Chapter 21: Crystal Structure and Reciprocal Lattice (Set-5)
In BCC, the planar atomic density is highest on which plane
A (100) plane
B (110) plane
C (111) plane
D (210) plane
For BCC, the densest plane is (110) because atoms pack most closely in that plane. This is why BCC slip often involves {110} planes and related directions.
In FCC, the planar atomic density is highest on which plane
A (100) plane
B (110) plane
C (210) plane
D (111) plane
FCC has close-packed {111} planes. Atoms form a hexagonal pattern in (111), giving maximum planar density and making slip easiest along {111} planes.
In FCC, the close-packed direction within the close-packed plane is
A ⟨100⟩ direction
B ⟨111⟩ direction
C ⟨110⟩ direction
D ⟨210⟩ direction
In FCC, close packing occurs along directions joining nearest neighbors in the (111) plane. Those directions are of type ⟨110⟩, central to FCC slip systems {111}⟨110⟩.
In BCC, the shortest lattice translation direction (nearest neighbor direction) is
A ⟨100⟩ direction
B ⟨111⟩ direction
C ⟨110⟩ direction
D ⟨210⟩ direction
In BCC, nearest neighbors lie along the body diagonal from a corner to the body-centered atom. That direction is ⟨111⟩, so it is the closest packed direction.
In BCC diffraction, the first three allowed N = (h²+k²+l²) values are
A 2,4,6
B 1,2,3
C 2,4,8
D 3,4,8
BCC allows reflections with h+k+l even. The smallest N values meeting this are (110) N=2, (200) N=4, (211) N=6. These generate first peaks in BCC.
In FCC diffraction, the first three allowed N values are
A 1,2,3
B 2,4,6
C 4,5,6
D 3,4,8
FCC allows all-even or all-odd. The smallest allowed are (111) N=3, (200) N=4, (220) N=8. This sequence is a standard FCC indexing signature.
A cubic powder pattern shows sin²θ ratios 2:4:6:8. The lattice type is most likely
A Simple cubic
B Body centered cubic
C Face centered cubic
D Hexagonal close packed
Ratios matching N=2,4,6,8 suggest allowed reflections (110),(200),(211),(220), typical for BCC due to h+k+l even rule. FCC would start at 3,4,8.
A cubic powder pattern shows sin²θ ratios 3:4:8:11. The lattice type is most likely
A Simple cubic
B Body centered cubic
C Face centered cubic
D Diamond cubic
3,4,8,11 correspond to (111),(200),(220),(311) allowed in FCC. BCC would show 2,4,6,8. Diamond would have additional absences beyond FCC.
For a diatomic basis at (0,0,0) and (1/2,1/2,1/2) with equal form factors, reflections vanish when
A h+k+l even
B h, k, l mixed
C h=k=l only
D h+k+l odd
Structure factor becomes f[1 + exp(iπ(h+k+l))]. For odd sums, the exponential is −1, giving cancellation and F=0. This is the same condition as BCC absences.
For a diatomic basis with unequal form factors f₁ and f₂ at (0,0,0) and (1/2,1/2,1/2), forbidden BCC reflections become
A Weakly allowed
B Always absent
C Always strongest
D Angle independent
F = f₁ + f₂ exp(iπ(h+k+l)). For odd sums, F = f₁ − f₂, not necessarily zero if f₁ ≠ f₂. Hence “forbidden” become weak, not fully absent.
In the Ewald sphere method, increasing wavelength λ makes the Ewald sphere radius
A Increase
B Stay same
C Decrease
D Become infinite
Sphere radius is |k| = 2π/λ. Larger λ reduces |k|, shrinking the sphere. This changes which reciprocal points intersect, affecting which reflections are accessible.
If lattice constant a increases, reciprocal lattice spacing in k-space
A Increases
B Decreases
C Stays same
D Becomes zero
Reciprocal lattice constants scale as 2π/a. A larger real-space lattice constant means smaller reciprocal spacing. Diffraction peaks move to lower angles because d-spacings increase.
