ADVANCED UNDERGRADUATE LABORATORY
EXPERIMENT 1, LAUE
Crystal Orientation
by Laue Back-Reflection of X-Rays
Revisons:
October 2013 (to reflect use of medical X-ray film) P.Krieger
August 2006 by Jason Harlow,
with suggestions from Jerod Wagman
Original by:
Derek Paul
1
Introduction
Production of X-rays
When electrons of a few tens of keV are incident on the metal target of an x-ray tube, about 2
percent of their energy is converted into X-rays and the balance is converted into heat in the target. As is
shown in Figure 1, the radiation consists of a continuous spectrum which is radiation spread over a wide
band of wavelengths and also a superimposed line spectrum of high intensity single wavelength
components.
Figure 1. X-ray Spectra From Tungsten and Molybdenum Targets (from Semat).
2
The former is the analogue of white light and is frequently called white radiation. The latter corresponds
to monochromatic light, and because the wavelength of each component is characteristic of the metal
emitting the rays, it is called characteristic radiation. The continuous spectrum can be produced without
the characteristic if the tube is operated at a low voltage, but as soon as the voltage is increased beyond a
critical value, the characteristic spectral lines appear in addition to the white radiation.
For this experiment we require white radiation so we use an X-ray tube with a tungsten target and
accelerating voltages are never greater than 50 kV. The maximum energy of a photon produced by an
electron of charge e accelerated through a voltage V in volts is given by
hc
(1)
= eV
hv max =
 min
where h is Planck's constant, c is the speed of light and vmax is the highest frequency X-ray produced
(which, of course, has the shortest wavelength λmin). Substitution of values for h, e and c gives
1239.8
(2)
 min =
V
where λmin is given in nm. We use voltages between 10 and 50 kV so that λmin ≤ 0.1 nm which is a
wavelength short enough to determine crystal structures with unit cell dimensions typically a fraction of
a nanometre. The material outlined is given in more detail in Barrett pp. 51-56.
Diffraction of X-Rays by Crystals
Bragg's law gives the relation between the X-ray wavelength λ and the interplanar distance d:
2dsinθ = nλ
(3)
where n is the order of the interference, and, as can be seen from figure 2, the scattering angle for
constructive interference is 2θ.
Figure 2. Diagram for Bragg Reflection.
3
The interplanar distance d depends upon the lattice constants and the Miller indices h, k, ℓ of the
crystal plane (see Wood, p.8)*. For example, in a cubic crystal,
a
d= 2
(4)
( h + k 2 + l 2 )1/2
and the expressions for other crystal systems are given in Wood p. 13.
However, (3) and (4) are marginally relevant to the Laue method which depends only on the fact
that the X-rays are reflected: the angle of incidence = the angle of reflection. The orientation of the
parallel crystal planes then determines θ uniquely, and hence the scattering angle, 2θ.
Laue Back Reflection Method
In this method a beam of white X-rays is incident on a stationary single crystal. For each crystal
plane, in a particular orientation, d and θ are fixed; thus, for a given n, some λ will be available in the
white radiation to give a Laue spot in a direction making an angle 2θ with the incident X-ray beam.
Since d and θ are fixed, all orders of interference n, for a given crystal plane will be superimposed on the
same spot. The symmetry of the Laue pattern corresponds to the symmetry of the crystal and directions
of the crystal axes are determined by the symmetry axes of the Laue pattern. The information given here
is expanded in Wood pp. 28-31 and Barrett pp. 211-217.
From the above it will be clear that the Laue method is useful for identifying crystal type and
orientation. It is easy in crystallography, however, to make a mistake, and the following remarks are
intended to help prevent such mistakes. When a crystal, for example, having a cubic structure, is
orientated along an axis having a high degree of symmetry, it is possible to identify the symmetry with
ease, and to calculate the expected positions of the spots. Suppose, for example, you have a tungsten
crystal aligned with a (100) plane perpendicular to the incident X-ray beam. The Laue spots will lie on a
series of lines passing through the point where the X-ray beam crossed the photograph. These lines form
a symmetric pattern which is predictable if you draw the two-dimensional lattice representing the front
(100) face of the crystal. Such a drawing enables identification of the main zones giving rise to the lines
of spots, and their angles are easily computed. If important zones are missing from a photograph, or if
the angles or symmetry are wrong, then the hypothesis of (100) orientation must also be in error, or
maybe the structure isn't cubic.
