International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.2, pp. 65-66

Table 1.2.6.11 

T. Janssena*

aInstitute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

Table 1.2.6.11| top | pdf |
Special points in the Brillouin zones in three dimensions

(a) Triclinic

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [\bar{1}] [k_8]
b [00{{1}\over{2}}] [\bar{1}] [k_7]
c [0{{1}\over{2}}0] [\bar{1}] [k_{6}]
d [{{1}\over{2}}00] [\bar{1}] [k_{5}]
e [{{1}\over{2}}{{1}\over{2}}0] [\bar{1}] [k_{4}]
f [{{1}\over{2}} 0{{1}\over{2}}] [\bar{1}] [k_{3}]
g [0{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{2}]
h [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{1}]

(b) Monoclinic P

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [2/m] [k_{7}]
b [00{{1}\over{2}}] [2/m] [k_{11}]
c [{{1}\over{2}}00] [2/m] [k_{12}]
d [0{{1}\over{2}}0] [2/m] [k_{13}]
e [0{{1}\over{2}}{{1}\over{2}}] [2/m] [k_{9}]
f [{{1}\over{2}} 0{{1}\over{2}}] [2/m] [k_{8}]
g [{{1}\over{2}}{{1}\over{2}}0] [2/m] [k_{14}]
h [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [2/m] [k_{10}]
i [00\gamma] 2 [k_{3}]
j [0{{1}\over{2}}\gamma] 2 [k_{5}]
k [{{1}\over{2}} 0\gamma] 2 [k_{4}]
l [{{1}\over{2}}{{1}\over{2}}\gamma] 2 [k_{6}]
m [\alpha\beta 0] m [k_{1}]
n [\alpha\beta{{1}\over{2}}] m [k_{2}]

(c) Monoclinic A

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [2/m ] [k_{6 }]
b [010] [2/m] [k_{8 }]
c [{{1}\over{2}} 00] [ 2/m] [k_{7 }]
d [{{1}\over{2}} 10] [ 2/m] [k_{9 }]
e [0{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{4 }]
f [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{5 }]
g [00\gamma] 2 [k_{2 }]
h [{{1}\over{2}} 0\gamma] 2 [k_{3 }]
i [\alpha\beta 0] m [k_{1 }]

(d) Orthorhombic P

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [mmm] [k_{19}]
b [{{1}\over{2}} 00] [mmm] [k_{20}]
c [00{{1}\over{2}}] [mmm] [k_{22}]
d [{{1}\over{2}} 0{{1}\over{2}}] [mmm] [k_{24}]
e [0{{1}\over{2}}0] [mmm] [k_{21}]
f [{{1}\over{2}}{{1}\over{2}}0] [mmm] [k_{25}]
g [0{{1}\over{2}}{{1}\over{2}}] [mmm] [k_{23}]
h [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [mmm] [k_{26}]
i [\alpha 00] [2mm] [k_{7 }]
j [\alpha 0{{1}\over{2}}] [2mm] [k_{12}]
k [\alpha{{1}\over{2}}0] [2mm] [k_{10}]
l [\alpha{{1}\over{2}}{{1}\over{2}}] [2mm] [k_{11}]
m [0\beta 0] [m2m] [k_{8 }]
n [0\beta{{1}\over{2}}] [m2m] [k_{15}]
o [{{1}\over{2}}\beta 0] [m2m] [k_{13}]
p [{{1}\over{2}}\beta{{1}\over{2}}] [m2m] [k_{14}]
q [00\gamma] [mm2] [k_{9 }]
r [0{{1}\over{2}}\gamma] [mm2] [k_{18}]
s [{{1}\over{2}} 0\gamma] [mm2] [k_{16}]
t [{{1}\over{2}}{{1}\over{2}}\gamma] [mm2] [k_{17}]
u [0\beta\gamma] m [k_{1 }]
v [{{1}\over{2}}\beta\gamma] [m11] [k_{2 }]
w [\alpha 0\gamma] [1m1] [k_{3 }]
x [\alpha{{1}\over{2}}\gamma] [1m1] [k_{4 }]
y [\alpha\beta 0] [11m] [k_{5 }]
z [\alpha\beta{{1}\over{2}}] [11m] [k_{6 }]

(e) Orthorhombic C

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [mmm] [k_{14}]
b [010] [mmm] [k_{15}]
c [01{{1}\over{2}}] [mmm] [k_{17}]
d [00{{1}\over{2}}] [mmm ] [k_{16}]
e [{{1}\over{2}}{{1}\over{2}}0] [2/m] [k_{12}]
f [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [2/m] [k_{13}]
g [\alpha 00] [2mm ] [k_{8}]
h [\alpha 0{{1}\over{2}}] [2mm] [k_{9}]
i [0\beta 0] [m2m] [k_{10}]
j [0\beta{{1}\over{2}}] [m2m] [k_{11}]
k [00\gamma] [mm2] [k_{6 }]
l [01\gamma] [mm2] [k_{7 }]
m [{{1}\over{2}}{{1}\over{2}}\gamma] [112] [k_{5 }]
n [0 \beta\gamma] [m11] [k_{1 }]
o [\alpha 0\gamma] [1m1] [k_{2 }]
p [\alpha\beta 0] [11m] [k_{3 }]
q [\alpha\beta{{1}\over{2}}] [11m] [k_{4 }]

