Chapter 21: Crystal Structure and Reciprocal Lattice (Set-2)
A conventional unit cell is mainly chosen because it
A Has one lattice point
B Minimizes cell volume
C Removes basis atoms
D Shows full symmetry
Conventional cells are selected to display the lattice symmetry clearly, even if they contain more than one lattice point. Primitive cells are minimal, but may hide symmetry.
In a cubic system, the unit cell angles are
A 90° each
B 60° each
C 120° each
D Unequal angles
Cubic crystals have α = β = γ = 90° and equal edges a = b = c. This high symmetry is why many planes and directions become equivalent.
A crystal system with a = b ≠ c and all angles 90° is
A Orthorhombic system
B Tetragonal system
C Hexagonal system
D Triclinic system
Tetragonal crystals have two equal axes and one different axis, with all angles 90°. This differentiates it from cubic (all equal) and orthorhombic (all unequal).
A crystal system with a ≠ b ≠ c and all angles 90° is
A Monoclinic system
B Rhombohedral system
C Hexagonal system
D Orthorhombic system
Orthorhombic crystals have three unequal axes but all angles remain 90°. This simple angle condition makes many calculations similar to cubic but with different spacings.
The basic repeating geometric arrangement of lattice points is a
A Crystal basis
B Structure factor
C Bravais lattice
D Ewald sphere
A Bravais lattice is a periodic arrangement of equivalent lattice points. Attaching a basis (group of atoms) to each point produces the full crystal structure.
In simple cubic, the nearest-neighbor distance equals
A a/2 value
B √2 a length
C √3 a length
D a edge length
In simple cubic, nearest neighbors lie along the cube edges. The distance between adjacent corner atoms is the edge length a.
In BCC, the nearest-neighbor distance is
A √3 a/2
B a/2 value
C √2 a/2
D a edge length
In BCC, nearest neighbors connect a corner to the body-centered atom. That distance is half the body diagonal: (√3 a)/2.
In FCC, the nearest-neighbor distance is
A a/2 value
B √2 a/2
C √3 a/2
D √3 a value
In FCC, nearest neighbors connect a corner to the face center. That distance is half the face diagonal: (√2 a)/2.
A close-packed plane in FCC is the
A (100) plane
B (110) plane
C (200) plane
D (111) plane
The densest plane in FCC is (111). Atoms in this plane form a hexagonal arrangement, giving maximum planar packing and strong slip behavior in metals.
The stacking sequence for FCC close-packed planes is
A ABCABC stacking
B ABAB stacking
C AABB stacking
D ABCC stacking
FCC has ABCABC stacking of close-packed layers. HCP has ABAB stacking; both are close packed but differ in layer order.
The packing fraction of BCC is approximately
A 0.52 value
B 0.74 value
C 0.68 value
D 0.80 value
BCC is more efficiently packed than simple cubic but less than close-packed structures. Its packing fraction is about 0.68, while FCC/HCP reach 0.74.
In XRD, the “2θ” value is the
A Bragg angle
B Plane tilt angle
C Crystal axis angle
D Scattering angle
Many diffractometers measure the angle between incident and diffracted beams, which is 2θ. Bragg’s law uses θ, the angle to the lattice planes.
When lattice constant increases, d(hkl) for fixed (hkl)
A Decreases always
B Increases directly
C Becomes zero
D Stays unchanged
For cubic crystals, d = a/√(h²+k²+l²). If a increases while indices remain same, spacing between those planes increases proportionally.
For cubic crystals, the plane (200) has spacing compared to (100)
A Half spacing
B Same spacing
C Double spacing
D Triple spacing
For (200), √(h²+k²+l²)=2, so d200 = a/2. For (100), d100 = a. Hence (200) planes are twice as closely spaced.
The direction [111] in cubic crystals points along
A Cube edge line
B Face diagonal line
C Plane normal only
D Body diagonal line
[111] connects opposite corners through the cube center, which is the body diagonal direction. It is important for close-packed directions and symmetry.
The direction [110] in a cubic cell is along
A Cube edge line
B Body diagonal line
C Face diagonal line
D Face normal line
[110] lies in a face of the cube and runs from one corner to an adjacent corner across the face. That is the face diagonal direction.
A zone axis is best described as a
A Common direction
B Family of planes
C Diffraction peak set
D Unit cell edge
A zone axis is a crystallographic direction common to a set of planes. In diffraction, it helps describe crystal orientation and relationships between plane families.
