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A conversion function for calculating elastic scattering angular distributions
NUCLEAR
INSTRUMENTS
AND
A CONVERSION
METHODS
FUNCTION
41 (1966)
89-92;
0
FOR CALCULATING
ANGULAR
NORTH-HOLLAND
ELASTIC
PUBLISHING
co.
SCATTERING
DISTRIBUTIONS
J. NLJRZYrjSKI Research School of Physical Sciences,
Australian National University, Canberra, Australia
Received 24 October 1965 A conversion function, denoted as F(x,fI) has been calculated and is tabulated for a range of values of x and for the laboratory angles tJ up to 180". The use of the tables makes possible an easy conversion of the measured intensity of the elastically scattered
particles to the ratio cr/nn, where (Tis the differential cross section for the elastic scattering and OH is the Rutherford scattering cross section.
1. Introduction
to perform these calculations during the course of the experiment in order to understand the results which are being obtained. The tables presented here were, therefore, compiled to simplify the procedure. The relevant transformation relations for these calculations are3): 1. for the scattering angle,
By taking into account the Coulomb interaction between any non-relativistic charged nuclear projectile and a target nucleus a formula for the elastic scattering differential cross section can be derived. It is’*2): zZe2
cTR(O’)= ~-
i 2m*v2
2
1
cosec”(+O’).
(1)
u*/v’ = {sin (O’- 19)) / sin 8 = MI/M,;
where ze and Ze are the charges of the incident particle and the target nucleus respectively, m* is the reduced mass of the interacting particles, v is the initial relative velocity and 8’ is the centre of mass scattering angle. The equation is referred to as the non-relativistic Rutherford or Coulomb scattering cross section. In general, even for low energies a departure from the pure Coulomb scattering is observed. For the higher energies, where there is a significant nuclear interaction, elastic scattering angular distributions display a characteristic diffraction structure which changes smoothly with the energy of the bombarding particles and with the atomic mass of the target nuclei. The departure from the pure Coulomb scattering due to the influence of the nuclear forces and the contribution of the other competing processes can be observed and discussed more easily if experimental results are presented in the form of the ratio o(&)/o,(~‘), where ~(0’) is the measured differential cross section expressed in the centre of mass system. This form is, therefore, used frequently.
2. for the intensity
of the scattered
(2)
particles,
Z’/Z = cos (W- 0) sin’ 0 i sin2 0’ = J(O,Q’).
(3)
In these equations v* is the velocity of the centre of mass, v’ is the velocity of the incident particles in the centre of mass system, M, and M, are the atomic masses of the incident particle and the target nucleus respectively, I’ is the intensity of the scattered particles in the centre of mass system, Z is the intensity in the laboratory system and J is the Jacobian for the considered transformation. It is clear that O/% = ar,,l(Q,lJ).
(4)
Using eq. (2) one can rewrite the formula for the Rutherford scattering cross section expressing it in terms of quantities measured in the laboratory system: crR(Q’)= aR(x,O) = (zZ/E[MeV])2R(x,B)
[mb/sr],
(5)
where 2. Conversion function
R(x,@E 1.2959964( 1 + x)‘cosec4{f[8 + sin- ‘(x sin@]}, x = Ml/M,, sin- I (x sin 0) < fn.
