Structure factor determination ofions from X-ray diffraction patterns

Item Type text; Thesis-Reproduction (electronic)

Authors Rothrock, Glenn Edgar, 1905-

Publisher The University of Arizona.

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STRUCTURE FACTOR DETERMINATION OF IONS FROM X-RAY DIFFRACTION

PATTERNS

by

Glenn Edgar Rothroek

Submitted In partial fullflllment of the requirements for the degree of

Master of Science

in the Graduate College of the University of Arizona

1 9 5 5

Approved;Major professor Tmte

;

11£979/ / 93S B S

z-

PREFACE

This thesis presupposes a sufficient knowledge of elementary crystal structure to enable the reader to under­stand references made to crystal planes In terms of Miller Indices and to the cubic symmetry of crystals as well as to the different lattices found In cubic atomic arrange­ments, The reader should also be familiar with the physics of wave motion and Its application to Bragg1 s law, namely n (),= 2d sin e.

The author wishes to express his appreciation for the Inspiration and guidance given him by Dr, Walter Soller of the Physics Department who suggested the problem of the thesis, and under whose supervision the work was done. Special mention should be made of Mr, Walter B, Ormsby who made It possible to photograph the charts and graphs appear­ing In this thesis, and of various students and assistants of the Physics Department for developing X-ray films and analysing Densitometer charts.

Glenn Edgar Rothrock

University of Arizona May 1935

99500

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TABLE OF CQ1TEKTS

Chapter P&RQ“T ^ . .

Prefae®......................... 11: - . : : ■ ■ ■ ■ ; -

I. Introduction.................... 1

II. Biepretlcal Bistmaglon.........* 6III. Experimental Procedure...... . 21

Tabulated Reaulta............... 35IV# Dlaeuaslon of Results........... 48V. Conclusion...................... 54

Bibliography...,................ 57

INDEX OF TABULATED RESUIgS

• PagoTable 1. Relative Int enaltl es of seven specimens

of CaP#s S5' , . . ' .. - * : - : ■ ' " ' -

Table 2, Relative Structure Factors and glancingangles of (hid) spectra for CaF^. 34

Table 3, Relative Intensities of seven specimensof KI» ... . 35

Table 4# Relative Structure Factors and glancingangles of (hid) spectra for KI. 36

Table 5* A. Computed actual F of fluorine#B. e * w « Iodine. 37

Plate 1# Double Exposure chart of CeFg# 38Plates 2 & 3* Single Exposure charts of CaFg. 39-40Plate 4. Double Exposure chert of KI. 41Plates 5 & 6. Single Exposure charts of KI. . 42-43Plate 7. Structure Factor Curves for CaFfi and KI. 44

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Plate 8. Graphical working sheets for KI, 46' ' - ■

Plate 9* Plan of fhotodensltometer. 46Plate 10. F(9) curve. 47

*1-

CHAPTER l— *Hf R0DIFC5I0S

It was only abtmt two decades ago that Isa® and Bragg began experimenting in the analysis of crystals by means of X-raySe At that time they became aware of the fact that a crystal mad© an excellent three-dimensional diffrac­tion grating for those short wave radiations and although they realized after a fashion that crystals had some system­atic arrangement of their atoms they could not verify this theory until they were able to secure diffraction patterns by means of X-rays* This knowledge of crystalline structure has grown until now it is known that practically all rigid objects, except glasses and some waxes, are crystalline by nature* Hence, the study of the properties of materials is necessarily a study of the crystalline state of matter.!

Because of the practicability of many phases of crystal study, this type of research Should be of interest not only as a pure science, but also for the fact that it promises to be of much practical use fo the metallographer and through him to almost every branch of industry*1 2

Even at the present time, the analysis of crystallinematerials has taken quite Important place in the Indus-

' • ' - . '1, Davey, W« P: A Study of Crystal Structure and Its

Applications, P* 1.2. Ibid* p* 2* , .

trial research world# This analysis may he made by several methods^, but the particular method uhich has been taught and used with success In the Physics Department of the Univ­

ersity of Arizona is the f,Hu!l-Deby@-Scherrer” method* better known as the Powder Method#

The analysis of a crystal, using the above method, will ordinarily be carried out through a scries of steps which will be much the same regardless of the complexity of the substance# The determination of any atomic arrangement may then be expected to follow somewhat these seven stages as enumerated by Soller#^

1# Determine the type of lattice and the crystalsystem.-

2# Find the parameter of the lattice#5# Determine the absolute size of the unit coll of

which the unit lattice consists#4# Calculate the number of chemical molecules in the

unit cell*5# Determine the possible arrangements of the atoms

in this unit cell using atomic or ionic radii as the case demands,

6# Correct the unit cell and check It by means of space symmetry groups. 3

3. Seller, W,, Unpublished material.

7.. Caloulatethe Intensity of the important reflee-: tiona, (by analytical or vectorial methods) using tho Imom structure factors of tiie elements com­posing the substances and comparing it with the actual Intensities of the reflections.

, By observing the last step above, one eon seo that the final cheeking of the atomic arrangement of a substance depends upon a comparison between two intensities— tho actual intensity of, reflection of X-rays from any certain plane of atoms and tho expected intensity from that plane which is secured by calculations using the tion that the arrangement of atoms which is under , tost exists#

The Intensity of X-rays that aro reflected

be given by the following relations4

where (n) la the number -of atoms per unit volume of the crystal* ((t) ia the wave length of the X-rays used, (e) is the charge mi the electron, (1) is the length of tho slit on the ionization chamber, (h) is tho thickness of the crystal mass, (jn) is the density of tho crystal mass, (a)

is tho mass of the electron, (e) la the velocity of light, 4

4. Bearden, J. A., Phys# Rev., vol. 29, 1927, p. 21#

■ r*W is the distance of the ionization chamber slit from the crystal mass, (j») is the density of individual crystals in the crystal mass, (P) is the structure factor, a quantity #ilch is dependent upon the electron distribution, , (j) is the number of planes of a given typo contributing to the intensity of one lino on the film, (6) la the reflection angle, (Ps) is the power in the beam reflected by the pmid^?ed crystals at an angle 0, and (P) is the power in the incident X-ray ben*

Since there are only, four variables in the above equatim for any given specimen, it is obvious that this expression can be used in tho following simple form, namely;

Ir <* P2jf(9), ' (2)to determine tho relative reflection intensity from any , • . '— “— — 1 ■ '

on® plane as compared with the intensities from the other planes represented in the X-ray powder diffraction pattern, (F) being known for certain atoms at particular values of (9), while j and f(g) se (1 + Cos2 2e)/ain2 26 can always be determined so long as the (hkl) plane under considera­tion is known* These relative Intensities can be cocpared with certain integrated areas which correspond to the den­sity of the line on the diffraction pattern* Rietz found

he compared theso areas with one of their own number, say the largest area, that the same ratio existed between them as between the actual intensities of reflected X-rays

eslculated from Bearden*s equation (1)-#®Thus, it is soon that if Tie can dot ermine tho Inte­

grated areas referred to above, we have in substance a comparatively simple method of finding the quantity F sinco the integrated areas are relative only# This quan-

‘ r

tity F, the Structure Factor of an atom, is now defined as a number which expresses the ratio of the amplitude of the X-ray waves reflected by all the electrons of the atom in their natur­al positions to that of a single electron lying in the geo­metrical plane of reflection#^

It is the purpose, then, of this thesis to substantiate a comparatively simple method for finding tho absolute value of the structure factors of the more common elements by deal­ing with the relative Intensity of X-ray reflections from - crystal planes as described by Riots, and to actually deter­mine the structure factors of a few Ions which, to the know­ledge of the author, have not as yet been found.

