Diffraction
Figure 15.2. single-crystal (a–c) diffraction patterns (right)
and corresponding sample orientations (left) and powder diffraction pattern (d) for tremolite. Compare this figure to that
of halite (Figure 15.1). The more complicated tremolite
structure produces more complex diffraction patterns. Note
that tremolite is monoclinic (2/m) while halite is isometric.
a. Tremolite projected onto its (100) plane. Because tremolite
is monoclinic, a is not perpendicular to the page. The symmetry (i.e., [010]2 and [010]m) is revealed in the diffraction pattern. b. Tremolite projected onto its (010) plane. In this case
we are looking down [010]2, revealing the 2-fold symmetry in
the plane. However, notice that the structure and the diffraction pattern appear to be rotated with respect to each other—
this is an effect of diffraction and will be discussed later in
this chapter (Figures 15.7 to Figures 15.9). c. Tremolite projected onto its (001) plane. This orientation is similar to that
in 15.2a above and shows the [010]2 and [010]m symmetry, as
well as the c axis being tilted slightly out of the plane of the
page. d. The powder diffraction pattern for tremolite.
Relative to the pattern of halite (Figure 15.1d), this one has
many, many more peaks because tremolite’s structure is
much more complex, and has lower symmetry, than halite.
Many more peaks appear at lower 1/d values, which correspond to the longer repeats (e.g., cell edges) in tremolite.
of the structures. You could use these patterns to calculate how much bigger the ionic radius of K is
than Na. In fact, Brady and Boardman (1995) do
just that, along with giving other nice teaching
examples of the use of powder X-ray diffraction.
These ideas are summarized in Table 15.1,
which gives equations relating the d-spacings in
minerals to their hkl values and cell parameters.
Notice how the isometric minerals (e.g., halite and
sylvite) have simple formulae. But as symmetry
decreases, the formulae get much more complicated. This “complication” with decreasing symmetry will be a theme throughout this chapter.
Light Diffraction
As previously stated, light diffraction is all
around us and can be observed in many places!
Diffraction occurs when light is bent by a periodic pattern. In mineralogy, we are concerned with
the periodic patterns made by atoms in minerals.
But first, let’s look at some other periodic patterns
that occur in nature so we can later relate them to
the patterns we cannot see with our own eyes.
Figure 15.4a shows rows of small Christmas
trees photographed from different angles. In the
top image, we are looking parallel to the rows.
The next two images show other “rows” that
occur from looking at different directions through
the trees. These still images do not do this analogy justice, so take a look on the DVD-ROM of the
c
a
a
a.
b
c
a
b.
a
b
c
c.
9.0
8.5
8.0
Intensity
366
7.5
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65
1/d (Reciprocal Angstroms)
d.
movie of this. As you drive along and look at a
row of trees, or corn, or anything in rows, they
align themselves in different directions. If you
drew lines to connect the trees in different orientations, you would arrive at different parallel
planes passing through the trees (Figure15.4b). In
the figure, each series of planes is labeled with
three numbers. These should remind you of the
Miller indices we learned about in Chapters 12
and 13. Noticed how the spacing between the
planes decreases as you go from (110) to (210) to
(310) to (410) to (510). If this was a mineral structure (instead of a field of trees!), X-rays would be
diffracted off these planes, resulting in the hypothetical unit cell shown in Figure 15.4c.
We can relate these spacings between planes
and their hypothetical X-ray diffraction patterns
to periodic arrays and light diffraction. It is easy
to observe diffraction with point sources of lights
(i.e., car headlights or streetlights) shining
through a screen window after dark. In Figure
15.5a, the car on the right is viewed through a
window screen, while there is no screen on the
left side. Figure 15.5b is a photograph taken after
dark showing how the car lights on the right are
diffracted through the screen compared to the car
on the left. This diffraction effect is somewhat
subtle and really appears as a smearing of the
light away from the headlights. In Figure 15.5c,
the screen is moved so it now covers the headlights of both cars, which now show the square
diffraction pattern. In Figure 15.5d the screen is
removed and neither car’s headlights exhibits
diffraction. Finally, Figure 15.5e shows the view
from the house with the screen completely
removed. Although you can’t see the wires in the
screen in these images, the spreading of light in
the vertical direction relates to the screen wire in
the horizontal direction, because the diffraction is
perpendicular to the periodic array that caused it.
