close

Вход

Забыли?

вход по аккаунту

?

Diffraction Scattering and Regge Pole Technique.

код для вставкиСкачать
A N N A L E N D E R PHYSIK
7. F O L G E
*
B A N D 21, H E F T 3-4
*
1968
Diffraction Scateering and Regge Pole Technique
By S. A. EL-WAKILand A. A. KRESNIN~)
With 9 Figures
Abstract
The diffraction scattering of spin zero nuclei was considered in Regge pole representation. The second pair of pole in the complex 2-plane was found to affect the small angle,
scattering. This can be shown in the analysis of a-particle scattering on the nuclei Mgz4,
Zns4 and S P . The scattering of two identical particle was also considered a t an energy
above the Coulomb barrier. Satisfactory result was obtained.
1. Introduction
The oscillatory structure of the elastic and inelastic cross-section was
assumed t o be associated with the diffraction effects. BLAIR introduced the sharp
cut off model [ l ] t o describe the &-particle elastic scattering but this model fails
t o reproduce the cross-section behaviour a t large angle of scattering and near
diffraction minima. A number of improvements were suggested by MCINTYRE
et al. [2] and FRAHN
et al. [3] which lead t o quite satisfactory results. The
inelastic scattering has been investigated by the adiabatic approximation
method [4],in which the relation between the elastic and inelastic cross-section
is easily understood [5]. AUSTERNand BLAIR [S] introduced a new mathematical technique for the adiabatic distorted waves inelastic scattering of arbitrary multipolarity. Some suitable approximations make it possible t o reexpress
the inelastic amplitudes in terms of derivatives of the partial wave elastic scattering function.
I n the present work it is attempted t o investigate the elastic and inelastic
scattering of spin zero nuclei with the aid of the complex angular momentum
method [7]. The contribution of the other two pair of poles nearest t o real axis
was consider t o analysis the scattering of a-particles on the nuclei Mg24, Zns4
and S F , from which one can see that it affects the values of the cross-sections
at small angle of scattering, also the method was used t o study the scattering
of two identical nuclei 01s-016a t E = 15, and 8.8 MeV quite good agreement
with experimental results was obtained.
2. Elastic Scattering
The elastic scattering amplitude of a spinless projectile incident on a spin
zero is:
l)
8
On leave from Physics-Technical Institute, Kharkov, USSR.
Ann. Phyaik. 7. Folge, Bd. 21
114
hna.len der Physik
*
7. Folge
*
Band 21, Heft 3/4
*
1968
The scattering function S (1) can be written as :
#(I)
=
~ ( 1e2is(l)
)
where 6 (1) is the phase shift which represents the effect of the nuclear potential
on the scattering. The nuclear phase-shift vanishes for large 1-partial waves and
increases as the 1-value decreases. With certain conditions on the analytical
properties of q ( I ) in the complex 1-plane the amplitude (2.1) can be expressed
by means of the WATSON-SOMMERFELD
transform [8] as the spm of a background integral on the line A B of the contour shown in fig. 1and an expansion
in terms of poles in the complex 1-plane. Using this transformation the amplitude
f ( 6 ) becomes :
i
(2v
1)
f ( e ) = ,kJmnG(i
- x ( Y ) ) p V ( - COB e) av.
(2.3)
+
I
Fig. 1. Paths of integration in the v-plane
The contribution of the background integral is small, it is of the order e-lola
if one takes q (1) in the form of WOODS-SAXON
shape i.e.
(2.4)
INOPIN
et al. [7] calculate f ( 0 ) in the case of two pole approximation of ~ ( 1 )
a n d they had obtained the differential cross-section in the form :
8nR
o(e) = k sin 0 l a @ )12e-zae[b2 + Cos2(kRe + y ) ] ,
where a(1) is the residue of ~ ( 1 at,
) 1 = lo
b = sh(2 I m S ( 1 ) ) = s h ~ and
,
n a c
21,
y = - +4-
(2.5)
+ ia,R is the interaction radius,
+ arg a @ ) .
