Cristal Momentum

The Bloch Theorem

In the most simplified version of the free electron gas, the true three-dimensional potential was ignored and approximated with a constant potential conveniently put at 0 eV.
The true potential, however, e.g. for a Na crystal including some energy states, is periodic and looks more like this:

Semiconducting properties will not emerge without some consideration of the periodic potential - we therefore have to solve the Schrödinger equation for a suitable periodic potential. There are several (for real potentials always numerical) ways to do this, but as stated before, it can be shown that all solutions must have certain general properties. These properties can be used to make calculations easier and to obtain a general understanding of the the effects of a periodic potential on the behavior of electron waves.

The starting point is a potential V(r) determined by the crystal lattice that has the periodicity of the lattice, i.e.

V(r) = V(r + T)

With T = any translation vector of the lattice under consideration.

We then will obtain some wavefunctions yk(r) which are solutions of the Schrödinger equation for V(r). As before, we use a quantum number "k" (three numbers, actually) as an index to distinguish the various solutions.

The Bloch theorem in essence formulates a condition that all solutions yk(r), for any periodic potential V(r) whatsoever have to meet. In one version it ascertains

yk(r) = uk(r) · exp (i · k · r)

With k = any allowed wave vector for the electron that is obtained for a constant potential, and uk(r) = arbitrary functions (distinguished by the index k that marks the particular solution we are after), but always with the periodicity of the lattice, i.e.

uk(r + T) = uk(r)

Any wavefunction meeting this requirement we will henceforth call a Bloch wave.
The Bloch theorem is quite remarkable, because, as said before, it imposes very special conditions on any solution of the Schrödinger equation, no matter what the form of the periodic potential might be.
We notice that exactly as in the case of the constant potential, the wave vector k has a twofold role: It is still a wave vector in the plane wave part of the solution, but also an index to yk(r) and uk(r) because it contains all the quantum numbers, which ennumerate the individual solutions.
Blochs theorem is a proven theorem with perfectly general validity. We will first give some ideas about the prove of this theorem, and then discuss what it means for real crystals. As always with hindsight, Blochs theorem can be proved in many ways; the links give some examples.

Here we only look ageneral outlines of how to prove the theorem:

*It follows rather directly from applying group theory to crystals. In this case one looks at symmetry properties that are invariant under translation.
*It can easily be proved by working with operator algebra in the context of formal quantum theory mathematics.
*It can be directly proved in simple ways - but then only for special cases or with not quite kosher "tricks".
*It can be proved (and used for further calculations), by expanding V(r) and y(r) into a Fourier series and then rewriting the Schrödinger equation. This is a particularly useful way because it can also be used for obtaining specific results for the periodic potential. This proof is demonstrated in detail in the link, or in the book of Ibach and Lüth.
Blochs theorem can also be rewritten in a somewhat different form, giving us a second version:

yk(r + T) = yk(r) · exp(ikT)

This means that any function yk(r) that is a solution to the Schrödinger equation of the problem, differs only by a phase factor exp(ikT) between equivalent positions in the lattice.

This implies immediately that the probability of finding an electron is the same at any equivalent position in the lattice since, exactly as we expected, because

[yk(r + T)]2 = [yk(r)]2·[exp(ikT)]2= [yk(r)]2

Since [exp(ikT)]2 = 1 for all k and T.

That this second version of Blochs theorem is equivalent to the first one may be seen as follows.

Implications of the Bloch Theorem

One way of looking at the Bloch theorem is to interprete the periodic function uk(r) as a kind of correction factor that is used to generate solutions for periodic potentials from the simple solutions for constant potentials.
We then have good reasons to assume that uk(r) for k vectors not close to a Brillouin zone will only be a minor correction, i.e. uk(r) should be close to 1.
But in any case, the quantity k, while still being the wave vector of the plane wave that is part of the wave function (and which may be seen as the "backbone" of the Bloch functions), has lost its simple meaning: It can no longer be taken as a direct representation of the momentum p of the wave via p = k, or of its wavelength l = 2p/k, since:
The momentum of the electron moving in a periodic potential is no longer constant (as we will see shortly); for the standing waves resulting from (multiple) reflections at the Brillouin zones it is actually zero (because the velocity is zero), while k is not.