In showing the meaning of the structure factor and Its place in tho equation for integrated reflection of X-rays from crystal planes, rigorous proofs of the mathematical relations involved will not be given#? * 7

®Riets, F. J., Thesis 1933, p. SI. U%yekoff, B. W# G#, "The Structore of Crystals", p# 90#7For such proofs seo the following refcrences:- Compton, A. H., "X-rays and Electrons", p.p. 119-131#Bragg, W. H#, "X-rays and Crystal Structure ", Opt. XIII

p.p. 196-228.Wyckoff, R, W. G., "The Structure of Crystals", p. p. 86-92.

CHAPTER II.— THEORETICAL DISCUSSION

Due to the fact that the cubic system of atomic arrangement is more symetrical than other systems, it will be convenient to consider a compound having this

3 % - '•crystal structure in introducing the discussion of struc­ture factor.

Take the face centered lattice of BaCl or KI as shown in figure 1* One characteristic of this structure is that all planes with even indices-*- are planes composed of both

en most authorities mention planes of higher order than one they do so hsing the Miller indices as follows: the first, second, third, and fourth orders respectively of

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metal and halogen atom, in #ilch cases all the atoms co­operate in their scattering* . However, the planes with oddindices are of a different nature* Here the successive■ ' ;planes contain^ atoms of the metal alone, and the chlorine atoms alone alternate. Consider now the reflections from the odd set of planes (111)* When reflection occurs from any set of planes, the true distance between such planes as given by Bragg*s law is from like piano to like plane, but in this case, half way between the planes containing chlorine atoms ore spaced planes containing metal atoms* These reflect waves which are, for the first reflection, just out of phase to the waves reflected from the halogen atoms. This effect is to tend to destroy the first re­flection and similarly every other reflection from planes of odd indices, while they strengthen the reflections from the planes of even order*

How notice figure 2 which Is a summary of the work

the (111) plane being indicated by (111), (111-2),(111-5), and (111-4). This is rather cumbersome to write

and confusing to deal with, and, if the order number is dropped, as is .so often done, one has difficulty in follow­ing the discussion* A simpler and more definite system of indices has been proposed and used to advantage by Dr. Boiler in hie courses at the University of Arizona.This system Is called "Reflection Indices", sad is essentially the some as that mentioned above except that the order number is included in the (Mel) Indices as a common factor* Thus, it will be noticed that evory reflec­tion has one, and only one, means of designation. The several orders of the above planes will now be written (111), (222), (355), (444), and this system will he followed through­out the discussion ofthe thesis*%ragg, W. H., "X-rays and Crystal Structure", p. 117.

8—

of Braggs showing how the intensity of the reflections of higher orders falls off with respect to the grazing angle# The lines are labled according to the planes which are reflecting the X-rays while their height is indicative of the intensity of the reflection of X-rays from the same

*

planes# It will be noticed that a line connecting thetops of the even numbered planes falls upon a smooth curve and that there Is a similar curve of a different slope for the odd planes^# It should bementioned here, perhaps, that this data was secured by working with single crystals# However, the above discus­sion applies to powdered crystals equally as well for simi­lar smooth curves are secured from powdered specimens#

%Compton, A. H#, "X-rays and Electrons", p# 117,

It has been ahosm by Bragg in hie earliest work thatthe smooth, curve joining the tops of these lines is repre-.seated approximately by the expression

H . pi. +' cosp. 29 e-B ain8 9 (.sin2©

In this expression (C) is a constant and 'depends upon ##' energy in the beam, the wave length of the X-rays, and the nature of the crystal specimen, Ihe f w t ^ ( I + c.os*'allows for the fact that the primary beam Is unpolarlzed, and the factor (e"B sin2 9) £S included to allow for thermal agitation* What seems most unusual, however, is that (sin2 9) in the denominator is an arbitrary factor ehoosMi by Bragg merely to make the calculated reflection fit the experimental data.4 .

It is known conclusively from optics that when the width of the lines of a diffraction grating are comparable to the distances between the successive lines that the intensities of the higher orders of the spectrum diminish rapidly. Since X-rays are fundamentally wave phenomena, and since a crystal serves appropriately in tee capacity of a diffraction grating, we can likewise expect tee higher orders of X-ray reflections to diminish rapidly. From an observation of figure 2 we notice that this is-indeed true.®

^Compton, A. H., "X-rays and Electrons," p, 119. ®Ibid., p. 119.

G. G, Darwin at this samo timo examined for the first time the intensities of X-ray reflections from crystal pianos as a function of electronic distribution, and ho found that if all the electrons were in the mid-planes of their atomic layers, then the Integrated intensity would he Inversely proportional to (sin 6 cos 6) instead of (sin8 9) as indi­cated by Bragg in equation (3). Thus there remains a factor of (l/tan 9) In the experimental formulae (3) which is pre­sumably due to the fast that the scattering electrons are not in the mid-planes • of their atomic layers; in other words, tiiat the aise of an atom is comparable with the distance from one atom to the next#6 - ' z

In the first part of this discussion one important approx­imation was made, namely, that each atom or ion was assumed to diffract like a single point whose diffracting ability was proportional to the atomic or ionic number. It would have been more in accordance with our pictures of the mochan- Ism of diffraction if we had assumed that when X-rays are diffracted, the actual diffracting centers are the electrons which compose the extra nuclear portions of the atoms in the crystal. Whether wo assume these electrons to be stationary or in motion in definite orbits around the nucleus, in any

ease it is hard to escape the conclusion that the X-rays

^Compton, A. H., "X-rays and Electrons0, p. 119.

diffracted from one electron must be somewhat out of phase with the X-rays diffracted by some other electron In the same atom. The total Intensity of any beam of X-rays dif­fracted by a crystal would, then, be expected to depend not only upon the configuration of atoms In the crystal, but also upon the distribution of. electrons In the atoms ofVwhich the crystal is composed*

It is more convenient and just as rigorous to assume that the eleetrons are In motion Instead of stationary and then the problem of determining the amount of X-rays reflected from an atom resolves itself into that of deter­mining the probability tia&t an electron at any instant will be a distance (8) above or below a geometrical plane passed through the centers of tho atoms. From this assump­tion an elegant mathematical discussion la given in several

texts® in determining the value of F, the structure factor of the atom* In mathematical symbols, then, tite structure factor F Is stated as follows;

F = m|p(b)cob[— ^ S L "jdg» (4)where (H) is the number of electrons in the'atom,, p (8) dS is the probability that any electron of the atom lies between (8) and (z + dz) above the geometrical lattice plane through its center, , (a) marks the greatest possible value of (8), and (4siS sin9)/^ measures the difference 7

7Davey, W. P., "The Study of Crystal Structure and It#Application#*, p# 292.