Another very common diffraction pattern
occurs in the rain when light shines through a car
windshield while the wipers are in action. Unlike
the two-dimensional diffraction pattern shown in
Figure 15.5, this type of diffraction pattern is only
one-dimensional. The next time you drive after
dark in the rain with your windshield wipers on,
you will notice this diffraction pattern, so drive
carefully! Figure 15.6 and the associated movie on
the DVD-ROM show this phenomenon. In 15.6a, I
have taken a photograph of the light on the outside of my garage while sitting inside my car in
daylight. In 15.6b, it is now dark and the garage
light is turned on, but it is not raining. In Figures
15.6c and d, the light looks streaky, and the effect
is different in the two separate images. Why? The
streaking occurs perpendicular to the directions of
the lines of water made by the windshield wiper.
In Figure 15.6c the left hand windshield wiper has
just passed between the camera and light, while in
367
Diffraction
200
a = 6.28Å
222
220
220
111
200
222
a. Sylvite
111
113
200
a = 5.63Å
222
220
111
113
220
200
222
b. Halite
111
113
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
1/d (Reciprocal Angstroms)
Figure 15.3. Diffraction patterns for (a) sylvite (KCl) and
(b) halite (NaCl). The single-crystal patterns result from diffraction looking down [110]2. The diffraction patterns are
“indexed” with their associated hkl values (i.e., the lattice
planes causing direction are added to the patterns). These
two minerals share the same structure, and differ only in the
length of the a unit cell repeat, which is larger for sylvite
because the ionic radius for K is greater than that for Na.
The larger spacing for sylvite shows up in the powder diffraction patterns by shifting the peaks to the left (i.e., smaller 1/d values, which correspond to larger d values). A similar phenomenon occurs for the single-crystal patterns in
that the diffraction spots are closer together for sylvite, indicating that the atoms are farther apart. Notice the lack of a
(100) diffraction spot or peak. The (100) peak “should”
occur halfway between the center of the single-crystal diffraction pattern and the (200) spot, but it is “absent;” these
“absent” spots will help further refine the symmetry elements of a mineral. Also, notice that the intensities of the
peaks differ between the two patterns; thus the intensities
must be related to the atoms. As is seen in these patterns of
two simple compounds, interpretations of diffraction patterns tell us: (1) symmetry (based on arrangements of diffraction spots), (2) atomic spacings (based on distances of
diffraction in 1/d space, and (3) atomic differences (based on
the intensity of the diffraction peaks and/or spots).
Figure 15.6d, the effect of the right hand windshield wiper is shown. These effects show up best
if you look though the area of windshield where
368
Diffraction
Table 15.1:
Formula, by crystal system, to calculate dhkl values given unit cell parameters and (h k l) followed by a matrix method
(discussed in Chapter 13) that is generalized to all crystal systems and easily programmed.
(
)
1
1
1
a2 —2 + —2 + —
h
k
l2
isometric: dhkl =
tetragonal: dhkl =
orthorhombic: dhkl
2
2
2
2
2
(—ah + —ak + —cl )
a
b
c
= (— + — + — )
h
k
l
1
3
1
1
c
—
a —+—+— +—
4 ( h hk l ) l
2
2
2
2
2
2
2
2
hexagonal: dhkl =
monoclinic: dhkl =
triclinic: dhkl =
2
(
2
2
2
sin2 b
b2
+—
k2
h2 l 2 2hlcosb
—2 + —2 –
ac
a
c
)
1 – cos2a – cos2b – cos2g + 2cosacosbcosg
h2 2
k2
2hk
2hl
2kl
l2
—2 sin a + —2 sin2 b + —2 sin2 g + —— cosacosb – cosg + —— cosacosg – cosb + —— cosbcosg – cosa
ab
ac
bc
a
b
c
(
)
(
)
(
)
or, more generally for all crystal systems:
a2
abcosg accosb
1
b2
bccosa
—2— = [ h k l ] abcosg
d hkl
accosb bccosa
c2
–1
h
k
l
which can be rewritten as:
1
dhkl =
a2
abcosg accosb
[ h k l ] abcosg
bccosa
b2
c2
accosb bccosa
–1
h
k
l
the two windshield wipers overlap. You can even
see them in daylight if you have an older car and
live in an area high in dust. Over time, the quartz
particles in the dust scratch the windshield as the
wipers move forward and backwards. So even in
the daylight you may be able to see the streaking
effect as sunlight is being diffracted through the
scratches in the windshield.