A zone boundary plane in k-space is where Bragg reflection occurs for electrons when
A k = 0
B k·G = 0
C G = 0
D 2k = G
At zone boundary, electron wavevector satisfies Bragg condition with reciprocal vector G, typically k = G/2. This causes standing waves and opens band gaps.
The first Brillouin zone of a simple cubic reciprocal lattice has the shape of a
A Octahedron
B Cube
C Dodecahedron
D Tetrahedron
For simple cubic reciprocal lattice, Wigner–Seitz cell is a cube. Boundaries lie halfway to nearest reciprocal points along ±x, ±y, ±z.
The first Brillouin zone of an FCC reciprocal lattice is a
A Rhombic dodecahedron
B Cube
C Truncated octahedron
D Regular octahedron
FCC reciprocal lattice corresponds to BCC real lattice. The Wigner–Seitz cell of FCC lattice is a truncated octahedron, commonly used in band-structure symmetry discussions.
The first Brillouin zone of a BCC reciprocal lattice is a
A Cube shape
B Rhombic dodecahedron
C Regular octahedron
D Truncated octahedron
The first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice. For a BCC lattice, this cell is a truncated octahedron, formed by bisector planes to nearest reciprocal points.
For cubic crystals, the zone axis [uvw] is perpendicular to plane (hkl) only when
A u=v=w
B u+h=0
C (uvw) = (hkl)
D u=v only
In cubic symmetry, the direction [hkl] is normal to the plane (hkl). Thus [uvw] is perpendicular to (hkl) when indices match, aside from scaling and sign.
In cubic crystals, the angle between planes (h1k1l1) and (h2k2l2) depends on
A Atomic number
B Dot product of normals
C Packing fraction
D Debye factor only
The plane normal is along [hkl]. Angle between planes equals angle between their normals: cosφ = (h1h2+k1k2+l1l2)/(|n1||n2|). This is purely geometric.
The d-spacing in tetragonal crystals depends on
A Only a
B Only c
C Only angles
D Both a and c
For tetragonal, d-spacing formula includes h,k with a and l with c: 1/d²=(h²+k²)/a² + l²/c². Hence both lattice constants matter.
If a cubic sample shows peak broadening increasing strongly at high angles, likely cause is
A Only size effect
B Only absorption
C Microstrain effect
D Only multiplicity
Strain broadening grows with tanθ, making high-angle peaks broader. Size broadening follows 1/cosθ and is less strongly angle-dependent, so strong high-angle growth suggests strain.
In powder diffraction, why are intensities corrected using Lorentz-polarization factor
A Fix peak positions
B Fix geometric bias
C Fix peak widths
D Fix lattice constants
Measurement geometry and X-ray polarization cause systematic angular dependence in recorded intensity. Lorentz-polarization correction removes this bias to better compare intrinsic reflection strengths.
For a crystal with basis atoms at (0,0,0) and (1/4,1/4,1/4), phase factor includes
A exp(iπh)
B exp[i2π(h+k)]
C exp(iπkl)
D exp[i(π/2)(h+k+l)]
Phase is exp[2πi(hx+ky+lz)]. With x=y=z=1/4, phase becomes exp[2πi(h+k+l)/4]=exp[i(π/2)(h+k+l)], controlling reflection strengths.
In a two-atom basis crystal, a systematic absence can occur even if lattice is simple because
A Basis cancels waves
B Bragg law fails
C λ becomes zero
D d becomes infinite
Even on a simple Bravais lattice, multiple atoms in the basis scatter with relative phases. For certain (hkl), destructive interference makes structure factor zero, removing that peak.
If a powder pattern fits FCC selection rules but some allowed peaks are extremely weak, a likely reason is
A Wrong wavelength
B Grain boundaries only
C Form factor drop
D Coordination number
Atomic scattering factor decreases with increasing scattering angle and depends on electron distribution. Some allowed reflections at high angle can be weak because individual atomic scattering becomes small.
