$6\mathrm{.}12$ As noted in Section $3\mathrm{.}15,$ for single crystals of

some substances, the physical properties are

an isotropic$$that is$,$ they depend on crystallo

graphic direction. One such property is the modulus

of elasticity. For cubic single crystals, the modulus

of elasticity in a general $[$uvw$]$ direction, Euvw, is

described by the relationship

$1$

Euvw

$=1$

E$100$

$3($

$1$

E$100$

$1$

$($

$22+22+22)$

E$111$

$)$

$(6\mathrm{.}30)$ where E$100$ and E$111$ are the moduli of elasticity

in $[100]$ and $[111]$ directions, respectively; $,,$

and $$ are the cosines of the angles between $[$uvw$]$

and the respective $[100],[010],$ and $[001]$ direc

tions. Verify that the E$110$ values for aluminum,

copper, and iron in Table $3\mathrm{.}4$ are correct.