A crystalline solid consists of atoms stacked up in a repeating lattice structur

Question

A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure P1.7a. The atoms reside at the corners of cubes of side L=0.200 nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal as shown in Figure p1.7b. Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves.

P1.7a P1.7b

OOOOOOO OO OOOOO
OOOOOOO OOO OOOOO
OOOOOOO (rectangular prism) OOOO OOOOO (spacing in between =d)

Explanation / Answer

given that, the length of the side of the corners of the cubes is                                    L = 0.2 nm                                       = (0.2 nm)(10-9 m / 1 nm)                                       = 0.2*10-9 m as shown given figure, diagonal distance between the planes is                           D = [L2 + L2]                           D = [2L2]                           D = [2L2]                                = (2)(L)    ...... (1) from given figure, we have the space between the diagonal planes is equal to half the distance between diagonally adjacent atoms on a plane. that is, the space between the diagonal planes is equal to half the distance between diagonally adjacent atoms on a plane. that is,                      d = D/2                         = (1/2)[(2)(L)]                         = (1/2)[(2)(0.2*10-9 m)]                         = 0.141*10-9 m (or)                      d = 0.141 nm

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