Nuclear Instruments and Methods in Physics Research B9 (1985) 89-96 89 North-Holland, Amsterdam

P R E D I C T I O N S O F S C A ' I T E R I N G F R O M AN A B S O R B I N G GAS T A R G E T

Fernando P A R E N T E

Departamento de Flsica and Centro de Ftsica de Fenbmenos de lonizaq~o lnterna da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, P-1699 Lisboa Codex, Portugal

Gene E. ICE *

Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA

Received 9 August 1983 and in revised form 22 October 1984

In gas scattering experiments using scattering chambers, the solid angle subtended by the detector is usually defined by a system of two apertures in parallel planes. Incident particles can be scattered into the detector from any point along a path that may be centimeters long and may have non-negligible height and width. Along this path the solid angle subtended by the detector changes. Two methods are described by which the number of scattered particles that reach the detector can be computed from theoretical cross sections. The solid angle is evaluated geometrically, for circular apertures. Absorption and polarization of the incident radiation are easily included in the calculations. In one of the approaches, the finite cross section of the incident beam, assumed homogeneous, is taken into consideration.

1. Introduction

In some measurements of scattering from gas targets, scattering chambers are used that are entirely filled with the target gas. These chambers minimize parasitic scattering f rom the chamber walls and windows, which is important for gases with small scattering cross sections. When such a scattering chamber is used, scattering can take place along the entire length of the chamber, instead of being confined to a well-defined region (fig. 1). To define the differential solid angle subtended by the detector, collimating apertures (usually circular) are mounted in the chamber scattering ports. Even so, the incident particles can be scattered into the detector f rom a path that may be centimeters long. Over this path length, the solid angle subtended by the detector, the mean scattering angle O, the differential scattering cross section do/d~2(O, ~), and even the intensity of the incident beam may change appreciably (fig. 1). A simple means of compar ing theoretical cross sections to experimental measurements is to assume that that the theoretical differential scattering cross section is related to the actual cross section by a constant. An experimental cross section is then determined by multiplying the value of the theoretical cross section by the ratio of the experimentally observed count rate to the predicted count rate

do do d N / d t E x p dl2 (0, ~, E)Exp = ~ -~ (0 , ¢p, E)TheOrdN/dtTheor , (1)

where the symbols have the usual meanings. There are several methods to compute, f rom theoretical differential scattering cross sections, the number

of photons or particles that can reach the detector. Computer programs have been written to compute the theoretical count rate based on a Monte Carlo method to determine the solid angle numerically at each

* Research sponsored by the Division of Materials Sciences, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

0168-583X/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

90 F. Parente, G.E. h'e / Predictions of scattering from absorbing gas target

r e a r ape r t u re > /

/ /

f r o n t aperture.

A Be>a m 0 B

Fig. 1. Scattering chamber geometry. Photons scattered from point A to point B on the incident beam axis may go through the two-aperture system.

point along the beam axis. Such calculations are costly, time-consuming and subject to large uncertainties [1]. An analytic method to compute the theoretical count rate for a scattering chamber with either rectangular or circular apertures in the scattering ports was presented by Silverstein [2]. Silverstein's method makes use of elliptic integrals, in the case of circular apertures, and leads to a very simple formula. In this method it is, however, rather difficult to introduce corrections for absorption before and after scattering, and for the finite dimensions of the incident beam.

Here we present two methods to compute the solid angle subtended by the detector at each point inside a scattering chamber with circular apertures at the scattering ports. The second of these methods is a variation of the first which allows us to treat the case of an incident beam with finite dimensions. Both methods lead to simple formulae for the predicted count rate and allow for the introduction of absorption corrections.

2. Geometrical calculation of the theoretical count rate

The theoretical calculation of the number of particles that are expected to reach the detector when a scattering chamber is used, is conceptually simple. The differential scattering cross section may be a function of the energy of the incident particles, the scattering angle 0, and of the azimuthal scattering angle q,, measured from an arbitrary reference plane containing the axis of the reference beam (fig. 1). The average scattering angle 0 depends of the position (x, y, z) at which the particle is scattered, and the angle of the scattering port into which the particle is scattered. Likewise, the solid angle subtended by the detector is a function of (x, y, z). If the apertures are well-matched to the experiment, the variation in the differential cross section is small and nearly linear over the range of scattering angle through which the particles can reach the detector from any point. Hence the total number of particles scattered into the detector is

Nsca,=Nookfff_ e-"'Ze-"2' [O(,,, y, z), , 10(x , z)dxdydz. (2)

Here, do /d l2 is the differential scattering cross section per particle of the scatterer, 12 is the solid angle, N o is the incident flux at the center of the chamber, P is the gas density, e -~,z is the incident beam attenuation factor and e -~2; is the attenuation of the scattered beam. The coordinate z is measured along the beam axis from the center of the chamber. The coordinates x and y are along two mutually perpendicular axes, horizontal and vertical respectively, perpendicular to the beam axis, and measured from the center of the chamber, l =- l ( x , y , z ) is the path length from the scattering site to the exit window of the chamber. The constant k includes corrections for counter efficiency and absorptions in the windows.

