Calculation of alpha particle elastic scatterin

Payne, G.

English

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Department of Physics and Astronomy 
THE UNIVERSITY OF IOWA
https://ntrs.nasa.gov/search.jsp?R=19680002496 2020-03-12T11:48:42+00:00Z
U. of Iowa 67-55 
Alpha Particle Elastic Scattering* 
by 
G. Payne 
Department of Physics and Astronomy
The University of Iowa 
Iowa City, Iowa 
*
This research was supported in part by the Atomic 
Energy Commission under contract ATC(11-1)-1190 and 
by the National Aeronautics and Space Administration 
under contract NsG-233-62.
ABSTRACT 
The elastic scattering of alpha particles by alpha 
particles is calculated from threshold to 100 MeV laboratory 
energy using both the Schroedinger equation and the N/B equa-
tions. The N/B equations do not yield very accurate results. 
The Schroedinger equation solution extends an earlier calculation 
of Preist to higher energies, higher partial waves and uses 
more recent values of meson nucleon coupling constants. With 
slight changes in their values we obtain good agreement with 
experimental phase shifts to 100 MeV for £ = 0, 2, -i..
2 
I. INTRODUCTION 
The elastic scattering of alpha particles by alpha 
particles has been measured experimentally up to one hundred 
and twenty MeV lab energy.' Several theoretical calculations 
have been made. 2 , 3 Okai and Park, using a Guassian type 
potential for the two nucleon interaction, with five of six 
parameters determined by low energy data, solve the integro-
differential equation which results when exchange forces are 
allowed for. Preist neglects exchange and uses the one boson 
exchange potential for the a and the w mesons, the coupling 
constants being taken from early phase shift fits of nucleon 
nucleon data. 8 A more recent fit 4 in which the 1=0 mesons 
are the a, the w and the tp mesons, gives very different values 
for the coupling constants, in particular the extreme repul-
sion of the w particle is considerably reduced. Using the 
more recent values, we calculate phase shifts for £ = 0, 2, i-i-
partial waves for lab energies between 0 and 100 MeV. 
Writing the potential as a Fourier transform, 
+ 
V(r)	 -1 1	 ikr =	 kdk e
	 v(k)	 (1)
3 
the central part of the two nucleon potential due to scaler or 
vector meson exchange is 
	
2	 2 
m	 c 2,	 ls	 1	 s V(k) = 
-[g l ---ì]	 2	 2	 2	 2 5
m	 k+m	 k+m 
	
S	 5 
	
2	 2 
c 2 (l ^l mv ) 21	 1	 -	 V V
	 =	 i:	 2	 2	 2	 2	 (2) V	 .
m	 k+m	 k+m 
	
V	 V 
This one boson exchange potential is a non-relativistic 
reduction of the field theoretic potential. It is stated to be 
accurate to order (meson mass/nucleon mass) squared. In the 
case of the w and the cp mesons this is certainly meaningless and 
we report only Values of c 2, c2 . To obtain the potential between 
two alpha particles we multiply (2) by the square of the form 
factor f2 of the alpha particle. We use the form factor of 
Pr ei st3
f(k2) =
	
	
(3) 
w2+k2)
II. 
for the mesons and 
f(k2	 w ) = 2	 2	 2)	 (1)+ k  
for the photon. The factors come from 4 nucleons and 2 protons in 
each alpha particle. In units in which the pion mass is one, 
w2 has the value 77.07. 
The ambiguity in the non-relativistic reduction of the 
potential might be avoided by solving the N/D equations and 
using the full relativistic form. We will conclude, however, 
that the N/D equations are not sufficiently accurate. 
Both the Schroedinger equation 
d2u ( r)  
+ [q2 -	 ( r ) - L(+1) u(r) = 0	 () 2 
r 
and the N/D e4uations4
r 
N e(s) = B
A (S)-	
p(s')ds' ___ 
-	
I '+	 BL(s) - BL ( s' )] NL(s')ds'(6) 
0	 t.	 0 
(s+s	 CO p(s')NL(s')ds'	
(ri.) 
Tf	 o (s'_s) (S/+s ) 
0
5 
p(s)_,'-;- 
s+-l-.M
q	 TT jl	
2n11 
2rr1 
e	 -1
(8) 
have been solved numerically. Here with M the mass of the alpha 
particle	 is the electrostatic constant 	
=	
= .032/ q); 
s = q2, the momentum squared; p(s) the phase space factor modified 
by coulomb effects 
and Bk(s) the Born ter7i6 
	