There is no unique wavelength to a plane wave modulated with some arbitrary (if periodic) function. Its Fourier decomposition can have any spectra of wavelengths, so which one is the one to associate with k?
To make this clear, sometimes the vector k for Bloch waves is called the "quasi wave vector".
Instead of associating k with the momentum of the electron, we may identify the quantity k, which is obviously still a constant, with the so-called crystal momentum P, something like the combined momentum of crystal and electron.
Whatever its name, k is a constant of motion related to the particular wave yk(r) with the index k. Only if V = 0, i.e. there is no periodic potential, is the electron momentum equal to the crystal momentum; i.e. the part of the crystal is zero.

The crystal momentum P, while not a "true" momentum which should be expressible as the product of a distinct mass and a velocity, still has many properties of momentums, in particular it is conserved during all kinds of processes.
This is a major feature for the understanding of semiconductors, as we will see soon enough!
One more difference to the constant potential case is crucial: If we know the wavefunction for one particular k-value, we also know the wavefunctions for infinitely may other k-values, too.
This follows from yet another formulation of Bloch's theorem:
If yk(r) = uk(r) · exp(ikr) is a particular Bloch wave solving the Schrödinger equation of the problem, then the following function is also a solution.

3-D Crystal Lattice Images
Simple lattices and their unit cells

Simple Cubic (SC) - There is one host atom ("lattice point") at each corner of a cubic unit cell. The unit cell is described by three edge lengths a = b = c = 2r (r is the host atom radius), and the angles between the edges, alpha = beta = gamma = 90 degrees. There is one atom wholly inside the cube (Z = 1). Unit cells in which there are host atoms (or lattice points) only at the eight corners are called primitive.

Body Centered Cubic (BCC) - There is one host atom at each corner of the cubic unit cell and one atom in the cell center. Each atom touches eight other host atoms along the body diagonal of the cube (a = 2.3094r, Z = 2).

FCC Primitive - It is also possible to choose a primitive unit cell to describe the FCC lattice. The cell is a rhombohedron, with a = b = c = 2r, and alpha = beta = gamma = 60 degrees. [A cube is a rhombohedron with alpha = beta = gamma = 90 degrees!]

Simple Hexagonal (SH) - Spheres of equal size are most densely packed (with the least amount of empty space) in a plane when each sphere touches six other spheres arranged in the form of a regular hexagon. When these hexagonally closest packed planes (the plane through the centers of all spheres) are stacked directly on top of one another, a simple hexagonal array results; this is not, however, a three-dimensional closest packed arrangement. The unit cell, outlined in black, is composed of one atom at each corner of a primitive unit cell (Z = 1), the edges of which are: a = b = c = 2r, where cell edges a and b lie in the hexagonal plane with angle a-b = gamma = 120 degrees, and edge c is the vertical stacking distance.

Closest Packing

Hexagonal Closest Packing (HCP) - To form a three-dimensional closest packed structure, the hexagonal closest packed planes must be stacked such that atoms in successive planes nestle in the triangular "grooves" of the preceeding plane. Note that there are six of these "grooves" surrounding each atom in the hexagonal plane, but only three of them can be covered by atoms in the adjacent plane. The first plane is labeled "A" and the second plane is labeled "B", and the perpendicular interplanar spacing between plane A and plane B is 1.633r (compared to 2.000r for simple hexagonal). If the third plane is again in the "A" orientation and succeeding planes are stacked in the repeating pattern ABABA... = (AB), the resulting closest packed structure is HCP.
HCP Coordination - Each host atom in an HCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r: six are in the planar hexagonal array (B layer), and six (three in the A layer above and three in the A layer below) form a trigonal prism around the central atom.

Cubic Closest Packing (CCP) - If the atoms in the third layer lie over the three grooves in the A layer which were not covered by the atoms in the B layer, then the third layer is different from either A or B and is labeled "C". If a fourth layer then repeats the A layer orientation, and succeeding layers repeat the pattern ABCABCA... = (ABC), the resulting closest packed structure is CCP = FCC. Again, the perpendicular spacing between any two successive layers is 1.633r.
CCP Coordination - Each host atom in a CCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r: six are in the planar hexagonal (B) plane, and six (three in the C layer above and three in the A layer below) form a trigonal anti-prism (also known as a distorted octahedron) around the central atom.