As both scattering angle and the intensity of the scattered particles are measured in the laboratory system, to calculate the ratio G(O’)/CT~(O’) it is necessary to transform the measured cross section rrla,,(Z3)from the laboratory to the centre of mass system and also to calculate (~~(0’) for a centre of mass angle corresponding to a given laboratory angle 8. Usually it is desirable
Similarly, using eq. (2) one can express the Jacobian as a function of the same variables x and 0 as used in the expression (5). Then considering (4) and (5) one obtains: 89
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90 x
=
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O(LAB)
SKI
.150
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F(X,Ol
‘3ICM)
FIX,'31
OICMI
.200
F(X,O)
5.0 10.0
5.25 10.50
3.5800 2.1461
5 4
5.50 lO.YY
3.5800 2.2461
15.ii
15.74
4.4649
3
16.48
4.464Y
i0.U
LO.Yt(
I.4254
1
Cl.Yb
1.4153
25.0 30.U
26.21 31.43
5.9054 Z.ti8dl
2 2
27;42
5;9052
2
2b.63
5.9049
2
32.67
L.tia7Y
i
34.30
2.8816
2
35.0
36.64
1.5850
‘
5tl.LY
l.584b
L
> 4
5.75
3.5800
5
11.4Y
2.2460
4
3
17.22
4.4649
3
3
22.94
1.4253
3
QICMI
F1X.Q)
3.5800 2.2460 4.4648 1.4253 5.9045 2.8871 1;5840 9.4607 6.0326
5 4 3 3 2 2 2 1 1
jY.Y4
1.5844
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40.0
41.84
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Y.4684
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9.4652
1
45.0
47.05
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1
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b.0403
1
51.03
6.0371
1
6.00 ll.YY 17.97 23.92 29.85 35.74 41;39 47.39 53.13
50.0
5L.iO
4.UbdO
1
24.5Y
4.0601
1
50.60
4.05b8
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58.#1
4.0523
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55.0
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2.8502
1
5Y.70
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62.06
2.8451
1
64.43
2.8405
60.0
62.48
2.~7~9
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2.0710
1
67.46
2.0678
1
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1.5544
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1
69.97 75.44 80.83
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1.3520
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122.81
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127.46
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132.06
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130.0
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3.5800
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1.4250 5.9016 2.8842 1.5811 9.4309 6.0024 4.3217 2.8095 2.0317
3 2 2 2 1 1 1 1 1 1
30.0
37.18
2.8866
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60.0 65.0
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78.10
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1.5313
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1.5128
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9.2725 7.4272
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7.3535
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151.43 154.90
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L.ObT>
0
2.9457 12Y.20 2.5543 133.4Y 2.2394 137.64 1.9842 l‘t1.6b 1.7762 145.55 1.6061 14Y.33 1.4667 15j;oo 1;3525
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l.jjUO l.Lbb3
LJ "
13d.63 lb2.28
l.LbZL 1.1311
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160.08 165.51
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161.54 164.73
1.0957 1.0369
1.1553
0
166.88 170.20
1.0766 1.0414
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167.86
9.9101-l
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l.iLUO
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9.3303-l
175.0
168.71 172.4Y 176.~2
176.75
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FU ‘NCTION
A CONVERSION
x
=
FOR
.450 @lii+l)
@(LAB)
ELAS’I ‘IC
CALCULATING
SCATTERING
. 550
.500 OICM)
FLX,Ol
ANGULAR
FtX,e)
DISTRIBUTIONS
.6o0
0lCMl
F(X,B)
QICM)
FLX,e)
7.75 15.48
3.5800 2.2460
5
8.00
3.5800
5
4
2.2460 4.4641
4 3
7 .L5
3.58bO
5
3.5800
5
2.2460
4
14.Y8
L.1460
4
15.u
14 .4tl 21 .b9
4.4644
3
22.44
4.4643
5
23.18
4.4642
3
15.98 23.93
20.0
28.85
1.424Y
5
LY.85
1.4148
3
30.84
1.4247
3
31.64
1.4246
3
25.U
35.Y6
2.YOb7
2
37.2c
5.899b
i
38.44
5.8986
2
3Y.69
5.8975
2
3U.U 95.u
43,ulJ
i.titc>i
L
44.48
i.tltiLi
L
45.96
2.8811
2