Interplanar spacing depends mainly on
A Atomic number only
B Temperature only
C Detector distance
D Lattice geometry
d-spacing is determined by lattice constants and plane indices. It is a geometric property of the crystal, and it sets the Bragg angles for diffraction peaks.
In reciprocal space, larger real-space spacing d means
A Larger |G|
B Smaller |G|
C Same |G| always
D Random |G| only
Reciprocal lattice vector magnitude is proportional to 1/d. So widely spaced real planes correspond to smaller reciprocal vectors, appearing closer to the origin in k-space.
The Laue condition for diffraction is essentially
A Δk equals G
B G = k only
C k equals zero
D d equals lambda
Diffraction occurs when the change in wavevector (scattering vector) equals a reciprocal lattice vector G. This condition is equivalent to Bragg’s law in reciprocal space form.
Bragg’s law is a statement of
A Destructive interference
B Thermal vibration
C Constructive interference
D Crystal defects
Bragg’s law gives the angles where waves reflected from successive planes add in phase. That constructive interference creates strong peaks in diffraction patterns.
For fixed d and θ, increasing order n requires
A Smaller wavelength
B Same wavelength
C Random wavelength
D Larger wavelength
From nλ = 2d sinθ, if d and θ are fixed, λ must increase with n. Higher-order reflections can also appear at different angles for fixed λ.
In powder diffraction, peak positions mainly depend on
A d-spacings
B Structure factor
C Grain boundaries
D Polarization only
Peak positions (2θ values) are determined by Bragg’s law, which depends on d-spacing. Intensity depends on structure factor and other factors, not the position.
Peak intensity is strongly affected by
A Lattice constant only
B Structure factor
C Wavelength only
D Bragg angle only
Structure factor sets how scattered waves from atoms add or cancel. Even if a reflection satisfies Bragg’s condition, its intensity can be weak or zero due to cancellation.
The atomic form factor generally decreases as
A Temperature decreases
B Grain size increases
C Density increases
D Angle increases
At higher scattering angles, the phase differences across the electron cloud increase, reducing coherent scattering. So atomic form factor typically drops with increasing sinθ/λ.
Systematic absences are most useful to identify
A Lattice type
B Crystal color
C Crystal density
D Thermal expansion
Absence rules (like BCC and FCC selection rules) reveal centering and symmetry. By checking missing reflections, one can distinguish between SC, BCC, FCC, and other lattices.
In BCC, the (100) reflection is
A Always strongest
B Always present
C Systematically absent
D Temperature dependent
For BCC, reflections require h+k+l even. For (100), sum is 1 (odd), so the structure factor cancels and the reflection is absent.
In FCC, the (100) reflection is
A Allowed peak
B Only at high T
C Only at low T
D Absent peak
FCC reflections occur only when h, k, l are all even or all odd. (100) has mixed parity, so it is systematically absent.
In FCC, the first allowed reflection is usually
A (111) reflection
B (100) reflection
C (110) reflection
D (200) reflection
FCC selection rules remove (100) and (110). The lowest-index allowed plane is (111), so it commonly appears as the first strong peak in FCC powder patterns.
In BCC, the first allowed reflection is usually
A (100) reflection
B (110) reflection
C (111) reflection
D (200) reflection
In BCC, (100) is forbidden because h+k+l is odd. (110) has sum 2 (even), so it becomes the first allowed reflection.
The Lorentz-polarization factor mainly corrects
A Peak positions
B d-spacing values
C Lattice constant
D Measured intensities
Instrument geometry and polarization of X-rays affect recorded intensity across angles. Lorentz-polarization factor applies corrections so intensities better reflect crystal scattering power.
A “basis” in crystallography means
A Atom group attached
B Lattice point grid
C Reciprocal vectors set
D Bragg planes set
A basis is the set of atoms associated with each lattice point. Lattice plus basis gives the actual crystal structure, like NaCl having two-atom basis on an FCC lattice.
The reciprocal lattice points correspond to
A Real lattice points
B Crystal defects only
C Allowed diffraction spots
D Grain boundaries
Each reciprocal lattice point represents a set of real-space planes. When scattering conditions match a reciprocal point, a diffraction spot or peak appears.
