固体物理作业题.doc

固体物理学Chapter 1Problem 1.1:Compute the packing fraction f for the bcc lattice.Problem 1.2:(a) Show that the packing fraction f for the diamond lattice is 3 /16 .(b) What is the packing fraction and coorination number of the honeycomb lattice?Problem 1.3:Consider the hexagonal close packed lattice. (a) Show that c = a 8 3 = 1.633a . Frequently a crystalstructure is called hcp even c is not exactly equal to the ideal value. (b) Show that the packing fraction forthe ideal hcp lattice is 2 / 6 = 0.7405Problem 1.4:The ionic compound A+B- crystallizes in the NaCl structure. Plot the packing fraction as a function of theratio + = r / r . Assume that 1.Problem 1.5:Repeat the calculation of problem 1.4 for the CsCl structure.Problem 1.6:Use the information in the textbook to calculate the densities (in kgm-3) of the following solids: (a)Aluminum, (b) Iron, (c) Silicon and (d) Zinc. Atomic weights of some common elements are listed in thetextbook.Problem 1.7:SrTiO3 crystallizes in the perovskite structure. The strontium atoms are at the corners of the cube with sidea, the titanium atoms are at the body center, while the oxygen atoms occupy the cube faces.(a) What is the Bravais lattice type?2(b) Verify that the primitive unit cell contains one Sr, one Ti and three O atoms.(c) Write down a set of primitive lattice vectors and basis vectors for the perovskite structure.Problem 1.8:The primitive lattice vectors of a certain Bravais lattice can be writtenR n n ax n by n zv r r r1 2 2 1 3( 2 ) 12= 1 + + +What is the lattice type?Problem 1.9:In each of the following cases indicate whether the structure is a Bravais lattice. If it is, give three primitivelattice vectors. If it is not describe it as a Bravais lattice with as small as possible basis. In all cases thelength of the side of the unit cube is a.(a) Base centered cubic (simple cubic with additional points in the centers of the horizontal faces of thecubic cell).(b) Side centered cubic (simple cubic with additional points in the centers of the vertical faces of the cubiccell).(c) Edge centered cubic (simple cubic with additional points at the midpoints of the lines joining nearestneighbors).Problem 1.10:指出体心立方晶格(111) 面与(100) 面， (111) 面与(110) 的交线的晶向。Chapter 2Problem 2.1:(i) What is a lattice? Express a lattice mathematically.(ii) What do you mean by a “basis”?(iii) How can you combine a lattice with a basis to obtain a crystal structure?Problem 2.2:(i) What are the 3 fundamental translation vectors?(ii) Show with 2 dimensional examples, how the fundamental translation vectors may defineeither a non-primitive (conventional) unit cell or a primitive unit cell.(iii) Express mathematically the size (area in 2 dimensional & volume in 3 dim) of a unit cell.Problem 2.3:(i) What is a Bravais lattice?(ii) Draw the five 2-dimensional Bravais lattices clearly showing the fundamental latticetranslation vectors. What is the difference between a centered rectangular lattice and asimple rectangular lattice?Problem 2.4:(i) How many 3 dimensional Bravais lattices are present?(ii) How many 3 dimensional crystal systems are there?(iii) Make a Table having the following columns: Lattice System - Bravais lattice - Diagramof Conventional unit -Name & Symbol unit cell - Cell characteristics.Problem 2.5:(i) What do you mean by packing fraction?(ii) Calculate the packing fraction in a simple cubic, base centered cubic and a face centeredcubic structures.(iii) In which structure are the atoms most closely packed?(iv) What do you mean by “coordination number” in a crystal structure?2(v) Explain with diagram the NaCl structure and the CsCl structure? What is the structure ofdiamond?Problem 2.6:(i) What are crystal planes?(ii) What do the Miller Indices represent?(iii) What do the following indices represent? (hkl), hkl,hkl and ?(iv) Draw the unit cell & the following planes in a simple cubic lattice: (100), (00), (200),(11), (201), (20), (122).(v) What do you mean by crystal lattice inter planar spacing (dhkl )?(vi) Write the formulae for dhkl for a orthogonal lattice and a cubic lattice. Also write theformula for the angle between 2 planes in a cubic lattice.Problem 2.7:(i) What is the density of atoms (number per unit area) on a (111) plane of a fcc lattice?