47.46
49.Yb
1.58vl
L
3i.b7
1.57YO
i
53.5Y
1.5779
2
55.13
2.8800 1.5767
2 2
bO.70 67.89
9.397Y 5.9682
1
62.69
Y.3855
1
1
5.Y552
1
74.92
3.Yljb4
1
70.10 77.36
3.9726
1
5.0
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7.50
40.u 45.U
56;61
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1
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I
63.55
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1
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1
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85.66
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1
88.44
1.9940
1
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1.9787
1
65.U
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1.5013 1.1431
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1.4883
1
114.90
1.4738
1
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1.4578
1
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1
101.12
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104.32
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1
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0
110.42
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70-u 75;o
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8;8886
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106.31
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5.36116
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162.69
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8.5017-l
165.13
7.3169-l
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163:Ou
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d:Y5lo-1
lbS.96
7.8547-1
167.46
6.7201-l
125.U
165.7'0
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ib7.zu
8.40?5-L
lbti.44
7.3451-l
169.69
6.2530-l
16U.l,
168.05
b.YIuti-L
ibY,o:,
I.YSII-1
170.84
6.9525-I
171.84
5.8950-l
lb5.0
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b.b4Yi-l
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173.18
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173.93
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174.*e
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174.Y8
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175.98
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178.00
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x
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=
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B(Citi)
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5
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L .1400
4
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34.80
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35.88
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2
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2
52.02
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i
38.67
1.5743
2
2
62.31
1.5720
2
i
bb.7‘+
Y.55Y7
1
60.48 63.82
1.5731
4O.U
56.tiY 64.7~
9.3468
1
70.95
9.3345
1
45.v
72.36
5.Y415
1
74.67
>:Y275
1
5.9134
1
79.45
5.8997
1
5U.L
7Y.8b
J.Y>,uti
I
ti~.43
J.Y4Lb
I
77.05 87.07
3.9273
1
117.79
3.9120
1
55.L
87.17
1.74iti
i
SY.YY
L,7Lb4
i
YL.91
2.7OY9
1
95.94
2.6919
1
6U.b
Y4.Lb
l.Ybdi
i
Y7.3L
L.Y441
1
.00.51
l.YL54
1
103.85
1.9057
1
b5.u
lOl.cJY
1.4401
*
104.Jb
1.4LOY
1
111.47
1.3779
1
107.05
l,U7Y4
*
ill.13
1.05Db
1
1.4001 1.0361
1
7d.U
107;w 114.81
1.0111
1
lid.itY
ti.L5lJl
0
111.42
7.7663
115.60
7.4909
o
llY.tiO
6.3646
0
a.o11> b;ijbO
n
0O.U
117.24 123.58
1 0
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0
127.61
5.8760
0
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5.5782
(I
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4.Y811 3.Y3Y5
0 0
1~9.21
4.7460
0
133.34
4.4762
o
137.84
4.1631
0
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lL5.Jb 13U.34
134.43
5.7021
0
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3.4289
O
135.3b
3.1461
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3.1104
p
'1'5.1,
ljY.il
i.YllO
143.34
2.6412
0
2.3281
0
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2.5365
0
143.58
L.>O7&
o o
147.61
2.0479
0
0
2.0650 l.OYbtJ 1.41L7
0 0 0
147.34
i.84b4
151.42
1.6012
o
151.13
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i)
174.81
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o
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143.89 147.b5 15l.UY
0
1.7501
105.1,
143.13 147.84 151.98 155.60 155.74
