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In Stock. Seller Inventory x Henri Alloul. If the form 1. The periodic boundary condition 1. We must therefore have. This leads to the quantisation condition 1. Equation 1. The condition 1. The energy eigenvalues 1. There are several points to note about this result which will also apply in more complex situations, in particular, in the case of dimensions higher than one: 1. The energy levels 1.
The LCAO approximation is only valid if all the states in the band lie below the first excited state Ee of the atomic potential, i. The state of energy Ek is given by. Energy band arising from the atomic valence level for the linear atomic chain. In practice, to represent this periodic function, it is convenient to choose k in the interval for Nn even. This interval is known as the first Brillouin zone. Clearly, other choices are also possible.
The motivation for this particular choice will be spelt out in more detail in Chap. A graphical representation of the band structure in the first Brillouin zone is shown in Fig. The form 1. This is illustrated in Fig. The first case corresponds to a situation in chemistry where the orbitals are bonding, while the second corresponds to antibonding orbitals. The same question is raised for any Bloch function. We shall see later that this relation is no longer valid, but that some features of the behaviour of plane waves in vacuum will survive.
Owing to the orthonormalisation relation 1. Furthermore, we have one orbital per atom, so the Hilbert space has dimension Nn.
Finally, the quantisation rule 1. The important point in 1. It is thus convenient to be able to rewrite 1. While it is already convenient for the 1D case, this ploy will become even more useful in the 2D and 3D situations. The number of corresponding states close to ki is therefore. Let us now consider the linear chain with Ek given by 1. The density of states is plotted versus the energy in Fig. Note the square root divergences at the band edges. In the last section, we made two approximations.
The first was the nearest-neighbour approxi- mation 1. The second assumed orthogonality of the orbitals at different sites, expressed by 1. If instead we retain only the orthogonality of the orbitals but drop the nearest-neighbour approximation, using the full equation 1. Here we have considered the case where Nn is even. Regarding the summation range, we ought to make the following comment.
In the sum over l, the upper bound is thus correct. Note that the quantisation condition for the k does indeed lead to a real value for the last term. Note finally that, at large distances, the hopping integrals tl decrease exponentially with distance, and the sum 1. In the usual case where the hopping integrals decrease quickly with the distance l, the general form of Ek does not change from what was obtained with the nearest-neighbour approximation, and in particular retains a global maximum and a global minimum, without intervening local extrema.
This gives rise to a similar density of states curve to the one shown in Fig. However, we can envisage some rather exceptional cases in which the tl are such that Ek has other, local extrema. In this case, the density of states would be changed more radically by the existence of further singularities, also involving square root divergences, in the middle of the band, and more precisely, at the locations of the local extrema.
The situation becomes somewhat more involved if we drop the orthonormalisation condi- tion 1. It is replaced by the general expression. To calculate eigenfunctions and eigenvalues, we return to 1. The coefficients ak,l have the same form as before, viz. Note that the result 1. Expanding in a Fourier series, 1. The wave functions take the same form as in the simpler case of 1. A similar calculation to the one leading to 1. Note that the norm of any wave function must be positive, which guarantees that the denominator in 1. So far we have been considering the case of the atomic valence level Ev of Fig.
Clearly, the same approach can be used for the deeper levels E1 , E2 ,. Note that this hopping integral is given by see Fig. Now the atomic orbital wave functions have the property that their spatial extents are narrower for more tightly bound electron levels. This can be seen in Fig. This choice of representations magnifies the degree of overlap between the orbitals. It increases continuously for the states 1s, 2s, 2p, and 3s Adapted from , p. The figure shows clearly that, in the relation 1. It thus follows that the tight-binding approximation leads to energy bands in the solid that get broader as the energy of the relevant atomic level gets higher.
This is depicted schematically in Fig.
As expected from the behaviour of t1 as a function of r, the bands broaden when the distance is reduced. The deep levels give rise to extremely narrow bands a few micro electronvolts or meV , while the valence levels give rise to broad bands in the electronvolt range. In the case of nat- ural Na, with spacing r0 in Fig. We shall see in Chap. Two subjects have been discussed in this chapter. To begin with, we described the quantum mechanics of a solid comprising a number of particles of the order of the Avogadro number.
It is easy to write down a Hamiltonian which takes into account the kinetic energies of the atomic nuclei and the electrons, together with their mutual Coulomb interactions. However, we soon realised that this problem would prove insoluble in its full generality. A more practical and realistic approach here is to begin by making two approximations: 1. We assumed that the atomic nuclei were static, which amounts to saying that the nuclei sit at fixed positions in space.
