X Laue 1
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, 2 nd 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, 3 rd
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

X Laue 2
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: n i
 i
= 2d i sin
 i .
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

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 d i
, 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 d i
are intrinsic properties of the crystal, and the
 i
are fixed (perhaps unknown), once the crystal is mounted relative to the X-ray beam direction. The various sin
 i
(and corresponding
 i
) are thus determined, and so are the
 i 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
 i
can be measured, giving the orientation of the planes relative to the beam direction
2. Powder diffraction

X Laue 3
In the single crystal case, the many
 i
’s are fixed and a continuous range of 
's is needed. With a polycrystal or powder sample all
 i
’s are present and a narrow range of

's will produce a pattern of circular rings centered on the beam direction, with angle

. The plane separation d can then be calculated from measurement of

, using the Bragg equation with a known X-ray wavelength

.
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
 i
’s gives only the ratios
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. Rotating the crystal a little to change the beam direction while tracking the planes does not help determine the d i
, but gives only the uninteresting ratio
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=
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.
A preliminary powder (fixed

) 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

( b and c ),

( c and a ) and

( 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

X Laue 4 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
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, 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 [h
1 k
1 l
1
] is not necessarily normal to the plane (h
1 k
1 l
1
). 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):


4 threefold axes a = b = c


=

=

= 90 o
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Cubic Al (fcc)
W (bcc)
Cu (fcc)
NaCl (fcc)
C
(diamond)
,
Si, Ge (fcc)
1 The assertion that it is possible to find such integers implies that the intercepts from which the M.I.s are derived are rational numbers (the law of rational indices, Hauy (1784) - see Sands, p66).
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X Laue 5
Tetragonal
Orthorhombic
Hexagonal
Trigonal
Monoclinic
Triclinic
Cd
One fourfold axis a = b ≠ c

=

=

= 90 o
3 twofold axes, or three mirror planes, or a ≠ b ≠ c 

=

=

= 90 o two mirror + 1 twofold
1 sixfold axis a = a' = b ≠ c 

=

= 90
0
,

= 120 o
1 threefold axis a = b = c


=

=

≠ 90 0
1 twofold axis, or 1 mirror plane, or a ≠ b ≠ c 

=

= 90
0
,

> 90 o
1 twofold normal to a mirror
Inversion center a ≠ b ≠ c


≠

≠

≠ 90 o
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+
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(see Wood)
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;

=

=

= 90 o for a primitive cubic system). (110) type planes are separated by the face diagonal
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and (111) type planes by
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. 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).
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2
2 +
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2
Ошибка!
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2
2

X Laue 6 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.
Zone axis
*
Laue spots
*
*
*
Zone planes
X-ray beam perpendicular to zone axis
*
Film
*
*
*
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.

Zone axis
X Laue 7
Film
*
* * *
*
X-ray beam
Zone axis
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
X Laue 8
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 h

= Ошибка!
= E
0
= 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 E
0
, target atomic number Z and electron beam intensity i by
I(E)dE

iZ(E
0
-E)E dE with total intensity
I tot

iZV
2
(E
0
= 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 (K  = L --> K, K  = M --> K, K  = N --> K orbit), and all appear simultaneously. This is because there is only one quantized electron state for ionization, 1s
1/2
. In contrast, there are three possible n = 2 quantized states of different binding energy (2s
1/2
, 2p
3/2
and 2p
1/2
), 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 3s
1/2
, 3p
3/2
, 3p
1/2
, 3d
5/2
and 3d
3/2
bound electrons where p
3/2
indicates, as usual, l = 1, spin s = 1/2 and l and s are parallel.

X Laue 9
Finally, according to Mosely's Law

 = 0.248x10
16 (Z-1) 2 where the (Z-1) represents the shielding of the ion nucleus by one K electron, as seen by the L electron, instead of the usual shielding of 2e seen in the neutral atom.
For a tube operating at 30 keV with a tungsten target, characteristic tungsten K Xrays will thus not appear, but those of energy less than 30 keV will.)
The variation of characteristic K X-ray intensity with operating conditions may be expressed as
I
K

