The Chemistry Student SOME ELEMENTARY PRINCIPLES OF X-RAY CRYSTAL ANALYSIS. m In the previous articles of this series,'*2we have discussed some of the principles of X-ray diffraction and; assuming known structures for some very simple crystals, have inquired into the nature of the characteristic d e c t s produced by the interaction of crystals and X-rays. It now remains to consider briefly the means whereby these prindples are applied to the analysis of patterns obtained from crystals of unknown structure. No detailed or exhaustive exposition of analytical methods will be attempted. A grasp of these methods sufficient for creative work may well be left to the advanced or specialized student, whom the writer does not aspire to inform. The objective in the present series of articles is quite as modest as the title implies-namely, to give the general elementary student some intelligent notion of what it is all about. Crystal Analysis a Process of Elimination Since we have traced in some detail the.&ects produced when a crystal of sodium chloride is submitted to several of the principal methods of X-ray investigation, the uninitiated may hate supposed off-hand that there exists some reverse operation whereby one may reason directly backward from an X-ray pattern of one sort or another to the aystal structure which produced it. In practice, however, analysis is not effected in quite this manner. The operation actually consists in selecting from among the number of theoretically possible structures that which must produce X-ray effectsin closest agreement with the effects actually observed. It should be made clear a t the outset that it is not always possible to determine the exact structtuc (i. r., the precise placement of each atom) of any and evcry crystalline matnial which may he submitted to examination. Sometimes it is imporrihle to obtain all the data necessary for so complete an analysis; sometimes other diEiculties intervene. The nature of the obstacles to wmplete analysis will become more apparent as our discussion proceeds. ~
~
Crystal Symmetry At first thought it might seem that there is an unlimited number of ways in which atoms may be arranged to form crystals. However, the symmetry conditions imposed definitely limit the number of possibilities. THISJOURNAL, 7,13&50 (Jan.,1930). Ibid., 7, 660-72 (April. 1930). 1373
F r c m 27
FIGURB26 The axes of referrnee of point oupn belonging t o t h e triclioi~system. three a r e are of vnequal lengths (ofbfc) and make any angles with one another.
z
The ares of reference of t h e orthorhombie p*nt groups. The three ares of vnequal umt lengths (ofbfc) are mutually perpendirular.
The
axes
of reference for the monoclinic
point groups are 01 unequal unit lengths (a#bfcl. Z in perpendicular t o t h e plane of X and Y which, howeucr, can makeany angle with one another.
z
The axes of rrlcrenee of the tctrn ona1 point groups. The unit length along J? and Y is different lrom t h a t along Z (a+ '1. -
I
FIGURE 30 The axes of reference of t h e cubic point groups are fhme mutually perpendiculnr lines of equal vnst lmpthn.
Fmum 31 The h e m onal ares of reference. Two of
:yziz~ ~ ~ ~ 2 ~ E $ ~ 4 ~ ~ z 2 f , z f $
the third (2)is of a ditiermt unit length and is normal t o t h e plane of the first two.
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It has been mathematically demonstrated that the number of space groups possible is precisely two hundredthirty. These two hundred thirty space groups are distributed among thirty-two classes of crystal symmetry which are, in turn, divided among six main coordinate systems. The characteristics which distinguish these systems are set forth in Figures 26 to 31, after Wyckoff. If the substance to be examined furnishes crystals large enough for optical examination, the system, and sometimes the class, of symmetry can be determined by crystallographic methods. Crystallography is the study of the e.xternalsymmetry of crystals. The fundamental symmetry characteristics of a aystal are revealed by the distribution of its plane faces and by various physical properties, such as growth characteristics, etc. E
This is not to say that the symmetrical characteristics of a crystal are always immediately obvious from superficial observation. A substance belonging to the cubic system does not always crystallize in perfect cubes. For example, Figure 32
=
DEFIGURE33.-SmW1~c THATTRE CRYSTAL PICTBo IN h X J R E
32 BELONGSTO
SYSTESoa SYUMETRY
THE CUBIC
mate form of a beveled triangular plate. Figure 33, however, indicates that it belongs to the
It is also possible to determine the system of symmetry 01 a macroscopic aystal by means of a stereographic projection of itS hue pattern, stereo. graphic projection is a geo-
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JOURNAL OF CHEMICAL EDUCATION TABLEI (1)
Crystnllografihic Symnzetry
(2) Symmetry of X-Ray Diffraction Effects
I. Triclinic System
11. Monoclinic System zc
1
111. Orthorhambic System
2Di ) IV. Tetragonal System 4Ci 4Ci
V. Cubic System
E
Ti
Ti
VI. Hexagonal System A. Rhombohedra1 Division
3Di
)
B. Hexagonal Division 6c