We also restricted to the case of crystalline solids, characterised by a regular and periodic arrangement of the nuclei. This is valid for a great many solids, but nevertheless rules out the interesting but complex category of disordered solids. The problem resulting from this approximation, which concerns only the dy- namics of the electrons, is still too difficult to handle. We would have to account for the interactions between some electrons, which would be strictly im- possible.
We thus further restricted to an average description of the Coulomb interactions between the electrons, which is often quite adequate. In this case, the problem involving a large number of electrons reduces to a large number of problems involving just one electron, which can be dealt with by well known methods of quantum mechanics and statistical physics. We then studied the problem of a single electron in a crystalline solid, where it feels a periodic potential. Instead of trying to investigate the general situation, for which there are in any case no explicit solutions, we considered the so-called tight-binding approximation, in which the electrons are tightly bound to the atoms.
Having introduced this approximation for an arbitrary solid, we focused on the specific case of a linear chain of atoms, for which we were able to carry out the calculation exactly. The main result is that the influence of neighbouring atoms in the solid leads to a broadening of the discrete atomic levels into a band of allowed energies, characterised by a function E k which gives the energy of a quantum state in terms of its crystal momentum k.
The spectrum arising from a given atomic level is thus bounded but quasi-continuous. Chapter 2 Crystalline Solids: Diffraction. X-ray diffraction pattern for a C60 single crystal obtained with the experimental setup known as a precession chamber. This directly visualises the Bragg spots corresponding to a plane of the reciprocal lattice of the crystal to solve Question 2.
Image courtesy of Launois, P. Our main concern in this book is to describe the electronic properties of crystalline solids. The existence of translation symmetries associated with such ordered crystal structures leads to specific features in the electronic structure and to a specific rep- resentation of the energy states in wave vector space.
In Sect. This periodic structure of matter causes diffraction of electromagnetic waves, or equiv- alently, of quantum particles. We shall see in Sect. It is usual to consider a crystal as a natural object with regular external geometric features, as found for example in rock salt, diamonds, and so on. By the end of the nineteenth century, the systematic study of the external shapes of such natural crys- tals led scientists to conclude that this regularity of the outer faces must be due to structural regularities on the microscopic scale.
The molecules or atoms had to be assembled in a periodic manner to make a crystal. In this chapter, we shall see how to specify the arrangement of a crystal structure. This structure can be ascertained experimentally either by direct observation, or by light diffraction X-ray crystallog- raphy. These experimental methods show that many solids actually have a crystal structure, even when their outer surfaces do not give this impression. They are in fact polycrystalline, i. We begin in Sect. We then illustrate some simple 3D systems in Sect. A crystal is an arrangement of atoms or molecules that is invariant under transla- tions in three space directions constituting a triad a1 , a2 , a3.
Many human con- structions have ordered structures exhibiting such characteristics, especially in two dimensions: wall paper and floor tiling often have periodic structures that can be considered as 2D crystals. In these cases, the primitive material basis is repeated through two translations. A molecular example, observed by a modern microscopic method, is shown in Fig. The right-hand figure was obtained with twice the resolution to reveal details of the atomic structure.
The distance between molecules is 4. Image courtesy of Cousty, J. Saclay, France. C33 H68 , deposited on a graphite surface. This set of points the lattice points or nodes consti- tutes the crystal lattice or Bravais lattice. For elementary solids, containing only one atomic species, the atoms may coincide with the lattice nodes, since the basis then often comprises a single atom. But as soon as the solid contains several atomic species, such a situation is no longer possible. Note that the lattice of points provides a way of defining a primitive unit cell for the material basis, namely, as the smallest volume that can tile the space by applying the translations Rl.
The primitive cell and the triad a1 , a2 , a3 are not unambiguously defined, as can be seen from Fig.
The triad a1 , a2 , a3 is often taken to be the set of vectors that best reveals the symmetries of the lattice, e. These can be specified by the two vectors a1 , a2 or the two vectors b1 , b2. The vectors a1 , a2 better reveal the symmetries of the lattice. Four primitive cells are shown for the rectangular lattice, those specified by a1 , a2 and b1 , b2 , an arbitrary cell, and the Wigner—Seitz cell containing the lattice point O.
Of particular importance is a primitive unit cell known as the Wigner—Seitz cell. This is constructed in such a way that every point of the cell is closer to one lattice point for example, O than to any other lattice point see Fig. It is bounded by the orthogonally bisecting planes of the vectors Rl with origin the chosen node. For an elementary solid, this volume constructed with one atom at the center represents in some sense the region of influence of this atom. These are called lattice planes. In two dimensions, they con- stitute parallel rows see Fig.
