ME 595M, T.S. Fisher

Density of Phonon States (Kittel, Ch5)• Consider a 1D chain of total length L carrying M+1 particles

(atoms) at a separation aFix the position of particles 0 and MEach normal vibrational mode of polarization p takes the form of a standing wave

Only certain wavelengths (wavevectors) are allowedλmax=2L (Kmin=π/L), λmin=2a (Kmax=π/a=Mπ/L)In general, the allowed values of K are

0 1 MM-1

L

a

~ sin( )exp( )n Kpx nKa i t− ω

2 3 ( 1), , ,..., MKL L L Lπ π π − π

=Note: K=Mπ/L is not included because it implies no atomic motion, i.e., sin(nMπa/L)=sin(nπ)=0.

ME 595M, T.S. Fisher

Density of States, cont’d• Thus, we have M-1 allowed, independent values of

KThis is the same number of particles allowed to moveIn K-space, we thus have M-1 allowable wavevectorsEach wavevector describes a single mode, and one mode exists in each distance π/L of K-spaceThus, dK/dN = π/L, where N is the number of modes

π/(M-1)a π/aπ/L

π/a

ME 595M, T.S. Fisher

Density of States, cont’d

• The phonon density of states gives the number of modes per unit frequency per unit volume of real space

The last denominator is simply the group velocity, derived from the dispersion relation

11 1 1 1( )

/dN dN dKDd L dK d d dKLα=

ω = = =ω ω π ω

1( )g

Dv

ω =π

ME 595M, T.S. Fisher

Periodic Boundary Conditions• For more generality, apply periodic boundary

conditions to the chain and find

Still gives same number of modes (one per particle that is allowed to move) as previous case

Often used in the study of higher-dimension systems (2D and 3D)

2 40, , ,..., MKL L Lπ π π

= ± ±

ME 595M, T.S. Fisher

2D Density of States• Each allowable

wavevector (mode) occupies a region of area (2π/L)2

• Thus, within the circle of radius K, there are approximately πK2/ (2π/L)2 allowed wavevectors

• Density of states calculated for homework

K-space

π/a

2π/L

K

ME 595M, T.S. Fisher

3D Density of States• Using periodic boundary conditions in 3D, there is

one allowed value of K per (2π/L)3 volume of K-space

• The total number of modes with wavevectors of magnitude less than a given K is thus

• The 3D density of states becomes

3 33

24

2 3 6L VKN K⎛ ⎞ ⎛ ⎞= π =⎜ ⎟ ⎜ ⎟π⎝ ⎠ ⎝ ⎠ π

2

3 21 1 1( )

/2dN dN dK KDd V dK d d dKLα=

ω = = =ω ω ωπ