyles, Bouricius, Freeman and Breit, Revs. Mod. Phys. 23 (1951) 147 R. R. Haefner, Revs. Mod. Phys. 23 (1951) 228 C. H. Humphrey, quoted by Nilson et al. 6) Breit, Wheeler and Yost, Phys. Rev. 49 (1936) 174 H. A. Bethe, Phys. Roy. 103 (1956) 1353
to be reproduced by photoprint or microfilm without written permission from the publisher
T H E I N T E R A G T I O N OF T W O A L P H A - P A R T I C L E S E. V A N D E R S P U Y and H. J. P I E N A A R
Department o] Physics, University, Stellenbosch Received 27 March 1958 A b s t r a c t : I t is investigated w h e t h e r the phase shift d a t a for a l p h a - a l p h a scattering up to b o m b a r d m e n t energies ~ 6 MeV can be represented b y a velocity-independent alphaa l p h a interaction potential, or a t w o - b o d y interaction. I t is found on the basis of square well potentials a n d of a general purely phenomenological analysis, which is here developed for interactions w i t h an infinite repulsive core, t h a t this is n o t possible even for this low energy domain. The results indicate, however, t h a t an outer interaction radius 4 × 10 -la cm w i t h a s t r o n g repulsive core of radius ~ 1.8 × 10 -13 cm is required. These dimensions are in line w i t h f u n d a m e n t a l g r o u p resonance calculations of the alphaa l p h a interaction s t a r t i n g from nucleon-nucleon forces, which also indicated t h a t the repulsive core of the interaction is velocity-dependent. This velocity-dependence may, for example, be phenomenologically indicated b y a lower effective repulsive core radius for higher I.
1. I n t r o d u c t i o n
The scattering of ~-particles on He 4 has recently been experimentally investigated with improved accuracy over an extended bombardment energy range, E ' = 0 to 22.9 MeV 1, 2, 3), and the results could be fitted with a unique set of phases for the S, D, F, G waves. It is therefore appropriate to investigate the phenomenological ~--~ interaction potential that will represent these results. The ~-particle is a saturated and tightly bound system. One might hence expect, in the collision of two such particles at sufficiently low energies, that the two interacting groups will act as two invariable, "rigid" units. For such a situation one would expect that polarization of the ~-particles, and exchange of its constituents would play no major role. At higher energies of collision the particles would interpenetrate more, and hence polarize each other, and call into play repulsive forces which will counteract the overlap of these two saturated groups, as is probably demanded b y the exclusion principle. One should, however, note that the exclusion principle does not necessarily demand repulsive forces in all cases of overlap of identical spin-½ particles, but rather that the overall wave-function for groups of such particles should be appropriately antisymmetrized. This antisymmetrization suffices in some cases 4) to secure the vanishing of the total wavefunction without the explicit intermediary of a repulsive force. 397
398
?.. V A N D E R
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PIENAAR
At sufficiently low energies one would therefore e x p e c t t h a t two such e-particles would i n t e r a c t according to a v e l o c i t y - i n d e p e n d e n t t w o - b o d y i n t e r a c t i o n potential. The velocity-independence would include independence of l, the orbital angular m o m e n t u m q u a n t u m number. I t is the purpose of the investigation r e p o r t e d in this p a p e r to examine w h e t h e r the e x p e r i m e n t a l results p e r m i t of the l a t t e r i n t e r p r e t a t i o n at all, and w h a t features of the interaction are phenomenologically indicated. A sufficiently low e n e r g y of collision in the a b o v e sense is p r o b a b l y one which is, say, an order smaller t h a n the high e x c i t a t i o n e n e r g y of an ~-particle ( ~ 2 0 - - 3 0 MeV). Since a b o m b a r d m e n t e n e r g y of E ' = 6 MeV (centre of mass e n e r g y E = 3 MeV) would seem to be a m p l y allowed on this score, it seems n a t u r a l to limit this investigation to the results of H e y d e n b u r g and T e m m e r 1) and Russell et al. 2). These results go up to E ' = 5.63 MeV and only involve S and D phases. The n e x t results 3) start at 12.3 MeV, a n d are only used here to check the range of application of some of the potentials. Nevertheless, it should be m e n t