Lab 31 in TFFM08 The Laue Method

IFM – The Department of Physics, Chemistry and Biology
Lab 31 in TFFM08
The Laue Method
NAME
PERS.
NUMBER
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DATE
APPROVED
Rev July 2014 Chung-Chuan Lai
1. Introduction
Note to students: Before this lab it is highly recommended that you read the first two chapters of
Kittel C. Introduction to Solid State Physics about crystallography and diffraction.
1.1 Backgrounds
Early in the 20th century, scientists speculated that crystalline solids are composed of periodically
arrayed atoms. In 1913, Max von Laue presumed that the periodicity of crystals should be suitable for
diffracting X-rays, which have wavelengths close to atomic scale. He was then able to demonstrate
that X-rays were in fact diffracted by crystals.
The discovery is thereafter followed by a British father-and-son team: the Braggs, who developed the
well-known Bragg’s law for crystals to relate incident X-ray wavelength to the diffraction angle and
crystal inter-planar spacing. The law enables scientists to determine the crystal structures of numerous
compounds with a simpler mathematical expression.
Yet another mathematical theory of X-ray diffraction in solid state was published in 1917 by a German
crystallographer P. P. Ewald who developed the basics for an imaginary concept that now known as
reciprocal lattice/space. The reciprocal space is a mathematically (Fourier) transformed physical
space, which has its dimensions inversely proportional to physical space. The reciprocal lattice is not
physically existed. However it contains a lot physical meaning in interpretation of X-ray diffraction
patterns. The patterns in Laue diffraction are set to intercept the diffracted beams as spots (beam crosssections), that contain visual records of reciprocal lattices from the sampled crystals directly. The very
useful illustration in visualizing reciprocal space, the Ewald’s sphere, is the most important proposal
of Ewald in all kind diffraction physics (not only valid for X-ray diffractions!) of solid states.
1.2 General crystallography
Crystallography is a subject concerning arrangements of atoms in crystals, while a crystal is a solid
material containing periodically repeated atoms under certain arrangement in space. A crystal consists
of two parts: a lattice and a base, see Figure 1. A lattice of a crystal is a set of mathematical dots
(volume-less dots!) which can represent the periodicity of the crystal. The chemical surroundings, e.g.
the number and element of nearest neighboring atoms, of any two dots in a lattice should be
completely identical. These dots are called lattice points.
A base, on the other hand, describes the chemical surroundings on a lattice point, including their
positions, the number and the element of atoms. It can be one or more atoms in a base, while the base
of a crystal is repeated on each lattice point, so that the combination builds up the whole crystal. As
can be seen in Figure 1, the pair of lattice and base is not definite for a crystal and can in fact cause
confusion in crystal description. In general, we will use the lattice with highest symmetry and lowest
numbers of atoms in the corresponding base. Therefore in Figure 1, the choise of (Lattice 1)+(Base 1)
is preferable compared to the (Lattice 2)+(Base 2) option in describing about the crystal AX.
It can be seen that the symmetry of a crystal is not determined solely by a lattice or a base, but a
combination of them. And the result from any combination should be the same in describing a crystal.
Therefore, the symmetry of a crystal will not necessarily be the same with the lattice of choice (but it
could also be the same as well).
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Lattice 1
Base 1
Crystal AX
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Lattice 2
X
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Base 2
A
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Crystal AX
A
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=
A
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Figure 1. Illustrations of a built-up crystal from different pairs of lattices and bases. The example is given in
2-dimensional space, however it is valid in 3-dimensional space as well.
The concept of using lattice to describe crystals helps material scientists in categorizing crystal
structures, as well as in describing displacements (axes) and volumes in crystal space. If an enclosed
volume can fill the whole crystal space with its exact repetition in all directions, the volume is referred
as a unit cell, which contains lattice points and base atoms within a cell. Just like the lattice and base
for a crystal, the shape and size of the unit cell for a crystal are also not definite, as long as the chosen
unit cell can fill the whole crystal. However, the unit cell is normally a parallelepiped, since (i) a
parallelepiped has three distinct edges that can be used as the three axes in a 3-dimension space, and
(ii) special parallelepipeds, such as cubes, rectangular cuboids, and rhombohedrons have high
symmetry to be easily visualized.