If you don’t own a car, you can simulate this
effect by dipping your finger in water and smearing it on glass. Next look at a point light source
through the glass. Notice how the diffracted light
pattern is perpendicular to the direction you
wiped the glass. Look around you and you’ll see
this effect in many, many places: polished metal,
scratched glass, etc.
Now we move on to diffraction in different sizes
of screen wire. To do this, we are using something
you may have seen in your sedimentary class—
sieves. Sieves come in various sizes (the spacings
between the wires are different) so that sediments
can be separated into different size fractions. A diffraction pattern can be created by shining a laser
through theses sieves and projecting these patterns
on a distance wall. Figure 15.7a shows such an
experimental setup where a green laser shines
through a sieve located a few inches away from it
and projects a diffraction pattern on a screen ten
feet away. Figures 15.7b–d show several diffraction
patterns and the sieves that produced them. Notice
how the square pattern of the sieve produces a
square diffraction pattern, which is similar to the
diffraction pattern of halite in Figure 15.1a. In
Figure 15.7c, a smaller sieve size is used. We might
predict that the diffraction pattern from a smaller
sieve would have more closely spaced diffraction
spots than a lager sieve. However, observation of
the resulting diffraction pattern in Figure15.7c
shows the opposite to be true (i.e., the closer the
spacing of the wires, the farther apart the dots are
in the diffraction pattern). Also, diffraction occurs
perpendicular to the repeating array that forms it
(recall the windshield wipers). In Figure 15.7d, the
screen in Figure 15.7b has been sheared so the
wires are no longer perpendicular. Now the horizontal wires of the sieve produce the vertical lines
of diffraction patterns, and inclined diffraction patterns are produced perpendicular to the inclined
wires in the sieve. Finally, Figure 15.7e shows two
superimposed diffraction patterns resulting from
the use of two separate lasers: a red laser with
wavelength of 632 nm, as used in the previous figures, and a green laser with wavelength of 532 nm
(please look at the color image on the DVD-ROM).
The red laser, with longer wavelength, is diffracted
at greater angles than the green laser. Thus, diffraction is not only related to the spacing of the wires,
but is also related to the wavelength of light that is
used.
There is one last light diffraction experiment
you can do yourself. In Figure 15.8, a series of diffraction gratings have been produced by making
Diffraction
369
closely-spaced lines with a graphical computer
program and then printing them on transparencies using a high-resolution laser printer. Johnson
(2001) discusses a similar demonstration, but he
uses dots instead of lines. In Figure 15.8a, a series
of closely-spaced horizontal lines results in a vertical diffraction pattern. In Figure 15.8b, the spacing of the horizontal lines has been increased, thus
causing the diffraction spots’ spacing to decrease.
In Figure 15.8c, the grating used in Figure 15.8b is
duplicated and the second grating (with the same
spacing) is placed perpendicular to the first. This
results in a square diffraction pattern similar to
that produced by halite in Figure 15.1a. Figure
(510)
(410)
(310)
(210)
(110)
(110)
(010)
b
a
(100)
b. Dots and planes
a. Trees
(210)
(310)
(410)
(510)
c. “Unit cell”
Figure 15.4. Different spacings in rows of planted trees (a) and their correspondence to different spacings in rows (b) of atoms
in minerals with an enlarged view (c) of the “unit cell.” a. The upper image is a view down rows of planted trees and the
next two photos show other “rows” that occur as we view the trees from different directions (it might help to notice the relationship between the shadows the trees cast and the trees). You will see this as you drive down the road and look at rows of
planted trees, corn, or anything lined up in equally spaced rows. (Check out the movie showing this on the DVD-ROM.) b.