(ii) What is the density of atoms (number per unit area) on a (110) plane of a bcc lattice?Problem 2.8:(i) Construct the Wigner-Seitz primitive cells for one-dimensional lattice.(ii) Construct the Wigner-Seitz primitive cells for square lattice and for honeycomb lattice(i.e., hexagonal lattice) (2D).(iii) Construct the Wigner-Seitz primitive cells for simple cubic, body-centered cubic andface-centered cubic lattices (3D), respectively.Problem 2.9:(i) Construct the reciprocal lattice for two-dimensional rectangular lattice, square lattice,oblique lattice and hexagonal lattice.(ii) Construct the reciprocal lattices for simple cubic, body-centered cubic and face-centeredcubic lattices.3Problem 2.10:(i) Construct the first Brillouin zone for two-dimensional rectangular lattice, square lattice,oblique lattice and hexagonal lattice.(ii) Construct the first Brillouin zone for simple cubic, body-centered cubic andface-centered cubic lattices.Problem 2.11:考虑晶格中的一个晶面hkl。(i) 证明倒格矢垂 1 2 3 G hb kb lbr v v v= + + 直于这个晶面。(ii) 证明晶格中两个相邻平行晶面的间距为Ghkl d r2 ( ) =(iii) 证明对于简单立方晶格有d = a/ h2 + k 2 + l 2Problem 2.12:证明第一布里渊区的体积为 (2)3/Vc。其中Vc 是晶格原胞的体积。提示：布里渊区的体积等于傅里叶空间中的初基平行六面体的体积，同时利用矢量恒等式(c a) (a b) = (c a b)aProblem 2.13:(i) Explain in short how X-rays can be diffracted by a crystal. A neutron beam can also beused instead of X-rays to study diffraction. Why? (State de Broglies hypothesis of matterwaves i.e. wave and particle duality: =h/p).(ii) Draw a neat diagram and deduce Braggs Law for diffraction by a crystal (2d sin = n).Visible light cannot be used to study diffraction by crystals, why?(iii) The Bragg angle for reflection from the (111) planes in Al (fcc) is 19.2 degrees for anX-ray wavelength of =1.54 . Compute: (i) the length of the cube edge of the unit cell;(ii) the interplanar distance for these planes. (Ans 4.04 angstrom and 2.33 angstrom).Problem 2.14:4An x-ray source emits an x-ray line of wavelength l = 0.154 nm. The lattice constant andcrystal structures of iron and aluminum are found in the tables listed in the textbook.(i) Find the Bragg angle(s) for reflections from the (111) planes of Al.(ii) Find the Bragg angle(s) for reflections from the (110) planes of Fe.Problem 2.15:The primitive lattice vectors of a 2-dimensional triangular lattice area air r = ;b a i ajr r r232= +where a is the nearest neighbor distance.(i) Find the reciprocal lattice(ii) Draw the Wigner Seitz cell and locate the coordinates of its corners.(iii) Draw the Brillouin zone and locate the coordinates of its corners.Problem 2.16:An X-ray reflection from a certain crystal occurs at an angle of incidence of 45o when thecrystal is maintained at 0oC. When it is heated to 150oC the angle changes by 6.4 minutes ofarc. What is the linear thermal expansion coefficient of the material?Chapter 3Problem 3.1On the origin of Van der Waals force(a) Give a qualitative interpretation on the origin of the Van der Waals force.(b) Give a quantitative interpretation on the origin of the Van der Waals.Problem 3.2An approximate way of combining the repulsive and attractive interactions between the atoms in amolecular crystal is the Lennard-Jones potential= = 12 612 6 ( ) - 4 -r r rBrV r A where A and B are constants which depend on which atom or molecule is involved. It is conventional toprameterize the potential in terms of an energy parameter and length parameter . Table 3.1 lists theLennard-Jones parameters for inert gases.Table 3.1 lists the Lennard-Jones parameters for inert gasesElement (angstrom) (eV)Ne 2.74 0.0031Ar 3.40 0.0104Kr 3.65 0.0140Xe 3.98 0.0200(a) Plot the Lennard-Jones potential and force for the inert gases Ne, Ar, Ke and Xe, respectively.(b) Derive the equilibrium distance ro for the inert gases in Table 3.1.(c) Derive the potential in the neighborhood of the equilibrium distance ro for the inert gases in Table 3.1(d) Compare with the result found in Table 4 (C. Kittel, p. 41).Problem 3.3The lattice parameters of KCl are given in table 5.1(a) Calculate the Coulomb energy between a K+ and a Cl- ion at the nearest neighbour distance in units ofeV.