Chapter 21: Crystal Structure and Reciprocal Lattice (Set-1)
In a crystal, the smallest repeating volume that builds the entire lattice is called
A Atomic radius value
B Grain boundary region
C Unit cell volume
D Defect cluster zone
A unit cell is the smallest repeating 3D block. By translating it along lattice vectors, the whole crystal structure is generated without changing the pattern.
A unit cell containing lattice points only at the eight corners is a
A Body centered cubic
B Face centered cubic
C Hexagonal close packed
D Simple cubic lattice
In a simple cubic cell, lattice points exist only at corners. Each corner point is shared by 8 cells, giving 1 atom per unit cell effectively.
The crystal structure that has an atom at the body center in addition to corners is
A Body centered cubic
B Face centered cubic
C Simple cubic lattice
D Rhombohedral lattice form
BCC has corner atoms plus one full atom at the center. This changes packing and diffraction selection rules compared to simple cubic.
A cubic unit cell with atoms at all corners and face centers is
A Simple cubic lattice
B Body centered cubic
C Face centered cubic
D Primitive tetragonal cell
FCC includes corner atoms and one atom at each face center. This gives 4 atoms per unit cell and a high packing efficiency.
The coordination number of a simple cubic structure is
A 8 nearest neighbors
B 6 nearest neighbors
C 12 nearest neighbors
D 4 nearest neighbors
In simple cubic, each atom touches along ±x, ±y, ±z directions. So it has 6 nearest neighbors at equal distance.
The coordination number of a BCC structure is
A 6 nearest neighbors
B 12 nearest neighbors
C 10 nearest neighbors
D 8 nearest neighbors
In BCC, a corner atom’s closest atoms are the body-centered atoms of surrounding cubes. This gives 8 equidistant nearest neighbors.
The coordination number of an FCC structure is
A 6 nearest neighbors
B 8 nearest neighbors
C 12 nearest neighbors
D 14 nearest neighbors
FCC has very dense packing. Each atom is surrounded by 12 nearest neighbors, which is why FCC metals often show high ductility.
The packing fraction is highest for
A Face centered cubic
B Simple cubic structure
C Body centered cubic
D Simple tetragonal cell
FCC is a close-packed arrangement with packing fraction 0.74. Simple cubic is 0.52 and BCC is about 0.68, so FCC is highest.
HCP and FCC structures share the same packing fraction because they are
A Both primitive lattices
B Both body centered
C Both low symmetry
D Both close packed
HCP and FCC are both close-packed structures with identical packing efficiency (0.74). They differ only in stacking sequence of close-packed planes.
The number of atoms per FCC unit cell is
A 1 atom per cell
B 2 atoms per cell
C 4 atoms per cell
D 6 atoms per cell
FCC has 8 corner atoms contributing 1 total plus 6 face atoms contributing 3 total. Together, it gives 4 atoms per unit cell.
The number of atoms per BCC unit cell is
A 1 atom per cell
B 2 atoms per cell
C 3 atoms per cell
D 4 atoms per cell
BCC has 8 corners contributing 1 atom in total and one full atom at body center. Hence total atoms per unit cell are 2.
For a cubic crystal, the lattice constant is the
A Angle between axes
B Face diagonal length
C Edge length a
D Body diagonal length
In cubic systems, all edges are equal and all angles are 90°. The lattice constant “a” is the unit cell edge length.
Density of a crystal mainly depends on
A Packing fraction only
B Temperature factor only
C Diffraction angle only
D Atoms per cell
Density uses mass in one unit cell divided by unit cell volume. Mass depends on how many atoms are effectively inside the unit cell.
A primitive cell is defined as the unit cell that contains
A Minimum lattice points
B Maximum atoms only
C Only face atoms
D Only edge atoms
A primitive cell is the smallest volume that still tiles space and contains exactly one lattice point. Conventional cells may contain more for symmetry convenience.
In Bravais lattice concept, lattice points represent
A Different atoms always
B Only defects sites
C Equivalent environments
D Only surface atoms
In a Bravais lattice, every lattice point has identical surroundings. The “basis” attached to each lattice point may contain one or more atoms.
Miller indices (hkl) of a plane are found from
A Plane normal angle
B Intercepts reciprocals
C Unit cell density
D Atomic form factor
Take the plane intercepts on axes in units of lattice constants, invert them, and clear fractions to smallest integers. Those integers are the Miller indices.
If a plane is parallel to the y-axis, its y-intercept is
A 0 always
B 1 always
C Negative one
D Infinity value
Parallel to an axis means it never cuts that axis, so intercept is ∞. The reciprocal becomes 0, making the corresponding Miller index zero.
