P. (Paul) Groth.

The optical properties of crystals, with a general introduction to their physical properties; being selected parts of the Physical crystallography online

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Online LibraryP. (Paul) GrothThe optical properties of crystals, with a general introduction to their physical properties; being selected parts of the Physical crystallography → online text (page 1 of 28)
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Professor of Mineralogy and Crystallography in the
University of Munich





University of Colorado

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UNTIL recently, in the higher institutions of learning, crys-
tallography has largely been taught only in connection with
mineralogy, as an aid to the characterization of minerals and
therefore in a purely descriptive manner. But this does not
accord with the present state of the science. Haiiy, the
founder of crystallography, had already made an attempt to
explain the forms of crystals, while the investigations of Brewster
and the later ones of Senarmont and Grailich, together with
those conducted recently by Mallard and others, have given
us a detailed knowledge of the regular connection between the
physical properties of crystals and the crystal form. As a
consequence of these discoveries the conviction has gradually
made its way that the form of a crystal is solely a consequence
of its interior structure, of its make-up from the smallest
crystal particles, which act on one another with definite forces
depending regularly on the crystallographic direction, and is
therefore a physical property of the substance in question.
Hessel, and later Bravais and Gadolin, independently, suc-
ceeded in determining the entire number of possible crystal
forms by purely geometrical methods; while reasoning based
on the physical properties of crystals leads to exactly the
same results. For the conclusions as to the interior structure of
crystallized media as set forth in the theories of Bravais,
Sohncke, Fedorow, Schonfliess, and others that necessarily
follow on this basis, point to the existence of exactly the same
kinds of symmetry. The consequence is that crystallography



must be regarded as a part of molecular physics; and, since
Voigt has made it seem probable that the so-called "amor-
phous" bodies are to be conceived of as aggregates of very
small crystalline particles, the science may be designated quite
generally as the "molecular physics of solids".

A scientific treatment of this subject, then, can proceed only
hand in hand with the entire physics of crystals. And in its
theoretical aspect the edifice of crystal lore stands at the present
day as one of the best established in the whole realm of physics,
of fundamental importance for an understanding of the
material world.

But not only in theoretical but also in practical respects has
the treatment of crystal science by physical methods attained a
constantly increasing importance. Our complete knowledge of
the regular relation between the optical properties and the
symmetry of crystals has given us the means to carry out, by
optical methods, the determination of crystallized substances in
microscopic preparations with a certainty which, only a few
decades ago, no one would have believed possible. It is com-
monly known what a revolution petrography has experienced in
consequence of this, and how important an aid to the chemist
"crystal- analysis" has become through the work of O. Lehmann,
C. Haushofer, and others; it is well known, too, how fruitful
have been the methods of crystal optics in botanical and his-
tologic-zoological investigations.

Under these circumstances students of natural science,
especially those devoting themselves to chemistry, miner-
alogy, and geology, can no longer be permitted to neglect the
study of crystallography in the sense indicated. The present
text-book presumes an acquaintance with general experimental
physics and chemistry, but with no mathematics beyond what
is afforded by secondary schools; it aims not only to lead the
student to an understanding of the laws to which crystallized
substances are subject, but also to enable him to turn the
methods of the science to their practical application. No


difficulties of any sort should be met with, especially by those
who have attended a lecture on mineralogy and gained thereby
an idea of the most common crystal forms.