®c.fe .Works of A. H* Compton and W. P. Davey as listed In bibliography.

in path or phase of the ray diffracted by an electron at (15) compared with a similar electron lying in tho lattice plane.® This factor F, then, should become a part of Bragg* s formulae for reflected X-rays since the intensity will surely depend upon the way the electrons of the atoms are situated with respect to the geometric planes.

It will probably enlighten this discussion somewhat at this point if we consider just how the structure factor.

and hence the intensity of reflected X-rays, varies with the glancing angle 9. Suppose the atom lies with its cen­ter at 0 on a crystal plane, (figure 5), and consider its contribution to a spectrum formed by reflection at a graz­ing angle 9 from the plane. It is clear that the radiation contributing to this spectrum has been scattered by the atom through an angle (29). The radiation scattered by an electron at (A), distant (x) from the plane, differs i n .

%y®hoff, W« G., ’’The Study of Crystalsn, p. 90.

phase from that scattered by electrons lying in the plane by (4n3 sin 0)/^ * How (8) may be greater than & , since the diameters of the atoms are in general several times the wave length of X-rays, so that the phase between the contri­butions to the scattered beam from different parts of the atom may be considerable if the angle (0) is at all large#10 11 If © ” 0, all the scattered components have traveled equal paths. The contributions from all the electrons are then in phase, and the total amplitude from the scattered beam is (H) times that scattered in the same direction from a single electron. However, as (0) increases, the phase differences increase, the contributions from the different parts of the atom no longer ©©-operate, and the total ampli­tude scattered is less than (H) times that due to a single electron# The factor (F) therefore approaches the total number of electrons in the atom for small angles of scatter­ing, but decreases rapidly as the angle of scattering increases13-

which is clearly seen from the structure factor curves as determined in this thesis#12

It will be recalled that in Bragg*s formulae for inte­grated reflection (equation 5) there was placed a factor (®“B oin2 e), to account for the thermal agitation of the

10c,f. Figure 211James, R. W#, "X-ray Crystallography"1S@#f. Plate 7.

cryotal atoms. It is to be expected that temperature Trt.ll diminish the intensity of reflection, and that the higher the temperature, tide greater will be the offset. Further­more, the effect of temperature should be more manifest in higher orders where the consequences of the departures of the atoms from their average positions are more serious.15 Tlieno expectations were borne out to some extent through data from certain experiments of Doybe. Compton is of theopinion, however, that his work docs not give great confi-

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dence in the necessity of this thermal factor In the reflec­tion formulae.14 The structure factor F, since it depends upon the distances of the electrons from the middle of the atomic layers, will take account of the thermal displace­ments of the atoms, and in view of the uncertainty of the thermal calculations, it is probably wiser to leave the expression for the reflected energy in its present form#The phenomena of primary and secondary extinction also influence the reflected intensities of X-rays from the planes of single large crystals, but by powdering the speci­men as is done in this experiment both, extinctions can usually be eliminated.15

Mow comparing Bearden1 s equation (1) with tho one Bragg originally proposed for Integrated intensities, we see that

^Bragg, W. H., "X-rays and Crystal Structure0, p. 207# 14Compton, A. H#, "X-rays and Electrons", p# 126#15Wyckoff, W# G#, "The Structure of Crystals", p. 92.

after adding the factor F and disregarding the thermal agitation factor (c”B sin2 9y that the two equations are substantially - the some. It is well to note here perhaps that the structure factor we Imre been discussing so far is called, by more recent authorities, the natomic structure factor” F or the scattering powers of the individual atoms of a crystal under conditions of reflection. The composite ef their several contributions expressed by the structure factor has been seen to measure the "reflection structure factor” or F*,1G Taking the simplified form of Beardenis equation (2), and transposing it for F one gets

F « F ‘ cc Vir/jf(ey » ?r. (5)

Since FjL is proportional to Ir, a term' that Is only rela­

tive in comparison to the other intensities on the diffrac­

tion pattern, then Fp will be merely a relative term, and it shall be treated as such for the time being#• In order to use the foregoing equations to calculate the structure factor, it is essential to know how they must he altered to apply to the various atomic groupings for it is clear, that since such arrangements merely pro­vide more elaborate electronic distributions about the geo­metrical reflecting planes, the structure changes will be function# of the factor F# Consider the figure below which

^6Wyckoff, W, G,, "The S tincture of Crystals ”, p, 92* ~

is a diagram of a crystal of two atoms with alternate pure planes. Hhen reflection occurs from some certain group of planes as A, A*, A", etc,, we know that the path differ­ence between these planes is 2m , i* #. one or more cmoplet# wave lengths,1 and the amplitude of this wave due to the con­tributions from the atoms when compared to that caused by

the seme number of electrons at the lattice points would be the atomic structure factor F* Row when a group of planes as B, B 1, Bn, etc,, are placed in between the A-planes, we

1 1 l i1 i i1 i l1 i i- ! iI !i i i i i t i ia b b ' a " e" er eut

3 H--

have something different. At the beginning of this section it was shown how waves from planes like those in the figure above tend to neutralize each other. The resulting ampli-

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tude of this wave is going to be the difference of the struc­ture factors of the A and B planes of atoms, or more generally a vector sum of the atomic structure factors. If the A and B atom happen to have the some number of electrons, and hence the same structure factors, their effects would be entirely neutralized. One can imagine more complicated crys­tals where there are several planes interposed between the unit distance planes instead of one as in figure 4, and all

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sorts of phase relationships may exist# Likewise, it will hold that even for the most complex plane and phase arrange­ments the reflection can be secured by a vector sum, although to use this method one must analyse each set of planes intervening and the phase relation of these planes, which for the most part Is rather laborious#

By means of an optical method the problem may be stated as that of compounding several wave motions all of the seme period, but differing in both amplitude and phase. Ho matter ho# complicated the crystal system is; a group of

s -planes containing all of its atoms can obviously be passed parallel to any possible face* In the most general case there will be as many different kinds of planes as there are atoms in the unit cell* In the diagram above lot o^,

* Pi, p8, — 8%, sg, — — , be the planes con­taining the different kinds of atoms and let F., p ' p ,° p - s'----, represent respectively the amplitudes of reflectionfrom the e-planes, p-planes, s-planos, etc* Then theposite reflection F* is the vector sum of F0,. p ,. Pc>, or

F'2 « A2 + B2 (<

where A - F0 cos + Sp cos ^ + P8 cob $b + — — a»d ; B« Fq mim ' p o , 4 Pp si* /fp P8 sin' ffB + •— -»which when interpreted in Cartesian coordinates can be shewn thus. These phases can readily he expressed in terms of the coordinate positions of the atoms lying in them.

