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to be able to correlate the physical structure and property of the solid ones is necessary to have a description of the situation of the valence electrons that they hold joined atoms in the solid one. of description: the electrons completely are delocalizzati, like it describes the theory to them of the bands, or are essentially localizes and come analyzed the local property to you of the system (as it happens, as an example, in the theory of the crystalline field). When there is appreciable superimposition between orbital of adjacent atoms is opportune to apply to the theory of the bands valence electrons is in common between all atoms of the same type. The localized description is applicable, instead, when the interatomic interactions are weak people and the electrons turn out strongly. For how much, but, the two distant visions are a lot is not always. Both have advantages and disadvantages and their use depends on the context and the examined problem. We will begin from the point of view of delocalizzati electrons, introducing some introductory aspects of the model of free electrons and of the theory of the bands, whose developments go sure to outside of the scopes of meaningful of the metals they are their excellent property of conduction. From such property it is capacities in natural way to think in the terms of a model in which electrons they are relatively free and they can move under the small influence. Many metals moreover are ductile and can be forged in order to produce useful objects. All the metallic elements introduce structures with high solid numbers of coordination regarding ionic (typical NC = 6) and to those. Us it cannot therefore be attended of the ties. There is from only thinking rather than the valence electrons occupy in uniform way the space between Ionian of the metal the space between the nearer atoms contain density electronic, but in. Therefore, regarding the covalenti, the metallic ties bending of ties, and door to the ductility in these materials the energy of tie for atom in the typical metals is of various. As an example, the energy of tie for atom in the molecola.6 eV, while the value for atom in. However we notice that, in according to case, the valore.8 eV the energy for tie derives from various numerous ties with atom near therefore is small in the metals. An other important aspect of the metals is that, if we consider their Ionian ones, there is much empty space interionico.04 Å, much greater one of the sum of the beams large available for valence electrons or conduction electrons can be distributed in uniform way in the greater part of the crystalline volume, in way much various one from how much happens in the solid ones. All that leads to what it is the model of the free electron. We can think next to the formation of a "tie. For an isolated atom the valence electrons are found in a hole of upgrade created them from the nucleus and from. To approaching itself of various atoms the superimposition of upgrades them atomic determines a situation in which the valence electrons are found in upgrade them effective that are inferior to that one of isolated atoms one useful assumption of great simplification can assume that the complicated one upgrades them effective, with minimums a lot emphasize to you in correspondence of Ionian the metallic ones, (Figure) are constant for the electron to the inside of the metal. If the electron tries to leave the metal it comes behind pulled from loads clean positive left on the metal. it is found in an energy hole upgrades them, of W. depth electrons of free valence are hour. their behavior can be dealt in various ways and the energy can be determined. The energy, that it is exclusively of kinetic type in the models that. We will consider two models of the theory published in 1900 and based on the model of the kinetic theory of electrons they have the same medium And integrated kinetic energy with the foundations of the quantistica mechanics, that is that one that often is called theory of Sommerfeld. These theories, also with many successes, do not succeed however to answer to some fundamental questions (like that one: why some materials are metallic and others of the solid ones it will be necessary to resort to the theory of the bands. This theory holds R-on account the traslazionale simmetria of the crystalline structure, what that it has, as we will see, important consequences electrons of valence of atoms are free to the inside of the metal. Every atom supplies its electrons of valence electrons moves like an electronic gas. We have seen the hole of upgrades them in which it is gone to Assumeremo that all the electrons have the same energy and. It is assumed moreover that a some mechanism of Ionian collisions between and electrons exists for which electrons they can be carried in equilibrium. The kinetic theory of gases is applied quidi to this gas. Drude published this theory [ Annalen der Physik 1, 566 and 3, 369 (1900) ] three years after the discovery of the electron gives. Benchè the equations of the perfect gas are applied must but