4.3. Brillouin Zones

Contents

• Brillouin Zones
  • Face Centered Cubic
  • Body Centered Cubic
  • Simple Cubic
  • Hexagonal
  • Rhombohedral for 2c<a ²
  • Rhombohedral for 2c>a ²
  • Body Centered Tetragonal c<a
  • Body Centered Tetragonal c>a
  • Simple Tetragonal
  • Face centered orthorhombic ¹/_a<(¹/_b+¹/_c), ¹/_b<(¹/_c+¹/_a) and ¹/_c<(¹/_a+¹/_b)
  • Face Centered Orthorhombic with 1/c²>(1/a²+1/b²)
  • Face Centered Orthorhombic with 1/b² > (1/a²+1/c²)
  • Face Centered Orthorhombic with 1/a² > (1/b²+1/c²)
  • Body Centered Orthorhombic for a>b>c or a>c>b
  • Body Centered Orthorhombic for b>a>c or b>c>a
  • Body Centered Orthorhombic for c>b>a or c>a>b
  • Base Centered Orthorhombic a>b
  • Base Centered Orthorhombic b>a
  • Simple Orthorhombic
  • Base Centered Monoclinic
  • Simple Monoclinic
  • Triclinic

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4.3.1. Face Centered Cubic

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(L\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2})\)
\(X\)	\((\frac{1}{2} 0 \frac{1}{2})\)
\(W\)	\((\frac{1}{2} \frac{1}{4} \frac{3}{4})\)
\(K\)	\((\frac{3}{8} \frac{3}{8} \frac{3}{4})\)

4.3.2. Body Centered Cubic

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(H\)	\((\frac{1}{2} \bar{\frac{1}{2}} \frac{1}{2})\)
\(P\)	\((\frac{1}{3} \frac{1}{4} \frac{1}{4})\)
\(N\)	\((0 0 \frac{1}{2})\)

4.3.3. Simple Cubic

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(X\)	\(( 0 \frac{1}{2} 0)\)
\(M\)	\((\frac{1}{2} \frac{1}{2} 0)\)
\(R\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2})\)

4.3.4. Hexagonal

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(A\)	\((0 0 \frac{1}{2})\)
\(M\)	\((0 \frac{1}{2} 0 )\)
\(L\)	\((0 \frac{1}{2} \frac{1}{2})\)
\(K\)	\((\bar{\frac{1}{3}} \frac{2}{3} 0)\)
\(H\)	\((\bar{\frac{1}{3}} \frac{2}{3} \frac{1}{2})\)

4.3.5. Rhombohedral for 2c<a ²

Condition \(\sqrt{2c} < a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Z\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2})\)
\(L\)	\((0 \frac{1}{2} 0 )\)
\(F\)	\((0 \frac{1}{2} \frac{1}{2})\)

4.3.6. Rhombohedral for 2c>a ²

Condition \(\sqrt{2c} > a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Z\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2})\)
\(L\)	\((0 \frac{1}{2} 0 )\)
\(F\)	\((\frac{1}{2} \frac{1}{2} 0)\)

4.3.7. Body Centered Tetragonal c<a

Condition \(c < a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(N\)	\((0 \frac{1}{2} 0 )\)
\(X\)	\((0 0 \frac{1}{2})\)
\(Z\)	\((\bar{\frac{1}{2}} \frac{1}{2} \frac{1}{2})\)
\(P\)	\((\frac{1}{4} \frac{1}{4} \frac{1}{4})\)

4.3.8. Body Centered Tetragonal c>a

Condition \(c > a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(N\)	\((0 \frac{1}{2} 0 )\)
\(X\)	\((0 0 \frac{1}{2})\)
\(Z\)	\((\frac{1}{2} \bar{\frac{1}{2}} )\)
\(P\)	\((\frac{1}{4} \frac{1}{4} \frac{1}{4})\)