In the fourth edition of " Physikalische Krystallographie",
in describing the properties of crystallized bodies in general, a
system has been introduced which makes it possible to place
clearly before beginners the laws governing the dependence of
the crystal properties on the crystallographic direction, in a
steady advance from simple to complex. While in respect of
their optical, thermal, electrical, and magnetic behavior, the
totality oi crystals fall into only Jive groups, this number is
increased by the properties of cohesion and elasticity to seven
and nine respectively; and the behavior of crystals with regard
to solution and growth gives us, finally, all the possible sym-
metry classes of crystals, numbering thirty-two. The most
important part of the subject from a practical standpoint is
that concerned with optics. Here, in consequence of Lord
Kelvin's researches and Fletcher's lucid exposition, the un-
tenable " elasticity" of the ether had already been laid aside
in the third edition. The present treatment of crystal optics
is based on its purely geometrical aspect, suggested by the
latter author. This method makes it possible, yet without
mathematical theory, to gain a correct insight into even the
most complicated phenomena (conical refraction, for example),
something which is indispensable for microscopical studies.
Other changes are limited essentially to particular amplifica-
tions an example of which occurs in the discussion of total
reflection and to making the wording more precise. Excerpts
(with slight adaptations by the translator) from the prefaces to the
third (1895) and fourth (1905) German editions.


THIS partial translation of the fourth edition of Groth's
" Physikalische Krystallographie " is made up chiefly of matter
contained in Part I of the original work, on "The Properties
of Crystals"; besides embracing the general introduction and all
that falls under the heading " Optical Properties " in this
part, it includes, also whatever may be found there on the influ-
ence of other properties on the optical properties. Short extracts
from Parts II ("Systematic Description of Crystals") and III
("The Methods of Crystal Investigation") have been introduced,
on occasion, for illustration and example.

The translation was undertaken at the instance of Professor
Russell D. George, of the University of Colorado; the translator
is indebted to him for his kindly interest and frequent advice,
as well as to Professor Oliver C. Lester for several important

While the author has elsewhere been closely followed, it was
found expedient, in discussing the influence of other properties
on the optical properties, to select the parts to be translated and
sometimes to condense them, as well as to make considerable
changes in their arrangement and classification. The confusing,
tautological expression " direction of " an optic axis, common in
both languages, has been avoided. Additional references and
cross-references have sometimes been given, and a complete table
of abbreviations is appended; any further amplifications by the
translator will be found enclosed in brackets or otherwise

March, 1910.



Abh. Ges. d. Wissensch. Gottingen. Abhandlungen der k. Gesellschaft der Wissen-
schaften zu Gottingen. Berlin.

Bull. soc. franc, de mineral. Bulletin de la Societe francaise de mineralogie.

Carls Repert. f. Exper.-Physik. (Carls) Repertorium fiir Experimental-Physik.

Centralbl. f. Min. Centralblatt fur Mineralogie. Stuttgart.

Leiss: Die op. lustrum. C. Leiss: " Die optischen Instrumente der Firma
Fuess." Leipzig, 1899.

Min. Mag. Mineralogical Magazine (and Journal of the Mineralogical Society).

Nachr. Ges. d. Wissensch. Gottingen. Nachrichten der k. Gesellschaft der Wissen-
schaften zu Gottingen. Gottingen.

Phil. Mag. London, Edinburgh, and Dublin Philosophical Magazine. London.

Phys. Kryst. P. Groth: " Physikalische Krystallographie." Leipzig; 3rd ed.
1895, 4th ed. 1905.

Pogg. Ann. d. Physik. (Poggendorff's) Annalen der Physik. Leipzig.
Proceed. Phys. Soc. Proceedings of the Physical Society. London.

Sitzungsber. Akad. d. Wissensch. Berlin. Sitzungsberichte der k. p. Akademie der
Wissenschaften zu Berlin. Berlin.

Tscher. min. u. petrog. Mitteil. (Tschermak's) mineralogische und petrogra-
phische Mitteilungen. Vienna.

Zeitschr. f. Kryst. Zeitschrift fur Krystallographie. Leipzig.

Zeitschr. f. Kryst. u. Min. (Groth's) Zeitschrift fur Krystallographie und
Mineralogie. Leipzig.




