Natural occuring Ball Packings of the Elements

lattice	absolute order	relative order	crystall class	Schönflies	Hermann-Mauguin	Space group
bcc			cubic	O_h⁹	4/m 3 2/m	Im3m
fcc	ABC	+ + +	cubic	O_h⁵	4/m 3 2/m	Fm3m
hcp	AB	+ -	hexagonal	D_6h⁴	6/m 2/m 2/m	P6₃mmc
dhcp	ABAC	+ - - +	hexagonal	D_6h⁴	6/m 2/m 2/m	P6₃mmc
Sm-type	ABABCBCAC	+ - + + - + + - +	trigonal	D_3d	3 2/m	R3m

2 layers are decisive

lattice	period length	rules	rule: yx	force	relative rule
fcc	3	1	-> z	repulsive 2nd layer	+ (same)
hcp	2	1	-> y	attractive 2nd layer	- (invert)

Elements which crystalize in fcc lattice: Al, Ni, Cu, Rh, Pd, Ag, Ce, Yb, Ir, Pt, Au, Pb, Ca and Sr.
Elements which crystalize in hcp lattice: Mg, Co, Re, Tc, Tl, Sc, Zr, Gd, Ti, Lu, Ru, Tb, Hf, Os, Dy, Y, Tm, Er, Ho and Be. All these elements have a lattice parameter quotient of c/a < 1.633 from the ideal case c/a=sqrt(8/3). Zn and Cd have such a strongly distorted hcp-lattice -- c/a > 1.8 -- that symmetry is lost.
Here a is the distance of neighboured atoms inside any layer and c/2 the perpendicular distance of neighboured layers (thus c is the height of the hexagonal base unit).

3 layers are decisive

lattice	absolute order	relative order	rules	zyx	xyx	fcc/hcp mixture	distinguishable
dhcp	ABAC	+ - - +	2	-> y	-> z	1:1	A layer

3rd layer controls repulsion/attraction of 2nd layer to new layer. Using relative order we can define this packing with only one rule: "n m -> -n" , meaning the next layer change will be inverse of the last but one layer change.
At ambient pressure and temperature, the elements La, Pr and Nd have their ground state in dhcp lattice. Also Ce, Pm and Sm at higher temperatures and/or pressures and the actinides Am, Cm, Bk and Cf have this lattice.

4 layers are decisive

relative order	absolute order	period length	rules	yzyx	zxyx	yxyx	xzyx	fcc/hcp mixture	distinguishable
+ + - -	ABAC	4	2	-> y	-> z			1 : 1	A layer
+ + + - - -	ABACBC	6	3	-> z	-> z		-> y	2 : 1	B layer
+ + - + + - + + -	ABABCBCAC	9	3	-> y	-> y	-> z		1 : 2	no layer
+ + + - + + + - + + + -	ABABCACABCBC	12	4	-> z	-> y	-> z	-> y	1 : 1	no layer

α-Sm (at room temperature) and Lithium below 80 K as well as Gd, Tb, Dy, Ho, Er Tm and Lu at high pressures have Sm-type lattice with this characteristic layer structure of period length 9.

If 5 layers are decisive

relative order	absolute order	period length	rules	zxzyx	yxzyx	yzxyx	xyzyx	xyxyx	zyzyx	xzxyx	zyxyx	fcc/hcp	distinguishable
+ + + + - - - -	ABACBABC	8	4	z	y	z	z					3 : 1	C layer
+ + - + - - + -	ABABACAC	8	4					z	y	y	y	1 : 3	A layer
+ + + + -	ABABC	5	5	z	y	y			z		z	3 : 2	C layer
+ + + - + - - - + -	ABABACBCBC	10	5	y		y		z	z		y	2 : 3	B layer
+ + + - + + - - - + - -	ABABCBACACBC	12	6	y		y	z		y	z	z	1 : 1	A layer
+ + + - - + - - - + + -	ABABCACBCBAC	12	6	y		z	y		z	y	z	1 : 1	B layer
+ + + + - + - - - - + -	ABABACBABABC	12	6	z	y	y		z	z		y	1 : 1	C layer
+ + + + - + + - - - - + - -	ABABCBACBCBABC	14	7	z	y	y	z		y	z	z	4 : 3	B layer
+ + + + - - + - - - - + + -	ABABCABACACBAC	14	7	z	y	z	y		z	y	z	4 : 3	A layer
+ + + - + - - + + + - + - - + + + - + - -	ABABACABCBCBABCACACBC	21	7	y		y	z	z	y	z	y	3 : 4	no layer
+ + - + - - - + + - + - - - + + - + - - -	ABABACBCACACBABCBCBAC	21	7	y		z	y	z	z	y	y	3 : 4	no layer
+ + + + - + - - + + + + - + - - + + + + - + - -	ABABACABCACACBCABCBCBABC	24	8	z	y	y	z	z	y	z	y	1 : 1	no layer
+ + - + - - - - + + - + - - - - + + - + - - - -	ABABACBABCBCBACBCACACBAC	24	8	z	y	z	y	z	z	y	y	1 : 1	no layer

If no layer is distinguishable in the sequence of the periodic layer-sequence, such a sequence is called symmetric. Only for period length of 2, 3, 9 and 12 such a layer sequence is unique:

List of short-periodic symmetric layers

period length   # of layers   sequence
         1           0        -
         2           1        AB
         3           1        ABC
         6           0        -
         9           1        ABABCBCAC
        12           1        ABABCACABCBC
        15           2        ABACABCBABCACBC  ABABABCBCBCACAC
        18           3        ABACABCACBCABCBABC  ABABABCACACABCBCBC  ABABCABCBCABCACABC
        21           6
        24          10
        27          19
        30          33
        33          62
        36         112

Except the special case AB the period length must be divisible by 3, if a symmetric sequence exists. The nontrivial subsequence of stepping 3 of this list is also known as sequence A165920 and may be computed as

1/(3n) ∑

_{d|n, n/d!=0 mod 3} μ(n/d) (2^d - (-1)^d) with μ(.) the moebius-function.

Achim Flammenkamp
Last update: UTC+2