CORRECTION FOR THE EFFECT OF BEAM HEIGHT

After the correction for the beam width has been made, we must correct for the effect due to a beam which is infinitely narrow but has a cl•rtain height along the y-axis. Let M again be the point of observation (Fig. 35b), and let the position of the point R in the direct beam be designated by the coordinate y = (2TTOR)/s).. The fraction of the total beam power contained in the segment of dimension dy, at the ordinate y, will be denoted by i2,0(Y) dy, where

. i2(y)

i2.o(Y)

=

^{Ji2(Y) dy}

If /(h) is the true scattering distribution (i.e., that obtained with a direct beam of infinitely small radius passing through 0), the observed intensity at the point M is

f1(hl

=

Ji2.o(y)J(Vh²

+

^y2J dy (10) where

h

=

(2TTOM)/s).

A very important fact must be pointed out here: when the true scattering distribution is represented by a Gaussian curve, the observed curve is proportionaltotherealcurve. This is easily seen, forif /(h) = Ae-.t"l•' f 1(h)

= [f

^{2. 0}(y)e-k'u' dy] Ae-k'h' =constant X I(h) (11) If the beam height is large and the intensity of the beam is a constant c, independent of y, equation 11 becomes

(lla) This demonstrates that the proportionality factor between the measured and true intensity distributions depends on the coefficient in the exponent.

This fact, first pointed out by Hosemann [79], is of considerable interest,

llll

since an exponential distribution is often a good approximation of the curves of low-angle scattering.

Shull and Roess [155] have pointed out a possible extension of this remark. If the experimental curve is not sufficiently well represented by an exponential function, one can try to separate it into a series of exponentials. Thus, if the measured intensity can be written as

$1(h)

=

2,T;e-klh¹

i

the transform of l(h) will be the sum of the transforms of the component exponentials. The expression for the true intensity distribution will then be

where

l(h)

= J.

^T/e-k/h•

i

I ki

T• =

y1;

Ti

This method is not generally usable, since not only is the separation of an arbitrary function into a series of exponentials sometimes impossible,¹ but also, even when it is possible, a general method for performing the separation is lacking.

When the equation of the scattering curve for an incident beam of point-like cross section is known, the scattering curve obsenred for a beam of infinite height can always be calculated by means of equation 10.

This was illustrated for an exponential scattering curve in equation 11.

Several other such results are found useful:

(i) If the asymptotic shape of the intensity curve is of the form of Kh-⁴ (equations 2.26 and 2.108), the observed curve varies asymptotically as h-3 . This is seen from the following:

I

^{C() 1}

J1(h) = K -co (h2

+

y2)2dy By making the substitution y = h tan ex

2K

f"'

^{2 dcx}

..?i(h) =

h3 Jo

-(I_+_t_a_n_2

_cx_)_

^{2 -co_s_2_:i:}

2K

f"'

^{2 7TK}

=

h3 Jo

^{cos2 a da =} _2h3 ^(llb)

or

1 The group of functions e -k/h' does not form an orthogonal group.

116 SMALL-ANGLE SCATTERING OF X-RAYS

(ii) In equations 2.28 and 2.111 the integral

i

^00h2/(h) dh appears. In terms of the observed intensity, this should be replaced by the integral

Ii"'

^{- h.F1(h) dh.} This can be shown as follows:

2 ⁰

["' h.F₁(h) dh = ["'

f

^{00 hl(vh2}

⁺

^{y2) dh dy}

Jo Jo

^-oo

By making the change of variables y = z sin ct and h = z cos ct so that

y2

+

^{h,2 = z2} and dh dy

=

z dz dct ["' h.F₁(h) dh

= f

⁰⁰

f "'

^{2 I(z)z2 cos ct dz drr..}

Jo Jo

^-,,,/2

= 2

L

^{00 z2I(z) dz (Ile)}

In the same way it can be seen that

(lld) These results are valid only if the beam can be considered as infinitely high even for the most distant parts of the curve.

(iii) Schmidt, Kaesberg, and Beeman (1954) have calculated numerically the scattering curve observed for spherical particles in a dilute system when the incident beam is of infinite height. (The scattering for an incident beam of point-like cross section is given by equation 2.31.)

3.4.2.1. Slit Correction for Infinite Height

A rigorous method of correction for this case has been described by DuMond [209] and by Guinier and Fournet [216), [217). It is rigorous only for the particular case in which the beam is of uniform intensity and infinite height. However, if the low-angle scattering decreases con-tinuously from the center, this method can be applied if the beam intensity i2(y), is constant up to a value of y such that .F1(y) is negligible. If the pattern consists of one ring of scattering, the beam must have a height at least equal to the diameter of the ring for this method to be applicable.

The true intensity distribution is determined by the following equation:

1

i""

^.F1'(Vh2

⁺

^{u2) du}

I(h)

=

-1TC O yh,2

+

u2 (12)

EXPERIMENTAL EQUIPMENT where c designates the constant value of i2.0(y},

.f1'(vh2

+

^{u2) =}

d.f1(V~)

d(vh2

+

^u2)

117

and u represents a variable of integration of no physical significance.

