Chapter 21: Crystal Structure and Reciprocal Lattice (Set-4)
In BCC, if atomic radius is r, the lattice constant a is
A a = 2r
B a = 2√2 r
C a = 4r/√3
D a = √3 r
In BCC, atoms touch along the body diagonal. Body diagonal is √3 a and equals 4r (corner to center to corner). Hence a = 4r/√3.
In FCC, if atomic radius is r, the lattice constant a is
A a = 4r/√2
B a = 2r
C a = 4r/√3
D a = √2 r
In FCC, atoms touch along a face diagonal. Face diagonal is √2 a and equals 4r. Therefore a = 4r/√2 = 2√2 r.
The packing fraction for FCC can be written as
A π/6 value
B π/(8√3)
C π/(3√2)
D π/4 value
FCC has 4 atoms per cell and a = 2√2 r. Packing fraction = (4×(4/3)πr³)/a³ = (16πr³/3)/(16√2 r³) = π/(3√2) ≈ 0.74.
The packing fraction for BCC can be written as
A π/(3√2)
B √3π/8
C √2/π value
D √3π/16
BCC has 2 atoms per cell and a = 4r/√3. Packing fraction = (2×(4/3)πr³)/a³ = (8πr³/3)/(64r³/(3√3)) = √3π/8 ≈ 0.68.
In a cubic crystal, planes (hkl) and (2h 2k 2l) are
A Parallel planes
B Same plane
C Perpendicular planes
D Random planes
Multiplying indices by the same factor keeps the plane orientation the same, but changes the intercepts. Hence (hkl) and (2h 2k 2l) are parallel with different spacing.
For cubic crystals, d(2h 2k 2l) compared to d(hkl) is
A Same d
B Double d
C Four times
D Half d
Since d = a/√(h²+k²+l²), doubling indices multiplies the denominator by 2. Therefore spacing becomes half: d(2h 2k 2l) = d(hkl)/2.
The interplanar spacing ratio d(100) : d(110) is
A 1 : √2
B 1 : 2
C √2 : 1
D 2 : 1
d(100)=a, d(110)=a/√2. So d(100)/d(110)=√2. Hence the ratio d(100):d(110)=√2:1.
A plane with Miller indices (0 1 1) is parallel to
A x-axis direction
B x-axis only
C y-axis only
D z-axis only
For (0 1 1), x-intercept is ∞ because h=0. So the plane is parallel to the x-axis, while it cuts y and z axes at finite intercepts.
A direction [2 4 2] is equivalent to which reduced direction
A [211] direction
B [121] direction
C [112] direction
D [221] direction
Direction indices are reduced to smallest integers by dividing by the greatest common factor. [2 4 2] divided by 2 gives [1 2 1].
For cubic crystals, the angle between [100] and [110] is
A 30° value
B 60° value
C 45° value
D 90° value
Use dot product. [100]·[110]=1, |[100]|=1, |[110]|=√2. So cosθ=1/√2, giving θ=45°.
For cubic crystals, the angle between [110] and [111] is
A 35.26° value
B 45° value
C 60° value
D 90° value
Dot product: [110]·[111]=2, magnitudes √2 and √3. cosθ=2/√6=0.8165, so θ≈35.26°.
The condition for a reflection to appear is that the structure factor must be
A Negative value
B Maximum always
C Temperature independent
D Nonzero value
Bragg condition sets possible angles, but intensity depends on structure factor. If structure factor equals zero due to cancellation, the reflection is systematically absent.
For a monoatomic simple cubic crystal, the structure factor is mainly
A Always zero
B Depends on h+k+l
C Constant amplitude
D Depends on parity only
With one identical atom at each lattice point, there is no basis interference within the unit cell. Structure factor is essentially atomic form factor times a phase factor.
In BCC, the structure factor becomes zero when
A h+k+l even
B h+k+l odd
C h, k, l all odd
D h, k, l all even
BCC has atoms at (0,0,0) and (1/2,1/2,1/2). Phase term adds with factor exp[iπ(h+k+l)]. When sum is odd, cancellation makes F = 0.
In FCC, the structure factor is zero for reflections with
A Mixed odd-even
B All odd indices
C All even indices
D h+k+l even
FCC basis points cause phase cancellation unless h, k, l are all even or all odd. Mixed parity gives destructive interference and systematic absence.
The intensity of an XRD reflection is proportional to
A d-spacing only
B θ only
C |F|² value
D λ only
Diffracted amplitude is given by structure factor F. Measured intensity is proportional to the square of amplitude, so intensity ∝ |F|², along with geometrical correction factors.
The reciprocal lattice vector G(hkl) is associated with
A Crystal directions
B Grain boundaries
C Vacancy defects
D Crystal planes
Each set of planes (hkl) maps to a reciprocal lattice vector normal to those planes. The vector magnitude relates to plane spacing, linking diffraction to reciprocal space.