22 M (,)2 S	 1 1 2iqr [F(L+l+i, 2L+2,_2iqr)]2
CO 
BL(s) 
= - ((2L+1))2
	
x	 V(r)r	 di'	 (9) 
where I F is the confluent hypergeometric function5 
	
F,(+1+i11, 2L+2, -2iqr)	 F(21+2) 
= F(L+1+ifl) F(L+l-il]) 
5 1 -2iqrt e	 (i-t)	 dt	 (10)
0
To evaluate the Born term 4 integrals must be performed (over r, 
k, t, t'). The r integration and one of the t integrations can be 
done analytically. Making a change of variables yields 
+ 
BL(s) = C	 k	 2 
	
k+p1 2 (22)8 J
	
(ii) 
- 
2	 ifl 
	
+1 r( t_i )	 '	 k ii	 (t2_l)L 
J	 - l)	 l L( t+l )	 2 t+l-	 )j	 k L+l (12) 2q (t+l---) 
2q 
=Mg 2 22L (1) L+l	 (Lt)2	 1	 (13) 2 (2q) 2.Q.+2 F(L+l+iT)  
We notice that for i=O (no coulomb effects) and w = 
J = 2(_l) L+l QL (l - 
k2 
2q 
2 
BL(s) 
= - 2q22 
QL( l +
2q
7 
(lii.) 
('5)
II. NUMERICAL PROCEDURE 
1. Schroedinger Equation 
The potential, equation 1, has been evaluated numerically. 
Because of the rapid decrease of the integrand due to the form 
factor, it is sufficient to do the integral from k = 0 to k = w. 
A twelve point per cycle of the trignometric function in the inte-
grand Sinipson t s rule is used. The wave function given by the 
Schroedinger equation is numerically integrated out to r = 7-1 
(, pion mass) and matched to 
u(r) = N(cosôL F(r) + sin 6L G(r))	 (16) 
the Coulomb wave functions being generated by the procedure described 
in reference '. A three point formula based directly on a second 
order differential equation is used. A step size .O2 from r=O to 
1 and a step size .06 from r=l.O to 7.0 is used. About one minute 
is required on a IBM 704 computer, most of the time being con-
sumed in evaluating the potential.
El;] 
2. Born term BL(s) 
The k integration,, equation 11, is trivial but in the t 
integrand, equation 12, there are singularities at the end-
points t = ± 1. The 2ilowing procedure is used to evaluate the 
t integral numerically. 
For I k 2	 2 / 2q j < 5 we make a polynomial fit 
[ (t+i- J)]	 n-i = a1 + a2 t+. . .+a t 	 (17) 
2q 
and 'do the residual integral 
'k analytically 
=±1 []u11 tk_J(t2_1)L dt
(18) k 2
	 .l t+]	 (t+l- k2 £+l 
- 2q 
-	 The Ikcan be expressed as a finite series of terms which 
can be generated by recursion relations. These recursion relations 
become unstable when lk 2 / 2q 2 I is greater than 5. Legendre poly-
nomials which are orthogonal on the integual t = -1 to +1 are used 
to make the fit, equation 17, and the a  can be generated by 
recursion relations. Then
10 
(19) 
For Ik2/2q2 > 5 we make a polynomial fit 
[t+l- A	 t+...+a _ L_1 =	
n t a1+ a2	
n-i (20) 
2q 
and do the residual integrals 
'k analytically 
=	
'	
tkt2_12 dt
	 (21) k 2L J t+1 
The I can be expressed as a finite series of terms which 
can be generated by recursion relations. To make the fit, equation 
20, we use a Taylor series about t = 0. Then 
- (	
2i e
	
E a '
	
(22) 
	
2q -
	 2'	
k k
 
Finally, for s = q2 = 0 a value can be obtained by using 
the limit as q - 0 of the confluent hypergeometric function 
(2qilr)k 
P 1 (L+l+ill, 2L+2,-2. 	 q0	 k=o
kr) —4 (2L+1)! E (2L+k+1) k!
	
