Exercise 5 | 06.11.2019

Electr. Structure & Properties of Solids Prof. J. Kortus, R. Wirnata - WS 2018

Page 1

1 Review: Classification of crystals

Each crystal is characterized by the symmetry operations that transforms the crystal to the exactsame initial state/view. We have already had a look at the elements of the translation and point group.Every possible operation on a Bravais lattice can be deconstructed in a composition of elements ofthese two groups. The point group is identical to the one from the group theory introduction.Combined, they build the space groups (or lattice point groups).

If we use adapted coordinate systems instead of cartesian ones, it can be shown that there are 7reasonable choices for crystals in 3D: triclinic, monoclinic, orthorhombic, tetragonal, trigonal,hexagonal and cubic. This allows us to find very easy matrix representations of the symmetryoperations. As a trade-off, we have to fix certain other lattice parameters (cf. fig 1). When addingtranslations symmetries to them, i.e. include certain centerings, we end up with 14 Bravais latticesthat stay invariant under translation by lattice vectors. On the other hand, the 32 possible pointgroups extend to 230 space groups for non-spherical bases.

Figure 1: 14 possible Bravais lattices in 3D and 7 corresponding crystal systems. (src: [1])

Notes

• A comprehensive overview of all point groups and related issues can be found here:https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions

• Two somewhat more catchy tables of the 32 crystallographic point groups and their connectionsto the 230 space groups as well as a listing of the common point groups in molecular physicscan be found in the corresponding German article:https://de.wikipedia.org/wiki/Punktgruppe

Exercise 5 | 06.11.2019

Electr. Structure & Properties of Solids Prof. J. Kortus, R. Wirnata - WS 2018

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2 Real space

• What is a Bravais lattice?• What is the difference between a conventional and a primitive cell?• What is the Wigner-Seitz-Cell?

Solution: A Bravais lattice is an infinite lattice of spatial points (lattice points) with an orien-tation and structure that stays identical independently of the chosen origin lattice point, or in otherwords, is invariant under translation by a lattice vector. Mathematically, the position of a latticepoint is given by

Rn = n1a1 + n2a2 + n3a3 =3∑

i=1

niai , (1)

with ai = primitive lattice vectors. Then the ideal infinite crystal lattice can be regarded as the setof all lattice points

Γ ={Rn|n = (n1, n2, n3) ∈ Z3

}(2)

Further, the condition V = a1 · (a2 × a3) ̸= 0 is required, i.e. the ai may not be linearly dependent.The magnitude of a lattice vector is called lattice constant (fig. 1.i). Primitive means (fig. 1.ii+iii):

(i) the lattice is invariant under translation by a lattice vector Tn = Rn =∑3

i=1 niai, ni ∈ Z

(ii) all possible lattice points can be reached using a combination of different translations Tn

↔ a single cell contains exactly 1 lattice point (that must not necessarily lie in its center)

In contrast, the conventional cell is usually larger but fills the entire space without any “holes” aswell. The Wigner-Seitz-Cell (cf. fig 1.iv) is a unique primitive cell in which the lattice point liesin the center of the cell or in other words, it is the region in direct space that is closer to the originthan to any other point of the direct/real lattice. Here, the lattice point sits at the origin.

(i) (ii)

(iii) (iv)

Figure 2: Some possibilities for primitive cells (i-iii) and the Wigner-Seitz-Cell (iv). (src: [1])

Exercise 5 | 06.11.2019

Electr. Structure & Properties of Solids Prof. J. Kortus, R. Wirnata - WS 2018

Page 3

Figure 3: Examples for primitive (blue) vs. con-ventional unit cells, here for bcc (a)and fcc (b) cells. (src: [1])

3 Reciprocal lattice

• What is the reciprocal lattice and how can its existence be motivated?

Solution: The charge density ρ(x) should comprise the same periodicity as the underlying lattice,i.e. be invariant under translations by lattice translation vectors ρ(x+Rn) = ρ(x) , ∀Rn ∈ Γ. Everyperiodic function can be expanded in a Fourier series, i.e.

ρ(x) =∑G

ρ(G) eiG·x =∑G

ρG eiG·x =∑m

ρGm eiGm·x , (3)

with an infinite number of in general complex expansion coefficients ρG. The G-vectors itself haveto fulfill the condition

eiGm·Rn = 1 (4)in order to respect the lattice periodicity (indices n and m left out for better readability),

ρ(x+R) =∑G

ρG eiG·(x+R) =∑G

ρG eiG·x eiG·R != ρ(x) . (5)

Obviously, this can only be true if Gm · Rn = 2πl, l ∈ Z. Taking the definitions from (1) andpostulating that every G may be constructed in a similar way, i.e.