notice that the electron gas is much dense one. The high density of the gas can also be appreciated calculating the P pressure with the law of ideal gas, PV = RT, that he is equal to 3381 atm (having can observe that density and pressure are therefore large that we cannot wait for us that it is worth the law of the ideal gas, but this is not a fundamental difficulty, poichè is possible to bring of the corrections to the equation of the ideal gas. As far as the question: that what holds electrons in the metal, that is what determines a great value of uniform of Ionian associated positive charges to the metallic ones, that they render the sufficiently large one to hold electrons in the solid one. The semplicità of the model is one of its main attractions. In a generalized manner the density electronic of the metals is expressed (in el. supplied free electron number from every obvious atom which it is the value of Z, if we consider metals as an example like. In the following Table All the values are given to the N. values are in interval 10. it is the beam of a sphere whose volume is equal to the volume for electron. Also these values are given in Table electron have much more volume to disposition in the alkaline metals that in come deal in the model of Drude medium energy and thermal speed to you. From the kinetic theory we expect free ones have kinetic energy and they do not have energy upgrades them, therefore that the energy correlated to a medium quadratic speed (rms) v. represents a thermal medium quadratic speed of electrons in the model of Drude we have assumed that the valence electrons can move liberations and therefore can be deals to you using. But before what must make other assumptions with respect to the collisions before being able to deal arguments like the conductivity, assumes that the electrons are subject to collisions (without to specify the nature of the interactions), or that they endure. These collisions are dealt like instantaneous events, that it means that the time in which the diffraction happens is much every minor of other time of our problem electrons can acquire a thermal equilibrium correspondent to. It is assumed therefore that the electrons riemergano from the collisions without memory of their previous speeds and that they move subsequently in accidental directions with speeds appropriated to. In the second place, between a collision and the other the electrons move in linear way following the laws of Newton applied to an electric field in direction x, we will have md x = - eE. therefore the electron will have a speed adds given them from -(eE/m)t until the field comes applied and until that not. However, in average, the electrons come diffratti after a time t, and poichè after every collision they are still in thermal equilibrium, a constant electric field determines the fact that the electrons have one speed thermal distribution to the T temperature, in the greater part of the metals. Third party point, all the information on the electronic collisions are. On the base of these assumptions many property of dimensions can be gained and shape we correlate the current density, J, to the field. We consider a piece of metal of uniform section, with [ carica/(tempo x area) ] and those of and are (voltaggio/lunghezza). but Drude ignored this complication and reasoned in terms of a metal. We consider the motion of free electrons speed, but the directions are accidental and the vectorial sum of all. If, but, for effect of an external electric field an additional average speed is created v. (= - eEt/m) it obtains the linear relation between J and and, and one equation for. Poichè are not had meaningful variations if we apply to an electric field alternated with the frequency of microwaves (10 achieves some that? much inferior to 10 must correspond to a time. The times? they are all of the order of 10 sec, and correspond to greater frequencies much of those of the microwaves consisting with the idea that the current flow constitutes only one small perturbation in the normal behavior of electrons. This simple theory, that we have shortly introduced, correctly interprets some phenomena (like the relation of linearity between J and and), even if the presence of the amount t, that it is difficult to measure, renders not easy to estimate the conductivity will consider here) will exceed this difficulty. This simple model introduces but many aspects. Between the others, it does not explain the dependency of s from. Two aspects, then, of the crystalline structure come. In the first place, in some not cubical metals they are found remarkable of the quantistica mechanics to the electron gas, imposing that theory of Sommerfeld of the metals is defined. We will neglect for hour the spin and on the effects spin electronic it will come added subsequently, in the customary way. of non-relativistica wave, for a particle of mass m with energy in order to determine all the possible values of and and y. the fundamental of the independent electrons, second approximation which the electrons interact between they solo through one energy. The approximation involves for