Citation
Friauf, James B. (1926) The crystal structure of MgZn2. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd11112004102634
Abstract
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Crystals of the intermetallic compound, MgZn2, were prepared and the crystal structure was determined from xray data furnished by Laue and rotation photographs. The crystal was found to have hexagonal axes with a. = 5.15A and c. = 8.48A. The unit cell contains four molecules. The effect of absorption in the crystal in determining the wavelength giving a maximum intensity of reflection in Laue photographs was used to confirm the dimensions of the unit cell. The atoms have the positions: [...] where u = .830 and v = 0.62. The least distance between two magnesium atoms is 3.16A, between two zinc atoms, 2.52A, and between a magnesium and a zinc atom, 3.02A.
The constitution diagram for the binary system, magnesiumzinc, has a pronounced maximum corresponding to the formation of an intermetallic compound, MgZn2, which forms eutectics with both constituents. Since both magnesium and zinc crystallize in the hexagonal close packed arrangement, a determination of the crystal structure of their compound was thought to be of interest.
Crystals of the compound were formed by melting together the calculated amounts of magnesium and zinc under a molten mixture of sodium and potassium chlorides to prevent oxidation. The melt was then allowed to cool slowly in the electric furnace, about four hours being taken to cool from ten degrees above to ten degrees below the melting point of the compound, 595° C. In this way a mass of crystals was obtained from which individual crystals were separated for the production of Laue and spectral photographs.
Two rotation photographs taken with the xrays from molybdenum water cooled tube on an xray spectrograph of the kind described by Wyckoff, furnished data for the determination of the size and shape of the unit cell. As no information on the crystal class or axial ratio of crystals of this compound was found in the literature, a consideration of the secondary spectra as well as of the principal spectrum was necessary in order to obtain the quadratic form which gives the spacings of the planes. These spacings can be computed from the positions of the reflections on the plate and furnish information of the same nature as that available from a powder photograph, the difference being that in a rotation photograph taken with the crystal turning about a definite axis, the reflections occur in spots instead of in complete circles as they would in a powder photograph taken on a plate. Furthermore, due to the limited rotation of the crystal (30° in this case) certain planes will never reach a position to reflect, while in a powder photograph reflections are to be expected from all planes having a suitable spacing.
Table I gives the data from a rotation photograph. The observed spacings are the means of those calculated from the reflections produced by the K[...] doublet and the K[...] line fo molybdenum for all the planes of the same form showing on the plate. These spacings were compared with the charts given by Hull and Davey, and were found to agree with the spacings for a hexagonal unit cell having a. = 5.15A and c. = 8.48A. since the crystal was rotated about one of the a. axes for this photograph, the assignment of indices obtained from the chart was checked by comparing the computed and observed values for the x and y coordinates of the spots. The fact that certain reflections could not occur due to the limited rotation of the crystal could also be used in some cases to distinguish between planes having nearly the same spacings. On another photograph taken with the crystal rotating about the c. axis, only the principal spectrum was measured. This gave a. = 5.15A which is in agreement with the value just given. The third column of the table gives the values for the spacings computed from the dimensions of the unit cell.
This unit cell agrees with the data from Laue photographs taken with the white radiaton from a tungsten target. When the wavelengths of the xrays producing the spots on symmetrical and unsymmetrical Laue photographs were calculated on the basis of this unit cell, no values were found less than the short wavelength limit, about .24A, of the xrays used. The curves showing the intensity of reflection from different planes of the same form reflecting at different wavelengths in unsymmetrical Laue photographs, start from the short wavelength limit, rise to a maximum between .36A and .40A, and then decrease for longer wavelengths. The presence of a maximum intensity so far below the wavelength of the silver absorption edge is due to absorption in the crystal. The photographic intensity, I, of the white radiation from a tungsten target operated at 50 kv can be approximately represented between the short wavelength limit, [...] and the wavelength of the silver absorption edge, .485A, by the equation [...] where B is a constant. This must be modified, however, if the crystal is strongly absorbing as is the case with MgZn2. For a first approximation it can be assumed that all the rays producing spots on a Laue photograph are absorbed for a distance equal to the thickness of the crystal. The absorption coefficient of the crystal can be computed from data given by Richtmeyer and Warburton for the atomic scattering and fluorescent absorption coefficients. Since the absorption due to scattering is small and nearly independent of the wavelength, it will have no other effect than to decrease the value of the constant, B, but the fluorescent absorption, which is proportional to the cube of the wavelength, will cause greater weakening of the longer wavelengths and the maximum intensity is accordingly shifted to the short wavelength side of the silver absorption edge. The density of the crystal, 5.16, its thickness, about .3mm, and the computed absorption coefficient give [...] where I’ is the photographic intensity of the white radiation after passing through the crystal, and B’ is the constant, B, multiplied by the factor which represents the common decrease in intensity of all wavelengths due to scattering. The curve given by this equation has a maximum at .36A and agrees in form with the curves showing the intensity of reflection as a function of the wavelength, thus furnishing additional evidence for the correctness of the unit cell chosen.
