Question B2 Consider a one-dimensional chain of atoms, each joined to its nearest neighbours by a bond of spring constant J. The unit cell, of length a, contains one large atom of mass M, which we label = 1, and two smaller atoms of mass m, which we label l = 2 and l = 3. We denote the displacement of the eth atom in the nth unit cell by ur (note that this does not mean exponentiation): n-1 u +1 mm mmmmm M m m M m m a a) Give an expression for the force on each atom, F1, F2 and F3, in terms of the atomic displacements u. b) Use the standard ansatz u = u exp (i(kna - wt)) to write the equations of motion for this chain in the form (1) 01 = Pū² 03 0 where P is a 3 x 3 matrix. (Do not attempt to diagonalise this matrix.) (2) c) Show that at the Brillouin zone boundary (k = π/a), one solution to your matrix equation (2) is u₁ = 0, u² = u³ = 1. Calculate the angular frequency w of this mode. Explain why this value is independent of M. Is your answer consistent with the solution for a diatomic chain of alternating masses? (You may take the diatomic result from your notes or elsewhere without justification.) d) There is similarly a solution to (2) at the Brillouin zone centre (k = 0) for which the angular frequency is independent of M. Find this solution and determine its angular frequency w.

Question B2 Consider a one-dimensional chain of atoms, each joined to its nearest neighbours by a bond of spring constant J. The unit cell, of length a, contains one large atom of mass M, which we label = 1, and two smaller atoms of mass m, which we label l = 2 and l = 3. We denote the displacement of the eth atom in the nth unit cell by ur (note that this does not mean exponentiation): n-1 u +1 mm mmmmm M m m M m m a a) Give an expression for the force on each atom, F1, F2 and F3, in terms of the atomic displacements u. b) Use the standard ansatz u = u exp (i(kna - wt)) to write the equations of motion for this chain in the form (1) 01 = Pū² 03 0 where P is a 3 x 3 matrix. (Do not attempt to diagonalise this matrix.) (2) c) Show that at the Brillouin zone boundary (k = π/a), one solution to your matrix equation (2) is u₁ = 0, u² = u³ = 1. Calculate the angular frequency w of this mode. Explain why this value is independent of M. Is your answer consistent with the solution for a diatomic chain of alternating masses? (You may take the diatomic result from your notes or elsewhere without justification.) d) There is similarly a solution to (2) at the Brillouin zone centre (k = 0) for which the angular frequency is independent of M. Find this solution and determine its angular frequency w.