An overhead view of a series of rows (vertical) of plants that would correspond to atoms in 2-D. The repeat along a (horizontal), would represent the spacing of the “rows” and b (vertical) would represent the spacing on the plants within the rows.
Next, a series of planes ((110), (210), etc.) has been added. Notice how the spacing changes between these planes (i.e., it
decreases as the numbers increase). c. An enlarged view of the single unit cell with the (110), (210), etc. planes placed inside
the cell. The distance from the origin of the cell to the various planes decreases as the numbers increase.
370
Diffraction
a.
a.
b.
b.
c.
d.
e.
Figure 15.5. A series of photos showing diffraction of car
headlights by a screen window taken from the front porch of
my house looking at cars in my garage. a. A screen window
is placed on the right side in front of the camera lens. The
camera is positioned so that the screen is in front of the car
on the right, but not the one on the left. Now we wait for
dark! b. After dark, the headlights of both cars are turned on
and diffraction is seen for the headlights on the right,
through the screen. c. In this photo, the screen is moved to
cover both sets of headlights. d. Now the screen is removed
and no diffraction occurs. e. Daylight with no screen.
15.8d simulates the orthorhombic diffraction pattern that results when the gratings in Figure 15.8a
and b are placed perpendicular to each other.
c.
d.
Figure 15.6. Diffraction through a car windshield after
dark, in the rain. a. This photograph shows the outdoor light
on my garage (same one as in Figure 15.5) at home viewed
through my car windshield during daylight. The garage
light is centered on the windshield in the area where the cars
windshield wipers overlap. b. Same image as in 15.6a,
except now it is dark and the light is turned on. c. Now the
windshield wipers are turned on and rain is simulated.
Notice the diffraction in one direction—the diffraction
streak is perpendicular to the water streaks created by the
wiper. d. This is the same perspective as in 15.6c, except the
other windshield wiper (the one on the passenger’s side) has
just moved over the light, causing the orientation of the diffraction pattern to change. Now it is perpendicular to the
second wiper. (Check out the movies of this on the DVDROM—they add the motion.) The next time you are riding
(not driving, this is too cool of a distraction!) in a car after
dark in the rain, you can clearly see this effect.
Figure 15.8e uses those same two gratings, but the
one that was vertical in Figure 15.8c is now rotated counterclockwise. This counterclockwise rota-
tion causes the horizontal diffraction in Figure
15.8c to be rotated, in a similar manner as the
sheared screen in Figure 15.7d.
Reciprocal Lattices and d-spacings
All the preceding demonstrations have shown
that there is a relationship between the crystal
structures of the materials (i.e., the periodic
arrangement of the atoms in minerals) and their
diffraction patterns. It is very important to understand that shorter distances within a periodic
array appear as longer distances in their diffraction patterns. Obtuse angles in the periodic arrays
appear as acute angles in the diffraction patterns,
while 90° angles in the periodic array appear as
90° angles in the diffraction patterns. These conclusions demonstrate the relationship between
the direct lattice and the reciprocal lattice. The
direct lattice is the crystal structure of the material, while the reciprocal lattice is the name given
to what is produced in the diffraction pattern.
The term reciprocal derives from the fact that
there is an inverse relationship between the distances in the crystal structure and the diffraction
pattern. We have now seen the phenomenon
Diffraction
371
demonstrated using light waves, and this concept
was also explained mathematically in Chapter 13.
Figures 15.9a–d shows a series of lattices for
isometric, orthorhombic, hexagonal, and monoclinic crystal systems. To represent the direct lattice, we used the familiar cell parameters (a, b, c
and a, b, g) and for the reciprocal lattices, we use
the same letters but with a “star” on each one.