In order to systematically categorize crystals with different lattices, French physicist Auguste Bravais
established a system to sort crystals into 14 Bravais lattices (Figure 2). The Bravais lattices make up
the basic symmetries of crystals, that the symmetries can sometimes been observed macroscopically
from crystals. By adding up a base of a crystal to its lattice, the symmetry can be reduced in a way that
can be seen in, e.g. X-ray diffraction patterns. This makes it possible to further divide crystals into 32
point groups or symmetry classes. The categorization is complete by including atomic configuration of
a crystal, which then defines a total of 230 space groups. For those that are interested, there is a lot to
discover in books about solid state physics, crystallography and mineralogy.
1.3 Crystal symmetries
An object or pattern is said to be symmetry if it stays unchanged after applying a spatial operation such
as “a rotation about an axis” or “a reflection across a plane”. If an object possesses rotational
symmetry, it will appear unmoved after rotation about the rotation axis. For example, a hexagon
appears unchanged after a rotation of 60 degrees about an axis perpendicularly passing through its
center, while a cube appears unchanged after a rotation of 120 degrees about an axis through its body
diagonal. The axis that a symmetric pattern rotates about is called a rotation axis, and any plane
perpendicular to a rotation axis is called a rotation plane or a plane of symmetry. The point where a
rotation axis and a rotation plane intersect is referred as a rotation center. In crystallography, we use a
number n to describe an n-fold rotational symmetry, where n equals 360 degrees divided by the least
angle to rotate to the next symmetric pattern. We commonly have 6-fold (60 degrees), 4-fold (90
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Figure 2. The 14 Bravais lattices and 7 crystal systems (Triclinic, Monoclinic, Orthorhombic,
Tetragonal, Hexagonal, Trigonal, and Cubic) in 3-dimensional space. The capital alphabets P (primitive),
C (bottom-centered), I (body-centered), F (face-centered) and R (rhombohedral) represent the initial
symbols of different lattices.
degrees), 3- fold (120 degrees), 2-fold (180 degrees), and at least a 1-fold (360 degrees) rotational
symmetry appears in crystals. However, it has also been discovered that 5-fold symmetry can exists in
some alloy materials, which is referred as quasi-crystals since they do not possess a strict periodicity
as common crystals.
If a pattern possesses mirror symmetry, the pattern is symmetric across a mirror plane, while the ‘halfpatterns’ on each side of the mirror plane are mirror images of each other. Again we referred any plane
perpendicular to a mirror plane as a plane of symmetry. On the other hand, we use letter m to describe
mirror symmetry in a crystal. One should never confuse a mirror symmetry with a 2-fold rotational
symmetry, though in some cases they can co-exists. A way to help memorizing is that: the left and the
right hand of a person have mirror symmetry, but have no 2-fold rotational symmetry since they will
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not completely overlap each over when rotate 180 degrees around a axis perpendicular to the plane of
the palm.
1.4 Reciprocal lattice
A reciprocal lattice in reciprocal space is a Fourier transformed physical lattice in physical space,
where a set of physical lattice planes is conserved into a point in reciprocal lattice. As described by the
name, the reciprocal space has dimensions in reciprocal order (length)-1, e.g. 1/m. It is a very useful
analytical method when studying in wave physics and crystallography.
