An Investigation of the Validity of X-Ray Crystal Method of Determining N and e Citation Miller, Park Hays, Jr. (1940) An Investigation of the Validity of X-Ray Crystal Method of Determining N and e. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/a0h7-tk08. https://resolver.caltech.edu/CaltechTHESIS:01272025-223541303 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): DuMond, Jesse William Monroe Thesis Committee: Unknown, Unknown Defense Date: 1 January 1940 Record Number: CaltechTHESIS:01272025-223541303 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:01272025-223541303 DOI: 10.7907/a0h7-tk08 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16960 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 29 Jan 2025 21:01 Last Modified: 29 Jan 2025 21:09 Thesis Files PDF - Final Version See Usage Policy. 14MB Repository Staff Only: item control page

.AN INVESTIGATION OF THE VALIDITY OF THE X-RAY CRY ST AL MEI'HOD OF DETEfil.lIINING N JlJIJD e Thesis by Park Hays Miller, Jr. In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1939 Table of Contents (continued) Fig. 1 The Seeman-Bohlin FocNsing Spectrograph 14 a 2 Effect of Film Thickness 16 a 3- Effect of Sample Thickness 16 a 4 Effect of Vertical Divergence 17 a 5 Diffraction by a Line Grating 17 a 6 Orientation of the Crystal Grating 23 a 7 Microphotometer Trace of Ag Sample 27 a 8 Schematic Drawing of the X-Ray Tube 28 a Table of Contents Introduction 1 Density Measurements 7 Al uminum Silver Quartz 9 11 13 Mea surement of the Lattice Consta..11.t 15 The spectrograph Errors of the camera 15 17 Thiclmess of the film Thi ckness of the sample Film shrinkage Vertical divergence of the x- ray beaTU Temperature of the sample Overlapping of the doublet s Finite Resolving Power of a Crystal Grating Experimental corrections 17 17 18 18 19 19 20 26 The X-ray Tube 29 Results of the X-ray Measurements 33 Aluminum Silver Quartz 35 b 38 39 Molecular Weight s of t he Samples 41 Calculation of the Extinct ion Coefficient 43 Effect of Distortion on the Met als 45 Calculation of N and e 46 Discussion of the Results 47 Abstract To test the validity of the x-ray crystal method of determining N and e the crystal structure and densities of t wo polycrystalline metals, silver and aluminum, and one 11 per£ect 11 crystal, powdered quartz, were determined. The densities were determined by using a specific gravity bottle filled with distilled water and the following re sults were obtained. Material fl at 25. o 0 c Silver Aluminum Quart z (macroscopic) Quartz (fine powder) 10. 48 70 ± 0.0008 gm cm- 3 2.69839 ± 0.00019 2.64822 ± 0.00005 2.64810 ± 0.00015 The lattice constants were measured in a Seeman-Bohlin focussing spectrograph and the following parameters were obtained. Silver Aluminwt1 Quartz a =4.08574 :t 0.00004 x 10-8 cm a = 4.04927 0.00002 a = 4.91263 :J:. 0.00009 C = 5.40454 :f: 0 .00011 * Using these results the following values on N and e were calculated. N Silver Aluminum Quartz (1) Quartz ( 2) 6.0318 ± 0.0006x 10 25 mol gm mol-l 6.0258 ±0.0007 6.0259 ±0 .0005 6.0280 ± 0.0005 e 4 . 7957 :;i; 0.0007 x 10-10 esu 4 .8005 ±0.0007 4. 8020 ± 0.0006 4. 7988 ± 0.0006 where the t wo different values for quartz are for t wo different values of the molecular weight of silicon. The effect of distortion of the metals on the crystal structure and the density was investigated. The values obtained for powdered quartz are free from the objection that the density determination may not be representative of the thin layer at the -aurface of a crystal involved in the x-ray reflection. Since the values of N and e obtained for silver and aluminum differ by many times the experimental error and the difference between a polycrystalline material and a 11 perfect 11 crystal is one of degree, and since after ta.~ing into consideration the unfortunate uncertainty in the molecular weight of silicon the values obtained using powdered quartz still seem to be in a slight disagr eement vdth those obtained using large calcite crystals I believe that the best value of e obtainable by the x-ray method is 4.802 ± 0.002 x 10-lO esu. 1 INTRODUCTION The problem of determining accurately Avogadro 1 s number, N, and its associated constar1t, e, the charge of the electron is an important one. There are two direct and accurate procedures, the oil-drop method and the x-ray method. The former involves the value of the coefficient of viscosity of air about which there is no universal agreement as yet, and the latter contains certain fundamental objections as to the validity of the method. It is with a critical examination of the x-ray method that this thesis is concerned. $ Information as to the submicroscopic absolute dimen,;ions of the lattice of a crystal which we can obtain by means of x-ray diffraction perwits the computation of Avogadro's number which is given by N =- M (,) r ( a.1.,a 2. )a.'3) 0,1..,c<:1,.) o<,> where M is the molecular weight of the crystal, f the volume associated with one molecule. e its density, and a~, a2 , a3 are the axial lengths and cf.1, , o< 2 , ~ 3 the intea;>-axial angles. Ne= Q, the Faraday, as known, we have ~ Or if we assume = efQ . M For face-centered cubic crystals f(a.s.,a2 ,a3 , c:i< ~, 0( 2 , a( 3) is 1 a3 • 4 Thus to determine N for such systems it is necessary to know the molecular weight which is best found by cf:lemical or massspectroscopic means, (.' which is best measured by immersion or a specific gravity bottle, and a, which is mea.sured by means of x-rays through the Bragg equation 2 ' sin G(l - sin2 G) where ~ h2 is the sum of the squares of the Miller indices designating the particular plane used. G i s the complement of the angle of incidence measured from the normal of the plane,1-$ is the index of refraction ~ i s of the order of 10- 6 and since in our work sin2 G ) 0.9 thi s term may be neglected. For )\ we use the ruled grating measurements of Bearden, B~cklin and Sodermann who independently found that the wavelength in absolute unit s is 1.00203 ± 0.000016 times the arbitrary Siegbahn wavelength. 14 Quartz is a hexagonal crystal with three Si02 molecule s in For such systems Oc'1 = ct. 2 = 90°, the conventional hexagonal unit cell. o( 3 = 120° , a 1 = d.l = a ,a 3 = c and f becomes a2 c/2 V3 . The Bragg equation takes the form ,\:: ~ { .!±,_ ( h1 -+ h-1,h:\ ,i. h~ ) 2. '3a.~ 'A h~ 1· •1~ +- .!!.1 -.,n e C~ The difficulty with the x-ray method is t wofold. In the f irst pl ace the x-rays measure only a periodic spacing so that if either a more or less Tegular superstructure, random holes or i mperfections in the lattice exist their presence will not affect the latti ce constants as measured by the x-rays, but the measured density, which i s the average density, will not be the true density of the periodic sections of the lattice. The existence of a superstructure (or mosaic structure) has been proposed by Zwicky22 to account for certain mechani cal properties and habits of growth. He propo sed that there are periodic variations in the grating space in the crystal along a direction normal to-the planes. Their period would be of the order of 10-6 cm and the superposition of such a long period on the i nterpl anar spacing of 10- 8 cm would not affect the relative positions of the x-ray diffraction maxima and would have negligible effects on their intensity. However, one would not expect the error i ntroduced in the density by the presence of the superEtructure or i mperfections of the lattice to be the sruae for all crystals. Tu1 measured the density and lattice parameters of calcite, rocksalt, potassium chloride, and diamond crystals and found that they gave a consistent value of N, . three . the largest deviation being~ parts in 10,000. Also, one would not expect, if t here is a regula~ superstructure, that the secondary component would have exactly the same coefficient of thermal expansion as the normal component. A.H. Jay 2 measured the thermal expansions of silver, quartz a.nd bismuth by the x-ray powder method, precision measurements being employed. The macroscopic thermal expansion was also measured by optical methods and tho saine values obtained indicating that the expansion of the atomic l attice does not differ from that of the crystal as a whole. Thus the presence of a large secondary component appears doubtful. However, it is one of the purposes of the present investigation to work with lattice s known to be distorted and imperfect and to find how much these distortions affect the value of 4 the lattice consta.rits of the periodic part of the lattice and the average density of the crystal. It will then be possible to correct for these distortions and to a.,."lswer once a.YJ.d for all t his criticism. The second objection to the x-ray method is that because of the extinction of the x-rays in passing through the crystal the lattice constarit of an extremely thin superficial layer (5 x 10- 5 cm for calcite in the first order with Mo Ko< ) is measured, and this may not be representitive of the interior of the crystal. Allison and Armstrong 3 measured the wavelength of Mo K«1. in various orders of reflection from calcite and found Order Half-depth of penetration cm 1 5 X 10-~ 4 X 10-"" 6 X 10- 4 1.2 X 10-:3 3 4 5 no. of planes 1.6 X 10 3 1.3 X 10 4 2 X 10 4 4 X 10 4 ,\ 0.707831 Ao o. 707902 o. 707850 0.707840 1-,,,, - >i~ >-.1. 