A second check on the orientation, which further checks the structure, is the distances of the
spots from the central point at which the X-ray beam crossed the film. A few of these distances have
been calculated in tables 1 and 2 of the following subsection (Tables 1 and 2 have been left incomplete)
for the (100) orientation of a cubic crystal. Table 3 gives a more complete set of distances for a (110)
orientation and the [001] node.
*All
pages of Wood which are referenced in this manual are included in appendix II.
4
Some computations of expected positions of spots
1.
For backscattering from a (100) plane in a cubic structure, very simple formulas exist for the
[001] and [011] zone axes. These can be deduced from three-dimensional geometry with the use
of the tangent formula as follows.
For the [001] zone axis the angle between the (100) plane and the (h10) plane leads to the spot
distance y from the incident X-ray beam:
y = 2x/(h-1/h), h>1
where x is the distance from the crystal face to the photographic film, assumed planar. For the
[011] zone axis the angle between the plane (100) and the plane (h11) leads to the corresponding
distance:
  2x 2 /( h  2 / h ) h  2
Table 1
Values of y in cm for x = 3cm, [001] zone axis
Plane indices
hkl
910
810
710
610
510
410
310
210
y
0.675
0.762
0.875
1.208
1.25
1.6
2.25
4.00
Table 2
Values of ρ in cm for x = 3cm, [011] zone axis
hkl
ρ
911
0.967
811
1.095
711
1.264
611
1.497
511
1.845
411
2.424
311
3.636
211
8.485
The extension of the above formulae to the more general plane indices (hk0) and (hk1) are quite trivial.
For reflections from a (110) cubic crystal face we consider the rotations about the [001] direction.
The angle between the plane (hk0) and (110) is calculated for k>h. The angle between the normal to
plane (hk0) and the incident beam, α, is given by
5
α = arctan(1) - arctan (h/k), k>h
but ρ = xtan(2α), therefore
ρ = xcot(2arctan(h/k))
= x(k/h-h/k)/2
Table 3
ρ(hk0) for the (110) plane and the [001] zone axis
hkl columns: k-h=constant
890
780
670
560
450
790
340
570
230
580
350
470
590
120
490
370
250
380
130
ρ
0.354
0.402
0.464
0.55
0.675
0.762
0.875
1.029
1.25
1.4625
1.6
1.768
1.863
2.25
2.708
2.857
3.15
3.4375
4.00
Table 3 indicates the paucity of low-index reflections at suitable angles in this case. The table for planes
(hkl) with the h and k values interchanged is identical, but experimentally the spots are then observed on
the other half of the line.
It is instructive to calculate y-values and ρ-values for planes other than the restricted sets of k-values
given in Tables 1,2, and for other zone axes and orientations of a cubic crystal.
6
Experimental
Radiation Monitoring and Protection
Anyone operating the X-ray machines must be wearing a radiation monitor, available from the
technician in Room 250.
Before turning on the unit, be certain that the cap covering the X-ray post is down, or the camera
is in place with the lead shutter closed or the large lead shields are in place so that no one is
inadvertently exposed to X-rays. Note: there is an interlock switch which renders the X-ray power
supply inoperative unless at least one of the lead radiation shields is in position. The leaded glass in the
lead shield walls is to enable you to observe visually the X-ray beam alignment when necessary.
Operating Procedure for the Norelco X-ray Machine
Turning the unit on.
The X-ray unit with the green front has a tube with a tungsten target and is to be used for this
experiment.
Beginning with the unit turned completely off:
a.
Push the lever on the electrical service box on the wall to ON.
b.
Open fully the tap (counter-clockwise) on the wall. This connects the cooling water. No
water will flow yet.
c.
Push the overload reset switch upward to ON if it is down.
d.
