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    HW 3

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    meanyack
    This document provides instructions for homework 3 in condensed matter physics. It includes 3 problems to solve: 1) Analyzing the interplanar separation and distance between planes for a crystal lattice, including proving relationships for a simple cubic lattice. 2) Examining the hexagonal structure of a space lattice, including finding properties of the primitive cell and relationships between the direct and reciprocal lattices. 3) Proving relationships between the primitive lattice and reciprocal lattice vectors, including the volume of the primitive cell and establishing a connection between the direct and reciprocal lattice volumes.

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    HW 3

    Uploaded by

    meanyack
    466 views1 page
    This document provides instructions for homework 3 in condensed matter physics. It includes 3 problems to solve: 1) Analyzing the interplanar separation and distance between planes for a crystal lattice, including proving relationships for a simple cubic lattice. 2) Examining the hexagonal structure of a space lattice, including finding properties of the primitive cell and relationships between the direct and reciprocal lattices. 3) Proving relationships between the primitive lattice and reciprocal lattice vectors, including the volume of the primitive cell and establishing a connection between the direct and reciprocal lattice volumes.

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    This document provides instructions for homework 3 in condensed matter physics. It includes 3 problems to solve: 1) Analyzing the interplanar separation and distance between planes for a crystal lattice, including proving relationships for a simple cubic lattice. 2) Examining the hexagonal structure of a space lattice, including finding properties of the primitive cell and relationships between the direct and reciprocal lattices. 3) Proving relationships between the primitive lattice and reciprocal lattice vectors, including the volume of the primitive cell and establishing a connection between the direct and reciprocal lattice volumes.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    466 views1 page

    HW 3

    Uploaded by

    meanyack
    This document provides instructions for homework 3 in condensed matter physics. It includes 3 problems to solve: 1) Analyzing the interplanar separation and distance between planes for a crystal lattice, including proving relationships for a simple cubic lattice. 2) Examining the hexagonal structure of a space lattice, including finding properties of the primitive cell and relationships between the direct and reciprocal lattices. 3) Proving relationships between the primitive lattice and reciprocal lattice vectors, including the volume of the primitive cell and establishing a connection between the direct and reciprocal lattice volumes.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
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    Phys 445 Condensed Matter Physics

    Homework 3
    Due to October 15th 2010, Friday 9:40 in class

    1) Interplanar separation. Consider a plane hkl in a crystal lattice. (a) Prove that the
    reciprocal lattice vector G = hb1 + kb2 + lb3 is perpendicular to this plane. (b) Prove that
    the distance between two adjacent parallel planes of the lattice is d (hkl ) = 2π / G (c)
    Show for a simple cubic lattice that d 2 = a 2 / (h 2 + k 2 + l 2 ) .

    2) Hexagonal structure: Using the primitive vectors of the hexagonal space lattice
    3a a
    a1 = xˆ + yˆ ;
    2 2
    3a a
    a2 = − xˆ + yˆ ;
    2 2
    a3 = czˆ
    (a) Show that the volume of the primitive cell is ( 3 / 2)a 2 c .
    (b) Find the primitive translation vectors of the reciprocal lattice and show that the
    reciprocal of the simple hexagonal Bravais lattice also simple hexagonal, with
    lattice constants 2π / c and 4π / 3a , rotated through 30o about c-axis with
    respect to the direct lattice.
    (c) For what value of c/a does the ratio have the same value both in direct and
    reciprocal lattice? If c/a is ideal in direct lattice, what is its value in the reciprocal
    lattice?

    3) (a) Prove that the reciprocal lattice vectors satisfy


    (2π )3
    b1 i(b2 × b3 ) =
    a1 i(a2 × a3 )
    (b) Prove that the volume of a Bravais lattice primitive cell is
    Vc = a1 i(a2 × a3 )
    where ai are the three primitive lattice vectors. Combining this with the relation
    showed in (a) establishes that the volume of the reciprocal lattic eprimitive cell is
    (2π )3 / Vc .
    (c) Consider the primitive vectors which are constructed from the reciprocal lattice
    vectors bi as in the same manner as the bi s are constructed from the ai .Prove that
    these vectors are just the ai thenselves; i.e. show that
    b2 × b3
    2π = a1
    b1 i(b2 × b3 )
    and so on.

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