Diatomic Oscillator Chains and Acoustic and Optical Branches

Diatomic Oscillator Chains and Acoustic and Optical Branches

M06-06 considered a monatomic chain with one displacement degree of freedom in each primitive cell.

The next step is to place two inequivalent masses in each cell.

This small change produces one of the most important new features of lattice dynamics: more than one frequency branch.

Consider a one-dimensional chain with

• alternating masses m₁ and m₂,
• identical nearest-neighbor springs of constant κ,
• primitive-cell length a,
• displacement u_n(t) for mass m₁ in cell n,
• displacement v_n(t) for mass m₂ in cell n.

The two masses in one cell provide two mechanical degrees of freedom.

For every allowed wave number

(1)

the normal-mode problem therefore has two eigenfrequencies.

These form:

• a lower acoustic branch,
• an upper optical branch.

The dispersion relation is

(2)

The lower sign is the acoustic branch.

The upper sign is the optical branch.

1 Choice of primitive cell

Let cell

(3)

contain:

• mass m₁ with displacement u_n,
• mass m₂ with displacement v_n.

Mass

(4)

is connected to:

• v_n on its right,
• v_n−1 on its left.

Mass

(5)

is connected to:

• u_n on its left,
• u_n+1 on its right.

This indexing makes translation by one primitive cell correspond to

(6)

2 Equation of motion for mass m₁

The right spring contributes

(7)

The left spring contributes

(8)

Thus

(9)

3 Equation of motion for mass m₂

The left spring contributes

(10)

The right spring contributes

(11)

Thus

(12)

4 Bloch-form normal modes

Use the traveling normal-mode form

(13)

and

(14)

Then

(15)

and

(16)

After canceling the common phase factor,

(17)

and

(18)

5 The two-by-two eigenvalue problem

The amplitudes satisfy

(19)

For a nonzero polarization vector,

(20)

the determinant must vanish.

6 Characteristic equation

The determinant condition is

(21)

Now

2	= (1 + eika)(1 + e−ika)	(22)
	= 2 + 2 cos ka	(23)
	= 4 cos 2.	(24)

Therefore

0	= m1m2ω4 − 2κ(m 1 + m2)ω2	(25)
	+ 4κ2 sin 2.	(26)

Thus the squared frequencies are roots of a quadratic equation.

7 Acoustic and optical dispersion relations

Solving for

(27)

gives

ω±2	= κ	(28)
	± κ.	(29)

Define

(30)

and

(31)

The names acoustic and optical refer to the different long-wavelength motions of the two masses.

8 Why there are two branches

A primitive cell has two independent displacement amplitudes:

(32)

and

(33)

For each fixed

(34)

the dynamical problem is therefore two-dimensional.

Its two eigenvalues are

(35)

and

(36)

This is the lattice version of the rule developed in M06-04 and M06-05:

The number of normal-mode branches equals the number of independent displacement degrees of freedom in each primitive cell.

9 The acoustic branch at k = 0

At

(37)

(38)

The lower branch becomes

(39)

The amplitude equations reduce to

(40)

Hence

(41)

Both masses move together.

No spring changes length in the exact

(42)

mode.

This is uniform translation of the entire chain.

10 The optical branch at k = 0

At

(43)

the upper branch has

(44)

The amplitude equation gives

(45)

Therefore

(46)

The two masses move in opposite directions.

The center of mass of each cell remains fixed:

(47)

11 Why the names acoustic and optical are different

The acoustic branch has

(48)

as

(49)

Long-wavelength disturbances on this branch behave like sound waves.

The optical branch remains at finite frequency as

(50)

In ionic crystals, opposite motion of differently charged ions can create an oscillating electric dipole that couples to electromagnetic radiation.

That historical physical context motivates the name optical branch.

In the purely mechanical model developed here, the important distinction is the relative motion inside the primitive cell.