Gm = m1b1 +m2b2 +m3b3 , (6)

then one possible choice for the primitive reciprocal lattice vectors bi could be (Einstein notation!)

bi = πϵijkaj × ak

|a1 · (a2 × a3)|= πϵijk

aj × ak

Vc, (7)

where the latter fulfill the Laue conditions

ai · bj = 2πδij , (8)

and thus, mi ∈ Z. If the Function’s periodicity is just the lattice periodicity of the crystal then theG-vectors form a grid called reciprocal or Fourier lattice

Γ−1 ={Gm

∣∣∣Gm =∑3

i=1mibi, mi ∈ Z ∧ eiGm·Rn = 1 ∀Rn ∈ Γ

}. (9)

Since there is no restriction for the number of G-vectors, i.e. there are infinitely many of them, thereciprocal lattice is also a Bravais lattice.

Figure 4: Reduced zone scheme for a—in principle—infinitely large Fourier grid. (src: [1])

Exercise 5 | 06.11.2019

Electr. Structure & Properties of Solids Prof. J. Kortus, R. Wirnata - WS 2018

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4 Dual lattice, general Fourier space & thermodynamic limit

• What is the direct lattice of a crystal and how is it related to the corresponding Bravais lattice?Solution:The direct lattice is the set of all lattice points in a piece of bulk material, i.e. a crystalsample of length Ni times the corresponding primitive lattice vectors ai,

Γ = {Rn|n = (n1, n2, n3), 0 ≤ ni < Ni (i = 1, 2, 3)} . (10)

The Bravais lattice is retained in the so called thermodynamic limit Ni → ∞ for i = 1, 2, 3.

• What is the second important type of periodicity used in theoretical solid state physics forcrystals? What is the dual lattice, how many grid points does it contain in general and whatchanges in the thermodynamic limit?Solution:Lattice periodicity is the inherent characteristic of an idealized crystalline material to fill

the entire space by repetition of building blocks known as unit cells. A direct result ofthis postulation is the Fourier lattice as the (infinitely large) Bravais lattice in reciprocalspace. When talking about bulk properties (i.e. properties which do not involve surfaceor boundary effects), ρ(x+Rn) = ρ(x) represents a good idealization to assume.

Periodic boundary condition reflects the fact that each real sample is finite. We have a lookat the N -particle wave function ΨN : The name already indicates that in the correspondingSchrödinger equation the number of particles N should remain finite1. On the other hand,we just agreed that for bulk properties, (5) may be regarded as a good approximation.The solution to this problem is stipulating the so called Born-von Karman boundarycondition for the electronic wave function,

ΨN(. . . ,x+Niai, . . .) = ΨN(. . . ,x, . . .) , i = 1, 2, 3 . (11)

This can be treated similar to (5):

f(x+Niai) = f(x) ⇔ eiNik·ai = 1 ⇔ Nik · ai = 2πmi,mi ∈ Z (12)

Note, that we chose k as variable in reciprocal space, because we used G already forthe Fourier grid. Like before, We state that each k-vector can be written as a linearcombination of the three primitive reciprocal basis vectors,

k = v1b1 + v2b2 + v3b3 =3∑

i=1

vibi , (13)

where we do again not restict the vi in any way. Now, inserting (13) into (12) we find theexplicit expressions for the expansion factors vi,

vi =mi

Ni

∈ Q , i = 1, 2, 3 . (14)

This restricts the entire Fourier space to only certain allowed vectors,

km =m1

N1

b1 +m2

N2

b2 +m3

N3

b3 , (15)

known as Born-von Karman vectors.1In fact, this is also quite important for the numerical treatment e.g. when calculating band structures, because

realizing infinitely large arrays in finite memory has proven to be quite hard.

