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advanced27 min studyLesson 501 in Physics

Use translation symmetry to label electronic states

Declare reciprocal conventions and distinguish crystal momentum from mechanical momentum.

Before you begin
On this page

The question for today

What you’ll understand

  • Derive the Bloch form from commuting lattice translations.
  • Use reciprocal vectors with an explicit factor of two pi.
  • Count independent wavevectors in a finite periodic crystal.

A periodic crystal does not let an electron conserve ordinary free-space momentum, but it does supply an exact translation symmetry in an ideal one-electron Hamiltonian. Bloch’s theorem uses that symmetry to organize states into bands labeled by a wavevector. This label is defined modulo reciprocal lattice vectors. Keeping the translation phase, the reciprocal convention, and the finite-crystal state count together prevents zone folding from being mistaken for an increase in the number of physical states.

Commuting translations supply a phase

Assume a static one-electron Hamiltonian with a potential periodic under every Bravais lattice translation and no magnetic field requiring magnetic translations. Lattice translations commute with the Hamiltonian and with each other, so energy eigenstates can be chosen as simultaneous translation eigenstates. Unitarity makes each translation eigenvalue a phase. Compatibility with successive translations forces those phases to have the exponential form of a wavevector dotted into a lattice vector.

Factor that phase from the position-space wavefunction. The remaining cell-periodic function has exactly the lattice periodicity, yielding the Bloch form. The theorem specifies a structure for eigenstates, not their energies or their orbital character. Even in a degenerate eigenspace one can choose a translation-adapted basis, but an arbitrary superposition of different crystal momenta need not itself be a single Bloch state. A localized wavepacket deliberately uses such a superposition.

ψnk(r+R)=eik⋅Rψnk(r),ψnk(r)=eik⋅runk(r),unk(r+R)=unk(r)\psi_{n\mathbf k}(\mathbf r+\mathbf R)=e^{i\mathbf k\cdot\mathbf R}\psi_{n\mathbf k}(\mathbf r),\qquad \psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),\quad u_{n\mathbf k}(\mathbf r+\mathbf R)=u_{n\mathbf k}(\mathbf r)
R is a real-space lattice vector in m, k a wavevector in m⁻¹, and n a band index. Translation is defined here by evaluating ψ at r+R; the Hamiltonian is periodic and uses ordinary commuting translations.↔ Scroll wide equations horizontally.

Reciprocal vectors identify equivalent labels

Choose primitive reciprocal vectors whose dot products with the direct primitive vectors equal two pi times a Kronecker delta. Then adding any reciprocal vector to k leaves every lattice-translation phase unchanged. The cell-periodic factor changes to compensate, so the same physical Bloch state can be represented with either label. Restricting labels to one Brillouin zone removes this redundancy; a band index is still required because many energies can share the same translation phase.

The physical crystal momentum is often written as hbar times k, but it is not generally the expectation of the mechanical momentum operator. The periodic part contains reciprocal Fourier components, and a Bloch state therefore contains several plane-wave momenta. A free particle is the special case with constant periodic factor. Changing from reduced to extended zone labels changes bookkeeping, not the actual spectrum or the number of available electron states.

ai⋅bj=2πδij,G=∑jmjbj,k∼k+G\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij},\qquad \mathbf G=\sum_jm_j\mathbf b_j,\qquad \mathbf k\sim\mathbf k+\mathbf G
ai are direct primitive vectors in m, bj and G reciprocal vectors in m⁻¹, and mj integers. This uses angular wavevectors rather than reciprocal-cycle coordinates, so the factor 2π is explicit.↔ Scroll wide equations horizontally.

A finite crystal makes the count concrete

For a one-dimensional ring with N primitive cells, require the wavefunction to repeat after the total length Na. The allowed wavevectors are spaced by two pi divided by that length. Exactly N distinct wavevectors fit in a half-open Brillouin interval of width two pi over a. Counting both endpoints separately would duplicate one state. Each orbital band has N states per independent spin channel; spin degeneracy is an additional physical assumption rather than a consequence of translation symmetry.