i o n e d t h a t the phase results from the different sources 1,2, 5) all fall on one s m o o t h continuous graph for a n y value of l. T h e analysis 3) of the S wave phases up to E ' = 5.63 MeV indicates t h a t these phases m a y be described b y 3 parameters. T h e S wave phases m a y therefore d e t e r m i n e 3 p a r a m e t e r s of the S wave i n t e r a c t i o n unambiguously. The D wave phases will d e t e r m i n e at least two f u r t h e r p a r a m e t e r s of a veloci t y - i n d e p e n d e n t interaction. If the l a t t e r is p e r m i t t e d , one m a y get not only the o u t e r i n t e r a c t i o n radius, core radius, d e p t h of the a t t r a c t i v e well, b u t also two shape-defining p a r a m e t e r s . T h e discussion of the e x p e r i m e n t a l results ~), as well as the theoretical discussion of H e r z e n b e r g 6), indicate an o u t e r radius of the ~ - - ~ nuclear i n t e r a c t i o n r N ~ 5/ (we express distance r in t e r m s of / = 10 -13 cm). T h e r e fore, in the phenomenological analysis, the logarithmic d e r i v a t i v e of the w a v e f u n c t i o n was calculated from the phases using Coulomb-functions 7), at assumed values of r N = 4, 4.5, 5[. R e a c t i o n data, b u t not clearly the scattering data, show an S wave resonance at 94.5 keV, at which e n e r g y it is permissible to t a k e the phase shift doN = 90 ° 1). Unless otherwise s t a t e d all potentials were selected to give 50N = 90 ° at E = 91.5 keV. It does not m a k e m u c h difference to the i n t e r a c t i o n w h e t h e r this point is c o u n t e d in or not, b u t t h a t actual i n t e r a c t i o n p o t e n t i a l m a y have a relatively large effect on the resonance energy. In section 2 it is investigated which square well p a r a m e t e r s give the best fit to the S and D wave results. This indicates the features of the interaction schematically. In section 3 we develop a general analysis for finding i n t e r a c t i o n param e t e r s from p a r a m e t e r s describing the calculated logarithmic derivatives at r N, for the case of a nuclear i n t e r a c t i o n with an infinite repulsive core.
THE
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399
ALPHA-PARTICLES
This analysis is a logical extension of a previously published 4) analysis relating phase parameters to interaction parameters. W i t h a repulsive core, the analysis can be put in a simple form, only needing sinusoidal radial wavefunctions, even for l > 0. The general analysis is applied in section 4 to the S wave d a t a to find a 3 or 4 parameter interaction giving a fit. These interactions are then tested for the D wave case. The general technique is finally applied to the x - - x interaction deduced b y Herzenberg s) from f u n d a m e n t a l considerations. In all cases for r < rN, the Coulomb-potential is understood as being added to the nuclear potentials.
2. Square Well Analysis If one starts from the SchrSdinger equation for two identical, spin-less, charged x-particles with velocity-independent interaction, one obtains in the usual way, after elimination of the angular dependence, the following equation for the radial part of the wave-function R~(r)/r:
d~Rz(r)
dr-----T - +
{
l(l+l) }
k s - V(r)
r2
R~(r) ---- 0,
(1)
where only even values of l need be considered; k S = 2#E/h 2, where = {m~ denotes the reduced mass of the two x-particles; and
U(r) : Ve(r)+UN(r), 2/~2r for r > r N Ue(r) : 0 forr
{S/ze 0
UN(r) :
for r > r N
2#VN(r)/?~~ for r < rN,
where VN(r) is the nuclear interaction potential. Now for the square well nuclear interaction of a sufficiently general t y p e for the present purposes one restricts V~(r) as follows:
VN(r) ---- + Go for r <_ rl, ---- - - V
for r 1 < r < rN, ~ 0
for r > r N,
(2)
which allows for an infinite repulsive core and has three parameters (rl, r N V) ; r 1 : 0 was always included as one possibility. The attractive outside well is d e m a n d e d b y the presence of a very low energy S wave resonance, the absence of b o u n d states, and the boN phase, which goes up to 180 °, at low energies, and then decreases from 180 ° and becomes negative at E ' ~ 19 MeV; as well as b y the positive D phases. Recognizing the possibility of velocitydependence of V Nat high energies, the fact t h a t SoNbecomes negative indicates a strong repulsive core at least at high energies. Such qualitative features have been noted b y Russell et al. ~).