Exercise 5 | 06.11.2019

Electr. Structure & Properties of Solids Prof. J. Kortus, R. Wirnata - WS 2018

Page 5

Dual lattice is obtained by restricting 0 ≤ mi < Ni instead of mi ∈ Z:

Γ∗ ={qm|m = (m1,m2,m3), 0 ≤ mi < Ni (i = 1, 2, 3) ∧ eiNiqm·ai = 1

}, (16)

i.e. it is the set of reciprocal vectors qm, a.k.a. Bloch vectors, that obey

Rn · qm = 2π

(n1m1

N1

+n2m2

N2

+n3m3

N3

), (17)

for any direct lattice vector Rn and which are of the form

qm =m1

N1

b1 +m2

N2

b2 +m3

N3

b3 , (18)

with the sameNi used in the definition of the direct lattice. Consequently, there are exactlyas many points in dual space as points that form the direct lattice, namely N = N1N2N3.Note, that for better distinction between arbitrary (allowed) wave vectors km and duallattice points qm, we use different symbols which should not be confused with each other.Unfortunetly, common literature and text books are quite inconsequent regarding thisdifferentiation.The above restriction actually picks only those vectors, which are unique up to a trans-lation by a Fourier vector G. In the so called thermodynamic limit Ni → ∞, wherethe direct lattice becomes a Bravais lattice (=infinite lattice), the dual lattice becomescontinuous and is known as the first Brillouin zone B. Thus, the thermodynamic limitin real space corresponds to a continuum limit in Fourier space, i.e.

Γ∗ → {q = v1b1 + v2b2 + v3b3, 0 ≤ vi < 1 (i = 1, 2, 3)} . (19)

From the definition of dual and reciprocal lattice vectors it is obvious that an arbitrarypoint in Fourier space can be constructed by taking any reciprocal vector and adding adual vector, i.e.

k = G+ q . (20)When drawing the band structure (=dispersion relation) ω(k) as a function of vectorsonly from within the first Brillouin zone, then for a particular q, its corresponding vectorskn = q+Gn from the n-th Brillouin zone do not simply disappear but show up as the socalled band index n. For Bloch functions, this is usally indicated as

ψnk(x) = ⟨x|nk⟩ . (21)

By suitably adding reciprocal lattice vectors Gm, the dual lattice can also be chosen tobe more symmetric w.r.t. the origin. This is what is usually illustrated in textbooks andalso shown in fig. 5.

A very thorough treatment of this entire direct vs. dual vs. Bravais vs. Fourier grid issue canbe found in App. A and §2.2.1 of Ref. [2].

Exercise 5 | 06.11.2019

Electr. Structure & Properties of Solids Prof. J. Kortus, R. Wirnata - WS 2018

Page 6

Figure 5: Schematic view on the partition of a one-dimensional crystal in direct and reciprocal space.Red lines symbolize q-points that are supported only in the first Brillouin zone and formthe dual lattice. Their number resemble the number of joint unit cells in real space andthey originate from the postulation of Born-von Karman periodic boundary conditions. Inthe thermodynamic limit, the set becomes continuous. By contrast, G-vectors, indicatedby blue green ticks, form the reciprocal or Fourier lattice. In principle, there are infinitelymany of them and they are virtually always discrete, but for computational reasons theirnumber is capped at a specific |Gmax|. From these two basic types of vectors, a generalpoint in reciprocal space can be composed as k = q +G.

• How large is the spacing between dual lattice points in each direction for a primitive orthorhom-bic lattice?Solution:In the primitive orthorhombic lattice (oP), all pairs of the three primitive direct lattice vec-tors are perpendicular to each other. By construction, this is also true for all three primitivereciprocal lattice vectors. Further, we can read off ai ∥ bi, i = 1, 2, 3 from (7), and since

2π(8)= ai · bi = |ai||bi| cos∠(ai, bi) , (22)

we can conclude|bi| =

2π

|ai| cos∠(ai, bi)=

2π

|ai|, i = 1, 2, 3 , (23)

and consequently, the spacing of q-vectors on the dual lattice is

ki =bi|bi|

· k =mi

Ni

bi · bi|bi|

=mi

Ni

|bi|2

|bi|=mi

Ni

|bi|(23)=

mi

Ni

2π

|ai|. (24)

or alternatively, directly from the definition in (18),

ki =mi

Ni

|bi| =mi

Ni

2π

|ai|. (25)

References

[1] Rudolf Gross and Achim Marx. Festkörperphysik. 2., aktualisierte Auflage. Berlin ; Boston: DeGruyter, 2014. isbn: 978-3-11-035869-8.

[2] R. Starke and G. A. H. Schober. “Response Theory of the Electron-Phonon Coupling”. In:arXiv:1606.00012 [cond-mat] (May 2016). arXiv: 1606.00012.