Rhombohedral (R) lattice - If, in the (ABC) layered lattice, the interplanar spacing is not the closest packed value (1.633r), then the primitive (Z = 1) unit cell is a rhombohedron with a = b = c <> 2r and alpha = beta = gamma <> 60 degrees. The non-primitive hexagonal unit cell (Z = 3).may also be chosen.

2- & 3-layer repeats - There is only one way to produce a repeat pattern (crystal lattice) in two layers of hexagonally closest packed planes: (AB) = HCP. Likewise, there is only one way to produce a repeat pattern in three layers of hexagonally closest packed planes: (ABC) = CCP.

4-layer repeats - However, there are two ways to produce a closest packed lattice in four layers: (ABAC) and (ABCB). By extension, there are increasing numbers of ways to produce closest packed lattices in five layers, six layers, etc., up to and including non-repeating random stacking. Thus, there are many closest (and pseudo-closest) packings in natural and artificial materials.

Holes ("Interstices") in Closest Packed Arrays

Tetrahedral Hole - Consider any two successive planes in a closest packed lattice. One atom in the A layer nestles in the triangular groove formed by three adjacent atoms in the B layer, and the four atoms touch along the edges (of length 2r) of a regular tetrahedron; the center of the tetrahedron is a cavity called the Tetrahedral (or Td) hole; a guest sphere will just fill this cavity (and touch the four host spheres) if its radius is 0.2247r.

Octahedral Hole - Adjacent to the Td hole, three atoms in the B layer touch three atoms in the A layer such that a trigonal antiprismatic polyhedron (a regular octahedron) is formed; the center of the octahedron is a cavity called the Octahedral (or Oh) hole. A guest sphere will just fill this cavity (and touch the six host spheres) if its radius is 0.4142r. It can be shown that there are twice as many Td as Oh holes in any closest packed bilayer.

Simple Crystal Structures

CsCl Structure - Each ion resides on a separate, interpenetrating SC lattice such that the cation is in the center of the anion unit cell and visa versa. The two lattices have the same unit cell dimension.

NaCl Structure - Each ion resides on a separate, interpenetrating FCC lattice. The two lattices have the same unit cell dimension.

Halite Structure - The sodium chloride structure may also be viewed as a CCP lattice of anions (Z = 4), with smaller cations occupying all Oh cavities (Z = 4).

Fluorite Structure - The structure of the mineral fluorite (calcium fluoride) may be viewed as a CCP lattice of cations (Z = 4), with the smaller anion occupying all of the Td holes (Z = 8). The Td cavities reside on a SC lattice which is half the dimension of the CCP lattice.

Zinc Blende Structure - The structure of cubic ZnS (mineral name "zinc blende") may be viewed as a CCP lattice of anions (Z = 4), with the smaller cations occupying every other Td hole (Z = 4). [Note: the other ZnS mineral, wurtzite, can be described as a HCP lattice of anions with cations in every other Td hole.]

Zinc Blende lattices - The lattice of cations in zinc blende is a FCC lattice of the same dimension as the anion lattice, so the structure can be described as interpenetrating FCC lattices of the same unit cell dimension. Note that the only difference between the halite and zinc blende structures is a simple shift in relative position of the two FCC lattices.

Referencias Bibliograficas:

http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_2_3.html

http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/backbone/r2_1_4.html

http://www.ece.rutgers.edu/~maparker/classes/581-chapters/Ch06-Structure-Phonons/Ch06Sec10XalMomentum.pdf

Bragg diffraction

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that in these crystals, for certain specific wavelengths and incident angles, intense peaks of reflected radiation (known as Bragg peaks) were produced. The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes.

W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter d. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively.

The Bragg condition

Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with wavelength comparable to atomic spacings are incident upon a crystalline sample, scattered in a specular fashion by the atoms in the system, and undergo constructive interference in accordance to Bragg's law. For a crystalline solid, the waves are scattered from lattice planes separated by the interplanar distance d. Where the scattered waves interfere constructively; they remain in phase since the path length of each wave is equal to an integer multiple of the wavelength. The path difference between two waves undergoing constructive interference is given by 2dsinθ, where θ is the scattering angle. This leads toBragg´s law which describes the condition for constructive interference from successive (h,k,l) of the crystalline lattice:

Where n is an integer determined by the order given, and λ is the wavelength. A diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in the diffraction pattern when scattered waves satisfy the Bragg condition.