a system of N electrons of sostiture to the Hamiltoniano total of the system the sum of Hamiltoniani monoelectronic. The energy total is the sum of monoelectronic energies, and the wave function total is the product of monoelectronic functions of wave. Hamiltoniani characterizes contains them terms that represent the equation for one single particle, assuming for V(r) a value. Our assumption consists in thinking that the interactions electron-electron and the interactions with the Ionian ones turn out in average in. As solution of the equation we will use the classic wave tentativamente. The wave carrier has an ulterior one meant, in how much for depends on the carrier k, than to part the constant? carrier coincides with the linear moment p of the free electron k also is called propagation vector. From the point of view of the autofunzioni?(r), the carrier k represents one label for the distinguished values of opportune energy to use one simbologia of the type. You notice yourself also that the autovalori of the energy are doubly degenerate, why have the same energy is the autofunzione exp(ik. correspondent to a moment +?k, it is the autofunzione exp(-ik. only differ for the sign of k, while the autovalori. that it cannot only depend on the direction of propagation of free electrons but on the module of their moment. From the relation of de Broglie, p = h /, we know that is one wavelength associated with the particle, and therefore. The autofunzioni are standardized placing the integral on the volume del. monodimensional example (two types of conditions to the contour) necessary for imposing of the opportune conditions to the contour two types of such conditions will come discussed, one that door to standing waves why in three dimensions door to functions that are much easier to use in the context of the theory of the bands. The first type of conditions is to demand that the wave function goes to in the box monodimensional it and corresponds to assume of. the common standing waves, in which various entire multiples of l/2 they are contained in one scato it of L. length situation that we will consider is the condition to the periodic contour. This demands that the solutions periodic are scato it monodimensional as withdrawn on if same concur the use of functions of wave of esponenziale type, that they are the natural functions when travelling waves are considered. of probability it is *? = 1/L, that is he is uniform in all the champion. In three dimensions the periodic condition to the contour becomes?(x + L, y. For a free electron members x, y, and z of the Hamiltoniano they are independent between they: the wave function is the product of the functions in the three directions and the energy is the sum of the three contributions allowed for the wave carrier k is. A unitary cell in I space-m is 1 x 1 x 1, with one volume. A unitary cell in I space-k has a volume is a fundamental result that depends on the periodic condition to. This volume in I space-k for concurred value uniform regular disposition of the states you concur yourself in I space-k. The bidimensional unitary cell, the quadratino (n), has one area of (2). You notice yourself that the propagation vector k (that moment, except that for a factor constant opportunely represented like a function of the carrier k in one along orthogonal aces, then, for every point of the space therefore defined, can be given, in via of principle represents the carrier effectively, a value of the energy that a such function is not easy to visualize does not have to worry to us with respect to the interpretation of the meant one of this. We must remember that this space is quantizzato, and that is that only discreet points in it correspond to physical states in fact, is one risen of three-dimensional quantico number that label the moments and the autovalori of the energy. E' easy to verify that the values admitted of k form simple a cubical reticulum, with equal reticular constant to 2?/L, whose points are the values allowed. The carriers in this space have dimensions of one length. You notice also one characteristic profit of this I space-k in quantizzata shape: every point corresponds to one be energetic allowed of. In practical but we are authorizes to you from the principle of Pauli to arrange two electrons with spin opposite in such state counting the states, therefore, we must make attention if the spin it is included. More than one be it can have equal energy. We have resolved the equation of wave for free electrons in. We will have to follow the principle of exclusion of Pauli. Therefore, for every value of the quantico number it spaces them m we will be able to only repair two electrons with spin opposite. two electrons can be only repaired. ±1, 0) or (0, 0, ±1) all the energies are degenerate tvi are one orbitalica degeneration of order 6 or one degeneration of order. Therefore we have repaired a total of 14 electrons. The energy for m = 0 is null, while that one of successes to you 12. The successive electron in this Aufbau will have to be placed in one be to greater energy, with m. = (±1, ±1, 0), with degeneration of