\(\( (*reciprocal\ BCC\ lattice - \ 
          XRD\ angles*) \)\(\ \)\(
\)\(\(lk[n_, \ m_, \ 
            l_]\  := \ kk[n, \ m, \ l]\  . \ 
            kk[n, \ m, \ l]\ ;\)


      \(kk[n_, \ m_, \ l_]\  := \ 
          n*b1\  + \ m*b2\  + \ l*b3;\)


      a\  = \ 
        Pi\ ; \  (*for\ integer\ reciprocal\ lattice*) ;



      \(bccr\  := \ {b1\  = \ \((Pi*\((ez\  + \ ey\  - \ ex)\))\)/a; \ 
            b2\  = \ \((\((ex\  + \ ez\  - \ ey)\)*Pi)\)/a; \ 
            b3\  = \ \((Pi*\((ex\  + \ ey\  - \ ez)\))\)/
                a;\ }\ ;\)


      \(xyz := {ex = {1, \ 0, \ 0}; \ ey = {0, \ 1, \ 0}; \ 
            ez = {0, \ 0, \ 1};}\ ;\)


      \(xyz;\)


      \(bccr;\)\)\)

\(\(Table[
          lk[n, m, l], \ {n, \(-2\), 2}, \ {m, \(-2\), 2}, \ {l, \(-2\), 
            2}];\)\(\ \)\(
\)
      \)

Flatten[%] // Sort

list[k_] := Range[Floor[2  k/Sqrt[b1 . b1]]]

list[22]

listbcc[k_] := 
      Block[{l = 
            Flatten[Table[
                  lk[n, m, l],  {n, -2, 2},  {m, -2, 
                    2},  {l, -2, 2}]] // 
              Sort}, 
Intersection[l,  
          list[k]]]

listbcc[20]

ang[
          k_]  :=  ((ArcSin[Sqrt[#1]/((2*k)/Pi)]*2*180)/
                Pi  & )[listbcc[k]] ;

ang[20. ]