From the supplied eight-atom silicon basis, calculate F and I* for (111), (200) and (220). Identify any cancelled reflection and demonstrate its cancellation by grouping the atomic phase contributions. Does cancellation mean the corresponding geometric lattice-plane family does not exist?

Use the supplied periodic conventional cubic cell, not an isolated XYZ cluster. Cartesian XYZ coordinates are in angstrom; 1 nm=10 angstrom. XYZ atom rows start at one after the two header lines. Fractional positions below are exact; each listed basis atom has occupancy 1. Cell vectors are (a,0,0),(0,a,0),(0,0,a); all angles90 degrees. Crystal images are r_j+a*(n_x,n_y,n_z) for integer n. No relaxation, defects, thermal motion or finite crystallite envelope is supplied.
silicon S0006: a=5.43070 angstrom. Row / element / fractional x,y,z:
1 Si 0 0 0
2 Si 0 1/2 1/2
3 Si 1/4 1/4 1/4
4 Si 1/4 3/4 3/4
5 Si 1/2 0 1/2
6 Si 1/2 1/2 0
7 Si 3/4 1/4 3/4
8 Si 3/4 3/4 1/4

Use fractional coordinates x_j,y_j,z_j from the table and F(hkl)=sum_j f_j exp[2*pi*i*(h*x_j+k*y_j+l*z_j)], where i^2=-1. All occupancies are1. Define I*=|F|^2. This is only the basis structure-factor contribution, not a measured powder peak intensity; omit multiplicity, Lorentz/polarization, thermal and absorption corrections. Set the same constant f_j=1 for every Si site. Use the positive phase convention written above and the supplied origin; retain exact fractional positions. Treat amplitude below10^-10 as numerical zero. Report complex F and I*; do not substitute a memorized selection rule without showing the relevant phase groups.

Attachment S0006:
8
Coordinates in angstrom; atoms numbered from 1.
Si 0.00000000 0.00000000 0.00000000
Si 0.00000000 2.71535000 2.71535000
Si 1.35767500 1.35767500 1.35767500
Si 1.35767500 4.07302500 4.07302500
Si 2.71535000 0.00000000 2.71535000
Si 2.71535000 2.71535000 0.00000000
Si 4.07302500 1.35767500 4.07302500
Si 4.07302500 4.07302500 1.35767500