The density of MgZn2 was determined by weighing in a specific gravity bottle after breaking the sample into small pieces in order to avoid, as far as possible, the inclusion of blowholes. Two determinations gave 6.164 and 5.155. Using the value 5.16 for the density, the computed number of molecules in the unit cell was found to be 3.93, the deficiency from the integral number, 4, doubtless being due to the fact that the density determined by the use of a specific gravity bottle is likely to be less than the density determined by xray measurements unless porosity of the sample can be completely eliminated.
Smaller unit cells, containing 1, 2, or 3 molecules were found to be inconsistent with the data available.
A Laue photograph taken with the incident beam of xrays parallel to the principal axis of the crystal had a sixfold symmetry axis intersected by six planes of symmetry. The spacegroup giving the arrangement of atoms in the crystal must consequently be isomorphous with one of the pointgroups [...] or [...]. Reference to a tabulation of the results of the theory of spacegroups shows the possible ways of arranging four magnesium and eight zinc atoms in the unit cell. If it is assumed that the magnesium atoms are equivalent and that the zinc atoms are likewise equivalent, the possible arrangements are those which can be obtained from the spacegroups [...] and [...] since these are the only spacegroups considered having a group of eight equivalent positions. All of these arrangements give zero for the amplitude factor of the first order reflection from 04.1. The data given in Table II for an unsymmetrical Laue photograph show, however, that 04.1 gives a strong first order reflection. These arrangements are consequently inadmissible and the assumption of equivalence of chemically like atoms must be relinquished. With the freedom of choice thus allowed there are numerous ways of arranging the atoms. The zinc atoms may be in two groups of four equivalent positions, two groups of six and two equivalent positions, or in some other combination giving the required number of atoms. The number of possible combinations for the magnesium atoms is somewhat less. The choice of the correct atomic arrangement is simplified by the observation that many of the groups of six equivalent positions lie in a plane parallel to the base of the unit cell. If, however, six zinc atoms which constitute more than half the reflecting power of all the atoms contained in the unit cell, are arranged in such a plane, the absence of odd order reflections from 00.1 and the observation that the fourth order reflection from 00.1 is stronger than the second order cannot be satisfactorily explained. Groups of six equivalent positions having such an arrangement are consequently excluded from further consideration.
No ways of arranging four magnesium and eight zinc atoms in the unit cell can be obtained from the space groups [...] or [...] since none of these spacegroups contains the requisite number of equivalent positions. All arrangements that can be derived from the spacegroups [...] and [...] can be readily eliminated since for each of these spacegroups, each group of eight or less equivalent positions (and hence any combination of them) gives zero for the amplitude factor of the first order reflection from 04.1. Such arrangements are consequently inconsistent with the data. With the restriction that has been made as to the character of the groups of six equivalent positions to be considered, all the arrangements that can be derived from the spacegroups [...], [...], and [...] give the same amplitude factors for 34.3 and 16.3, and since the data show that the more complicated plane, 16.3, gives a stronger reflection, such arrangements are inadmissible. A number of three or four parameter structures which cannot be so readily eliminated may be obtained from the spacegroups [...] and [...], but none of these seems to offer the slightest possibility of accounting for the observed intensity relations.
The only structures remaining to be considered are those arising from the spacegroups [...], [...], [...], and [...]. Of the structures which can be obtained from the spacegroups [...] and [...], the only one not conflicting with the data is that in which the atoms have the following positions: [...]. This arrangement is obtained by placing the magnesium atoms in one group of four equivalent positions and the zinc atoms in two groups of six and two equivalent positions. A consideration of the type of structure involved shows that it is sufficient to consider only values of the parameters satisfying the conditions [...] and [...]. If it is assumed that there is a reasonable distance between the two magnesium atoms in the same vertical line, v will be restricted to the middle half of this range.
