Nearest Neighbor Distances
in Cubic
The number of atoms or ions close to a given atom or ion in a crystal lattice is important for many applications - for example, the Madalung constant used in lattice energy calculations. The cubic lattice is used to illustrate the idea because the distance equation is simple. The calculations can be generalized to any crystal system.
Let r be the vector which defines a lattice point, relative to a chosen orign, in a cubic crystal:
where a, b, and c are the orthogonal basis vectors all of length a (the cubic lattice constant). For a primitive (simple cubic, SC) lattice, u,v,w must all be even integers. For a face-centered cubic (FCC) lattice, two of these integers must be both odd or both even; for a body-centered cubic (BCC) lattice, the three integers must be all odd or all even. For interpenetrating FCC lattices (e.g., NaCl), lattice points in the other lattice have either one or three odd integers.
Lattice points surround the chosen origin in concentric spherical "nearest neighbor" shells, and the reduced radius of each shell is
where m is the sum of squared integers and d is the length of vector r. Note, however, that not all values of m represent shells populated by lattice points. For example, the sum of squared integers cannot equal 7, 15, 23, 28, 31, etc., so these virtual shells are empty.
The nearest-neighbor distance formula above may be simplified further for each lattice due to the restrictions placed on the parity of u,v,w. For example, m = 4nsc, where nsc is the index number of the nearest neighbor shells in a simple cubic lattice:
Note, however, that the "seventh" shell (m = 28) is not populated by lattice points. Similarly, m = 2nfcc and
That is, the nearest neighbor shells in an FCC lattice have m even (except for m = 28, etc). In an interpenetrating FCC lattice like NaCl, the counterions would populate the m odd shells (except for m = 7, 15, 23, 28, 31, etc.). For BCC lattices, the parity restrictions are more complex: let h be a set of integers 0,1,2,3,etc.; then
for nbcc = 3h + 1 (= 1,4,7,10,13,...), m = 8h + 3
for nbcc = 3h - 1 (= 2,5,8,(not 11),...), m = 8h - 4
for nbcc = 3h + 0 (= 3,6,9,12,...), m = 8h + 0
The lattice points which populate a given shell form a coordination
polyhedron. The actual population (P) of lattice points on a shell is not
easy to calculate. The table below lists the populations of all shells (real
and virtual) with 
. To see images of each coordination polyhedron (to m = 16),
click on the shell number. If you have the browser application CHIME installed,
you can interact with the polyhedron.
|
m |
d/a |
P |
nsc |
nbcc |
nfcc |
|
0.500 |
6 |
|
|
(1) |
|
|
0.707 |
12 |
|
|
1 |
|
|
0.866 |
8 |
|
1 |
(2) |
|
|
1.000 |
6 |
1 |
2 |
2 |
|
|
1.118 |
24 |
|
|
(3) |
|
|
1.225 |
24 |
|
|
3 |
|
|
7 |
1.323 |
0 |
|
|
(4) |
|
1.414 |
12 |
2 |
3 |
4 |
|
|
1.500 |
30 |
|
|
(5) |
|
|
1.581 |
24 |
|
|
5 |
|
|
1.658 |
24 |
|
4 |
(6) |
|
|
1.732 |
8 |
3 |
5 |
6 |
|
|
1.803 |
24 |
|
|
(7) |
|
|
1.871 |
48 |
|
|
7 |
|
|
15 |
1.936 |
0 |
|
|
(8) |
|
2.000 |
6 |
4 |
6 |
8 |
|
|
17 |
2.062 |
48 |
|
|
(9) |
|
18 |
2.121 |
36 |
|
|
9 |
|
19 |
2.179 |
24 |
|
7 |
(10) |
|
20 |
2.236 |
24 |
5 |
8 |
10 |
|
21 |
2.291 |
48 |
|
|
(11) |
|
22 |
2.345 |
24 |
|
|
11 |
|
23 |
2.398 |
0 |
|
|
(12) |
|
24 |
2.449 |
24 |
6 |
9 |
12 |
|
25 |
2.500 |
30 |
|
|
(13) |
|
26 |
2.550 |
72 |
|
|
13 |
|
27 |
2.598 |
32 |
|
10 |
(14) |
|
28 |
2.646 |
0 |
7 |
11 |
14 |
|
29 |
2.693 |
72 |
|
|
(16) |
|
30 |
2.739 |
48 |
|
|
15 |
|
31 |
2.784 |
0 |
|
|
(16) |
|
32 |
2.828 |
12 |
8 |
12 |
16 |
|
33 |
2.872 |
48 |
|
|
(17) |
|
34 |
2.915 |
48 |
|
|
17 |
|
35 |
2.958 |
48 |
|
13 |
(18) |