The reciprocal lattice is most directly used to find
A Coordination number
B Diffraction spot positions
C Packing fraction
D Vacancy count
Reciprocal lattice points represent plane families and diffraction conditions. Their intersections with the Ewald sphere determine allowed diffraction directions and spot/peak positions.
In cubic crystals, a reflection with indices (hkl) corresponds to reciprocal vector magnitude proportional to
A a/√(h²+k²+l²)
B (h+k+l)/a
C a(hkl)
D √(h²+k²+l²)/a
Reciprocal vector magnitude |G| is proportional to 1/d, and for cubic d=a/√N. Thus |G| ∝ √N/a, where N=h²+k²+l².
If a crystal is rotated, which changes most directly in Ewald construction
A |k| magnitude
B X-ray wavelength
C Reciprocal lattice orientation
D Unit cell volume
Rotating the crystal rotates the reciprocal lattice relative to the fixed Ewald sphere. This changes which reciprocal points intersect the sphere, enabling different reflections without changing λ.
In Laue diffraction, multiple spots appear mainly because
A Many crystal grains
B Many wavelengths present
C No Bragg condition
D Only defects scatter
White radiation contains many wavelengths. For a fixed crystal orientation, different λ satisfy Bragg condition for different planes, producing many simultaneous diffraction spots.
The condition for systematic absence in FCC can be derived by summing phase factors at lattice points
A Four face points
B One point only
C Two body points
D Eight corners
FCC has equivalent points at (0,0,0), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0). Summing their phase factors cancels for mixed parity indices.
A peak indexed as (311) in FCC is allowed because
A All even indices
B Mixed parity indices
C h+k+l odd
D All odd indices
FCC allows reflections when h,k,l are all even or all odd. For (311), indices are all odd, so it is allowed and often appears after (220) in FCC patterns.
In BCC, (222) is allowed and compared to (111) of simple cubic, it appears because
A BCC allows odd sums
B BCC has only corners
C BCC forbids odd sums
D BCC has face centers
BCC forbids reflections with h+k+l odd. (111) is absent in BCC, but (222) has even sum and becomes the “parallel-plane” allowed reflection at higher angle.
If a reciprocal lattice point is farther from origin, the corresponding real-space plane spacing is
A Larger spacing
B Smaller spacing
C Same spacing
D Undefined always
Reciprocal distance is proportional to 1/d. So points farther from origin represent higher spatial frequency, meaning more closely spaced lattice planes in real space.
The most common cause of “extra” weak peaks in alloy XRD beyond fundamental peaks is
A Detector noise only
B Packing fraction change
C Coordination change
D Superlattice ordering
Ordering creates a larger real-space periodicity, introducing additional reciprocal points at fractional positions. These generate weak superlattice reflections not present in the disordered lattice.
If a reflection is forbidden by centering but appears weakly, a likely physical reason is
A Different basis atoms
B Absorption change
C Lattice constant error
D Multiplicity error
“Forbidden” assumes equal scattering amplitudes. If atoms have different form factors or occupancies, cancellation becomes incomplete, and weak intensity can appear at otherwise absent reflections.
A key difference between Bravais lattice and crystal structure is that crystal structure includes
A Only translations
B Only symmetry points
C Atomic basis
D Only reciprocal vectors
Bravais lattice is a mathematical grid of equivalent points. Crystal structure is lattice plus basis (atoms attached to each point), which determines actual diffraction intensities and chemistry.
In reciprocal lattice terms, Bragg’s law is equivalent to conservation of
A Mass and charge
B Wavevector change
C Crystal temperature
D Unit cell volume
Elastic scattering requires momentum conservation: k’−k must equal a reciprocal lattice vector. This statement in reciprocal space is equivalent to Bragg’s law in real space.
The Scherrer equation may overestimate size if peak broadening includes
A Only lattice constant
B Only multiplicity
C Only absorption
D Instrument broadening
Measured peak width includes instrument contribution. If not subtracted, the width appears larger, giving smaller calculated size or wrong size. Proper analysis corrects instrumental broadening first.