The amplitude factor, S, is computed from [...] where A and B have their usual significance of sine and cosine summations, and it is zero for first order reflections from planes of the forms hh.2p+1 irrespective of the values of u and v. No first order reflections from any such planes were found on any of the Laue photographs although planes of the forms 22.1, 33.1, 44.3, 55.3, and 44.5 were in position to give first order reflections at a favorable wavelength. Another characteristic feature of this structure is tht the magnesium atoms contribute nothing to the amplitude factors for first order reflections from planes of the forms 03h.2p+1. The intensities of such reflections are consequently useful in determining the positions of the zinc atoms. Of the observed planes of this kind, 03.1 was found to give a weak first order reflection while no reflections were found from 06.5 and 09.5 although planes of both these forms were in a position to reflect at favorable wavelengths and reflections were observed from more complicated planes. The value of the parameter, u, must consequently be such as to give only a small value to the amplitude factors for these planes. By plotting these amplitude factors as a function of the parameter it is readily seen that the only values of u which will reduce them all to zero are 0, 1/6, 1/3, and 1/2. While u cannot be equal to one of these values because of the weak reflection from 03.1, it can hardly be much different from them because of the rapidity with which the amplitude factors change. Consideration of the amplitude factors for other planes shows that the only values of u giving general agreement with the requirements of the Laue data are those in the neighborhood of u = 1/6.
With restriction on the value of u, some information concerning v can be obtained from a consideration of the intensity of relations for the planes 26.3, 26.5, and 26.7. The amplitude factors for the first order reflections from these planes are: [...]. The first part, due to the zinc atoms, is the same for all three planes, positive for values of u near 1/6, and changes only slightly for small changes in the value of u. As the data show that 26.5 is stronger than 26.3 it must have a larger amplitude factor since it is a more complicated plane. Also, since 26.7 is a more complicated plane than 26.3 and gives an equally strong reflection, it must have a greater amplitude factor than 26.3. These conditions are satisfied by giving v a small negative value. A consideration of the intensity relations for the planes 08.3, 08.5, and 08.7 leads to the same conclusion.
With the values of u and v restricted in this way it was found by trial that satisfactory agreement with the data was obtained for u = .170 and v = .062. The extent of the agreement is shown in Table II which gives the data from an unsymmetrical Laue photograph. The table shows the spacing of the plane producing the reflection in A. U., the intensity as estimated visually by comparison with a plate which had been given a series of graduated exposures, the product of the order of reflection by the wavelength producing the reflection, and the amplitude factor computed for the values of the parameters given on the assumption that the reflecting powers of the zinc and magnesium atoms are proportional to their atomic numbers. In comparing the intensities of two planes, if the plane with the smaller spacing gives the grater intensity under comparable conditions of wavelength, it must have a greater amplitude factor. As previously stated, the maximum intensity falls between .36A and .40A and the intensities in the table have been given in this region when possible.
This two parameter structure is the simplest which will give agreement with the data available. The only other possible structures are a three parameter structure derived from [...] and a five parameter structure derived from [...]. Neither of these can be eliminated since suitable values for the parameters reduces each to the two parameter structure which has been found to give agreement with the data. A consideration of these two more general structures indicates, however, that neither will give satisfactory agreement with the data except for values of the parameters which reduce them to forms closely approaching that of the two parameter structure, and it is consequently concluded that this structure of more general structure so similar as to be indistinguishable from it, represents the crystal structure of MgZn2 when u and v have the values given.
This structure can be described in an alternative way without the use of a negative parameter by setting u = .830 and v = .062. Figure 1 shows the arrangement of atoms in the unit cell. The least distance between two magnesium atoms is 3.16A, between two zinc atoms, 2.52A, and between a magnesium and a zinc atom, 3.02A. The values computed from the atomic radii determined from the crystal structures of magnesium and zinc are respectively 3.22A, 2.67A, an
Item Type:  Thesis (Dissertation (Ph.D.)) 

Degree Grantor:  California Institute of Technology 
Division:  Chemistry and Chemical Engineering 
Major Option:  Chemistry 
Thesis Availability:  Public (worldwide access) 
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Defense Date:  1 January 1926 
Record Number:  CaltechETD:etd11112004102634 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd11112004102634 
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Deposited On:  11 Nov 2004 
Last Modified:  26 Dec 2012 03:09 
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