Chapter 21: Crystal Structure and Reciprocal Lattice (Set-3)
When calculating atoms per unit cell, a face-centered atom contributes
A 1 atom
B 1/4 atom
C 1/2 atom
D 1/8 atom
A face-centered atom is shared by two neighboring unit cells, so only half belongs to one cell. This sharing is why FCC totals 4 atoms per unit cell.
An atom located at an edge center contributes to one unit cell by
A 1 atom
B 1/4 atom
C 1/2 atom
D 1/8 atom
Each edge is shared by four unit cells meeting along that edge. Therefore, an atom at an edge center contributes one-fourth to a single unit cell count.
A corner atom contribution to a unit cell is
A 1/8 atom
B 1 atom
C 1/2 atom
D 1/4 atom
A corner atom is shared by eight surrounding cubes. So only one-eighth of that atom is considered part of one unit cell when counting atoms per cell.
For a cubic crystal, the spacing d(110) equals
A a value
B a/2
C a/√2
D a/√3
In cubic crystals, d(hkl)=a/√(h²+k²+l²). For (110), √(1+1+0)=√2, hence d(110)=a/√2.
For a cubic crystal, the spacing d(111) equals
A a/√2
B a/√3
C a/3
D a/2
Using d=a/√(h²+k²+l²), for (111) the denominator is √(1+1+1)=√3. So planes (111) are closer than (100) in cubic crystals.
In cubic crystals, (222) planes have spacing compared to (111)
A Same spacing
B Double spacing
C Triple spacing
D Half spacing
d(222)=a/√(4+4+4)=a/√12, while d(111)=a/√3. Since √12=2√3, d(222)=d(111)/2.
A direction [100] in cubic crystals is along
A Body diagonal
B Face diagonal
C Cube edge
D Zone boundary
[100] points along the x-axis direction of the cubic cell, which is simply the cube edge. It is one of the principal axes directions.
The plane (100) in a cubic cell is perpendicular to
A [100] direction
B [110] direction
C [111] direction
D [010] direction
In cubic crystals, plane (hkl) has normal direction [hkl]. Therefore, plane (100) is perpendicular to [100], which is along the x-axis.
If a plane cuts x at a and is parallel to y and z, its Miller indices are
A (010) plane
B (100) plane
C (001) plane
D (110) plane
Intercepts: x=a → 1, y=∞ → 0, z=∞ → 0. Taking reciprocals gives (1,0,0). Hence the plane is indexed as (100).
If a plane cuts x at a, y at a, and is parallel to z, its indices are
A (101) plane
B (111) plane
C (110) plane
D (200) plane
Intercepts: x=a (1), y=a (1), z=∞ (0). Reciprocals give (1,1,0). So the plane is (110).
If a plane cuts x at a, y at a, and z at a, its indices are
A (110) plane
B (200) plane
C (210) plane
D (111) plane
The plane intercepts all three axes equally at one lattice constant. Reciprocals of (1,1,1) remain (1,1,1). So its Miller indices are (111).
In diffraction, the “scattering vector” is defined as
A k + k’
B k’ − k
C d − λ
D 2πa only
The scattering vector is the change in wavevector from incident to scattered beam. Diffraction happens when this vector matches a reciprocal lattice vector.
The reciprocal lattice basis vectors have units of
A meter (m)
B kilogram (kg)
C 1/m length
D joule (J)
Reciprocal vectors represent spatial frequencies, so they have inverse length units. In physics convention they often include a 2π factor, but units remain 1/length.
In physics convention, the magnitude of a reciprocal vector for planes spacing d is
A 2πd
B 2π/d
C d/2π
D πd only
A set of planes with spacing d corresponds to reciprocal vector magnitude |G|=2π/d. This links real-space periodicity to reciprocal-space distances.
The reciprocal lattice is constructed from
A Real lattice vectors
B Atomic radii values
C Defect concentration
D Grain boundary maps
Reciprocal vectors are defined mathematically from real-space primitive vectors using vector products. They form a lattice in k-space that represents allowed diffraction conditions.
In Ewald construction, the sphere radius equals
A 2π/a
B d-spacing
C |k| value
D Miller index sum
The Ewald sphere has radius equal to the magnitude of the incident wavevector k=2π/λ. Reciprocal points on this sphere satisfy the diffraction condition.
For electron waves in crystals, zone boundary occurs when
A k = 0
B k equals G/2
C λ equals d
D h+k+l odd
At a zone boundary, the Bragg condition for electron waves occurs, often when k lies halfway to a reciprocal lattice point. This causes standing waves and band gaps.