Details about construction of the reciprocal lattice will not be discussed here, but its main features are
worth noting for conveniences of the lab:
i)
A lattice vector in a physical lattice is normally denoted as R = ma1 + na2 + pa3, while in
reciprocal lattice is normally denoted as G = qb1 + rb2 + sb3, where a1, a2, a3 is primitive
vectors of physical lattice, b1, b2, b3 is primitive vectors of corresponding reciprocal lattice and
m, n, p, q, r, s are integers. The primitive vectors in both spaces have following relation:
ai · bj = 2πδij where δij = 1 when i = j otherwise δij = 0
ii)
If a vector Ghkl = hb1 + kb2 + lb3 is a reciprocal lattice vector pointing from origin to a
reciprocal lattice point hkl, then Ghkl is always perpendicular with crystal plane (h k l), where
(h k l) is the miller index of the plane, h, k, and l are integers.
iii)
A reciprocal lattice point represents a set of lattice planes in crystal lattice, where the distance
between the origin and the reciprocal lattice point is reciprocal of the plane spacing in physical
space. That is, |Ghkl| = 2π/dhkl where dhkl is the inter-planar spacing of plane (h k l).
1.5 Diffraction conditions
Assuming the photon beams before and after scattering have wavelength λ and λ’, their wave vectors
are then equal to k (|k| = 2π/λ) and k’ (|k’| = 2π/λ’). The difference between the incoming and the
outgoing beams Δk = k’ – k is therefore called as scattering vector. In X-ray diffraction study of
crystals, the scattering is often elastic, in which the photon energy is conserved. In other words, λ = λ’
and |k| = |k’|.
When Δk = Ghkl for any hkl set of index, the diffraction condition is fulfilled for a set of planes (h k l).
As the diffraction condition is fulfilled, a constructive diffraction is created. At the mean time a
constructive diffraction pattern of certain planes can therefore be seen in X-ray diffraction spectrum as
a region with higher intensity, i.e. diffraction peaks with high intensity at certain angles can be seen.
1.5.1 Bragg’s condition
For a crystal fixed in space, an incident X-ray beam will hit a number of sets of parallel planes
containing atoms in the crystal. According to Bragg’s law, constructive interference of diffracted
beams from atomic planes occurs when
2d · sinθ = nλ
(1)
Here, d is the inter-planar spacing, λ is the wavelength of the X-ray beam, θ is the angle between the
incident beam and the diffracted planes, and n is an integer. Since sinθ ≤ 1, any radiation with a
wavelength λ longer than 2d will not satisfy the Bragg’s equation (Eq. (1)) and, hence, cannot be used
for diffraction experiments. Moreover, radiations with too long wavelength will have very low
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penetration depth, i.e. only the surface is radiated and sampled (Can this also be a benefit? why?). On
the other hand, any wavelength λ shorter than 2d will satisfy Eq. (1). However radiations with too
short wavelength will result in very localized small θ values, which are very difficult for analysis
(What does this mean? Can short wavelength somehow be used?). In most cases, radiations with
wavelength comparable to the inter-planar spacing are the most preferable. Since the lattice constants
in solid crystals are in order of a few Å, the best radiation to be used for diffraction experiments would
be X-ray, which has the wavelength in the range ~0.5-2.5 Å.
1.5.2 Laue Equations
In fact, Bragg’s diffraction condition is a simplified equation from the diffraction condition in general
(Δk = G) in order to interpret diffractions at a set of periodic planes (such as crystals) in the physical
space. Before the Bragg’s condition, Laue has proposed another expression for the diffraction
condition by considering it in wave vector, |k| = 2π/λ, and reciprocal lattice space:
R · Δk = 2π × (integer)
or
a1 · Δk = 2πh
a2 · Δk = 2πk
a3 · Δk = 2πl
(2)
Although equation (1) and (2) look very different, they describe the same conditions required for
diffraction. Ewald provided a very helpful visualization of diffraction conditions in reciprocal space.
The constructed visualization is called Ewald’s sphere, as shown in Figure 3.
When constructing Ewald’s sphere, the wave vector k (AO) is drawn on the corresponding reciprocal
lattice in direction of incidence, with its length = 2π/λ and with its end at any reciprocal lattice point.
The origin O on this lattice is therefore selected as the lattice point where k ends. We then draw a
sphere with radius = |k| = 2π/λ about A, where k starts.
The diffraction condition is fulfilled at any reciprocal lattice point which intersects with the
Figure 3. Example of diffraction conditions from the (130) plane in (a) physical space (Bragg’s
condition) and (b) reciprocal space (Ewald’s sphere).
