Table Of ContentELECTRON DIFFRACTION IN GASES
A Thesis
Submitted to the Faculty
of
Purdue University
by
Walter Malcomson Barss
in partial fulfillment of the
requirements for the Degree
of
Doctor of Philosophy
May, 1942
PURDUE UNIVERSITY
THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION
BY Walter Mai com son Bnrss
ELECTRON Ll'ah'RhCTlOii IN GASES
ENTITLED
COMPLIES WITH THE UNIVERSITY REGULATIONS ON GRADUATION THESES AND IS APPROVED
BY ME AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE DEGREE OF
Doctor of Philosophy
\WAi-
Professor In Charge of Thesis
\UrC^- W
"^ ^ 3 Head of School or Department
June 13, 19 42
TO THE LIBRARIAN :-
THIS THESIS IS NOT TO BE REGARDED AS CONFIDENTIAL.
V^rCv_4r
Professor in Charge
Registrar Form 10—2-39—1M
TABLE OF CONTENTS
I Introduction . 1
A The Scattering of X-rays by Gases 1
B The Scattering of Electrons by Gases 5
C General Analysis Methods for Gas Diffraction 9
II Experimental Equipment. . . .. 13
A Construction and Operation of Camera 13
B Radial Intensity Compensation 16
C A Mechanical Aid to Intensity Synthesis 19
III Practical Methods of Analysis 24
A Measurements of the photographed diffraction pattern . . .. 24
B Derivation of intensity curves 26
C The Pauling and Brockway Method 3°
D The Fourier Analysis Method 37
E Synthesis of Intensity Curves 41
IV Application of Analysis Methods 44
A Comparison of Radial Distribution Methods for Carbon
Tetrachloride. ..44
B Determination of the Structvire of Chloropicrin CCI0NO2 . .. 45
C Partial Analysis of an Unknown Methane Derivative 63
V Conclusions
A Discussion of the Calculated Structure of Chloropicrin . .. 64
B Summary of Analysis Methods. 67
Bibliography
List of Illustrations and Tables
following page
Figure 1 The Electron Diffraction Camera 13
Figure 2 Correction for Background Intensity by a Rotating shutter. . 18
Figure 3 The aln x Synthesizer 20
Figure 4 Radial Distribution Curves for Carbon Tetrachloride 44
Figure 5 Microphotometer and Intensity Curves for Chloropicrin. . .. 46
Figure 6 Intensity Curves, s^ I(s), for Chloropicrin. 46
Figure 7 Radial Distribution Curves for Chloropicrin 47
Figure 8 Use of the Radial Distribution Function in Selecting a Model 48
Figure 9 Model of the Chloropicrin Molecule, CCIoN0 49
2
Figure 10, 11, 12 Comparison of Observed and Computed Intensities for
Chloropicrin . . . . . 58
Figure 13 Intensity Curves for an Unknown Methane Derivative . . . .. 63
Figure 14 Radial Distribution Curve for an Unknown Methane Derivative. 63
following page
Plate I The Electron Diffraction Camera and Auxiliary Equipment. . . 12
Plate II The sin x Synthesizer 22
x
Plate III Electron Diffraction Pattern for Chloropicrin 45
page
Table 1 Parameters for CCIoN0 for which Models were Calculated. . .. 57
2
Table 2 Correlation of best Calculated Curve vri.th the Observed Curve . 62
ABSTRACT
A brief review has been given of the theory of scattering of
electrons by gases.
The equipment used for this research has been described in some
detail, with particular reference to new or improved devices. The first
of these is a continuously driven rotating shutter placed in front of the
photographic plate to compensate for the rapidly increasing intensity of
scattering at small angles. The second is a convenient mechanical aid
to the calculation of intensity curves which facilitates the construction
of the curves and the choice of model parameters.
The standard methods of treatment of electron diffraction pic
tures have been discussed and some improvements introduced; in particular,
a computation scheme adapted to the production of Radial Distribution
functions for a preliminary choice of parameters.
The structure of chloropicrin has been determined by the new
methods and the results discussed in relation to other chemical and physi
cal data. The molecular parameters C — Cl = 1.75^0.0/A° angle Cl — C — Cl =
110.8°df2° and C — N = 1.59±0.04 A have been determined and the parameters
N — 0 = 1.21 A°, angle 0 — N — 0 = 127° shown to be compatible with the
observed diffraction pattern.
1
I INTRODUCTION
A. The Scattering of X-rays by Gases
According to the classical theory of the interaction of electric
charges with electromagnetic radiation, whenever radiation such as x-rays
passes through a region containing electrons which may be forced into os
cillation by the electromagnetic field, the radiation is scattered, with
an intensity distribution given by the Thomson formula
I = Io ( eT (1 + cos8s6 )
{1 r
where I = the initial intensity
0
R = the distance of observation from the scatterer
<fi = the angle of scattering.
Each factor in this expression has a simple physical significance. The
first indicates the common inverse square law of decrease of intensity
with distance from a point source of radiation. For scattering by elec
trons the factor (e2/mc2)2 is the square of the electron radius and
hence represents the cross-section of the electron for radiation. The
last factor accounts for the effects of polarization of the x-rays and
gives the angular variation of scattered intensity.
Since the specific charge e/m is much greater for electrons
than for other particles, the electrons in any material account for by
far the greatest part of the scattered x-ray intensity. When e and m
are taken as the charge and mass of the electron the formula gives the
intensity of x-r-iys scattered by a single electron, frequently referred
to as the Thomson scattering and given the symbol I .