4.3.9. Simple Tetragonal

Condition \(c > a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(M\)	\((\frac{1}{2} \frac{1}{2} 0 )\)
\(Z\)	\((0 0 \frac{1}{2})\)
\(A\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2} )\)
\(R\)	\((0 \frac{1}{2} \frac{1}{2} )\)
\(X\)	\((0 \frac{1}{2} 0 )\)

4.3.10. Face centered orthorhombic ¹/_a<(¹/_b+¹/_c), ¹/_b<(¹/_c+¹/_a) and ¹/_c<(¹/_a+¹/_b)

Condition \(\frac{1}{a^2} < (\frac{1}{b^2} + \frac{1}{c^2} ) \cap \frac{1}{b^2} < (\frac{1}{a^2} + \frac{1}{c^2} ) \cap \frac{1}{c^2} < (\frac{1}{a^2} + \frac{1}{b^2} )\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((0 \frac{1}{2} \frac{1}{2})\)
\(X\)	\((\frac{1}{2} 0 \frac{1}{2})\)
\(Z\)	\((\frac{1}{2} \bar{\frac{1}{2}} 0 )\)
\(L\)	\((\frac{1}{2} 0 0 )\)

4.3.11. Face Centered Orthorhombic with 1/c²>(1/a²+1/b²)

Condition \(\frac{1}{c^2} < (\frac{1}{a^2} + \frac{1}{b^2} )\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((0 \bar{\frac{1}{2}} \bar{\frac{1}{2}})\)
\(X\)	\((\frac{1}{2} 0 \frac{1}{2})\)
\(Z\)	\((\frac{1}{2} \frac{1}{2} 0 )\)
\(L\)	\((\frac{1}{2} 0 )\)

4.3.12. Face Centered Orthorhombic with 1/b² > (1/a²+1/c²)

Condition \(\frac{1}{b^2} < (\frac{1}{a^2} + \frac{1}{c^2} )\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((\frac{1}{2} \bar{\frac{1}{2}} \bar{\frac{1}{2}})\)
\(X\)	\((\frac{1}{2} 0 \frac{1}{2})\)
\(Z\)	\((\frac{1}{2} \frac{1}{2} 0 )\)
\(L\)	\((\frac{1}{2} 0 )\)

4.3.13. Face Centered Orthorhombic with 1/a² > (1/b²+1/c²)

Condition \(\frac{1}{a^2} < (\frac{1}{b^2} + \frac{1}{c^2} )\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\(( 0 \bar{\frac{1}{2}} \bar{\frac{1}{2}})\)
\(X\)	\((\frac{1}{2} 0 \bar{\frac{1}{2}})\)
\(Z\)	\((\frac{1}{2} \frac{1}{2} 0 )\)
\(L\)	\((\frac{1}{2} 0 )\)

4.3.14. Body Centered Orthorhombic for a>b>c or a>c>b

Condition \(a > b > c \cup a > c > b\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((0)\)
\(X\)	\((\frac{1}{2} \bar{\frac{1}{2}} \bar{\frac{1}{2}})\)
\(R\)	\((\frac{1}{2} 0)\)
\(S\)	\((\frac{1}{2} 0 \bar{\frac{1}{2}} )\)
\(T\)	\((\frac{1}{2} \bar{\frac{1}{2}} 0 )\)
\(W\)	\((\frac{3}{4} \bar{\frac{1}{4}} \bar{\frac{1}{4}} )\)

4.3.15. Body Centered Orthorhombic for b>a>c or b>c>a

Condition \(b > a > c \cup b > c > a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((\frac{1}{2} \bar{\frac{1}{2}} \bar{\frac{1}{2}})\)
\(R\)	\((\frac{1}{2} 0 0)\)
\(S\)	\((\frac{1}{2} 0 \bar{\frac{1}{2}} )\)
\(T\)	\((\frac{1}{2} \bar{\frac{1}{2}} 0 )\)
\(W\)	\((\frac{3}{4} \bar{\frac{1}{4}} \bar{\frac{1}{4}} )\)