12 Long-wavelength acoustic limit

Define

(51)

For

(52)

(53)

The square root in the dispersion relation becomes

		(54)
	≈ S −.	(55)

Thus the acoustic branch satisfies

(56)

Therefore

(57)

where

(58)

This is the long-wavelength sound speed of the diatomic chain.

13 Long-wavelength optical limit

At small

(59)

the optical branch remains near

(60)

To leading order,

(61)

Thus the optical branch bends downward away from

(62)

Its group velocity vanishes at

(63)

14 Brillouin-zone boundary

At

(64)

(65)

Suppose

(66)

so

(67)

is the heavier mass.

Then

(68)

and

(69)

15 Zone-edge polarization

At

(70)

(71)

The off-diagonal coupling terms in the two-by-two amplitude matrix vanish.

The equations become

(72)

and

(73)

For

(74)

the lower-frequency zone-edge mode has

(75)

Only the heavier sublattice moves.

The upper-frequency zone-edge mode has

(76)

Only the lighter sublattice moves.

16 Frequency gap

Again assume

(77)

The top of the acoustic branch is

(78)

The bottom of the optical branch is at the same zone edge:

(79)

Since

(80)

(81)

Thus there is a frequency interval with no propagating normal mode in this ideal infinite lattice.

The gap width in angular frequency is

(82)

The gap vanishes when

(83)

17 Example 1: branch frequencies for unequal masses

Let

(84)

At

(85)

(86)

while

ωop(0)	=	(87)
	=	(88)
	≈ 17.32 rad∕s.	(89)

At

(90)

(91)

and

(92)

The zone-edge frequency gap is therefore

(93)

Notice that the optical branch decreases from

(94)

at

(95)

to

(96)

at the zone edge.

18 Example 2: sound speed

For the same masses, let

(97)

Then

cac	= 0.40	(98)
	≈ 1.63 m∕s.	(99)

This is the limiting slope of the acoustic branch near

(100)

19 General polarization ratio

The first amplitude equation gives

(101)

Therefore

(102)

This complex ratio contains:

• the relative amplitude of the two sublattices,
• their relative phase.

Different points along a dispersion branch therefore have different mode polarizations.

The eigenfrequency alone does not fully specify the normal mode.

20 A phase convention that makes the dynamical matrix real

The two masses are physically offset within each cell.

It is often convenient to absorb half the cell phase into the

(103)

amplitude by writing

(104)

Then the mode equation can be written with a real symmetric coupling term:

(105)

This form makes the Hermitian nature of the lattice normal-mode problem especially transparent.

21 Group velocity of the two branches

Define

(106)

Then

(107)

Differentiation gives

(108)

and

(109)

Within

(110)

the acoustic branch rises with

(111)

while the optical branch falls.

Both group velocities vanish at the zone boundary.

The optical group velocity also vanishes at

(112)

22 Equal-mass limit

Set

(113)

Then

(114)

Inside the first Brillouin zone,

(115)

so

(116)

Therefore

(117)

and

(118)

The gap closes at

(119)

23 Zone folding

When

(120)

the physical chain is actually monatomic with nearest-neighbor spacing

(121)

Its natural primitive cell could therefore be half as long as the two-mass cell.

Using the artificially doubled cell halves the Brillouin-zone width.

The original monatomic dispersion curve is then folded back into the smaller zone.

The two apparent branches are simply two pieces of the same monatomic branch.

24 Why mass contrast opens the gap

With equal masses, the folded branches meet at the reduced-zone boundary.

When the masses become unequal, the translational symmetry of the smaller

(122)

cell is lost.

The true primitive cell has length

(123)

The degeneracy at the crossing is lifted.

The two modes separate into:

• a lower heavy-sublattice zone-edge mode,
• an upper light-sublattice zone-edge mode.

This creates the frequency gap.

This simple mechanical model is an important prototype for band-gap formation in periodic media.

25 Mode counting

Suppose there are

(124)

primitive cells with periodic boundary conditions.