Set the timer to some nominal figure (e.g. two hours) to cover the duration of the testing.
e.
Turn the key-operated line switch clockwise to ON.
f.
Push the line START button. The fluorescent panel light will come on and you will hear
the cooling water flowing.
g.
Turn the KILOVOLTS and MILLIAMPERES control knobs to their full counter-clockwise
position
h.
Push the X-RAY ON button. The rectifier pilot light will light, indicating that the filament
circuit of each high voltage rectifier tube is receiving power. After a few seconds warm-up
delay, the X-RAY light will come on, indicating that the X-ray tube high voltage circuit is
receiving power.
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Using the X-ray unit.
Do not exceed the maximum load rating for the tube with a tungsten target which is 50 kV and
20 ma. Lowering the voltage (kV) does not permit increasing the current (ma) above the maximum rated
limit, even though the product kV  ma does not exceed the maximum power limit.
With a sample in position and all shields properly in place, apply power to the tube as follows:
a.
Turn the KILOVOLTS control knob slowly, in increments of 5 kV, to the desired value as
indicated on the KILOVOLTS meter.
b.
Turn the MILLIAMPERES control knob slowly, in increments of 5 mA, to the desired
value as indicated on the MILLIAMPERES meter.
Turning the unit off.
a.
Turn the MILLIAMPERES control knob fully counterclockwise to a minimum.
b.
Turn the KILOVOLTS control knob fully counterclockwise to a minimum.
c.
Push the X-RAY OFF button.
d.
Push the LINE OFF button.
e.
Turn the key operated line switch clockwise to OFF.
f.
TURN OFF THE COOLING WATER TAP ON THE WALL (CLOCKWISE).
g.
Push the lever on the grey electrical service box on the wall to OFF.
Laue Photographs of Oriented (001) and (110) Tungsten Crystals
Do not touch the faces of the tungsten crystals. They have been etched clean.
1.
Uncover the X-ray port and turn the filter holder to OPEN (no filter in the beam).
8
2.
Place the camera in the position as shown in figure 3.
Place the track adapter on the camera track of
the X-ray unit. Open the X-ray tube port safety
cover and slide the track adapter forward until
the radiation cap is in contact with the port.
Tighten the knurled locking knobs (arrows)
The radiation cap should fit against the tube
port as shown. it may be necessary to adjust
the position of the camera track to get proper
alignment of the collimator and the port.
The radiation cap is shown here in the CLOSED
position. With General Electric equipment
the end of the radiation cap should seat in the
circular recess of the tube port.
Figure 3. Position of the Camera on the X-ray Machine.
3.
Without touching the faces of the crystal, mount the (001) crystal on the top of the pin at
the centre of the goniometer so that the (001) face will be perpendicular to the x-ray beam.
Use a small dab of Duco cement and be very careful not to get Duco cement on any part of
the goniometer except the pin.
4.
Mount the crystal position jig at some arbitrary point on the camera track with the point
facing away from the camera and secure it by hand, turning the locking bolts.
5.
When the Duco cement has dried, mount the goniometer on the camera track with the
crystal face to be photographed facing the point of the crystal position jig. Adjust the
goniometer so that the bases of the goniometer and jig are in contact and the point of the jig
is at the centre of the crystal face and is as close to it as possible without touching it. Be
careful not to scratch the crystal.
6.
Slide the goniometer back from the jig, remove the jig and then slide the goniometer
forward until its base is in contact with the base of the camera track adapter. The crystal
face is now 3.0 ± 0.1 cm from the photographic film plane and the X-ray beam will fall on
the centre of the crystal face.
7.
Load the film package as outlined in appendix I.
8.
Turn on the X-ray unit as outlined in the section above called Turning the unit on. For a
setting of 40 kV and 10 ma a typical exposure time (when medical X-ray film is used) is
about 30-40 minutes.
9
9.
After the correct exposure time, turn down the current and then the voltage and press the
X-RAY OFF button. Then reach in to close the shutter. It is not necessary to turn the
machine entirely off as outlined in the section above called Turning the unit off if you are
going to take more pictures.
10.
Develop the picture as outlined in appendix I.