[(
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- 1] 1.67 .
3. X-ray fluorescence and filtering
A sharp increase in absorption of characteristic radiation occurs for a filter with atomic number (Z-2), where Z represents the emitter. For several metal electron targets, use of a (Z-1) filter discriminates strongly against K  radiation, while the
K  radiation passes freely. For example, with a Cu target the K 
2
and K 
1 characteristic energies are 8.976 and 8.904 keV respectively. Since both are slightly greater than the K X-ray absorption edge for nickel, Kab = 8.331 keV, they will be strongly absorbed (the absorption cross section falls off above threshold). The Cu K 
1
and K 
2
characteristic energies, on the other hand, are
8.047 and 8.027 keV, which cannot photo ionize Ni by ejection of a Ni K electron. (See AN34, Appendix, p167) They can, however, eject the more weakly bound Ni L electrons. (See Wood, p13, for filter thicknesses needed to reduce the K  /K  ratio to 1/600.)
Since production of characteristic X-rays from atoms requires prior ionization, energy conservation makes production of characteristic K radiation form tungsten
(W, Z = 74) impossible with electron acceleration potential below ~ 70 keV. (For tungsten the critical absorption energy (K binding energy) for appearance of the characteristic K X-rays is 69.508 keV. (See AN34, p168.)
The critical absorption energies for appearance of L tungsten X-rays are L
Iab
=
12.090, L
IIab
= 11.535, and L
IIIab
= 10.198 kev; 30 kV operation will produce significant quantities of L X-rays, and also of M. The five characteristic tungsten
L X-ray energies are
L 
1
= 11.283 L 
2
= 9.959 L 
1
= 9.670 L 
1
= 8.396 L 
2
= 8.333 keV.
A Ni filter (K ab
= 8.331) would thus filter all of the tungsten L lines. (Even small angle prior Compton scattering of the 8.333 kev line could produce a secondary
Compton photon below the 8.331 keV filter K threshold.) A Cu filter (K ab
=

X Laue 10
8.980 keV) would pass both of the tungsten L  lines, while discriminating against the threshold-exceeding L  and L  . However, use of a tungsten electron bombardment target with nickel filter in the Phillips Laue X-ray generator does not give the characteristic pattern of circles from a polycrystalline target.
Determination of values of d with known wavelength

from powder or polycrystalline sample is better accomplished with a copper target and nickel filter, using either the Phillips or the other (Japanese) machine.
III. Procedure
1. Camera and sample mount
Inspect the extra (copper) X-ray tube. Note the filament, black wax vacuum seal, four port X-ray filter windows. The target block is not visible. The filters pass characteristic X-rays and attenuate the remaining bremsstrahlung (continuous) Xradiation. The assembly is water cooled (beam power at 30 kV and 30 ma is 900 watts, in vacuum).
Inspect one of the unused beam filter wheels. Set the filter to be used to zero
("white light"). Orange bands will be seen to left and right of the filter. Inspect the camera. (Do not remove.) Note the orange expose tab, and the film clamp lever on the side facing the X-ray tube. (Do not touch the collimator if it is already aligned!)
If necessary, insert the cylindrical sample holder in the goniometer, flat face on the goniometer axis. (Loosen the clamping screws and spread the clamp with a screw driver as necessary.) Mount the sample on the cylinder face with doublesided Scotch tape. Slide the goniometer onto the dovetail with the sample facing the X-ray beam and clamp gently about 3 to 5 centimeters from the film position
(for silicon crystal). Record the distance D from sample surface to camera film position.
2. Film loading and exposure
Note the "don't touch" warning markings on the film packet. The slide labeled
"open" and "close" on either end latches the hinged roller shield. Open this and inspect the rollers (two metal, one rubber) which squeeze and distribute the developer onto the film when it is removed after exposure. Note the effect of the film clamp lever
The metal rollers sometimes become coated with dried developer, which creates bumps on the roller surface. These can cause uneven distribution of developer, resulting in evenly spaced white marks on the developed film, which confuse and partially obliterate the Laue pattern. The rollers can be cleaned with alcohol, using kimwipes. First place the slide in the "open" position and raise the cover, then press the spring clips on the other side and release the cover latch from the ends of the spindle. Both rollers can then be exposed for cleaning.