The lattice planes group together points that can be obtained from one another by two of the translations Rl.
In the above, we illustrated the idea of a crystal by 2D representations. Since real crystals are three-dimensional, these 2D representations may appear rather. One might think that this would be that of the lattice plane corresponding to the infinite crystal. How- ever, in many cases, the translational symmetry breaking associated with the ex- istence of a surface leads to a significant modification of the surface structure. We refer to this as surface reconstruction. An example is graphite, or the high-Tc cuprate superconductors.
The structural and electronic properties of these materials are strongly affected by the 2D nature of the material. In , it became possible to peel off graphite sheets and hence study isolated layers of graphene, which is an almost ideal 2D crystal form of carbon with highly original electronic properties. In two dimen- sions, there are only five types of Bravais lattice. In the centered rectangular lattice of Fig. In the first case, we observe the conventional centered rectangular cell specified by a, b and the primitive cell a1 , a2.
In fact the latter contains two lattice points, viz. These two points are indeed equivalent, as required by the notion of a Bravais lattice, since each one is the center of the rectangle formed by its four nearest neighbours. The conventional cell must be considered as a cell with one basis the two lattice points , and the crystal will be obtained by introducing twice the material basis of the primitive cell around these points. Question 2. Determine the Bravais lattice and the primitive cell for the alkane crystal observed by scanning tunneling microscopy in Fig. Determine the Wigner—Seitz cells associated with the centered rectangular and hexago- nal lattices of Fig.
The 2D structures attracting most attention since are the graphene honeycomb structure of Fig. Determine the Bravais lattice and a primitive cell for each of the two structures of Fig. In three dimensions, the simplest lattice to visualise is the cubic lattice. The three primitive translation vectors a1 , a2 , a3 form an orthogonal triad, and each of them has length equal to the side a of the cube constituting a primitive cell see Fig.
Chemical elements crystallise scarcely into such a simple cubic lattice, but it is encountered in many polyatomic crystals we shall discuss the example of CsCl below. However, many chemical elements crystallise into body-centered cubic bcc crystal lattices see Table 2. This Bravais lattice shown in Fig. Table 2. In the bcc lattice, the conventional cell contains two lattice points, one at the origin and the other at the center of the cube. These two lattice nodes are indeed equivalent as each is the center of a cube formed by its eight nearest neighbours see Fig.
The most common lattice for elementary solids is the face-centered cubic lat- tice fcc. This is what is usually obtained when we try to stack hard spherical balls. It is a common structure for many metals see Table 2. The face-centered. The conventional cell contains four lattice nodes. The atoms are all chemically identical in diamond, Si, and Ge. In the case of InP or GaAs, the two species differ, and are represented by empty spheres and full spheres. Note that the primitive translations of the Bravais lattice are the three vectors joining the cube vertex to the centers of the adjacent faces, so a rhombohedral primitive cell can be constructed see Fig.
All atoms within the conventional cell are shown in Fig. Check that the crystals in Fig. When we consider polyatomic crystals, the primitive cell necessarily contains sev- eral atoms. A simple illustration is given in Fig. Although Cs is at the center of a Cl cube, the associated Bravais lattice is the. The basis comprises one atom of Cl at the vertex of the cube and one atom of Cs at the center of the cube. Many mixed oxides of transition metals crystallise into a cubic structure called the perovskite structure, with primitive cell ABO3 , in which A and B are cations with different size and valence.
These oxides can exhibit a wide range of physical properties, from ferromagnetism in the manganites LaMnO3 and cobaltites LaCoO3 to antiferromagnetism in iron-based perovskites like LaFeO3. Families of metal oxides with a highly 2D structure can sometimes be obtained by combining planes with perovskite structure with square MO planes, where M is a third metal cation. An example of such a structure is found in HgBa2 CuO5. This is shown in Fig. The prim- itive cell of this structure is a right-angled parallelepiped whose sides a and b are equal tetragonal structure. Ba O Cu Hg. Below the 3D representation designed to show the octahedra surrounding the A cations are two primitive cells centered respectively on the A and B cations.
Naturally, there are many other 3D crystal systems with even fewer symmetries than the simple lattices considered above. There is no question here of undertaking an exhaustive study: there are 14 Bravais lattices in three dimensions! Not all natural or artificial solids are crystalline. In many cases, there is no long range order in the atomic arrangement. In particular, when a liquid is suddenly cooled down below its solidification temperature, we can obtain a solid state which simply freezes in the arrangement of atoms as it occurred in the liquid state.
