Reciprocal System of Vectors
Reciprocal System
Description:-
In Physics and solid Chemistry we come across various lattice structures ... In this applet you will see that how to form a reciprocal system of a basis system of 3 vectors in R^3 space ...
consider that a,b,c are the three non coplanar non zero vectors in R^3,
if we construct a system of a',b',c' such that
a.a'=b.b'=c.c'=1
then a',b',c' is the reciprocal system of basis a,b,c ...
we can find them by using the Scalar Triple Product...
as
a.(b x c) = b.(c x a)=c.(a x b)=[a b c] ,
=> a.((b x c)/[a b c]) = b.((c x a)/[a b c]) = c.((a x b)/[a b c]) = 1
(as [a b c] is scalar..)
hence a'=(b x c)/[a b c] , b'=(c x a)/[a b c] , c'=(a x b)/[a b c]
observe that a' is perpendicular to plane of b and c ,
b' is perpendicular to plane of c and a , and hence so c'.