Question

2. Planes and the reciprocal lattice. In the lectures I claimed that the reciprocal lattice vector Gnın2n3 = nībı + n2b2 + n3b3 = is perpendicular to the plane (nın2n3). This is always true, and does not require the assumption that the primitive lattice vectors are orthogonal. Here you will prove this claim. (a) First prove this statement for the case where two of the integers (n1n2n3) are vanishing, e.g. n2 = П3 0. Hint: the plane is spanned by the vectors a2 and az. Find a vector perpendicular to a2 and az (i.e. normal to the plane), and show that it is parallel to the reciprocal lattice vector bl. (b) Repeat this calculation for the case where only one of the integers is vanishing, e.g. n3 = 0. Hint: The vector az is clearly parallel to the plane. To find a second vector, consider the intersections of the plane with the axes defined by aj and a2. (c) Finally, using the approach in part (b), prove this statement for the case where all the integers are nonzero.

Answer 1

Solution:
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NOW, considen a plane with Voctors, + J. b) D there all are lies in the plane. and also the reciprocal lattice vectors. Gehikid) - ktk boy + 2 Here, Check their dot product G Chic,d) a ( 62 ) = Church х 1+Ia while, G Chow,9). C)-Ch6+kk td ) 1-1=0 B Gentil) - 5). Chob+) 1-120

Hence, 3 The planes ( - )* ( 32 - bb are perpendicular to the reciproca lattice & I is parallel to the Reciprord lattice vector 6 Repeat this calculation for the case where only one of the integers is lcanishing e.g ₂ = 0 let us consider, A plane halang vectori C5). + baje (5) Now, it we cheats for na Ghice). ( - 6 - Chutkatli) x (6) 1-1=0 Hence, only the vector Chwa

lattice TY to the reciprocal Vector. © finally, using the approach in part (b), prove this statement for the case where all the integer are non- tero Her for, All integer To be non-zero we need to construct a llector which is parallel axis to the reciprocal laltice veckor - Cabi ta beta) so that & Chee). Carrãe, Qc) 70 coresponding vectors are, +2), C where, & Chine. (b) to nito)

& Chitil). 63 t &0 (2220) I and Schiere). 5) + Ce +53) + 0) (3+0) o Thank you.