(f) Orthorhombic I

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [mmm] [k_{17}]
b [001] [mmm] [k_{18}]
c [0{{1}\over{2}}{{1}\over{2}}] [2/m11] [k_{13}]
[1{{1}\over{2}}{{1}\over{2}}] [2/m11] [k_{10}]
d [{{1}\over{2}} 0{{1}\over{2}}] [12/m1] [k_{14}]
[{{1}\over{2}} 1{{1}\over{2}}] [12/m1] [k_{11}]
e [{{1}\over{2}}{{1}\over{2}} 0] [112/m] [k_{15}]
[{{1}\over{2}}{{1}\over{2}} 1] [112/m] [k_{12}]
f [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [222] [k_{16}]
g [\alpha 00] [2mm] [k_{7 }]
h [0\beta 0] [m2m] [k_{8 }]
i [00\gamma] [mm2] [k_{9 }]
j [{{1}\over{2}}{{1}\over{2}}\gamma] [112] [k_{6 }]
k [{{1}\over{2}}\beta{{1}\over{2}}] [121] [k_{5 }]
l [\alpha{{1}\over{2}}{{1}\over{2}}] [211] [k_{4 }]
m [0\beta\gamma] [m11] [k_{1 }]
n [\alpha 0\gamma] [1m1 ] [k_{2 }]
o [\alpha\beta 0] [11m] [k_{3 }]

(g) Orthorhombic F

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [mmm] [k_{14}]
b [100] [mmm] [k_{15}]
c [010] [mmm] [k_{16}]
d [001] [ mmm ] [k_{17}]
e [\alpha00] [2mm] [k_{4 }]
f [\alpha10] [2mm] [k_{5 }]
g [0\beta0] [m2m] [k_{6 }]
h [1\beta0] [m2m] [k_{7 }]
i [00\gamma] [mm]2 [k_{8 }]
j [01\gamma] [mm2] [k_{9 }]
k   [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{10}]
[-{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{11}]
[{{1}\over{2}} -{{1}\over{2}}{{1}\over{2}}] [\bar{1}] [k_{12}]
[{{1}\over{2}}{{1}\over{2}} -{{1}\over{2}}] [\bar{1}] [k_{13}]
l [0\beta\gamma] [m11] [k_{1 }]
m [\alpha 0\gamma] [1m1] [k_{2 }]
n [\alpha\beta0] [11m] [k_{3 }]

(h) Tetragonal P

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [4/mmm] [k_{17}]
b [00{{1}\over{2}}] [4/mmm] [k_{19}]
c [{{1}\over{2}}{{1}\over{2}}0] [4/mmm] [k_{18}]
d [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [4/mmm ] [k_{20}]
e [0{{1}\over{2}}{{1}\over{2}}] [mmm] [k_{16}]
f [0{{1}\over{2}}0] [mmm] [k_{15}]
g [00\gamma] [4mm] [k_{13}]
h [{{1}\over{2}}{{1}\over{2}}\gamma] [4mm] [k_{14}]
i [0{{1}\over{2}}\gamma] [mm2] [k_{12}]
j [\alpha\alpha 0] [2mm] [k_{10}]
k [\alpha\alpha{{1}\over{2}}] [2mm] [k_{11}]
l [0\beta 0 ] [m2m] [k_{8 }]
m [0\beta {{1}\over{2}}] [m2m] [k_{9 }]
n [\alpha{{1}\over{2}}0] [2mm] [k_{6 }]
o [\alpha{{1}\over{2}}{{1}\over{2}}] [2mm] [k_{7 }]
p [\alpha\beta 0] [11m] [k_{1 }]
q [\alpha\beta{{1}\over{2}}] [11m] [k_{2 }]
r [\alpha\alpha\gamma] m [k_{5 }]
s [0\beta\gamma] [m11] [k_{3 }]
t [\alpha{{1}\over{2}}\gamma] [1m1] [k_{4 }]

(i) Tetragonal I

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [4/mmm] [k_{14}]
b [001] [4/mmm] [k_{15}]
c [{{1}\over{2}}{{1}\over{2}}0] [mmm] [k_{13}]
d [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{4}m2] [k_{12}]
e [00\gamma] [4mm] [k_{10}]
f [{{1}\over{2}} 0{{1}\over{2}}] [12/m1] [k_{11}]
g [{{1}\over{2}}{{1}\over{2}}\gamma] [2mm] [k_{9 }]
h [\alpha\alpha 0] [2mm] [k_{7 }]
i [\alpha 00] [2mm] [k_{7 }]
j [\alpha (1-\alpha )0] [2mm] [k_{8 }]
k [{{1}\over{2}} \beta{{1}\over{2}}] [121] [k_{5 }]
l [\alpha\beta 0] [11m] [k_{2 }]
m [\alpha \alpha \gamma] m [k_{3 }]
[\alpha(1-\alpha)\gamma] m [k_4]
n [\alpha 0\gamma] [1m1] [k_{1 }]