(2.6)
In the above calculation the LEGENDRE
function P, (- cos 0 ) is replaced by
its asymptotic form; for I Y I
1 that is:
P+
1015
and
>>
cos e) = f&
cos((y
+ +) (Jc - e) -
e 5 - 1;'
a(1) = ja(1)I e i a r g @ ) .
(2.7)
Now, we proceed t o consider the effect of the other two pair of poles. I n this
case the position of the poles are assumed t o be a t 1, = lo ial and 1, = 1; ia,
+
+
S. A. EL-WAHIL
and A. A. KRESNIN:
Diffraction Scattering and Regge Pole Technique 115
+
+
such t h h lo
16 and a1 a2. Following the same way described in [7] the
differential cross-section in the four poles representation is written as :
The elastic scattering cross-section of identical particles of spin-zero can be
expressed in terms of the scattering amplitude :
where f ( 0 ) is defined by (2.1).
I n the two pole representations the differential cross-section can be written as:
y = kR8
=
+ y,
~ c ~-( e)n
+
y =
y.
n
+ arga(Z) + 5
2
10
(2.13)
3. Inelastic Scattering
The scattering amplitude f o r single excitation of arbitrary multipole order
(I)as was derived by AUSTERN
and BLAIR [6] in the absence of the COULOMB
8*
116
Annalen der Physik
field has the form:
jlMr,OO(e)
=
+
*
*
7. Polge
Band 21, Heft. 3/4
*
1968
+ i p c,(I)5iz-q2r +
(3.1)
m(u,
OO~EO)(I'I-M~~M~~EO)Y~~~(~,O)
(21
x
i)1/2
ail
i n which the differential inelastic scattering cross-section is :
o(@=
3
(3.2)
/fIM,00(w/2>
where (jlml, j2m, 1 j m ) is the CLEBSCH-GORDANcoefficient. I n the limit that
I'
I the sum over E in (3.1) can be done, and the result is:
>
jlHI.OO N
(i)
(21+
1)1/2
c,m [I:JGIq (21'
+1
~
as (1')
aF
2 yF'(e, 01,
where
,
S(1')= q (2') e'WI')
and
laI
.I
[ I : M * ]=
+
+
[ ( I - HI)!
( I M')!]1'2
(I
( I - HI)!!
( I HI)!!
'
+ M I ) even
(3.3)
0
, (I MI) odd.
C, ( I )is the reduced matrix element which depends on the details of the nuclear
function as well as on the order of excitation (n) and on the final angular momentum I.
For single excitation n = 1
+
+
C , ( I ) = ( 2 1 l)-l/ZR/31,
(3.4)
where
is the deformation parameter.
C, (I)can be obtained by comparing experimental and theoretical yalues of
cross-section a t particular angle. This introduce an independent method for
determining the nuclear deformation parameters. Now, the scattering amplitude
in (3.1) is transformed into contour integration using SOMMERFELD-WATSON'S
transformation as in the elastic scattering. The result of calculation in the case
of two pole representation is :
d e ) = 5 ( 2 1 + 1) [c,(0i2
- 2 0 [sin yjl sin
v1 + (1+ b,)2 COS (yl- ~ , ) 1 ) ,
where a (E) is the residue of q (1) a t I ,
b, = sh (2 I m (I,)) = sh,
y1 =
la(4I2
= ,2
+ ia,,
(lo + 31) e - $(& + 1) +
y11
y1=
arga(E,)
n
+ 2zoa + 4
and
1
v,=(zo+&++Y;j
y,'=y,-I,.
a
(3.6)
I n the four pole reprFsentation, we consider that we have two poles a t
1, = I,
ia,, 1, = 1;
ia2 and their complex conjugate, the cross-section in
+
+
S. A. EL-WAKIL
and A. A. KRESNIN:
Diffraction Scattering end Regge Pole Technique
117
this representation is :
+ +
0 = 01
02
(3.7)
where ( T ~ has
, ~
the same form as in ( 3 . 5 ) with y,, 9, and b, as in (3.6). y2,yz are
defined by :
b, = sh (2 I m S(Z,)) = shx,.