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metrical operation which converts the zone ellipses of the Laue pattern into true circles. The pattern so obtained may take any one of eleven characteristic forms. Table I, copied from Wyckoff, shows how it is possible, by determining the symmetry of the X-ray diffraction effects, to at once eliminate from consideration a large number of classes of symmetry. The letters and figures in the left-hand column indicate the thirty-two classes of crystal symmetry; the letters and figures in tbe right-hand column indicate the eleven types of X-ray diffraction effects. I t is also possible to assign indices t o the spots in a stereographic projection of a Laue pattern but the procedure is rather complicated and the projection itself must be U 0 made with great accu- P racy. Since the symmetry of any crystal large enough to furnish a good N Laue photograph may be determined optically, and since indices may be more readily assigned by means of the gnomonic projection, we shall, not consider the stereographic projection in any detail. Fundamentally, the FIGURE 34.-THE GNOMONIC PROJECTION OF A gnomonic projection is a LAUESPOT. geometrical construction The X-rays incident at right angles to the plane of which registers the nor- projection and of the photograph (MNOP)are reflected mals to the by a plane RSTU (when extended) in a crystal at C, giving rise to a spot at F. By the gnomonic projecplanes upon a projection tion of the Laue spot F is to be understood the gnomonic projection of the rdecting plane RSTU. This plane, ~h~ projection point is G'. plane may he the plane of the Laue photograph itself or some plane parallel to it, which means that the incident X-rays producing the photographs were directed normal to the plane of projection. Figure 34, after Wyckoff, clarifies the explanation. One of the significant characteristics of this projection pattern is that the projected points associated with the planes of any one zone all fall on a straight line and that they are regularly spaced along that line. A little consideration of the planes illustrated in Figure 15 of the second article of this series, together with the construction of a few appropriate diagrams, will reveal the geometrical causes which give rise to this happy circumstance. In practice one does not bother to go through the details of an actual
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geometrical construction for each spot. A specially designed rule shortens the process and makes i t quite easy and simple. Figure 35 shows how the rule is applied. In Figure 36 we see one quadrant of the gnomonic projection of a Laue photograph of rock salt. (Incidentally, this diagram contains some secondand third-order spots which were not shown in Figure 14 of the preceding article.) As is the case with all cubic crystals, the coordinate network is
E is the projection of F; A is the projection of B ; C" is the center of the Laue pattern, and also of the gnomonic projection. The spots have been exaggerated in size and reduced in number for the sake of simplicity. square. The method of assigning indices is almost obvious from the diagram. It is evident that the values of h and k indices are determined by the coordinates and that the 1index is always one for projections which fall a t the intersections of coordinates. Employing the same system we would write the indices for the spot marked 173 as 5, 2$, 1. Since all indices must be whole numbers, however, we multiply these figures by three and obtain 173. Likewise, in the case of the spot marked 512
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we would originally write 2$, 4, 1, which must be multiplied by two throughout. The form of the coordinate network and the method of assigning indices varies somewhat with the symmetry of the crystal examined but the principles are the same throughout.
The Mass Associated with a Possible Unit Cell Assuming, then, that we have by some means established the system of symmetry of the crystalline material under investigation, the next step is to determine the mass associated with a crystallographically possible unit cell. I n other words we must ascertain the number of molecules t o a possible unit cell. For this determination we may assume a unit cell which has a shape defined by the customary axial ratios and angles.
For each system of symmetry there is a simple mathematical expression which gives the volume of the cell in terms of its axial lengths or of its axial lengths and angles. The simplest expression of all is that for the cubic system:
v = ad
(1)
where ao is the length of the cube edge. The volume of the cell (V) is related to the mass of the cell and the density of the crystal as follows: p =
mM/V
(2)
Here p is the density of the crystal, m is the number of molecules associated with the unit cell, and M is the mass of its chemical molecule. M is obtained by multiplying the molecular weight of the compound by 1.65 X lo-%' grams. Combining equations (1)and (Z),we obtain: p =
mM/ao'
or
m = paoa/M
(3)
However, our equation still contains one unknown quantity (ao) beside the one for which we hope to solve (m). The quantity a. cannot be determined directly but it can be calculated if the spacing for any set of planes (dhkz)has been determined. The spacing of a set of planes can be calculated if a reflection of monochromatic rays of known wave-length from a corresponding crystal face has been obtained. (Modification of the Bragg method.) The familist equation, nX = 2dhkl sin 8, is employed.
The experimentally obtained quantity, Rhbl, can then he placed in the following equation, which allows us to solve for ao: d n= ~ dIm/(h'
+ kP + lP)'/* = aa/(hz + ka + P)'"
(4)
We are then able to solve equation (3). Similar hut somewhat more complicated equations have been derived for the other coordinate systems.