order 4, but two are others. Figure schematically represents full the energetic levels of a metal in the model of free electrons. Firm, Density of the states and Surface of Firm. We will see hour like using it in the filling, what that will concur us to introduce new profits concepts hour, solo the situation to the zero absolute therefore that we will be able to ignore the thermal excitations of electrons. In the filling of the states part from those more low energy. These correspond to the smallest values than k. therefore, space-k come filled up spherically, leaving from the origin electrons of a metal, generate a great sphere in space-k, that it contains all the values of k correspondents to occupied states. Every state included in the sphere corresponds then to two electrons of spin. For a sure given electron number we define a value k. Quale is the energy, the temperature and the speed of electrons with. All the electronic states that they have |k| smaller are contained in a sphere of beam k in space-k, and the number of such states is given from the relationship from the two possible values of spin for orbitalico state. If N and therefore the electron number totals, will be, valid is for one, than for two and three dimensions, we calculate the correspondent energy. Poichè we are considering the system to the zero absolute. to define an equivalent temperature, the temperature of Firm. similar way we can define a speed, the speed of Firm, poichè all the energy and of kinetic type such largenesses are given in Table for some metals are approximately 5 eV, much greater of the energy. The correspondents speed of Firm are dell' order cm/sec, much greater delle thermal speed high values derive give it to quantization delle energies and give it to restrictions del principle of Pauli, that she forces electrons to occupy advanced energetic levels much of those previewed for one. The surface of Firm is the energetic surface that represents the separation between occupied states and occupied surface of constant energy, with value And free it is not a sphere (said also sphere of Firm or level of Firm) but, like we will see, for real metals it can have a temperature, and also to high values much of T it remains almost. This surface is a lot important why only those electrons that are energetically near to it can contribute to the property magnetic electrical workers and sphere of Firm is contained allowed an enormous number of states (10) and the density is much large one in spaces-k. For crystals of normal dimensions the states allowed correspond to. We indicate here with g(E), but we will be able to call it r(E) or DOS later on. The density of the states is defined like the number number of states (comprised the spin) with energy until to a value. K the states are full until And total of the free electron system bulk modulus), can also be calculated in this model, brought back to comparison the values calculate to you and experience them of B for various between values calculate to you and measured of bulk the modules (in 10. the agreement are not much bond, are relatively satisfactory that the order of magnitude is corrected also having totally ignored the presence. Remarkable E' that for the case 3D the density the case 2D it is constant, independent from and, and for the case 1D is obtains various dependencies much of the DOS (and, analogous, of the Energy of Firm and the other connected largenesses). dimensionalità plays an important role in several areas of science. Physical effects 2D are important as an example in science of. Effects 1D are observed experimentally in ultrathin cables and sure monodimensional conductors (organic and not). There are observations that show as the quantistico model of the free electron gas is in reasonable agreement with. We consider like only example the density of the states. If the metallic aluminum is bombed with electrons of sufficient energy in order to drive away an electron 2p from the inner shell of To they come emitted beams X from the champion against the energy of i beams X is shown in Figure. In the metallic state the electrons 3s the "free electron gas", while the other electrons tear an inner electron 2p, leaving a vacation of the gas (Firm sea) can fall back in the vacant level emitting cancellations (in the region soft x-ray of the phantom, ca. of such cancellation is proporziona them to the density of the states. Therefore the diagram of the intensity represents (at least the agreement is discreet, but above all it is important that the calculate shape of the DOS is in reasonable agreement with the experiment. Analogous it turns out to you (with some greater articulation of the curves) are obtained from the analysis of the various simple metals of the block sp, or also of alloys between these metals (you see Figures). Drude came used one only medium energy of electrons. Subsequently Lorenz applied to the electron gas the law of distribution of Maxwell-Boltzmann, without meant to you. We have seen, instead, that the use of the quantistica mechanics and of the principle of exclusion of Pauli door to K, we must establish one