ISSN:
1600-5724
Source:
Crystallography Journals Online : IUCR Backfile Archive 1948-2001
Topics:
Chemistry and Pharmacology
,
Geosciences
,
Physics
Notes:
From a given standard setting (O, A, B) (conventional unit-cell origin and vectors) of a two-dimensional space group G(p4), it is possible, for each isomorphic subgroup g(p4), to select exactly one standard setting (o, a, b) subject to the following conditions. (1) Vector conditions: a = p1A + p2B, b = -p2A + p1B, P1 〉 0, P2 ≥ 0 (P1, P2 ∈ \bb Z). (2) Origin conditions: (a) if (P1 + P2) is odd, then the coordinates X, Y of o, with respect to (O, A, B), obey the next conditions: X, Y integers, 0 ≤ X 〈 GCD(P1, P2), 0 ≤ Y 〈 (P21 + P22)/ GCD(P1, P2), GCD = greatest common divisor; (b) if (P1 + P2)/GCD(P1,P2) is even, then 2X and 2Y are both even or odd, 0 ≤ X 〈 GCD(P1, P2), 0 ≤ Y 〈 (P21 + P22)/2GCD(P1, P2); (c) if P1, P2 are even and (P1 + P2)/GCD(P1, P2) is odd then 2X and 2Y are both even or odd, 0 ≤ X 〈 GCD (P1, P2)/2 and 0 ≤ Y 〈 (P21 + P22) /GCD(P1, P2). In any case there are exactly (P21 + P22) subgroups relevant to the same vector conditions. Tables of isomorphic subgroups p(4) are given for indices up to 25.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1107/S010876738400129X
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