A glass is obtained in this way by quenching the liquid, whereupon the atoms arrange them- selves in a way that suffers only one constraint, namely that atoms are not allowed to interpenetrate. If the atoms are thought of as hard spheres, the resulting glass struc- ture looks like what would be obtained by putting beads in a container and shaking them up.
This glassy state is generally metastable, in the sense that the system can have a lower free energy when the atoms are arranged into an fcc or hcp crystal structure, which correspond to the closest packing of the beads. Crystallisation can then be obtained by heat treatment, which amounts to shaking the box of beads in our analogy.
But many other situations can be observed, with varying degrees of order. Con- sider for example what happens for some alloys of two metals. A structure close to a crystal structure can often be seen in these materials. The atoms distribute them- selves randomly at the lattice points of a perfect crystal structure. This is called a solid solution. The Cu and Au atoms distribute them- selves randomly over the fcc lattice sites of pure Au, and the lattice spacing varies slightly depending on the Cu concentration.
This structure is not strictly speaking a perfect crystal structure, although it can be treated as such in many respects. For some values of x, a heat treatment allows the atoms to arrange themselves into a perfect crystal structure on the lattice. What would then be the associated primitive cell and Bravais lattice? Imperfect crystal arrangements are observed in many other cases, in particular for complex molecular structures. For example, the real crystals of cuprate supercon- ductors shown in Fig. A novel illustration of disorder in crystals is shown below in the case of the fullerene C60 , a molecule discovered in , which has a football shape.
Right: Rb3 C60 crystal. These lattices are face-centered cubic. In Rb3 C60 the rhombohedral primitive cell contains a C60 molecule and three Rb atoms. It is easy to insert cations between the C60 molecules and thereby create compounds of the form An C The compound Rb3 C60 has attracted considerable attention as it happens to be a metal that becomes a superconductor be- low 27 K.
Its fcc structure contains 3 rubidium atoms and one molecule of C60 per primitive cell. The rubidium atoms have two different types of position: one, located in the middle of an edge of the cube, has an octahedral C60 environment, while the other two have a tetrahedral C60 environment one vertex of the cube and three face centers. Note that the C60 molecule has symmetries that are not compatible with the face-centered cubic lattice. Indeed, there is no way of orientating the C60 molecule so that it can map onto itself under all the symmetries of the lattice.
There are not really any 3D crystal structures whose primitive cells are given by those shown in Fig. These structures can nevertheless be considered as crystalline, but with orientational disorder of the C60 molecules. At high temperatures, the C60 molecules are not immobile, but have rotational motions. These rapid rotational movements are such that, on average, the C60 molecules behave like spheres, and one can consider that the symmetry of the C60 molecule is no longer relevant. The average structure is as shown in Fig. Since the molecules are not fixed, we do not strictly have a crystal.
Such systems in which molecular motions occur are called plastic crystals. For pure C60 , a phase transition takes place at K from the face-centered cubic high temperature structure of the plastic crystal to a body-centered cubic plastic crystal, as the rota- tional motions of the C60 molecules occur in a correlated manner about particular axes relative to the crystal axes. At low temperatures, these rotational motions freeze, but the relative orientations of the C60 molecules in low temperature phases are not yet perfectly understood.
Although one can speak of an average face-centered cubic structure in Rb3 C60 , the state of relative disorder or order of the C60 molecules has not yet been completely characterised. These diffraction con- ditions are used in Sect. This effect was originally demonstrated by von Laue and the Braggs father and son in — The latter quantities are related by. In specific cases, the radiation may also be in the form of neutrons or electrons. To understand how diffraction works, consider first the scattering of an arbitrary plane electromagnetic or matter wave by some obstacle, usually an atom located at the origin see Fig.
Here, a0 is a vector-valued amplitude in the case of electric or magnetic fields and a complex-valued amplitude in the case of quantum matter waves. This amplitude de- pends only on the intensity of the incident radiation.
In general, the scattered wave will have the form. Scattered amplitude ascat k1. However, the exact expression for these coefficients is not needed to understand the underlying principle of the diffraction methods described below. Let us now ask what happens when the object is displaced through u from the origin. Clearly, the scattered amplitude must be taken from the point u rather than from the origin.
Finally, we use a detector that only detects waves scattered in a specific direction k1. The amplitude in this specific direction will then be. There is therefore a phase difference between the waves scattered in the direction k1 by the objects located at O and u. However, as the Braggs and von Laue observed, if all the phase factors are the same in certain directions k1 , the resulting phase coherence between the amplitudes scattered in these directions will lead to a high diffracted intensity.
