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
2
Also: → − → − − a1 × (→ a2 × → a3 )| T = |− − = |→ a × a a sin α n ˆ| 1
⇒ 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
1
Volume of primitive cell of monoclinic lattice − − − V = |→ a1 .→ a2 × → a3 | → − = |a . a a sin(90◦ ) n ˆ| 1
2
3
− − (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
Thank you for interesting in our services. We are a non-profit group that run this website to share documents. We need your help to maintenance this website.