4.3.16. Body Centered Orthorhombic for c>b>a or c>a>b

Condition \(c > b > a \cup c > a > b\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((\frac{1}{2} \frac{1}{2} \bar{\frac{1}{2}})\)
\(R\)	\((\frac{1}{2} 0 0)\)
\(S\)	\((\frac{1}{2} 0 \bar{\frac{1}{2}} )\)
\(T\)	\((\frac{1}{2} \bar{\frac{1}{2}} 0 )\)
\(W\)	\((\frac{3}{4} \bar{\frac{1}{4}} \bar{\frac{1}{4}} )\)

4.3.17. Base Centered Orthorhombic a>b

Condition \(a > b\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((\frac{1}{2} \frac{1}{2} 0)\)
\(Z\)	\((0 0 \frac{1}{2} )\)
\(T\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2} )\)
\(S\)	\(( 0 \frac{1}{2} 0 )\)
\(R\)	\(( 0 \frac{1}{2} \frac{1}{2})\)

4.3.18. Base Centered Orthorhombic b>a

Condition \(b > a\)

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((\bar{\frac{1}{2}} \frac{1}{2} 0)\)
\(Z\)	\((0 0 \frac{1}{2} )\)
\(T\)	\((\frac{1}{2} \frac{1}{2} \frac{1}{2} )\)
\(S\)	\(( 0 \frac{1}{2} 0 )\)
\(R\)	\(( 0 \frac{1}{2} \frac{1}{2})\)

4.3.19. Simple Orthorhombic

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(Y\)	\((\bar{\frac{1}{2}} 0 0)\)
\(X\)	\((0 \frac{1}{2} 0 )\)
\(Z\)	\((0 0 \frac{1}{2} )\)
\(U\)	\(( 0 \frac{1}{2} \frac{1}{2} )\)
\(T\)	\(( \bar{\frac{1}{2}} 0 \frac{1}{2})\)
\(S\)	\(( \bar{\frac{1}{2}} \frac{1}{2} 0 )\)
\(R\)	\(( \bar{\frac{1}{2}} \frac{1}{2} \frac{1}{2} )\)

4.3.20. Base Centered Monoclinic

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(A\)	\((\bar{\frac{1}{2}} 0 0)\)
\(Z\)	\((0 \bar{\frac{1}{2}} \frac{1}{2} )\)
\(M\)	\((\bar{\frac{1}{2}} \bar{\frac{1}{2}} \frac{1}{2} )\)
\(L\)	\(( \bar{\frac{1}{2}} 0 \frac{1}{2})\)
\(V\)	\(( 0 0 \frac{1}{2} )\)

4.3.21. Simple Monoclinic

Point	\((g_{1}g_{2}g_{3})\)	as well as
\(\Gamma\)	\((000)\)
\(B\)	\((\bar{\frac{1}{2}} 0 0)\)
\(Y\)	\((0 \frac{1}{2} 0 )\)
\(Z\)	\(( 0 0 \frac{1}{2} )\)
\(C\)	\(( 0 \frac{1}{2} \frac{1}{2})\)
\(D\)	\(( \bar{\frac{1}{2}} \frac{1}{2} )\)
\(A\)	\(( \bar{\frac{1}{2}} \frac{1}{2} 0 )\)	\(( \bar{\frac{1}{2}} \bar{\frac{1}{2}} 0 )\) or \(( \frac{1}{2} \frac{1}{2} 0 )\)
\(E\)	\(( \bar{\frac{1}{2}} \frac{1}{2} \frac{1}{2} )\)	\(( \bar{\frac{1}{2}} \bar{\frac{1}{2}} \frac{1}{2} )\) or \(( \frac{1}{2} \frac{1}{2} \frac{1}{2} )\)

4.3.22. Triclinic

Point	\((g_{1}g_{2}g_{3})\)
\(\Gamma\)	\((000)\)
\(B\)	\((\frac{1}{2} 0 0)\)
\(F\)	\((0 \frac{1}{2} 0 )\)
\(G\)	\((0 0 \frac{1}{2} )\)

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