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0
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7.8988-l
154.2b
l.ltJc?tl
0
157.31
1.0105
0
1.0122 0 R.L154-1
161.47
1LU.O
163.85
6.1482-l
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i.UldY " 8,74>4_1 7.64YY_I 6.78bcj-1
15Y.YY
il.4a>r_1
6:77,1-l
165.94
5.0364-1
ibL.43
7.Ll5t)_l
5.6774-l
167.79
4.1403-l
14u.u
157.17 159.00 162.50 164.7~
164.07 lbb.74
b.2476-1 5.4851-l
145.0
166.89
6.1045-l
lu8.t.7
4.tiYO7-1
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168.Y7 170.Y4
5,2bY5-1
170.4Y 172.21
4.4ZtiY-1 4.0735-l
5.u 1d.U
1lO.U
8.25
_
-
1.3257
o
1.0158
0
160.~
172.84
4;tij71-1
173.85
j.&io41-1
160.51 16~.Yl 163.07 167.03 168.8L 170.48 172.02 175.48 174.86
165.0 17U.U
174.6Y
4.6046-l 4.4455-l 4..>5L7-1
>.b07Y-1 3_474r-1
17b.lY 17.7.4b
175.0
176.48 178.L5
175.44 ilo.Y8 178.50
;).>964-1
178.75
2.5008-l
179.00
1.6943-I
180.0
180.00
4.jZL2-1
18O.UU
5,.37OY-1
180.00
2.4806-l
180.00
1.6796-1
1jU.d 135.U
5.15>8-1
4.8409-l
169.45
3.4726-l
4.1VY7-1
170.95
2.9713-1
3.7072-l
172.31
2.5930-1
3.32Y2-1 3.0412-l 2.a252-1
173.58 174.76
2.3070-I 2.0917-l
175.88
2.6685-l 2.5623-l
176.95 177.99
l.Y318-1 1.8167-l 1.7391-l
91
.I. NURZYtiSKI
92
f’i:,+
= G&O) / {(zZ/E[MeV])2F(x,0)
},
(6)
R
where
F(x,O) E R(x,e)/J(x,o) = 1.2959964(1 +x)*[x~0~U+(1-x*sm~~)*]*~ *cosec4 {+[0+sin-‘(xsin0)]}(1
-x*sin*0)-).
From the relation (6) it is apparent that once the numerical value of the function F(x,O) is known the ratio a/aR can be obtained simply. This function, therefore, has been calculated and the numerical values are tabulated. 3. Description
are the exponents of the numbers in the second column. The following notations are used in the tables: @LAB) - 0 (scattering angle in the laboratory system), U(CM) - 0’ (scattering angle in the centre of mass system), F(X,U) - the conversion function. To use the tables one should first calculate the value of the parameter x = M,/M,. The calculation of the angular distribution in the centre of mass system is accomplished simply by dividing the measured intensities of the scattered particles in the laboratory system Z(Q) by an appropriate value of the function F(x,U) for a given x and 0. The normalization factor of the angular distribution is given by
and method of use of the tables
The calculations were performed on the Australian National University IBM-1620 computer, the tables being copied directly from the computer output sheets. In order to conserve space and obtain data in a simple form, a special subroutine for searching the exponents of the calculated numbers was devised and included into the main computer programme. The function &(x,0) calculated for x between 0.005 and 1.000 in intervals of 0.005 and for the laboratory angles 0 from 2.5” to 180” in intervals of 2.5” is tabulated elsewhere4), reduced tables being presented here. For a given x, the centre of mass angles, corresponding to the laboratory angles listed in the very left column on each sheet, are printed in the first column and the numerical values of the function F(x,8) in the second and third columns. Integers in the third column
C(E[MeV])*
/(z’Z’),
where C is a factor depending on the conditions experiment and is defined by the relation:
of the
clab(0) = CI(0) [mb/sr]. The author is greatly Titterton for his interest discussions. References 1) E. Rutherford,
indebted to Professor E. W. in this work and for valuable
Phil. Mag. 21 (1911) 669. 2) L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955). 3) A. M. Baldin, V. I. Goldanskii and I. L. Rosental, Kinematics of Nuclear Reactions (Oxford U. P., 1961). 4) J. Nurzyl:ski, Austr. Nat. Univ. Rep. ANU-P/349 (unpublished).
INSTRUMENTS
AND
A CONVERSION
METHODS
FUNCTION
41 (1966)
89-92;
0
FOR CALCULATING
ANGULAR
NORTH-HOLLAND
ELASTIC
PUBLISHING
co.
SCATTERING
DISTRIBUTIONS
J. NLJRZYrjSKI Research School of Physical Sciences,
Australian National University, Canberra, Australia
Received 24 October 1965 A conversion function, denoted as F(x,fI) has been calculated and is tabulated for a range of values of x and for the laboratory angles tJ up to 180". The use of the tables makes possible an easy conversion of the measured intensity of the elastically scattered
particles to the ratio cr/nn, where (Tis the differential cross section for the elastic scattering and OH is the Rutherford scattering cross section.