The derivation of this equation may be outlined as follows:

Equation 10 can be written 118

J.fh) = 2ci ⁰⁰J(Vh•

+

y•) dy Differentiation with respect to h gives the following:

.f1'(h) = 2c

i

0 ^oo J'( V h'

+

^{y1 ) •} v ^I h' ^h

+

y• ^dy Dividing by h and then changing the variable h' into h'

+

u• gives

.f_--~~~=2c 1'(Vh¹

+

u')

i""

^l'(Vh'

⁺

^u•

⁺

^y•) _dy

Vh'+u• o Vh'+u•+y•

Integrating with respect to u

i "" .f1'(Vh'

+

u') iooioo l'(Vh'

+

u'

+

y')

--===~ du = 2c du dy

o v'h•

+

u' o o Vh'

+

u•

+

y•

A second change of variables, u = r cos 0, y = r sin 0, then gives

i

^{oo .f}

_---====--

^1'(Vh'

⁺

^u") du = 2c

i"'

²

i""

^l'(Vh'

⁺

^r') I r I dr dO

o v'h•

+

u• o o Vh'

+

r•

= C7r[J(Vh

2 + r')]F-+OO

= -CWJ(h) r=O

on condition that l(h) approaches zero as h approaches infinity, which is true for scattering experiments at very small angles.

The use of equation 12 requires the determination of the derivative of the experimentally determined function 5 1(h). Then, for each value of h , a curve o f ..1'i'(Vh²

+

^{u2) •} is race ^{t d} an a grap ca m egra 10n is ^{d hi l . t t' .}

vh²

+

^u2

performed. The process is tedious, but it can be precise, as has been verified by Guinier and Fournet [217].

When the function.f1(h) differs only slightly from a Gaussian function, the calculation may be shortened considerably by writing

(13) where T, k, and f(h) are determined by means of an auxiliary graph of

ll8 SMALL-ANGLE SCATTERING OF X-RAYS

log.f1(h) as a function of h2• Application of equation 12 to equation 13 then gives

I _(h)

_{= ---=}

Tk _e -k'h' _{- -}1

f"'

^f'(Vh2

⁺

^u2) _du

cV7T 7TC o Vh²

+

u²

This approach leads to a. much greater accuracy in the determination of the function l(h).

3.4.2.2. Case of a Beam of Arbitrary Height

If i2(y) depends on y and, in particular, if the beam is of finite height (for example, i2(y) is a constant between -y0 and Yo and is zero outside of this interval), the property discussed above of curves of the form e _.,..,,. is still valid. For this type of curve the measured distribution can be used directly, whatever the intensity distribution of the direct beam, if only the shape of the curve is important. If absolute measurements of intensity must be made, equa1rion 11 must be employed to determine the proportionality factor.

The general problem of the arbitrary curve that cannot be resolved into a sum of exponentials remains to be considered. The solution we shall outline here was pointed out by Kratky, Porod, and Kahovec [111].

The equation to be solved is

(10) in which i2.0(Y) is some arbitrary function of y. Porod has shown that the solution for this general equation, a relation analogous to equation 12 but containing another function, g(u), is

/(h)

= - ~ f "'.f1~Yh2 +

u2) g(u) du

1TC Jo Jt2

+

u2 ^(12a)

The function g(u) is determined by the following condition: on making the change of variables u

=

r cos() and y

=

r sin 0, the integral

("'2

Jo i2.0(r sin O)g(r cos 0) d() should be a constant, i.e., independent of r.

If i2•0 is a constant, g(u) is a constant, and equation l2a reduces correctly to equation 12.

The authors did not indicate a method, even numerical, for determining g(u) in the general case.

When the beam is of uniform intensity and limited to the interval between -y0 and y0 , these authors indicated a solution which, though not rigorous, should certainly be a good approximation. An exact

119 solution of the function g(u) was determined for values of u less than Yo

v2.

For large values of u they found that the function tended towards the value g(u) = u/y0 , oscillating about this quantity. For their applica-tion they adopted the following values:

In practice, the method of application of this correction is the following:

the function

.f

'(vh2 + ^u2)

F(u) = -1

-yh2 + u2

is constructed as in the previous case for an infinite slit. The integration is more complicated because of the presence of the function g(u). First the integration of the function F(u) is performed over value's of u ranging from 0 to y0 • To this is added twice the value of the integration over the accuracy obtained is of the same order as that found in the correction for beams of infinite height by means of equation 12.

It is not yet possible to say that definite, complete solutions exist for the problem of correction for the height of the incident beam. Thus in practice one should either employ very high beams of uniform intensity and make an important, but calculable, correction, or else diminish the height of the beam as much as possible to diminish the error. We believe that this last solution is to be recommended, provided that the study is not extended to very small angles.

Actually, if the correction is small, an approximate method pointed out both by Fournet and Guinier [216] and by Franklin [214] can be used.

120 SMALL-ANGLE SCATTERING OF X-RAYS

This method still requires the determination of the slopes of the experi-mental curve, but these enter only in the corrective terms.

The above relation can be \I-Tit.ten, to a slightly poorer approximation, as

2cy0l(h) = f¹

(J

^{h2 -}

^~~:)

In document OF SMALL-ANGLE (pagina 123-129)