The Correct Unit Cell Having established the symmetry characteristics of a crystal by optical examination or by the study of Laue patterns, and having calculated the mass associated with a possible unit cell, we must next choose the correct or "true" unit cell. This is defined as the smallest prism which by simple repetition along the coordinate axes will build up the entire crystal. If our crystal belongs to the cubic system we are not faced by any embarrassment of choice as to the shape of the cell, for cubic axes can be drawn in only one way. Our sole problem is concerned with the proper size of the cell. In actual practice we choose the smallest cell which agrees with the experimental data. In some of the other systems of symmetry
the choice is much more difficult, and, while it is often possible to choose a cell consistent with the experimental data, an enormous amount of labor is involved in eliminating all other possibilities. Selecting the Correct Structure Assuming that we have now arrived a t a knowledge of the symmetry of our crystal, of the number of molecules associated with the unit cell, and of the correct unit cell, it now remains to apply the theory of space groups, and to choose from among the possible structures that one which best agrees with the diffraction data. For the sake of being specific let us take sodium chloride.' Optical examination and the study of projections of Laue patterns, tell us that the sodium chloride crystal must belong te the class of symmetry designated as Oi in Table I. There are no data of any kind which would lead us to think that there are more than four molecules associated with a unit cell, and the unit cell must, of course, be a cube. By referring to such tables as those compiled by Wyckoff6we find that for these conditions there are only four conceivable sorts of arrangement. The four sodium ions in the unit (1) might be similarly related to one another, or it may be imagined (2) that three of them are alike but diierent from the fourth, (3) that two of them are different from the other two, or (4) that all four are unlike. (1) If the four sodium atoms are equivaleht and the four chlorine atoms are also equivalent, the following atomic arrangements are seep to he possible: (a) (Figure 37) Na: 000; 04:; $01; H0. Cl: 444: too; o w ; not.
By transferring its origin t o the point i $ f the second arrangement is, however, shown to be identical with the first. (2) If three of the atoms of a chemical kind are equivalent but different f r o b the fourth: (c) (Figure 37) Na: 000; 410; +Of; 041. CI: $44; OOf; oqo; ;on. (d) (Figure 39) Na: 000; 004; 010; f00. Cl: $4:; $40; $04; 01%
I n the positions of its atoms arrangement (c) is indistinguishable from (a) and (b). (3) If the chemically like atoms fall into two groups of two geometrically equivalent atoms each: Since there is hut one group of two equivalent positions possessing cubic symmetry, no arrangement of this sort is possible. 4 The succeeding analysis is practically an abstract of that given by Wyckoff. "The Structure of Crystals." 6 "An Analytical Expression of the Results of the Theory of Space Groups." Publ. 318, Carnegie Institution of Washington.
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J ~ B1930 ,
(4) If the four chemically like atoms are nonequimlent to one another: Since there are only two groups of singly equivalent positions in space groups with cubic symmetry, no such arrangement is possible.
Structure a Na: 000. OMM: MMO CI: M M ~ ~; o o o ; m ; on%.
Structure c Na: 000: MMO; MOM: OK%. CI: WMX; 00%; OMO: MOO.
F I G U38 ~ Structure b Na:
MMM: %%Xi %%Xi %%%:
%%X. c1: %%%; %%%; %%%.
Hence we arrive at the conclusion that we have only to choose between the structures illustrated in Figures 37 and 39. Again we return to our X-ray diffraction data. Mathematical formulas, with which we shall not bother here, have been derived for the calculation of the "reflections" which may be expected from these types of structure. Applying such formulas we find that if sodium and chlorine are assumed to have appreciably different refraction powers (an assumption well justified by experiment) we shall get for: Structure a.Relatively faint odd-order reflections for all sets of planes for which h, k, and 1 are all odd; No other odd-order reflections; Strong even-order reflections for all sets of planes. Structured.~elativelyrfaintodd-order reflections when h, k, and 1 are all odd. Relatively faint odd-order reflections when h, k, and I are two even and one odd. Strong even-order reflections for all sets of planes. Since the experimental data are actually those which would be predicted for structure a (Figure 37) we choose that structure.
Intensities of Reflections I
I
FIGURE39
Structure d Na: 000; 00%; 0x0; naa. C1:
MMM;
OW%.
MXO;
MOM;
We should not conclude without a brief note concerning the significance and determination of relative intensities of reflections. The following generalizations are roughly but not exactly true. (1) The diffracting power of a crystal element (ion, atom, or molecule) is proportional to the number of extra-nuclear electrons.
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(2) The intensity falls off more or less regularly toward higher orders. The apparent exceptions to these rules are explainable upon the basis of a more detailed study of the relation of structure to diiraction effects. For simple problems like the one we have just considered it is sufficient to estimate intensities visually as strong, medium, or weak. A more exact estimation is sometimes attempted on a basis of zero to ten, ten being the highest intensity. Where differentiations of this order are necessary it is usually desirable to employ the ionization spectrometer. (Photometric methods are also possible and are often employed.) The ionization spectrometer takes advantage of the fact that X-rays ionize gases and that the degree of ionization is proportional to the intensity of the radiation. The degree of ionization is readily measurable by means of the conductivity. In the investigation of intensities we would, then, bring an ionization chamber successively into positions necessary to trap and measure the "reflections" from a number of important, previously determined sets of planes.
