DETERMINATION OF THE SPECIFIC SURFACE

In the preceding sections we have tried to use the low-angle part of the scattering curve. But in systems composed of particles of various sizes, that low-angle part which can be represented by an exponential approxi-mation is often inaccessible to experiment. Also, if the particles are packed together the low-angle part is perturbed by interference effects.

It is therefore more advantageous under these conditions to exploit the other possible approximation, that for the tail of the scattering curves. It would undoubtedly be possible to determine the characteristic function y{r) { §2.4.3) from the experimental curve without making any hypotheses, for, mathematicaJly, ry{r) is the Fourier transform of hl(h). But in practice the function y{r) can be used to obtain one important quantity, the value of the specific surface of the specimen, and this is deter-mined by the tail of the scattering curve.

The fundamental equation, equation 2.108, is valid both for systems composed of arbitrary particles distributed in an arbitrary manner and for an arbitrary distribution of matter of constant density p. method requires that the measurements of the scattered intensity be extended up to rather large angles, where the intensities are rather weak.

That the product h⁴l(h) becomes constant must be verified. Finally, the intensity of the direct beam should be measured in order to obtain an absolute value for S. However, if samples of the same matter but of different structures are compared, then, by examining samples all of which have the same mass, the ratio of the products h⁴I(h) gives the ratio of the specific surfaces of the specimens.

The assumption is made in these equations that the electronic density in the interior of the particles is rigorously constant, and this, by virtue of the atomic structure of matter, is never the case. This atomic structure gives rise to a high-angle scattering pattern such as that for an amorphous body, for example. The intensity to be expected from this effect in the low-angle scattering region is weak but not zero. It is negligible in the small-angle part of the region of scattering arising from the heterogeneity of the sample, but it can become perceptible towards the outer part of the curve, the part in which we are interested at this point. Other sources of extraneous radiation must also be added: fluorescence radiation, Compton scattering, thermal scattering by crysta.ls, various parasitic scatterings,

INTERPRETATION OF RESULTS 157 etc. The product h⁴/(h) thus will begin to increase when h becomes too large. The asymptotic law is also not valid for too small values of h.

There is therefore only a certain range of validity, whose extent depends on the particular sample, in which this relation can be used in making measurements of specific surfaces. These measurements lmve significance only if it is verified tlmt the product h⁴I(h) remains constant over a reasonable interval.

If the cross section of the incident beam is not pointlike but rather has a large height, then it is the product h3J1(h) which should be constant at large values of h (equation 3.llb).

We can avoid the measurement of the intensity of the direct beam in determining the absolute value of the specific surface by making use of the normalization relation, equation 2.111. If Vis the total volume of the specimen and c is the fraction of this volume occupied by matter (of electronic density p),

L''

^{h2J(h) dh = 2112}1.(h)pSVc(I - c) (33) Ifs.pis the specific surface per unit volume, then 8 =

vs,p,

and

S _ ^lim [h41(k)] _ lim [h'l(k)]

•P - • 2 - 7TC(l - c)

f

^m

V211p l.(k)

Jo

^{h•I(k) dk} (34)

When the scattering experiment is made with an incident beam of large height andJ1(h) is the measured intensity, equations 3.llb and 3.llc give

lim [h3J1(k)] =

i

^{lim [h4I(h)]}

and

i

^{m h2J(k) dh}

^{= -}

¹

i"°

h.F1(h) dk

0 2 0

Therefore

s.p

= 4c(l - c) lim [h3f1(h)]

f"'

k.F1(h) dh <³⁵>

The difficulty in evaluating the integrals

Lm

^{h21(h) dk or}

L""

hJ1{h) dh is that J(h) or J 1(h) is known only beyond a certain minimum value of h.

The contribution of the unknown small-angle part of the curve is certainly relatively small, particularly in the integral L""h²J(h) dk, since here the

158 SMALL-ANGLE SCATTERING OF X-RAYS

measured intensity is multiplied by h2 • Nevertheless, there is still some uncertainty which causes this method to be less precise than the method employing the intensity of the direct beam.