400
E. VAN DER
SPUY
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H. I. PIENAAR
The d e r i v a t i o n of Rz(r ) for the inside domain, r < r N, follows in the wellk n o w n w a y from (1), (2), a n d one i m m e d i a t e l y deduces for the inside solution the logarithmic d e r i v a t i v e d R , (r) / = I r dr / R~(r) } ~= rN
/o = KrN cot K(rN--rl) , /2 =
xN~
-2+
(3)
3_1)
XN 2
with 1
cot ( K r l + ¢ ) - -
Krl
Krl/3,
= KrN,
= ks+
Now f o r b o t h S a n d D one can select a set (rN, r1, V) to give the best fit to the values for the outside logarithmic derivative as calculated in the usual w a y from the observed phases at the selected values of r N = 4, 4.5, 5/. The results were as follows: (a) F o r the S wave, the best c o m p r o m i s e fit for r N -= 4 / i s near r 1 = 1.7/, V = 7.2 MeV, if one includes the 94.5 keV resonance point; a n d near r 1 = 1.8/, V = 7.9 MeV, if one ignores the 94.5 keV resonance point. The corresponding p a r a m e t e r s for r N = 5/, are rl = 0.7/, V = 2.4 MeV (including resonance point); r I =: 0.8/, V ~ 2.4 MeV (not including resonance point). There is, if a n y t h i n g , a s o m e w h a t b e t t e r fit w h e n the curve is n o t m a d e to pass t h r o u g h the exact resonance point, a n d for rx = 4/. (b) F o r the D wave, the best c o m p r o m i s e fit for rN = 4/, is p r o b a b l y reached for r 1 = 1.S/, V = 10.5 MeV. L o w e r radii rl for lower V values are possible if one i m p r o v e s the fit at low energies at the expense of t h a t in middle e n e r g y range• F o r r N = 5 / t h e fit is generally m u c h worse; r~ = 0 to / a n d V--- 3.9 MeV give a good fit in the middle e n e r g y range (here E' = 4 to 5 MeV) ; b u t the well is too shallow at the higher e n e r g y end (here E' = 5.63 Me\ T) b y m 0.1 MeV, and m u c h too deep at low energies (here E' ~ 3 MeV, where 62x first deviates from zero). (c) Since we f o u n d a b e t t e r fit for r N = 4 / f o r b o t h S a n d D waves, the value r N ~-- 6 / w a s not tried• The conclusion was checked b y t a k i n g r N = 4.5/. Actually, the fit at r N = 4 / a l s o e x t r a p o l a t e s more satisfactorily to energies as high as E ' = 12.3 MeV. Hence one deduces, for b o t h the S and D wave, t h a t the best fit requires r N as low as 4/, a n d d e m a n d s a repulsive core of considerable radius: rl ---- 1.8/, b u t indicates V s = 7.9 MeV, V D ~ 10.5 MeV. The d i s p a r i t y between the latter two will increase for lower r N on a c c o u n t of the centrifugal potential. The investigation thus indicates for this square well t y p e t h a t even for
THE
INTERACTION
OF TWO
ALPHA-PARTICLES
401
E ' < 6 MeV one has a velocity-dependent interaction with a large radius core (r 1 = 1.8/). F r o m the f u n d a m e n t a l discussion of Herzenberg e) it appears t h a t the two aspects: repulsive core and velocity-dependence, can be m a d e one, because his calculations show t h a t the velocity-dependence resides in the repulsive core, while the outside a t t r a c t i o n is m a i n l y velocityindependent. If one m a y schematize the velocity-dependence of the core as a p u r e / - d e p e n d e n c e of rl, the outside a t t r a c t i o n V being common to all l, then, for example, interpolation indicates t h a t at r N ---- 4/, for rls = 2.0[, rXD = 0.8/; V ---- 9.5 MeV a reasonable fit is obtained. It is outside the scope of the present investigation to examine velocity-dependence, which a n y w a y can only be done most arbitrarily w i t h o u t more precise f u n d a m e n t a l indications, b u t if the correct t y p e of velocity-dependence is t h a t with V common to all l, a n d r~ velocity-dependent, then the above potential will probably give v e r y nearly the best compromise fit. Continuing the search for a velocity-independent potential, one notices t h a t the best solution given above has the D attractive well deeper t h a n the S well. On account of the centrifugal potential the D wave wants considerable attractive potentials at large radii. One would therefore t h i n k t h a t a compromise could be reached b y means of a stepwise variation of the potential: Vs(r ) - - o o
for r _ < r l ,
=--V1
for r l < r < r
2
-~--V~ for r 2 < r < r
N
--
0
for r > r N,
with (--V1) > (--V2). The domain r < r~ would then have relatively less effect on the D wave t h a n on the S wave; V2 can perhaps be selected large enough to give a sufficiently large 62N, while (--V1) would still be shallow enough to give the right b0N. This potential is m u c h more a r b i t r a r y (for one thing it has 4 free parameters) and was hence only investigated in one example. Take r N ---- 4/, r I = 1.8/ and V~ ---- 10.5 MeV, to agree with the best D well above; further, V1 ---- 5 MeV (note Vs = 7.9 m ½(10.5+5)), and select r 2 to give an S wave fit: one finds r~ ---- 3.3/. However, this well does not fit the D wave d a t a at all. It is too shallow, or the filling up for rl < r < r 2 still has a considerable D wave effect. One notes, however, t h a t this device brings the best fitting S well somewhat nearer fitting the D wave d a t a t h a n for the square well w i t h o u t step, b u t not so m u c h as to make this a promising t y p e of compromise. W i t h reference to the types of interaction potentials discussed b y Haefner s) and b y Herzenberg ~), and which we shall consider in section 4, one should note here t h a t the t y p e of change from the pure square well to the above step well works in the same sense as the change to the Haefner well, b u t in the opposite direction to the change to Herzenberg's interaction.
