Characters and symmetries of SHG in magnetic point groups

SHG casesEven SHGOdd SHG            MPGs
1. Original MPG without P (20)✓
χe(G0)
Same as odd		✓
χo(G0)
Same as even		1(M), 2(M), m(M), 222, mm2, 4(M), 4¯(M), 422, 4mm, 4¯2m, 3(M), 32, 3m, 6(M), 6¯(M), 622, 6mm, 6¯m2, 23, 4¯3m
2. Gray MPG without P (20)✓
χe(G0)		×11′, 21′, m1′, 2221′, mm21′, 41′, 4¯1′, 4221′, 4mm1′, 4¯2m1′, 31′, 321′, 3m1′, 61′, 6¯1′, 6221′, 6mm1′, 6¯m21′, 231′, 4¯3m1′
3. BW MPG without P but with PT (20)×✓
χo(S0)		1¯′, 2′/m, 2/m′ , m′m′m′, mmm′, 4/m′, 4′/m′, 4/m′mm, 4′/m′m′m, 4/m′m′m′, 3¯′, 3¯′m, 3¯′m′, 6′/m, 6/m′, 6/m′mm, 6′/mmm′, 6/m′m′m′, m′3¯′, m′3¯′m
4. BW MPG without P nor PT (27)✓
χe(G0)
Different with odd		✓
χo(G1)
Different with even		2′(M), m′ (M), 2′2′2(M), m′m′2 (M), m′m2′ (M), 4′, 4¯′, 4′22′, 42′2′(M), 4′m′m, 4m′m′(M), 4¯′2′m, 4¯′2m′, 4¯2′m′(M), 32′(M), 3m′(M), 6′, 6¯′, 6′22′, 62′2′(M), 6′mm′, 6m′m′(M), 6¯′m′2, 6¯′m2′, 6¯m′2′, 4′32′, 4¯′3m′

Note: (1) The corresponding SHG tensor are shown in origin and gray SHG tabel and black-white SHG tabel. (2) The numbers in the first column means the amount of MPGs with SHG effect. (3) “M” in the bracket forth column means the MPGs also have LMO effect.

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