The interference term o3 has the form :
(3.8)
x cos 2(Re 6, - Re 6,) - [sin (y, - y 2 )sh x1 ch xz
(3.9)
- cos y1 sin y z sh (x, - x2)]sin 2 (Re 6, - Re 8 , ) ) .
a&) and a (Z2) are the residues of 7 (I) a t I , and I,. If the nuclear potential phase
shift S (1) is a smoothed function, we can write Re 6 (I,) = Re 6 (I2) and hence
the interference term reduced t o :
(3.10)
4. Discussion
4.1. The elastic scattering
The comparison between experimentally observed elastic scattering angular
distribution and equation (2.5), (2.12) gives us the possibility t o obtain
all parameter of the model for the case of two poles. The effective interaction
radius R and the phase shift y are obtained by matching the location of the
maxima of eq. (2.5) and (2.12) t o the experimental angular distribution. Similarly, values of N, and b! can be obtained from the observed values of the
-
83CR
cross-section in the maxima and minima. The factor N =
( a( I ) can be
k sin 0
obtained from the absolute value of the experimental cross-section a t the same
angle. Comparison was made for elastic scattering of a-particle on the nuclei
MgZ4, ZnB4and S@*,from which one can see that, the computed cross-sections
are generally in quite good agreement with the experimental angular distributions for larger angles, Moreover, the relative depth of minima in the experimental angular distribution is not constant but rather strongly depends on the
angle of scattering. It seems that t.aking into account the second pair of poles
we introduce five additional parameters (R', a@,), arg a ( / & ,b,, a z ) .
Actually we have only one independent parameter a2 as it is resonable t o
suppose t h a t the values of R', a(Z) and arg a(Z) are determined by the proper-
l2
118
Annalen der Physik
*
7. Folge
*
Band 21, Heft 3/4
*
1968
ties of a given nucleus and are just the same for all poles. Also one can see that
b2 is a dependent parameter, since b, = sh (2 Im S(Z,)) and we have approximately :
+
I m 6 (Z,
iol)
= Im
[6 (2,) +
iol
-
On the other hand, in the quasiclassical approximation:
where 81, is the angle of scattering for particles with a n angular momentum lo.
So we have :
-xz_--
a2
XI
011
.
Moreover, we have that the best agreement with the experimental angular
distribution is obtained when the ratio is the same for all investigated nuclei.
10'
10
t
def
1
10-1
10
20
30
40
50
60
-+eO
Fig. 2.
Angular distributions for elastic scattering
of 41 MeV a-particles by MgZ4
20
30
40
50
60
70
80-ten
Fig. 3.
Angular distribution for elastic scattering of
21 MeV a-particles by ZnS4
The result of calculations is shown in fig. 2, 3, 4 for the case of two poles and
4 poles from which one can see that the next two poles improved the values of
the cross-section a t small angles.
S . A. EL-WAKILand A. A. KRESNIN:Diffraction Scattering and Regge Pole Technique 119
The expression (2.12) was also used to calculate the angular distribution for
the elastic scattering of 0l6and 0l6at 8.8 MeV. where the MOTTpredictions [9]
are good and a t the energy 15MeV., where the angular distribution show
10'
10
10-1
Fig. 4. Angular distribution for elastic scattering of 41 MeV &-particlesby Srs8
10
lo-*
20
40
30
50
8 O
60
70
80
-+
------l
I
16 16
0 to
7Y
e ,$
5
La
\
E
.C
%I$
*
10'
10
M
50
70
90
ecm.