3 , 4 V ,

c';For a cubic crystal it can be done as follows. The dist&aeebetween geometrical like planes (figure 5) is known to be

dhkl = + ka + 1S = oio2 =. P1P2 = e1o2 — — (7)

The equation of the (hkl) plane containing the origin is hx1 + ky* > la1 = hxaQ + kya0 > lza0 » a0(hx+ ky + Iz) » 0, (@)

where ao is the edge of the unit crystal cube. The equation of a plane parallel to this plane through any other point xsyBss is

t'“ * » * - > 'sr1%% IS,where the distance from any point s to a plane is given by

the expression.

f(hXs + # 6 + 130)* h2 + 3c2 + 12 ao = oisi (figure 5) (10)

Bering refleetiee# the waves from geometrically like planes • • • • _ . ‘ * ' -

reinforce one another and ihus, as was discussed above, thephase difference between the waves coming from ox. arid og or 02 and 03 is 2an. The phase difference between o% and sx» however, is some fraction of 2nn or

(hxs + kyB 4- Iza)A „ ^ + 122sn ' an_______

' W T W 7 ~ &

* hXg + # B + 1*B* (11)

‘ = 2im (hxs + kyB + lzs). (12)Substituting these values In the expression for P*2 ®A2 + B2, we finally have the relation we were after, namely:

F*2 a £p0 cos 2tm (hx0 + ky0 + lz0) + Fp cos Son (hxp > kyp + lzp) + j 2 + |Fo ein 2fih (ta©- + # 0 + ls0) + Fp sin 2sn (hxp + kyp + lzp) + — -•-j2» (13)17

This expression applies only to crystals of cubic symmetry hut one can readily change it to apply for other symmetries, and even oblique axis* The quantities F©* Fp* in.equation (15) are evidently proportional to the average

the le St S r r i « M f * W 5 8 . iB

amplitude scattered by Warn o, p» a, , single atoms res­pectively in the direction of the film. If the relative scattering powers of the atoms, in ether words the F factors, and their positions In the unit coll are known, we can cal­culate the reflection F* and hence the amplitude of the (hkl) spectrum* Conversely, if we measure the relative intensities of the different spectral lines on the film we may by exam­ining the way in which they vary with each other be able to determine the coordinates of the atoms. All methods for deter­mining the structures of complex crystals are based on this idea.#

Using the method of Reitz1 s thesis, as stated before,§'■wo can evaluate FP by knowing Ip, the relative intensity of

any particular (hkl) reflection. By equation 15 above w© can determine absolute F 1* in terms of the atomic Fa. Let us suppose now that we have a powdered compound containing two kinds of atoms and the structure factor of one of these atoms is known over a sufficient variation of 9, the glanc­ing angle. If now we can change Fp to F f, then we can, by equating the two values of F*, solve the resulting equation for the unknown atomic F. The exact graphical procedure of this process will be taken up in the next section.

•^Jameo, R. W., "X-ray Crystallography", p. 52.

CHAPTER III. — EXEERIMEMTAI, IROCEDURE

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In ealeul&ting Fr » */lr/jf(©) we must evaluate the Irfor each (hkl) spectral line, and determine the glancing angle for each particular speetrwiae well. The first X-ray diffraction from an atomic plane occurs for most crystals at an angle in the neighborhood of 5® to 10°. It is possible # the means at hand to measure the intensity of spectral lines out as far as 25® or 30®, and thus it is seen that the degree of freedom of the variable 0 in the above equation lies somewhere between 15° and 30® which is sufficient for our purpose.

Bietz in his thesis has itemized a process for the ana­lysis of "Densitometer Charted consisting of six steps. He makes the technique quite plain and definite for securing the areas under the peaks on the Densitcmeter charts, and the reader is referred to his work to become familiar with this process. Briefly, however, he translates the densities ef the lines on the film into areas on the charts and then converts these areas into relative intensities of the spectra■ • ■ ■ . . " : v - ' , ,by merely taking the ratio of each area with respect to any

%letz, ?. J., Thesis 1933, p. 9.

one certain area of the group, say the largest.Several suggestions might further be given the reader,

however, in reference to this analysis. In view of the fact that the crystals of the powder are not infinite in number, it is quite possible that in some specimens more planes will be lined up for certain reflections than others, and by taking the results of from five to ten different specimens of the same substance better results are bound to be obtained. Another quite important step that could be added to the six steps as given by Rietz is that of correl­ating the chart peaks with the (hkl) planes of the crystal, 'and this will be discussed quite in detail.

Correlating the line® can be done directly from the...XT"

film by the use_of the 0, E, Angstrom log Scale (especially adapted tp the dimensions of the oasettes on the X-ray diffraction apparatus), and the logarithmic charts in the

Instruction Book^, It can also be done conveniently from the Densitometer charts by means of an improvement on the film carriage® and an instrument designed and built in the Physics Laboratory which, for lack of a better name, will be called the "Densitometer chart Protractor11,

^General Electric Company Instructions for operation of X-ray diffraction apparatus. ®See plate 9,

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The Improvement on the film carriage referred to above is a film holder which slides up and down, perpendicular to the motion of the film carriage and permits a separate trace of both specimens on the film to be made on the seine Densi­tometer chart, and at the same time superimposing the zero lines of each specimen. Since the zero lines are identical

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on the chart, we know that there is a direct correspondence between the angular distances on both traces. Hence, by choosing one of the specimens as standard, (such as N&C1), about which all crystal Information is known, we have a check on the peaks of the trace of the unknown and thereby the angle at which each spectral line of the unknown spe­cimen was diffracted.

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The protractor mentioned above was used to interpret these lines in angular distances from the zero line. The protractor consisted of a brass sheet 12” x 56” upon which arcs ©f circles were scratched by means of a sharp steel in th® manner Indicated by figure 7, The figure only shows scale divisions every 10°, but on the instrument every 0,1° was marked so that the protractor could be read to hundreths of a degree quite accurately.