General Introduction to the Properties of Crystals 3


The Nature of Light n

Combination (Interference) of Plane-polarized Light 16

Optically Isotropic (Singly Refracting) Bodies

Propagation of Light 21

Reflection of Light 26

Refraction of Light 30

Polarization of Light by Reflection and Refraction 49

Double Refraction of Light 50

Polarization-colors of Doubly Refracting Crystals 63

Polarization Apparatus 74

Optically Uniaxial Crystals

Double Refraction of Light in Calcite 81

Double Refraction of Light in Other Uniaxial Crystals 101

Behavior of Uniaxial Crystals in the Polarization Apparatus 106

Optically Biaxial Crystals

Deduction of the Optical Properties of Crystals from a Surface of

Reference (Optical Index-surface or Indicatrix) 121

Ray-surface of the Optically Biaxial Crystals 129

Determination of the Principal Refractive Indices of Biaxial Crystals 144

Interference Phenomena of Biaxial Crystals in Parallel Polarized Light 152

Interference Phenomena of Biaxial Crystals in Convergent Polarized Light 161
Determination of the Optic Axes in Biaxial Crystals and Measurement

of their Angle 183

Recapitulation: Classification of Crystals According to their Optical Properties 196

Combinations of Doubly Refracting Crystals

Determination of the Character of the Double Refraction of Uniaxial
and Biaxial Crystals by Combination with Other Doubly Refract-
ing Crystals 198

Optical Behavior of Combinations of Doubly Refracting Crystals of the

Same Kind 211

Rotation of the Polarization Plane of Light in Crystals 220




Absorption of Light in Crystals 232

Brush Phenomena '. 246

Surface Colors 249

Fluorescence 251

Phosphorescence 252

Influence of Other Properties on the Optical Properties of Crystals

Thermal Properties 253

Elastic Strain by Mechanical Forces

Homogeneous Strain 261

Elastic Strain Not Homogeneous 262

Optically Anomalous Crystals 279

Elastic Strain by Electrical Action 282

Permanent Strain

Plasticity 283

Gliding 284

Twinning . 288


Supply Houses for Apparatus, Models, Crystals, and Preparations 293



Page 5, sixth line from bottom of page, for " composition " read
constitution." *

Page 264, next to last line of footnote, for " negatively " read



Absorption of Light in Crystals 232

Brush Phenomena '. 246

Surface Colors 249

Fluorescence 251

Phosphorescence 252

Influence of Other Properties on the Optical Properties of Crystals

Thermal Properties 253

Elastic Strain by Mechanical Forces

Homogeneous Strain 261

Elastic Strain Not Homogeneous

Optically Anomalous Crystals

Elastic Strain H ^'



WHEN a body has the same constitution at all points, so that
any two equal, similar, and similarly oriented parts of it are
undistinguishable from each other by any difference in quality,
that body is said to be homogeneous.

Homogeneous bodies fall into two classes:

1. Bodies in which not only all points, but also all directions,
are equivalent; i.e. in which the different directions are undis-
tinguishable from one another by any physical property of the
body. These bodies are spoken of as amorphous, because they
have no peculiar shape, or as isotropic, because they transmit
every kind of motion in the same way in all directions. Here
belong all gases and vapors, and nearly all liquids; also a num-
ber of so-called "solids", as colloids, resins, glasses. But the
latter bodies are not sharply separated from the liquids; for
example, with an increase of temperature they pass through the
softened, or viscous, state gradually over into the liquid.

2. Homogeneous bodies whose properties depend on the
direction, so that the value of any one property attains in certain
directions a maximum, in others a minimum. (This may be
the case only for particular physical properties, while for others
there may exist, grounded in the nature of the property in ques-
tion, an equality of the value for all directions.) Bodies of this
kind are capable of crystallization, i.e. of assuming a regular
form which is peculiar to the body and which stands in a regular
relation to the non-equivalence of the directions within it
and are therefore called crystallized or crystalline bodies.



The physical properties are distinguished into scalar and

The scalar properties are properties represented by a single
quantity independent of the direction, as temperature, density,
specific heat, etc.

