Question

Consider a one dimensional monatomic lattice with lattice constant 4.91A and atoms of mass 28amu. The speed of sound in the lattice is 11.6kms-1 . A) calculate the maximum frequency of waves that can be transmitted by the lattice.B) calculate the interatomic force constant of the lattice

Answer 1

Molecular Physics | Thermodynamics

We need to calculate the maximum frequency of waves and calculate the inter atomic force

Solution:

Given

Mass of atoms = $m = 28 \space amu = 28 \times 1.67 \times 10^{-27} kg$

speed of sound in the lattice =$v_s= 11.6 \space km /s = 11.6 \times 10^3 m/s$

lattice constant = $a = 4.91A^o = 4.91 \times 10^{-10} m/s$

A).

 The maximum frequency of waves which was transmitted by the lattice is

$$w_m =\frac {2}{a} \times v_s = \frac {2}{4.91 \times 10^{-10} } \times 11.6 \times 10^3$$

$$w_m =4.725 \times 10^{13} rad/sec$$

B).

$$w_m = \sqrt {\frac {4\beta}{m}}$$

  Squaring on both sides

$$(w_m)^2 = {\frac {4\beta}{m}}$$

$$\beta = \frac {m \times \beta^2 } {4} = \frac {28 \times (4.725 \times 10^{13})^2 } {4}$$

$$\beta = \frac {28 \times 1.67 \times 10^{-27} \times 22.325 \times 10^{26}} {4}$$

$\beta = \frac {1043.917 \times 10^{-1}}{4} =26.0979 N/m$

Answer: $\beta = 26.0979 \space N/m$