1. Introduction
to perform these calculations during the course of the experiment in order to understand the results which are being obtained. The tables presented here were, therefore, compiled to simplify the procedure. The relevant transformation relations for these calculations are3): 1. for the scattering angle,
By taking into account the Coulomb interaction between any non-relativistic charged nuclear projectile and a target nucleus a formula for the elastic scattering differential cross section can be derived. It is’*2): zZe2
cTR(O’)= ~-
i 2m*v2
2
1
cosec”(+O’).
(1)
u*/v’ = {sin (O’- 19)) / sin 8 = MI/M,;
where ze and Ze are the charges of the incident particle and the target nucleus respectively, m* is the reduced mass of the interacting particles, v is the initial relative velocity and 8’ is the centre of mass scattering angle. The equation is referred to as the non-relativistic Rutherford or Coulomb scattering cross section. In general, even for low energies a departure from the pure Coulomb scattering is observed. For the higher energies, where there is a significant nuclear interaction, elastic scattering angular distributions display a characteristic diffraction structure which changes smoothly with the energy of the bombarding particles and with the atomic mass of the target nuclei. The departure from the pure Coulomb scattering due to the influence of the nuclear forces and the contribution of the other competing processes can be observed and discussed more easily if experimental results are presented in the form of the ratio o(&)/o,(~‘), where ~(0’) is the measured differential cross section expressed in the centre of mass system. This form is, therefore, used frequently.
2. for the intensity
of the scattered
(2)
particles,
Z’/Z = cos (W- 0) sin’ 0 i sin2 0’ = J(O,Q’).
(3)
In these equations v* is the velocity of the centre of mass, v’ is the velocity of the incident particles in the centre of mass system, M, and M, are the atomic masses of the incident particle and the target nucleus respectively, I’ is the intensity of the scattered particles in the centre of mass system, Z is the intensity in the laboratory system and J is the Jacobian for the considered transformation. It is clear that O/% = ar,,l(Q,lJ).
(4)
Using eq. (2) one can rewrite the formula for the Rutherford scattering cross section expressing it in terms of quantities measured in the laboratory system: crR(Q’)= aR(x,O) = (zZ/E[MeV])2R(x,B)
[mb/sr],
(5)
where 2. Conversion function
R(x,@E 1.2959964( 1 + x)‘cosec4{f[8 + sin- ‘(x sin@]}, x = Ml/M,, sin- I (x sin 0) < fn.
As both scattering angle and the intensity of the scattered particles are measured in the laboratory system, to calculate the ratio G(O’)/CT~(O’) it is necessary to transform the measured cross section rrla,,(Z3)from the laboratory to the centre of mass system and also to calculate (~~(0’) for a centre of mass angle corresponding to a given laboratory angle 8. Usually it is desirable
