ATIEMPTS AT DETERMINING THE STATISTICAL DISTRIBUTION OF THE PARTICLES

It is easy to calculate the intensity curve by means of equation 2.93, J(h) = I.(h)NJ..pkFk²(h)

k

if the distribution of particles is given, that is, if the form, the size, and the relative fraction Pk of the particles of each type are given.

Conversely, several authors (Hosemann [79], [81]; Shull and Roess [155]) have tried to deduce the distribution of the particles from the experimental curve. But this problem does not have a unique solution.

Hypotheses must be made, and the results depend on these hypotheses.

Several of the methods that have been proposed will be discussed in this section.

Hypotheses are not made on the form of the particles, but it is assumed that they are all geometrically similar, so that the number of electrons nk

in a particle of type k, defined by a radius of gyration R0k, is proportional to Raok·

152 SMALL-ANGLE SCATTERING OF X-RAYS

It is assumed that the particles of type k give a scattering curve represented by the exponential law, e-<h'R'o.>13• We have already discussed the validity of this (p. 26). This approximation must be valid for the largest particles of the group for all values of the parameter h at which the scattered intensity is important; if small particles are present this scattering will extend to large angles, and in this case the complete calculations are correct only if the shapes of the particles are such that the exponential relation is a very good approximation up to rather large values of hR0•

Next it is assumed that the distribution of the radii of gyration of the particles is represented by a Maxwellian distribution of the form (Whit-taker anil Watson (1927))

(30)

where m(R0)dR0 represents the proportion of the particles whose radius of gyration is between R0 and R0

+

^dR0 • By means of the two para-meters, r ₀ and n, we can satisfactorily describe a wide range of distributions.

The arithmetic mean of the radii of gyration is given by

r (~ ⁺ 1)

RoJtI = ro

(n 1)

r - +

-2 2

and the fractional standard deviation of this value,

V

^t1R0 2/R_0, is l/V2(n

+

lJ.

The scattered intensity is given by equation 2.93 as proportional to the integral

Ok 2 - 3 -dR

I

^{m(R ) - a -}_R ^{nk e _ h2R'ot Ok}

Ok

As a result of the form chosen for m(R0k) the integration can be carried out, giving

constant

l(h) = __

[_1

_+_(h-~0-)--2]_,..(n_,+-..,.4)"""/2 ⁽³¹⁾

The problem is now to determine r0 and n in such a manner that this equation will suitably represent the experimental curve.

INTERPRETATION OF RESULTS ₁₅₃ Several methods of calculation have been given. The method due to Hosemann is based on the use of a curve of h²I(h) as a function of h. This curve always presents a maximum at some position hM greater than zero.

A second position, hT, is defined by the intersection with the h-axis of the tangent to the curve at the inflection point Won the high-angle side of the curve (Fig. 49).

h²l(h)

h

Fig. 49. The Hosemann method for analyzing the intensity distribution from a mixture of particles rs41.

Hosemann derived the following relation:

hp 1

2-h-M - 1

~

V-;=2(=n=+=l)

From this equation the parameter n can be determined. Then the arithmetic mean of the radii of gyration can be found by the relation

( n

+ 2)

1

J ^{- r -}

6 2

RoM = h M n

+ 2 ^{r (}

^{n ;}

l)

The mathematical proof of these results, as drawn from equations 30 and 31, will be found in an article by Hosemann [84].1

In the method of Shull and Roess, equation 31 is first "\\Titten in the form

n+4 (

3)

log I = constant - - - log h²

+

₂

2 r0

(32)

1 Hosemann considers systems of spheres instead of general particles defined by their radius of gyration. His formulas have been changed accordingly.

154 SMALL-ANGLE SCATTERING OF X-RAYS

Then the experimental intensity curve is constructed with the coordinate system log I, log (h²

+

oc), and a value of°' is sought for which the curve

It can be stated that, although it may be easy to determine the average dimension of the particles, it is difficult to select the form of the distri-bution law, because the difference between theoretical curves for groups with the same average dimensions but with different distribution laws are very small (see Fig. 50).

Roess and Shull [148] have made calculations for spherical particles and particles in the form of ellipsoids of revolution with a variable axial ratio, v, in which they employed the true structure factor instead of the exponential law. They obtained exact results by expressing them in terms of generalized hypergeometric functions. 'They were thus able to obtain a family of curves (Fig. 50) plotted with the coordinates log I and log a²h2 , in which each curve was defined by two parameters, n and v.

The parameter v described the form of the particles; n, the statistical distribution; and a, the dimensions of the average particle. The important conclusions of this distinguished work are that there is little variation in the form of the curves and also that the same forms are found for different pairs of the parameters n and v. If then we consider the experimental uncertainty of measurements, particularly at high angles, it appears that it is difficult to choose from the family of curves that par-ticular one which gives the best coincidence with the experimental curve.

Therefore the parameter n and, consequently, the statistics of the distri-bution are poorly determined. Again, very different values can be found, depending on the choice of v, that is, the form of the particle. This demonstrates the important limitations of low-angle scattering methods when the particles are not homogeneous.

Let us now briefly mention other contributions to this subject. Shull and Roess have repeated their calculations using a "rectangular"

distribution (uniform distribution between a minimum and a maximum size). It has also been shown (Riseman [145]) that, for a system of spherical particles, it is mathematically possible to determine the statistical distribution of sizes from the experimental curve without imposing a priori some form on this intensity curve. However, there seems to be little physical interest in such calculations.

I

I 1.0 0.6 0.4

0.2

0.1 0.06 0.04

0.02

0.01

0.1 0.2 0.4 0.6 1.0 2 4 6 10 40 60 100 Ro2h2

(a)

1.0 0.7 0.4

0.11---+--t---t---t---t---t---";c't"~:-'l<~'t---t---r---1

O.o? l----+--t---t---+----c---t--~-"<i'.--"<c-'<-t--'~-t---r---1

(b) Fig. 50. Calculated curves of I vs. (hR_{0 )1•} (a} Maxwellian distri-butions of spheres characterized by the pa.re.meter n; (b) Maxwellian distributions of oblate ellipsoids (axial ratio: v = 0.25). (Roess

and Shull [148].)

156 SMALL-ANGLE SCATTERING OF X-RAYS

In document OF SMALL-ANGLE (pagina 160-165)