Volume of Primitive Cell

February 26, 2017 | Author: Arooj Mukarram | Category: N/A
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Volume of Primitive Cell Triclinic and Monoclinic Lattice Volume of primitive cell of triclinic lattice − − − V = |→ a1 .→ a2 × → a3 | → − = |a . a a sinα n ˆ| 1

2

3

=a1 a2 a3 sin α cos θ − − − − − (1) − (θ: angle between n ˆ and → a1 ) Consider vector triple product: → − → − − T =− a1 × (→ a2 × → a3 ) → − → − → − − − − =a (a .a ) − → a (→ a .→ a ) − − − − − (2) 2

1

3

3

1

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Also: → − → − − a1 × (→ a2 × → a3 )| T = |− − = |→ a × a a sin α n ˆ| 1

2

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=a1 a2 a3 sin α sin θ − − − − − (3) Using (2) and (3): − − − − − − − − − − − − − − − − − − − − [→ a2 .→ a2 (→ a1 .→ a3 ) − → a2 .→ a3 (→ a1 .→ a2 )](→ a1 .→ a3 ) + [−→ a2 .→ a3 (→ a1 .→ a3 ) + → a3 .→ a3 (→ a1 .→ a2 )](→ a1 .→ a2 ) =(a1 a2 a3 sin α sin θ)2 −→ − → − − − − − − − a22 (→ a1 .→ a3 )2 − 2(→ a2 .a3 )(→ a1 .− a2 )(→ a1 .→ a3 ) + a23 (→ a1 .→ a2 )2 =a21 a22 a23 sin2 α sin2 θ a21 a22 a23 (cos2 β − 2 cos α cos β cos γ + cos2 γ) =a21 a22 a23 sin2 α sin2 θ cos2 β − 2 cos α cos β cos γ + cos2 γ = sin2 α(1 − cos2 θ)

⇒ sin2 α cos2 θ =1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ p sin α cos θ = 1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ Use in (1): V =a1 a2 a3

p

1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ

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Volume of primitive cell of monoclinic lattice − − − V = |→ a1 .→ a2 × → a3 | → − = |a . a a sin(90◦ ) n ˆ| 1

2

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− − (ˆ n: Unit vector perpendicular to plane containing → a2 and → a3 .) → − ◦ Angle between n ˆ and a1 is 90 + β. V = |a1 a2 a3 cos(90◦ + β)| = |−a1 a2 a3 sin β| ⇒ V =a1 a2 a3 sin β

2

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