Crystal Structure and Lattice Vibrations - Solid State Physics - Physics Previous Year Questions

1

2007 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2007

Consider the atomic packing factor (APF) of the following crystal structures:
P. Simple Cubic
Q. Body Centred Cubic
R. Face Centred Cubic
S. Diamond
T. Hexagonal Close Packed
Which two of the above structures have equal APF?

A

P and Q

B

S and T

C

R and S

D

R and T

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2

2007 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2007

In a powder diffraction pattern recorded from a face-centred cubic sample using x-rays, the first peak appears at \( 30^0 \). The second peak will appear at

A

\(32.8^0 \)

B

\(33.7^0 \)

C

\(34.8^0 \)

D

\(35.3^0 \)

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3

2008 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2008

Metallic monovalent sodium crystallizes in body centered cubic structure. If the length of the unit cell is \(4 \times 10^{-8}\) cm, the concentration of conduction electrons in metallic sodium is

A

\(6.022 \times 10^{23}\) cm^-3

B

\(3.125 \times 10^{22}\) cm^-3

C

\(2.562 \times 10^{21}\) cm^-3

D

\(1.250 \times 10^{20}\) cm^-3

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4

2008 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2008

The primitive translation vectors of the body centered cubic lattice are \(\vec{a} = \dfrac{a}{2}(\hat{x} + \hat{y} - \hat{z})\), \(\vec{b} = \dfrac{a}{2}(-\hat{x} + \hat{y} + \hat{z})\) and \(\vec{c} = \dfrac{a}{2}(\hat{x} - \hat{y} + \hat{z})\). The primitive translation vectors \(\vec{A}, \vec{B}\) and \(\vec{C}\) of the reciprocal lattice are

A

\(\vec{A} = \dfrac{2\pi}{a}(\hat{x} - \hat{y}), \; \vec{B} = \dfrac{2\pi}{a}(\hat{y} + \hat{z}), \; \vec{C} = \dfrac{2\pi}{a}(\hat{x} + \hat{z})\)

B

\(\vec{A} = \dfrac{2\pi}{a}(\hat{x} + \hat{y}), \; \vec{B} = \dfrac{2\pi}{a}(\hat{y} - \hat{z}), \; \vec{C} = \dfrac{2\pi}{a}(\hat{x} + \hat{z})\)

C

\(\vec{A} = \dfrac{2\pi}{a}(\hat{x} + \hat{y}), \; \vec{B} = \dfrac{2\pi}{a}(\hat{y} + \hat{z}), \; \vec{C} = \dfrac{2\pi}{a}(\hat{x} - \hat{z})\)

D

\(\vec{A} = \dfrac{2\pi}{a}(\hat{x} + \hat{y}), \; \vec{B} = \dfrac{2\pi}{a}(\hat{y} + \hat{z}), \; \vec{C} = \dfrac{2\pi}{a}(\hat{x} + \hat{z})\)

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5

2008 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2008

The structure factor of a single cell of identical atoms of form factor \(f\) is given by \(S_{hkl} = f \sum \exp[-i2\pi(x_j h + y_j k + z_j l)]\) where \((x_j, y_j, z_j)\) is the coordinate of an atom, and \(hkl\) are the Miller indices. Which one of the following statement is correct for the diffraction peaks of body centered cubic (BCC) and face centered cubic (FCC) lattices?

A

BCC : (200); (110); (222) FCC : (111); (311); (400)

B

BCC : (210); (110); (222) FCC : (111); (311); (400)

C

BCC : (200); (110); (222) FCC : (111); (211); (400)

D

BCC : (200); (210); (222) FCC : (111); (211); (400)

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6

2008 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2008

The lattice specific heat C of a crystalline solid can be obtained using the Dulong Petit model, Einstein model and Debye model. At low temperature \(\hbar\omega >> k_B T\), which one of the following statements is true (a and A are constants)?

A

Dulong Petit : \(C \propto \exp(-a/T)\) ; Einstein : \(C = .\) constant ; Debye : \(C \propto (T/A)^3\)

B

Dulong Petit : \(C = .\) constant ; Einstein : \(C \propto (T/A)^3\) ; Debye : \(C \propto \exp(-a/T)\)

C

Dulong Petit : \(C = .\) constant ; Einstein : \(C \propto \frac{e^{-a/T}}{T^2}\) ; Debye : \(C \propto (T/A)^3\)

D

Dulong Petit : \(C \propto (T/A)^3\) ; Einstein : \(C \propto \frac{e^{-a/T}}{T^2}\) ; Debye : \(C = .\) constant

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7

2008 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2008

A linear diatomic lattice of lattice constant a with masses M and m (M > m) are coupled by a force constant C. The dispersion relation is given by \[\omega_\pm^2 = C\left(\frac{M+m}{Mm}\right) + \left[C^2\left(\frac{M+m}{Mm}\right)^2 - \frac{4C^2}{Mm}\sin^2\frac{ka}{2}\right]^{1/2}.\] Which one of the following statements is INCORRECT?