(j) Trigonal R (rhombohedral axes)

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [\bar{3}m] [k_{7}]
b [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{3}m] [k_{8}]
c [\alpha\alpha\alpha] [3m] [k_{6}]
d [00{{1}\over{2}}] [2/m] [k_{4}]
e [{{1}\over{2}}{{1}\over{2}}0] [2/m] [k_{5}]
f [\alpha(-\alpha )0] [2] [k_{2}]
g [\alpha(-\alpha){{1}\over{2}}] [2] [k_{2}]
h [\alpha\beta\beta] m [k_{1}]

(k) Hexagonal P

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [6/mmm] [k_{16}]
b [00{{1}\over{2}}] [6/mmm] [k_{17}]
c [{{1}\over{3}}{{1}\over{3}}0] [\bar{6}m2] [k_{13}]
d [{{1}\over{3}}{{1}\over{3}}{{1}\over{2}}] [\bar{6}m2] [k_{15}]
e [00\gamma] [6mm] [k_{11}]
f [{{1}\over{2}} 00] [mmm] [k_{12}]
g [{{1}\over{2}} 0{{1}\over{2}}] [mmm] [k_{14}]
h [{{1}\over{3}}{{1}\over{3}}\gamma] [3m] [k_{10}]
i [{{1}\over{2}} 0\gamma] [2mm] [k_9]
j [\alpha 00] [2mm] [k_5]
k [\alpha 0{{1}\over{2}}] [2mm] [k_7]
l [\alpha\alpha 0] [2mm] [k_6]
m [\alpha\alpha{{1}\over{2}}] [2mm] [k_8]
n [\alpha 0\gamma] m [k_3]
o [\alpha\alpha\gamma] m [k_4]
p [\alpha\beta 0] m [k_1]
q [\alpha\beta{{1}\over{2}}] m [k_2]

(l) Cubic P

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [m\bar{3}m ] [k_{12}]
b [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [m\bar{3}m ] [k_{13}]
c [{{1}\over{2}}{{1}\over{2}}0] [4/mmm] [k_{11}]
d [00{{1}\over{2}}] [4/mmm] [k_{10}]
e [00\gamma] [4mm ] [k_{8 }]
f [{{1}\over{2}}{{1}\over{2}}\gamma] [4mm ] [k_{7 }]
g [\alpha\alpha\alpha] [3m] [k_{9 }]
h [{{1}\over{2}} 0\gamma] [mm2 ] [k_{6 }]
i [\alpha\alpha 0] [2mm] [k_{4 }]
j [\alpha\alpha{{1}\over{2}}] [2mm] [k_{5 }]
k [\alpha\beta 0] [11m ] [k_{1 }]
l [\alpha\beta{{1}\over{2}}] [11m] [k_{2 }]
m [\alpha\alpha\gamma] [m ] [k_{3 }]

(m) Cubic F

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [m\bar{3}m] [k_{11}]
b [001] [4/mmm] [k_{10 }]
c [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{3}m] [k_{9}]
d [10{{1}\over{2}}] [ \bar{4}m2 ] [k_{8}]
e [\alpha 00] [4mm] [k_{6}]
f [\alpha\alpha\alpha] [3m ] [k_{5}]
g [\alpha 01] [2mm] [k_{7}]
h [\alpha\alpha 0] [2mm] [k_{4}]
i [\alpha(1-\alpha){{1}\over{2}}] [2 ] [k_{3}]
j [\alpha\beta ] [11m] [k_{1}]
k [\alpha\alpha\gamma] m [k_{2}]

(n) Cubic I

[{\bf k}][K_{{\bf k}}]Kovalev
a [000] [m\bar{3}m] [k_{11}]
b [001] [ m\bar{3}m] [k_{10}]
c [{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}] [\bar{4}3m] [k_{10}]
d [{{1}\over{2}}{{1}\over{2}}0] [mmm] [k_{9}]
e [\alpha 00] [4mm] [k_{8}]
f [\alpha\alpha\alpha] [3m] [k_{7}]
g [\alpha{{1}\over{2}}{{1}\over{2}}] [ 2mm] [k_{6}]
h [\alpha\alpha 0] [2mm] [k_{4}]
i [\alpha(1-\alpha) 0] [2mm] [k_{9}]
j [\alpha\beta ] [11m] [k_{1}]
k [\alpha\alpha\gamma] m [k_{2}]
[\alpha (1-\alpha)\gamma] m [k_3]