0.01 % 0.003 0.001 and concluded that the lattice constant is uniform throughout the cr-.rst al. Dulviond and BoJ.lman 4 measured the density and lattice para- meters of extremely fine powdered calcite, where t he size of the individual crystals was of the order of the depth of half penetration of the x-rays. They found that within eXperimental error ~ the constants ru1d density were identical with those of macroscopic calcite. While t hese experiments indicate the absence of any imperfections in the crystals used, the value of N would be on much surer ground if vrn k..ri.ew what i mperfections are found and just vrhat t heir effect is on the grating constant aJ1d the average density. present experiment s was t wofol d . The purpose of the First, to answer the above question, and secondly to make a preci sion determination of t he l attice constants and densfty of t hree new crystal s by methods different from t hose employed by Tu 2 and Bearden 5 . Both measured t heir 'densities by i mmer- sion and thei r lattice parruneters with a t wo crystal spectrometer. In the present r esearch the densities were measured with a specific gravity bottle a..YJ.d t he lattice parameters with a Seeman-Bohlin focusing spectrograph. Silver and aluminum were the i mperfect crystals used and quart z was t aken as a 11 perfect 11 crystal. Powder sampl es of alwninum and silver were prepared by f iling or sawing and these sampl es along with solid pi eces which were as l arge as would enter through the neck of the specific gravity bottle were annealed together in a vacuuyi. The powder sample of quartz was prepared by breaking up optical clear quart z in a steel mort ar by pounding with a heavy sledge hammer. The codrser grains were re j ected by the fine screen t hrough which the sample was sifted. The lattice parameters and density of t hese samples were measured. I f we think of a sample of the mat erial as made up of many small perfect crystals or crystal.,. lites , randomly situated relative to each other , then the resolving power of these units for x-rays will depend on their size, that is on the average number of perfectly periodi cally spaced planes t haj they .6 contain. Thus from the wia.th of the diffraction pattern (Debye- Scherrer rings) it is possible to measure the average length of the periodic sections of the lattice or ti1e size of the crystallites. In the case of the met als the powder and large pieces were then distorted by mechanical working and the new density, grating constant and size of crystallite were'determined .. Actually the shape and dimensions of the m1i t cell were found to be unaffected by worlcihg but the 1sngth of the regular periodicity was found to decrease. The density was found to change slightly, an increase for silver and a decrease for aluminum. On the basis of these measurements it is possible to estimate the density of 11 perfect 11 crystal s of aluminrun and silver. 7- THE DENSITY iv1EASUREi\tL"Si:lr s 1 The weights used were calibrated against both a 50 grarn and 1 gram standard ment. obtained from Dr. Swift of the Chemistry Depart- The method of double weighing vii.th an analytical balance was employed throughout the experiments. upon to within 0.1 mg. The weighings could be depended In fact the specific gravity bottle was weighed ma.ny times over a pyriod of several months and its weight ( 22 gm) corrected for air buoyancy never varied by more than 0.1 mg . The specific gravity bottle was of conventional design and included a thermometer v,hich could be read directly to fifths of a degree and estimated to twentieths. Thi s thermometer as well as a second one used in the temperature bath were checked against a Bureau of Sta.ndards thermometer obtained from Mr. Weis of the Chemistry Department . The ground glass joint was reground with 600 carborundum for at first there was found to be considerable evaporation dtu·ing the course of weighing. After it was reground the loss by evaporation was less than 5 mg in 24 hours. empty bottle. Let W be the apparent weight of the The bottle was then filled with freshly boiled redis- tilled water and pl aced in a temperature bath. The height of water in the capillary was roughly adjusted by removing some of it with a fine glass tube placed dovm t he capi~lary. The final adjustment was accomplished by changing the temperature of the wat er bath. When the t emperature was the same in the bath and the bottle and the top of the 8 liquid in the capillary was opposite the fiducial mark, then the bottle was removed, dried and weighed. WB. Let the apparent weight be The volume of the specific gravity bottl e is where f'~ is the absolute density of water for the particular temper- ature, 6 ~ is the density of air, and 8.6 the density of the brass weights. The laboratory distill ed water was redistilled in an all glass still, freshly boiled, corked and cooled before using. This was found necessary for if this cooled water was exposed to the air for about f ive minutes and t hen corked again after twenty four hours the density increased by 0.005%. The result of 15 measurements when r educed by least squares gives as the volume of the bottle, at the t emperature t Vt == 24.92735 ± 0.00008 + [t - 25.0 1 ( 4. 85 ± 0 .30) The probable error in thi s and succeeding measurements was calcu..lated by the method recommended by Birge. Let Ws be the apparent wei ght of the sample whose density is to be measured; its mass is where fs is the approximate density of the sample. The sample is 9 then pl aced in the specific gravity bottle a.11.d enough water added to cover it. The bottle is evacuated until the water boils. This makes sure that all occluded gases and bubbles are removed from the surface of the sample. The bottle is then filled with water, the temperature ad j usted, and wei ghed giving the apparent weight Wv, The vollm1e of water added is ~ (-1- _ l )} 1.00 8.6 Thus the density of the sample is r= The aluminum was annealed for 4 hours at a temperature of 350°C in a vacuum furnace and allowed to cool in the furnace. The furnace was made of quartz, and the sample was placed in a porcelain dish. The following results were obtained Density of Annealed Aluminum '.fl 23.20°C 24.25 24.50 24. 70 25.20 25.40 25.80 26.40 26.70 Ws 10 gm 10 8 10 8 10 10 10 8 (;) 2.69867 gm cm- 3 2.69877 2.69880 2.69829 2.69812 2.69816 2.69845 2.69801 2.69803 p at 25.o 0 c 2.69835 gm cm- 3 2.69863 2.69871 2.69824 2.69816 2.69823 2.69859 2.69826 2.69834 Av. 2.69839 ± 0.00005 1-0 The thermal coefficient of linear expansion of aluminum was taken as 2.2 x 10-5 per degree centigrade in reducing both the density and the lattice constant to 25.o 0 c. Some annealed aluminu.m was :placed in an arbor press and compressed without sudden impact so that the dimensions were changed by a factor of two. This should not change the lattice as much as hammering or severe working, but if there are any hol es in the lattice they should be closed and the density increased. The following results were obtained. Density of Pressed Alumin1llil T 23.oo 0 c 24.05 25.20 Ws 14 gm 14 14 p p at 25.o 0 c 2.69835 gm cm-3 2.69813 2.69839 2.69871 gm cm-3 2.69830 2.69835 Av. 2.69829 ± 0.00004 One may conclude from the fact the density actually decreased a slight amount that there was no effect due to holes in the annealed crystals. The samples were severely worked by hammering and the density redetermined with the following results.• Density of Worked Alu.lllinum T Ws 24.60°C 25.15 25.70 12 gm 12 12 E' 2.69827 gm cm- 3 2.69791 2.69778 p at 25.0°C 2.69820 gm cm- 3 2.69794 2.69790 2.69801 ± 0.00006 ll The silver was annealed in the same furnace for 24 hours at 0 600 C, and allowed to cool i n the furnace. ations in the following table were made. The first five determinThe sample was worked and reannealed, and the last three measurements are the resulting density determinations. Density of Annealed Silver 24.45°c 25.20 25.20 26.10 26.20 Ws 40 gm 40 40 40 40 10. 