(23) 
to obtain
11 
( 
B(0) = C $ kdk 2 2
	
2 2
	
J	 (2)-i-) 
-	 k+	 W+k 
1	
CO	 k	 1 (3	 (L+2+2K\ 
= (_ik)22 k=O	
\ 
	
-ik 1	 (2L+l+k)!	 K	
)	
(25) 
= _Mg2 (,) 2 22L (.:)	 (26) 
TT 
The inner integral, equation 12, need be done only once. 
It has the same value for each of the three mesons and the photon. 
It does not need to be evaluated at each point desired in solving 
equation 7 as it varies smoothly and can be fit, to better than 
one percent, to a set of pole terms of first and second orders 
5	 a.	 5	 a'. 
BL(s)=E + E	
1 
i=l	 i	 i	 2 ±=l 	(s+b ) 
In the calculations below, B is evaluated at 11 points between 0 
and 100 MeV laboratory energy and equation 27 is used to interpolate 
40 points. 
The accuracy of the above procedure for evaluating the Born 
term can be checked by using the Schroedinger equation. If the 
coupling constants are multiplied by a small constant, say .001,
(27)
both the Schroedinger equation and the N/D equations should yield 
essentially the same result, the first Born approximation. It 
is found when this is done, that the resulting phase shifts agree 
to better than one per cent. 
As an aid to the numerical accuracy of the above procedure, 
the k integration equation 9 is done off the real axis. This 
places a lower limit on the magnthude of the quantity in parenthesis 
in equations 17 and 20 facilitating their fit to a polynomial. 
Further, it simplifies the part of the coulombic potential due to 
the non-point nature of the alpha particle. With the k inte-
gration due off the axis (that is, a pole at k=0 removed), this 
is just given by equation 11 with p=O. This coulonibic residue is 
unimportant and could be ignored altogether. 
3. The N/D Equations 
The integral equation, equation 6, for the N function is 
non-singular. The integral is treated by Sinxpson?s rule: 40 
points are chosen between 0 and 100 MeV laboratory energy, 10 
between 100 and 200 MeV, and 10 above 200 MeV; 60 points in all. 
The resulting 60 by 60 matrix equations are solved by Guass Jordan 
reduction. The D function is then obtained by a quadrature, the
12 
singular point of the principle value integral being passed over 
by making an analytic approximation at that point. The phase 
shift is determined by 
1	
e'L sin8	
NL(s) 
2 +1 Cq	 = D) L(s 
£	 2 
where C = ii (1 +
	 )	
2,-r'1 
j=l	 j	 e2'_1 
The calculation takes approximately one minute on a IBM 7O4I-
computer, most of the time being consumed is generating the Born 
term.
As mentioned, the accuracy of the two programs, the Schroedinger 
solution and the N/D solution can be checked against one another 
by multiplying the coupling constants by a small number. When this 
is done the phase shifts agree to within a per cent. 
Comparing the two methods, the Schroedinger equation is 
slightly quicker and presumably more accurate; the N/D equation 
yields phase shifts at a much larger number of points.
13 
III. RESULTS 
Preist used the early values 8 g2 = 5.61, g2 = 16 . 7 . To 
obtain a fit he had to reduce the attractive g2 to 2.95. We 
show his results in Figure 1 for the £ = 0,2 waves. The 2 = 
phase shift only rises to 600 at higher energies for these 
values while the experimental value is 140°. Using a simple 
gradient search program, we tried to improve the fit below 
40 MeV for 2 = 0,2. The attractive g 2 climbed back to 5.45 
Cr 
and the repulsive g2 nearly tripled to 45.4. The large changes 
in the coupling constants needed to fit y-c data indicates they 
are not good values. 
In Figure 2 we show the results using the more recent 
coupling constants of Scotti and Wong. We take as the masses 
of the mesons m ) = 780 MeV, ma = )
437 MeV, m
w
 = 780 MeV. Scotti 
Ii  
and Wang give the values 
g2 = 3.05	 g2 = 2 . 77	 9 = 2.26 
for their fit to nucleon nucleon data. To obtain the fit shown 
in Figure 2, we use the values
lii. 
15 
c
c2 =2.66	 c2=3.07
	