Summary To summarize, the complete analysis of a crystal involves: 1. Assignment to a system and, if possible, a class of symmetry by means of optical examination or Laue patterns or both. 2. Calculation of the mass associated with a possible unit cell (which leads to the number of molecules per unit cell). 3. Selection of the proper unit cell. , 4. Application of the theory of space group to ascertain to which of the possible 230 groups the crystal actually belongs. 5 . Choice of correct structure after consideration of the "reflections" present and their relative intensities. We have not dealt with any of these operations exhaustively nor in any detail, and we have considered only an extremely simple and easy example. More complicated cases and more detailed study of methods are for the advanced or special student. We should note that not all the data necessary for a complete analysis are always obtainable. (Often, for instance, it is impossible to grow crystals large enough for optical examination or for Laue or Bragg treatment.) In the case of cubic crystals and occasionally of simple hexagonal crystals it is possible to effect an analysis without complete data. In other cases incomplete data almost inevitably means that incomplete analysis, only, is possible.
Acknowledgment The writer desires to express gratitude to Dr. Emil Ott of The Johns Hopkins University for reviewing this manuscript and to Miss Marguerite Little of the JOURNAL staff for aid in preparing some of the illustrative diagrams employed.
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He also acknowledges his indebtedness to the following references: W. P. Davey, "Study of Crystal Structure and Its Applications," Gen. Elec. Reu., 27,742-8,795-802 (1924); 28,129-37,258-65,342-8, 586-99, 721-30 (1925); 29, 118-28, 274-87, 580-9 (1926). Sir William H. Bragg, "An Introduction to Crystal Analysis," D. Van Nostrand Co., Inc., New York, American edition, 1929. Sir William H. Bragg and W. L. Bragg, "X-Rays and Crystal Structure," G. Bell & Sons, Ltd., London, fourth edition, 1924. R. W. G. Wyckoff, "The Structure of Crystals," The Chemical Catalog Co., Inc., New York, 1924. George L. Clark, "Applied X-Rays," McGraw-Hill Book Co., Inc., New York, first edition, 1927. F. K. Richtmyer, "Introduction to Modern Physics," McGraw-Hill Book Co., Inc., New York, first edition, 1928, Chapter XII.
Britain Honors 1929 Nobel Prize Winners. So little notice was taken by the public press of Great Britain of the award of Nobel prizes, except the prizes for Literature and peace, that the banquet given an February 3rd by the Biochemical Society in honor of the Nobel Laureates in medicine and chemistry last year is of particular interest. I n Sweden the prizes are rightly regarded as of outstanding and world-wide importance, and when they are presented on the anniversary of Alfred Nobel's death, the head of the State is present, and the ceremony is conducted with impressive solemnity and dignity. The Swedish people show that they are proud of the great foundation with which Nobel entrusted them, and the nation delights to associate itself with the progrrssive thought and work of the recipient of the prizes. Btitish fhreates find a very different atmasphere prevailing when they return to their own shores; and it is not surprising to know that this public indiierence is not understood in Sweden. The three Laureates in medicine and chemistry who were present a t the dinner were Sir Frederick Gowlaud Hopkins, Prof. Hans von Euler, and Prof. Arthur Harden. Prof. C. Eijkman was unfortunately unable to attend, owing to ill health. Sir Charles Martin presided a t the banquet, which was largely attended, among the company being the Swedish Minister and Sir Ernest Rutherford, president of the Royal Society. I n proposing the toast of "The Nobel Laureates," Sir Charles Martin gave a very interesting survey of the growth of knowledge of accessory food factors from the time when Eijkman became director of the pathological laboratory a t Weltevreden to the present period of marvelous activity. Beriberi was then regarded as an infection, and i t was Eijkman's experiments which showed that the disease was similar to a disease in fowls and due to a dietetic deficiency. Turning to Sir Frederick Hopkins, the chairman hailed him as a prince of biochemists who has enriched physiology for forty years by his discoveries. From his experiments with synthetic diets he was led to conclude that for nutrition certain minute quantities of hitherto unknown substances were essential; and so began our knowledge of vitamins. Prof. Harden's researches began with the fermentations of bacteria by accurate quantitative methods, and developed into his brilliant work on alcoholic fermentatioi. His discoveries have brought about completely new conceptions of the chemistry of these changes. Prof. von Euler's explorations in the same field have been most extensive, and there is scarcely any aspect of the subject that he has not illuminated by his researches.-Chen. & Ind.