In cubic crystals, the spacing d(hkl) is given by
A √(h²+k²+l²)/a
B a(h+k+l)
C a/√(h²+k²+l²)
D a²/(hkl)
For cubic crystals, interplanar spacing depends on lattice constant and Miller indices. Larger (hkl) values mean planes are closer, reducing d.
A bar over an index like (1̄ 0 1) indicates
A Negative intercept
B Double plane spacing
C Zero direction component
D Orthogonal plane only
A barred index means the plane intercept on that axis is on the negative side. It is a standard notation for negative Miller indices.
The notation [uvw] generally represents
A Plane family set
B Brillouin zone
C Diffraction intensity
D Crystal direction
[uvw] are direction indices along a crystallographic direction. They describe a vector in the lattice, unlike (hkl) which labels lattice planes.
The notation {hkl} usually represents
A One specific direction
B One lattice constant
C Family of planes
D One defect type
{hkl} denotes all symmetry-equivalent planes in a crystal. For cubic crystals, many planes share the same spacing and properties due to symmetry.
In cubic crystals, planes (100), (010), (001) belong to
A Same plane only
B {100} family
C {111} family
D {110} family
These planes are equivalent by cubic symmetry (just rotated). Hence they are grouped as the {100} family and have equal interplanar spacing.
The reciprocal lattice is useful because it represents
A Diffraction conditions
B Real atom sizes
C Crystal color only
D Defect density only
Diffraction is naturally expressed using reciprocal space vectors. Bragg conditions and scattering vectors match reciprocal lattice points, simplifying XRD and electron diffraction analysis.
A reciprocal lattice vector is perpendicular to
A Crystal edge only
B Grain boundary only
C Crystal defect line
D Real lattice plane
Each reciprocal lattice vector corresponds to a set of real-space planes. Its direction is normal to those planes, and its magnitude relates to plane spacing.
For a simple cubic real lattice, the reciprocal lattice is
A Always BCC type
B Always FCC type
C Also simple cubic
D Always HCP type
A simple cubic lattice in real space produces a simple cubic reciprocal lattice, with reciprocal lattice constant 2π/a (using physics convention).
For a BCC real lattice, the reciprocal lattice is
A FCC reciprocal lattice
B Simple cubic lattice
C BCC reciprocal lattice
D Hexagonal lattice form
BCC and FCC are reciprocal pairs. This is why diffraction peak selection rules differ, and why reciprocal space helps identify the underlying real lattice type.
For an FCC real lattice, the reciprocal lattice is
A FCC reciprocal lattice
B Simple cubic lattice
C Orthorhombic lattice
D BCC reciprocal lattice
FCC and BCC are reciprocal to each other. This relationship is widely used in crystallography and band theory when constructing Brillouin zones.
The Ewald sphere construction is mainly used to
A Measure crystal density
B Compute packing fraction
C Visualize diffraction
D Find coordination number
Ewald sphere gives a geometric way to see when diffraction occurs. A reciprocal lattice point lying on the sphere surface satisfies the scattering condition.
The scattering vector magnitude is closely related to
A Crystal temperature only
B Reciprocal lattice spacing
C Grain size only
D Atomic number only
The scattering vector connects incident and diffracted wavevectors. Its values match reciprocal lattice vectors for allowed reflections, linking measured angles to lattice geometry.
Bragg’s law for constructive interference is
A nλ = d cosθ
B nλ = 2d cosθ
C nλ = d sinθ
D nλ = 2d sinθ
Bragg’s law states that path difference between waves reflected from successive planes equals an integer multiple of wavelength. This gives sharp diffraction maxima at specific angles.
In Bragg’s law, θ is the
A Scattering half-angle
B Crystal tilt angle
C Bragg angle
D Detector rotation only
θ is the angle between the incident beam and the crystal planes. Many XRD setups measure 2θ, but Bragg’s law uses θ.
Increasing the order n in Bragg reflection generally
A Requires larger angle
B Increases plane spacing
C Reduces wavelength needed
D Changes lattice constant
For fixed d and λ, larger n demands larger sinθ, so θ increases until physically possible. Higher orders may be weak or absent due to intensity factors.
Powder XRD is useful mainly because it
A Needs single crystal
B Avoids Bragg condition
C Ignores lattice planes
D Uses random orientations
In powder samples, tiny crystallites are oriented randomly. Some grains always satisfy Bragg condition for each set of planes, producing rings/peaks for phase identification.
