3. Consider a monatomic linear with equilibrium separation a. Suppose the outer electrons (of mass m) in a given atom move with a displacement different from that of the corresponding ion core (of mass M). Let the displacement of the ion core s be: = uei(Ksa-wt) Us and the displacement of the center of mass of the outer electrons associated with ion s be: V = vei(Ksa-wt) Each ion core is assumed to interact only with its own outer electrons with a force proportional to the displacement of the electron distribution from the nucleus, and the force constant is C₂. However, neighboring electron distributions interact with a force constant C₁. a) Show that -w² Mus = C₂ (vs - Us) -w²mvs = C₂ (us- Vs) + C₁ (Vs+1 + Vs-1 -2vs) b) Substitute for the displacements, and solve the resulting simultaneous equations. Find an expression for w². c) Take the limit as m→ 0 (the mass of electrons is much smaller than that of the ion core), and show that the dispersion relation for the acoustic mode is given by: w² 4C₁ M Ka 2 sin² 4C₁ sin². C₂ (1 + Ka)
3. Consider a monatomic linear with equilibrium separation a. Suppose the outer electrons (of mass m) in a given atom move with a displacement different from that of the corresponding ion core (of mass M). Let the displacement of the ion core s be: = uei(Ksa-wt) Us and the displacement of the center of mass of the outer electrons associated with ion s be: V = vei(Ksa-wt) Each ion core is assumed to interact only with its own outer electrons with a force proportional to the displacement of the electron distribution from the nucleus, and the force constant is C₂. However, neighboring electron distributions interact with a force constant C₁. a) Show that -w² Mus = C₂ (vs - Us) -w²mvs = C₂ (us- Vs) + C₁ (Vs+1 + Vs-1 -2vs) b) Substitute for the displacements, and solve the resulting simultaneous equations. Find an expression for w². c) Take the limit as m→ 0 (the mass of electrons is much smaller than that of the ion core), and show that the dispersion relation for the acoustic mode is given by: w² 4C₁ M Ka 2 sin² 4C₁ sin². C₂ (1 + Ka)
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Transcribed Image Text:3. Consider a monatomic linear with equilibrium separation a. Suppose the outer
electrons (of mass m) in a given atom move with a displacement different from that
of the corresponding ion core (of mass M). Let the displacement of the ion core s be:
Us = uei(Ksa-wt)
and the displacement of the center of mass of the outer electrons associated with ion s
be:
V = vei(Ksa-wt)
Each ion core is assumed to interact only with its own outer electrons with a force
proportional to the displacement of the electron distribution from the nucleus, and the
force constant is C₂. However, neighboring electron distributions interact with a force
constant C₁.
a) Show that
-w² Mus = C₂ (vs - Us)
-w²mvs = C₂ (us - Vs) + C₁ (Vs+1 + Vs-1-2vs)
b) Substitute for the displacements, and solve the resulting simultaneous equations.
Find an expression for w².
c) Take the limit as m → 0 (the mass of electrons is much smaller than that of the ion
core), and show that the dispersion relation for the acoustic mode is given by:
w² =
=
4C₁
M
Ka
sin 2.
2
4C₁
(1 + 4C sin² Ka)
C₂
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