At the positions which correspond to the glancing angles of the SaGl spectral lines other scratches were then made. These were filled with black ink so that they would easily stand out from the other scratches. Lines were then drawn on a narrow strip of paper from each spectral peak of the unknown crystal traco perpendicular to the galvanometer zero line,* Analytically similar lines of a different tex­ture (colored, dotted, etc,) were also drawn on the some strip of paper from the Ha€l spectral peaks. How, by plac­ing this strip upon the protractor and shifting it slightly to and from the center of the arcs, a perfect agreement could he secured between the colored HaCl linos on the paper and the blackened scratches on the protractor. This can

best be done by matching only two of the HaCl lines at a time and with sufficient accuracy. When the strip of paper

*Seo plates 1 and 4«

iB in thio position it ie simple enough to read off the •agles eorresponding to the unknown spectral peak lines#

SisS&aees" can be seen "by an inspection of Table 2. Efom Bragg*slaw, sin 0 * a&/2d» it will be noticed that sin 9 varies directly as the order (n) of the reflection, or that sin 9 for n « 4 will be four times as large as for m * 1. How notice the sine function for the (hkl) reflec­tions (220) and (440) or for (111) and (444) and it will*• • • ? i* - • *

be seen that the comparison is remarkably close.This accuracy is due in part, no doubt, to the mag­

nification of the distance between the spectral pe&ks by the Densitometer, but also to a large extent as a result of the fact that by directly matching the lines of the unknown with those of Hadl, which are standard, all errors due to'a shrinking of the chart or the film or an uneven- ees of the cassettes are eliminated.

Wing Bragg*o law again and equation (7), one can get the following relation, namely*

d » ni/2 bin 9 * ao/^2 ^ ^ + 12 (14)

or, ' ■ ' ■ • • " • •(h8 t k2 + 12) = (2ao oin 9)z/f. (15)

When the type of lattice of a cubic structure is known, St is likewise knotm what reflections will appear. For the face centered lattice then, a table can be prepared similar to the one below as complete as we wish.

*26-

reflection 111 200 220 311 222 400 331 420 422h2 + k2 + 3.2 3 4 8 XI 12 16 19 20 24

Since a0 and J are known in equation (15), it follows that each value of (0) dtermines a (hkl) reflection plane# and by comparing (h2 > k2 + l2 ) calculated from (15), with these secured from the table above, it can easily be seen which (hkl) reflections correspond to any certain spectral peaks This method may be a little more tedious than the one using the G. E, charts but it is generally more accurate and will determine reflection planes out as far as one can make out peaks on the Densitometer charts, whereas the G. E. charts are limited to the visibility ef the lines on the film and are not made to read much beyond the fourth or sixth orders#

To cheek this method under consideration of finding atomic F's it was thought best to take a compound with a

comparatively simple structure for which both atomic F'swere known and to check the calculated F for one of them

kwith the accepted value as given in a table of Wygsff*stext on pages 100 and 101. The face centered lattice ef

Of*S.)fluorspar, CaFg, was chosen for this cheek.^ The spectra of fluorspar has an especial interest for the following reason. One can easily observe that the (200) and (222) reflections have no obvious peaks on the Densitometer

charts for CaFg, (Plates 2 and 3). Mow it is knwon that for the particular atomic arrangement of fluorspar, the (200) planes containing calcium atoms are interposed*with planes containing twice as many fluorine atoms and the observed

-27-

weakneBB of this line shove that calcium planes and fluorine planes are nearly equivalent in regard to reflect­ing power# By using the equation Ix * , one candetermine the theoretical values of the relative intensi­ties of any(hkl) reflection if the 3? factors are known for

the atoms composing the crystal. This was done for the(220), (200), (222), and (420) reflections. The (220) reflection was the largest observed on the CaFg Densitometer charts, and the other three were so small that they were neglected. Theoretically, however, the relative intensi­

ties of the (200), (222), and (420) as compared to the (220) reflection are respectively 1,6, 0.2, and 0,1 per- centum. It would appear, then, that one calcium ion con-

eighteen electrons balances two Xluorine ions of ten electrons each. The F*-factor as determined from the phase relation of the (Ca-2P) planes and the atomic F-factor curves gave values of 1,2, 2*0, 2.8, for the (200), (222)# and (420) planes which shows that the neutralizing factor varies directly with the glancing angle*

To determine the reflection F* it was stated in the previous section that it was necessary to know the atomic position* These are given in \vyclcoff* s text# p. 230, ae follows# '

Cat 000;

-ji -Mkt m t m i m i rnt m * m t "Substituting these values in (13) for (%@yc%o)# (xpypzp),etc;# and eliminating n, since we are dealing with reflect

•' . . . . ■ . . ■ ' ' .

tion indices* we secure for (F*)2 the following#(F' )2 » a 2 + B2 wh«re

mm , ' • • 1 •

. X *: Ca icos 2tr(o) + cos is^h + k) + cos s(h + 1) +cos tr(k + ijj + f Jcos rtfZ (h * k > 1) * cos v/2 (h f 3k + 31) + cos %/2 (3h + k + 31) + cos -fl/2 (Sd + 3k + 1) + cos ?%/2 (3h + 3k + 31) * cos t^2

(h + k f 31) t cos n/2 (h + 3k + 1) + cos tr/2 (Sh > k > l)j,

B = Ca oj ± p|sih rt/Z (h + k + 1) t sin %/2 (h + 3k + 31) + sin V 2 (3h + k + 31) + sin %/2 (3h + 3k + 1)+ sin wf2 (3h + 3k + 31) + sin 7^2 (h + k + 31)

-29-

♦ sin ji/z (h * 3k ♦ 1} * sin n/2 (3h .+ k + 1) . Ca = 0 in 3 above since h, k, and 1 are always Integers and hence we will always have' the sine of n times an integer which is zero*

By taking each one of the (hkl) spectra which appear in the reflection from a face centered cube and substituting each h, k, and 1 separately in the above relations for A and B we will get three groups of planes» an example ©f each beingthe (111), (200) and (220)* The relative constituents of the amplitude of each group is shown vectorially as follows*

'Si(200)(0)

(220) • (Ca + 2F)

. Cl1r

- ■ .....■■w

We now have two equations of Z* which can be set equal

to each other and from which we get these two types of equations,

(Ca + 23?) . Vlr/Jf(8) and (Ca) = (16)The right hand members of these equations are known for each (hkl) reflection and hence we can form a table of values for (Ca + 2F) and (Ca) as in Table 2. These two groups of data can then be plotted against sin 9/*71 to form a (Co) and A (Ca + 2F) curve.