4861 gm cm-3 10.4858 10. 4877 10. 4866 10.4866 p at 25. o 0 c 10 .4859 gm cm-3 10 .4859 10. 4878 10 .4872 10. 4873 23.60 24.45 25.90 40 40 40 10. 4887 10.4869 10 .4869 10. 48 79 10. 4867 10. 4874 T p Av. 10. 48 70 ± 0.0002 The t hermal coefficient of linear expansion was taken as 1 .9 x 10- 5 per degree centigrade in reducing both the density and l at tice c onstant to 25. 0°0. The silver was then compressed without sudden impe.ct in an arbor pr ess and the densit y determined. Density of Pressed Silver T w 23.35°G 24.20 25.40 22.95 24.25 25.10 40 gm 40 40 29 29 29 p at 25. o0 c p 10.4891 gm cm- 3 10.4888 10. 4866 10.4878 10.4878 10.4894 10 .4881 gm cm- 3 10. 4883 10. 4868 10 .4866 10.4874 10. 4895 Av. 10.4878 ± 0.0003 12 One concludes from the small increase in density that the annealed silver is not spongy nor does i t contain large holes. The silver was then severely worked by hammering and its density measured. Density of Worked Silver T w 23.60 °C 24 .15 25.20 26.00 40 gm 40 40 40 p at 25.o0 c p 10.4937 gm cm-~ 10.4894 10.4901 10.4916 10.4929 gm cm· 5 10.4889 10. 4902 10.4922 10.4911 ± 0.0006 I attempted to measure the density of fine silver powder prepared by chemical precipitation. Amend. It was obtained from Eimer and The density of the powder as it ca.rrefrom the maker was about 10.470 gm cm-3, but after it was dried and remeasured it had fallen to 10.410 gm cm-3. a.ri As the particles were almost colloidal in size extremely thin surf ace impurity picked up while drying would be sufficient to explain the decrease in density. The powder had different 11 wetting 11 properties after it was dried and acted "greasy" and water refused to wet it. This surface coating was probably oxide for care was taken to prevent chemical con-l;amination. The crystal structure of the fresh and dried samples was the same. In fact the demsity of silver prepared by chemical reactions seems to be quite variable. Kohlschutter and Eydmann 7 found values ranging from 9.945 to 10.499 for fine powder prepared by precipitation or decomposition 13 When the quartz powder was being prepared some pieces as large as would conveniently enter the neck of ·the specific gravity bottle were saved and their density determined. Density of Macroscopic Quartz T w 22.65°C 22.80 23.50 24.50 12 gm 12 12 12 E' at 2s.0°c 2.64824 gm cm- 3 2.64814 2.64808 2.64841 <? 2.64846 gm cm- 3 2.64834 2.64822 2.64846 Av. 2.64822 ± 0.00005 The thermal coefficient of volume expansivity was taken as 3.5 x 10- 5 per degree centigrade in reducing the density to 25.o 0 c. The density of the finely poVIdered qu?,rtz could not be determined as accurately as the density of the large pieces. The powder had a tendency to contain small air bubbles, but these were probably removed by boiling the water under vacuum. However, since some of the sample was almost colloidal in size' it formed a suspension in the water and wien the thermometer was inserted in the neck of the specific gravity bottle a small a.'!lount of water containing some quartz mu st be lost. Density of Powdered Quartz T 23.os 0 c 23.70 24.25 24.25 24.50 24.65 24.80 w 14 gm 10 14 16 10 14 10 p 2.6483 gm cm- 3 2.6484 2.6480 2.6471 2.6489 2.6480 2.6484 p at 2s.0°c 2.6482 gm cm- 3 2.6483 2. 6Ll 79 2.6470 2.6489 2.6480 2.6484 Av. 2.64810 ± 0.00015 14 The density of the powdered quartz seems to be identical with that of the macroscopic sample. •For comparison we include the results of densi·ty determinations made by other observers. Density of Silver, Alu.rninum and Quartz at 25.0°C Material Silver (electrolytic) Silver (vacuum distilled) Aluminum (cold rolled 99.97%) Quartz Quartz p 10.4914 <3m cm- 3 10.4893 2.698 2.6483 2.6479 Observer Timofeiev 22 Kahlba 23 Edward, 4 Various I.C.T ~- 20 Tu1 The quartz was an optically clear sample and apparently a perfect specimen. It was , I believe, found in, California. The aluminum was obtained through the courtesy of Wm. L. Fink of the Aluminum Company of America. Aluminum Silicon • Copper Iron T<iltanium He gave the following analysis 99.980% 0.004 0.011 0.004 0.001 The silver was obtained from Handy and Harmon, Bridgeport, Connecticut. The Applied Research Laboratories of Los Angeles gave the following analysis. , Silver Calcium Silicon Lead Iron Copper 99.957% 0.009 0 . 014 0.004 0.006 0.010 Corrections will be made for the presence of t hese impurities. B A FIG I THE SEEMAN BOHLIN FOCUSING SPECTROGRAPH :15 MEP.SUREIY1filIT OF THE LATTICE CONSTA.NT A Seeman-Bohlin focussing type of powder spectrograph was used because it gives strong lines with a relatively short exposure and if the wavelength of x-rays is chosen so that the Bragg angle for some plane or planes is nearly 90°it is capabl e of very great accuracy for sin G varies slowly with Gin this region. of this focussing type of camera is as follows. The theory In Fig. l let the photographic film be placed on the arc ASB with a hole in its center so that divergent x-rays of wavelength slit S. \ can pass through the fine These x-rays will fall upon the powder samples placed along the arc CDE diametrically opposite the slit S. Since the powder sample contains a large number of tiny crystals randomly oriented there will be some crystals properly aligned, no matter where they lie on the arc CDE, to diffract the x-rays of wavelength A from a given set of planes of grating spaced to a particular point F according to the Bragg equation A = 2d sin G, where G is the angle which the incident and diffracted rays each make with the atomic pl ane normal to the lattice distanced. That the camera is focussing is apparent when we consider that the arc SF subtends the same angle for every point along CGE. There will be cr-y-stals oriented in such a way as to diffract the x- rays of wavelength arcs F 1 S and FS are equal. ,>,. to the point F 1 such that the The spectrograph gives a photograph which is symmetric with respect to the slit and by measuring F 1 SF we can find 16 2s without knowing the position of the slit. Now 2G = 1i' - p G = !! - i and if Dis the camera diameter~ will be given by 2 2 s/D and Bragg's equation will become or and we see that when s is small ~ a large change in s makes a small change in d . For example, in our camera where D = 1 7. 5 cm and whens= 11 cm, which is as big as it is in any of the photographs used, then a 1% change in .s means a 0.10% change ind. M. U. Cohen 8 has discussed the errors introduced by vertical divergence of. the beam, slit wi dth, thickness of the powder sample, shrinkage of the film and errors in the measurement of the ca.mera r adius and he has shmm that for cubic crystals ~ d ~ k ¢ tan ¢/2 and suggests plotting a against¢ tan ¢/2 which when extrapolated to ¢ = 0 would give the correct value of the lattice constant. three parameter crystals can be treated similarly. Two and However, in the present work since only t wo values of¢ were obtained from a photograph of the metals it was felt that extrapolation using onlytwo points would be a dangerous procedure. Furthermore, such a procedure might mask systematic errors of measurement. each error was made separately. Therefore, the correction for In the case of the quartz ten lines from six different pl anes were used, but the same method was used for by examining the residuals in the least squares calculation we would have an excellent check on the validity of ou'f corrections. \ rlG 2 EFFECT or F"ILM THICKNE:SS FIG 3 EFFEC T OF SAMPLE THIGKI\ESS 1:7 ERRORS OF THE C.Ai"\/lERA As the errors are all small they will be treated as independent. Thickness of the photographic film. The effect is to broaden the lines and shift the maximum toward the slit. From Fig. 2, letting f be the thickness of the photographic film, and ¢' = _s_'_ and X D but ¢' = X + ¢ so s1 and since 3 tan 'Pr1. = -s + - s 3 + .. 2f = D -f 2f tan¢ = s(l - 2Df) + f tan ¢ D 3D ther:efore The x-ray film has emulsion on both sides and if they are equally affected the maximum is shifted by s - s ' The correction to be added ,b,. Sd = : (1 + ~). This error if uncorrected 2 to the measured s is would introduce at most an error ind of 0.007%. Thickness. of the saTJ1ple. towards the slit S. The effect i.s to shift the maximum Using Fig. 3 we see that ¢ is the measured angle but ¢' is the angle that should be used in the calculation. ¢' = ¢ + X film thickness, then where X =D t - t tan ¢, ¢, = ¢ + _t_ tan .