The result is the reciprocal lattices parameters (a*,
b*, c* and a*, b*, g*). You might be able to see that
there is a geometric relationship between a, b, c
and a*, b*, c*. Notice how a* is perpendicular to
the plane defined by b and c; b* is perpendicular
to the plane of a-c; and the c* is perpendicular to
the plane defined by a-b. Likewise a is perpendicular to the plane defined by b*-c*; b is perpendicular to the plane defined by a*-c* and c is perpendicular to the plane defined by a*-b*.
So, for the case of an isometric lattice a, b, and c
coincide with a*, b*, c* (Figure 15.9a), but the distances would still be inverted between direct and
reciprocal space. This relationship was shown for
the diffraction patterns of sylvite and halite in
Figure 15.3. A similar situation of coincidence of
the directions of the direct and reciprocal axes
occurs for the orthorhombic system (Figure 15.9b),
though now there is an inverse relationship
Diffraction
pattern
e.
Sieve
Laser
a.
b.
c.
d.
Figure 15.7. A series of photos showing diffraction patterns created by shining a laser through sieves. (Two sieves are used:
a 100 mesh (the one with the greater spacing) and a 300 mesh. (the mesh size refers to the number of holes in the sieve per
inch.) a. The experimental setup shows the laser shining through a sieve with the resultant image projected from about ten
feet away on a white screen. b. The upper image shows the diffraction pattern created from the sieve pictured above. c. Now
the diffraction spots are farther apart than in 15.7b and surprisingly, the sieves holes are closer together! Recall the inverse
relationship between atom spacing and diffractions back in Figures 15.1–15.3. d. In this pattern, the 100 mesh sieve has been
distorted to represent a monoclinic lattice. At first glance it would appear that the sieve and pattern have been rotated (just
like in Figures 15.1c and 15.2b), but it is the horizontal wires that produce the vertical set of diffraction spots (just like the
windshield wipers in Figure 15.6). Notice how the wires that are inclined from vertical produce diffraction spots that are perpendicular to them. The perpendicular relationships of planes to diffraction effects are seen in materials that do not have perpendicular axis sets. e. This is a repeat of the 300 mesh pattern, except now two separate lasers are used to form the pattern—
a red laser (l = 632 nm) and a green laser (l = 532 nm). The red laser with a longer wavelength produces more widelyspaced diffraction spots. (It would be helpful to look at the color image of the DVD-ROM to see this better.)
372
Diffraction
a.
b.
c.
d.
e.
Figure 15.8. A series of diffraction patterns (upper image of each layer), and associated computer-generated gratings (lower
image) printed on clear transparencies with a high-resolution laser printer. a. A one-dimensional diffraction pattern is formed
perpendicular to a closely-spaced set of lines. This setup simulates the windshield wiper effect in Figure 15.6. b. Another onedimensional diffraction pattern produced at right angles to a closely-spaced set of parallel lines. In this case the lines are farther apart than in 15.8a and thus produce diffraction spots that are closer together. c. A two-dimensional diffraction pattern.
To make this grating, the pattern from 15.8b was used with a duplicated copy rotated 90° and placed over it. This simulates
the car headlights in Figure 15.5 and the sieves in Figures 15.7a and 15.7b. d. A two-dimensional diffraction pattern is produced by taking the grating from 15.8a and placing the one from 15.8b, after a 90° rotation, on top of it. This represents the
diffraction pattern from an orthorhombic lattice. The longer repeat in the grating results in closer spacing of points in the diffraction pattern. Again, an inverse relationship exists between the distance between the lines in the grating and the spacings
between points in the diffraction pattern. e. The two-dimensional diffraction pattern produced by placing two gratings from
15.8b on top of each other at a non-90° angle. This pattern simulates the distorted sieve shown in Figure 15.7d, except the
inclined lines are tilted to the left of vertical, thus producing a different orientation in the rows of diffracted spots.