In the methods that are described below, the beam is divided into a series of short segments (method I)

F Parente, G.E. lee / Predictions of scattering from absorbing gas target 91

or small volume units (method II), and the relevant quantities (average scattering angle, average solid angle, beam path lengths before and after scattering, etc) are determined analytically for a point at the center of each segment or volume unit, respectively. The integral in eq. (2) is then evaluated numerically. When the integrated function changes smoothly with position z along the axis or with the position (x, y, z) in the beam, the integration process is fast.

3. Analytic calculation of the solid angle

3.1. Method I

In this method, the assumption is made that the incident beam lies in the same plane as the axis of the scattering windows, and the beam has infinitesimal height and width (x = y -- 0). The solid angle subtended by the detector window is defined by the scattering point z and the nearest and farthest of the circular collimating apertures set in the scattering port. One must determine the intersections of the two projections of these apertures on the unit sphere. If the angles subtended by the collimating apertures are < 10 °, the area of their projection on a flat surface tangent to the unit sphere is within 1% of the true solid angle. When first order corrections are carried out and integrated over the beam path, results typically differ by less than 0.1% from the zeroth-order results. The problem of determining the solid angle therefore is reduced to finding the overlap of two projections on a flat surface which is normal to the line connecting the scattering point z and the center of the intersection. The two projections are ellipses, the dimensions of which can easily be determined by the scattering geometry. The geometry is shown in fig. 2.

Because it is assumed that the beam is "on axis", the minor axis of the two elliptical projections will lie along the same line (fig. 2). The area of overlap for any two ellipses with dimensions shown in fig. 2 has been calculated analytically. The easiest cases clearly are those in which there is no intersection or one ellipse is completely contained in the other. These situations occur under the following conditions:

D >__ B 1 + B 2 Area = 0, (3)

D < B 1 - B E Area = ~rA2B 2. (4)

D, B a, and B 2 are defined in fig. 2. Intermediate cases are more complicated. For the point of intersection, (X, Y), it is possible to solve for Y. This yields

2 D + ~ 4 D 2 - 4 [ 1 ( B2A1 (A1B2] 2]

Y= 2[l_(BzA1/B1A2)2 ] (5)

XY

Fig. 2. Elliptical projections on a plane perpendicular to the unit sphere.

92 I': Parente. G.E. Ice / Predictions of scattering from absorbing gas target

_ - ~-2 vA~ d x ; (6)

F o r Y > D A r e a = L ( D + B 2 , / ~ x 2 B~,/a2 x 2) A2 V~'2- - A1 V'*l - dx. (7)

These integrals are listed in standard integral tables.

BI[X~ l l_X2+A2s in_ I ( X ) ] B2[X~22_X2+A2s in_ l (~2) 1 Y<=D: A r e a = ~ - 1 ~ +~22 -2DX; (8)

Y > D : A r e a = 2 D X + ~ X ~ + A ~ s i n 1 ~--~2 -~11 X~/A~-X2+A~sin-1 ~ " (9)

The area of overlap for the two ellipses is thus found, and hence we obtain the solid angle subtended by the detector window when the two ellipses represent the projections of the collimating slits on the unit sphere.

3.2. Method H

This second method differs from the first only in the way in which the effective solid angle subtended by the detector at each point of the scattering chamber is computed. One advantage of this method over I is that all points, both on and off the beam axis, are treated equally. A finite-dimensional beam can thus be considered without recourse to corrections.

We consider two elliptical cones having a common vertex inside the scattering chamber and defined by the inner and outer circular apertures in one scattering port. These two cones can be called "circular inclined cones" because the intersections with any plane parallel to the planes of the apertures are circles. The core of the method consists in computing the overlap of one of the apertures (e.g. the inner one) with the intersection of its plane and the other cone, this intersection being also a circle, and projecting this overlap on the unit sphere centered at the vertex of the cones. One must compute now the overlap of two circles instead of two ellipses for any point inside the chamber. The "center of mass" of the overlap area is computed and defines the "effective" or mean scattering angle at each point inside the chamber.

/ / /

~1///

I / / ~" OJ f / i ~ / / t

f //p/t ,e . . . . . . . . I

> 1

Fig. 3. Perspective view of the scattering chamber arrangement. Scattering occurs at point O in the figure.

