The document discusses key concepts regarding solid states and crystallography. It defines crystalline and amorphous solids, and describes the ordered crystal lattice structure of crystalline solids. It then outlines three fundamental laws of crystallography: 1) the law of constancy of interfacial angles, 2) Hauy's law of rationality of indices, and 3) the law of constancy of symmetry. The document also discusses techniques for determining crystal structures, including Bragg's law of diffraction and X-ray diffraction methods like the Laue method and powder method.

1. TEJASVI NAVADHITAMASTU
“Let our (the teacher and the taught) learning be radiant”
Let our efforts at learning be luminous and filled with joy, and endowed with the force of purpose
Paper III: PHYSICAL CHEMISTRY
Dr. Prabhakar Singh. D.Phil. Biochemistry
Department of Biochemistry, VBSPU, Jaunpur
Unit-III: A. Solid States: Definition of space lattice, unit cell; Laws of crystallography - (i) Law
of constancy of interfacial angles, (ii) Law of rational of indices (iii) Law of symmetry,
Symmetry elements in crystals. X-ray diffraction by crystals, Derivation of Bragg’s equation,
Determination of crystal structure of NaCI, KCI and CsCl, Laue's method and powder method.

2. Gases and liquids can flow and take up the shape of their container. Solids, on
the other hand, have a definite volume and shape. They are rigid and lack the
ability to flow.
In both gases and liquids, atoms, ions and molecules continually move. They
translate randomly as well as rotate and vibrate. This determines the ability of
gases and liquids to flow. In solids, atoms, ions and molecules are held together
by relatively strong chemical forces-ionic bond, covalent bond, or by
intermolecular van der Waals’ forces. They do not translate although they
vibrate to some extent in their fixed positions. This explains why solids are rigid
and have definite shape.
TYPES OF SOLIDS
Broadly solids are of two types :
(a) Crystalline solids; also called true solids
(b) Amorphous solids

3. A crystalline solid exists as small crystals, each crystal having a characteristic geometrical shape.
In a crystal, the atoms, molecules or ions are arranged in a regular, repeating three dimensional
pattern called the crystal lattice. Sugar and salt are crystalline solids.
An amorphous solid (Gr amorphous = no
form) has atoms, molecules or ions
arranged at random and lacks the
ordered crystalline lattice. Examples are
rubber, plastics and glass. In their
disordered structure, amorphous solids
resemble liquids. Thus glasses are to be
regarded as super-cooled or highly
viscous liquids. The liquid nature of glass
is sometimes apparent in very old
window panes that have become slightly
thicker at the bottom due to gradual
downward flow.

7. LAWS OF CRYSTALLOGRAPHY
The geometric crystallography is based on the three fundamental laws::----
1)The laws of constancy of interfacial angles: the crystals of a substance
can have different shapes depending upon the number and size of the faces
but the angle at which the two adjacent faces intersect remains always
constant.
2) Hauy's law of rationality of indices: the intercepts of any face of plane
of a crystal on suitable crystallographic aces can be expressed by small
multiples of three unit distances a,b,c or some simple integral multiple m,n,p
of these unit distances , i.e; ma, nb, pc or fraction of whole numbers.
3) The law of constancy of symmetry: according to this law, all crystals of
the same substance possess the same elements of symmetry.

8. 1. THE LAWS OF CONSTANCY OF INTERFACIAL ANGLES
The law of the constancy of interfacial angles (or 'first law of crystallography') states that the
angles between the crystal faces of a given species are constant, whatever the lateral extension
of these faces and the origin of the crystal, and are characteristic of that species. It paved the
way for Haüy's law of rational indices.

9. The law of rational indices states that the intercepts, OP, OQ, OR, of
the natural faces of a crystal form with the unit-cell axes a, b, c are
inversely proportional to prime integers, h, k, l. They are called the
Miller indices of the face. They are usually small because the
corresponding lattice planes are among the densest and have
therefore a high interplanar spacing and low indices.
The law of rational indices was deduced by Haüy
(1784, 1801) from the observation of the stacking laws
required to build the natural faces of crystals by piling
up elementary blocks, for instance cubes to construct
the {110} faces of the rhomb-dodecahedron observed
in garnets or the ½{210} faces of the pentagon-
dodecahedron observed in pyrite, or rhombohedrons
to construct the {21.1} (referred to an hexagonal
lattice, {21¯0} , referred to a rhombohedral lattice)
scalenohedron of calcite.
2. HAUY'S LAW OF RATIONALITY OF INDICES

11. 3. THE LAW OF CONSTANCY OF SYMMETRY
According to this law, all crystals of the same substance possess the same
elements of symmetry.

13. X-RAY DIFFRACTION

19. SODIUM CHLORIDE CRYSTAL

20. POTASSIUM CHLORIDE (KCl)

23. CESIUM CHLORIDE CRYSTAL

24. LAUE METHOD
The Laue method is mainly used to determine the orientation of large
single crystals. White radiation is reflected from, or transmitted
through, a fixed crystal.
The diffracted beams form arrays of spots, that lie on curves on the
film. The Bragg angle is fixed for every set of planes in the crystal. Each
set of planes picks out and diffracts the particular wavelength from the
white radiation that satisfies the Bragg law for the values of d and q
involved. Each curve therefore corresponds to a different wavelength.
The spots lying on any one curve are reflections from planes belonging
to one zone. Laue reflections from planes of the same zone all lie on
the surface of an imaginary cone whose axis is the zone axis.
There are two practical variants of the Laue method-
1. THE BACK-REFLECTION AND
2. THE TRANSMISSION LAUE METHOD

25. BACK-REFLECTION LAUE
In the back-reflection method, the film is placed between the
x-ray source and the crystal. The beams which are diffracted
in a backward direction are recorded.
One side of the cone of Laue reflections is defined by the
transmitted beam. The film intersects the cone, with the
diffraction spots generally lying on an hyperbola.
TRANSMISSION LAUE
In the transmission Laue method, the film is placed behind the
crystal to record beams which are transmitted through the
crystal.
One side of the cone of Laue reflections is defined by the
transmitted beam. The film intersects the cone, with the
diffraction spots generally lying on an ellipse
Crystal orientation is determined from the position of the spots. Each spot can be indexed, i.e. attributed
to a particular plane, using special charts. The Greninger chart is used for back-reflection patterns and
the Leonhardt chart for transmission patterns.
The Laue technique can also be used to assess crystal perfection from the size and shape of the spots. If
the crystal has been bent or twisted in anyway, the spots become distorted and smeared out.

26. THE POWDER METHOD
The rotating crystal method could only
be used if a single undistorted crystal is
available. To overcome this limitation,
the powder method was devised. In
this method the crystalline material
contained in a capillary tube is placed
in the camera containing a film strip.
The sample is rotated by means of a
motor. The X-rays pass through the gap
between the ends of the film.
The powdered sample contains small crystals
arranged in all orientations. Some of these will
reflect X-rays from each lattice plane at the same
time. The reflected X-rays will make an angle 2θ
with the original direction. Hence on the photo
are obtained lines of constant θ. From the
geometry of the camera, θ can be calculated for
different crystal planes.