Similarly, using eq. (2) one can express the Jacobian as a function of the same variables x and 0 as used in the expression (5). Then considering (4) and (5) one obtains: 89
.I. NURZYrj
90 x
=
.u50 QLCM)
O(LAB)
SKI
.150
.lUO
F(X,Ol
‘3ICM)
FIX,'31
OICMI
.200
F(X,O)
5.0 10.0
5.25 10.50
3.5800 2.1461
5 4
5.50 lO.YY
3.5800 2.2461
15.ii
15.74
4.4649
3
16.48
4.464Y
i0.U
LO.Yt(
I.4254
1
Cl.Yb
1.4153
25.0 30.U
26.21 31.43
5.9054 Z.ti8dl
2 2
27;42
5;9052
2
2b.63
5.9049
2
32.67
L.tia7Y
i
34.30
2.8816
2
35.0
36.64
1.5850
‘
5tl.LY
l.584b
L
> 4
5.75
3.5800
5
11.4Y
2.2460
4
3
17.22
4.4649
3
3
22.94
1.4253
3
QICMI
F1X.Q)
3.5800 2.2460 4.4648 1.4253 5.9045 2.8871 1;5840 9.4607 6.0326
5 4 3 3 2 2 2 1 1
jY.Y4
1.5844
L
40.0
41.84
Y.4704
1
43.bY
Y.4684
1
45.5>
9.4652
1
45.0
47.05
b.U4LL
1
4Y.05
b.0403
1
51.03
6.0371
1
6.00 ll.YY 17.97 23.92 29.85 35.74 41;39 47.39 53.13
50.0
5L.iO
4.UbdO
1
24.5Y
4.0601
1
50.60
4.05b8
1
58.#1
4.0523
1
55.0
57.35
2.8502
1
5Y.70
L.d48j
1
62.06
2.8451
1
64.43
2.8405
60.0
62.48
2.~7~9
I
b4.Y7
2.0710
1
67.46
2.0678
1
65;0
67.bcl
1.5544
I
1
69.97 75.44 80.83
1 1 1 1 0 0 0 o 0 0 o o o o o 0 0 0 0 o o o o 0 0 o
71J.2ti
1.5524
1
ii.81
1.5492
70.0
72.bY
l.lYbd
1
75.jY
l.lY4ic
1
7b.10
1.1915
1
75.0
77.77
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0
80.54
Y.4106
0
83.33
9.37bO
0
8O.U
02.82
7.5851
0
b5.b5
7.565b
0
88.4Y
7.5330
0
85.3
87.86
6.2147
pi
YO.72
6.1952
0
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6.1625
0
9OIU
92.d7
3.1715
0
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>.15bO
0
Y8.63
5.1253
0
95.u
97.86 102.42
4.37Yb 3.7270
0 "
100.7i 105.62
4.5601 5.7373
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103.59
4.3275
0
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107.77
j.Lb+Y
U
110.34
3.145>
0
lOcr.‘+Y 113.33
3.704d 3.LlLtI
o 0
110.0 115.u
111.bY
L.b719
0
115.jY
i.aiL4
0
118.10
2.81Y8
0
117.bii
1.3520
u
1ZO.ZO
2.5355
0
122.81
2.5030
0
1~0.0
112.48
I.LT72
U
1~4.Y7
L.L7bL
i,
127.46
2.2456
0
125.0
127.55
2.11871
ii
iLY.70
210677
j
132.06
2.0353
o
130.0
132.20
l.Yl44
(1
154.5Y
l.tlY50
0
136.60
1.8626
0
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0
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0
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1.7207
o
1 411 . iI
141.b4
1.6556
0
l‘+j.bY
i.O5bL
0
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0
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1.36UO
I,
14U.zY
1.24ub
0
14T.94
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0
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0
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0
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0
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0
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0
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o
16ti.u
lbO;Yd
ii5714
"
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0
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1.3201
0
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lb5.14
i.JJ‘t8
U
lbb.‘tb
1.515>
0
167.~2
1.2830
0
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1.2044 i.iY45