11.
The centre of symmetry should be close enough to the centre of the film so that the axis of
symmetry of the crystal can be determined. Write the axis of symmetry of the crystal at the
bottom of the film.
12.
Remove the (001) crystal mount the (110) crystal and proceeding as before, photograph
and identify the prominent planes. Using an arrow, indicate the direction toward one of the
planes of {111}.
13.
Verify the result of (12) by turning the crystal through 35 so that the beam is parallel to
(111) and take a photograph. Identify prominent spots and write down the symmetry at the
bottom of the photograph. The stereogram p. 59 and photograph p. 67 in Wood may be
useful.
Laue Photograph of the CaF2 Crystal
1.
Remove the tungsten crystal and mount the CaF2 crystal, shown in figure 4, with the
sanded bottom on the top of the pin and the cleavage face perpendicular to the x-ray beam.
10
Figure 4: Top and Front Views of the CaF2 Crystal.
2.
Determine the direction of the cleavage face. For 40 kV and 10 ma, exposure times
(when using medical X-ray film) are typically about 3 hours.
3.
Confirm (2) by finding two other directions of high symmetry. Turn the crystal so that one
of these two faces is perpendicular to the x-ray beam. Take a photograph. If the symmetry
of the picture is what you would have predicted then this confirms your findings in (2).
QUESTIONS
[Note: The MATLAB script available on the website is a good source to gain intuition for how to answer
some of the below questions.]
1.
To what voltage would you have to go in order to see the characteristic spectral lines Kα
and Kβ for tungsten? Would both lines appear simultaneously or would one appear and
then the other only after the voltage had been further raised? See Semat p. 293 and Barrett
p. 624.
2.
What is a zone? What is a form? See Wood** p. 30 and p.8.
3.
What is the difference between (012), (0,1,2), [012], {012}, <012>? See Wood p. 8.
4.
What is the crystal structure of tungsten and also of calcium fluoride or fluorite (CaF2).
Refer to "Crystal Structures", Vol. I by R.W. G. Wyckoff.
5.
Using a 3 cm Greninger net, which may be obtained from the instructor, mark in pen on the
(001) photograph, the {103} and {114} spots. See Wood p. 32 for an explanation of how
to use the Greninger net. See Barrett pp. 40-42 or Wood pp. 41-43 for a set of tables
giving angles between planes in cubic crystals. It may be helpful to compare your
photograph with that in Wood p. 31.
6.
In this question a two dimensional lattice is used to give insight into the corresponding
three dimensional lattice.
**All
a.
Draw a two dimensional square cell with an atom at the centre of the square, as well
as at the corners of the square.
b.
Now, by repeating this cell several times, demonstrate that the lines containing the
centring points contain just as many atoms as the lines containing the primitive
lattice points.
c.
In three dimensions, a 100 reflection develops from a primitive lattice when the path
difference between rays from successive (100) planes is equal to one wavelength.
pages of Wood which are referenced in this manual are included in appendix II.
11
Show that for a face centred lattice, reflection from the planes containing the centring
points (i.e. reflections from the centred lattice) will cause cancellation and hence the
total intensity will be zero for first order reflections (see Nuffield p. 89).
7.
8.
The argument of the previous question is now generalized.
a.
Draw a two dimensional face centred square lattice with several cells. By analogy to
(hkℓ) planes in three dimensions we have (hk) lines in two dimensions. On the
square lattice draw in several (31) lines.
b.
Draw another two dimensional face centred square lattice with several cells. Draw in
several (21) lines.
c.
Using (a) and (b), be able to demonstrate that if h + k is even (i.e. h + k = 2n), then
the centring lattice points fall on lines containing the corner points and hence the
spacing of the lines is unaffected by centring. If h + k is odd then the effect of
centring is to halve the spacing between the lines. If we were to think in terms of
reflections from lines, then there would be no reflections from the lines in this latter
situation.
d.
If all faces are centred, as in a face centred cubic lattice, be able to generalize to the
rule for allowed reflections. The starting point is h + k = 2n, k + ℓ = 2n, h + ℓ = 2n,
see Nuffield pp. 89,90.