¢ or D - t As t being the / ---- - -- - - - ' ..... / / / I I s H l Gl I I ' '--- ,, ' ..... ..... -, ..... ..... F"IG 4 EfFECT Of VERTICAL DIVERGENCE • -.1 FIG 5 ' DIF"FRAC TION Of A LINE GRAT ING ,, / 18 ¢' = .f D {1 + 1(1 D - + 317° 2 s )] As the radiation is strongly absorbed in the smnple it is only the surface layer that matters. ~ -St = st ( 1 + 1 ) . D 3IJ2 The correction to be added is This error if uncorrected viOuld introduce at most an error ind of 0.040%. Film shrinkage is co:r-rectecl by maki.,ig two marks on the film with 2. templet just after the exposure is completed and before developin.g. The distance between t hese marks is measured after developing and the shrinkage assumed to be lmiform. The shrinkage was found to vary considerably from photograph to photograph and would on the average introduce ind an error if uncorrected of 0.033%. Vertical Divergence of the Beam. The effect is to broaden the lines and shift the maximum away fro m the slit. that since L SKH ::= 1r/2 ~ 112 SK = (h2 + D2 ) sin (¢ + ~ ) SK D Using Fig. 4, vrn see = (1 + h2) i/2 sin n< ~ D2 sin ¢ (I + but 2 h ) sin ¢ = sin ¢ + ~ cos¢ 2D2 2 ltan ¢ Therefore 2D2 And as the image of the focal spot is approximately a 1L11.iformly illmn- inated rectangle, the maximum is shifted by .6.s V ~/2. 2 tan ¢ = h~s ? [ I + ~ (.f) 21 = h.... 4D 4D2 .'.JD Then 19 is the correct ion to be subtracted. This error if negl ected would introduce at most an error of 0.021% ind. The height of the illum- inated sample is 2h . Temperature of th§'_ sar11ple. Because of the t hermal expansi on of the lattice a change of one degree centigrade introduce s an error of about 0.00 2°/4 in d. The temperature of the camera was checked several times during every exposure and it never varied more than half a degree. The camera dimension$ were not affected by temperature by an appreciabl e amount. Overl apping of the ~ doublets. If the l attices 111ere l ess than 1000 A0 on a side the diffraction patterns of t he components of the K d. doublet would begin t o overlap due to the finite resolving power of t he crystal grating and the apparent maximum would be ' shifted in position. A method of calcu_lating the true maximum from the apparent ma.ximwn will be given. in d~ould be introc:hice~ For a lattice 300 A0 long, an error f 0.0:?0%) if one used the position of the apparent maximum r ather than of the true maY-imum. 0 When t he CI"IJStal grating was longer than 2000 A on a si de the diffraction pat tern consisted of distinct spots. As the length of grating decreased the spot s became less disti nct and began to over-lap and when the l ength of grating was less than 1000 A" the l ines on the photographic pl ate became quite uniform 2nd began to broaden out consider abl y. From the arno1.mt of t his broadening it is possibl e to calculate the length of the lattice or the size of the crystal. 20 FINITE RESOLVING POTIER OF A CRYSTJli We shall now derive an expression which connects the shape of the diffraction pattern with the length of the l attice. During the analysis we sh2.ll make the following assumptions. 1) The crystals are rectangular parallelopi peds with a simple orthorhombic l attice with axial lengths ax, e-y, az and 2Nx + 1, 2Ny + 1, 2Nz + 1 particles on a side respectively. The formul a derived, however, will be more general than this, for the shfipe of the line depends only on the length of the lattice perpendicular to the Bragg plane. 2) The crystals are oriented at random. 3) The crystals are undistorted and act independently. 4) The radiation is monochromatic. 5) Absorption and extinction are negligible Consider first normal incidence of a plane wave on a line gr ating (Fig. 5) composed of 2Nz + 1 identical particle s with the distance az between adjacent neighbors. Each poi nt particle will send out a spherical wave of amplitude F, rt, . C .:,.ri-"i-(1?J1.._.:£) Real -..:Le - R,,_, where T is the period and >- r >. the wavelength of the incident radiation, Cz is a scattering constru1t, •and t the time. Since (2Nz + l)az L..L Ro we have Frau.nhofer diffraction and at p the vector amplitudes add al gebraically. Also (Ro - Rr) = raz cos o< z so the amplitude at Pis + = 21 and if the incident plane radiation is not normal but makes an angle f ·with the grating we have ! Summing the geometric series we find Ci: '° f -- Real f["L- :J.ri'Jfo _ L-'>, Nr (coso(. +cosB~) _ Cos~fi'J.'t(fllv:+ 1) rt)[ Cos .;i.(i'c1,T: A i; C A I - Cos .:i:c2i. (c.OSo(i::+ cosf-t) The average intensity over a period is independent oft and is proportional to the square of the amplitude. ], -. c',1{"°' ~ '4N, (~+CD'(', ) _ co 5 -¥,-.i. (N,.,' )(Co so<, 1- cos f•) j :, I - Cos eJ-o~(coso(:/: + Cosf~) Furthermore, siI1ce Nz is large almost all the energy will be found in the range ¥ -1'!: ( cos oe~ -t CoS(!~ ) == .2.. m-2- Ti' (the maximUI!l) , (the first minima on either side). Hence differing from m~ 'It' ¥.:3-r(cos o<t-t cos (3~) may be considered as by a small amount, and expanding we find 22 For a three di mensional grating the total intensity will be I"'- .J_-Z, I"J I,, := c~ c'j - II I/cl/ 1< sit/ f ~ N"f.a.."'1-{ GoS O<.,x-t cos ra~J] [.tM,,.:ii' _ -?j-a..,y.( e,0 s <>t-,,+ Cost3?. and. must sati sfy The maximum value of I occurs when the three denominators vani sh simultaneously. when D~ For a general wavelength the Bragg maximum will occur Co.so(~=· cos (3~ ::: .l~ ( m~=-.1.) , 0 ('1 = - (1 "1 = 'f -oit ( ;n"1 ==- 0 ) (or any permutation of x, y, and (If t here is a particular wavel ength such t hat z). ,J cos......,?<+ C os(d-,, =- ~l... a."' Cos o("1 + Cos (3"1 ::; si multaneously where ma.Yj_mum). fYl...., .,., , I)) "1 I m.,.. -c: ;t o , we will have a Laue We are i nterested in the diffraction patt ern along the arc SF in the plane SGF and the incident and diffracted wave normals lie in this plane. Since nearly all the energy goes into a region clo se to the central maximum, we will set o( ?< = (3-x : :. ~ - c where and ,:) •LL 1 Co5(3 0 = , and finally let . o(¾: o(0 - 5.+.w = f-,o -d + w Cos c,< = (~ d ,~y 1 o where =-:r-x7 &u LL.. Then since .1. i ndependently. c and f are small they can be varied The cosine condition then requires that When we make these substitutions I becomes This is the intensity in the direction UJ from the central maximum F (Fi g . 6) in the plane SGH when t he crystal is oriented in the range tM de: . Then by integrating over S and c we will t ake into consideration the random orientations of the crystals. substitution s have been so constructed t hat J and C Our > can be inte- grated independently and during the integration the angle SGF 11 remains constant (Fi g. 6). mc1.ximum . = Since It i:s sufficient to integrate over the central s G FIG 6 ORIENTAT(ON OF THE ---- CRYSTAL GRATING - M (ci.)(== °'-a :o n!J.) 24 and vre see that the shape of the line is independent of Nx,ax,tt, ai.-id a -, and the se factors affect only the total intensity . However, since x , y, z can be interchanged the diffraction pattern will consi st of the sum of three patterns of different widths and intensities. · For the moment then let us consider the crystal s as cubic with a cubic unit cell. Experimentally we determine the • shape I (w ) and from it we wish to determine 2N~a~ the depth of the lattice. It is most conven- ient to do this by measuring the average width which we define as the area under the curve divided by the maximum hei ght. and and diffraction pattern is :81- p.,..~ . , I..,"" ~-84 X tJ. 'it'a...,_Nt: 9>11'.X. be the l ength of the cryst al in t he z direction; t hen =8 =- O. qo-" A Sit') o<o where Bis expressed in radians. If we had u sed the breadth at t he point of half maximum we would find Scherrer 9 gives 1!/ = 0.1~ _A ..IL sn, ~" similar assumptions. .:B 1 = 0. 8'1 A ...J\_ s i11 o(0 in his derivation based on Various investigator s have calculated the value of the numerical coefficient for various shaped cry stal s and found t hat the line breadth is not particularl y sensitive to variations in shape , 25 giving values ranging from .9 to 1.1. When the cubic lattice does not have the same munber of particles on each side, our formula for B when appl i ed to the observed diffraction pattern give s for .A.. the arithmetic mean value, ; (N"'+N.,•111:)d Furthermore, let us consider the case where the cubic crysta.ls have N particles on each side but l et the crystal s be of different si ze s. -f (N) . Let the number of size N be [N :f (N) dN 4 find )\_ -:. a. lN~f(N) dN 1't Then using our formula for B we [(P;) "f(%;)d{~L ;;:;. Oley,{~) '3f (~)d (t) .: k: 1No d, where N0 is the maximum value of f(N). and if we knovr f ( 1'L) No calculate the value of the numerical conste.nt K. we can For distr ibutions of particles we would experimentally expect t hi s numerical con:stant will not be greatly different from unity. On the avere.ge I think our formula for B will give the most probable value of the l ength of the l attice within 25%. It i s necessary to correct for slit width and vertical diver gence which give apprecie.ble symmetric line broadening. correction s it will be neces sary to expand f In making these <i 51::t-L c{4.. 