Reciprocal space

More elegant is the description in reciprocal space. Reciprocal lattice vectors describe the set of lattice planes as a normal vector to this set with length G = 2π / d. Then Bragg's law is simply expressed by the conservation of momentum transfer (G = k_f − k_i) with incident and final wave vectors k_i and k_f of identical length. The precedent relation is also called laue difracttion and not only gives the absolute value, but a full vectorial description of the phenomenon. The scanning variable can be the length or the direction of the incident or exit wave vectors relating to energy- and angle-dispersive setups. The simple relationship between diffraction angle and Q-space is then.

Selection rules and practical crystallography

Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation:

where a is the lattice spacing of the cubic crystal, and h, k, and l are the Miller indices of the Bragg plane. Combining this relation with Bragg's law.

One can derive selection rules for the Miller indices for different cubic Bravais lattices; here, selection rules for several will be given as is.

Nobel Prize for Bragg diffraction

In 1915, William Henry Bragg and William Lawrence Bragg were awarded the Nobel Prize for their contributions to crystal structure analysis. They were the first and (so far) the only father-son team to have jointly won the prize. Other father/son laureates include Niels and Aage Bohr, Manne and Kai Siegbahn, J. J. Thomson and George Thomson, Hans von Euler-Chelpin and Ulf von Euler, and Arthur and Roger Kornberg, who were all awarded the prize for separate contributions.

We can think of the crystal as being composed of thousands of mirrors that reflect the X-rays. These "mirrors" are called Bragg planes. When light is reflected from a mirror, the angle of incidence (the angle at which it strikes the plane of the mirror) is equal to the angle of reflection. The same is true of Bragg planes, and the reason is that when the angle of incidence is equal to the angle of reflection, light rays hitting the plane (mirror) in phase exit in phase, regardless of where they hit the mirror. Notice, by the way, that the incoming and outgoing rays differ in direction by a total angle of 2θ. If rays reflected from a plane have identical pathlengths, then rays reflected from different planes must have different pathlengths. We can easily work out how far apart the planes must be for the difference in pathlength to be equal to the wavelength of the incoming radiation, so that the scattered rays from the two planes would again be in phase. It turns out that the difference in pathlength depends on the angle of incidence (and reflection). The image below shows how to work out the relationship, which is called Bragg's law.

The difference in pathlength between the rays reflected from the two planes is twice the distance l. Simple geometry tells us that the upper angle in the little triangle must be q, because the sum of the angles inside a triangle is 180 degrees, and the other two angles are 90 degrees and 90-θ. Then simple trigonometry tells us that the distance l is equal to d sinθ. For the two rays to be diffracted in phase, twice l must be equal to the wavelength, so we have the relationship:

λ = 2 d sinθ

In Bragg's law, as the angle increases, d must become smaller for the pathlength to remain equal to one wavelength. We can show this by various common rearrangements of the equation:

sinθ/λ = 1/(2 d)

d = λ/ (2 sinθ)

We can obtain same phase of different waves by calculating the optical path deference between two waves. If we want that two waves will have the same phase and will get maximum intensity we need that the condition that the relative phase will be a multiple of 2π :

2dsinθ=λm

If we want that the intensity of the diffracting waves will be 0 we need to demand Δφ=π so the optical path deference between two waves will satisfy the condition:

2dsinθ=λ(m+1/2)

Finally ,in the general case we have the condition that the optical path deference between two waves will be arbitrary angel:

2dsinθ=λ(m+Δφ/(2π))

the Bragg scattering from two Bragg planes can be visualize in the function: scanning_angels_func(h,f,d_teta,final_teta) ,which can be downloaded in the download section. Here we can see an example of waves diffracting in phase for cubic latice with a=3 and wave with λ=3 : An example of waves diffracting in of phase