The vector properties are such as are denned by a numerical
value and a direction. When the numerical value is necessarily
the same in the opposite direction, i.e. when the two directions
that pass out from any point and belong to the same straight
line are always absolutely equivalent in respect of a property,
then that property is designated as bi-vector.*

From the foregoing definitions it follows that the amorphous
bodies can possess only scalar properties, while the crystallized,
on the other hand, have, besides these, vector and bi-vector

To set forth the regular relations that exist among the crystal
properties depending on the direction and with those proper-
ties belongs the geometric' form is the object of physical
crystallography. A plausible explanation of the laws of this
branch of science is supplied us by the molecular hypothesis, if
we assume that within crystals, while the molecules are indeed
in motion, yet their motion consists of vibrations about certain
intermediate loci, and that these loci are regularly arranged in
space. | Then the manner of this arrangement, which is known

* Instead of the usual designation "bi-vector", Voigt has proposed the use
of "tensor"; but for the purposes of crystallography the former term seems the
more representative. ,

t In contradistinction from this, for the amorphous bodies there must be
assumed an irregular distribution of the molecules in space, something which for
gases and liquids is at once clear. An amorphous body, according to this second
assumption, is only apparently homogeneous. That is, its unhomogeneousnesses,
because of their too rapid succession within the smallest compass, are no longer
accessible to physical examination; and the latter therefore yields for every
property only a mean value, which, naturally, is found to be the same for all
directions. For this reason crystals have even been designated as the only really
homogeneous bodies, and the terms "crystallized" and "homogeneous" as equiv-


as crystal structure, corresponds to a state of stable equilibrium
among the interior forces. But, since this equilibrium is influ-
enced by the vibrations depending on the heat content of the
body, the arrangement corresponding to the more stable equilib-
rium may for other temperatures be different. In this case the
crystallization under other conditions of temperature and pres-
sure results in an arrangement other than the first ; and if a certain
critical temperature is transgressed there will come to pass a
transformation of the first crystallized body into a second, chem-
ically the same as the first but physically different, just as at the
so-called melting, or freezing, point a transformation takes place
from the crystalline state into the amorphous, or vice versa.
Precisely as the melting, or freezing, point can be overstepped
without transformation, the state of the body then becoming
labile, so too does the same occur in the transformation of the
different crystalline states, or "modifications", into one another.
The property of a substance to present itself in several modifica-
tions is termed polymorphism (dimorphism, trimorphism, etc.),
or, to distinguish it from the chemical isomerism, physical
isomerism. The differences among the polymorphous modifi-
cations of a body exist only in the crystalline state: the transition
into the amorphous state (by fusion or vaporization, and also by
solution) necessarily does away with them.

Since, according to the foregoing assumptions, the crystal
structure of a body depends on the nature of its molecules, there
must exist a regular relation between the crystal structure and

the chemical eem^SBfE&i of the body; to set forth this relation is
the object of chemical crystallography.* This study teaches
that two chemically related bodies can present themselves in
crystalline modifications whose respective structures stand in
close relationship to one another, so that it is possible to deter-
mine the variation in crystal structure that is produced by

* A condensed statement of the subject is given by the author in his "Intro-
duction to Chemical Crystallography ". Authorized translation by Hugh Mar-
shall. Edinburgh and New York, 1906.


a change in the composition of the molecule; such relation-
ships are designated as morpho tropic. They are most intimate
(always supposing that corresponding modifications of the
different substances are taken for the comparison) when it is a
matter of two bodies whose respective molecules differ only in
this: that into the place of an atom in the molecule of the first
body there has entered, in the case of the second, an atom of
different element, which has the same valence as the first ele-
ment and is very closely related to it. In such cases the crystal
structure of the two bodies is so similar that their crystal form
is very nearly the same, under some circumstances even abso-
lutely identical; such crystallized substances are therefore said
to be isomorphic. Isomorphic bodies are capable of crystalliz-
ing together; i.e. of mixing together in various proportions to
form crystals (so-called isomorphic mixtures) which behave,
physically, like homogeneous bodies and whose properties vary
continuously with the composition.

In order to present the regular connection among the crystal
properties depending on the direction, these properties must be
brought into a system from which the individual regularities
follow as a matter of necessity. The basis of this system is
symmetry. When a body is so constituted that the two opposite
directions passing out from any point are absolutely equivalent,
we say it has a "center of symmetry"; therefore the bi-vector
properties are designated also as "centrally symmetrical", and,
in contradistinction from them, the vector as "acentric".