The Bragg condition in reciprocal space is written as
A k’ − k = G
B k = G
C k’ = k
D d = λ
The scattering vector equals the difference between diffracted and incident wavevectors. Diffraction occurs when this equals a reciprocal lattice vector, matching the Laue condition.
In Ewald construction, a diffracted beam exists when a reciprocal point lies
A Inside the sphere
B Outside the sphere
C On the sphere
D At the origin
A reciprocal lattice point on the Ewald sphere satisfies momentum conservation for elastic scattering. That condition produces a diffracted beam at a specific direction.
In powder diffraction, why do rings appear on a film
A Single crystal rotation
B Many random grains
C No Bragg reflection
D Only defects scatter
Powder contains crystallites in all orientations. For each (hkl), some grains satisfy Bragg condition, producing a cone of diffracted rays that forms a ring on a detector.
If the order n in Bragg law increases with fixed λ, d, then θ must
A Decrease
B Stay fixed
C Increase
D Become random
Bragg law nλ=2d sinθ. For fixed λ and d, higher n requires larger sinθ. Therefore θ increases until sinθ ≤ 1, beyond which reflection cannot occur.
For constant λ, the quantity proportional to (h²+k²+l²) in cubic indexing is
A sin²θ value
B sinθ value
C cosθ value
D tanθ value
From Bragg law and d=a/√N (N=h²+k²+l²), we get sin²θ ∝ N/a² for fixed λ. This is why sin²θ ratios help assign indices.
A common way to detect whether a lattice is BCC from powder peaks is that
A (111) always absent
B (100) always strongest
C (110) first peak
D Peaks equally spaced
BCC selection rule requires h+k+l even. So (100) is absent and (110) becomes the first allowed low-angle peak, a strong signature of BCC patterns.
A common signature of FCC powder pattern is that
A (110) first peak
B (111) first peak
C (100) first peak
D No absences occur
FCC forbids mixed parity reflections like (100) and (110). Therefore (111) is typically the first allowed reflection, helping identify FCC structures quickly.
The atomic scattering factor decreases mainly because
A Electron cloud spread
B X-rays lose energy
C Planes become fewer
D Lattice constant changes
Scattering from different parts of the electron cloud interferes. At higher angles, phase differences within the cloud increase, reducing coherent amplitude, so the form factor falls.
The Debye–Waller effect is stronger at
A Small scattering angles
B Zero temperature only
C Low multiplicity only
D Large scattering angles
Thermal vibrations cause phase smearing that increases with scattering vector magnitude. High-angle reflections have larger scattering vectors, so their intensities reduce more strongly.
The Wigner–Seitz cell in reciprocal space is the
A Primitive real cell
B Ewald sphere surface
C First Brillouin zone
D Miller index set
The first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice. It contains all k-points closer to the origin than to any other reciprocal point.
Zone boundary planes are perpendicular to the vector connecting
A Origin and G
B Two real atoms
C Two grain centers
D Two dislocations
Brillouin zone boundaries are perpendicular bisectors of vectors from the origin to nearby reciprocal lattice points. They mark where Bragg reflection of waves becomes important.
In reduced zone scheme, a wavevector outside first zone is
A Discarded always
B Folded back inside
C Made zero always
D Treated as defect
Reduced zone scheme maps k values into the first Brillouin zone by subtracting a reciprocal lattice vector. This helps compare bands within a single fundamental k-space cell.
The extended zone scheme is useful because it shows
A Only first zone
B Only energy gaps
C Band periodicity
D Only density changes
Extended zone scheme keeps k values across many zones, clearly showing periodicity of E(k) and how bands connect across zone boundaries without folding.
A diffraction peak may be weak even when allowed because atomic form factor may be
A Large always
B Negative always
C Constant always
D Small at angles
At higher scattering angles, atomic scattering factor decreases, reducing intensity even if structure factor is nonzero. That is why high-angle peaks are often weaker.
The selection rule “all even or all odd” is associated with
A FCC lattice
B Simple cubic
C BCC lattice
D HCP lattice
FCC has extra lattice points at faces, producing phase cancellations for mixed parity indices. Hence only all-even or all-odd (hkl) reflections appear in FCC diffraction.
In Bragg’s law, if λ increases while d fixed, the Bragg angle θ
A Decreases
B Increases
C Stays constant
D Becomes negative
With nλ=2d sinθ, increasing λ requires larger sinθ to satisfy equality. So θ increases, shifting diffraction peaks to higher angles for the same planes.
A lattice constant calculated from multiple peaks is more reliable because it reduces
A Atomic number effect
B Systematic absences
C Random measurement error
D Form factor drop
Using several peaks averages out small reading errors in 2θ and peak position. This gives a more accurate lattice constant than relying on a single reflection.