An example of waves diffracting out of phase

Von Laue condition for scattering and Ewald construction

Ewald came up with a geometrical construction to help visualize which Bragg planes are in the correct orientation to diffract. In the Ewald sphere, we have two origins (which can make you uncomfortable until you realize that it is just a geometrical construction that makes the mathematics of diffraction easy to picture). The origin of the crystal is at the center of the Ewald sphere, and the incoming X-rays are diffracted from that crystal. The origin of reciprocal space is at the point where the incoming X-ray beam would exit the Ewald sphere. If we rotate the crystal, we rotate the Bragg planes and so we rotat

Direct lattice and reciprocal lattice

A typical direct Bravis lattice vector in 2D can be written as : R=n1a +n2b ,where a,b are the direct lattice primitive unit cell vectors.
The typical reciprocal lattice vector can be written as: G=kA+hB , where A,B are the reciprocal lattice primitive unit cell vectors.
The relation between the direct lattice primitive unit cell vectors and the reciprocal lattice primitive unit cell vector : A=2π/D(by,-bx) , B=2π/D(ay,-ax)
The relation between direct lattice vector R and reciprocal lattice vector G: eiGR=1 From the last relation we can see that R and G are vertical.
We describe Bragg planes with the direct lattice vector R and or with the reciprocal lattice vector G and the relation between the indexes n,m,k and h is : n:m=1/k:1/h.
We can write Bragg law 2dsinθ=λm in terms of G and k (k=2π/λ): 2ksinθ=Gm.
Notice that if we take two G vectors with the same relation between k and h (for example-G1=(1,1), G2=(2,2) ),the distance between two Bragg planes d is calculated according to the smallest G vector, but the other G vector also satisfies the Bragg equation but for higher order n.
We define K as the vector of the incoming wave and K' as the vector of the outgoing wave and their relation with G as: G=K'-K or G=2ksinθ. definitions:
reciprocal latice:-in SC(Simple cubic)the recuiprocal latice looks ezactly like the real latice in change of the atoms scale G vector-G vector is difined as and it always "nizav" to the bragg plane which the scattering is taking place. the realion between G K and K':
the reciprocal space in the same direction. Since diffraction from a crystal is confined to points on the reciprocal lattice (corresponding to planes that can be specified by integer indices), we can think of rotating the reciprocal lattice when we rotate the crystal. The following figure shows this schematically, illustrating planes of points in the reciprocal lattice. The planes of points in the reciprocal lattice intersect the Ewald sphere to give a circle of points in the diffracting condition. When the planes are aligned perpendicular to the X-ray beam, these circles on the Ewald sphere will project onto circles of spots surrounding the direct beam position but, as we rotate the crystal (and the reciprocal lattice) the circles on the Ewald sphere will be distorted and will project into what are called lines of spots.

When executing the function ewald(k), which can be downloaded in the download section, we can see selection of Bragg planes and representations of vectors G K and K' in the reciprocal lattice (lower right window), Bragg planes and waves phases(upper left window), and the interference phase(upper right window).

Here are some examples of Bragg scattering from different planes:

Reciprocal Lattice

DEFINITION:

The reciprocal space lattice is a set of imaginary points constructed in such a way that the direction of a vector from one point to another coincides with the direction of a normal to the real space planes and the separation of those points (absolute value of the vector) is equal to the reciprocal of the real interplanar distance.

Any repetitive and periodic distribution of an object (or motif) can be characterized, or described, by the translations that repeat the object periodically. This set of translations generate what we call a direct lattice.

The translations that describe the periodicity in crystals can be expressed as a linear combination of three basic translations, not coplanar, ie independent, known as reticular or lattice axes (or unit cell axes). These axes define a parallelogram (in 2 dimensions), or a parallelepiped (in 3 dimensions) known as a unit cell(or elementary cell). This elementary area (in 2-dimensional cases), or elementary volume (in 3-dimensional cases), which holds the minimum set of the periodic distribution, generates (by translations) the full distribution which, in our 3-dimensional case, we call crystal.

In general, inside the unit cell there is a minimum set of atoms (ions or molecules) which are repeated inside the cell due to the symmetry elements of the crystal structure. This minimum set of atoms (ions or molecules) which generate the whole contents of the unit cell (after applying the symmetry elements to them) is known as the asymmetric unit.