0 in a series and it will be very inconveni ent to u se more than t Fo t er ms. As fasin~ic du..'=e, «.,.;I. a _ 4-13 + ~ 8 - S" q ~2 5" • • • if we keep a L 5/2 t wo t erms will r epresent t he integral V'i ithin 5% error. 26 Each section of the slit of width d(3 will give rise to a separate pattern with its maximum at a different position along SFF 11 (Fig . 6) but the resultant pattern which is the sum of all the separate patterns will be symmetric and the position of its maximum will be unchanged ( w ::: o ) . Let the angular width subtended at the cryst al sample by the rectangular slit be 2 ~ . The area under the curve will be the sum of the areas due to each section of the slit of wi dth d@ . The intensity di stribution is given by f I(wJ =- C f siJ1tz.,;ifi' Na.(w+~)siho<.6 ,A df -~ ( ~(1-d{w+(B) sino(o).J. a.nd whil e this cannot be integrated. directly its maximum value can be calculated. Before we do this let us include the correction due to the vertical divergence of the beam. We saw before that its effect is to spread out the intensity curves and shift t he maY.imurn to the point w ::: Si . The area under the curve is ::J. . B4CN).. ~ 1i' a. sin o<o x~A :J.5 -r and the intensity distribution is given by by expanding si11:2.k. /tt,~ in a series and integrating term by term. Using only t wo terms the result is 27 For this formula to be a good approY,imation The largest crystals we will try to measure have _A = 1000 A, the largest value of sin o<0 we will use is 0. 25, the shortest wave0 leng~h 1.54 A , and the large st value of This requires that (3 ~ J6 used was 1.5 x 10- 3 . 2. 3 x 10- 3 or in our c amera the slit should 0 be equal or less than 0.8 mm. A slit was built which had this width. When the expression for Bis put into the practical form for the camera used, it becomes At first the second term in the denominator is neglected and a preliminary value of .JL found. This is then inserted into the correction term and a final value of .A. calculated. in centimeters. B and s are both expres sed The shape of the diffraction pattern wa s experimentally determined by taldng a microphotometer trace of the photograph. As the lines on the photographs were curved, a set of circular defining slits having approximatel y the same r adius of curvature as the lines on the photograph was built. Because of this curvature the height of ... - .,__ s -- -~- - - / / / f(s)' Koi.1. / \ / \ \ ,'\ \._ / I / " "'- .;J.. ''- / 2.,.f(~t-~) I ?..'.3 ,..10-~ oA· - // \ I K01.2. - FIG 7 THE • \11,,--- Of AG SAMPLE (51 I) l333) PLANES TRACE A~ 280 °A FOR CU KO( FROM MICROPHOTOMETER 28 the illuminated portion of the photograph (perpendicular to the axis of resolution) was kept as small as possible. The density of the film was low so. that the galvenometer deflection shou.ld be proportional to the opacity of the film. Fig. 7. A typical microphotometer trace is shovm in The overlapping doublets were decomposed by using the doublet separation taken from other photographs where the overiapping was very slight and taldng the intensity ratio of :Ehe Ko<1.. to Ko<,_ as 2. Let F(s) be the observed curve10 , then F(s) = f(s) + 1 f(S + $ ) and 2 f(s) = F(s) - 1 F(s +·a ) + 1 F(s + 2t5) •••• 2 4 However, because of the geometry of the spectrograph f> is not a constant but proporj,ffional to 1/s, so care must be taken in applying the above equation. Also because of this the curves will be slightly unsymmetrical, being stretched out slightly on the long wavelength side ( smaller s). If it is only desired to fix the maximum of f( s) this is done by plotting f'(s) = F 1 (s) - 1 F'(s + S) + 1 F 1 (s + 2 &) •··• 2 4 for several points in the region of the suspected maximll1Il, finding some positive and some negative values. will give the maximum of f(s). Then the point f 1 (s) = 0 \ J I l r=-; ~ -:-~-- - ______LJ__ -9,) ·~ r ~ -~ : FIG 8 SCHEMATIC 'I -- ~ ! t = - -· / DRAWING OF THE X-RAY watey c.ooh\'\g coil \ = TUBE , '-... -~ p1c,e11 . ~ ._ I pyrex tub•n8 ~J ___ _ rubber- 1h!.ulat1'ng filameY"tt o.ssembl'1 =====~'.:=:i+i7 sl1din~ r==== _r:- , le~ -t,a,y~et ·- ~-=f - --- " -_ =.I=; . •/ / nJJJ- ----_- rubber- gcH,ket 5olde.-ed jo·•nt ·- ----- ':=r=========~ _ ____i_;:::: _ u \ ~f 1·- ·'· /· t ' ~- aluminum window / solderQd JOlftt !:>\'\uv\der // ~a~ket~ 29 THE X-RAY TUBE The source of radiation was an interchangeable target x-ray tube designed and constructed by Dr.H. Kirkpatrick. I assembled the tube for the first time and made minor revisions in the design. tube is shown schematically in Fig. 8. The The heated filament assembly was of commercial design and the current was supplied by an insulation transformer and adjusted by a series resistance in the secondary. Its distm1ce from the target could be adjusted by sliding the whole assembly in its holder, and 3/4 of an inch distance between the end of the fila,ment assembl y end thB target gave a good focal spot. The target and body of the tube were grounded and unrectified hi gh voltage applied to the cathode. The x-rays could em~rge through three holes in the body of the tube and close to the t arget so that a line f ocus was obtained. The se holes were made vacuum tight by covering with 0.001 inch aluminum foil and sealing with picein. The foil was hel d firmly in pl ace by the slotted bras s cleats shown in Fig . 8. well as the target wer e water cooled. to prevent stray radiation. copper rod. The body of the tube as The body was covered with lead Two t argets were built using ordinary One served as a source of copper r adiation while the other was chrowium plated. These two material s were selected because the .K component of their characteristic r adiation gave possible r eflections from silver and aluminum crystals with the Bragg angl e equal to nearly 90°. 30 The tube was evacuated by a singl e sta.ge brass oil pump built at; Stanford University and used n-Butyl Phthlat e. was used as a forepump. A Cenco Megavac The pressure was determined by means· of a Knudsen Gauge, and once the tube was made 11 hard 11 the pressure could be kept about 5 x 10-6 rmn . Hg, even while running. There was a low t emper- ature trap between the pump and the tube and a mixture of acetone and solid carbon dioxide was employed as a r efri gerant. The actual operation of interchanging a t ar get took about ·fifteen minute s but if required t welve hours pumping before it was hard enough to run again. One burned out f ilament was r eplac ed. The heat er wire is unsoldered at the extreme end of t he cathode, and t he filament assembly was removed through the t arget end. assembly was inserted. A new fil ament Thi s seemed preferabl e to t aking apart t he cathode end of the tube which would involve r epl acing a rubber gasket which could not be conveni ently painted if desired. The camera is the srune one used by Bollman and i s fully described in his t hesis11 . A new rectangular slit of 0.80 mrn wi dth and 1.20 mm height was construct ed by making a new slit plug of the same design but with a very small pin hole which was filed into a rectangul ar opening with a jewel er I s file . cumference within 0.001 inches. The slit was repl aced on the cir- The camera was made vacuum tight by pl acing a piece of 0.0005 i nch aluminum foil over t he window, painting generously with glypt41, and clamping it firmly in pl ace with t he face pl ate. The camera was pl aced on a movabl e t abl e.drilled with three 31 holes into which the three legs of the ca.'llera. fitted. The position of the table was ad justed until the image of the f ocal spot was as intense as pos sibl e and the rectangular image was symmetric across the equatorial pl ane of the camera. The t abl e was then waxed in place so that the c aJUera could be removed and then repl aced in the identical position. A film was pl aced diametrically opposite the slit and an exposure of the image of the focal spot t aken . The image was rectangu- l ar,. being 3.2 by 0.8 cm with the great er di men sion