Similar questions

Consider the equation a=6πb/q√17 relating the variables a, b and q. If a scientist does an expirment in which b is held constant, q is varied, and a is measured. A. Identify the independent and dependent variables. B. What functional relationship is being testing? C. What should be plotted in order to confirm this relationship? (order matters)
Bryce, a mouse lover, keeps his four pet mice in a roomy cage, where they spend much of their spare time (when they are not sleeping or eating) joyfully scampering about on the cage's floor. Bryce tracks his mice's health diligently and just now recorded their masses as 10.3 g,18.5 g,11.3 g, and 18.3 g. At this very instant, the x‑ and y‑components of the mice's velocities are, respectively, (0.115 m/s, −0.527 m/s), ,(−0.237 m/s, −0.391 m/s), ,(0.843 m/s, 0.455 m/s), and (−0.897 m/s, 0.135 m/s). Calculate the x‑ and y‑components of Bryce's mice's total momentum, px and Py. px____kg.m/s py ____kg.m/s
1%A9 li. t LTET 4G+ Vol) O A X O U 9:-7 Second H.W.pdf 1. Draw and label the vector with these components. Then determine the magnitude and angle of the vector. a- Az = 3 ,A, = -2 b- B, = -2 ,B, = 2 2. What is the speed of horse in meter per second that runs a distance of 1.2 miles in 2.4 minutes? 3. A car moving along X- axis according on the function: X (t) = 2 t2 + 5t +7 on [1,2] Find: a- The average velocity. b- The acceleration at 3s. 4. Julie is walking around a track at a 2m/s for some exercise. She then decides to start jogging so she accelerates at a rate of 0.5m/s? for 3 seconds. How far did Julie travel from the time she started to accelerate to the end of the 3 seconds? 5. Ralph uses a 2000 to do this. force to push his car along a road for 1000 m. It takes him500 s a- Calculate the amount of work that Ralph did on the car. b- Calculate the amount of power that he generated. II
A uniform density sheet of metal is cut into the shape of an isosceles triangle, which is oriented with the base at the bottom and a corner at the top. It has a base B = 49 cm, height H = 15 cm, and area mass density σ. a) Consider a horizontal slice of the triangle that is a distance y from the top of the triangle and has a thickness dy. Write an equation for the area of this slice in terms of the distance y, and the base B and height H of the triangle. b) Set up an integral to calculate the vertical center of mass of the triangle, assuming it will have the form C ∫ f(y) where C has all the constants in it and f(y) is a function of y. What is f(y)? c)Integrate to find an equation for the location of the center of mass in the vertical direction. Use the coordinate system specified in the previous parts, with the origin at the top and positive downward. d)Find the numeric value for the distance between the top of the triangle and the center of mass in cm.
1 if we have (s) = ) In(Z), and kgT2 we substituted ß by show the steps kgT until you get this (s) = kgln (Z)
The table data: 15 1.75 13 1.63 11 1.5 9 1.35 7 1.19 5 1.01 3 0.78 A. Graph these data B. Describe the graph C. Using the data, calculate 1/2t^2 and complete table 2 below
One mole of Gold (Au) atom has a mass of 197 g. We know that the density of gold is 19.30 g/cm3 in room temperature. Using those numbers, answer the following questions: a) What is the mass of a single gold atom in kg? b) Assume that gold atoms sit in a perfect cubic crystal structure. Estimate the distance between two gold atoms in meters. c) If we have a gold cube of 6 cm on each side, how many atoms can we fit along the edge of each side of the cube? d) What would be the mass, in kilograms, of the gold cube we discussed in part (c) (measuring 6 cm on each side)?
The nucleus of an atom can be modeled as several protons and neutrons closely packed together. Each particle has a mass of 1.67 x 10-27 kg and radius on the order of 10-15 m. (a) Use this model and information to estimate the density of the nucleus of an atom. 3.986E177 Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. kg/m³ (b) Compare your result with the density of a material such as iron. What do your result and comparison suggest about the structure of matter?
Evaluate the cross products A⃗ ×B⃗ A→×B→ and C⃗ ×D⃗ C→×D→? What is the magnitude of the cross product A⃗ ×B⃗ A→×B→? Express your answer using two significant figures. What is the direction of the cross product A⃗ ×B⃗ A→×B→? What is the magnitude of the cross product C⃗ ×D⃗ C→×D→? Express your answer using two significant figures. What is the direction of the cross product C⃗ ×D⃗ C→×D→?
The density of platinum is 21.45 x 103 kg/m3. a. Calculate the volume (in m3/atom) occupied per platinum atom b. Estimate the atomic diameter (in m);(The estimate uses the approximation that it is a cubic volume) c. Using this estimation, calculate the thickness of a metal foil (in m) containing 2.0 x 101 atomic layers of platinum.
Eleven molecules have speeds 16, 17, 18, 19,20,21,22,23,24,25, 26 m/s. Calculate the root-mean-square of this group of molecules. in meters per second. Please give your answer with one decimal place.
Determine the indices for the direction shown in the following cubic unit cell: X Answer: Z For negative indexes use in front of the number. Use the proper type of brackets and no spaces. Answer: NIL Determine the Miller indices for the plane shown in the following unit cell: NIH y WIT For negative indexes use in front of the number. Use the proper type of brackets and no spaces.