Structural combinations from 4 IVM (fcc) to Bcc lattices

Last update 2-24-2003

from A, B, C, D to A+B+C+D

(This is a Chart of the pattern of patterns)

I. Combinations of 4 IVMs (A, B, C, D)

[Combinations of 2, tetrahedral edges]

1a. 2 Cubic nets

A+B

AB+CD 2 interpenetrating cubic nets (bcc)

C+D

1b. 4 Diamond nets

A+C = AC

B+C = BC

A+D = AD

B+D = BD

[Combinations of 3, tetrahedral faces]

2. 4 Rhombic Dodecahedral nets ABC, ABD, ACD, BCD ivm (see II-1b below)

A+B+C = ABC

A+B+D = ABD

A+C+D = ACD

B+C+D = BCD

3. Combination of (I-1a), 2 cubic nets

AB+CD = 2 interpenetrating cubic nets = bcc lattice

II. Combinations of 4 Diamond nets (AC, BC, AD, BD) [I-1b]

[Combinations of 2]

1a. Two Interpenetrating Diamond nets

AC+BD

ACBD+BCAD 4 interpenetrating diamond nets (bcc)

BC+AD

1b. 4 Rhombic Dodecahedral nets (RD nets), (see I-2)

AC+BC = D’ RD net = ABC ivm

AD+BD = C’ RD net = ABD ivm

AC+AD = B’ RD net = ACD ivm

BC+BD = A’ RD net = BCD ivm

[Combinations of 3]

2. 4 Half-filled RD nets (of 3 diamond combinations)

AC+BC+BD = D’ RD net ½ filled by BD

AD+BD+AC = C’ RD net ½ filled by AC

AC+AD+BC = B’ RD net ½ filled by BC

BC+BD+AD = A’ RD net ½ filled by AD

3. Combination of (II-1a), 2 interpenetrating diamond nets

ACBD+BCAD = 4 interpenetrating diamond nets = bcc

III. Combinations of 4 RD nets (ACBC, ADBD, ACAD, BCBD) [II-1b]

[Combinations of 2]

1a. 2 Coupler nets

ACBC+ADBD = D’+C’= odd coupler

ACBC,ADBD+ACAD,BCBD =“Siamese Couplers”

=2 interpenetrating coupler nets = RITE net = (bcc)

ACAD+BCBD = B’+A’= even coupler

1b. 4 Half-filled RD nets (of 2 RD combinations), note the filling difference between III-1b and II-2

D’+B’ = ACBC+ACAD = D’ RD net ½ filled by AD

C’+A’ = ADBD+BCBD = C’ RD net ½ filled by BC

C’+B’ = ADBD+ACAD = B’ RD net ½ filled by BD

D’+A’ = ACBC+BCBD = A’ RD net ½ filled by AC

[Combinations of 3]

2. 4 Three RD nets (degenerated? into pairs of odd and even coupler nets)

A’B’C’ = BCBD+ACAD+ADBD = 2 interpenetrating RD, B’+A’= even coupler net (ADBD is redundant)

A’B’D’ = BCBD+ACAD+ACBC = 2 interpenetrating RD, B’+A’= even coupler net

A’C’D’ = BCBD+ADBD+ACBC = 2 interpenetrating RD, D’+C’= odd coupler net

B’C’D’ = ACAD+ADBD+ACBC = 2 interpenetrating RD, D’+C’= odd coupler net

3. Combination of (III-1a), 2 coupler nets

ACBC,ADBD+ACAD,BCBD = 2 interpenetrating coupler nets = RITE net = bcc

IV. Combinations of 4 Half Filled RD nets [III-1b]

1a.

1b.

2.

3.

I. Combinations of 4 IVMs (A, B, C, D)

[Combinations of 4]

3. Combination of all 4 IVMs

A+B+C+D = 4 IVM = bcc, including all the above pattern combinations.

Notes on main patterns:

· The various patterns obtained through the possible combinations of the 4 IVMs or 4 fcc lattices are limited between the fcc lattices and the bcc lattice.

· With 4 different lattices we can have a total combination of 6 pairs. And they are differentiated into two groups, group 1a with 2 combinations and 1b with 4 combinations.

· The 2 pairs in I-1a, II-1a, III-1a, are always complementary, and their combination lead directly to the final stage the bcc lattice.

· The 4 pairs in I-1b, II-1b, III-1b, are the new 4 lattices that can combine in the same way as the previous 4 lattices.

· I do not know if this repeating pattern goes on indefinitely, it appears that at III-2 we start seeing a redundancy where it suggest an end to the 4 pairs pattern. The 4 pairs become in essence 2 pairs due to redundancy.