0 0
171.00 175.>0
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0 0
171.4Y
l.L58i
0
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170.20 175.L5
175.75
1.2433
0
180.0
180.00
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0
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I.2702
0
180.00
1.2383
0
X
.251;
z
BILABI
@(CM)
QlCi4)
.400
.350
.300
F(XIBI
2.0652 1.5446 1.1870 86.14 9.3321 91.36 7.4870 516.49 6.1164 101.54 5.0792 106.49 4.2814 111.36 3.6588 116.14 3.1669 120.83 2.7740 125.44 2.4573 129.97 2.2001 134.43 1.9899 138.81 1.8175 143.13 1.6757 147.39 1.5592 151.59 1.4639 155.74 1.3864 15Y.85 1.3244 lbJ.YZ 1.2759 lb7,Y7 1.2395 lll.YT 1.2142 176.00 1.1993 180.00 1.1944
F(X,B)
B(CM)
FlX,O)
5.11 1U.U
3.5tluo
5
b.20
3.5800
>
6.75
3.5800
5
lL.4Y
i.L4bO
+
IL.YY
L-L400
't
13,.4d
2.L4bO
4
15.0
18.71
4.4b4a
I,
lY.45
4.4647
3
~0.20
LO.J
24.YI
1.4222
3
i>.tiY
1.4251
3
25.3
31.U6
5.YO5,Y
2
32.28
>.YO3L
L
4.4646 1.4251 5.Y025 2.8851
3 3 L 2 2
6.L!J
@(CM)
F(Xve)
7.00 13.Ya
3.5800
5
1.2460
4
20.94
4.4645
3
1.4250 5.9016 2.8842 1.5811 9.4309 6.0024 4.3217 2.8095 2.0317
3 2 2 2 1 1 1 1 1 1
30.0
37.18
2.8866
i
5b.63
I.H85Y
2
Lb.88 53.51 40.08
35.0
43.14
L.5d34
L
44.Y1
1.582s
L
46.58
1.5aZO
40.0
49.25
9.4250
1
51.12
Y.4481
1
53.00
9.4401
1
45.0
55.18
6.02611
i
5.7.25
b.0198
1
5Y.33
6.0117
1
5o;o
61.04
4.3465
1
63.LY
4.OjT5
1
65.55
4.0312
1
55.0
66.82
2.8347
1
6Y.23
2.8276
1
71.66
2.H192
1
60.0 65.0
72.>0
L.G573
1
75.06
2.0501
1
17.64
2.04lh
1
L;186 34.73 41.54 48.26 54iYO 61.43 67.84 74.13 80.27
78.10
1.5307
1
tiO.ld
1.5313
1
03.4Y
1.5LL8
1
86.16
1.5128
70.0
83.2Y
1.1810
1
8b.37
l-175,1
l
MY.20
l.lb50
1
Y2.08
1.1548
1
75.u
9.2725 7.4272
0 0
Yllt14
Y;lYYo
0
Y7,ld
7.3535
0
103.20
7.1606
0
b.v>bb
0
lOL.JY
5.41825
0
0 0 0
0
99.4L
9;llOY 7.2647 5.8935
9.0077
85.0
108.48
5.7888
0
9o;o
104.4&l
5.UlY4
U
107.4b
4.Y45L
0
Y4;76 100.16 105.41 110.49
97.73
80.0
88.97 Y4.25
4.85bl
0
113.58
4.7512
0
95.0
109;42
4;2216
0
112.jY
4.1475
0
115.41
4.0585
0
118.48
3.9538
0
114.25
3.59Yl
0
117.18
5.5252
0
lLO.lb
3.4366
0
123.20
3.3325
r)
ila.Y7
3.1074
0
1i1;t(4
jIO33d
0
1L4.7b
127.73
2.8426
0
123.5Y
2.7147
i,
L'
i:
0 0
2.4524
L.5T83
L.b41b 2.3257
132.08
128.10
126.37 130.78
0 0 0 0 0 0 0
13b.26
2.1390
0
140.27
1.8854
0
144.13
1.6793
0
147.84
1.5111
0
0
151.43 154.90
1.3735 1.2611
0 0
1.1697
0 o 0
132.50
2.1414
c
135.ob
L.ObT>
0
2.9457 12Y.20 2.5543 133.4Y 2.2394 137.64 1.9842 l‘t1.6b 1.7762 145.55 1.6061 14Y.33 1.4667 15j;oo 1;3525
136.dL
l.'IJib
U
15Y.13
l.abO>
0
141.U4
1.75y:
U
14j.LV
L.bbYO
0
145.18 149.13
l.OldL 1.5011
0 c,
147.1>
1;54b4
0
140.0
l>l.ld
1.4330
"
145.0
15J.L4
1.4071
"
l>Lt.Yl
l.>jUI
"
15o.5b
1.25Y3
0
158.26
15o.u 155.0
157.18 16l.Ub
l.jjUO l.Lbb3
LJ "
13d.63 lb2.28
l.LbZL 1.1311
0 0
160.08 165.51
1.1838 1.1236
0 0
161.54 164.73
1.0957 1.0369
1.1553
0
166.88 170.20
1.0766 1.0414
0 0
167.86
9.9101-l
135.0
160.0
164.Yl
l.iLUO
U
Ib5.GY
165.0
1.1839 1.158d 1.1440
0 0 "
lbY.45
1.1175
0
170.94
9.5677-l
l/i.YY 170.50
l.OYLI 1.0781
0 0
173.4tl
1.0170
0
173.98
9.3303-l