A

The atoms vibrating in transverse mode correspond to the optical branch.

B

The maximum frequency of the acoustic branch depends on the mass of the lighter atom m.

C

The dispersion of frequency in the optical branch is smaller than that in the acoustic branch.

D

No normal modes exist in the acoustic branch for any frequency greater than the maximum frequency at k = \pi/a .

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8

2009 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2009

In a cubic crystal, atoms of mass \( M_1 \) lie on one set of planes and atoms of mass \( M_2 \) lie on planes interleaved between those of the first set. If \( C \) is the force constant between nearest neighbour planes, the frequency of lattice vibrations for the optical phonon branch with wavevector \( k = 0 \) is

A

\(\sqrt{2C \left( \frac{1}{M_1} + \frac{1}{M_2} \right)} \)

B

\(\sqrt{C \left( \frac{1}{2M_1} + \frac{1}{M_2} \right)} \)

C

\(\sqrt{C \left( \frac{1}{M_1} + \frac{1}{2M_2} \right)} \)

D

0

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9

2009 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2009

The primitive translation vectors of the fcc reciprocal lattice are

A

\(\vec{b}_1 = \left(\frac{2\pi}{a}\right)(-\hat{i}+\hat{j}+\hat{k}); \vec{b}_2 = \left(\frac{2\pi}{a}\right)(\hat{i}-\hat{j}+\hat{k}); \vec{b}_3 = \left(\frac{2\pi}{a}\right)(\hat{i}+\hat{j}-\hat{k})\)

B

\(\vec{b}_1 = \left(\frac{\pi}{a}\right)(-\hat{i}+\hat{j}+\hat{k}); \vec{b}_2 = \left(\frac{\pi}{a}\right)(\hat{i}-\hat{j}+\hat{k}); \vec{b}_3 = \left(\frac{\pi}{a}\right)(\hat{i}+\hat{j}-\hat{k})\)

C

\(\vec{b}_1 = \left(\frac{\pi}{2a}\right)(-\hat{i}+\hat{j}+\hat{k}); \vec{b}_2 = \left(\frac{\pi}{2a}\right)(\hat{i}-\hat{j}+\hat{k}); \vec{b}_3 = \left(\frac{\pi}{2a}\right)(\hat{i}+\hat{j}-\hat{k})\)

D

\(\vec{b}_1 = \left(\frac{3\pi}{a}\right)(-\hat{i}+\hat{j}+\hat{k}); \vec{b}_2 = \left(\frac{3\pi}{a}\right)(\hat{i}-\hat{j}+\hat{k}); \vec{b}_3 = \left(\frac{3\pi}{a}\right)(\hat{i}+\hat{j}-\hat{k})\)

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10

2009 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2009

The volume of the primitive cell of the fcc reciprocal lattice is

A

\(4\left(\frac{2\pi}{a}\right)^3\)

B

\(4\left(\frac{\pi}{a}\right)^3\)

C

\(4\left(\frac{\pi}{2a}\right)^3\)

D

\(4\left(\frac{3\pi}{a}\right)^3\)

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11

2010 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2010

Consider X-ray diffraction from a crystal with a face-centered-cubic (fcc) lattice. The lattice plane for which there is NO diffraction peak is

A

(2, 1, 2)

B

(1, 1, 1)

C

(2, 0, 0)

D

(3, 1, 1)

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12

2010 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2010

The Bloch theorem states that within a crystal, the wavefunction, \(\psi(\vec{r})\), of an electron has the form

A

\(\psi(\vec{r}) = u(\vec{r})e^{i\vec{k}\cdot\vec{r}}\) where \(u(\vec{r})\) is an arbitrary function and \(\vec{k}\) is an arbitrary vector

B

\(\psi(\vec{r}) = u(\vec{r})e^{i\vec{G}\cdot\vec{r}}\) where \(u(\vec{r})\) is an arbitrary function and \(\vec{G}\) is a reciprocal lattice vector

