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Dispersion diagram VS dispersion surface

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Feri4030

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hi, every body
1. as it is known to investigate the bad-gap characteristics of 2-D periodic structures a dispersion diagram along Γ-X-M path is used



2. dispersion of a 2-D periodic structure is actually a "surface" not a 1-D diagram, however in a dispersion diagram along Γ-X-M path we just consider the edges of Brillouin zone

74_1257970256.jpg


3. apparently it can be shown that a existence of a bandgap in 1-D dispersion diagram is "sufficient condition" to have a band gap in dispersion surface, any body can prove that?
 

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