Use the modular lab to inspect the corresponding index bookkeeping. Reset and set modulus to 8 and additive step to 1; predict orbit period 8. Reset and set modulus to 8 and step to 2; predict period 4. This illustrates labels identified modulo an integer and how a stride can skip labels. It computes additive residue orbits only, not Bloch eigenfunctions, band energies, or a reciprocal lattice in physical units.

kj=2πjNa,Δk=2πNa,j modulo Nk_j=\frac{2\pi j}{Na},\qquad \Delta k=\frac{2\pi}{Na},\qquad j\ \text{modulo}\ N
a is the primitive spacing in m and N the number of cells. Periodic Born–von Karman boundary conditions give N independent k labels per band and per spin channel.↔ Scroll wide equations horizontally.

Work it through

Eight cells on a ring

A one-dimensional crystal has spacing a=0.5 nm and eight cells with periodic boundaries. Find reciprocal period G, allowed k spacing, and the reduced-zone representative of k=1.5π/a in the interval [−π/a,π/a).

  1. The reciprocal period is G=2π/a=4π nm⁻¹. The Brillouin interval has width G and endpoints −2π and +2π nm⁻¹, with the upper endpoint excluded to avoid duplication.

  2. The full ring length is 4 nm, so allowed spacing is Δk=2π/(4 nm)=π/2 nm⁻¹. The zone therefore contains eight independent labels, consistent with eight real-space cells for one orbital and one spin.

  3. The proposed k is 3π nm⁻¹. Subtract one reciprocal period to obtain −π nm⁻¹, or −0.5π/a. Its one-cell translation phase remains exp(i1.5π)=−i, equal to exp(−i0.5π).

The result

G=4π nm⁻¹ and Δk=π/2 nm⁻¹. The reduced representative is −π nm⁻¹, with unchanged translation phase.

The living notebook

Explore the model

Interactive lab

Interactive lab · modular

Track labels modulo a finite count

Reset and set m=8, additive step=1; predict period 8. Reset and set m=8, additive step=2; predict period 4.

Track labels modulo a finite countAddition modulo m starts at zero: rₙ = na mod m. Playback advances two additions per second. The orbit has period m/gcd(m,a), including period one for a = 0. This is not a cryptographic security test or a primality algorithm. Current readings: Residue r0: 0; Orbit period · m/gcd(m,a): 12; Distinct residues visited: 1 of 12. All parameters are adjustable with the labeled controls below.Repeated addition modulo 12: the current residue is 0, and the orbit period is 12.01234567891011+ 5mod 12An additive orbitn = 0rₙ = 0Period = 12Cycle starts at 0:0 → 5 → 10 → 3 →8 → 1 → 6 → 11 →4 → 9 → 2 → 7 → 0Exact residues · two additions per second
Additive orbitCurrent residue / next edge
Residue r0
0
Orbit period · m/gcd(m,a)
12
Distinct residues visited
1 of 12
t = 0 s

Addition modulo m starts at zero: rₙ = na mod m. Playback advances two additions per second. The orbit has period m/gcd(m,a), including period one for a = 0. This is not a cryptographic security test or a primality algorithm. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThese are additive residue orbits. They illustrate label equivalence and skipped indices, not electronic eigenvalues or the phases of a Bloch wave.

Pause & reason

Check your understanding

With the stated angular-wavevector convention, what defines a reciprocal lattice vector?

Your turn

Try it yourself

Use twelve cells with a=0.25 nm. Find the wavevector spacing and number of independent states in one spin-degenerate orbital band. Explain whether identifying zone endpoints halves the count.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First count k labels per spin, then include the independently assumed twofold spin degeneracy.

Compare with the solution

The ring is 3 nm long, giving Δk=2π/3 nm⁻¹. There are twelve independent k labels in a half-open zone of width 8π nm⁻¹. With twofold spin degeneracy the band holds 24 electrons. Identifying the two endpoints removes a duplicated label only; it does not halve the twelve distinct labels.

Keep the idea

Take it with you

  • Bloch form follows from commuting lattice translations.
  • Crystal wavevector is defined modulo reciprocal vectors containing 2π in this convention.
  • One band has one k state per primitive cell per independent spin channel.

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