The lattice, which is a purely mathematical concept, can be selected in various ways in the same real periodic distribution. However, only one of these lattices "fits" best with the symmetry of the periodic distribution of the motifs.

As is shown in the left figure above, any lattice that describes the repetition of the motif (triangle + circle) can be decomposed into two identical equivalent lattices (one for each object of the motif). Thus, the concept of lattice is independent of the complexity of the motif, so that we can use only one lattice, since it represents all the remaining equivalent ones.

Once we have chosen a representative lattice, appropriate to the symmetry of the structure, any reticular point (or lattice node) can be described by a vector that is a linear combination (with integer numbers) of the direct reticular axes: R = m a + n b + p c, where m, n and p are integers. Non-reticular points can be reached using the nearest R vector, and adding to it the corresponding fractions of the reticular axes to reach it:

r = R + r' = (m a + n b + p c) + (x a + y b + z c) Position vector for any non-reticular point of a direct lattice

where x, y, z represent the corresponding dimensionless fractions of axes X/a, Y/b, Z/c, and X, Y, Z the corresponding lengths.

From a geometric point of view, on a lattice we can consider some reticular lines and reticular planes which are those passing through the reticular points (or reticular nodes).

Just as we did with the lattices (choosing only one of them from all the equivalent ones), we do the same with the reticular lines and planes. A reticular line or a reticular plane can be used as a representative of the entire family of parallel lines or parallel planes.

A unique plane, as the one drawn in the top right figure, defined by the numerical triplet known as Miller indices, represents and describes the whole family of parallel planes passing through every element of the motif. Thus, in a crystal structure, there will be as many plane families as possible numerical triplets exist with the condition that these numbers are primes, one to each other (not having a common divisor). The Miller indices are generically represented by the triplet of letters hkl.

If there are common divisors among the Miller indices, the numerical triplet would represent a single family of planes only. For example, the family with indices (330), which are not strictly reticular, can be regarded as the representative of 3 families of indices (110) with a geometric out-of-phase distance (among the families) of 1/3 of the original (see the figures below).

Thus, the concept of Miller indices, previously restricted to numerical triplets (being prime numbers), can now be generalized to any triplet of integers. In this way, every family of planes, will "cover" the whole crystal. And therefore, for every point of the crystal we can draw an infinite number of plane families with infinite orientations.

Of course, interplanar spacings can be directly calculated from the Miller indices (hkl) and the values of the reticular parameters (unit cell axes). The table below shows that these relations can be simplified for the corresponding metric of the different lattices.

Any plane can also be characterized by a vector (σ_hkl) perpendicular to it. Therefore, the projection of the position vector of any point (belonging to the plane), over that perpendicular line is constant and independent of the point. It is the distance of the plane to the origin, ie, the spacing (d_hkl ).

If from this normal vector σ_hkl of length 1/d_hkl, we take another vector, n times (integer) longer (n.σ_hkl), the above mentioned product (|σ_hkl| d_hkl = 1) would imply that the new vector (n.σ_hkl) will correspond to a family of planes of indices nh,nk,nl having an interplanar spacing n times smaller.

Therefore, it appears that the moduli (lengths) of the perpendicular vectors (σ_hkl) are reciprocal to the interplanar spacings. The end points of these vectors also produce a periodic lattice that, due to this reciprocal property, is known as the reciprocal lattice of the original direct lattice. The reciprocal points, obtained in this way, are identified with the same numerical triplets hkl (Miller indices) which represent the corresponding plane family.

Geometrical construction of some points of a reciprocal lattice (green points) from a direct lattice. To simplify, we assume that the third axis of the direct lattice (c) is pependicular to the screen. The red lines represent the reticular planes (perpendicular to the screen) and whose Miller indices are shown in blue. As an example: the reciprocal point with indices (3,1,0) will be located on a vector perpendicular to the plane (3,1,0) and its distance to the origin O is inversely proportional to the spacing of that family of planes.

It should now be clear that the direct lattice, and its reticular planes, are directly associated (linked) with the reciprocal lattice. Moreover, in this reciprocal lattice we can also define a unit cell (reciprocal unit cell) whose periodic translations will be determined by three reciprocal axes that form reciprocal angles among them.