We consider the crystal lattice as an ensemble of lattice planes as shown in Fig. The Bragg condition O u obtains when the path dif- ference is a multiple of the d wavelength. If in addition we wish to observe phase coherence between the amplitudes scattered by the different parallel lattice planes, examination of Fig. This diffraction condition, known as the Bragg condition, is given here in a form that corresponds to the representation of the crystal lattice in lattice planes.
We now consider the scattering by a very large number of atomic bases of the prim- itive cell arranged in positions Rn. The total scattered wave will simply be the su- perposition of many terms of the form 2. If K is chosen such that. Under these conditions, the diffracted intensity in the direction k1 is. Equations 2. Here we interpret 2.
The relation 2. As a consequence, the vectors K satisfying 2. This is the reciprocal lattice associated with the Bravais lattice in po- sition space, called hereafter real or direct. Here the denominator is precisely the volume of the prim- itive cell of the direct lattice. Note that, according to 2. In particular, the reciprocal lattice of the reciprocal lattice is just the direct lattice. However, the real crystal is a lattice of atoms or molecules, or more generally, a lattice of what we have called material bases, whereas the reciprocal lattice is a lattice of points that are independent of the bases of the real crystal.
Conversely, the reciprocal lattice of a body-centered cubic lattice is an fcc lattice, e. The notion of reciprocal lattice can be used to relate the Bragg and von Laue rep- resentations of the diffraction conditions. As illustrated in Fig. The direction of diffraction k1 therefore corresponds to a Bragg diffraction on the lattice planes of the crystal parallel to the orthogonally bisecting plane of the vector K.
There is thus a one—one correspondence between the lattice planes of the crystal and the orthogonally bisecting planes of the reciprocal lattice vectors, which are known con- ventionally as Bragg planes. Once they have been found, the crystal lattice can be determined. This will be exemplified for the case of X-ray diffraction in Sect.
Consider now the scattered intensity if the Bragg condition is not satisfied. To sim- plify the notation, we discuss the case of a simple cubic structure with lattice con- stant a and length La in each of the three space directions. The sum in 2. Figure 2. If the Bragg condi- tion is not satisfied, the scattered intensity will therefore remain very low. We thus conclude that the diffraction conditions are very precisely specified, and this will only be limited experimentally by the size of the diffracting crystal and the wavelength dispersion of the incident radiation.
To determine the reciprocal lattice, and hence the Bravais lattice, it remains only to determine all the space directions in which Bragg diffraction occurs. Experi- mentally, it is not totally obvious how to determine the directions in which Bragg diffraction will occur. The diffraction directions are obtained by determining the points of the reciprocal lattice on a sphere, known as the Ewald sphere, with radius k0 and centered at the. For an incident vector k0 , diffraction directions such as kd are obtained using the Ewald construction as shown in a.
For an incident vector k0 and a detector in the direction k1 , no Bragg diffraction will generally be detected b , except for certain specific orientations of the crystal, such that the associated reciprocal lattice is oriented as shown in c , for example. Diffraction directions such as kd which join O to these points of the reciprocal lattice are few and far between.
If we have only one detector set in the direction k1 , for example, there will gen- erally be little diffraction in this direction. If the crystal is then rotated in space, according to 2. As can be seen from Fig. Given that we are working in 3D space, it is clear that, for a randomly chosen orientation of the crystal relative to the incident radiation, we will only very rarely observe a Bragg diffraction peak or spot.
There are several experimental methods to get round this difficulty, e. Knowing the orientations at which diffraction occurs, we may then characterise the reciprocal lattice, and hence also the crystal symmetries and the size of the primitive cell in the crystal lattice itself. So far we have not considered the structure of the diffracting object in any detail. Suppose now that we observe the diffraction by a crystal whose primitive cell com- prises Na atoms, possibly of different chemical nature, at positions rl inside the unit cell.
There are thus two multiplicative factors in the expression for the diffracted inten- sity. The second,. Note that these coefficients will be different for X-ray diffraction, the X rays being scattered by electrons, and neutron diffraction, since neutrons are scattered by the atomic nuclei. Indeed, at the high frequencies associated with X rays, the electric field of the electromagnetic wave couples predominantly with the electrons in the atom.
The amplitude of the wave scattered in the direction k1 by the electronic density at O is proportional to the electronic density at this point:. Consider now an arbitrary point at position u. In an analogous way to Fig. The diffraction pattern produced by an arbitrary object thus contains information about the structure of the diffracting object.
Going back to the case of a single atom and comparing 2. For a crystal whose primitive cell contains Na atoms at position rl , the diffraction amplitude is the Fourier transform of the total electronic density, which is the peri- odic reproduction of the electronic density of the primitive cell. The latter is given by 2. It can be determined once we know the form factors and the positions of the different atoms in the primitive cell. We thus find that the experimental determination of the intensities of the Bragg diffraction peaks or spots will be extremely useful for ascertaining the arrangement of atoms in the primitive cell of the Bravais lattice.