C

\(\psi(\vec{r}) = u(\vec{r})e^{i\vec{G}\cdot\vec{r}}\) where \(u(\vec{r}) = u(\vec{r} + \vec{A})\), \(\vec{A}\) is a lattice vector and \(\vec{G}\) is a reciprocal lattice vector

D

\(\psi(\vec{r}) = u(\vec{r})e^{i\vec{k}\cdot\vec{r}}\) where \(u(\vec{r}) = u(\vec{r} + \vec{A})\), \(\vec{A}\) is a lattice vector and \(\vec{k}\) is an arbitrary vector

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13

2011 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2011

For a three-dimensional crystal having \(N\) primitive unit cells with a basis of \(p\) atoms, the number of optical branches is

A

3

B

\(3p\)

C

\(3p - 3\)

D

\(3N - 3p\)

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14

2011 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2011

A metal with body centered cubic (bcc) structure shows the first (i.e. smallest angle) diffraction peak at a Bragg angle of \(\theta = 30^\circ\). The wavelength of X-ray used is 2.1 \(\text{\AA}\). The volume of the PRIMITIVE unit cell of the metal is

A

\(26.2\ (\text{\AA})^3\)

B

\(13.1\ (\text{\AA})^3\)

C

\(9.3\ (\text{\AA})^3\)

D

\(4.6\ (\text{\AA})^3\)

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15

2012 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2012

Which one of the following CANNOT be explained by considering a harmonic approximation for the lattice vibrations in solids?

A

Debye’s \(T^3\) law

B

Dulong Petit’s law

C

Optical branches in lattices

D

Thermal expansion

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16

2012 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2012

A simple cubic crystal with lattice parameter \(a\) undergoes transition into a tetragonal structure with lattice parameters \(a_t = b_t = \sqrt{2} a\) and \(c_t = 2a\), below a certain temperature. The ratio of the interplanar spacings of (1 0 1) planes for the cubic and the tetragonal structures is

A

\(\sqrt{\frac{1}{6}}\)

B

\(\frac{1}{6}\)

C

\(\sqrt{\frac{3}{8}}\)

D

\(\frac{3}{8}\)

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17

2012 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2012

The dispersion relation for a one dimensional monatomic crystal with lattice spacing \(a\), which interacts via nearest neighbour harmonic potential is given by \(\omega = A \left| \sin \frac{Ka}{2} \right|\), where A is a constant of appropriate unit. The group velocity at the boundary of the first Brillouin zone is

A

0

B

1

C

\(\sqrt{\frac{Aa^2}{2}}\)

D

\(\frac{1}{2} \sqrt{\frac{Aa^2}{2}}\)

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18

2012 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2012

The force constant between the nearest neighbour of the lattice is (M is the mass of the atom)

A

\(\frac{MA^2}{4}\)

B

\(\frac{MA^2}{2}\)

C

\(MA^2\)

D

\(2MA^2\)

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19

2013 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2013

Group I contains elementary excitations in solids. Group II gives the associated fields with these excitations. MATCH the excitations with their associated field and select your answer as per codes given below.

Group I	Group II
(P) phonon	(i) photon + lattice vibration
(Q) plasmon	(ii) electron + elastic deformation
(R) polaron	(iii) collective electron oscillations
(S) polariton	(iv) elastic wave

Codes
(A) (P-iv), (Q-iii), (R-i), (S-ii)
(B) (P-iv), (Q-iii), (R-ii), (S-i)
(C) (P-i), (Q-iii), (R-ii), (S-iv)
(D) (P-iii), (Q-iv), (R-ii), (S-i)

A

B

C

D

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20

2013 · Physics · Solid State Physics · Crystal Structure and Lattice Vibrations

Physics (PH) 2013

A lattice has the following primitive vectors (in Å): \( \vec{a} = 2(\hat{j} + \hat{k}) \), \( \vec{b} = 2(\hat{k} + \hat{i}) \), \( \vec{c} = 2(\hat{i} + \hat{j}) \). The reciprocal lattice corresponding to the above lattice is

A

BCC lattice with cube edge of \((\frac{\pi}{2}) \) Å\(^{-1}\)

B

BCC lattice with cube edge of \((2\pi) \) Å\(^{-1}\)

C

FCC lattice with cube edge of \((\frac{\pi}{2}) \) Å\(^{-1}\)

D

FCC lattice with cube edge of \((2\pi) \) Å\(^{-1}\)

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