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constructed sphere. Assuming that we have an intersection at lattice point B, the out-coming diffracted
beam will have its wave vector k’ (AB). The scattering vector Δk (OB) is equal to the reciprocal lattice
vector Ghkl since both O and B are points in reciprocal lattice, i.e. Δk = Ghkl, where hkl is the Miller
indices of lattice point B with respect to O. A diffraction angle 2θ is defined as the angle between k
and k’, or angle OAB. Remember that |Ghkl| = 2π/dhkl where dhkl is the inter-planar spacing of plane
with Miller indices (h k l). You can remind yourself about reciprocal lattice from section 1.4.
1.6 X-ray diffraction methods
1.6.1 Powder diffraction
(Plane spacing of polycrystalline sample investigated with single wavelength)
X-ray powder diffraction is a common method to analyze crystal phases within either polycrystalline
or powdered samples, where it is ideal to use monochromatic X-ray with a single wavelength λ. The
polycrystalline or powdered sample provides a collection of crystal planes oriented in (almost all)
different directions. The diffraction condition will then be fulfilled for different sets of planes in the
sample, giving rise to diffraction peaks at corresponding angles θ.
1.6.2 Laue diffraction
(Plane directions of fixed single crystal studied with multiple wavelengths)
In the Laue method, the single crystal sample is fixed and X-rays with different wavelengths are used.
Ideally, one would like to use a continuous spectrum of X-rays with various wavelengths that will
fulfill the Bragg’s law for all sets of planes. However, in reality available wavelengths in X-ray
spectral region are limited and, hence, only certain range of d-spacing can be measured. The diffracted
X-ray beams are then projected onto a screen of a charge-coupled device (CCD) for all different
planes at the same time, whose patterns are thereafter analyzed for determination of the crystal
orientation and symmetry.
Typically, there are two common configurations of optics used in Laue-diffraction experiments, one is
called “back-reflection Laue” and the other “transmission Laue”. In our lab, we use back-reflection
Laue arrangement with the X-ray CCD detector placing in between the X-ray source and the sample.
Figure 4 shows a detailed illustration of two set-ups for back-reflection Laue diffraction.
An X-ray beam from X-ray tube radiates on a crystal through a capillary collimator at the center of a
CCD screen. The screen can either be (a) a flat screen or (b) a dome-shape screen. For a flat screen
set-up, the incident beam is perpendicular with the surface of the screen, while for a dome-shape
screen, the incident beam is pointed to the center of the dome. The incident beam is then
scattered/reflected back to the CCD screen with certain distortion depending on the shape of the screen,
as shown in Figure 4. The pattern in flat screen will be more diffused (more scattered) away from the
center of the screen. In our lab, a flat screen set-up for Laue diffraction is used. The screen is in an
area dimension of 163×109 mm2, and the distance L between the sample and the screen is variable
(normally in between 3 – 5 cm).
When the diffraction condition is fulfilled, the X-ray beam diffracted from a set of planes in the crystal
will result in an image (or a dot) with higher intensity on the screen. While multiple sets of planes
fulfill the Laue condition at the same time, a diffraction pattern of corresponding dots can be observed
(shown as an example in Figure 4). The distance between the center of the screen and a dot is given by
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Figure 4. Different set-ups of Laue diffractometer and resulting patterns for a 4-fold symmetry.
Ltan2α, where α is the reflected angle of the beam. 2α is also the supplementary angle of the
diffraction angle 2θ, i.e., 2α + 2θ = π.
The resulting diffraction pattern provides necessary information for determination of the orientation of
the planes and the symmetry of the studied crystal orientation. For example, a crystal in cubic system
should, in general, has a four-fold symmetry along its [100] direction. From Laue diffraction of this
cubic crystal along its [100] direction (when its (100) plane facing perpendicularly to the incoming Xray), it will give 4 dots at the same distance from the center of the screen. Therefore, Laue diffraction
is a useful and fast tool for determination of the crystal structure of an unknown crystal, or the
orientation of a crystal with known structure.