Let us consider a specific example to illustrate how the basis of the primitive cell affects the intensity of the Bragg peaks. Consider X-ray diffraction by a crystal of C60 whose fcc primitive cell is shown in Fig. A suitable experimental setup records on a photographic film all the diffraction spots corresponding to the vectors K in one plane of reciprocal space see the image on p. The pattern observed can thus be used to directly visualise a plane of the reciprocal lattice and its symme- tries. In the image on p.
Identify the body-centered cubic reciprocal lattice plane visualised in the image of p. Deduce the dimensions of the primitive cell for C Some of the diffraction spots are very faint. Which vectors of the reciprocal lattice do these correspond to? To understand this, we take into account the fact that the X rays are scattered by electrons.
We may consider the C60 molecule as a uniform charge dis- tribution p r over the surface of a hollow sphere of radius R0 , viz. Calculate the structure factor for the C60 crystal. Deduce the radius R0 of C This aspect of X-ray diffraction was very important in determining the spatial struc- ture of complex molecules, such as biological molecules, which are commonly conserved in solution.
By crystallising N molecules, one then benefits from the fact that, in a crystal, the Bragg diffraction spots have intensities that increase as N 2 , whereas the intensity only increases as N when the molecules do not have a crys- talline arrangement. This method was used to determine the structure of DNA and many other biologically important molecules. Note that disorder or atomic and molecular motions modify the intensities of the Bragg diffraction spots.
The effect of lattice vibrations is discussed in Problem 1: Debye—Waller factor. A crystal lattice can be described as a combination of two entities: the Bravais lat- tice, which is a periodic arrangement of lattice points in space, and hence an abstract construction, and a material basis which is the actual physical entity associated with each node of the Bravais lattice. These are given by. In the simplest cases, the material basis is a single atom, but it may be a much more com- plex physical entity, such as an arrangement of atoms, one or more molecules, and so on. The structures of arbitrary molecular entities can be determined using diffraction methods.
An incident wave of wave vector k0 is elastically scattered by the various objects making up the molecular entity. Interference between the scattered waves leads to a diffraction pattern. In the case of a crystal, the diffracted intensity is only significant in specific directions k1 satisfying. This relation is the Bragg diffraction condition. The vectors K of the reciprocal lattice are defined by the condition.
The vectors k1 satisfying the Bragg diffraction condition can be used to determine the reciprocal lattice. The amplitude of the diffraction is related to the basis of the primitive cell of the crystal.
The structure factor of the primitive cell is then. The alkane molecules are arranged in parallel rows. Note that two consecutive rows are staggered in a quincuncial arrangement. This is checked by looking at Fig. The lattice is therefore centered rectangular. This is shown on part of the image and also on a molecular model. It is an irregular polyhedron for the centered rectangular lattice and a hexagon for the hexagonal lattice see Fig.
These are shown on an enlarged portion of the image of Fig. The honeycomb structure of graphene corresponds to a hexagonal lattice in which one in three sites have been removed. It contains two carbon atoms placed at the nodes of two hexagonal sublattices of types A and B, which differ in the opposite orientations of their nearest neighbours. Each atom A has three nearest neighbours B and vice versa. Its Bravais lattice is hexagonal with twice the lattice constant of the initial structure. The crystal in Fig. There are thus two atoms per polyhedral primitive cell in Fig.
There are 8 atoms per fcc unit cell: the vertex, the three centers of adjacent faces, and the 4 empty spheres in Fig. The atoms located on the two empty spheres represent the basis for this crystal. The atoms must be placed at the sites of the fcc lattice in a periodic manner to obtain a perfect arrangement. With two different atoms, viz. It should be fairly clear that it is not possible to associate the sites of the fcc struc- ture two by two to define a new Bravais lattice.
However, if we take 4 atoms with 3 the same, one of the atoms can be placed at the vertex of the fcc conventional unit cell and the three others at the centers of the adjacent faces empty in Fig. We thereby construct a crystal with simple cubic Bravais lattice of side a and a basis of 4 atoms per cell. Full spheres represent Au and empty spheres Cu. There are therefore two simple possibilities, those corresponding to Au3 Cu or Cu3 Au, i.
Naturally, we can imagine ordered solutions for lower concentrations of one of the metals, but there is no particular physical reason why they should ever occur. Note that, in the ordered Cu3 Au structure, the gold atoms have no Au nearest neighbour. This structure is found to be stable on thermodynamical grounds, which indicates that Au atoms strongly repel one another. However, the Au3 Cu structure is not obtained experimentally, probably because the repulsion between copper atoms is not strong enough.
