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Exercise 1: Dispersion of longitudinal phonons in a row of atoms of type C=C-C%=C-C= Consider a linear lattice with constant "a" having as a basis two identical atoms situated in a line and spaced from equilibrium by b (b < a/2), see Chapter I, Ex. 15. The instantaneous position of the atoms is x1, X2, ... X2n-1, X2n X2n+1 ..., and the distance along the line relative to their equilibrium position is u1, U2,.. We consider only interactions between nearest neighbors which are characterized by the force constants B, and B, as shown in Fig. 4. Find the equations of motion of the two species of atoms constituting the basis. U2n-1, U2n U2n+1• Starting from solutions of form: Uzn = Aexpi(@t-k x2,); Uzn+1 = Bexpi(@t-k x2n+1), %3D %3D find the dispersion relations o = longitudinal branches as a function of B, B2, m (the mass of an atom), and a. Also determine the ratio of the amplitudes A/B for each of these branches in the center of the BZ (k = 0). Comment on the result. f(k), the acoustic and optical %3D B, U2n-2 U2n-1 U2n U2n+1 U2n+2 Figure 4

Exercise 1: Dispersion of longitudinal phonons in a row of atoms of type C=C-C%=C-C= Consider a linear lattice with constant "a" having as a basis two identical atoms situated in a line and spaced from equilibrium by b (b < a/2), see Chapter I, Ex. 15. The instantaneous position of the atoms is x1, X2, ... X2n-1, X2n X2n+1 ..., and the distance along the line relative to their equilibrium position is u1, U2,.. We consider only interactions between nearest neighbors which are characterized by the force constants B, and B, as shown in Fig. 4. Find the equations of motion of the two species of atoms constituting the basis. U2n-1, U2n U2n+1• Starting from solutions of form: Uzn = Aexpi(@t-k x2,); Uzn+1 = Bexpi(@t-k x2n+1), %3D %3D find the dispersion relations o = longitudinal branches as a function of B, B2, m (the mass of an atom), and a. Also determine the ratio of the amplitudes A/B for each of these branches in the center of the BZ (k = 0). Comment on the result. f(k), the acoustic and optical %3D B, U2n-2 U2n-1 U2n U2n+1 U2n+2 Figure 4

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Chemistry
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ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
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Exercise 1: Dispersion of longitudinal phonons in a row of atoms of type C=C-C=C-%3D Consider a linear lattice with constant "a" having as a basis two identical atoms situated in a line and spaced from equilibrium by b (b < a/2), see Chapter I, Ex. 15. The instantaneous position of the X2n-1, X2n X2n+1, .., and the distance along the line U2n-1, Uzn, Uzn+1• atoms is x1, X2, relative to their equilibrium position is u1, U2, We consider only interactions between nearest neighbors which are characterized by the force constants B, and B, as shown in Fig. 4. Find the equations of motion of the two species of atoms constituting the basis. Starting from solutions of form: Uzn = Aexpi(@t - x2,); uzn+1 = Bexpi(@t -k x2n+1), %3D %3D find the dispersion relations o = f(k), the acoustic and optical longitudinal branches as a function of B,, B2, m (the mass of an atom), and a. Also determine the ratio of the amplitudes A/B for each of these branches in the center of the BZ (k = 0). Comment on the result. %3D Bi a Un-1 U2n U2n+1 U2n+2 Figure 4 Knowing that the sound velocity along the row is v, = 5000 m/s (experimental value) find the frequency of atomic oscillations in the center and at the limit of the BZ with a = 5 Á, b = 1.25 Å, B1/B2 = b/(a – b). Show the corresponding dispersion curves and indicate the forbidden frequency interval. %3D %3D %3D %3D the
Transcribed Image Text:Exercise 1: Dispersion of longitudinal phonons in a row of atoms of type C=C-C=C-%3D Consider a linear lattice with constant "a" having as a basis two identical atoms situated in a line and spaced from equilibrium by b (b < a/2), see Chapter I, Ex. 15. The instantaneous position of the X2n-1, X2n X2n+1, .., and the distance along the line U2n-1, Uzn, Uzn+1• atoms is x1, X2, relative to their equilibrium position is u1, U2, We consider only interactions between nearest neighbors which are characterized by the force constants B, and B, as shown in Fig. 4. Find the equations of motion of the two species of atoms constituting the basis. Starting from solutions of form: Uzn = Aexpi(@t - x2,); uzn+1 = Bexpi(@t -k x2n+1), %3D %3D find the dispersion relations o = f(k), the acoustic and optical longitudinal branches as a function of B,, B2, m (the mass of an atom), and a. Also determine the ratio of the amplitudes A/B for each of these branches in the center of the BZ (k = 0). Comment on the result. %3D Bi a Un-1 U2n U2n+1 U2n+2 Figure 4 Knowing that the sound velocity along the row is v, = 5000 m/s (experimental value) find the frequency of atomic oscillations in the center and at the limit of the BZ with a = 5 Á, b = 1.25 Å, B1/B2 = b/(a – b). Show the corresponding dispersion curves and indicate the forbidden frequency interval. %3D %3D %3D %3D the
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