GRE Physics Companion: Discrete Oscillator Chains and Dispersion

GRE Physics Companion: Discrete Oscillator Chains and Dispersion

For a monatomic nearest-neighbor chain,

(1)

The traveling normal modes satisfy

(2)

1 High-value GRE facts

1. ω = 2| sin(ka∕2)|.
2. Distinct wave numbers may be restricted to −π∕a ≤ k < π∕a.
3. k and k + 2π∕a are equivalent on the lattice.
4. ω_max = 2.
5. The zone-boundary mode has neighboring masses in opposite phase.
6. In the long-wavelength limit, ω ≈ c|k| with c = a.
7. v_p = ω∕|k|.
8. v_g = dω∕dk.
9. For 0 < k < π∕a, v_g = c cos(ka∕2).
10. A fixed-end chain of N masses has exactly N normal modes.

Part I: Original GRE-style problems

Problem 1: lattice equation

For identical masses with nearest-neighbor springs,

1. mü_n = κ(u_n+1 + u_n−1)
2. mü_n = κ(u_n+1 − 2u_n + u_n−1)
3. mü_n = −κu_n
4. mü_n = κ(u_n+1 − u_n−1)
5. ü_n = 0

Problem 2: dispersion relation

The monatomic-chain dispersion relation is

1. ω² = (κ∕m)ka
2. ω² = (4κ∕m) sin ²(ka∕2)
3. ω² = (4κ∕m) cos ²(ka∕2)
4. ω = κk∕m
5. ω² = κ∕m for all k

Problem 3: maximum frequency

The maximum branch frequency is

1. 
2. 2
3. π
4. κ∕m
5. unbounded

Problem 4: Brillouin-zone edge

The positive edge of the first Brillouin zone is

1. 1∕a
2. π∕(2a)
3. π∕a
4. 2π∕a
5. a∕π

Problem 5: zone-boundary pattern

At

(3)

the displacement pattern is proportional to

1. 1
2. n
3. (−1)ⁿ
4. e⁻ⁿ
5. n²

Problem 6: continuum speed

In the long-wavelength limit,

(4)

equals

1. 
2. a
3. 
4. κa∕m
5. aκ∕m²

Problem 7: group velocity

For

(5)

the group velocity is

1. c
2. c sin(ka∕2)
3. c cos(ka∕2)
4. c tan(ka∕2)
5. zero for all k

Problem 8: group velocity at zone boundary

At

(6)

(7)

is

1. c
2. c∕2
3. 2c
4. 0
5. infinite

Problem 9: periodic chain wave numbers

For a periodic chain of

(8)

sites,

(9)

is

1. sπ∕(Na)
2. 2πs∕(Na)
3. s∕a
4. 2πs∕a
5. independent of N

Problem 10: fixed-end frequencies

For

(10)

moving masses between fixed ends,

(11)

is

1. 2 sin[rπ∕2(N + 1)]
2. r
3. 2 cos[rπ∕2(N + 1)]
4. for every r
5. zero

Problem 11: numerical group velocity

If

(12)

then at

(13)

the group velocity is closest to

1. 0.71 m/s
2. 1.00 m/s
3. 1.41 m/s
4. 2.00 m/s
5. 2.83 m/s

Problem 12: shortest distinct wavelength

The shortest distinct wavelength of the monatomic lattice is

1. a∕2
2. a
3. 2a
4. 2πa
5. arbitrarily small

Part II: Complete worked solutions

Solution 1

The two spring forces give the centered second difference:

(14)

Answer: (B).

Solution 2

Substituting

(15)

gives

(16)

Answer: (B).

Solution 3

The sine magnitude reaches one:

(17)

Answer: (B).

Solution 4

One complete nonredundant interval is

(18)

The positive edge is

(19)

Answer: (C).

Solution 5

At the zone boundary,

(20)

Answer: (C).

Solution 6

For

(21)

(22)

Thus

(23)

Answer: (B).

Solution 7

Differentiate

(24)

for positive

(25)

Then

(26)

Answer: (C).

Solution 8

At

(27)

(28)

Therefore

(29)

Answer: (D).

Solution 9

Periodic boundary conditions require

(30)

Therefore

(31)

Answer: (B).

Solution 10

For fixed ends,

(32)

Substitution into the infinite-chain dispersion relation gives

(33)

Answer: (A).

Solution 11

First,

(34)

Then

(35)

Answer: (C).

Solution 12

The zone boundary has

(36)

Thus

(37)

Answer: (C).

2 GRE checklist

1. Start with the centered second-difference force law.
2. Insert the lattice-wave phase factor carefully.
3. Reduce wave numbers modulo 2π∕a.
4. Use the long-wavelength approximation only when |ka|≪ 1.
5. Distinguish phase velocity from group velocity.
6. At the zone boundary, remember the alternating pattern and zero group velocity.
7. Match the wave-number quantization to the actual boundary conditions.
8. Count exactly one normal mode per independent scalar displacement degree of freedom.

References

References