If X-ray wavelength is decreased, the Bragg angle for same planes
A Increases always
B Becomes zero always
C Decreases generally
D Becomes random only
From nλ = 2d sinθ, smaller λ means smaller sinθ for fixed d and n. So θ decreases, shifting peaks to lower angles.
The “atomic scattering factor” mainly depends on
A Crystal grain size
B Electron distribution
C Miller indices only
D Brillouin zone only
Atomic scattering factor describes how an atom scatters X-rays due to its electron cloud. It decreases with higher scattering angle because electrons are spread out spatially.
The Debye–Waller factor is linked to
A Miller index sign
B Unit cell volume
C Coordination number
D Lattice vibrations
Thermal motion reduces coherent scattering intensity. Debye–Waller factor accounts for this temperature effect, making high-angle reflections weaker as vibrations increase.
The structure factor determines mainly the
A X-ray wavelength range
B Crystal density only
C Peak intensity pattern
D Bragg angle only
Structure factor sums scattering from atoms in the basis with phase differences. It controls which reflections are strong, weak, or absent, shaping the diffraction intensity pattern.
Systematic absences occur in diffraction because of
A Basis interference
B Detector noise only
C Grain boundary scattering
D Thermal expansion only
For certain (hkl), phase cancellation between atoms in the basis makes structure factor zero. Then reflections are absent even though Bragg condition could be satisfied.
For BCC crystals, allowed reflections satisfy
A h+k+l odd
B h+k+l even
C h, k, l all odd
D h, k, l all even
In BCC, the body-center atom introduces a phase factor. If h+k+l is odd, cancellation occurs and intensity becomes zero; even sums give allowed reflections.
For FCC crystals, allowed reflections satisfy
A Mixed odd-even indices
B Only h+k+l even
C All odd or even
D Only h, k zero
FCC basis causes cancellation unless h, k, l are all even or all odd. Mixed parity reflections are systematically absent, which helps identify FCC structures from XRD.
The multiplicity factor in powder diffraction accounts for
A X-ray absorption only
B Thermal vibration only
C Unit cell density
D Multiple equivalent planes
Many different planes have the same d-spacing due to symmetry, contributing to the same peak. Multiplicity counts these equivalent planes, affecting observed peak intensity.
The first Brillouin zone is defined as the
A Smallest real cell
B Defect-free region
C Wigner–Seitz in k
D Maximum intensity peak
The first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice. It contains points in k-space closer to the origin than any other reciprocal point.
Brillouin zone boundaries are formed by planes that are
A Perpendicular bisectors
B Parallel to real planes
C Same as Miller planes
D Always at k = 0
Zone boundaries are perpendicular bisector planes between the origin and nearby reciprocal lattice points. They mark where Bragg reflection of electron waves can occur.
Energy band gaps in solids are strongly linked to
A Real-space density
B Crystal color only
C Defect concentration only
D Zone boundary diffraction
At Brillouin zone boundaries, electron waves can satisfy Bragg-like conditions, causing standing waves and splitting energies. This creates band gaps central to band theory.
The reduced zone scheme is mainly used to
A Measure XRD wavelength
B Plot bands within first zone
C Compute crystal density
D Find Miller indices
Reduced zone scheme folds all k-states into the first Brillouin zone. It helps compare band energies within a single zone and visualize gaps clearly.
A vacancy defect means
A Extra atom inserted
B Shifted grain boundary
C Missing lattice atom
D Crystal becomes amorphous
Vacancy is a point defect where an atom is absent from its normal lattice site. Vacancies increase with temperature and influence diffusion and electrical properties.
An interstitial defect means
A Missing atom in lattice
B Only surface imperfection
C Only edge dislocation
D Extra atom between sites
Interstitial defect occurs when an atom occupies a space between regular lattice positions. It distorts nearby atoms and can strongly affect mechanical strength and diffusion.
A dislocation is best described as
A Line defect in crystal
B Point defect only
C Plane defect in lattice
D Volume defect region
Dislocations are line defects where atomic planes are misaligned. They allow plastic deformation at much lower stresses than would be needed for perfect crystal slip.
The Debye–Scherrer method is a
A Single crystal rotation
B Powder diffraction method
C Optical microscopy method
D Density measurement method
Debye–Scherrer uses powdered crystals and X-rays to produce diffraction rings/peaks. It is widely used for phase identification and lattice parameter calculation.
The Scherrer equation is mainly used to estimate
A Lattice constant a
B Coordination number
C Crystallite size
D Atomic form factor
Peak broadening in XRD can come from small crystallite size. The Scherrer equation relates peak width to average crystallite size (approximate), assuming strain broadening is small