The particular method which the author employed in dealing with these curves was to first standardize, so to

speak, tho calculated P-curve for calcium Tilth, the F-curve, secured from Wyckoff^s text... This was done by finding the average of the ratios of the ordinates of the calculated curve to the ordinates of the curve from Wyckoff.. V«hen this average was found, another curve was drawn by multi­plying the ordinates of Wyckoff’a curve by this, average ratio. Thus we have an F-curve for calcium from the ex­perimental data that Is relative to the <Ca +2F) curve, and still has tho seme general shape as the standard of comparison,, namely, Wyckoff *8 curve. Theoretically,, of

course, the calculated F-curve should have the same general shape as the standard curve* However, when the points do

not describe a smooth curve,, dae perhaps to faulty diffrac­tion films or certain mechanical errors in determining the peak areas,, this process of standardizing tho points is a necessary step in the method.; . V

Consider now those two curves, namely,> the standar­dised Ca curve end the (Ca -h £F) curve. By subtracting theordinates of the smaller curve from those of the larger/ ; ■ - V ' : - ■ . ° .curve we get (ca + 2$)- Ca « 2F. Dividing by 2 we get the relative F-curve for fluorine at once. Then, by dividing this curve by the same ratio as was multiplied into Wyokoff »s Ca curve we should got a set of values forming a smooth curve which Indicate the actual atomic structure factor of fluor-

SEE, of-cable s^romtry whose structure Is that of Ba6l»Tho atomic P curve for K'ls k n o m and that for X is to be dotomined, hence, the structure factor for I xti.ll be the unknown in the equations to follow. The s mne' general pro­cedure is followed in dealing with this crystal as with Oafg in all but a few steps which will be discussed.

placed as follows, (figure 1);

I: 000; 250; M i o M ,Ki : 222; 002; oio; 4oo»

and the equation for the reflection F* as secured from equation (1) is again

f (P* )8 » A2 > B2 whereA * ijcos 2a (o) + cos a (h + 3c) + cos tf (h + 1)

■V cosiir (k + 1)] + ic[cos %r (h + 3c + 1) + cos rr (1) -t cos TT (k) > cos n (h)j,

*■ * 0 t *H;With this crystalt only two groups of planes were

found. The even planes formed one group of (I + K) P*s where the reflections assist each other and the odd planes formed another group of (I-X) P's where the reflections counteract each other. A table, (table 4), was filled out for H2 with the data secured from seven specimens of the substance in a manner similar to that for CaP2.

-32-

Siaee there was no pure atomic F-curve to begin with in the case of KI, it was necessary to rely upon the aver­age value of a number of specimens to give as smooth a curve as possible. In any case, if the points do not deter­mine a definite smooth curve, it is advisable to draw the best possible estimated curve through, or as near to, as many as possible. VZhen the (I + K) and (I -K) curves were drawn in this manner it was possible to secure the relative values of I and K by the following relation of curve coordinates.

I = 1/2[(I f K) + (I - E)| andZ i * (17)K = 2/2[(I i- K) - (I - K| .

There will be two curves now which represent the relative P factors of both K and I. Values for S are already known, however, and this known curve can he drawn on the same graph with the relative X points. How it is possible to standardize this group of points by means of the known curve In the same manner as was done with CaPg and in doiig this an average ratio of deviation will necessarily have to be found. By multiplying this same ratio into the relative

F-curvo for Iodine a curve will be secured that corresponds to the actual atomic structure factor curve for Iodine.

In much the same manner it is possible to find the

F-curve for any atom of a binary compound when the structure f&eter is known for one of the atoms.

Table 1.— Relative Intensities of the Spectra of each of the CaFo Specimens Used to Secure ff-curve in Plate 7t Graph 1

l I 8 ' : r> s ■ , . : s ■ . :s ? .Planesi Si 8 S2 8 s3 8 84 : S5 8 S6 8

: ■ 8 - ' '% .: : s ' S' . . ....

• ' - . (Ga + 2F) planes ’ ’ ’ • " :220 1100.0 i100.0 s100.0 s100.0 s100.0, s100.0 s100.0 ,400 8 17.658 21.3 8 16.2 8 21.75s 22#42# 19.1 s 24.0422 8 27.3 s 34.9 s 26.8 s 42#50$ 26.78s 33.9440 s 3.20s 6.23s 6.52s 10.30s 11.22 $ 6.25s 8.55620 : 8.51s 8.37 s 6.45s 10.62i 11.60s 8.05s 8.23,444 s 1.66s 1.86 s 0.82s 1.98s 2.17 s 1.78: 1.49642 1 5.71s 6.02s ,4.08; 6.38 s 6*96?’ is48s_ , 6.97

Ca + (f -X '1 , planes

111 % 83.1 t 87.5 $ 78i4 s 72.4 s 94.2 s- 82.2 * 89.2311 s 41.9 i 45.4 t 37.8 s' 43.0 s 49.6 s 40.2s 40.3331 * 17.2 s 23.8 i 17.48s 18.7 s 25.2 t 20.5 s 20; 15333 I 10.1 8 10.8 8 9.62s 11.62s 13.5 s 10.82s 12.47531 % 3.83s 10.72s 6.59 s 10.15s 12.28$ 9.80s 10.16553 % 2.81; 3.26s 2.24* 2.47* 3. 30$ 1.87: 1.99551 % 3.12s 3.32s 1.90s 2.54: 4.85s 2.19 s 3.05553 : 2.39% 2.63s - j y m . — 2.74 s _ 3.90s 2.97 s-3*62

-34

Table 2.— RelativeCaFp ComputeA:from an Average of Seven Diffraction- . . . : , ' '

Pattern PilmB.

Planes 5 ^ ^egrecs8_____ _5__ ;_____1__________ _$__

■./— 5-

(Ca + 2P)5

220 i 10.76 8 110*7 t .260 100.00 t 95*05400 $ 15.27 l 40.5 $ .368 20.19 8 70*61422 8 18.79 8 104.9 : .451 32.16 i 55.37440 8 21.80 8 38.5 s .521 8.1® 8 46.09620 8 24.50 8 ' 60.3„s .681 '. 8.83 8 38*27444 8 27.00 8 16*5 8 .637 1.68 i 31.91642 s 29.33 s 83.4 ; .687 5.94 s 26.69 ’

Ca t (P - T) .8

111 8 6.65 8 302.3 8 .158 81.81 8 52*02311 8 12.60 8 240.7 i .306 42.60 8 42.07331 i 16.68 8 134.7 8 .401 20.40 8 38.92

333-5118 19.95 8 123.6 s .478 11.28 8 30.21- 531 8 22.80 : 140.0 ; .544 9.79 i 26.44

533 s 25.32 l 56.3 8 .602 2.56 s 21*32551-7118 27.90 3 92.4 % .657 2.99 8 17*99553-7318 50.26 s 118.4 8 .706 2.85 ■ 8 ’ 15.51 _- (Ca - 23?)