402
E. VAN D E R S P U Y AND H. J . P I E N A A R
3. G e n e r a l A n a l y s i s
A former analysis 4) relating phase parameters to potential parameters is extended bere to the case of an interaction with an infinitely repulsive core. F o r convenience one writes (1) in a different form: d2
dr---2 Rz(r)+{k2--U,(r)--Uec(r)}Rz(r)
-= O,
(4)
where
u.o(r) = l
Ul(r) =
t| l ( l + l ) ~ / ;
8/~e~ + ~---;
for r >
rN,
for r <
rN;
0 for r > r s, 2/~VN (r) l(l+ 1) 2#Vs, (r) h2 + ~ r -?~2 = UN~(r) for r 1 < r < rN, Vc:
+oo
for r < r l ;
here r 1 is the radius of the repulsive core, and UN,(r) ~ 0 outside the domain
rl < r < r~. In the usual w a y the b o u n d a r y conditions for Rz(r ) are 1
R~(r) , - . - ~ sin (kr--ll~+~zc+b~N),
R~(r <_ rl) : O,
where ~, is expressed as the sum of 6, s, a n d the usual Coulomb phase shift ~ze ---- --~7 log 2kr + arg P(l+l+i~?), with
~7 ~ 4#e2/hZk. In the above case the outside solution (r > r~) is 1 R~ (r) = - - {cos ~ Nl
F , (r) + sin ~lNG~(r)},
where F~ (r), G~(r) are the regular a n d irregular Coulomb-functions as defined b y Breit et al. 10); the normalization c o n s t a n t Nz is now chosen so t h a t Rz(rs) ---- 1: N z ---- cos ~NF~(rN) + sin ~G~(rN). Introduce now the auxiliary function R',(r) defined as follows: d2
~r 2 R'z(r)+{k2--U'ce(r)}R',(r)
= O,
where
U'ec(r) =
Ve = + o o for r < r 1 0 for r 1 < r < rN, l ( l + 1) 8/ze~ for r > r N.
r~
+ ~Wr
(5)
T H E I N T E R A C T I O N OF T W O A L P H A - P A R T I C L E S
403
The normalization of R'~(r) is effected in the same w a y to that for R,(r), except for the substitution of 60 for ~N above. It follows in the usual way that the Wronskian R'z(r)
dR,(r) dr
dR',(r) Rz(r)
dr
--const.
for r > r
N.
(6)
In general from (4) and (5),
{R'z (r) dRz(r) -- dr
Rz(r)
dR',(r)t~° = dr ]0
Therefore 1
(1 _1o) • -rN- =
f
o
Um(r)Rz(r)R'~(r)dr.
,, UNz(r)Rz(r)R',(r) dr,
(7)
using the definition of Um(r ) and the boundary conditions for Rz(r) and R'z(r), and eq. (6); [z,/0 are the logarithmic derivatives of R~(r), R'z(r ) at
r~rN. Now R', (r) clearly is only required for the domain r 1 < r < rN, when, with the boundary conditions,
R',(r) = sin k(r--rl) cosec k(rN--rl).
(8)
Similarly p m a y be found b y continuity at r = r N from (8):
]o = kr N cot k(rN--rl).