Unlike X-ray powder diffraction, indexing a Laue diffraction pattern can be more difficult, since
multiple/continuous wavelengths and multiple planes are included. The Bragg’s law (2dsinθ = nλ)
becomes difficult to solve since both d and λ are unknown parameters in Laue method (θ is known
from α). In cubic crystal systems, where the plane (h k l) has its plane normal [h k l], it is possible to
calculate the angle between two vectors (and also the angle between two planes) to estimate the Miller
indices of a Laue diffraction dot. Also, a double of the reflection angle, 2α, is equal to the angle
between two planes.
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2. Pre-lab exercises
(If you do not finish them BEFORE the lab started, you are NOT allowed to do the lab.)
1.
(VERY IMPORTANT) Remember the rules for Laue lab.
(i) Follow the instructions from your lab assistant. If you are uncertain about any operation
in the lab, please do not try to perform it before asking the lab assistant.
(ii) When the system is in operation, do not open the doors of the diffractometer, otherwise
the system can be damaged.
(iii) It is absolutely not allowed to touch the CCD screen with hands or any object since you
may completely destroy the detector.
2.
Verify that the diffraction condition, Bragg’s condition and Laue equations are equivalent.
Familiarize yourselves about Ewald’s sphere.
3.
Calculate the angle between the (100) and (110) planes as well as between the (100) and (111)
planes of a cubic crystal. (It is important for you to remember how to do it!)
4.
(i) Determine the maximum angle 2αmax visible in a Laue image of size 163×109 mm2 with the
set-up described in Figure 4(a). Assume the distance L as 4 cm.
(ii) For a cubic crystal, how can the (513) spot be visible when the (100) plane is parallel with the
CCD screen by varying only the distance L (i.e. what is the shortest L needed)?
5.
Find out, from your own references, the general curve of intensity as a function of wavelength for
a typical X-ray source commonly used in the laboratory. What determines the high and low
wavelength limit?
6.
Read carefully through lab tasks, though you may not completely understand all of it. Also, bring
with you a calculator with sin and cos functions which you may need for quick checking the
obtained data in your experiment.
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3. Lab tasks
1.
A single crystalline Si is placed at sample stage then a Laue image is taken from the sample.
(i) What symmetry of the sample can be seen from the image? Does it reflect the crystal structure
of Si? What is the family of the planes parallel to this image?
(ii) Given the Miller indices of a diffraction dot and the distance from that dot to the center, can
you find out the conversion factor between the length in computer image and the corresponding
length in reality (in unit pixels/cm)?
Note: A conversion factor is needed since computer programs save image data in unit of pixels,
therefore the length measured from the program is always in pixels. However we need to find out
the real distance between a diffraction dot and the center in order to index that diffraction dot.
(iii) Use the conversion factor obtained from 1. (ii) and index few more points. Do they seem to
be reasonable in terms of diffraction physics and crystallography?
(iv) Participate in the demonstration for a simulation and fitting program of Laue diffraction
patterns. Compare the simulated pattern with the Laue diffraction pattern of Si.
2.
Another piece of single crystalline Si is placed at sample stage for taking a Laue image.
(i) What symmetry of the sample can be seen this time? What is the family of the planes parallel
to this image?
(ii) Depending on your answer in 2(i), what is the angle between this plane and the plane in task 1?
3.
Repeat the same procedure for the third single crystalline Si sample.
(i) What symmetry of the sample can be seen this time? What is the family of the planes parallel
to this image?
(ii) Depending on your answer in 3(i), what is the angle between this plane and the plane in task 1
and 2?
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4. Lab report
Some groups of you are requested to do a lab report. You can work as a team and submit one copy
from the group. This report will then be evaluated by the lab assistant.
The report should be summarized with activities and observations during the lab, and should be
organized as a formal scientific report under following headings:
Purpose: A 1-2 sentence statement on the purpose of your experiment, i.e. what are you trying to
measure and determine.
Equipment: State the major piece of equipment using a model number.