being the vertical one. The di amet er of the crunera was checfoed and found to be i dentical • with that given by Bollman. (D = 6.905 i nches). The camera slit was as close to the target as pos sible, the distance between the center of the target and the slit being about 4.5 cm. The camera was evacuated to prevent scattering of the x-r ays by the air. The high voltage was controlled by means of an autotran sformer in the primary of the high voltage t ransformer . It was found that the graat est contrast in the photographs was obtained when the copper target was operat ed at 30 ,000 volts (r. m.s .) and t he chromi um ti;i.rget at 20,000 volt s (r .m . s. ). A nickel f ilter was tried with the copper radiation but the contrast was not i ncreased, copper K c< A normal exposure with radiation was one hour with 10 milliamperes current, and with chromium Ko< two hours with 10 millia.mperes current. The povsrder sample was pl aced on the camera circumference by making a past e of it with a little Duco cement diluted about three to one with acetone. This made an excellent binder with very little body. The foil, and 32 aluminum evaporated on mica were attached to the camera wall with a small a.mount of stopcock grease. The templet for making the fiducial marks on the film consisted of t wo phonogrn.ph needles fimJ.y imbedded in a brass bar. The film was removed from the camera and pl aced between two plane and par allel blocks of wood with the top block having two small holes through which the needles could pass, and a light impression of the needle points was made. After the film was developed it was pl aced on a ground glass viewer and a faint line ruled on each maximum of the ciiffraction pattern. It was easy to see the point of maximum intensity with the eye but the lines were so broad that they could not be distinguished under the microscope of the .comparator. The distances between the ruled lines and between the needle points were measured on the comparator, the ruled lines appearing quite sharp. Actually with such broad line s and the crude way of marking the points of maximum intensity the accuracy of measurement :possible on the comparator is an unnecessary refinement. But this method of measurement was used because after the line s have been ruled on the photogr aph the comparator is an extreiµely impartial instrument. I found that in ma..."king several measurements on the distance between two points with a pair of di11iders ( as well as in recording the swings of the point er of the balance in the weighings) there was a strong tendency to repeat the first observation. I11 the case s where a microphotometer trace was taken the ruled lines appeared as very 33 narrow peaks and did not distort the generaJ. shape of the curve. Comparing the position of the ruled maximum with r eal maximum I estimate the error in using the ruled mai.imum to be for a single measurement 0,01 cm which r epresents at most an error of 0.009% in the lattice constant. Furthermore with the large number of photographs used, such erro r s , which I do not believe were systematic, should cancel out. RESULTS OF THE X.:..RAY ivIEASUREMENTS Both aJ.uminurn and silver are face-centered cubic crystals, and for such systems the interplanar distance for the ( 511) and ( 333) pl anes (and aJ.l permutations of the indices) is the same. If there i s any change of shape of the unit cell the inter pl anar di st ance s for the ( 333), (333), (333) and (3 33) might all be different and these four planes would give rise to four separ ate diffraction patterns. If the distortion is small the patterns will overlap and give the appear ance of a single but broader line. In the sa.me way, with distortion, t he permut ations of the ( 511) indices ·will give rise to as many as t welve non equivalent interplanar distances. ]!'or a generaJ. lattice with a ~, a2 , a 3 being the axial lengths the interaxiaJ. angl es, the inter pl anar dist ance in t erms of the reciprocaJ. lattice (defined as ~ ~ • E .i = ~J )25 i s and 34 and from our definition of b s1.n'"<=><, ~ ~ = ~.A( =, I - Co5 .2. , ~ 1- c 5 .2. , _ cos 'c,z,, + .;2. coso(, Go.s ~ :zCos d 5 ) 0 o<.~ .;, Cos o<', cosd ::i.. cos o( 9 Now we shall consider the case of cubic crystals which have undergone a small distortion where d,--;.a.+b:l,) d.,.=-8-+ 4 8..J /2.9::.d.+Ad.3> and d. ~10 = 7):11' <£.3 • Then We see that for the (511) and the (333) planes the first term is the same. For any change in the length of the lattice, the second term always has the same sign for both positive and negative values of the h's. However, the third term is sometimes positive and sometimes negative and its average value is zero. Thus when we find that our diffraction patterns are broadened when the lattice is distorted by working it may be caused by either the finite resolving power of the shorter crystal grating or by the fact that we have for.med several planes with nearly the same interplanar distance. However, if the position of the central maximum is unchanged we can say posit ively that t here is no change in the axial lengt hs for h~.AB., + h:Lla.2. + h; h cl. !3 cannot vanish for ( 333) , ( 511) , (151) , and (115) simultaneously except .-Cl a , = ..d 82 = ~ ¾ = 0. Thi s 35 assumes that all the crystallit-es are distorted in the same manner. Furthermore, since a cubic face-centered lattice is a close packed configuration it is difficult to see hov1 the inter/ axial angles could be changed without altering the lengths of the sides. Since the volume of the unit cell is then when the unit cell is distorted as on the previous page the volume becomes, considering only first order terms, This second term is proportional to the average value of 2. ii 2. h, 6a 1 + h2. A a , + h3 Ao 3 when we consider all permutations of the Miller indices. Therefore, i f the latter term is zero (average value of the interplanar distance unchanged), then the distortion we have pictured changes the volume of the unit cell by second order terms only. As the largest first order change in some inte~planar distance required to explain the observed width of line is 0.2%, any second order effects are negligible. The wavelengths used in the cc:dculations were 35 b ,a Siegba.hn ur1its Absolute Angstroms Cu K ct 2 1. 541232 Cu K~ 1 1. 537395 1.544361 1. 540516 2.28891 2.28503 2.29356 2.28967 The results of the powder photographs of alrnninum are as fo llows : Lattice Constant of Annealed Aluminum Powder ( 99. 98~;) at 25.0° C Radiation and Planes a Gu Kd- ( 511) ( 333) 4.04919 A (absolute) 4 .04925 4.04926 4.04933 4.04935 4.04942 Gr Kct (222) 4.04914 4 .04919 4.04924 4.0493 7 Av. 4.04927 ± 0.00002 The estimated length of l attice was 2000 A0 • 36 Lattice Constant of lUlllealed Aluminum Foil (99.80%) at 25 .0 C Radiation and Planes a Cu K ( 511) (333) Cr K ( 222) 4 . 04904 A 4 .04905 4.049 10 4.04917 4.04922 4.04933 4.04876 4.04895 4 . 04905 4 . 04913 Av. O (absolute) 4.04908 0 The estimated length of lattice was 1500 A • Lattice Constant of Worked Aluminum Powder ( 99 .98%) at 25.0 °C Radiation and Planes Cu Ko< Cr K o1. (511) (333) ( 222) a 4,04930 Ao (absolute) 4 .04872 Lattice length 530 A• 740 % change in a 0.001 -0.014 Lattice Constant of Worked .AJ.uminun Foil (99.80%) at 25.0 °C Radiation and Planes a Cu Ko< ( 511) (333) 4 .04878 A 4 .04917 Cr K"'- ( 222) 4.04878 0 (absolute) Lattice length % change 1200 Ao 490 -0.007 0.002 740 -0.007 in a One may conclude that when the alurainum is worked by hammering smaller diffracting units having the same unit cell as the diffracting units in the carefully annealed sample are formed, and since a difference of 0.2% in some interplanar distances which were identical before the distortion would be required to explain the observed width and as in every case but one the maximum was shifted 37 by less than a 0.007% change in the average interplanar distance one concludes that the distortion occurs mainly in the layers separating the diffracting units, In all cases the microphotometer traces were symmetrical within the limitations mentioned on page .:i.8. Powder photographs were taken of aluminuJn evaporated on mica. The lines were extremelJr weak and difficult to measure for the sample was of the order of a wavelength of light thick. 