In Bragg’s law, if θ increases while d fixed, λ must
A Decrease
B Stay same
C Become zero
D Increase
From nλ=2d sinθ, with fixed d and n, λ is proportional to sinθ. So larger θ gives larger sinθ and thus requires a larger wavelength.
If d decreases and λ fixed, the Bragg angle θ generally
A Decreases
B Becomes constant
C Increases
D Becomes random
With λ fixed, sinθ = nλ/(2d). If d becomes smaller, the right side increases, but for allowed reflections typically lower-index planes have larger d and appear at lower θ.
X-ray wavelengths commonly used in XRD are of order
A Angstroms range
B Millimeters range
C Micrometers range
D Meters range
Typical X-ray wavelengths (like Cu Kα) are around 1–2 Å, comparable to interatomic spacings. That matching scale makes Bragg diffraction from crystal planes possible.
If a peak is observed at higher 2θ, the corresponding d-spacing is
A Larger d
B Smaller d
C Same d
D Random d
Higher 2θ means higher θ. From Bragg’s law, larger θ requires smaller d for fixed λ (and fixed order). Thus high-angle peaks correspond to closely spaced planes.
The form factor is also called
A Multiplicity factor
B Lorentz factor
C Atomic scattering factor
D Zone axis
Atomic scattering factor (form factor) measures how strongly an atom scatters X-rays. It depends on electron distribution and decreases as scattering angle increases.
Temperature rise usually causes diffraction peak intensities to
A Increase always
B Decrease generally
C Become zero
D Shift randomly
Increased temperature increases atomic vibrations. Debye–Waller factor reduces coherent scattering, especially at high angles, so peak intensities typically decrease while positions shift slightly.
A “systematic absence” means a reflection is missing due to
A Structure factor zero
B Detector failure
C Wrong wavelength
D Large grain size
Even if Bragg condition is satisfied, destructive interference among atoms in the basis can make the structure factor zero, causing that reflection to be absent consistently.
For BCC, the reflection (211) is
A Forbidden peak
B Always strongest
C Allowed peak
D Always weakest
In BCC, reflections are allowed when h+k+l is even. For (211), sum = 4 (even), so it is allowed and can appear in diffraction.
For FCC, the reflection (200) is
A Forbidden peak
B Only at high T
C Only at low T
D Allowed peak
FCC allows reflections when h, k, l are all even or all odd. (200) has all even indices, so it is allowed and often appears after (111).
For FCC, the reflection (210) is
A Allowed peak
B Forbidden peak
C Always strongest
D Angle independent
(210) has mixed parity (2 even, 1 odd, 0 even). FCC requires all indices all even or all odd, so structure factor cancels and the peak is absent.
Multiplicity affects intensity because it counts
A Beam energy loss
B Thermal vibrations
C Equivalent plane sets
D Lattice constant change
In powder patterns, many symmetry-equivalent planes share the same d-spacing and contribute to one peak. Higher multiplicity means more contributors, raising peak intensity.
The Lorentz factor mainly depends on
A Crystal density
B Atomic number
C Unit cell edges
D Diffraction geometry
The Lorentz factor corrects for how long a set of planes stays in diffraction condition and for angular measurement geometry. It changes intensity as a function of θ.
In a cubic lattice, if you know d(111) and λ, you can find
A Atomic number
B Lattice constant a
C Defect density
D Grain boundary angle
Measure θ from Bragg’s law to get d. For cubic, d(111)=a/√3, so a=√3 d(111). This is a standard method to determine lattice constant.
In reciprocal space, the first Brillouin zone boundary is formed by
A Perpendicular bisectors
B Real lattice planes
C Unit cell edges
D Diffraction peaks
The zone boundary is the set of planes that are perpendicular bisectors between origin and nearby reciprocal lattice points. It defines where Bragg reflection of waves begins.
The point X in cubic Brillouin zone lies on
A Zone center
B Zone corner
C Zone face center
D Random position
In simple cubic BZ, X is at the center of a face along a principal axis direction. It is a key symmetry point used in band structure plots.
The point L in cubic Brillouin zone lies at
A Zone center
B Zone corner
C Face center
D Edge center
For the FCC reciprocal lattice (BCC real), L is typically at a zone corner direction like [111]. Symmetry points differ by lattice type, but L indicates a corner point.
The extended zone scheme means k-values are
A Shown across many zones
B Folded into first zone
C Removed at boundaries
D Replaced by energy only
Extended zone scheme keeps k values in their original repeated-zone positions, showing dispersion across successive zones. It helps visualize band crossings and periodicity in k-space.
