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    Kittel Chapter 9 Solutions

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    AbulLais89
    This document contains homework problems from Chapter 9 of Kittel's solid state physics text. The problems involve calculating Brillouin zones for different lattice structures, sketching Fermi surfaces, and calculating properties of open electron orbits in metals. One problem asks the student to sketch the first two Brillouin zones of a rectangular lattice. Another problem has the student calculate the period and real space motion of an electron on an open orbit in a monovalent tetragonal metal.

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    Kittel Chapter 9 Solutions

    Uploaded by

    AbulLais89
    2K views23 pages
    This document contains homework problems from Chapter 9 of Kittel's solid state physics text. The problems involve calculating Brillouin zones for different lattice structures, sketching Fermi surfaces, and calculating properties of open electron orbits in metals. One problem asks the student to sketch the first two Brillouin zones of a rectangular lattice. Another problem has the student calculate the period and real space motion of an electron on an open orbit in a monovalent tetragonal metal.

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    Solution Manual to Kittel's Solid State Physics introductory textbook!

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    This document contains homework problems from Chapter 9 of Kittel's solid state physics text. The problems involve calculating Brillouin zones for different lattice structures, sketching Fermi surfaces, and calculating properties of open electron orbits in metals. One problem asks the student to sketch the first two Brillouin zones of a rectangular lattice. Another problem has the student calculate the period and real space motion of an electron on an open orbit in a monovalent tetragonal metal.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    2K views23 pages

    Kittel Chapter 9 Solutions

    Uploaded by

    AbulLais89
    This document contains homework problems from Chapter 9 of Kittel's solid state physics text. The problems involve calculating Brillouin zones for different lattice structures, sketching Fermi surfaces, and calculating properties of open electron orbits in metals. One problem asks the student to sketch the first two Brillouin zones of a rectangular lattice. Another problem has the student calculate the period and real space motion of an electron on an open orbit in a monovalent tetragonal metal.

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    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
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    You are on page 1of 23

    Homework # 8

    Chapter 9 Kittel Phys 175A Dr. Ray Kwok SJSU

    Prob. 1 Brillouin zones of rectangular lattice

    Daniel Wolpert

    9.1 Brillouin zones of rectangular lattice. Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a, b=3a 2/a 2/3a

    9.1 Brillouin zones of rectangular lattice. Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a, b=3a

    9.1 Brillouin zones of rectangular lattice. Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes a, b=3a 2/a 2/3a
    2nd BZ First BZ

    Prob. 2 Brillouin zone,rectangular lattice


    Gregory Kaminsky

    This is a Wigner-Seitz cell.

    A two-dimensional metal has one atom of valency one in a simple rectangular primitive cell a = 2 A0 ; b = 4 A0.

    a) Draw the first Brillouin zone. Give its dimensions in cm-1. b) Calculate the radius of the free electron fermi sphere. c) Draw this sphere to scale ona drawing of the first Brillouin zone.

    Calculation of the radius of the Fermi sphere


    2 2 * * kF

    2 * kF 2* 2 = N 2

    4 * 0 2 4*2*(A )
    2

    =1

    kF = 2 * A0 =

    * 1012 cm 1

    Brilloin zone Radius of free electron fermi sphere =

    *1012 cm 1

    * 1012 cm 1

    Make another sketch to show the first few periods of the free electron band in the periodic zone scheme, for both the first and second energy bands. Assume there is a small energy gap at the zone boundary.

    This is the first energy band

    Second energy band

    Prob. 4 Brillouin Zones of Two-Dimensional Divalent Metal


    Victor Chikhani
    A two dimensional metal in the form of a square lattice has two conduction electrons per atom. In the almost free electron approximation, sketch carefully the electron and hole energy surfaces. For the electrons choose a zone scheme such that the Fermi surface is shown as closed.

    Hole Energy surface

    Electron Energy Surface

    BZ periodic scheme

    Second Zone periodic scheme

    Prob. 5 Open Orbits

    John Anzaldo

    An open orbit in a monovalent tetragonal metal connects opposite faces of the boundary of a Brillouin zone. The faces are separated by
    G = 2 108 cm 1.

    A magnetic field B = 10 1T

    is normal to the place of the open orbit. (a) What is the order of magnitude of the period of thek motion in
    8 Take v = 10 cm / s

    space?

    (b) Describe in real space the motion

    of an electron on this orbit in the presence of the magnetic field.

    9.5
    From Eq. 25a we have , where I have decided to use SI units. v v v G d r q = e dt = h = ev B Letting we get = v , setting dk = G dt because v B since B is normal to the Fermi surface. Solving for gives Gh = . Plugging in the givens we evB get
    v v dk dr v h = q B dt dt

    2 2 108 100cm 34 kg m 6.62 10 Gh cm m s = = 1.315 10 10 s evB 2 1.602 10 19 C 108 cm 1m 10 1 kg s 100cm Cs

    Part b)
    The electron will travel along the Fermi surface as shown. The velocity will change as the electron moves along the Fermi surface.

    Mike Tuffley 5/12/09

    U(x)

    -a/2

    a/2 x

    -U0

    Chapter 9 Problem 7

    Adam Gray

    1 (a) Calculate the period ( B ) expected for

    potassium on the free electron model. (b) What is the area in real space of the extremal orbit, for B = 10kG = 1T ?

    Starting with equation 34:


    1 2e ( ) = B hcS

    Where

    S = K

    2 f

    Using Table 6.1 on pg. 139, for potassium we find kf=0.75x108cm-1 .

    Plugging in:

    1 2e ( ) = B hc(K 2 f )

    1 2e ( ) = B hcK2 f
    Note: The equation 34 was for cgs units, so all values used with this equation must be in this form. c=3x1010 cm/s h=1.05459x10-27 erg s e=4.803x10-10 erg1/2 cm1/2

    This results in

    1 ( ) = 5.55109 G 1 B

    (b) To solve this part of the problem, go back to the equations we used for the cyclotron.

    Be c = mc
    Solve for r

    r =

    P = mv = hk

    vf v f mc hk f c r = = = = c Be Be Be mc vf

    Plugging in values from before and B=10kG r = 4.94x10-4 cm The orbit is circular, so the area is

    r = 7.67 10 cm

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