I200 8222 s420 $ This information not included

442-6008 beoause the areas for these622 8 planes were practically zero.640 8800 s

-*3S- •

Table 3.— Relative Intensities of the Spectra of each of the IQ Specimens Used to Secure ff-curve in Plate 7, Granh 3.

t 8 %' ' f ' 8 . - .Planes$ Si 8 Sg 8 S3 8 S4 ; S5 8 Sg 8 S7

* % 8 8 8 8 ' 3 '(I + K) plane b

. 200 s . ■ • ; ......$ t i • - t • * ■1100*0 *100.0 :100.0 *100.0 »100*0 i loo.-O f 100*0220 : 74.3 s 92.1 s 97.8.: 79.7 f 91.4 1 92.0 s 83.8222 24.0 i 38.0 i 27.6Vs 22.2 37.70s 29.3:s 27.3400 11.8 8 21.4 : 19.7. s 10.4 1 19.55a 20.6 s 20.#420 8 29.8 s 40.0 8 40.4 . s ’ 26.4 s 42.70s 46.5 s 43,6422 8 15.6 * 20.2 i 23.0.8 14.6 : 25.22s 39.0:: 30.0440 s 2.89 s 6.2 s 3.94:V 2.61; 6.15; 8.75s, 6.9442 9.83* 11.4 : 10.96s . 7.61: 17.85s 22* 0 •• &- ■ 21. 3620 8 4.92s 5.0 a 5.48s 3.41: 6.528 11.0 r 9.05622 3.76 s ' 5.47a 5.92s 3.01: ■ 7.33: n * r $ 7.62444 8 2.61s 2.56s 1.54: .80s 1.52s 2.16: 1.78640 8 2.34* 5.09s 2.84s 2.80; 5.90s 6.49s 7.50642 2.01s 3.33% 3.51s ■2.21* 4.90: 5.45s 3.81800 8 1.45: 2.34: .88: 1,60: .63; 1.13s 2.14644 1.45* 1.67s 2.19s 1.01: 2.24; 2.58s 2.74660 8 1.67s 1.75: 1.01s 1.25: __2_. 16 8 ; _ Iglg,

(I - K) planes

s a 3 8 • ■

111 s 49.7 s 57.4 1 48.9 t 42.9 s 44.5 « 48.3 s 49.9311 1 29.8 : 35.7 s 28.4 s 25.6 1 35.9 8 39.7 a 28.9331 8 10.4 8 16.2 s 19.2 s 8.2 s 12.2 : 22.2 s 14.3333 8 4.92s 5.0 s 4.83s 6.0 1 9.10* 9.99: 8.58531 8 4.62: 4.26 s 6.80s 5.2 1 9.01s 8;35: 12.5553. 8 2.018 3.81s 3.07 s 2.21s 1.79a 4.53 s 2.03551 ; 2.34: 3.33s 2.41; 1.818 •-■2,86* 2.78* 4*86553 : 2.01: 1.45: 1.971 1.21: 1.34: — & 2 1

f

“36— •

Tattle 4„— Relative Intensities and Structure Factors forO goraputed from an Average of Seven Diffraction Pattern Films,

Planesslt Qdegreec

81 f(9)j ! &in Q ! ^-Relative

1 .710 ’Intensities_3_______ $_________:___: i3Pr3 y w

(I + K)

200 i 5.7888 293 # *142 % 100.00

38 58.42

220 : 8.25 8 286 3 .202 : 87.30 8 55.25222 s 10.10 8 126 8 .247 % 29.44 8 48.34400 : 11.63 8 71 s .284 8 17.76 50.01420 t 12.96 227 8 .316 t 38.49 8 41.18422 s 14.26 186 8 .347 | 23.66 35.67440 3 16.62 8 68 8 .403 : 5.33 28.00

442-6008 17.62 3 150 8 .426 j 14.42 31.01620 8 18.66 3 110 8 .451 3 6.48 8 24.27622 8 19.58 96 8 .472 3 6.32 25.66444 8 20.48 8 29 3 .493 3 1.86 1 25.26640 3 21.31 3 80 3 .512 8 4.42 8 23.51642 8 22.21 3 144 3 .532 t 3.60 15.81800 8 23.74 3 15 s .567 3 1.45 S 31.09

820-6443 24.62 3 119 1 .587 3 1.98 12.90660 : 25.33 : 28 : .602 : 1.51 8 23.22

(I - ^

111 3 5.0183 519 8 .123 i 48.80

88 30.66

311 8 9.68 413 8 .237 i 32.00 8 27.84331 3 12.66 8 238 8 .309 : 14.65 8 24.81

333-5118 15.21 218 3 .370 3 6.92 3 17.82531 8 17.50 3 244 : .423 8 7.25 17.24533 8 19.50 8 99 s .466 : 2.64 3 16.33

551-7113 21.16 8 164 8 .508 3 2.87 3 13.23553-7318 22.8 220 3 .546 3 1.87 3 9.22

fable 5. Coianuted Structure Factors Compared withmeters given in

= • , A - Fluorine B - Iodine

sine; Computed I Data from Computed• 710 . from Table 2 j V/yckoff, from Table 4

.2 t 6.5 s 6.30 ' 30.75

.26 i 5.8 I 5.35 . 27.00

.5 - ; * 5.2 i 4.50 23.50

.35 i 4.5 ; 8 3.70 . 20.40•4 s 3.9 . 8 3.00 17.70.45 s : 3.3 8 2.47 , 15.40.5 8 2.8 8 2.07 13.40.55 8 2.4 t 1.77 ,11.50.S 2 1.9. s. 1.50 . 9.80.65 s i.s . : 8 1.30 , 8.30.7 t 1.1 2 1.07 6.90

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Plate 10

CHAPTER V*— DISCUSSION OP RESULTS

There seem to he certain sources of error in the determination of atomic structure factors by the X-ray Powder Method which, if brought to the attention of the one doing similar research, the author believes will serve to bring about more consistent results.

First, it is of utmost importance that the diffrac­tion patterns on the X-ray film be in sharp contrast to the background. In merely analysing a substance for the symmetry and the lattice one can tolerate dark films for it is still possible to see the spectra even with the naked eye, and the photodensitometer charts will record #ll#t peaks, or at least flattened knobs, but when it comes to getting tho area under the peaks, one wants the largest, sharpest peaks possible, and this comes only from very contrasted films. Furthermore, it is possible to get the spectral lines themselves too dark, the reason being that the logarithmic "T-square* which is used to transfer the logarithmic areas under tho peaks to linear areas,(the process Reitz described in detail In his thesis) will function only if the peaks keep 5 mm. distant from the Galvanometer zero line (o.f. Plate 2 and 3).

The details for securing contrasted films would bo a

chapter by Itself, hut suffice It to say here that if the proper combination of crystal mass, voltage intensity, and length of exposure to X-rays is obtained, it can be done.The more Intense X-ray beam secured from the highest voltage terminal seems generally to fog the film, at least for some of the lighter substances, and this destroys the contrast. Often times one must experiment with several trials before the right combination is secured. Care should also be taken to use only clear, clean solutions for developing and wash­ing, otherwise, the sediment deposited is sure to destroy some of the contrast.

Secondly, the film should be placed in the film holder on the Densitometer so that the light which falls on the photo-electric cell goes through the central portion of the spectral lines of each specimen. The reason for this Is obvious if any ordinary exposed film is observed, for it will be noticed that the spectral lines are fogged on the end, but have much sharper edges in their centers. This latter condition strives for a better Densitometer peak.