The first Brillouin zone contains all k-points
A Farthest from origin
B On zone boundary
C Outside reciprocal space
D Closest to origin
The first Brillouin zone is the set of points closer to k=0 than to any other reciprocal lattice point. It is the basic cell for wavevectors in periodic solids.
The Γ point in band diagrams represents
A Zone corner point
B Zone center point
C Edge midpoint only
D Random symmetry point
Γ is the center of the Brillouin zone where k = 0. Many band structures are plotted along symmetry directions starting at Γ.
A simple way to build Brillouin zones is using
A Wigner–Seitz method
B Real space cell
C Scherrer equation
D Miller direction rule
Construct perpendicular bisectors between origin and nearby reciprocal points. The region nearer to origin than any other point forms the first Brillouin zone.
Grain boundaries are categorized as
A Point defects
B Line defects
C Volume defects
D Planar defects
Grain boundaries are interfaces between crystals of different orientations. They are two-dimensional defects that strongly affect strength, diffusion, and electrical behavior.
Broadening of XRD peaks can be caused by
A Large crystal size
B Small crystallite size
C Perfect lattice only
D Zero strain only
Small crystallites lead to limited coherence length, widening diffraction peaks. Microstrain can also broaden peaks, but Scherrer broadening is a classic size effect.
If microstrain increases in a sample, XRD peaks typically
A Broaden more
B Sharpen more
C Disappear always
D Shift to zero
Microstrain causes slight variations in d-spacing across crystallites. This spreads Bragg angles around the ideal value, resulting in broadened peaks.
Phonons are quanta of
A Electron motion
B X-ray photons
C Defect diffusion
D Lattice vibrations
Phonons represent quantized normal modes of lattice vibration. They explain heat capacity behavior, thermal conductivity, and temperature effects like Debye–Waller intensity reduction.
Thermal expansion occurs mainly because
A Miller indices change
B Bragg angle increases
C Interatomic potential asymmetry
D Density always rises
At higher temperature, atoms vibrate with larger amplitude. Because the interatomic potential is not perfectly symmetric, the average separation increases, causing lattice expansion.
Crystal anisotropy means properties depend on
A Direction in crystal
B Only temperature
C Only atomic number
D Only grain size
In crystals, bonding and spacing vary with direction, so properties like conductivity, elasticity, and refractive index can change depending on crystallographic orientation.
A symmetry operation that leaves a crystal unchanged can be
A Random translation
B Electron scattering
C Grain boundary slip
D Rotation about axis
Rotations by specific angles around symmetry axes can map the lattice onto itself. Along with translations and reflections, these operations define the crystal’s symmetry.
Indexing diffraction peaks mainly means assigning
A Crystal colors
B (hkl) values
C Unit cell density
D Debye factor
Indexing is matching each observed peak to a set of planes (hkl). Using d-spacings and selection rules, one can identify lattice type and calculate lattice constants.
For cubic crystals, a common indexing check is using
A h+k+l only
B Atomic number ratios
C (h²+k²+l²) ratios
D Density ratios
In cubic systems, 1/d² ∝ (h²+k²+l²)/a². So ratios of 1/d² values match integer sums, helping assign correct (hkl) indices.
In reciprocal space, Bragg reflection corresponds to
A k equals zero
B Zone folding only
C Density matching
D Ewald intersection
Diffraction occurs when a reciprocal lattice point lies on the Ewald sphere surface. This geometric intersection gives the same condition as Bragg’s law.
The term “k-space” is another name for
A Reciprocal space
B Real space lattice
C Defect space
D Grain space
k-space describes wavevectors of electrons or waves in a periodic lattice. It is essentially reciprocal space and is central to diffraction and band structure analysis.
A reflection labeled (hkl) comes from planes that are
A Parallel to [hkl]
B Randomly oriented
C Always close packed
D Perpendicular to [hkl]
For cubic crystals, the direction [hkl] is normal to the plane (hkl). So the planes producing that reflection are perpendicular to the corresponding direction.
The basis causes different crystals to have
A Same intensities
B Different intensities
C Same selection rules
D Same packing always
Even with the same lattice, different basis atoms and positions change the structure factor. That changes which peaks are strong or weak, giving distinct diffraction fingerprints.
A key practical use of XRD is to
A Identify crystal phase
B Measure atomic radius only
C Remove grain boundaries
D Create unit cells
XRD peak positions give d-spacings and lattice parameters, while intensities reflect structure factor. Together they identify crystal structure and phases in unknown materials reliably