Actually there are two different conventions which are used. I am using he convention most commonly used in crystallography text. Here the reciprocal lattice vectors and the wave vectors are defined without the factor 2 Pi. Thus reciprocal basis vector a*=bXc/V, b*=cXa/V, c*=aXb/V and k=1/lambda. Correspondingly, G*_hkl = 1/d_hkl. The other convention is used most commonly in physics texts where the factor 2 Pi is introduced in the reciprocal lattice vectors and the wave vectors. Thus a*= 2 Pi bXc/V, b*= 2 Pi cXa/V, c*= 2 Pi aXb/V and k=2 Pi/lambda. Correspondingly G*_hkl = 2 Pi/d_hk. So physicist's reciprocal lattice vectors, wave vectors and the radius of Ewald sphere are all linearly expanded by 2 Pi in comparison to the corresponding quantities of the crystallographer. However, if a crystallographers reciprocal lattice vector lies on the his/her Ewald sphere then a physicist's reciprocal lattice vector will also lie on his/her Ewald sphere. Thus both will agree whether in a given experimental setting diffraction happens or not.

@@rajeshprasadlectures okkay, good

Yes, I made a typo here. You are right. Thanks for pointing this out.

For this one I have just used powerpoint.

Since the primitive vectors of a BCC are along the body diagonal of a cube, these angles can be easily calculated as cos^-1 (-1/3) =109.5 Degrees.

BCT with c/a = Sqrt [2] will actually be an FCC. This is another example of why crystal systems should not be defined simply by their lattice parameters. So, to avoid this case you will have to say that for tetragonal system is one with a=b.NE.c as well as for BCT c.NE. Sqrt [2]. This sort of conditions will soon make definitions based on lattice parameters very messy.

@@rajeshprasadlectures thank you sir

Kronecker's delta is just another name for the unit matrix. In the unit matrix only the diagonal terms are unity and all off-diagonal terms are zero. Similarly for kronecker's delta delta_11=delta_22=delta_33=1 and delta_12=delta_21=delta_13=delta_31=delta_23=delta_32=0. Thus ai. a*j=delta_ij is just a short way of writing nine dot products a1.a*1=1; a1.a*2=0; a1.a*3=0; a2.a*1=0; a2.a*2=1; a2.a*3=0; a3.a*1=0; a3.a*2=1; a3.a*3=1; The above relations are true for all systems and not only for cubic system. Note that we are not requiring a1.a2=0 which is true for orthogonal system (cubic, tetragonal and orthorhombic) but not for a general parallelopiped. We are requiring a1.a*2=0 where a*2 is a basis vector of reciprocal lattice and is not the same as a2. So by the relation a1.a*2=0 we are requiring that the second basis vector (a*2) of the reciprocal basis is orthogonal to the first basis vector (a1) of the real lattice. Hope this clarifies.

Sorry indeed. I fail.

This is purely a question of convention. The convention used by me is preferred by crystallographers where ai*aj=delta_ij. The convention often used by physicists is ai*aj*=2pi delta_ij. One can use either convention.

@@rajeshprasadlectures thank you a lot, now everything is clear

Crystals (lattice+motif) can have 32 point groups. Lattice alone can have only 7 point groups.