F. Parente, G.E. Ice / Predictions of scattering from absorbing gas target 93

To help in visualizing what has been said, fig. 3 shows in perspective the beam axis and the two slits that define the exit port. In this figure the important quantities that pertain to the problem are defined. The distance from the center of the chamber to the center of the first aperture is R 1. The distance between the centers of the two apertures of the scattering port is R 2- R 3 and R 4 are the radii of the inner and outer apertures, respectively. The axis of the scattering port makes an angle 0 with the incident beam axis. The point where scattering occurs is defined by the coordinates p (along the beam axis), o and "r (along a horizontal and a vertical axis, respectively, perpendicular to the beam axis). In fig. 4 are shown the projections on a vertical plane through the axis of the scattering port (fig. 4a) and onto a horizontal plane (fig. 4b) of the scattering chamber apertures for a 45 ° scattering port. The inner aperture is the circle passing through points S and T, and the edge of the outer aperture passes through points M and N. (Double primed letters in fig. 3 refer to projections onto the vertical plane, and single primes refer to projections onto the horizontal plane). The line GU (with projections G " U " onto the vertical plane and G 'U ' onto the horizontal plane, respectively) defines the axis of the scattering port.

Our objective is to determine the solid angle subtended by the two apertures at the point O, defined by

i f ~ U "

. ' " J ~ . V "

IS ° g ° O"

(b) o.-- F'

J . . . . . . . . . . - . . . . . o 1

(c)

Fig. 4. Projections in a vertical (a) and an horizontal (b) planes of the scattering chamber apertures for 45 ° scattering angle, showing the solid angle subtended by the apertures at one point in the interior of the chamber. In (c) is shown the intersection of the solid angle by the plane of the first aperture.

94 F. Parente, G.E. Ice / Predictions of scattering from absorbing gas target

the coordinates (0, o, ~) relative to the center of the chamber G. For this purpose the intersection of the circular inclined cone subtended by the point O and the outer aperture with the plane of the inner aperture is determined• This intersection is also a circle, with radius R, shown in true magnitude in fig. 4c. The area of the overlap regions and its center of mass are determined. The mean scattering angle at point O is the angle/3 defined by the incident beam axis and the line OV, passing through the center of mass CM of the overlap region. The solid angle subtended by the two apertures at the point O is obtained by multiplying the area of the region of intersection by the cosines of the angles defining the direction OV relative to the plane of the inner aperture, and dividing by the square of the distance from the point of scattering O.

We must compute the overlap area of two circles defined analytically by the equations

X 2 + y 2 = R 2 '

( x - L )2 + y2 = R 2,

where we have assumed R < R 3. The calculation possibilities can be considered in first place,

L + R < R 3 Area=~rR2,

L - R > R 3 A r e a = 0 .

is lengthy but

(10)

(11)

straightforward. Again, two simple

(12)

(13) The intermediate cases occur for

R 3 - R < L < R 3 + R. (14)

Here we can still separate two cases. In the first, we have L > ~/R 2 - R 2 , and the overlap area is

a • -1 Y • -1 Y A r e a = ( f - £ - 2 L + ~ ) y + R 2 s m ~ 3 + R 2 s m ~ , (15)

where

r = ~R 2 -o~2 /4L 2 '

and

a = R 2 - R 2 + L 2. (16)

The second case corresponds to L < ~ - R 2 , and the overlap area is

( ° ) Y - R 2 Y (17) Area=TrR 2 - 2 L - ~ + ~ Y+R~sin-1~--~3 s in- I .

The coordinate of the center of mass of the overlap area, along the line joining the centers of the two circles, in the system with origin at the center of the larger circle, is

X C M = [ ( R 2 - R 2 - L 2 ) y + L Y ~ - y2+LR2sin- l%](over laparea) -1 (18)

for the first case, and

• - 1 Y - 1 XCM= [TrR2L-( L2 + R 2 - R 2 ) y - LYvFR f - y2 - LR2 sln -~] (over laparea) (19)

for the second case. The mean scattering angle/3 at point O(o, o, T) is given by the following equation:

/3 = tan-l~/tan 2 7' + ta n2 ~ " . (20)

Here,

XCM COS f COS 0 + R 1 sin 0 - T tan 7' = (21)

O + Ra cos 0 - X c i cos 0 sin 0 '

F. Parente, G.E. Ice / Predictions of scattering from absorbing gas target 95

and

tan 77" =

where

o -- XCM sin

O + R1 cos 0 -- XCM COS ~ sin 0 '

= t a n - l ( o l / 0 1 ) ,

~ ( R s _ ~ . ) 2 + ( p + R c ) 2 _ p + R , c o s 0 + p , s i n 0

c°sltan t

o, ~/(Rs_,r)2 + ( P + Rc)2

(22)

(23)

(24)

and

R2 (25)

tan[tan l( "S Here,

Rs = (R, + R2) sin o, (26) and

R c = ( R 1 + R 2) cos 0. (27)