For cubic indexing, which quantity should be constant across peaks for a correct a value
A sinθ only
B intensity ratio only
C a from each peak
D form factor only
If indexing is correct and sample strain is small, lattice constant computed from each peak should match. Large variations suggest misindexing, systematic errors, or strain/zero-shift issues.
A typical sign of zero-shift error in powder XRD is that calculated lattice constant
A Decreases with θ
B Increases with θ
C Constant with θ
D Random only
A constant 2θ offset affects high-angle peaks differently in lattice constant calculation. Often, uncorrected zero shift causes a systematic trend of calculated a with θ, not constant.
In reciprocal space, smaller wavelength allows access to
A Smaller |G| only
B Only zone center
C Only low angles
D Larger |G| points
Smaller λ increases |k|, expanding the Ewald sphere radius. A larger sphere can intersect reciprocal points farther from origin, enabling observation of higher-order, higher-|G| reflections.
For an HCP lattice, the reciprocal lattice is
A Hexagonal type
B Cubic always
C BCC type
D FCC type
The reciprocal of a hexagonal lattice remains hexagonal in symmetry, though lattice parameters invert. This is important for constructing Brillouin zones and interpreting diffraction patterns of HCP metals.
A crystallographic direction in hexagonal lattice uses four-index notation because it
A Changes Bragg law
B Separates basal axes
C Removes all negatives
D Matches cubic only
Hexagonal lattices have three equivalent a-axes in the basal plane. Four-index Miller–Bravais notation (hkil) makes symmetry clear by explicitly including the third basal index.
The “zone axis rule” relates a plane (hkl) in a zone [uvw] by
A hu+kv+lw=1
B hu+kv+lw=2
C h+k+l=0
D hu+kv+lw=0
A plane (hkl) belongs to zone [uvw] if the direction [uvw] lies in that plane. In cubic systems this gives hu+kv+lw=0, a key indexing relation.
If two planes (h1k1l1) and (h2k2l2) intersect, the zone axis direction is along
A Their dot product
B Sum of indices
C Their cross product
D Difference of indices
The line of intersection is perpendicular to both plane normals. Since plane normals are [hkl] in cubic, the zone axis direction is given by the cross product of the normals.
In BCC, the most common slip plane is not close-packed, so plasticity is harder mainly because
A Coordination too low
B No close-packed planes
C No diffraction peaks
D d-spacings vanish
BCC lacks truly close-packed planes like FCC {111}. Slip requires higher stress because atomic packing in slip planes is less dense, increasing the barrier for dislocation motion.
The main reason FCC metals are very ductile is that they have
A Many easy slip systems
B Only one slip system
C No grain boundaries
D Zero vacancies
FCC has close-packed {111} planes and ⟨110⟩ directions, giving many equivalent slip systems. Dislocations move easily, leading to high ductility under applied stress.
In diffraction, “structure factor link to basis” means basis determines
A Peak positions
B Lattice constants
C Bragg angles only
D Peak intensities
Peak positions come from lattice geometry (d-spacings). Basis affects phase sums inside the unit cell, which controls structure factor and therefore intensities and systematic absences.
For a given lattice, changing X-ray wavelength changes
A Reciprocal lattice points
B Crystal symmetry
C Ewald sphere size
D Miller indices set
Reciprocal lattice is fixed by crystal. Changing λ changes |k|=2π/λ, resizing the Ewald sphere, altering which reciprocal points satisfy diffraction condition at given orientation.
A key limitation of simple Bragg peak indexing is that peak overlap becomes serious when
A Few peaks exist
B Many phases present
C Crystal is perfect
D λ is very large
In mixtures, peaks from different phases can overlap, confusing indexing and lattice parameter extraction. Accurate phase identification may require pattern fitting methods beyond simple peak assignment.
If a crystal has strong preferred orientation in powder XRD, the pattern mainly shows
A Wrong peak positions
B Wrong lattice constants
C No peaks at all
D Wrong peak intensities
Preferred orientation changes the fraction of grains aligned to satisfy Bragg condition for certain planes. Peak positions remain tied to d-spacing, but relative intensities become biased.