Q
The intensity of x-rays scattered coherently by a single atom i>
given by the expression
I = I t2 (1.2)
e
2
where f, the atomic structure factor for x-rays, is defined as the
ratio of the amplitude scattered in a given direction by the atom to that
scattered in the same direction by a single "point" electron. The factor
f may be calculated theoretically if the distribution of electrons in the
atom is known.
Although it is possible to derive an expression for f assuming
that the electrons are discrete point charges the results so obtained do
not agree with experimental observation. Better results are obtained us
ing the quantum mechanical picture of smeared-out electrons, in which the
orbital electrons are considered as a diffuse spherically symmetrical
cloud of charge about the atomic nucleus. The form of the electron cloud
is best characterized by the density function u(r) which is a function
only of the distance from the nucleus. If u(r) is the true charge den
sity in electrons per unit volume, it is convenient to define the func
tion v(r) by the relation
v(r) = 4nr2 u(r) (1.3)
i.e. v(r) is the number of electrons in a spherical shell of unit
thickness and radius r.
The atomic structure factor calculated assuming a continuous
charge distribution is of the form
f(B) = ^Rv ( r ) ^ £ dr (1.4)
where R = the atomic radius.
This form permits the calculation of f(s) for any desired atom since v(r)
may be derived theoretically. Hartree2 has calculated v(r) for the light
er elements, of atomic number less than 20, and Thomas-* and Fermi4 have
calculated v(r) for the heavier elements. James and Brindley5 have com
piled tables of the function f(s) based on these calculations.
In the scattering of x-rays by a monatomic gas it is possible
to obtain the curve f(s) experimentally. Application of a Fourier trans
formation to equation (1.4) then permits calculation of v(r) by the for
mula
v(r) = J sr f(s) sin srds. (1.5)
Q
This provides a possible experimental means of checking on the calcula
tions of Hartree, Thomas and Fermi.
In the scattering of x-rays by groups of atoms it is necessary
to consider interference between rays scattered by the individual atoms.
In the case of a crystalline solid the regular arrangement of the atoms
gives discrete directions of reflection corresponding to various planes
of atoms in the crystal. In the case of liquids and amorphous solids of
monatomic composition there are no constant separations of neighbouring
atoms and no preferred directions.
Zernike and Prins treated the problem of scattering by liquids
by introducing a density function g(r) which is the number of atoms per
unit volume at a distance r from any one atom. They obtained an expres
sion for the fluctuating part of the intensity of the form
I(s) = NI f2 /^4TTr2g (r) .5i£-2£ dr (1.6)
e 0
where g (r) = g(r) - g~
Q
g = average density; atoms per unit volume
N = total number of atoms irradiated
f = atomic structure factor
Zernike and Prins then wrote (1.6) in the form
s p(s) = 4rry g (r) r sin sr dr (1.7)
o 0
Application of the Fourier Integral Theorem then gives
T g0(r) ~ 2$? JQ s p(s) sin sr ds (1.8)
Here again it is thus possible to calculate a distribution function from
the fluctuating part of the observed scattered intensity.
In treating the scattering of x-rays by gases the method is to
calculate the intensity scattered by a molecule in a fixed orientation,
then consider the effect of allowing the molecule to take all possible
orientations. The intensity of the x-rays scattered by a single gas mole
cule is given by the Ehrenfest' formula
I = I I k fi f j 5in 5 *M (1.9)
e
l s r^
i= j=i
where r^j = distance between atoms i and j
f^ = atomic structure factor for atom i
In diffraction of x-rays by gases it is necessary to consider
only the molecular structure and the form factors for the constituent
atoms, because the large distances between molecules make the intermole
cular effects negligible. In contrast, the scattering by a liquid whose
molecules are polyatomic depends upon atomic, molecular and intermolecular
interference with all three effects contributing to the observed pattern.
It is important to note that the factor (sin sr)/sr occuring in
formulas (1.4), (1.6) and (1.9) does not have the same significance in all
three cases. In (1.4) it is due to consideration of simultaneous scatter
ing by elements of charge in different parts of the atom and appears in
the expression for the amplitude. In (1.6) and (1.9) the same factor
occurs in the expression for the scattered intensity and is due to con
sideration of all possible orientations of a group of atoms with respect
to the direction of the original x-ray beam.
It should also be noted that only in diffraction by crystals
is there a one-to-one correspondence between a diffracted ray and a spe
cific spacing between atoms. In diffraction by liquids and gases there
is no such correspondence between rings of the pattern and specific inter
atomic spacings because each pair of atoms contributes a continuous inten
sity curve of the form (sin sr)/sr.
B. The Scattering of Electrons by Gases
g
In 1925 Elsasser suggested that it might be possible to obtain
interactions between a beam of electrons and the regularly spaced atoms of
a crystal. Two years later Davisson and Germer' succeeded in diffracting
a beam of slow electrons from the face of a nickel crystal.
Since then it has been found possible to treat the diffraction
of electrons by considering the interference of their associated de Broglie
waves scattered by the particles of an atomic or molecular scatterer.
The de Broglie wavelength associated with an electron whose mo
mentum is mv, is
X = h/mv (1.10)
where h = Planck's constant
The relativistic expression for this wavelength, in terms of the potential
in volts through which the electron has fallen, is
/v I em V/ V 1200m c2/
0 o
where m = rest mass of the electron.
Q
An approximate value, for lower voltages, is
X = /l50/V x 10-8 cm. (1.12)
The essential form of the scattering of electrons by charged
particles can be derived classically by the treatment Rutherford used
for the scattering of alpha-particles. The effective cross section of a
scatterer is given by rrb2 where b is the collision parameter. From con
sideration of simple mechanical laws and the Coulomb interaction between
