X Laue XXX
Laue X-ray Spectroscopy
THK93; MRM97
Purpose: To illustrate the principles involved in the application of X-rays to the determination of single crystal type and orientation
References
1. Serway, Moses and Moyer: Modern Physics, pp 58-61 (X-ray spectrum, Bragg scattering)
2. Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles: R. Eisberg and R. Resnick, Wiley, 2nd ed., 1985 pp 40-43 (bremsstrahlung); 337-342 (characteristic X-rays); Appendix Q (Crystallography)
3. Crystal Orientation Manual: E.A. Wood, Columbia University Press, 1963 (QD905.W58)
4. The Structure of Crystals: A.M. Glaser, IOP Publishing, 1987 (QD921.G52)
5. Introduction to Crystallography: D.E. Sands, Benjamin, 1969 (QD905.S25)
6. X-ray Energy Spectrometry, R.Woldseth, Kevex Corp., Burlingame CA, 1973
7. Experiments in Nuclear Science: AN34, 3rd ed. revised, EG&G Ortec Inc., 1984. Experiment 8, pp 53-58; Experiment 11, pp 71-74; Experiment 12, pp 75-80; Experiment 23.1, p133; Appendix, p 167.
8. Solid State Physics: N. W. Ashcroft and N.D. Mermin, Saunders 1976, Holt, Rhinehart and Winston 1976; Chapter 6, Determination of Crystal Structure by X-ray Diffraction
9. Introduction to Modern Physics: Richtmyer and Kennard: , Chapter X
10. Structural Inorganic Chemistry: A.F. Wells, Oxford, 1975
11. Cullity: Elements of X-ray Diffraction
12. Neutron and Synchrotron Rdiation for Condensed Matter Studies, v1 ; J. Baruchel, J.L. Hodeau, M.S. Lehmann, J.R. Regnard and C. Schlenker, eds.,Springer-Verlag (1993)
Equipment: Polaroid X-ray camera XR-7 (Land Diffraction Cassette 57-1), Polaroid X ray film, Type 57, 4"x5" land Film 3000 ASA/36DIN, Phillips X ray generator with tungsten (Cu) water cooled target , MgO, Si, MgCl single crystals
I. Theory
1. Bragg scattering
In order to observe the constructive interference spots from the parallel planes of a single crystal, the Bragg condition must be satisfied:
nili = 2disinqi .
This expression represents two types of coherent interference of scattered waves from the atoms of a single crystal:
a. Scattering from the atoms of a single plane of atoms (angle of incidence = angle of reflection, the "mirror" condition), and
b. Coherent addition, at angle q between the X-ray beam and the scattering plane, of the net waves of parallel single planes with spacing d.
The first condition requires that the coherent scattering direction from a particular plane must lie in a plane determined by the normal and the X-ray beam direction, which is the mirror condition. The second limits the photon wavelengths that can produce coherent scattering from stacked, parallel planes.
Because there are many sets of parallel planes in a crystal, with many orientations relative to the beam direction, the result is a set of Laue spots. The pattern is characteristic of the crystal symmetries and spacings di, and of the direction of the X-ray beam relative to the principal crystal axes. If the crystal is mounted such that the X-ray beam is incident in a principal direction, the spot pattern will immediately manifest any rotational symmetry of the crystal about that axis. If the mounting is in a random direction, the symmetry will determine the pattern, which may be difficult to decipher. In many cases the crystal structure is known, and a face has been cut or cleaved and polished roughly perpendicular to a desired crystallographic axis. In this case, the pattern will show the approximate symmetry, and will indicate the degree of error in the facial orientation, relative to the desired symmetry axis.
The di are intrinsic properties of the crystal, and the qi are fixed (perhaps unknown), once the crystal is mounted relative to the X-ray beam direction. The various sinqi (and corresponding qi) are thus determined, and so are the li required for the constructive interference. In order to see many spots in a single exposure, it is therefore necessary to have a range of X-ray wavelengths ("white light").
For any single crystal the qi can be measured, giving the orientation of the planes relative to the beam direction
2. Powder diffraction
In the single crystal case, the many qi’s are fixed and a continuous range of l's is needed. With a polycrystal or powder sample all qi’s are present and a narrow range of l's will produce a pattern of circular rings centered on the beam direction, with angle q. The plane separation d can then be calculated from measurement of q, using the Bragg equation with a known X-ray wavelength l.