'Th.e results of the best photographs are Lattice Constant of Evaporated Alu.minlll11 at 25.0 °C Radiation and Plane Cu Ko< ( 511) (333) Cr Ko< (222) a 0 4.0487 A 4.0465 4 .0 525 (absolute) The last two measurements were from photographs where the lines were only visible on only one side of the slit, and may not be due to aluminum at all but to the mica onto which the aluminum was evaporated. It is also interesting to mention that in order to obtain the first photograph, which was the only really successful one, it was necessary to talce the powder photograph irrm1ediately after the evaporation was finished or no lines were obtained. Furthermore, when a second exposure was tried one day later with the same sample used in the successful photograph the lines were no longer present. This I interpret as showing the aluminum crystals are no longer randomly orientated, but have taken up some preferred orientation on the mica surface which was freshly cleaved just before the evaporation took place. The results of the powder :photographs of silver are as follows. Lattice Constant of Annealed Silver (99.96%) at 25.0 ° 0 Radiation and Planes ., 'lype a Cr K « (222) Cu K o( ( 511) Cr K o< (222) Cu K o< (511) (333) 0 ( 333) Av. 4.08540 A 4.08553 4.08569 4.085'78 4.085'78 4.08582 4.08586 4.08589 4.08596 4.08603 4.08621 4.08623 4.08531 4.08549 4.08553 4.08553 4 .085'74 4.08503 4.08604 4.08604 4 .08543 4 .0854'7 4.08549 4 .08560 4.08566 4.08570 (absolute) Powder Foil 4 .08574 ± 0.00004 0 The estimated lattice length for both foil and :powder is 2000 A • It will be noted that the results for silver are not as consistant as those for aluminum. This is mainly, I suspect, due to nonlinear film shrinkage, coupled with t he fact that larger values 39 of s were used, 1'11 th the large number of measurements used this error should cancel out. Lattice Constant of Worked Silver (99.96 ?i) at 25.0 °C Radiation and Planes Cu K (511) (333) Lattice Length 280 Ao 4.08562 A0 4 .08530 (absolute) 230 185 4.08551 a % change in a -0.004 -0 .011 -0.006 TtJPe Powder Foil Foil Again the evidence is that the unit cell is not distorted by working by halllnl3ring. Six photographs using two different samples of quartz were taken with Cu K"' radiation. The lines were indexed from similar photographs taken by Bradley and .Jay 26 . lines were used. Six planes and ten different When lines from two different planes overlapped no attempt to resolve them was made and they were omitted from the calculation. Interplanar Distances for Quartz at 26.0 °C Plane and wavelength (216) (216) (234) (234) (421) (315) (420) (420 ) ( 33Ji) ( 502) Cu Ko<.,_ Cu Ko<., Cu K°'.:i.. Cu Ko<., Cu Ko<. .,_ Cu KO\, Cu Ko<.:i. • CU Ko<, CU Ko(,_ Cu Kol- i 9 79.297° 78.563 77 .509 76.765 76.159 75.113 73.843 73.330 72.510 71.636 Interplanar Distance .78429 A .78427 . 78961 .78967 .79366 . 79539 .80230 .80242 .80797 .80994 0 {Siegbahn) 40 If we let a/c equal k these interplanar distances (see page 2) give rise to the following set of simultaneous equations which were solved by least squares Residuals 21.8667 23.5005 24.0382 24.0310 10.9659 23.5163 15.7974 15.7949 5.7408 5 .7410 giving a2 + + + + + + + + + + 2.6240 k2 0.6528 k2 0.0000 k2 0 .0000 1<2 15.8162 J.t2 0.6299 k2 9.9773 k2 9 ,9757 J.t2 22,1429 k2 22,1440 k2 <:> = 24.0368 ± 0.0009 and k2 is 0.82626 ± 0.000025 and -20 X 10-4 31 14 -58 -26 - 1 = 0 - a2 = 0 - a2 a2 = 0 - a"'2 = 0 - a = 0 = 0 - a2 - a2 = 0 = 0 - a2 - a2 = 0 - a2 = 0 44 6 2 g a= 4.90275 ± 0.00009; also c = 5.39362 ± 0,00011, When these measurements are changed into absolute Angstroms and corrected to 25.o 0 c using the following thennal coefficients of linear expansion, along the~ axis 1.34 10-5 per degree centi grade and along the c 10-5 per degree centigrade, we find axis 0.7 Lattice Constants of Quartz at 25.o 0 c a = C = 6 4.91263 ± 0.00009 A 5,40454 ± 0.00011 (absolute) There is no systematic trend to the residuals so our method of correcting for the errors of the camera is justified. For comparison we include the results of other observers on the lattice constants of altuninum., silver and quartz. Silver Aluminum Q,uartz Lattice Constants at 25.0° c a 4.0861 A0 (absolute) a 4.0496 a 4,9132 C 5.4044 Observer Neugebauer 27 !I Bradley and Jay 26 41 IvIOLECTJLAR WEIGH'IS OF S.AlvlPLES For a.llll11inum the molecular weight as determined by mass spectroscopic means was used. On the basis of 016 = 16.0000 its molecular weight is 26.9917 17 . (Aston's 17 vaJ,u.e corrected to C12 = 12.00388). From the relative abundance of the isotopes of oxygenl 8 the conversion factor of mass spectroscopic atomic weight to chemical atomic weight is 0.99973. When a small amount of impurity is present these foreign atoms will fit into the aluminum lattice without distorting it. Therefore the effective atomic we i ght is 26 .989. Al Si Fe Cu Ti 26 .9917 X 0,99973 X 0,99980 = 26.979 28.06 X 0.00004 = ',.; 0'~0Clll 55.84 X 0.00004 = 0.002 63.57 X 0.00011 = 0.007 47.90 X 0.00001 = 0.000 26 .989 ± 0.001 In the case of silver the chemical molecular weight is used, and the effective molecular weight is found to be 107,858 ~ Ag Pb Fe Si Cu Ca 107.880 X O.99957 = 107.833 207 .21 X 0.00004 = 0.008 = 55.84 X 0.00006 0.003 28.06 X 0.00014 = · 0,004 63.57 X 0,00010 = 0.006 40.08 X 0.00009 = 0.004 10'1. 858 :1:. 0.002 The probable errors are estimated on the bases of the probable error of the chemical analyses. 42 Unfortunately there is no universal a greement about the atomic weight of silicon. The value accepted by the International Committee on Atomic Weights is based on the determination of Baxter 28 , Weatherill, and Scripture who compared SiC1 4 and SiBr4 with Ag and formd the molecular weight of silicon to be 28.063 ± 0.003. h However, more recently Honigs~d 2g and Steinheil by comparing SiC14 with Ag found the molecular weight to be 28.105 = 0,003 and Weatheri11 30 and Brundage using the same method found 28,103 ± 0.003 as the molecular weight. Because of the large differ- ence between these observers it is unsatisfactory to average their values so we shall carry through the calculati on for quartz using t wo different values for its molecular weight, 60.063 ± 0.003 and 60.104 ± 0.003. The mass spectroscopic determination of the molecular we i ght of silicon gives 28 .125 but this must be regarded as quite inaccurate for the calcuiation is based on some preliminary values of the relative abundances of t.11.e isotopes as me asured by McKe llar 31 using the band spectrum of Si N. CALCULATION OF 'IHE EXTINCTION CO.i~'FICIENT Equation ( 6. 55) in Compton and .Allison19 giving the approximate extinction coefficient (along the actual path) is where Z is the number of electrons in a unit cell, A is the inci- dent wavelength, 9 0 is the Bragg angle, and Fis the crystal structure factor which is calculated by F = ~ f . e27Ti (h Xj + ~Yj + ¾Zj) L_ J j where h , ,h2 ,¾ are the Miller indices of the plane far the Bragg angle 9 0 , Xj,Yj,Zj are the pos itions of the j th atom in the unit cell, and fj can be found from Pauling and Sherman 20 in terms of scattering from a free J. J. 'Ihompson electron. The scattering from a single electron is from Compton and Allison (3.04) e4 2 rl 2 2 4 (1 + cos f) 2r ill C where¢ is the angle between the primary and scattered rays and i'1e u»it oe~reme»Z- of S which is~the index of refraction is found from (4. 46) in Compton and .Allison to be S = ne2 >, 2 21TI!lc2 where n is the total number of electrons :i;:er cubic centimeter. 44 n = f. x B. x Z where is the density in grams per cubic M V centimeter, M is the molecular weight, N is Avogadro's number, and But Vis the number of molecules per unit cell. + Thus cos .:i. 2G sing where Fj is the number taken from Table VII of Pauling and Sherman. and if we express )... )le = 3.78 X 10 in .Angstrom units 3 o,\ ~ IVN 1 + cos2 29 sin g Then for Cu K o{ radiation reflected from the (333) (511) planes the depth of half penetration is for Aluminum Silver For Cr K o( 1.4 x 10- 4 cm 5.8 X 10- 5 ~m or or 1.8 x 10 4 planes 7.3 X 10 3 radiation reflected from the ( 222) plane the depth of half penetration is for Aluminum Silver 6.1 x 10- 5 cm 1.6 x 10- 5 cm or or 5.2 x 10 3 planes 1.4 X 10 3 Thus in all cases the radiation penetrated throughout the diffracting units. For ~uartz with Cu K o( radiation the depth of half pene tration varies from plane to plane but is on the average about 4B 1.3 x 10-4 cm or 1.6 x 104 planes. The largest particles that could get through the screen were 5 x 10- 3 cm but when a sample was viewed under a microscope the average size appeared to be about 5 x 10-4 cm. As the radiation could penetrate from either side this is equivalent to an effective thiclmess of 2.5 x 10-4 cm so that the x-r~y information is fairly representative of all the sample used. EFFECT OF DIS'IDRTION ON 'I'.K:i: :METALS \'.le have seen that the unit