c2=2.95 
a	 w 
The small change in the coupling constants particularly 
when the ambiguity in the non-relativistic reduction is recalled, 
is further confirmation of their values. Also shown in Figure 2 
is the N/D result. The potential, it would seem, is too strong 
for the approximation of replacing the left hand cut by its 
nearest singularity to be very accurate. 
We have neglected exchange effects in this calculation. 
Expecting exchange effects to be more important at lower energies 
than at higher, we mention that we have difficulty in fitting 
the d wave below 10 MeV (we obtain 62 = 11° at 5 MeV against an 
experimental 62 = 3750) and the L=-i- wave below 30 MeV (we have 
= 50 at 20 MeV with an experimental 8 )4 27 . 7 0 ) . In Figure 3 
we show an attempt to improve the fit for £=O and 2 at lower 
energies. There is no improvement.
16 
L.
ACKNOWLEDGMENTS 
The author would like to thank Dr. J. A. Van Allen for his 
kindess in helping him obtain funds for the computer work.
REFERENCES 
1 Darriulat, Igo, and Pugh, Phys. Rev. 137, B 315 (1965). 
2 Okai and Park, Phys. Rev. 145, B787.(1966). 
3 Preist, Ii Nuovo Ciniento 37, 166 (1965). 
L Scotti and Wong, Phys. Rev. 138, B1)45 (1965). 
5 Handbook of Mathematical Functions, Ed. by Abramiountz and 
Stegun (Formula 13.2.1) 
6 Wong and Noyes, Phys. Rev. 126, 1866 (1962). 
7 ??Qp.j] Model of Elastic Scattering" Hodgson (Sec 2.3). 
8	 Scotti and Wong, Phys. Rev. 10, 142 (1963). 
9	 Bryan and Scott, Phys. Rev. 135, 434 (1964).
17 
FIGURE CAPTIONS 
Figure i s and d phase shifts below 40 MeV. The solid lines 
are Preist results, the dashed lines the result of 
a gradient search. The crosses indicating the trend 
of experimental data are the values fitted. 
Figure 2 £ = 0, 2, 4 phase shifts below 100 MeV. The solid 
lines are the Schroedinger equation result, the 
dashed, the N/D equations results. The crosses 
indicate the trend of experimental data and are the 
values fitted. 
18 
Figure 3 s and d waves below 40 MeV.
?, 
/ I I II / I
/ / / / 
OD 
w
N 
(D 
II 
ci3
It) 
CP 
II 
cg3 
o	 0	 0	 0	 0 
0	 0	 0	 0 0) Go 
I	 I 
FIGURE 1
0 
x
	
'p 
OD 
F-
F-
0 
0
x 
cCIx
x	 0 N-
c'J	 .-	 0 
CO 0> CO 
/ w 
/ 
/
0 
/ 0-
/ / 
/ 
/ 
/
/ 
0 / 
/ 
/ 
/
/
	
iL	 OI• 
	
(.D-.	 - 
qo 
II 
	
Lx	 II I	
Ir() 
cab ci3 NO-
	
x	 0	 at 
I	
0
I 	 V 
o 	 0 
0	 0	 0
x 
0
0 
co
0
0 
OD 
FIGURE 2
0 
OD 
OD 
o	 0	 0	 0	 0 
o	 0	 0•	 0	 0 
co	 0)	 o•)	 Sco 
-.	
I	 T 
FIGURE 3


Alpha particle elastic scattering

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