Under what conditions did you use it (e.g. V, A settings, mode)?
What sample did you study (e.g. 2000 Å V film on a <111> MgO substrate)?
Experiment: What did you do?
Results: Raw data, calculations, graphs
Discussion: Explain your results
Conclusions: One statement usually.
Comments: Were you lacking background information? Was the lab too easy, difficult, long, short,
boring...? Did you learn something? Do you want more hands-on or more theory?
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5. References
1. Giacovazzo, C., Fundamentals of Crystallography (1992)
2. Holden, A., and Morrison, P. S., Crystals and Crystal Growing (1982)
3. Hurle, D. T., Handbook of Crystal Growth (1993)
4. Janot, C., Quasicrystals (1992)
5. Mullin, J. W., Crystallization, 3d ed. (1993)
6. Rhodes, G., Crystallography Made Crystal Clear (1993).
7. Berry, L., Mineralogy: Concepts, Descriptions, Determinations, 2d ed. (1983)
8. Blackburn, W. H., and Demmen, William H., Principles of Mineralogy (1988)
9. Carmichael, R. S., CRC Handbook of the Physical Properties of Rocks and Minerals (1988)
10. Deer, W. A., et al., An Introduction to the Rock-Forming Minerals (1966)
11. Desautels, P. E., Rocks and Minerals (1982)
12. Frye, K., ed., The Encyclopedia of Minerals (1982)
13. Klein, C., Minerals and Rocks (1989)
14. O'Donoghue, M., American Nature Guide to Rocks and Minerals (1990)
15. Pough, F. H., A Field Guide to Rocks and Minerals, 4th ed. (1976)
16. Roberts, W. L., The Encyclopedia of Minerals, 2d ed. (1989)
17. Sinkankas, J., Mineralogy (1975).
18. Abruna, H. D., The Study of Solid/Liquid Interfaces with X-rays, Science, Oct. 5, 1990
19. Elter, M. C. et al., Solid State NMR and X-ray Crystallography (1988)
20. Taylor, C. R., Diffraction (1987).
21. P. E. J. Flewitt and R. K. Wild, physical methods for materials characterisation (1994)
22. C. Kittel, Introduction to Solid State Physics about crystals and diffraction. (1986)
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Appendix A (refs. 2, 3, 5, and 7-17)
(Not necessary to read for the lab, only for the interested.)
A.1 Max von Laue
The German physicist Max Theodor Felix von Laue, b. Oct. 9, 1879, d. Apr. 24, 1960, received the
1914 Nobel Prize for physics for his discovery (1912) of the diffraction of X-rays by crystals. He
showed that X-rays are very short electromagnetic waves that can be used to study the structure of
materials. He thus helped establish the field of X-ray structural analysis, an important branch of
physics and chemistry. Laue, who played an essential role in the golden age of German physics, was
professor of theoretical physics at Berlin from 1919 until his retirement in 1943, and was instrumental
in rebuilding German science after World War II.
A.2 Minerals
A mineral is a natural, homogeneous, inorganic solid with a crystalline atomic structure. Crystallinity
implies that a mineral has a definite and limited range of composition, and that the composition is
expressible as a chemical formula. Some substances that do not satisfy all these conditions, such as
metallic liquid mercury, are commonly considered in the mineral realm but should more properly be
designated mineraloids. The word mineral may have different meanings in nonmineralogical sciences.
In nutrition, it may mean any non-organic element. In economics and economic geology, minerals
may be practically anything of value extracted from the Earth, including petroleum and natural gas
(which are not minerals according to the above geological definition, being neither inorganic nor
solid).
Minerals comprise the vast majority of the material of the solid Earth. Aside from air, water, and
organic matter, practically the only non-minerals in the Earth as a whole are molten rocks (magmas)
and their solid glassy equivalents. Crystalline rocks themselves and even soils for the most part,
consist of aggregates of minerals. Almost all inorganic substances that are used by or of value to
humans are derived from minerals. Over 3,000 minerals are currently known, and about 50 new ones
are now discovered each year. Most Gems are minerals, though some, such as opal, are mineraloids.