The bi-vector properties are further distinguished into those
of higher symmetry and those of lower symmetry.

The former have in common that for any and every direction
their numerical value is determined by three, at the most,
numerical quantities, which refer respectively to three definite,
mutually perpendicular directions; and the numerical value of
a definite property of this kind, for any direction, is proportional
to the radius vector corresponding to that direction, of a triaxial
ellipsoid whose three semi-axes are proportional to the three


numerical quantities holding good with the property. The
"bi-vector properties of higher symmetry" are therefore called
also ellipsoidal properties. In the special case where the throe
principal axes of the ellipsoid are of equal length, the ellipsoid
passes over into a sphere; that is, the numerical value of the
property in question is independent of the direction, like that of
a scalar property. For this reason it is with the ellipsoidal
properties that the relations are simplest, wherefore the pres-
entation of the subject begins most properly with them; and, of
them, with the optical properties. These, on account of their
practical importance, will not only be developed from the funda-
mental ideas, but also treated so far in detail as is demanded by
the practical application of optical methods to the determination
of crystalline forms in chemical crystallography, mineralogy,
petrography, and other natural sciences. The remaining ellip-
soidal properties, i.e. the thermal, the electrical, and the magnetic,
are so absolute analogous to the optical that they need be
presented only in brief.*

With the properties of elasticity and cohesion of crystals the
relations are more complex: the dependence of these prop-
erties on the direction cannot be represented by a surface of
such simple form as an ellipsoid or a sphere, but only by one
of more complicated form and of an in general lesser degree
of symmetry. These properties shall therefore be distinguished
from those of higher symmetry, the ellipsoidal properties, as
properties of lower symmetry.

The lowest degree of symmetry, finally, is possessed by the
vector properties, the properties in respect of which even the
two opposite directions pertaining to the same straight line are
not necessarily equivalent. Among these crystal properties
belong those of solution and growth; the latter are the propert'es
in virtue whereof the crystal assumes its definite geometric form.

Those bi-vector and those vector properties with which it is

* [This refers of course to the original work, as the present translation deals
only with the optical properties.]


a matter of the action of mechanical forces on the crystal, are of
especially high theoretical significance for the reason that, with
the crystal under such conditions, the forces must be overcome
that are exerted by its smallest particles on one another. The
consideration of these properties therefore leads to that of
the causes of crystal structure, and thus to the consideration of
these particles and forces themselves. For treating the theories
of crystal structure that come into consideration there are
requisite certain conceptions from geometry*; and these con-
ceptions are of additional importance for the reason that, being
applicable likewise to the theories respecting the configuration
of the atoms in the molecules, they may be considered as the
basis of stereochemical views. These conceptions shall there-
fore first of all be elucidated, and then, with their aid, all the
laws deduced that govern the geometric form of crystals; while
the particulars of the several classes of symmetry, which
follow from all those general laws, and the description of a
number of crystallized bodies forming especially important or
interesting examples of the several classes, are reserved for
Part II.|

* According to this, geometry is an auxiliary science to crystallography, not
the latter a part of the former; for geometry deals with the form, crystallography
with the contents of the same, i.e. with the material of the crystal as carrier of
the properties, among which the form belongs.

t ** Systematische Beschreibung der Krystalle." (Cf. footnote on p. 7.)




To explain the properties of light we assume it is a periodic
motion of the smallest particles of the luminiferous ether, a form
of matter which pervades universal space and likewise all bodies,
but which within the latter, under the influence of ponderable
matter, takes on certain peculiarities.

If we imagine the ether at rest, i.e. the forces acting among
its particles in equilibrium, and if by an impulse a particle be

Online LibraryP. (Paul) GrothThe optical properties of crystals, with a general introduction to their physical properties; being selected parts of the Physical crystallography → online text (page 1 of 28)