When a band gap opens, it is most likely at
A Zone center Γ
B Random k only
C Zone boundary
D k equals zero only
Band gaps commonly form at Brillouin zone boundaries due to Bragg reflection of electron waves. Standing waves create energy splitting, producing a gap between bands.
A Burgers vector is used to describe
A Plane intercepts
B XRD wavelength
C Unit cell volume
D Dislocation magnitude
Burgers vector gives the magnitude and direction of lattice distortion around a dislocation. It is obtained from a Burgers circuit and is fundamental in dislocation theory.
Edge dislocation is associated with an
A Missing plane only
B Extra half-plane
C Point vacancy only
D Grain boundary only
An edge dislocation occurs when an extra half-plane of atoms terminates inside the crystal. This creates compressive and tensile regions and controls plastic deformation.
Screw dislocation is characterized by
A Extra half-plane
B Vacancy cluster
C Helical distortion
D Planar mismatch
In a screw dislocation, atomic planes form a spiral (helical) ramp around the dislocation line. The Burgers vector is parallel to the dislocation line.
A grain boundary mainly affects properties by increasing
A Perfect periodicity
B Scattering centers
C Lattice constant
D Packing fraction
Grain boundaries interrupt periodicity and act as scattering sites for electrons and phonons. They also influence diffusion paths and mechanical strength due to misorientation.
Scherrer formula uses peak width to estimate
A Crystallite size
B Density value
C Coordination number
D Miller indices
The Scherrer equation relates XRD peak broadening (after instrumental correction) to average crystallite size. Smaller crystallites broaden peaks due to limited coherent domain size.
In Scherrer equation, larger peak broadening implies
A Larger crystals
B Higher density
C Smaller crystals
D Lower wavelength
Peak width is inversely related to crystallite size. If peaks are broader, the coherent diffracting domains are smaller, assuming strain broadening is not dominating.
Phonon scattering increases with temperature mainly because
A Fewer vibrations
B Lattice becomes rigid
C Planes become absent
D More vibrations
Higher temperature increases phonon population and vibration amplitude. This boosts scattering of electrons and reduces coherent X-ray scattering intensity through Debye–Waller effects.
A common practical step in indexing cubic powder peaks is to calculate
A sinθ only
B sin²θ ratios
C cosθ only
D tanθ only
For fixed λ, sin²θ is proportional to (h²+k²+l²)/a² in cubic crystals. Ratios of sin²θ values become simple integers, helping assign correct (hkl) indices.
If two different planes have same d-spacing in cubic crystals, they are
A Always identical
B Always forbidden
C Often symmetry related
D Always defects
Cubic symmetry makes many different (hkl) sets equivalent in spacing and properties. Such planes produce the same peak position, and multiplicity accounts for their number.
In reciprocal space, a smaller lattice constant a gives
A Larger reciprocal spacing
B Smaller reciprocal spacing
C No change in G
D Random reciprocal points
Reciprocal lattice distances scale as 1/a. So when real lattice constant decreases, reciprocal lattice points spread farther apart, changing diffraction peak positions.
Bragg peaks become broader in nanosamples mainly due to
A Larger unit cell
B Finite domain size
C Higher symmetry
D Lower multiplicity
Nanocrystals have small coherent diffracting regions, leading to size broadening. This is distinct from peak shifts (lattice change) and intensity changes (structure factor effects).
In Laue method, the crystal is typically exposed to
A Monochromatic X-rays
B Visible light
C White X-rays
D Gamma rays
Laue diffraction commonly uses polychromatic (white) X-rays with a stationary single crystal. Different wavelengths satisfy Bragg condition for different plane sets, forming a spot pattern.
In single-crystal diffraction, rotating the crystal helps because it
A Changes atomic number
B Removes systematic absences
C Changes lattice type
D Satisfies more Bragg conditions
Rotation changes the orientation of reciprocal lattice relative to Ewald sphere, allowing different reciprocal points to intersect and produce reflections. This collects more structural information.
The concept of “reciprocal lattice points” is closest to representing
A Real atom positions
B Plane families in real
C Vacancy locations
D Grain boundary planes
Each reciprocal lattice point corresponds to a set of real-space planes with specific spacing and orientation. Diffraction peaks occur when scattering conditions meet these points.
A reliable way to distinguish BCC from FCC using XRD is to check
A Missing reflection rules
B Only peak heights
C Only sample color
D Only grain size
BCC allows h+k+l even, while FCC allows all-even or all-odd. Comparing observed missing peaks against these selection rules quickly identifies whether the lattice is BCC or FCC.