There are

(125)

independent scalar displacements:

(126)

values of

(127)

and

(128)

values of

(129)

There are

(130)

distinct allowed wave numbers in one Brillouin zone.

At each

(131)

there are two branches.

Therefore

(132)

normal modes, exactly matching the number of degrees of freedom.

26 Acoustic branch as collective translation

For very small

(133)

neighboring cells move almost together.

The two masses inside each cell are approximately in phase.

The cell behaves approximately as a composite mass

(134)

That is why the acoustic branch approaches a continuum sound wave.

Its frequency tends continuously to zero as wavelength tends to infinity.

27 Optical branch as internal cell vibration

For the optical branch near

(135)

the two masses within each cell move against one another.

This is an internal vibration of the primitive cell.

Even at infinite wavelength, the internal relative coordinate continues to oscillate.

Therefore

(136)

This distinction between collective translation and internal vibration generalizes to much more complicated lattices.

28 Relation to the small-oscillation eigenvalue problem

For every fixed

(137)

the diatomic chain reduces to a two-coordinate generalized normal-mode problem.

The Bloch phase handles translation from one primitive cell to the next.

The remaining vector

(138)

describes the internal mode shape within the cell.

Thus lattice dynamics separates naturally into:

• spatial phase between primitive cells,
• polarization inside each primitive cell.

This is the periodic-system extension of the eigenvector ideas developed in M06-05.

29 Common mistakes

1. Using the nearest-neighbor spacing a∕2 as the primitive-cell length after defining a as the cell length.
2. Forgetting that there are two displacement amplitudes for every wave number.
3. Using the monatomic dispersion relation for a diatomic lattice.
4. Losing the complex phase factors e^±ika in the neighboring-cell terms.
5. Forgetting to take the determinant of the full two-by-two amplitude matrix.
6. Confusing the acoustic and optical signs in the quadratic solution.
7. Assuming the optical branch goes to zero frequency as k → 0.
8. Assuming the acoustic k = 0 mode stretches the springs.
9. Forgetting the mass-weighted amplitude relation m₁U +m₂V = 0 for the optical k = 0 mode.
10. Assuming both sublattices move at the unequal-mass zone boundary.
11. Reversing which zone-edge frequency belongs to the heavier and lighter masses.
12. Assuming a frequency gap exists when the masses are equal.
13. Confusing a frequency gap with the Brillouin-zone interval in wave number.
14. Forgetting that equal masses make the two-mass cell nonprimitive.
15. Describing an eigenfrequency without also considering the polarization eigenvector.

30 Practice exercises

1. Derive the two lattice equations for u_n and v_n.
2. Insert the Bloch-form normal modes and derive the two-by-two amplitude matrix.
3. Evaluate its determinant and derive the quadratic equation for ω².
4. Solve for the acoustic and optical dispersion relations.
5. Show that the acoustic k = 0 mode has U = V .
6. Show that the optical k = 0 mode satisfies m₁U + m₂V = 0.
7. Derive the long-wavelength acoustic speed.
8. Derive the small-k optical frequency correction.
9. For m₁ > m₂, derive both zone-edge frequencies and polarizations.
10. Derive the frequency-gap width.
11. Reproduce the numerical frequencies in Example 1.
12. Derive the general polarization ratio V∕U.
13. Derive the group-velocity formulas for the two branches.
14. Take the equal-mass limit and derive the two reduced-zone branch equations.
15. Explain mathematically and physically why the equal-mass branches are a zone-folded version of a monatomic dispersion curve.

31 Summary

The diatomic chain obeys

(139)

and

(140)

Bloch-form normal modes lead to two branches:

ω±2	= κ	(141)
	± κ.	(142)

The acoustic branch has

(143)

and

(144)

at

(145)

The optical branch has

(146)

and

(147)

For

(148)

the zone-edge frequencies are

(149)

The mass contrast opens a frequency gap between the branches.

In the equal-mass limit the gap closes, and the two branches become a zone-folded representation of the monatomic chain.

References

References