The rule for allowed reflections in a body centred cubic crystal is now considered.
a.
Show that for a body centred cubic crystal the centring point is midway between
(111) planes and thus there is no reflection from the (111) plane.
b.
In general it can be shown that a set of (hkℓ) planes divides the diagonal into h + k +
ℓ parts. From this, deduce the rules for allowed reflections in body centred cubic
systems.
9.
In general, low index spots are the stronger ones. On the (001) tungsten photograph,
compare the relative intensities of the (116) or (118) spots with that of the (115) spot.
Suggest an explanation for the relative intensities of these spots. Alternatively, compare
the intensities of the (013), (014) and (015) spots. Do all of these spots exist? See Wood
p. 65, or for more detail, Barrett pp. 80-84.
10.
In figure 4 of Wood p. 67 and the (111) photograph you have taken:
a.
Find the zone that (312), (211) and (321) belong to.
b.
Similarly, find the zone that (213), (112) and (123) belong to.
c.
These two zones seem to intersect at some point off to the left of the photograph.
What plane do they intersect in? You can check your answer by using figure 4a on
12
page 59 of Wood.
d.
Show that the zone containing (313), (212) and (323) goes through this plane.
Useful Hints: (also see Wood p. 8)
i.
If the zone [uvw] contains the planes (ℓmn) and (ℓ1m1n1), then you may
obtain [uvw] by using the rule for cross products. That is, treat (ℓmn)
and (ℓ1m1n1) as if they were vectors and take the cross product to obtain
[uvw]. Note that multiplying by a constant (positive or negative) does
not change things i.e. [ 111 ]  [ 333 ]  [ 111 ].
ii.
Similarly a plane (ℓmn) that is common to two zones [uvw] and [u1v1w1]
can be obtained by proceeding as though you were taking the "cross
product" of [uvw] and [u1v1w1].
iii.
If a plane (hkℓ) belongs to a zone [uvw], then the dot product of [hkℓ]
and [uvw] treated as vectors is zero (i.e. hu + kv + ℓw = 0).
11.
Find a reflection (215) and indicate it on your (001) photograph of the tungsten crystal (any
one of the planes in the form {215} is acceptable). You may find the locus of possible
positions for (215) by finding the angle between (125) and (001) using the table in Wood p.
43. Why is the angle between (125) and (001) the same as the angle between (215) and
(001)? Positive identification of (215) can be made in the following way. Using the
stereographic projection in Wood p. 55, identify correctly on your photograph (013), (114),
(103) and ( 114 ). Indicate on the photograph the arbitrary fiduciary direction perpendicular
to the sample axis which you have chosen to allow you to label the spots correctly. The
stereographic projection is explained in Wood p. 49. HINT: Use the hints in the previous
question and the fact that (013) and (114) belong to one zone and (103) and ( 114 ) to
another. What spot (plane) is common to both zones? See also Wood p. 31.
12.
In the Laue photographs on pp. 66-71 in Wood we see quantities like [100] and [101] etc.
which are placed next to arrows. What do these mean?
13.
On a stereographic projection, plot the cleavage face, the other two directions of high
symmetry and the bottom of the CaF2 crystal (sanded side). Also plot the directions of
high symmetry you used in step (3) of the instructions in the section "Laue Photograph of
the CaF2 Crystal". Plot your results on tracing paper with the cleavage face at the centre of
the projection. A stereographic net (or Wulff net) may be obtained from the instructor.
References
1.
2.
3.
4.
Barrett C.S. and Massalski T.B., Structure of Metals, 1966. (TN 690 B3) ON RESERVE
Nuffield E.W., X-ray Diffraction Methods, 1966. (QD 945 N83).
Semat H. and Albright J.R., Introduction to Atomic and Nuclear Physics, 5th edition, 1972. (QC
173 S298).
Wood E.A., Crystal Orientation Manual, 1963. (QD 905 W57) ON RESERVE.
13
5.
Wyckoff R.W.G., Crystal structures, Volume 1, 1963. (QD 951 W82) ON RESERVE.
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