between lengths of a and b vs. a* and b*. Figure
15.9c shows a hexagonal lattice and the associated
g angle of a 120° between a and b. The associated
reciprocal lattice again shows the aforementioned
relationship of the directions between the direct
and reciprocal lattice. Notice how the row of atoms
that defines the a direction is perpendicular to the
b* direction, while the row of atoms defining the b
direction is perpendicular to the a* direction. This
is the same phenomenon that we saw in Figure
15.2b in tremolite, which made the structure of
tremolite appear rotated from the diffraction pattern. Also notice that in the hexagonal lattice, g*
becomes the supplement (i.e., they total to 180°) of
g. Finally, for the monoclinic lattices (Figure 15.9d),
the lengths of a and b are shown inverted as a* and
b*, while the obtuse g angle becomes an acute
angle denoted as g* that is the supplement of g.
“Reflection” of X-rays
We can now begin to explain what causes some of
the diffraction effects that we have been observing. To do this, we will relate light reflection from
a material’s surface to X-ray diffraction by the
b g
a
Diffraction
material. However, while light reflects off of a
surface at all angles, X-rays will only “reflect”
from the structure of the mineral at certain angles,
where these angles relate to the wavelength of the
X-ray and the spacings of the planes in the mineral. As an example, refer to Figure 15.10 and the
associated movies on the DVD-ROM.
The setup of the experiment shown in Figure
15.10 places a sample of halite in the center of a
powder X-ray diffractometer (Figure 15.10a). The
X-ray source is on the left, the sample is in the
center, and the detector is on the right. Figure
15.10b shows the same setup as in Figure 15.10a,
except that the X-ray tube and detector have
been rotated to an angle that is labeled u. The Xray diffractometer has a mechanical system that
simultaneously rotates the X-ray tube and the
detector. Figure 15.10c shows the crystal structure of halite that was placed in the diffractometer. The horizontal, parallel lines drawn through
b g
a
b g
a
b* g*
a*
a. Isometric
b*
g*
a*
b. Orthorhombic
373
b g
a
b* g*
a*
c. Hexagonal
b* g*
a*
d. Monoclinic
Figure 15.9. Two-dimensional projections of direct (upper) and reciprocal (lower) lattices for the isometric (a), orthorhombic
(b), hexagonal (c), and monoclinic (d) crystal systems. The direct lattices shown here relate to the gratings shown in Figure
15.8, with atoms (small circles) replacing the intersections of the grating lines. The reciprocal lattices are related to the diffraction patterns shown in Figure 15.8. The direct lattices are represented by the all-familiar cell parameters (i.e., a, b, c and
a, b, and g) and projected on the a-b plane (i.e., down c). The reciprocal lattices are represented by the “star” cell parameters and listed as a*, b*, c* and a*, b*, and g*. Thus, they are projected on the a*-b* plane looking down c*. While the direct
lattice dimensions are fixed, those for the reciprocal lattice vary as a function of the wavelength that is used and the distance
between the sample (i.e., the sieve location in Figure 15.7a) and the plane in which the diffraction image is formed (i.e., the
screen in Figure 15.7a). Finally, the general observation that should be gained from these projections is that a reciprocal axis
(e.g., a*) is perpendicular to two other direct axes (e.g., b and c). a. For the isometric systems, the geometries of the direct
and reciprocal lattices are identical. b. For the orthorhombic systems, the shorter of the two axes in direct space becomes the
longer axis in reciprocal space. c. For the hexagonal systems, the lengths of a and b are similar to those of a* and b*. Here g
is the obtuse angle 120°, while g* is the acute angle 60°; thus these two angles are the supplement of each other (i.e., they
total to 180°). Thus non-90° angles also change between direct and reciprocal space, as shown in the case of the gratings in
Figure 15.7. Diffraction occurs perpendicular to the lattices rows. So a is perpendicular to b* and b is perpendicular to a*.
d. For the monoclinic systems, the lengths of the repeats along a and b are, as expected, inverted between the direct and reciprocal lattices. Also, g and g* are supplements, a is perpendicular to b*, and b is perpendicular to a*.