Condensed Matter
1) Obtain expressions for the heat capacity due to longitudinal vibrations of a chain of
identical atoms;
(a) in the Debye approximation;
(b) using the exact density of states (Eq. below).
With the same constants K and M, which expression gives the greater heat capacity and
why? Show that at low temperatures both expressions give the same heat capacity,
proportional to T.
g(w)
Hint:
L (M\1/2
ла K
&F
sec (ka)
2N (4K
1/² - 0²)
M
2) Estimate the Fermi temperatures of:
(a) liquid ³He (density 81 kg m-³), and
(b) the neutrons in a neutron star (density 10¹7 kg m−³).
<-1/2
h² (3π²N\2/3
2m
N(M¹/2 2(K/M)¹/2
T K (4K/M-²)¹/2
3) The Bragg angle for a certain reflection from a powder specimen of copper is 47.75° at a
temperature of 293 K and 46.60° at 1273 K. Calculate the coefficient of linear thermal
expansion of copper.
4) The crystal basis of graphene and of diamond is composed of two
carbon atoms in nonequivalent position. Thus, their dispersion curves are composed of
acoustical branches and optical branches and their acoustical branches are assumed to obey
to the Debye approximation: = vs|k| and their optical branches are assumed to obey to the
Einstein model @= @E = Cst. Deduce the numerical values of their Debye and Einstein
temperatures from their crystal structure and their common sound. velocity, Vs = 18,000 m/s
with also VE (Einstein frequency) at about 4 *10¹3 Hz. From the Cv graph shown in below,
evaluate the specific heat of diamond at room temperature, 290 K. (h, kB) 100
75
50
25
0
C, (%)
x 3;2;1 NkB unit
0
1D
-2D
-3D
T/0₂
1
Heat capacity, C, as a function of T/05 for 1D, 2D, and 3D solids.
The vertical scale is in Nkg unit to multiply by 1, 2, or 3 as a
function of the degree of freedom for the atom vibrations.
Note the initial evolution in T, T2, or 73 as a function of the
dimensionality of the solid.
5) (1) Knowing that lithium crystallizes in a cubic system with lattice considering its atomic
mass (7) and its volumetric mass (546 kg⋅m-³), find which one is it ([simple cubic, body-
centered cubic (bcc), or face-centered cubic (fcc)]?
(2) Knowing that the valence electrons of this metal (1 per atom) behave as free electrons,
find the shape of the Fermi surface and its expression and then calculate its characteristic
dimension KF.
(3) Compare KF obtained in (2) to the distance dm, which in reciprocal space separates the
origin from the first boundary of the first Brillouin zone nearest the origin. (Evaluate dm using
simple geometric considerations without having to sketch the first Brillouin zone.)
(4) Find the Fermi energy of lithium EF, the Fermi temperature TF, and the speed of F of the
fastest free electrons.
(5) Knowing that the resistivity p of lithium is of the order 10-5 cm at ambient temperature,
find the time of flight, t, and the mean free path A of conduction electrons.
(6) Find the drift velocity vd of conduction electrons subject to a electric field of 1 V/m and
compare it with the Fermi velocity VF.
(7) Starting from the relation ke =1/3 CeVFA (or with the help of the Wiedemann-Franz
expression), find the thermal conductivity due to electrons Ke of lithium at ambient
temperature T = 300 K.
Most Viewed Questions Of Condensed State Physics
4.5 Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is 60.0°. If dog A exerts a force of 270 N and dog B exerts a force of 300 N, find the magnitude of the resultant force and the angle it makes with dog A’s rope.
The acceleration of a rocket traveling upward is given by a =(6 + 0.02s)m/s², where s is in meters. Determine the time needed for the rocket to reach an altitude of s =100m. Initially, v=0 and s= 0 when t = 0.
A daring 510 N swimmer dives off a cliff with a running horizontal leap, as shown in the figure. What must her minimum speed be just as she leaves the top of the cliff so that she will miss the ledge at the bottom, which is 1.75 m wide and 9.00 m below the top of the cliff?
II) A high-speed drill reaches 2000 rpm in 0.50 s. ! What is the drill's angular acceleration? Through how many revolutions does it turn during this first0.50 s?
An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s-!. This speed is then maintained forT seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 min 75 s. (a) Sketch a speed-time graph to illustrate the motion of the athlete. (a) Sketch a speed-time graph to illustrate the motion of the athlete. (b) Calculate the value of T.
A farmer hitches her tractor to a sled loaded with firewood and pulls it a distance of 27 m along level ground. The total weight of and load is 12,000 N. The tractor exerts a constant 6000-N force at an angle of 35.0° above the horizontal. There is a 8600 N nor force on the sled from the ground, and a 3500 N friction force opposing the sled's motion. Match each force with the expression or value that corresponds to the work it does on the sled. Each force has a only one correct match. Incorrect expressions must be matched to "No matching force". 1. Tension on the chain pulling the sled 2. Weight of the sled 3. Normal force from the ground on the sled 4. Kinetic friction on the sled 5. No matching force
2/123 During a short interval the slotted guides are designed to move according to x =16 – 12 t + 4t² and y = 2 + 15t – 3t², where x and y are in millimeters and t is in seconds. At the instant when t = 2 s determine the radius of curvature p of the path of he constrained pin P.
5. A car is 2.0 km west of a traffic light at t= 0 and 5.0 km east of a traffic light at t=6 min. Assume the origin of the coordinate system is the light and the positive x direction is eastward. (3 points) а.What are the car's position vectors at these two times? b. What is the car's displacement between 0 min and 6 min? What is the car's average velocity between the two points? Record your answer in terms of kilometres per hour.
9. Three charges with q = +7.5 × 10-6 C are located as shown in Figure 7, with L = 25 cm. Determine the magnitude and direction of the total electric force on each particle listed below. (a) the charge at the bottom (b) the charge on the right (c)an electron placed at the origin . Two pith balls, each with a mass of 5.00 g, are attached to non-conducting threads and suspended from the same point on the ceiling. Each thread has a length of 1.00 m. The balls are then given an identical charge, which causes them to separate. At the point that the electric and gravitational forces balance, the threads are separated by an angle30.0°. Calculate the charge on each pith ball. K ASS
In one type of computer keyboard, each key holds a small metal plate that serves as one plate of a parallel-plate, air-filled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard,the area of each metal plate is 44.5 mm, and the separation between the plates is 0.71n mm before the key is depressed. Calculate the capacitance before the key is depressed. If the circuitry can detect a change in capacitance of 0.300 pF, how far must the key be depressed before the circuitrydetects its depression?