In interpreting certain results, Porod introduced the length

l

0=4Vc/S

which he designated as a "range of inhomogeneity." This length repre-sents an average1 of the diameters of the parts of the specimen occupied by matter; for instance, for a collection of spheres of radius R,

1

0

=

!R.

Another parameter derived from the characteristic function of the specimen which can be rather easily determined experimentally is the distance of heterogeneity or extent of coherence, l0 (Porod [137]), given by the integral 2

f

"'y(r) dr. Equations 2.109 and 2.110 determine l0

respec-Jo f"'

tively in terms of the integral

Jo

hl(h) dh and the total scattered energy E, the quantity obtained by measuring all the small-angle scattering outside of the direct beam in a counter or ionization chamber. These relations are the following:

and

l = _{• 277 V p}2c(I -₁ c)l8(h)

i""

o hlhdh ( )

l = 1

~

• V p2c(l - c)I.(h) A_2p2

where p is the distance from the sample to the receiver.

(36)

(37)

By making use of the normalization relation, equation 33, these become:

i"'hl(h) dh

l =71'=0

-• L""h²I(h) dh

(38)

and

271² E

z. =

A_2p2

f"

^{h2J(h) dh}

1 There are several ways of calculating the "average value" of the diameters of a particle. The average value considered here is not the same as the average used in §2.1.2.4 (IR for the sphere).

159 For incident beams of large height, equations 38, 3.llc, and 3.lld give

f')

..F1(h) dh

l = 2

°

⁽³⁹⁾

c

L''

h..F1(h) dh

Direct measurements of the total scattered energy have not yet been employed, but, if the sample scatters rather strongly, the method devised by Warren (§3.3.7) could be used. The evaluation of the integrals

4

3

\ \

\

\

2 ^\

\

\

\. \.

\.

\.

\.

\.

o~~~"--~-'

...

~~~~~~~~__,~

0 0.5 _logx

Fig. 51. Log f 1 vs. log x for a sa.mple of naphthalene. black. f 1 is the scattered intensity for an incident beam of infinite height, and x iB the distance on the film. The dashed line indicates a line of slope, -3.

(L. Kahovek, G. Porod, H. Ruck [97'].)

fhf1(h) dh and particularly

ff

1(h) dh cannot be made very accurately because of the unknown small-angle part of the curve. If the small-angle scattering increases rapidly with decreasing small-angle (Fig. 51) the extrapolation of the curve to zero-angle is very uncertain.

As an illustration of the application of these methods let us discuss the results obtained by Kahovek, Porod, and Ruck (97'] for a sample of naphthalene-black in which c = 0.12. The experiments were made with monochromatized Cu Koc radiation. The vertical dimensions of the slits defining the primary beam were large, so the beam could be considered as infinitely high. The intensity measurements were made with a Geiger counter. Figure 51 shows the curve of the variation of log./ 1 as a function of log x, where x is the distance between the point of observation and the direct beam. The curve verified the relation lim x3f1(x)

=

A, a constant.

160 SMALL-ANGLE SCATTERING OF X-RAYS

In addition the following quantities were determined:

f

^{.F1(X) dx}

⁼

^E

f

^{x.F1(x) dx}

⁼

^Q

From these it was found that for this sample,

S h A

V

=

4 ; c (1 - c)

Q =

7.5 m.²/cm.³

l

0

=s

4Vc ^=650A

l _• =2--=600A xE _h Q

An electron-microscope study of this specimen showed spherical grains with a diameter of approximately 1000 A.

REFERENCES FOR CHAPTER 4 Debye, P., and Menke, H. (1931), Physik. Z., 31, 797.

Ehrenfest, P. (1915), Proc. Amsterdam Acad., 17, 1132 and 1184.

Fournet, G. (1955), J. phys. radium, 16, 395.

Perutz, M. F. (1946), Trans. Faraday Soc., 42, 187.

Verwey, E. J. W., and Overbeck, J. T. G. (1945), Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam.

Whittaker, E. T., and Watson, G. L. (1927), A Course of Modem Analysis, Cambridge, London.

5. COMPARISON OF THE RESULTS FROM

In document OF SMALL-ANGLE (pagina 165-170)