(9)
Eq. (7) is clearly an integral equation for
G'z(r) = Um(r)R,(r),
(10)
in terms of the experimental function [z (for assumed values of the parameters rl, rN) and the analysis can now proceed substantially as detailed in a previous paper 4); it will therefore only be briefly sketched. The only differences are that now [,, rather than the phase function k 2z+1 cot ~zN of the previous analysis, is the experimental function; that one now has a repulsive core; and that the analysis need now only use the simple l = 0 type radial functions for all l: this effects a great simplification in practice. The analysis, starting from ]z but without core, follow analogous lines and will not be given. The final form of the analysis below apply also if one has no Coulomb forces for r > r N. Since G',(r) = 0 for r ---- r 1 and r = rN, the solution of the integral equation in (7) m a y be obtained b y expanding in terms of orthogonal functions:
G'~(r) = ~ A~ sin kn(r--rl) ,
(11)
n=l
where k~ = nk 1 is defined by kl(rN--rl) = ~. The integer n denotes in fact the nth harmonic of the representation. A~ depends of course on l, but the
404
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PIENAAR
dependence has no formal significance. Hence eq. (7) reduces to { / z _ k r N c o t k(rN--rl)}'---1 = ~ (--1) n+~ k n A s rN , =1 (ks~--k ~) Now in the domain r l < r < r
(12)
N eq. (4) reads
d2 dr 2 R t ( r ) + k ~ R~(r) = U m ( r ) R z ( r ) = G',(r) =
A n sin kn(r--rl). n=l
Thus, on account of the boundary conditions, _
R~(r)
sin k ( r - - r l ) sin k(rN
~ 2_
+ s
=
1
As (k~--kn 2) for
sin kn(r--rl), r~
N.
With (10) and (11) this leads to UN~ (r)
sin k (r--r1) ~ An I + 2, sin ks(r--r1) ) sin k ( r N - - r l ) ~=1 (k2--k~ 2) =
~ A s sin ks(r--r1).
(13)
•=1
Eq. (13) clearly shows t h a t the A n, functions of k 2 or the energy, are "interaction parameters" defining U m (r). If one k n o w s / , as a function of energy, the parameters defining the latter are related to the interaction parameters by (12), and hence to the interaction by (13). Again, (12) and (13) m a y be used to calculate/, if Urn(r) is known. The practical way of effecting these connections is to expand both sides of equations (12) and (13), as one may ~), in power series in k 2 oc E, and to equate coefficients of the same powers. The equations thus obtained do not contain k 2 and hence probably apply over a larger energy range than t h a t determined by the radius of convergence of the relevant expansions. To illustrate the procedure: As --= ]~12
C~o+C,np2+ C~2p~+ . . . .
with p = k/k s.
(14)
Experimentally one has cot 6tN and using Coulomb functions here one may calculate /l for any chosen r N. This m a y be fitted with a polynomial in powers of E' /~ = c¢+flE'-t-TE'2+ . . . . (15) Since k 2 = 2#E/'h 2 = / , E ' / h 2, p2 = #E,/f~2 k2 = / ~ (r N_r~)2 E,/h2 ~2; and /~ = a + ~ 2 bp2+~ a cp4+ . . . . where b = h2fl/tz(rN--rl) 2,
c
=
]i4y/#2(rN--rl)4
.....
THE
INTERACTION
OF TWO
ALPI-IA-PARTICLES
40~
The a, b, c are thus experimental parameters determining the logarithmic derivatives for a given set (rl, rN), from the parameters ~,/3, ~ . . . . which are generally applicable for a given r N. Eq. (13) now yields a number of equations, of which the following comes from the lowest order term in the expansion in powers of k 2 (it m a y be directly deduced from (13) by letting k - + 0):
UNl(r) l ( r _ r l I
oo C
~ ~n°sin
- -
kl 2
t\rN~rl!
n~l
} k~(r--rl) :
n 2
~ Cno sin k n(r--rl).