appropriate description of the thermal occupation of the quantici states allowed. Particles as the electrons with spin 1/2 obey to Fermi-Dirac statistics, and. For an ideal gas of electrons to T temperature, the probability f that one is popolato be permission, of energy and, is to consider this the definition of the energy of Firm for T. of the temperature, and therefore much neighbor can itself be used in the arguments for the greater part of metals = 1 takes place only a lot to and = And for states many electron and neighbors only can be excite to you in states not occupied thermally, giving to important contributions to the physical property energy of the distribution can themselves be measured experimentally mediating electrons, dN, with energy between and and and + dE multiplying the density of the states, g(E), for the probability that one be it can be with the condition that the integral of dN is N, the deducted number total of from the model of the electron gas exclusion principle very catches the greater part of electrons of the electronic gas of a metal in Firm energetic levels under the level of (surface of Firm), and therefore such electrons cannot contribute to the property of the metal property are determined therefore not as well as from the density electronic total, how much more rather from the density of the states to the level of Firm. We shortly cite (without demonstrations) some of these property specific heat of the electronic gas is given from:. This is the dominant contribution to the specific heat of a metal to low temperatures, and renders available one of the ways magnetic H, you would have to be a fraction of electrons with magnetic moment parallel to the field that exceeds the fraction of electrons with electronic would have to introduce a paramagnetic susceptibility experiences them damage an independent value from the T that turns out. Sure the paramagnetic behavior of electrons of a metal is more complex than how much it can be discussed to this point. Figure (a) extension a diagram of the energy total against the density. The Figure separates electrons with m and antiparallel regarding the direction of the magnetic field applied. For the principle of exclusion of Pauli, solos the electrons with energies can be reorient, changing their quantico number to you m. The deeper electrons occupy completely be occupied and they cannot riorientarsi. wait for a paramagnetic susceptibility of the T/T. order that is independent from the temperature and approximately 1/100 of the value for free electrons, poichè T/T. The result is in qualitative agreement with the experiments. The Figure (b) extension the same curves, but moved for effect of a magnetic field applied, that before every spin total of the occupied states has been reoriented number is the same one with and without field. However, this is a situation equilibrium poichè some electrons higher energy with m. electrons is not accessible empty states that they occupy changing. Only the electrons with kinetic energy of the order interesting point are that the application of Fermi-Dirac statistics does not alter this equation a bidimensional section of spaces-k to a time 0 and a successive time t, after the application of an electric field and nella. I space-k are moves to you like shown in Figure, with that happen in a medium time, the movement of the sphere of Firm of the model of Drude, and door therefore to the same expression for. Which electrons can have collisions and change their values k. Regarding the Figure, are the electrons with values of k on the left that they must collidere and change to values of k on the right of the sphere of Firm, restoring the equilibrium with one movement of the sphere of. (you notice yourself that it is to right that is available. Benchè the formula of Drude is equal to that one obtained here, the attended values of t in the two various cases is Fermi-Dirac statistics, the allowed collisions regard only electrons to the surface of Firm and these electrons have greater speeds much of those used from Drude. The collisions are essentially interactions of electrons with the impurities of the crystal and the reticular vibrations (fononi). last they increase with the temperature and the conductivity of a metal. The happened one of the simple model of the electron gas depends on the great superimpositions that are come true between diffuse orbital the atomic ones. The attractive semplicità of the theory renders it talora convenient also in cases in which the conditions of which over they are not totally. It is used, in these cases, a species of mathematical artifice, that the mass of electron one consists in rendering. In the solid ones that introduces tightened bands, deriving from insufficient superimposition of the orbital ones, the electrons are little furnitures and they are behaved like if they had one proof mass (m *) greater. Instead, in case of bands the much wide appearing behavior is that one of particles of inferior mass. Many of the equations that we have seen for free electrons can therefore be generalized replacing to the true mass of the electron.