X Laue 11
With the film clamp lever in the load position and the orange expose tab down, insert a piece of Type 57 Polaroid film in the camera with the emulsion side
(labeled "This side faces lens") facing the camera phosphor, i.e., facing the X-ray beam. Push down until the metal frame at the bottom clicks into place. Pull the film envelope up carefully until the oid of the word Polaroid is exposed. The film remains below; it should be still attached to the film frame. (Don't pull up too far
- the film may detach.) Pull up the orange expose tab to press the film against the phosphor.
Turn on the circuit breakers for the coolant chiller and pump and for the electron beam power supply (high voltage and filament). Let the coolant cool down to about 12. Close the lead glass shielding windows until the magnetic strips are aligned with the interlock switches. (A radiation safety check with beam on found no increase over natural background external radiation under these conditions.)
Turn the two power supply knobs down to zero (fully counterclockwise). Turn on the AC power to the X-ray tube filament. Wait a minute for warm-up, then turn the high voltage knob slightly clockwise. A slight click can be heard and an amber pilot light will come on. Turn the acceleration high voltage up to about 30 kilovolts by discrete click positions (first about 23 kV). Turn the filament up
(clockwise) to about 30 milliamperes. Expose the film to the camera phosphor for around 30 minutes (silicon), or 20 minutes (tungsten). The phosphor will fluoresce from the X-rays back-scattered from the various sample planes according to the Bragg equation above.
Turn down the filament supply to zero. Turn down the high voltage to zero. Turn off the AC (red button).
Push down the orange expose tab. Push down the film envelope to re-engage the film. (If it balks, see if it is cocked. If so, orient vertically and try again.) Flip the load lever 180 o
to the develop position and pull the film out. Allow 15 seconds for developing (Type 57), open film and let dry if damp. Apply protective coat if the pattern is good.
(If the film frame will not push down, it may be detached from the film; then the film will have to be removed and abandoned. Pull out the film frame. Dismount the camera, with attached collimator. Do not separate the collimator from the camera. Pry off the two chrome plated clips with a screwdriver and separate the camera back. Remove the film. Note the whitish phosphor. Replace the back and remount the camera, lifting the hinged filter cover for collimator placement.
Allow 30 minutes for the phosphor to recover before taking another picture.
When finished for the day, let the target cool down for five minutes or so, then turn off the chiller.
IV. Analysis

X Laue 12
1. Symmetry
Inspect the spot pattern from tungsten, magnesium oxide and silicon targets for reflection and rotation symmetry. Record all such observed, noting the orientation of reflection plane symmetry and the order (number of pattern repetitions in 360 o ) of rotational symmetry. Confirm that the order of rotational symmetry agrees with that expected for the direction of the X-ray beam. Try to identify intense spots with major crystal zones.
2. Face misalignment check
Locate four dark fiducial marks left on the film by the phosphor (2 vertical and two horizontal, about 2 cm from the bright center spot). Carefully draw a vertical and horizontal line thru these; the intersection is the nominal beam center.
Carefully draw symmetry lines for the bright spot (Laue) pattern. The intersection is the pattern center.
Any vector offset
 s between beam and pattern centers indicates the direction of misalignment between the crystal characteristic lattice orientation and the polished face, with magnitude given by tan

=
Ошибка!
.
The crystal face can be repolished to remove this discrepancy, provided the mounting is not disturbed.
V. Laue appendix
1. Vector relations
The scaler (dot) product and vector (cross) product of two vectors each have both a geometric and an analytic form, providing ease of conversion between the language of rectangular coordinates (components) and that of polar (lengths and angles):
Dot product ( scaler) A .
B = |A||B|cos

AB
= A x
B x
+ A y
B y
+ A z
B z, where |A| = A x
2 + A y
2 + A z
2 etc., which provides an easy way to determine the angle between two vectors whose rectangular components are given.
Cross product (vector) A x B = |A||B|sin

AB
(perpendicular to A & B )
= i (A y
B z
-A z
B y
) + j (B z
A x
-B x
A z
) + k (A x
B y
-A y
B x
) where i , j and k are orthogonal unit vectors of a rectangular coordinate system.

X Laue 13
The magnitude of the cross product is the area of the parallelogram formed by the two vectors. The direction is, by convention, the advance of a right hand screw when the first vector is rotated into the second.
Further, the quantity A .
B x C [ = A .
( B x C ) ] gives the volume of the parallelepiped formed by the three vectors A , B and C .
The x (unit vector i ) and y (unit vector j ) coordinate plane of a rectangular space coordinate (sc) system can be chosen as the plane of the cc a and b vectors. a and b can then be written in terms of the i and j unit vectors in terms of

, the angle between a and b . Then the third sc base vector c can be expressed in terms of a and b (using angles , hence i and j , components, ( a , b and c are not necessarily orthogonal), and a third component in the k direction, completing the expression of the cc base vectors a , b , c in terms of the sc orthonormal set i , j and k . The analytic forms of vector analysis can then be utilized to calculate such quantities as the direction normal to a plane.