110 130
Fig. 5 . Elastic scattering angular
distribution of 01* Ox6 a t
= 8.8 MeV. The solid curve is the
calculated curve. The spots is the
experimental points
+
'h
50
30
70
90
ll0
130
flcm.
Fig. 6. Elastic scattering angular
distribution of OI6 0 I 6 a t Ec.m.
= 15 MeV. The solid curve is the
calculated curve. The spots is the
experimental points
+
120
h n a l e n der Physik
8
7. Fo!ge
*
Band 21, Heft 3/4
*
1968
characteristic decrease in the cross-section below the MOTTprediction [lo]. The
results of calculation given in fig. 5 , 6 a t both energies show satisfactory
agreement with experimental results.
4.2. The inelastic scattering
The inelastic scattering of &-particles on the nuclei Mg24, Z d 4 and Eras with
the excitation of levels I = 2+ are considered. I n those calculation we have
used just the same parameters as for the case of the elastic scattering. The
parameter C,(I)which appears in eq. ( 3 . 5 ) and (3.7) can be obtained by comparing experimental and theoretical values of cross-section a t a definite angle.
The result of calculation for both cases of two and foul poles are shown in figs. 7,
8 and 9. As in the case of the elastic scattering, taking into account the second
10
20
40
30
50
60
+en
20
Fig. 7. Angular distributions for inelastic
scattering of 41 MeV a-particles by MgZ4
30
40
50
60
70
so-+eO
Fig. 8. Angular distribution for inelastic
scattering of 41 MeV a-particles by ZnB*
pairs of poles improves the agreement between experimental and theoretical
angular distributions especially for small angles. The values of the deformation
parameter obtained are listed in table 1 from which one see t h a t they are in
a close a,greement with the values obtained in ref. [5].
Table 1
target
Mg24
Zn64
Srn8
I
2 poles
1
1 1
0.254
0.18
0.041
4 poles
0.28
0.16
0.042
i
~
1
ref. [5]
0.24
0.19
0.08
S. A. EL-WARILand A. A. KRESNIN:Diffraction Scattering and Regge Pole Technique 121
1
t
O(S)
10-1
10-2
10-8
Fig. 9. Angular distribution for inelastic scattering of 41 MeV &-particles by Sras
30
40
50
60
70
80
9O-f
References
[l] BLAIR,J. S., Phys. Rev. 95 (1954) 1218.
[2] MCINTYRE,
J. A., K. H. WANGand L. C. BEKER,Phys. Rev. 117 (1960) 1337.
131 FRAHN,
W. E., and R. H. VENTER,Ann. Phys. (N.Y.) 24 (1963) 243.
S. I.,J E T P (USSR) 28 (1955) 734, 736 [translated JETP 1 (1955) 588, 6911.
[4] DROZDOV,
INOPIN,
E. V., JETP (USSR) 31 (1956) 901 [translated JETP 4 (1957) 7841.
BLAIR,J. S., Phys. Rev. 115 (1959) 928.
[ 5 ] BLAIR,J. S., et al., Nucl. Phys. 17 (1960) 614, 641.
[6] BLAIR,J. S., and N. AUSTERN,
Ann. Phys. (N.Y.) 33 (1965) 15.
[7] INOPIN,
E. V., and A. A. KRESNIN,
J E T P (USRR) 48 (1965) 1620.
A., partial differential equation in physics (Academic Press, New York
[8] SOMMERBELD,
(1942) p. 282.
[9] MOTT,
N. F., Proc. Roy. SOC.(London) A 126 (1930).
[lo] BROMLEY,
D. A., J. A. KTTERNER
and E. ALMIQVIST,
Phys. Rev. 123 (1961) 878.
Cairo (U.A.R.),Atomic Energy Establishment.
Bei der Redaktion eingegangen am 11.Juli 1967.
Документ
Категория
Без категории
Просмотров
0
Размер файла
381 Кб
Теги
scattering, diffraction, pole, techniques, regge
1/--страниц
Пожаловаться на содержимое документа