redes de bravais
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Redes de Bravais. Auguste Bravais (1811-1863). Rede de Bravais. conjunto de pontos obtidos como combinação linear inteira de vetores primitivos todos os pontos são equivalentes. Rede Triangular. Rede Honeycomb. a 2. a 1. vetores primitivos não são únicos (ver A&M Fig. 4.4). - PowerPoint PPT PresentationTRANSCRIPT
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Redes de Bravais
Auguste Bravais
(1811-1863)
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Rede de Bravais
• conjunto de pontos obtidos como combinação linear inteira de vetores primitivos
• todos os pontos são equivalentes
)(,332211 Znnnn i aaaR
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Rede Triangular Rede Honeycomb
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vetores primitivos não são únicos(ver A&M Fig. 4.4)
a1
a2
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5 redes de Bravais em 2D
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rede cúbica simples (SC)
• fig 4.2
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rede cúbica de corpo centrado (BCC)
• 4.5 e 4.6 table 4.2
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rede cúbica de face centrada (FCC)
• 4.8 4.9 table 4.1
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número de coordenação
• número de primeiros vizinhos (i.e. de sítios mais próximos)
• SC = 6
• BCC = 8
• FCC = 12
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célula unitária primitiva (CUP)
• volume que, transladado por todos os vetores na rede de Bravais, enche todo o espaço sem sobreposição
• não é única
• volume = Vtotal / NRB
• fig 4.10
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célula unitária não-primitiva (convencional)
• fig 4.12 e 4.13
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célula de Wigner-Seitz
• única CUP com todas as simetrias rotacionais e de reflexão da rede de Bravais
• RBs en 2D
• 4.15 e 4.16
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cristal real = rede de Bravais + base
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GrafenoRB: hexagonal
base: C + C
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diamante
• 4.18
• table 4.3
RB: FCC
base: 2 C
C Si Ge
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Zincblende
• table 4.7 filme
RB: FCC
base: Ga + As
(Zn,Fe)S GaAs
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NaCl
• Fig 4.24 Table 4.5 filme
CaO (cal virgem)
RB: FCC
base: Na + Cl
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CsCl
• CsCl (4.25 Table 4.6) filme
RB: SC
base: Cs + Cl
137CsCl foi o material do acidente radioativo de Goiânia em 1987
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CaF2 (Fluorita)
1 filme
RB: FCC
base: Ca + 2 F
principal fonte natural de F
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TiO2 (Rutila)
filme
RB: Tetragonal
base: 2 Ti + 4 O
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Hexagonal Close-Packed (HCP)
• hcp (4.19 4.20 Table 4.4) 1 filme
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Cubic Close-Packed (FCC)
• 4.8 table 4.1 1 filme
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7 sistemas cristalinos em 3d
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32 grupos pontuais em 3d
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32 grupos pontuais em 3d
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já em 2d ...
• 4 sistemas cristalinos
• 10 grupos pontuais
quadrado retângulo hexágono oblíquo
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os 6 subgrupos do quadrado
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os 8 subgrupos do hexágono
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• o grupo do retângulo é o (2mm).
• o grupo da figura oblíqua é o (2).
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as simetrias pontuais da célula de WS (com a base) são as simetrias pontuais do cristal
• isso decorre da correspondência biunívoca: (cristal) ↔ (célula de WS)
• aplicando ao cristal as operações de simetria da célula o cristal fica invariante e portanto R R´ (vetores da RB são mapeados em outros vetores da RB)
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simetrias pontuais levam R em R´
grupo (2mm)