The ordered AuCu alloy exists, but crystallises into a cubic lattice which bears no relation to the fcc structure of the pure metals. The ordered arrangement of the Au and Cu atoms can be detected by a diffraction method. The ordered lattice of Cu3 Au shown in Fig. There are there- fore many diffraction spots for the ordered alloy. In Fig.
This textbook sets out to enable readers to understand fundamental aspects Graduate Texts in Physics Introduction to the Physics of Electrons in Solids. This textbook sets out to enable readers to understand fundamental aspects underlying quantum macroscopic phenomena in solids, primarily through the.
The pres- ence of order can thus be revealed by the appearance of diffraction spots associated with these new vectors in reciprocal space. The C60 lattice is face-centered cubic with conventional unit cell of side a. Spots are therefore expected for. In the observed plane of the reciprocal lattice, the spots form a square lattice.
The only planes of the body-centered cubic reciprocal lattice with this property are the planes passing through one of the faces of the cube, i. See answer to Question 2. Note that the spots h, 0, 0 and 0, k, 0 are rather faint. In fact their intensity is found to be at least times lower than the intensity of the spots 2 2 0. The total number of electrons is per C In fact, using 2. As illustrated on the cover, the Fermi surface of a metallic material can be represented in reciprocal space using angle-resolved photoemission experiments. In this image, the Fermi surface of the CuO2 plane of the cuprate superconductor BiSr2 CaCu2 O8 is reconstructed experimentally in the reciprocal space plane kx , ky.
Question 3. Image courtesy of S. Borisenko, constructed from experimental results in Kordyuk, A. In this chapter, we introduce the generalisation needed to discuss the electronic properties of solids in more detail. We restrict the discussion to features that can be described using the approximations presented in Chap. Considering fixed atomic positions, and averaging electron—electron interactions, the task boils down to solv- ing a one-electron problem. We also restrict to the case of crystalline solids, having introduced the basic ideas in Chap.
While the tight-binding approximation provided a first handle on the notion of energy band, we shall see that this notion can also be introduced in the framework of quite the opposite approximation, in which the periodic potential is weak. This will reveal the importance of the reciprocal lat- tice and lead to a general definition of the Brillouin zones outlined in our discussion of the linear atomic chain in Chap. We also establish a link between the limiting cases of weak and strong atomic potentials.
We can then establish certain general rules about the band structures for the chain, but also for two- and three-dimensional solids see Sect. We find that this model implies only two possible states for a solid, viz. In addition, without carrying out any detailed calculation, we can make several quantitative predictions that can be compared with experiment.
A discussion of these points will bring out the strengths and weaknesses of the band model. Finally, in Sect. We saw in Chap. We shall see here that this organisation of the structure of the electronic states into energy bands is in fact a general property of the electronic energy spectrum in the presence of a periodic potential. We shall then see how the energy band structure can be obtained in a simple way from that of the free electrons in the weak limit of the periodic potential see Sect.
The notion of Brillouin zone can then be extended in a very general way see Sect. In an infinite crystal, or more precisely, in a crystal subject to periodic boundary conditions, electrons located at positions separated by a direct lattice translation Rl [of the form 2. They have the form. A state is thus specified by four quantum numbers, namely the three components kx , ky , kz of the vector k and an integer n, because 3. One consequence of 3.
Furthermore, the periodic boundary conditions for a rhombohedral solid contain- ing many primitive cells see Fig. For a macroscopic solid, these values of k will be extremely close together on the scale of the primitive unit cell of the reciprocal lattice see Fig. We can then apply the argument in the box in Sect. The vectors k play a key role when describing the properties of crystalline solids.
Indeed, using the definition 2. The function 3. Since the quantum states are only defined up to a vector of the reciprocal lattice, they can all be specified by restricting to a particular primitive unit cell of the reciprocal lattice. The form 3. For reasons of energy conser- vation, umklapp effects can often be neglected. It should be noted that, in a crystal, the average of the true momentum p can never be treated as a conserved quantity.
In fact, p is not conserved, due to the presence of the potential in the Hamiltonian. Band structure calculations based on the tight-binding approximation, as discussed in Chap. In other cases, for example, the case of valence electrons of certain metals, which may have a high kinetic energy, the periodic potential can be treated as weak.
For these nearly free electrons, the crystal potential may even be treated as a perturbation. We demonstrate below that the wave function of the electron is then close to a plane wave, and that the energy varies to a first approximation as it would for free electrons. However, even though the perturbation due to the periodic potential may be weak, it may become important for specific values related to the crystal periodicity.