3. Laue ambiguity
For an unknown crystal or for an unknown beam direction relative to the axes of a crystal, there is no scale factor for length with "white light" , as there is for the powder method. The Laue determination of the qi’s gives only the ratios . Rotating the crystal a little to change the beam direction while tracking the planes does not help determine the di, but gives only the uninteresting ratio
= .
A preliminary powder (fixed l) Bragg spectrum can thus be a very helpful preliminary adjunct to Laue spot interpretation.
The ratio of various elements (or molecules) in the crystal can be determined independently. Then the crystal density gives the unit cell volume and unit cell dimensions, or ratios of dimensions, can be calculated. These can be checked against the Laue patterns
4. Crystal lattice structure
a. Coordinate systems
A crystal is characterized by spatial repetition of a "unit cell". The repetitive displacement may differ in different directions, and the directions of repetition may not be mutually normal (right angled). For many descriptive purposes it is convenient to use crystallographic coordinates (cc), with axes along the repeat directions of repeat units a,b,c making relative angles a (b and c), b (c and a) and g (a and b).
For other purposes, spatial (Cartesian) coordinates (sc) are useful.
b. Specification of planes and directions. Miller indices.
With the cc origin at an atom of a unit cell, a plane will intercept the cc axes at a distance given (in characteristic spacing units) by an integer (or some rational fraction of an integer, since (by hypothesis) there will eventually be another atom along a parallel direction to any cc axis). For example, in a rectangular parallelepiped lattice with unit cell a,b,c (orthorhombic) there is a plane (with a set of parallel planes) which intercepts the cc axes at 2,1,1 (i.e., 2a, 1b and 1c). There is another plane parallel to the a axis which intercepts at 1,1,∞ .
These planes are specified, not by these intercepts, but by the reciprocals, rationalized to the smallest possible integers[1] (hkl), e.g.
intercept 2,1,1 --> reciprocal , 1,1 --> M.I. (122) and
intercept 1,1,∞ --> reciprocal 1,1,0 --> M.I. (110 ),
where the parentheses indicate planes. The sets of planes with Miller indices (220), (330) etc. are thus subsets of the set (110).
a,b,c are thus cc basis vectors, whereas the M.I.s (hkl) specify planar orientation in crystal coordinates.
The direction [122] (note square brackets) is that from the origin to the point 122 in cc's, i.e. toward the point 1a, 2b, 3c in sc's. The direction [h1k1l1] is not necessarily normal to the plane (h1k1l1). It is so for equal, orthogonal axes (cubic crystal), (hk0) planes of tetragonal crystals, or planes parallel to two crystal axes and perpendicular to the other.
c. Systems and lattice types
Certain standard types of crystalline axes are found, classified by symmetries, with properties as follows (d = spacing of adjacent, parallel planes):
Crystal System / Example / Symmetryelements / cc axes and angles
Cubic / Al (fcc)
W (bcc)
Cu (fcc)
NaCl (fcc)
C (diamond) ,
Si, Ge (fcc) / 4 threefold axes / a = b = c
a = b = g = 90o
Tetragonal / One fourfold axis / a = b ≠ c
a = b = g = 90o / + ( )2
Orthorhombic / 3 twofold axes, or three mirror planes, or
two mirror + 1 twofold / a ≠ b ≠ c
a = b = g = 90o / ( )2 + ( )2 + ( )2
Hexagonal / Cd / 1 sixfold axis / a = a' = b ≠ c
a = b = 900, g = 120o / + ( )2
Trigonal / 1 threefold axis / a = b = c
a = b = g ≠ 900
Monoclinic / 1 twofold axis, or 1 mirror plane, or
1 twofold normal to a mirror / a ≠ b ≠ c
a = g = 900, b > 90o / (see Wood)
Triclinic / Inversion center / a ≠ b ≠ c
a ≠ b ≠ g ≠ 90o
For cubic crystals (hkl) planes are perpendicular to vector normals with components h,k,l along the three orthogonal crystal axes. The (100), (010) and (001) planes are, perpendicular to the a, b and c unit vectors, respectively. The (110), (101), 011), (11'0), (101') and (011') (two equal indices in an eqi-axis system) cut thru diagonals on opposite cube faces. (1' indicates a planar intercept at -1 on the relevant crystallographic axis. The (111), (111'), (11'1) and (1'11) planes (three equal indices) are perpendicular to cube body diagonals.