(b) Assume that the parameters s and e of the van der Waals attraction between the ions (the termproportional to 1/r6 in the Lennard-Jones potential) are the same as for Ar (table 3.1). Calculate thevan der Waals energy between a K+ and a Cl- ion at the nearest neighbor distance of KCl. Comparewith the result found under (a).Problem 3.4Calculate the Madelung constant for the crystal structure of NaCl and compare your results with thoselisted in table 3.2.Crystal structure of NaClTable 3.2 Madelung constant for some crystal structures listed in tableStructure NaCl 1.7476CsCl 1.7627ZnS 1.6381Chapter 4Problem 4.1Consider a linear chain of atoms. Each atom interact with its nearest neighbor on either side via aLennard-Jones potential. Assume parameter values approximate to krypton (table 3.1)(a) Find the equilibrium spacing between the atoms.(b) Find the sound velocity.(c) What is the maximum frequency?Problem 4.2The harmonic chain model can be solved also when the interaction between the masses extends beyondthe nearest neighbors. Consider the case when the nth mass is connected to masses n+1 and n-1 with thespacing constant K1 and to masses n+2 and n-2 with the spacing constant K2. The equation of motion forthe nth mass is now( n n n ) ( n n n ) mu K u u -2u K u u -2u 1 1 -1 2 2 -2 = + + + + + &Assume periodic boundary conditions and solutions of the formu Aei(kan-t) n =where a is equilibrium lattice spacing.(a) Find a formula for w as a function of k.(b) Plot 1 m/K vs ka for the special case K1=K2.Chapter 5Problem 5.1In sodium metal each ion contribute one conduction electron. Using the data in Table 5.1 calculate forsodium(a) The Fermi energy(b) The Fermi velocity(c) The Fermi temperatureProblem 5.23He atoms can be considered as Fermi particles. At low temperatures 3He forms a liquid with a volumeof 4.6210-29 cm3 per helium atom. The mass of a 3He atom is 510-27 kg. Estimate the Fermi temperatureof 3He.Problem 5.3Estimate the electronic and lattice specific heats for Al at temperatures 1K, 10K, 100K in units of Jmol-1K-1. Use the free electron model for the electrons and the Debye model for the phonons. Aluminum istrivalent.Table 5.1 Crystal structure of some common substance. Unless specified the temperature is 300K.Substance Structure a (angstrom) c (angstrom)Ag fcc 4.09Al fcc 4.05AgBr NaCl 5.77Ar (at 4K) fcc 5.31Au fcc 4.08Be hcp 2.27 3.59C(diamond) diamond 3.57CdS ZnS 5.82Co hcp 2.51 4.07Cr Bcc 2.88CsCl CsCl 4.11Cu fcc 3.61CuCl ZnS 5.41Fe Bcc 2.87Ge diamond 5.66InSb ZnS 6.46K bcc 5.23KCl NaCl 6.29Kr(4K) fcc 5.64Na bcc 4.23NaCl NaCl 5.63Ne (4K) fcc 4.46Pb Fcc 4.95Si Diamond 5.43Xe(4K) Fcc 6.13Zn Hcp 2.66 4.95ZnS ZnS 5.41Table 5.2 Atomic weights of selected elementsElement Symbol Atomic weightAluminum Al 26.982Argon Ar 39.948Copper Cu 63.55Iron Fe 55.847Krypton Kr 83.80Silicon Si 28.086Zinc Zn 65.38Atomic mass unit amu = 1.6604210-27 kgChapter 6Problem 6.1 Calculate the kinetic energy of N electrons in 3D system zero temperatureU NEF530 = .Problem 6.2 Calculate the Fermi energy of a metal. What is the relationship between the electron density and theFermi energy in the free electron model? Explain how this relationship would be derived. Hint: Thevolume in k-space corresponding to one allowed k-state is (2)/L), where L is the length of thecrystal. The total volume of the crystal is L. There are twice as many electrons states as k-states dueto spin.Problem 6.3 Derive the relationship connecting pressure and volume of an electron gas at 0 K. The result may bewritten as p = 2(U0/V)/3. Show that the bulk modulus B = -V(dp/dV) of an electron gas at T = 0 isB = 5p/3 = 10U0/9V.Problem 6.4 Average electron energy in 2-D. For a metal at a temperature of T = 0 K, the conduction electrons atthe bottom of the band have an energy E = 0 and the conduction electrons with the highest energyhave an energy E = EF. For free electrons in two-dimensions, what is the average energy of theconduction electrons?Problem 6.5Find the density of states as a function of energy for a non-interacting free electron gas in twodimensions. For this system it is possible to find an analytic expression for the temperaturedependence of the chemical potential. Show that( ) = ln