This idea will lead to a more general definition of the first Brillouin zone. In a crystal, the one-electron Hamiltonian also includes the potential due to the ions given in 1. We know that, if we consider the non-degenerate states of the unperturbed Hamiltonian, the effect of the perturbation is to alter the energy levels and wave functions in the following way [2, Chap.
This effect can be neglected, since it does not change the parabolic shape of the variation of Ek as a function of k. Let us consider the case of a linear chain, where it will be easier to represent Ek graphically. The existence of the periodic potential means that periodic energy eigenvalues must be obtained in reciprocal space. This generates curves Ek that are peri- odic in reciprocal space. We thus obtain the extended zone band structure in k space see Fig. According to 3. To determine this matrix element, we use the fact that the poten- tial V r is periodic.
However, in the quasi-momentum space, a the 1D parabolic band is translated in such a way as to reveal the periodicity associated with the potential extended zone band structure.
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We may thus write 3. In this case, even a very weak periodic potential will have a significant effect on the electron energy levels. This is not surprising, be- cause we know from the discussion in Chap. We shall see more precisely in Sect. The Bragg planes are thus of singular importance for understanding the electronic structure of a solid. These planes, which orthogonally bisect the reciprocal lattice vectors taken from the origin, bound a particular primitive cell of the reciprocal lattice, namely the Wigner—Seitz cell constructed around the origin, as defined in Chap.
It is a primitive unit cell of the reciprocal lattice which specifies the first Brillouin zone. It contains all points k such that the line segment Ok does not intersect any orthogonally bisecting plane of the vectors of the reciprocal lattice leaving O. The volume of the first Brillouin zone is the same as that of any primitive unit cell of the reciprocal lattice, i. Find the expression for the vectors of the reciprocal lattice in two dimen- sions. Determine the reciprocal lattices for the hexagonal, centered rectangular, and oblique lattices, together with their first Brillouin zones.
The first Brillouin zone of the cubic lattice, e. The first Brillouin zone for a face-centered cubic Bravais lattice, e. The coordinates of equivalent points under the symmetries of the cube are easily deduced. The first Brillouin zone is shown. To obtain a better understanding of the energy level distribution of the Bloch states for a linear chain, in Sect.
In 2D, this would be a Bragg line, and in 3D, a Bragg plane, as we have already referred to it. We will then be able to show in Sect. To find out how these eigenstates are modified, we keep only those terms corresponding to the Fourier series expansion of the potential, viz. The above perturbation calculations show that the only free electron states to be modified significantly by the weak periodic potential are those close to a Bragg plane, i. The perturbing potential only couples states for which the vectors k differ from a vector of the reciprocal lattice, i.
Here the first correction term can be neglected as its energy denominator corre- sponds to free electron states far apart in energy. In the energy interval. Figure 3. The free electron parabolas in Fig. In a , the states coupled by the Fourier component V1 x of the potential are joined by a double-headed arrow: the first 1 couples states that are non-degenerate in the absence of the periodic potential, while the second 2 couples states that are degenerate in the absence of V1 x.
Band gaps appear. This calculation, carried out only in the 1D case for didactic reasons, shows that the periodicity modifies the parabolic dispersion relation of the free electron, causing band gaps to appear at the points which correspond in the reciprocal space to the midpoints of vectors in the reciprocal lattice. These marked singularities in the band structure, obtained for states correspond- ing to the vectors k close to a Bragg plane, arise because the corresponding plane wave states suffer a Bragg diffraction on the periodic potential of the ions, even if this potential is very weak.
The eigenstates for the wave vectors 3. It thus remains to understand how the above approach based on the idea of nearly free electrons can be related to the tight-binding approximation discussed in Chap. Naturally, the energy eigenvalues for the model in the last section can be obtained numerically to any required accuracy, whatever the amplitude of the periodic po- tential.
The results of such a calculation, in which only the term V1 x has been retained, are shown in Fig. We observe that, for a weak potential, the results are indeed those obtained us- ing the nearly-free electron approximation. However, for a stronger potential, the shape of the lower band approaches the cosine shape obtained in the tight-binding approximation.
For even stronger potentials, this trend becomes more pronounced and extends to higher bands, provided we take into account the Fourier components Vp x of the potential. We thus move smoothly from the case of nearly free electrons, corresponding to a weak atomic potential, to the LCAO approximation, which cor- responds to a strong atomic potential. The free electron parabola is 0 shown by a dashed curve. We now extend the tight-binding and nearly-free electron methods Sects. The notion of reciprocal space becomes essential here for representing the structure of the energy states in two dimensions.