Solution for this question in solid state physics (crystal lattice)

[lovephy85]

please solution for this question in solid state

A crystal has a basis of one atom per lattice point and a set of primitive translation vectors
a = 3 i , b = 3 j , c = 1.5 (i + j + k)
where i, j and k are unit vectors in the x,y and z directions of a Cartesian coordinate system. 1)What is the Bravais lattice type of this crystal
2) what are the Miller indices of the set of planes most densely populated with atoms? 3)Calculate the volumes of the primitive unit cell and the conventional unit cell

 

[jdwood983]

(1) So you have a vector a 3 units in the x direction, a vector b 3 units in the y direciton, and a vector c 1.5 units in the x+y+z direction. What does this look like? Use graph paper if you have to, but you should be able to visualize this.

(2) To find Miller Indices, do this:
* Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.
* Take the reciprocals
* Clear fractions
* Reduce to lowest terms

(3) The volume is found using

\tau=\left|\mathbf{c}\cdot\left(\mathbf{a}\times\mathbf{b}\right)\right|

So use your given vectors, plug them in and see what you get.

 

[lovephy85]

thx
i solved as
i considered the bravais lattice is bcc
but for miller indices
hkl = 112 respectively
V=|a.b*c|=27/2

are u think that is right?

 

[jdwood983]

thx
i solved as
i considered the bravais lattice is bcc
but for miller indices
hkl = 112 respectively
V=|a.b*c|=27/2

are u think that is right?

This looks right too me, but note that the volume is found by multiplying the base area by the component of c along the axis perpendicular to the base. This means you need the dot product of the cross product of the base with the c axis:

<br /> \tau=\left|\mathbf{c}\cdot\left(\mathbf{a}\times\mathbf{b}\right)\right|<br />

In this case, you do get the same answer, but you should be aware that the correct formula is the one I wrote, not the one you used.