The Solid State

  • Question
    CBSEENCH12010420

    Calculate the packing efficiency of a metal crystal for a simple cubic lattice.

    Solution

    In a simple cubic lattice, the particles are located only at the corners of the cube and touch each other along the edge.

    In a simple cubic lattice, the particles are located only at the corners of the cube and touch each other along the edge.

    Let the edge length of the cube be ‘a’ and the radius of each particle be r.

    So, we can write:

    a = 2r

    Now, volume of the cubic unit cell = a3 = (2r)3 = 8r3

    We know that the number of particles per unit cell is 1.

    Therefore, volume of the occupied unit cell =  4 over 3 pi r cubed
    Hence comma space packing space efficiency space equals space fraction numerator Volume space of space one space particle over denominator Volume space of space cubic space unit space cell end fraction space straight x space 100 percent sign

space equals space fraction numerator begin display style 4 over 3 end style πr cubed over denominator 8 straight r cubed end fraction space straight x space 100 percent sign
equals 1 over 6 straight pi space straight x space 100 percent sign
equals 1 over 6 space straight x 22 over 7 space straight x 100 percent sign
space equals 52.4 percent sign

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