Question about Bloch function in Reduced Zone Scheme

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In summary, the classical textbook "Introduction to Solid State Physics" explains that if a Bloch function is written as ψ_{k’}(r)=exp(i{k’}r) u_{k’}(r), where k’ is outside the first zone, a reciprocal lattice vector G can be found such that k=k’+G lies within the first Brillouin zone. This leads to the relation ψ_{k’}(r)=exp(ikr) u_k(r)=ψ_k(r). The reason for this is that u_k(r) is defined as u_{k’}(r) multiplied by the complex phase exp(-iGr).
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ck00
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The classical textbook, Introduction to solid state physics by Charles Kittle said:
"If we encounter a Bloch function written as [tex]ψ_{k’}(r)=exp(i{k’}r) u_{k’}(r)[/tex], with k’ outside the first zone, we may find a suitable reciprocal lattice vector G such that k=k’+G lies within the first Brillouin zone. Then
[tex]ψ_{k’}(r)=exp(ik’r) u_{k’}(r)=exp(ikr) [exp(-iGr) u_{k’}(r)][/tex]
[tex]=exp(ikr) u_k(r)=ψ_k(r)[/tex]"
I wonder why [tex]exp(-iGr) u_{k’}(r)=u_k(r)[/tex], how to derive this relation?
 
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He probably defines [itex]u_k(r)[/itex] this way. It is just the old u times a complex phase.
 

FAQ: Question about Bloch function in Reduced Zone Scheme

1. What is a Bloch function in Reduced Zone Scheme?

A Bloch function is a mathematical representation of the wave function for electrons in a periodic crystal lattice. In the Reduced Zone Scheme, the Bloch functions are defined and calculated in a smaller region of the Brillouin zone, making the calculations more efficient and accurate.

2. How is the Reduced Zone Scheme used in electronic structure calculations?

In electronic structure calculations, the Reduced Zone Scheme is used to simplify the calculations by only considering the electronic states in a smaller region of the Brillouin zone. This reduces the computational complexity and makes the calculations more efficient.

3. What is the significance of the Brillouin zone in the Reduced Zone Scheme?

The Brillouin zone is a mathematical construct that represents the allowed energy states of electrons in a periodic crystal lattice. In the Reduced Zone Scheme, the Brillouin zone is divided into smaller regions, and the calculations are only performed in one of these smaller regions, known as the reduced zone.

4. Can the Reduced Zone Scheme be applied to any crystal lattice?

Yes, the Reduced Zone Scheme can be applied to any crystal lattice with periodicity, including simple lattices like the square or hexagonal lattice, as well as more complex lattices like the face-centered cubic or body-centered cubic lattices.

5. How does the Reduced Zone Scheme improve upon the Full Zone Scheme?

The Reduced Zone Scheme improves upon the Full Zone Scheme by reducing the computational complexity and making the calculations more efficient. This is because the Full Zone Scheme considers all electronic states in the entire Brillouin zone, while the Reduced Zone Scheme only considers a smaller region of the Brillouin zone, leading to faster and more accurate calculations.

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