(17)
n=l
Note that the function in the curly brackets is the zero-energy form of Rz (r). This equation read at a sufficient number of r values in the domain r 1 < r < rN, yields the C~0 in terms of Um(r)/kl ~ or vice versa. Having Cno, the next order in the expansion of (13) yields an equation for the C,1 and so on. All such equations shall be collectively indicated by (17). Expanding (12) and using certain identities between C,o, C,~, etc. derived from eqs. (17), one readily arrives at
__1{l_a(rN_rl)/rN ) = ~ 7/7
(_1) C~0
n=l
=
n
__X n
(-
+ -"
1 oo
2
.2C,,o
~a(--1/ 45--c (rN--rl) /rN} : ~= l ( (--1) --~-
--
,
Os)
n 4
,,C,,1
~ C~o
- ~ - - I J"
One notices that (18) relates the experimental parameters a, b, c to the interaction parameters C,,~. If the experiments can be described by only 2 parameters, the first two equations m a y suffice, in which case we only need calculate the C~0. If the next order coefficient is important, or has to be checked, the Cni must be calculated from the next order equation of (17) above. Note that (17) only uses sinusoidal and power functions of (r--r1). It is clear from (17) that on the left hand side the effect of the higher harmonics is reduced in the term in the curly brackets which means that this term is usually a smooth function of (r--r~). Hence C~0 will only be appreciable for higher n, if U m (r) has sharp or rapid changes with (r--r1). Since in all the equations (18), the higher harmonics are relatively weighed down, one need, except for very rapidly changing potentials, only determine the lower harmonics to any accuracy. One further has a natural way of selecting a number of potential parameters: namely to use just the lower harmonics or C,o. The only arbitrariness involved herein is the perhaps natural one of the consequent limitation of the investigation to smooth potentials which do not show rapid fluctuations. The advantage of introducing r 1 explicitly is that all l m a y be dealt with by one set of simple functions and that the
406
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AND
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PIEPIAAR
infinite repulsive core need not be represented by the C,0--which would have called for m a n y higher harmonics.
4. Application of the General Analysis The q u a n t i t y / ~ is determined from the phases for r N = 4 and 5/ (found useful in the square well analysis) for both S and D waves. These functions are then fitted by least squares with polynomials as in (15), using the least number of parameters demanded to fit the results within the experimental errors, and letting the S curve pass through the 94.5 keV resonance. This yields 3 parameters for the S wave, and at least 2 more for the D wave. The former number agrees with that calculated by Russell et al. 2), using another phase function. One therefore has 4 to 5 parameters altogether if a velocityindependent potential is possible; because one must recognize that one of the parameters for the S wave, y, has relatively little effect and m a y perhaps not be insisted on. The general analysis is first of all applied to the S wave to find, for a given (rN, rx) combination, the Ci0, C20 from (18) which will give the correct ~, fl of (15), (16). This gives 4 parameters (rN, r 1, Clo, C2o), or 3 when one takes the case r I = 0, which fixes a potential by the first of eqs. (17). The C,1 follow for the latter potential from the second of the eqs. (17). The y corresponding to this potential m a y then be calculated by (16), (18), and hence the corresponding [0 compared with the experimental/0. One m a y then t r y other sets of r N, r 1 (and C10 and C2o) till the ~ is equal to that required by the experimental fit, or if this is not accurately possible, till the [o calculated from the potential, found in this way, agrees as nearly as can be with the experimental data, over the chosen energy range. The above procedure always ensures a fit at low energies for a start. Potentials which are found in this way to fit the S wave data well, are then tested for their fit of the D wave data. In this way, the 4 to 5 parameters from the S and D phases m a y suffice to determine the 4 parameters of the potential uniquely, if the latter can be velocity-independent at all. There is little reason to start with 5 interaction parameters (adding C30, say), since one of the experimental parameters (y from/0) gives a small contribution in the chosen energy range, and it is doubtful whether it m a y be insisted on. In any case, C3o for the class of smooth, non-rapidly varying potentials will contribute little to/o, but would tend to make the analysis prohibitively cumbersome for a desk calculating machine. Since the infinite repulsive core is explicitly introduced (and need hence not be represented by the harmonics), and since one knows from the phases that R o(r), R 2(r) have only part of a loop over the chosen low energy range, two harmonics can permit of a U m (r) with one sign reversal in the domain r 1 < r < r N (see eq. (17)), which should suffice to cover as general a class
THE
INTERACTION
OF
TWO
