Solving Algebraic Relations of Reciprocal Lattice Vectors

[dingo_d]

Homework Statement

Basically I have reciprocal lattice vectors:

$$a'=\frac{b\times c}{a\cdot(b\times c)}$$
$$b'=\frac{c\times a}{a\cdot(b\times c)}$$
$$c'=\frac{a\times b}{a\cdot(b\times c)}$$

And I have to prove that these relations hold:

$$a=\frac{b'\times c'}{a'\cdot(b'\times c')}$$
$$b=\frac{c'\times a'}{a'\cdot(b'\times c')}$$
$$c=\frac{a'\times b'}{a'\cdot(b'\times c')}$$

The Attempt at a Solution

I really dk where to start :\

Do I try with the direct component expansion or can I do it with Levi-Civita symobol:

$$a\cdot(b\times c)=\varepsilon_{ijk}a_ib_jc_k$$ and $$b\times c=\varepsilon_{ijk}b_jc_ke_i$$

And then it would be:

$$a'=\frac{\varepsilon_{ijk}b_jc_ke_i}{\varepsilon_{ijk}a_ib_jc_k}$$

but what can I do with it?

 

[sachinism]

from given data u can find thata`xb`=b`xc`=c`xa`=1so they are a set of orthogonal vectors

now use this to solve the required thing

 

[dingo_d]

Well I found in Arfken Weber that this is used in reciprocal lattice. Where it's also the information you gave.

In my problem I only have the three primed vectors $$a',\ b',\ c'$$ with the given forms and I have to proove the following. There is no mentioning of that information, not even a hint :\

 

[fzero]

Note that

$$a\cdot b' = a \cdot c' =0,$$

so we can write

$$a = \alpha ( b'\times c')$$

for some scalar $$\alpha$$. Find similar expressions for b and c, using the symmetry to relate the scalars. You can compute the proportionality by computing

$$a\cdot b\times c.$$

You will need the identity

$$(A\times B)\times (C\times D) = (A\cdot B\times C)D - (A\cdot B\times D)C.$$

 

[dingo_d]

Thanks for the hint :)

 

[dingo_d]

Am I doing this right?

$$a\cdot b'=0\Rightarrow\frac{b'\times c'}{a'\cdot(b'\times c')}\cdot b'=\frac{b'\cdot(b'\times c')}{a'\cdot(b'\times c')}=\frac{c'\codt(b'\times b')}{c'\cdot(a'\times b')}=0$$

I used scalar triple product for the numerator - either $$b'\times b'=0$$ or I use the determinant and see that I have two same rows - therefore determinant is 0.

EDIT:

I've proved it! Yay for me XD

Basically you transform everything and in the end just show that a=a, b=b and c=c ^^ Thank you all for help ^^