The quan t i ty L which appears in eqs. (11) through (19) is also a funct ion of Pa and o 1

L = (P l + Ol) 1/2- (28)

4. E x a m p l e s

C o m p u t e r rout ines have been wri t ten for the calcula t ion of the number of par t ic les that arrive at the de tec tor after being scat tered in the gas inside the scat ter ing chamber , based on theoret ical values of scat ter ing cross sections using me thod I [3], and method II [4], descr ibed above. The ma in s t ructure of the rout ines is the following: s tar t ing f rom the center of the scat ter ing chamber , the solid angle and the mean scat ter ing angle for each po in t a long the axis are computed . In me thod II, all the relevant pa rame te r s for absorp t ion correct ions and other correct ions and other correct ions (as for polar iza t ion , for instance) are

Table 1 Predicted scattered intensities for one test case a) obtained according to the methods of Silverstein, and method I and II of this work, respectively, for selected parameters of the scattering chamber and unitary incident flux. R 1 and R 2 are defined in the text. R 3 and R 4 are the aperture diameters.

R 1 R 2 R 3 R 4 O Silverstein Method I Method II (cm) (cm) (cm) (cm) (deg)

0.5 10.0 0.1 0.1 60 3.148 × 1 0 - lo 3.147 × 1 0 - lO 3.148 × 1 0 - lo 135 6.418 × 1 0 - lo 6.415 × 1 0 - lo 6 .417 × 1 0 - lo

5.57 8.13 0.1 0.1 60 4 .209 × 1 0 - 1 o 4 .208 x 1 0 - to 4 .209 × 1 0 - lo 135 8.581 × 1 0 - 1 o 8.578 × 1 0 - lo 8.579 × 1 0 - 1 o

5.57 8.13 0 .974 0 .974 60 3.833 × 10 - 7 3.827 × 10 - 7 3.841 × 10 - 7 135 7.821 × 10 - 7 7.816 × 10 - 7 7 .844 x 10 - 7

a) Values of the neon X-ray inelastic scattering cross section according to Peixoto et al. [5], for incident energy of 4 keV.

96 F Parente, G.E. Ice / Predictions of scattering from absorbing gas target

also computed. These parameters are used to compute the number of particles expected to reach the detector after scattering, based on theoretical values of the scattering cross section. The calculation is repeated for a number of points in the incident beam axis (and in two mutual perpendicular directions, also perpendicular to that axis, in method II), until the solid angle vanishes (or the height and width of the beam are reached, in method II). The results are finally integrated and normalized. Although it may seem lengthy at first glance, either of these approaches represent a tremendous improvement in precision and computation time over Monte Carlo methods of computing the effective cross section.

The results obtained with the two methods described in this work were compared with the results of Silverstein's method, for different parameters of a scattering chamber, using the Ne X-ray inelastic cross section of Peixoto et al. [5], for 4-keV incident photon energy and two scattering angles. It was assumed that the beam had negligible height and width, and no absorption corrections were made. These results are compared in table 1. In cases 1 and 2, the radii of the apertures are very small compared with the distances of the apertures to the center of the chamber, while case 3 corresponds to a more realistic situation. In all cases the agreement between the three approaches is better than 0.3%.

5. Conclusions

Two methods are proposed to compute the theoretical count rate in gas scattering experiments where a scattering chamber is filled with the target gas. In the two methods the solid angle defined by two circular apertures in the scattering port is computed geometrically for each point inside the scattering chamber, and the expected number of particles that reach the detector is then calculated, based on theoretical scattering cross sections. These approaches represent an improvement over time-consuming Monte Carlo techniques. The most important advantage is that they allow for easy inclusion of corrections for absorption and polarization. The second method is specially designed for finite-dimensional incident beams. The results obtained in 3 cases from these methods are compared, for the absorptionless limit, with results computed from an earlier analytical method of Silverman [2].

This work was performed during the stay of the authors in the Physics Department of the University of Oregon with support from the Air Force Office of Scientific Research through grant AFOSR-79-0026. Thanks are due to B. Crasemann, who suggested the work, and to M.H. Chen and M. Breinig, for helpful discussions and advice.

One of the authors (F.P.) acknowledges the support from Instituto Nacional de InvestigaqSo Cientifica, Portugal.

References

[1] W. Aman, PhD Thesis, University of Oregon (1971) unpublished. [2] E.A. Silverstein, Nucl. Instr. and Meth. 4 (1959) 53. [3l G.E. Ice, PhD Thesis, University of Oregon (1977) unpublished. [4] F. Parente, PhD Thesis, University of Oregon, 1979 (unpublished). [5] E.M.A. Peixoto, C.F. Bunge, and R.A. Bonham, Phys. Rev. 181 (1969) 322.