ALPHA-PARTICLES
407
of potentials as one can perhaps expect from the n a t u r e of the problem in hand. One finds then: (a) Neither r N ----- 4/, nor r N = 5/, with r 1 ----- 0, allows of a n y real solutions for Cxo, C2o. This does not m e a n t h a t there are no solutions possible if one were to t a k e sufficiently strong higher harmonics to describe the sharply changing potentials, with large values for low r, which seem to be d e m a n d e d b y the results given below for rl > 0. In this sense this finding is in line with the requirement of a strong repulsive core. 0 (b) W i t h r N = 5 / n o value for r I > 0 permits of a real solution for C10 , C20, or a smooth non-steep potential. (c) W i t h r N = 4/, the values r I : 1.2, 1.4, 1.8/ all permit of solutions; 2 each, in fact. The one solution could be selected as being of the same order of general depth as for the square well potentials; the other solution corresponds to m u c h deeper wells a n d hence to a bound state. The two solutions come from the quadratic n a t u r e of (18) for 2 harmonics C~0. W i t h this selection the complete interaction potential curve can be drawn and the corresponding ]0 calculated. As higher values of r a are t a k e n in the above set the fit becomes slightly better. Together with the D wave findings below, r 1 ~ 1.8] is perhaps the best compromise for the S well. Comparing the corresponding wells in the order r a = 1.2, 1.4, 1.8/, the first two show, for r just greater t h a n rl, a repulsive part which is strongest for r 1 ---- 1..2/. The effect of this is to indicate clearly the need for a bigger repulsive core radius, which supports the finding t h a t r~ ~ 1.8[ is probably the best compromise. F u r t h e r m o r e for r~ = 1 . 8 / t h e potential (for r 1 < r < rN) is purely negative, and up to radii r ~ 3], resembles the square well of best fit in being nearly of c o n s t a n t depth, a n d of its order of depth. For larger r the potential rises gradually to 0 at rE = 4/. (d) Testing these three potentials for the D wave one finds t h a t none nearly represents the results, the larger r 1 being further from a fit. E v e n potentials of this kind with lower rl do not come close to a D wave fit. I t is for this reason t h a t in (c) above, the compromise S wave fit is t a k e n as corresponding to r~ ~ 1.8/. The lack of D wave fit is not unexpected from the square well results (especially in view of the potential for r = 1.8] being so similar to the square well); a n d as an estimate from the latter, one would say t h a t the wells required b y the S wave results are too shallow for the D wave phases b y ~ 2.7 MeV. These results indicate, more generally, t h a t a velocity-independent potential fitting b o t h S a n d D results is not possible, even for the low energy domain. For the S waves, the results confirm the demand, for an infinite repulsive core of the considerable radius r 1 ~ 1.8]. T h e y also indicate r N m 4]. It is outside the scope of the present analysis to determine the na-
408
E. V A N
DER
SPUY
AND
H.
J. P I E N A A R
ture of the velocity dependence, but it is nevertheless clear that if the outside attractive well is to have the same shape up to the core, the velocity dependence could consist in r: depending on l, being in fact smaller for the D wave than for the S wave (for which, say, r: = 1.8/). The order of the compromise core dimension is near the suggestion of Herzenberg 6), rl = 2 to 3/. The latter analysis in any case shows from fundamental arguments that the core interaction is repulsive and velocity-dependent. Since it is possible to test any interaction with a core and an effectively finite range rN, the analysis was used to test the potential given in the paper b y Herzenberg 6), with r 1 = 2.5/. Using this potential for the S wave, the/0 was nowhere near a fit, the attractive well being, effectively speaking, too shallow, or alternatively the core radius being too large, to a considerable degree. Although no calculations were made, this m a y also be said with great probability of /2. Since the Herzenberg potential s) is uncertain in actual detail and velocity dependence below r = 2.5/, this analysis cannot usefully be pursued further at this stage. The fact is that if one permits of velocity-dependent effects a purely phenomenological analysis becomes impossible or at least quite arbitrary, unless a fairly precise indication of the type of velocity-dependence is furnished from other considerations (for example, fundamental considerations). Even so the question m a y well arise of the possible use of such a phenomenological potential, apart from being a mere shorthand for scattering results (which m a y be done more simply with square wells). Indeed, the mere fact that, in spite of the considerations in the introduction, the interaction is velocity-dependent indicates that the problem is not precisely enough a two-body problem. It is hence natural to expect that a velocitydependent phenomenological potential deduced for the ~--x interaction would only apply strictly to it, and not to x - - x cluster interaction in the Be 9 and Cn problems, for example. The need for such care is further indicated b y the consideration that whereas the a - - n interaction 4) involves no sensible velocity-dependence (apart from the spin-orbit coupling, of course), the ~ - - a interaction does. There is another difficulty in the w a y of an application of velocity-dependent forces such as is likely here: namely, that the strong repulsive forces would require due allowance for cluster correlation effects in problems involving ,t-clusters, in line with the work of Brueckner :1). Since one would then have to deal with repulsive velocity-dependent forces within a region of large radius, involving in general a few clusters which are themselves compound particles, the use of a phenomenological potential looks very unpromising. However, if it is possible to formulate the velocity-dependence in a reasonably simple form (/-dependence of core radius, say), connecting it with nucleon-nucleon forces b y means of a group-resonance calculation, then the
THE
INTERACTION OF T~,VO ALPHA-PARTICLES
409
phenomenological potential could provide an avenue for testing these forces and the ~--~ interaction derived from such a calculation. It m a y then furnish information on the properly velocity-independent part of the ~--~ interaction, and indicate features of the velocity-dependence which m a y qualitatively elucidate the application of the ~-cluster model. With electronic computers a direct calculation of ~--~ scattering from nucleon-nucleon forces m a y be possible, using the group-resonance formalism. The equivalent velocity-dependent potential m a y then be calculated from this solution. This would be another way of arriving at the x--~ potential, and would presumably have the restricted use indicated above. The trend of the above phenomenological conclusions, arrived at by using square wells and more general smooth potentials, is confirmed by Nilson etal.'s 5) t fitting of the experimental data to the Haefner potential 8) q2 h~ VN(r) ---- 0
for
r > rN,
= --D+
2~tr 2
for r < rN,
where q~ = 30 for convenience. To fit the 94.5 keV resonance, D may be calculated as a function of r~. The rest of the data then provide r N, and hence D. Since these fits cover a wider energy range, the experimental points naturally have a somewhat wider scatter round the theoretical curves than was found in the present investigation over a more limited energy range. Nilson et al. 5) found with the Haefner potential (the subscripts S, D, G refer to the partial wave): (a) r~s = 3.49/ for E' up to 20 MeV. (b) r•D :- 4.44! for E' up to 6 MeV. No fit here could represent D wave phases at E' > 6 MeV as well, the well with r~D = 4.44! being too shallow for higher energy data. (c) rNG just smaller than 4.44/would represent the G wave data up to E' = 20 MeV, but rapidly becomes too shallow at larger energies up to E ' : 22.9 MeV. These results thus also indicate velocity-dependence, inasmuch as r~ is not common to all the l values, and as a fit at lower energies deteriorates at higher energies. Similar results have been presented in a different form by Humphrey 9): he used a modified Haefner potential and found a general fit up to E' = 22 MeV with r~ = 3.75/ and D :
21 MeV for the S wave,
D = 25 MeV for the D wave, D = 32 MeV for the G wave. ? The result of this w o r k became available w h e n the calculations of the p r e s e n t p a p e r were nearly completed; the a u t h o r s are indebted to Prof. J e n t s c h k e for c o m m u n i c a t i n g these results to t h e m before publication.
410
E. VAN" DER SPUY AND H. J. PIENAAR
The value of r N and the trend of D agrees with the general findings of the present paper. These modified Haefner potentials go through zero at a radius between 1 and 2], and rise rapidly for smaller r. This indicates a repulsive core with an effective radius comparable to that, r 1 ~ 1.8/, found in this paper, and this effective radius also becomes smaller for higher I. It should be noted that the Haefner potential, like the square well potential, is merely a convenient form for expressing repulsion between overlapping systems, which to some extent can fit the results, but has no theoretical foundation as yet. One of the authors (H. J. P.) wishes to acknowledge the benefit of a bursary of the South African Council for Scientific and Industrial Research during the present investigations. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll)
N. P. Heydenburg and G. M. Temmer, Phys. Rev. 104 (1956) 123 J. L. Russell, G. C. Phillips and C. W. Reich, Phys. Rev. 104 (1956) 135 Nilson, Kerman, Briggs and Jentschke, Phys. Rev. 104 (1956) 1673 E. v a n der Spuy, Nuclear Physics 1 (1956) 381 Nilson, Jentschke, Briggs, Kerman and Snyder, to be published A. Herzenberg, Nuclear Physics 3 (1957) 1 Bloch, Hull, Broyles, Bouricius, Freeman and Breit, Revs. Mod. Phys. 23 (1951) 147 R. R. Haefner, Revs. Mod. Phys. 23 (1951) 228 C. H. Humphrey, quoted by Nilson et al. 6) Breit, Wheeler and Yost, Phys. Rev. 49 (1936) 174 H. A. Bethe, Phys. Roy. 103 (1956) 1353