cell is unchanged by working. Let us assurae that samples of aluminllill and silver are made up of blocks of undistorted unit cells separated by ammorphous or darnaged layers of constant thickness. The measured density will be affected by these layers of constant thickness 2...1 D. f = d3t' 0 'Ihen the measured density will be + 3d2D D ( d + A D) f I 3 or approximately where f" is the true density, f ' the density of the a:mmorphous or discontinuous layers, and d the length of one of the blocks. Thus for aluminllill, since f = 2.69839 gm cm-3 ford= 2000 A0 at 25.o 0 c f' = 2.69801 we find (o = 2.69853 gm cm- 3 . for d = 500 However, due to the sketchy nature of this calculation the value we will use in calculating N and e will 46 be the value found for the annealed sample with the probable error o.hd (->• eQual to the difference between the annealed value~plus the probable error of the annealed sample. Vie shall use as the density of alumin- um at 25.0° C 2.69839 + 0.00019 gm cm- 3 Following the same procedure for silver since 10 .4870 gm cm-~ 10.4911 10.4864 then for d = 2000 A for d = 300 0 0 at 25.o c and we will use 10.4870 ± 0.0008 gm cm- 3 as the density of silver at 25.0°C. CALCULATION OF N .A1TD e In making the calculations we shall give two probable errors, the first will represent the errors in the lattice constant as computed from the inner consistency of the experimental data, the error in the density and the error in the molecular wei ght . The second will include the probable error in the absolute measurement of the wavelength of the x-rays and the error in Q. If our assu.mptions are correct all the values of N and e should a gree within the first probable error, since the same value of Q, and the wavelength was used for all the substances. The value of Q15 used is 2.89270 ± 0.00021 x 10 14 esu gm mo1-l For aluminum, using cm- 3 at 25.o c f = 2.59839 ± 0.00019 gm gm gm moi-2 Iv1 = 0 26.989 a ± 0.001 = 4.04927 ± 0.00002 x 10-8 cm at 25.o 0 c 47 N = 6.0258 ± 0.0006 x 10 25 mol gm moi-1 ± 0.0007 e = 4.8005 ± 0.0004 x 10-lO esu ± 0.0007 we find For silver, using f = 10.4870 ± 0.0008 gm cm- 3 1 at 25 .o 0 c = 107.858 ± 0.002 gm gm moi- a we find 4,08574 ± 0.00004 x 10-8 cm at 25.o 0 c N = 6.0318 + 0,0005 x 10 23 mol gm moi- 2 + 0.0006 e = 4 .7957 + 0.0004 x 10-lO esu + 0.0007 For quartz, using cm- 3 at 25.o 0 c a = 4.91263 ± 0.00009 x 10-8 cm at 25 .o 0 c C = 5.40454 -;;, 0.00011 (Baxter, Weatherill & Ivl = 60.063 ± 0,003 gm gm mo1 -1 f = 2.64810 ± 0,00015 we find gm N = 6 ,0239 ± 0,0004 10 23 mol gm mol Scripture) ± 0.0005 e = 4.8020 ± 0, 0003 ± 10-lO esu 0.0006 but i f iJ = 60 .104 ± O,003 gm gm moi-1 we find N = 6 ,0280 ± 0 . 0004 10 23 mol gm moi- 1 ± 0.0005 e = 4.7988 ± 0.0003 -±. 0,0006 10-10 esu ;(Honigs9hmid & Steinheil, Weatherill & Brundage) DISCUSSION OF '.IRE RESULTS If the molecular weight of q_uartz is in the range of 60 , 063 then t he value of e obtained is in agreement wit h t hat g iven by Bearden14 , 4 ,8036 ± 0.0005 10-lO esu. Bearden's value is based on the measurement of the lattice constant of large samples of calcite 48 with a two crystal spectrometer and the density of the large samples were determined by immersion. Both the density and lattice para- meters have been measured by a great many observers and there is no doubt as to their value. However if the molecular weight of quartz is in the range of 60.104 then the value of e will be in agreement with that obtained for a}1,7nmum. In such an event it is impossible to reconcile the values obtained using quartz and ,.J.wrtJIWJJ'l wi th those obtained using calcite without resorting to the concept of a more or less regular superstructure in the crystals. Considering the results of silver and aluminum. alone where there is no uncertainty in the molecular weights we see that the difference in the value of N obtained for the two materials is 0.10 % while the largest experimental error is 0.001%. Thus some of our assumptions as to the nature of the polycrystalline metals must be the. invalid. In order to obtain agreement withAvalue of N obtained using calcite the value of the density must be increased. This suggests the presence of holes or gaps in the metals which would render the measured density less than the true density . However, the little me tal crystals must part together in such a way that there is a reproducible percentage of very small spaces between the cTystals for the density of different samples was the same and when an annealed sample was worked and reannealed the density returned to its original value. But the density of silver increases with working and that of aluminum 49 decreases. This I interpret as showing that working changes some of the crystalline structure into amorphous material which for silver and aluminum is less dense than the crystals (silver and aluminum expand on melting) and which can fill the holes and make the density greater, as in silver, or it can more than fill the holes and make the density less, as in aluminum. These experiments show that there is certainly not a mosaic structure, even in the metals, which renders the value of N and e obtained by the x-ray method incorrect by more than 0.1%. Using the results of Tu1 on the glancing angle and density of calcite, rocksalt, potassium chloride, and diamond and the latest value of their molecular weights we find the following values of N and e where the probable error neglects the uncertainty in the ruled grating measurements and the value of the Faraday. For Calcite f = 2.71003 ± 0.00005 gm cm-83 at 18° C d = 3.03557 ± 0.00001 x 10- cm at 18° C 9'{(3) = 1.09602 ± 0.00001 at 18° C M = 100 . 090 ± 0.005 gm gm•mol-1 33 N = 6.0235 .± 0.0003 x 1023 mol gm. mo1-l e = 4.8024 ± 0.0003 x 10-10 esu Fillr KCl e = 1. 989:SO ± 0. 00014 gm cm-3 at 18° C = 3.14541 ± 0.00004 x 10-8 cm at 18° C M = 74.553 ± 0.003 gm gm- mo1-l 34 N = 6.0214 ± 0.0005 x 1023 mol gm-mo1-l e = 4.8040 ± 0.0004 x 10-10 esu d For Rocksalt d = 2.81962 ± o.oooos x 10-8 cm at 18° c = 2.16418 ± 0.00014 gm cm- 3 at 18° C (NaCl) M = 58.456 ± 0.002 gm gm- mol~l 34 N = 6.0247 ± 0.0005 x 10 23 mol gm-moi-1 e = 4.8014 ± 0.0004 x 10-10esu e 50 For Diamond e = 3.5141 ± 0.0001 gm cm- 3 at 18° C ( .89k) a= 3.56692 ± 0.00002 x 10-8 cm at 18° C M = 12.0115 ± 0.0005 gm gm- moi-1 33 N = 6.0255 ± 0.0003 x 10 23 mol gm -moi-1 e = 4.8008 ± 0.0003 x 10-lO esu (1.78) 3.5142 ± 0.0001 gm cm- 5 at 18° C d= 3.56683 ± 0.00002 x 10-8 cm at 18° C N = 6.0258 ± 0.0005 x 10 23 mol gm- mol-1 e = 4.8006 ± 0.0003 x 10-10 esu (" = In the calculations we have taken 0.709268x 10-8 cm as the absolute wavelength of MoK,0 32. We see that the values of N and e obtained for these 11 perfect 11 crystals differ by many times the experimental error. We have no assurance that 11 perfect 11 crystals are not going to behave like the metals to some extent for the difference between a polycrystalline material and a "perfect" crystal is one of degree. I believe, therefore, that the assumption that calcite is a "perfect" crystal might invalidate the x-ray value of e by 0.04%. It is also significant that the value of e obtained for crystals other than calcite, with the single exception of KCl, is lower than the calcite value. Thus the best value of e obtainable by the x-ray method is 4.801 ± 0.002 x 10-lO esu. This value is a personal estimate. Actually the average value using my results and those of Tu1 (weighing the metals as one-half and using the mean of the two values obtained for quartz) is 4.8013 ± 0.0004 x 10-lO esu. However, as the individual values differ by many times the experimental error the least squares probable error means nothing for the spread in values is not due to 51 statistical fluctuation but to something fundBJllental in the method. The values of the lattice constants I deteYmined were slightly lower than tho se given by other observers (page 40), the average being 0.009% lower . If this difference were real it would mean raising my value of e by 0.03%. Neither this nor the uncertainty in the molecular weight of quartz would change the conclusion reached . ACKNOWI.EDGEivIEN'IS I wish to express my grat itude to Dr. J. W. Iv!. DuMond for suggesting the problem and for his constant gui dance and assistance during this research . I arn also indebted to Dr. J. H. Sturdivant for numerous helpful suggesti ons. Bibliography 1. Y. C. Tu, Phys. Rev., 40, 662 (19 32) 2. A. H. Jay, Zeits. f. Kri st., 89, 3- 4, 282 (1934) 3. Alli ,s on and Armstrong, Phys. Rev . , 35, 1 4 76 (1930 ) 4. 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