175.0
168.71 172.4Y 176.~2
176.75
1.3027
0
177.00
9.1906-l
18O.b
18ti.lJO
1.13Yl
L
ltir;.OL
1.0731
0
lt10.00
9.97Y3-1
180.00
9.1446-1
170.u
FU ‘NCTION
A CONVERSION
x
=
FOR
.450 @lii+l)
@(LAB)
ELAS’I ‘IC
CALCULATING
SCATTERING
. 550
.500 OICM)
FLX,Ol
ANGULAR
FtX,e)
DISTRIBUTIONS
.6o0
0lCMl
F(X,B)
QICM)
FLX,e)
7.75 15.48
3.5800 2.2460
5
8.00
3.5800
5
4
2.2460 4.4641
4 3
7 .L5
3.58bO
5
3.5800
5
2.2460
4
14.Y8
L.1460
4
15.u
14 .4tl 21 .b9
4.4644
3
22.44
4.4643
5
23.18
4.4642
3
15.98 23.93
20.0
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91
.I. NURZYtiSKI
92
f’i:,+
= G&O) / {(zZ/E[MeV])2F(x,0)
},
(6)
R
where
F(x,O) E R(x,e)/J(x,o) = 1.2959964(1 +x)*[x~0~U+(1-x*sm~~)*]*~ *cosec4 {+[0+sin-‘(xsin0)]}(1
-x*sin*0)-).
From the relation (6) it is apparent that once the numerical value of the function F(x,O) is known the ratio a/aR can be obtained simply. This function, therefore, has been calculated and the numerical values are tabulated. 3. Description
are the exponents of the numbers in the second column. The following notations are used in the tables: @LAB) - 0 (scattering angle in the laboratory system), U(CM) - 0’ (scattering angle in the centre of mass system), F(X,U) - the conversion function. To use the tables one should first calculate the value of the parameter x = M,/M,. The calculation of the angular distribution in the centre of mass system is accomplished simply by dividing the measured intensities of the scattered particles in the laboratory system Z(Q) by an appropriate value of the function F(x,U) for a given x and 0. The normalization factor of the angular distribution is given by
and method of use of the tables
The calculations were performed on the Australian National University IBM-1620 computer, the tables being copied directly from the computer output sheets. In order to conserve space and obtain data in a simple form, a special subroutine for searching the exponents of the calculated numbers was devised and included into the main computer programme. The function &(x,0) calculated for x between 0.005 and 1.000 in intervals of 0.005 and for the laboratory angles 0 from 2.5” to 180” in intervals of 2.5” is tabulated elsewhere4), reduced tables being presented here. For a given x, the centre of mass angles, corresponding to the laboratory angles listed in the very left column on each sheet, are printed in the first column and the numerical values of the function F(x,8) in the second and third columns. Integers in the third column
C(E[MeV])*
/(z’Z’),
where C is a factor depending on the conditions experiment and is defined by the relation:
of the
clab(0) = CI(0) [mb/sr]. The author is greatly Titterton for his interest discussions. References 1) E. Rutherford,
indebted to Professor E. W. in this work and for valuable
Phil. Mag. 21 (1911) 669. 2) L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955). 3) A. M. Baldin, V. I. Goldanskii and I. L. Rosental, Kinematics of Nuclear Reactions (Oxford U. P., 1961). 4) J. Nurzyl:ski, Austr. Nat. Univ. Rep. ANU-P/349 (unpublished).