In cubic indexing, if observed sin²θ ratios are 3:4:8, it suggests first peaks could be
A (111)(200)(220)
B (100)(110)(111)
C (110)(200)(211)
D (200)(210)(211)
For cubic, sin²θ ∝ (h²+k²+l²). Values 3,4,8 correspond to sums for (111)=3, (200)=4, (220)=8, matching a typical FCC allowed sequence.
A vacancy concentration in thermal equilibrium increases mainly with
A Lower temperature
B Higher density
C Higher temperature
D Larger grain size
Creating vacancies costs energy, but higher temperature increases entropy contribution. So equilibrium vacancy concentration rises rapidly with temperature, affecting diffusion and resistivity.
Interstitial defects generally cause stronger lattice distortion because they
A Remove an atom
B Reduce symmetry
C Increase grain size
D Add extra atom
An interstitial atom squeezes into spaces between normal sites, pushing neighbors and producing large local strain. This can strongly affect diffusion, hardness, and electrical properties.
The Burgers vector is perpendicular to dislocation line for
A Edge dislocation
B Screw dislocation
C Mixed dislocation
D Grain boundary
In an edge dislocation, Burgers vector is perpendicular to the dislocation line. In a screw dislocation, it is parallel. This difference controls slip behavior and stress fields.
XRD peak broadening due to strain is often distinguished from size broadening because strain broadening
A Affects low angles only
B Is constant with θ
C Increases with tanθ
D Depends on density
Microstrain broadening grows roughly with tanθ, while size broadening depends on 1/cosθ. Comparing angle dependence helps separate strain and size contributions in analysis.
A Laue pattern is most useful for determining
A Atomic radius
B Crystal orientation
C Grain size only
D Vacancy concentration
Laue diffraction from a stationary single crystal produces many spots. The symmetry and arrangement of these spots directly reveal crystal orientation and symmetry directions.
In Debye–Scherrer method, the sample is usually
A Large single crystal
B Thin film only
C Fine powder
D Liquid phase
The Debye–Scherrer technique uses a powder to ensure random orientations. This produces diffraction cones for each (hkl), allowing phase identification and lattice parameter measurement.
The reciprocal lattice provides a simple interpretation of which concept
A Diffraction condition
B Packing fraction
C Vacancy formation
D Crystal density
Diffraction is easiest in reciprocal space: scattering occurs when the scattering vector equals a reciprocal lattice vector. This unifies Bragg’s law and Laue conditions neatly.
A reflection labeled “second order” usually means
A d doubled
B λ doubled
C θ halved
D n = 2
In Bragg’s law, order n counts how many wavelengths fit the path difference. Second order means n=2, so the path difference equals 2λ for that reflection.
If the same planes give first order at θ₁, the second order peak for same λ appears at
A Same θ
B Smaller θ
C Larger θ
D Random θ
For n=2, sinθ must be twice the n=1 value, so θ increases. The second-order peak appears at a higher angle, if sinθ stays ≤ 1.
The reciprocal lattice point density increases when real-space unit cell volume
A Decreases
B Increases
C Stays same
D Becomes zero
Smaller real-space cell means larger reciprocal lattice spacing, but more points per unit reciprocal volume arrangement depends inversely on real cell volume. Overall, reciprocal cell volume is inversely proportional.
Brillouin zones are most directly related to
A Electron wavevectors
B Real space defects
C Crystal density
D Atomic radii
Brillouin zones partition reciprocal space into regions for electron k-values. They are central to band structure and explain zone boundary effects like band gaps.
The Fermi surface is a surface in
A Real space
B Defect space
C k-space
D Unit cell space
The Fermi surface separates occupied and unoccupied electron states at zero temperature in reciprocal space. Its shape influences conductivity, effective mass, and many electronic properties.
Crystal symmetry reduces the number of unique diffraction peaks because
A It increases absorption
B It changes wavelength
C It creates defects
D It makes planes equivalent
Symmetry operations map planes and directions into equivalent ones with same d-spacing. Many planes contribute to the same peak position, and multiplicity captures their count.
In a crystal with a two-atom basis, some reflections can vanish because of
A Packing fraction
B Phase cancellation
C Grain size only
D Unit cell angles
Atoms at different basis positions scatter with phase differences. For certain (hkl), those contributions cancel, giving structure factor zero and producing systematic absences.
A reliable practical method to find lattice constant from powder XRD is to plot
A sin²θ versus N
B θ versus λ
C d versus intensity
D F versus θ
For cubic crystals, sin²θ = (λ²/4a²)N, where N = h²+k²+l². A straight-line fit gives a from slope, improving accuracy using many peaks.