Question

given the acoustic and optical branches for a phonon dispersion of 1-d chain

1) Find expressions for the angular frequency w(k) at the points a, b, c and d shown on the plot

2) Calculate the ratio of u/v at each of the points. Note that u and v correspond to the magnitudes of the displacements for the masses m1 and m2, respectively, used to calculate the dispersion relation at the right. What do the ratios physically imply?

3) What happens if m1 = m2? sketch the dispersion relations

ω(k) 0 -Tt/a n/a

Answer 1

Given data m_1 > > m_2 let O be repeat distance of the lattice Unit cell have two basis with mass m_1 and m_2 The force constant (c) identical between all atoms equation of motion for M_1 implies M_1 d^2 us/dt^2 = c[vs - Vs - (u5 - v5 - 1)] = C [v_s - v_s - 1 - 2 u_s] rightarrow (1) for m_2 M_2 d^2 v_s/dt^2 = cp[u_r + 1 + u_r - 2V_s] rightarrow 2 Let the unseats travelling b/w & V_B = Ve^(i8 k a) e^-wt where u and V are amplitude d^2 us/dt^2 = (-i w)^2 u & d^2 U/dt^2 = (-i w)^2 v_s = -m_2 w^2 ve^is ku e^-iwt = C [u e^ika + u - 2v]e^is ka e^I w t -m_2 w^2 v = (Ce^ika tc) u - (2 C) v C (e^ika + 1) u + (m_2 w^2 - 2C) v = 0 \r\n By eliminating u and V from the equation (3) & (4) Vertical-bar (-m_1 w^2 + 2c) c(e^ika + 1) - C(1 + e^-ika) (m_2 w^2 - 2C) Vertical-bar = 0