- #1
torehan
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[tex]f(\vec{r}) = f(\vec{r}+\vec{T}) [/tex]
[tex]\vec{T}= u_{1} \vec{a_{1}} + u_{2} \vec{a_{2}}+u_{3} \vec{a_{3}} [/tex]
[tex]u_{1},u_{2},u_{3}[/tex] are integers.
[tex]f(\vec{r}+\vec{T})= \sum n_{g} e^{(i\vec{G}.(\vec{r}+\vec{R}) )}= f(\vec{r}) [/tex]
[tex]e^{i\vec{G}.\vec{R} }= 1 [/tex]
[tex] \vec{G}.\vec{R} = 2\pi m[/tex]
we call [tex] \vec{G}=h\vec{g_{1}} + k \vec{g_{2}}+l \vec{g_{3}} [/tex] reciprocal lattice vector.
but what about the primitive lattice vectors [tex]\vec{g_{1}} , \vec{g_{2}} , \vec{g_{3}}[/tex] ?
To simplify the discussion consider [tex]\vec{T_{1}} [/tex] in 1D;
[tex]\vec{T_{1}} = u_{1} \vec{a_{1}} [/tex]
[tex] \vec{G}.\vec{T} =(h\vec{g_{1}} + k \vec{g_{2}}+l \vec{g_{3}}) . ( u_{1} \vec{a_{1}}) = 2 \pi m [/tex]
Is there any definiton that indicates direct primitive lattice vectors and reciprocal primitive lattice vectors orthogonalities?
i.e
[tex]\vec{g_{1}} . \vec{a_{1}} = 2 \pi [/tex]
[tex]\vec{g_{2}} . \vec{a_{1}} = \vec{g_{3}} . \vec{a_{1}} = 0 [/tex]
[tex]\vec{T}= u_{1} \vec{a_{1}} + u_{2} \vec{a_{2}}+u_{3} \vec{a_{3}} [/tex]
[tex]u_{1},u_{2},u_{3}[/tex] are integers.
[tex]f(\vec{r}+\vec{T})= \sum n_{g} e^{(i\vec{G}.(\vec{r}+\vec{R}) )}= f(\vec{r}) [/tex]
[tex]e^{i\vec{G}.\vec{R} }= 1 [/tex]
[tex] \vec{G}.\vec{R} = 2\pi m[/tex]
we call [tex] \vec{G}=h\vec{g_{1}} + k \vec{g_{2}}+l \vec{g_{3}} [/tex] reciprocal lattice vector.
but what about the primitive lattice vectors [tex]\vec{g_{1}} , \vec{g_{2}} , \vec{g_{3}}[/tex] ?
To simplify the discussion consider [tex]\vec{T_{1}} [/tex] in 1D;
[tex]\vec{T_{1}} = u_{1} \vec{a_{1}} [/tex]
[tex] \vec{G}.\vec{T} =(h\vec{g_{1}} + k \vec{g_{2}}+l \vec{g_{3}}) . ( u_{1} \vec{a_{1}}) = 2 \pi m [/tex]
Is there any definiton that indicates direct primitive lattice vectors and reciprocal primitive lattice vectors orthogonalities?
i.e
[tex]\vec{g_{1}} . \vec{a_{1}} = 2 \pi [/tex]
[tex]\vec{g_{2}} . \vec{a_{1}} = \vec{g_{3}} . \vec{a_{1}} = 0 [/tex]
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