An item which probably accounts for more error than any other is that of putting in the bases of the spectral peaks that enclose the relative area. It is quite obvious

as to where the base lines go under some peaks for some sub­stances, but not enou^i work of this type has been done to say definitely where they should bo placed in every instance,

9850©

Often times there ore Indications at the haso of the peak of a more or less general X-ray radiation, and this will take the shape somewhat of a discontinuous dome. If the outline of this dome is followed, one has a base line that is fairly correct.1 The main idea is to get just that part

disregard all that due to general radiation. Again, the

Is by seeing which one gives the most consistent results.As a fourth precaution one should watch closely for

double peaks. On plates 5 and 6 such peaks in the moro obscure cases are the (531-600), (533-622), (551-640), etc.

distanco

should appear to within a few tenths of a degree, it is much easier, and in most cases fairly definite, to find where they are located; Some single peaks have more than one set of planes contributing to the intensity of their diffracted Xrays, and this should not be overlooked in determining the (j) factor in the P-factor formulae.

It will be well now to discuss somewhat the graphical work sheets, (plates 7 and 8). The process of drawing the graphs is all straigit forward with the possible exception of fitting the first curve to the points when they happen

Tc.f. Plate 2, (311) spectra; Plate 5 & 6(222) and (420). ■spectra.. / • ' :

to be rather scattered, and this can he determined quite definitely after a few trials.

It is seen that the (Ca) curve in plate 7 la twice as large as found in Wyckoff • a text, hut since these quanti­ties are all relative, introducing a factor does not change their relativenesa, and the factor can he removed later.The particular reason for multiplying by the factor two in this case was to get the curves more nearly the some place on the graph thus avoiding large ratios in standard­ising the curve. The fluorine F-factor curve from Wyckoff was inserted for sake of comparison. The F-f actors for values of sin 9/.710 s 0 were chosen on the assumption that the structure factor approaches the atomic number of the ionized atom when tho glancing angle approaches zero.

On the KI working sheets (plate 8) It can he noticed that the (14- K) and (I - K) points are scattered rather promiscuously along the general slope of the expected theoretical curves. This departure from the expected curve is a result of inadequate diffraction patterns, according to the opinion of the author. It so happened that films of two extremes were taken to secure the data for this compound. One kind was exposed to relatively intense X-rays, and the other to relatively week rays.

On the former there was not much contrast hut pretty fair spectral peaks of the higher orders, while in the other :

one there was good contrast hut the higher orders of the

film wore almost absent. The author was unable to get the happy medium between these two extremes, but there is no doubt that with sufficient experimenting it could be done.

It was observed during the process of acquiring data for this thesis that such substances as CaFg, liaCl, KCl, eta., produced good X-ray diffraction patterns regardless, it seemed, of how few precautions were taken. If the ease of the production of these X-ray diffraction patterns was in any way due to the comparatively equal number of electrons in the two atoms, then we could easily say that here was one cause of the heavy background, for the ratio of the ionic number of I to that of K is 5.4. There is, no doubt, some connection between these, two phenomena, as was observed from running such compounds as FbO and FbS where one atom was so much heavier than the other. , ,

Some of the heavy background on the films might be due also to an improper design in the casettes, as there is a great deal of secondary radiation when hard X-rays, such as are used in diffraction work, strike metal.

Plate 10 is a graph of the function of the glancing angle for different reflections. This graph, or one similar, can be used to determine f (9) as used in the relative inten­sity formulae. However, for very small values of (9), the values of f (9) become very large, and It is not possible to estimate these values from the curve with sufficient accuracy,

53-

Hence the graph waa included in the thesis merely to show graphically this importsnt function#

Figure 92 was included so that it would be possible to compare the working sheets of the two method a for deter­mining structure factors# These graphs were determined by using single crystals Which method has been accepted as

t o - * ^

1

being more or less standard# The F values by the powdermethod (Plate 7), are sll^itly higher than those from

#yekaff*e curve, which fact Is consistent with other data secured by this method#

%'rom data of Bragg, James and Bosanquet# c,f# Conpton, A# H#, ^X-rays and Electrons**, p# 136#

CHAPTER V*— GONCLUSIOM

In general the pov/dor mothod of Crystal Analysis is a most powerful tool In the study of crystals, although it has its limitations* For the determination of actual atomic structures, it is of only real value in the simpler crystal systems, namely cubic, tetragonal, and hexagonal# However, these types of crystals include most of the metals and alloys and a large number of simple binary compounds, the structures of which, owing to their habit of never crystallising in large crystals, could not have been deter­mined but for the powder method,1 and It is tho analysis of these materials which offers the greatest hope of imme­diate usefulness to science and industry#

As is seen in the discussion above, a great deal of importance is placed upon the powder meidiod and particu­larly upon the structure factor as related to ...spectral in­tensities which give a most delicate arid accurate indica­tion of the relative positions of the atoms# Hence,it is the opinion of the author that any method which will simp­lify the determination of these structure factors is worthy of time spent in research;

1 James, R. W., "X-ray Crystallography®, p, 50#

-55-

It is apparent then, ‘by way of final conclusion, that in comparing the computed F-curves frcmi diffraction pat­terns with those from Y/yckoff for fluorine; and In consider­ing that a slight error in dealing with curves graphically might he quite large in comparison with the mall structure factor values as of fluorine, that, although the curves do not coincide exactly, still there is enough agreement be-, tween the two to substantiate this method employed for deter­mining the F values of ions*

It is known that the method of using powdered crystals will perhaps produce results that are more indicative of

the true state of ###s#SMS0m''an atom than will that of using a single crystal. This is due to certain extinctions found in perfect crystals. Furthermore, the slight disagree­ment between the two F-curves for fluorine (plate 7) agrees exactly with J. A. Bearden* s results where he secured the

reflected intensities of X-ray. beams from powdered specimens by using an electrometer.2 This indicates beyond a doubt

that tho method under consideration can produce satisfactory results if the data is secured from several well contrasted diffraction patterns. i

The F-curve for Iodine is possibly in error 5$ or more some places, but it would be impossible to get any closer to tii© expected results by using the diffraction patterns that

^Bearden, J. A., Fhys. Rev., Vol. 29, 1987, p. SO. ~

wer® uscd> and again let It be said that the limitations of this method are first, the securing of good films, and second, that of putting in the proper base line bn the spectral peaks.

BIBLIOGRAPHY

Bearden, J. A*; Measurements and Interpretation of the Inten­sity of Xrays Reflected from Sodium Chloride and Aluminums Phys. Rev., vol. 29, 1927, p# 20.

Bragg, W. H., and W, L. Bragg? X-rays and Crystal Structure, Hew York; Harcourt Brace and Co., 1924.

Compton, A. H. Company;

Davey, V/. P.>

General Electric Instruction Book; G.E.I. 2082.James, R.^W.; X-ray Crystano^ranhy; Hew York; E. P. Dutton

f J fr J H

>s ofLversity of Arizona,

Wyckoff, W. G.; The StructureCatalogue C<

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