Be (beryllium) has an atomic mass of 9.0 amu (1 amu = 1.7 middot 10^-27 kg.) It

Question

Be (beryllium) has an atomic mass of 9.0 amu (1 amu = 1.7 middot 10^-27 kg.) It crystallizes in the hexagonal structure shown in 3 with lattice parameters ||a|| = a = ||b|| = b = 0.23 nm ||c|| = c = 0.36 nm and angles alpha = beta = 90 degree, y = 120 degree (c points vertical out of the plane). (A) Express the components of a, b, c in the Cartesian coordinate system xyz. (B) What is the theoretical density (mass per unit volume) of Be in this structure? (C) Optional: Should Be float on water? Cos (phi) = sin (90 degree - phi), sin (30 degree) = 1/2, sin (60 degree) = pi/2 = 0.87. Numerical precision is not important.

Explanation / Answer

|a| = |b| = 0.23 nm

b is along y axis

b^ = 0.23 j nm

larger angle between a and b = 120 deg.

    -ve y-component      = 0.23sin(30) = 0.165 nm

a^ = 0.20 i - 0.165 j

smaller angle between and b = 180-120 = 60 deg

c is along z -axis and makes 90 deg

c^ = 0.36 k nm

volume of the hexagonal   =a2*cSin(60) = 0.0165 nm3

   there are 2 atoms in a unit cell for Be

mass   = 2 *9*1.7e-27 kg = 183.6 e-27 kg

density of Be = 2*183.6 e-27 kg/ (0.0165 nm3) = 1854.5 kg/m3

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