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Electron CrystallographyElectron Microscopy and Electron Diffraction$

Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov

Print publication date: 2011

Print ISBN-13: 9780199580200

Published to Oxford Scholarship Online: January 2012

DOI: 10.1093/acprof:oso/9780199580200.001.0001

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(p.279) Appendix 3: Characteristics of the 17 plane groups

(p.279) Appendix 3: Characteristics of the 17 plane groups

Source:
Electron Crystallography
Publisher:
Oxford University Press

After Hovmöller (1986), but corrected and extended by Zou 1995.

Symbol

a and b axes

Equivalent positions

Systematic absences

Amplitude relations |F(hk)| = |F(−hk)| and …

Phase relations ϕ(h k) = −ϕ(−hk) and …

Phases 0° or 180°

p1

(x y)

p2

(x y)(−xy)

(h k)

pm

γ =90°

*(xy)(−x y)

|F(hk)| = |F(−hk)|

ϕ(h k) = ϕ(−h k)

(h 0)

pg

γ =90°

*(x y)(−x 1/2+y)

(0 k) : k = 2n + 1

|F(hk)| = |F(−hk)|

ϕ(h k) = ϕ(−h k) + k ∙ 180°

(h 0)

cm

γ =90°

*(x y)(½+x ½+y)

(hk): h + k = 2n + 1

|F(hk)| = |F(−hk)|

ϕ(hk) =ϕ(−hk)

(h 0)

(−x y)(½+x ½+y)

p2mm

γ = 90°

(x y)(−xy)

|F(hk)| = |F(−hk)|

ϕ(h k) =ϕ(−hk)

(h k)

(−x y)(xy)

p2mg

γ =90°

*(xy)(−x ½+y)

(0k) :k = 2n + 1

|F(hk)| = |F(−hk)|

ϕ(h k) =ϕ(−hk) + k ∙ 180°

(h k)

(−x −y)(x ½+y)

p2gg

γ = 90°

(x y)(½−x ½+y)

(h0) : h = 2n + 1

|F(hk)| = |F(−hk)|

ϕ(h k) =

(h k)

(−xy)(½+x ½−y)

(0k) : k = 2n + 1

f(−h k) + (h + k) ∙ 180°

c2mm

γ = 90°

(x y)(½+x ½+y)

(hk) : h + k =

|F(hk)| = |F(−hk)|

ϕ(h k) = ϕ(−h k)

(h k)

(−xy)(−x y)

2n + 1

(½−x ½−y)

(xy)(1/ −x 1/+y)

(½+x ½− y)

p4

a = b

(x y)(−y x)

|F(hk)| = |F(−kh)|

ϕ(h k) = ϕ(− k h )

(h k)

γ = 90°

(−xy)(yx)

p4mm

a = b

(x y)(−y x)(−xy)

|F(hk)| = |F(−kh)| =

ϕ(h k) = ϕ(−kh) =

(h k)

γ = 90°

(yx)(−x y)(y x)

|F(−hk)| = |F(kh)|

ϕ(−hk) =ϕ( kh)

(xy )(−yx)

p4gm

a = b

(x y)(½+y ½+x)

(h0) : h = 2n + 1

|F(hk)| = |F(−kh)| =

ϕ( hk) =ϕ(−kh) =

(h k)

γ = 90°

(−xy)(−y x)

(0k) : k = 2n + 1

|F(−hk)| = |F(kh)|

ϕ(−hk) + (h+k) ∙ 180° =

(½−x ½−y)

ϕ(k h) + (h+k)∙180°

(yx)

(½−x ½+y)

(½+x ½−y)

p3

a =b

(x y)(−y xy)

|F(hk)| =

ϕ(h k) =

γ = 120°

(y−x −x)

|F(khk)| =

ϕ (khk) =

|F(−hk h)|

ϕ(−hk h)

p3m1

a =b

(x y)(−y xy)

|F(hk)| =

ϕ(h k) =

(h h)

γ = 120°

(yxx)(−yx)

|F(khk)| =

ϕ(khk) =

(h −2h)

(x xy)(yx y)

|F(−hk h)| =

ϕ(−hk h) =

(−2h h)

|F(kh)| =

ϕ(−kh) =

|F(h+kk)| =

ϕ(h+kk) =

|F(−h h+k)|

ϕ(−h h+k)

p31m

a =b

(x y)(−y xy)

|F(h k)| =

ϕ(h k) =

(hh)

γ = 120°

(yxx)(y x)

|F(khk)| =

ϕ(khk) =

(h 0)

(−x yx)(xyy)

|F(−hk h)| =

ϕ(−hk h) =

(0 h)

|F(k h)| =

ϕ(k h) =

|F(h+kk)| =

ϕ(−hk k) = ϕ(hhk)

|F(−h h+k)|

p6

a =b

(x y)(−y xy)

|F(h k)| =

ϕ(h k) =

(h k)

γ = 120°

(yxx)(−xy)

|F(khk)| =

ϕ(khk) = ϕ(−hk h)

(y yx)(xy x)

|F(−hk h)|

p6mm

a =b

(x y)(−y xy)

|F(hk)| =

ϕ(h k) =

(h k)

γ = 120°

(yxx)(y x)

|F(khk)| =

ϕ(khk) =

(−x yx)(xyy)

|F(−hk h)| =

ϕ(−hk h) =

(−xy)(y yx)

|F(kh)| =

ϕ(k h) =

(xy x)(−yx)

|F(h+kk)| =

ϕ(−hk k) =

(x xy)(yx y)

|F(−h h+k)|

ϕ(hhk)

(*) For the plane groups pm, pg, cm and p2mg there are two possible settings. Here, only the recommended setting is given.

(p.280)