(100), (010) and (001) planes are separated by distance d = a (a = b = c; a = b = g = 90o for a primitive cubic system). (110) type planes are separated by the face diagonal and (111) type planes by . For face centered and body centered cubic lattices there are intermediate planes, and the separations are less.
Among these seven systems are 14 Bravais lattice types (see Eisberg, Appendix Q; Sands, pp.-63; Glazer, pp20-22).
d. Zones
A zone consists of the set of all scattering planes parallel to a common direction, known as the zone axis. Equivalently, the normals to the planes of a zone set all lie in (parallel to) a common plane.
If the X-ray beam is incident perpendicular to a zone axis, all Laue interference maxima for that zone will lie in the same plane, and the corresponding Laue spots will lie on a radial line. If the beam is not incident normal to the zone axis, the set of zone spots will form a curved line.
4. Symmetry
a. Point groups
These include those 32 classes (groups) of operation which leave at least one point undisturbed. (See Sands, Chapter 3). A group includes an identity operation, associative combination of members, the property that the product of any two members (successive operations) is a member, the existence in the group of an inverse to each member. The defining operation of the group need not be commutative (order immaterial.) They include:
a. Reflections
Identify nine planes of reflection symmetry in a cube.
b. Rotations
Identify three four-fold, four three-fold and six two-fold axes of a cube which restore the shape to its original space position. (Four-fold = four times, including the original, etc.)
c. Combined rotation and reflection
The net effect of these successive operations is to restore the original appearance. The separate operations need not be symmetry operations.
II. X radiation
1. Continuous X-radiation
The "white light" X-ray continuum can be conveniently provided by the "bremstrahlung" radiation resulting from impact of an intense electron beam on a cooled target. This radiation represents all wavelengths energetically possible (maximum given by hn = = E0 = eV), where V is the accelerating potential. In a thick target most electrons will scatter and emit several times; thus the average scattering energy is considerably less than eV. To obtain appreciable X-ray energy at energy E it is therefore necessary that eV be considerably greater than E.
The X-ray bremstrahlung energy distribution can be expressed in terms of the electron energy E0, target atomic number Z and electron beam intensity i by
I(E)dE µ iZ(E0-E)E dE
with total intensity
Itot µ iZV2 (E0 = eV).
Another source of continuous X-radiation is that from intense electron beams in synchrotrons. This is again a form of radiation due to acceleration of a charged particle. The energy for peak intensity increases with increasing beam energy. Beam "undulators" and "wigglers" are sometimes placed in the beam lines to produce greater acceleration, and thus higher electron energies.
2. Discrete X-radiation
Superposed on the bremstrahlung continuum resulting from electron bombardment will be sharp, intense peaks of discrete energies corresponding to the difference in the quantized bound energies of atom (Z+1), where Z is the atomic number of the target. The lowest energy target electron shells are generally filled. A K shell electron in tungsten cannot be "promoted" to the filled L (n = 2) shell; practically speaking, in order to see characteristic K radiation (final state n = 1), the exciting body (electron, proton, X-radiation) must ionize the atom. There is thus an ionization threshold greater than the characteristic X-ray energy for appearance of K radiation (Ka = L --> K, Kb = M --> K, Kg = N --> K orbit), and all appear simultaneously. This is because there is only one quantized electron state for ionization, 1s1/2 . In contrast, there are three possible n = 2 quantized states of different binding energy (2s1/2, 2p3/2 and 2p1/2), so three different energy thresholds will occur for appearance of L X-rays. Similarly, for ionization of an n = 3 bound electron there will be five thresholds, corresponding to the different ionization energies of the 3s1/2, 3p3/2, 3p1/2, 3d5/2 and 3d3/2 bound electrons where p3/2 indicates, as usual, l = 1, spin s = 1/2 and l and s are parallel.