A.3 History of mineralogy
Mineralogy is the study of the nature and origin of minerals. Although it is not one of the fundamental
sciences, it was one of the first scientific fields to be developed, and curiosity about minerals led to
many discoveries in physics and chemistry. Georgius Agricola (Latinized name of Georg Bauer),
sometimes considered the founder of mineralogy, summarized most early knowledge of minerals in
his works De Natura Fossilium (On the Nature of Fossils, 1546; Eng. trans., 1955) and De Re
Metallica (On Metals, 1556; Eng. trans., 1912). The discovery by Nicolaus Steno, Arnould Carangeot
(1742-1806), and Jean Baptiste Rome de l'Isle (1736-90) in the 17th and 18th centuries that interfacial
angles of crystals of a given mineral are constant, laid the basis for the science of crystallography. The
founder of crystallography, however, is considered to be Rene Just Hauy, whose most important works
were published in 1784 and 1801. He proposed correctly that crystals are formed by stacking of
identical structural blocks, now called unit cells, and he showed that, as a consequence, the intercepts
that the crystal faces make on a set of carefully chosen axes are always rational numbers when divided
by an appropriate common factor. Other early-19th-century scientists who contributed to the
development of crystallography include Johann F. C. Hessel (1796-1872), William H. Miller (1801-
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80), and Auguste Bravais (1811-63). Hauy's contemporary Abraham Gottlob Werner and his followers
developed methods for identifying crystals by physical characteristics.
The science of chemistry developed rapidly in the early 19th century, and the determination of the
compositions of minerals proceeded apace, especially through the efforts of Jöns Jakob Berzelius and
his followers. The first lasting classification of minerals according to chemical composition was
published by James Dwight Dana in 1837.
The elucidation of the optical properties of crystals took place in the early 19th century, with
noteworthy work being done by Jean Baptiste Biot and Sir David Brewster. The invention of the
polarizing microscope and the use of thin sections by William Nicol (1768-1851) made it possible to
identify minerals by means of their optical properties, and Henry Clifton Sorby, who is considered to
be the founder of petrography, later applied these methods to the study of rocks.
The simultaneous discovery in the late 19th century by Arthur M. Schoenflies (1853-1928), William
Barlow (1845-1934), and E. S. Fedorov (1853-1919) of the 230 space groups (see below) essentially
completed the development of classical crystallography. With the discovery (1912) of X-ray
diffraction by Max von Laue, Walter Friedrich, and Paul Knipping, the determination of the internal
atomic structures of crystals became possible. Sir William H. Bragg, Sir William L. Bragg, and others
immediately began to use this tool to determine crystal structures of minerals, and this work continues
to the present. X-ray diffraction also provided a rapid and objective means for identifying minerals.
The chemical analysis of minerals advanced importantly with the invention (1949) of the electron
microprobe by R. Castaing. The development of high-speed electronic computers has greatly
facilitated the determination of the crystal structures of minerals and other substances.
A.4 Synthetic mineralogy
A very important part of mineralogy in the 20th century has been the reproduction of minerals in the
laboratory, with the principal objective of ascertaining the conditions of temperature and pressure and
the nature of liquid or gaseous phases present during the formation of natural minerals. Other reasons
for synthesizing minerals are to obtain pure end-member specimens for determination of physical
properties, and to supply substitutes for scarce minerals of economic importance.
Synthesis has traditionally been achieved using furnaces with tungsten- or platinum-wire heating
elements, and large steel presses or hydraulic apparatus for attaining high pressure. The sample is
taken to high pressure and temperature, held for a time until the reaction is complete, and then
quenched and examined.
Extremely high pressures corresponding to those in the Earth's mantle have recently been attained with
the diamond anvil, a compact apparatus in which the sample is placed between the faces of two
diamond crystals and compressed with a thumbscrew. The device is small enough to fit on a
microscope stage, and the sample can be observed through the transparent diamond faces. A laser
beam can be projected through the microscope to heat the sample.
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