Free electron model

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In solid-state physics, the free electron model is a simple model for the behavior of the load carriers in solid metal. It was developed in 1927, primarily by Arnold Sommerfeld who incorporates the classical Drude model with Fermi-Dirac statistics of quantum mechanics and hence also known as Drum-Sommerfeld's model.

Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, in particular

• Wiedemann-Franz's law that connects electrical conductivity and thermal conductivity;
• temperature dependence of electron heat capacity;
• form of country electronic density;
• the range of binding energy values;
• electrical conductivity;
• Seebeck coefficient of thermoelectric effect;
• thermal electron emissions and terrain electron emissions from bulk metals.

The free electron model solves many of the inconsistencies associated with the Drude model and gives insight into some other properties of metal. This model assumes that the metal consists of a quantum electron gas in which the ion plays almost no role. Models can be highly predictive when applied to alkali and noble metals.

Video Free electron model

Ide dan asumsi

In the free electron model, four key assumptions are taken into account:

• Free electron approach: The interaction between the ion and the valence electron is largely ignored, except under boundary conditions. Ions only maintain the neutrality of the charge in the metal. Unlike in the Drude model, ions are not always the source of a collision.
• Independent electron approach: Interactions between electrons are ignored. Electrostatic field in metal is weak due to filtering effect.
• Time-relation approach: There are some unknown scattering mechanisms so the electron probability of collisions is inversely proportional to the relaxation time ${\ displaystyle \ tau}$ , which represents the average time between collisions. Collisions do not depend on electronic configuration.
• Pauli exclusion principle: Each quantum state of the system must be occupied by an electron. The restriction of available electron states is taken into account by Fermi-Dirac statistics (see also Fermi gas). The main predictor of the free electron model comes from Sommerfeld's expansion of the Fermi-Dirac dwelling for energy around the Fermi level.

The model name comes from the first two assumptions, because each electron can be treated as a free particle with the quadratic relationship between each energy and momentum.

The crystal lattice is not explicitly taken into account in the free electron model, but the justification of quantum mechanics is given a year later (1928) by the Bloch theorem: unbound electrons move in periodic potentials as free electrons in a vacuum, except the mass of electrons m into effective mass m * which may deviate considerably from m (one can even use a negative effective mass to describe conduction by hole electrons). Effective mass can be derived from calculations of band structures that were not initially taken into account in the free electron model.

Maps Free electron model

From the Drude model

Many physical properties follow directly from the Drude model because some equations do not depend on the statistical distribution of particles. Taking the classic speed distribution of the ideal gas or Fermi gas speed distribution only changes the results associated with the speed of the electron.

Terutama, model elektron bebas dan model Drude memprediksi konduktivitas listrik DC yang sama ? untuk hukum Ohm, yaitu

${\ displaystyle \ mathbf {J} = \ sigma \ mathbf {E} \ quad}  ÂÂ$ dengan ${\ displaystyle \ quad \ sigma = {\ frac {ne ^ {2} \ tau} {m_ {e}}},}  ÂÂ$

di mana ${\ displaystyle \ mathbf {J}}  ÂÂ$ adalah kerapatan saat ini, ${\ displaystyle \ mathbf {E}}  ÂÂ$ adalah medan listrik eksternal, ${\ displaystyle n}  ÂÂ$ adalah densitas elektronik (jumlah elektron/volume), ${\ displaystyle e}  ÂÂ$ adalah muatan listrik elektron, dan ${\ displaystyle m_ {e}}  ÂÂ$ adalah massa elektron.

Other quantities that remain the same under the free electron model as under Drude are the AC susceptibility, plasma frequency, magnetoresistance, and Hall coefficients associated with the Hall effect.

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Electron gas property

Many properties of the free electron model follow directly from the equations associated with Fermi gas, since the independent electron approach leads to an uninterrupted electron ensemble. For three dimensional electron gas we can define Fermi energy as a

where ${\ displaystyle \ hbar}$ are Planck's reduced constants. Fermi energy defines the Fermi level, ie the maximum energy of an electron in a metal can have a zero temperature. For metals, Fermi energy is in the order of electronvolts.

Country density

Kepadatan 3D dari status (jumlah keadaan energi, per energi per volume) dari gas elektron yang tidak berinteraksi diberikan oleh:

${\ displaystyle g (E) = {\ frac {m_ {e}} {\ pi ^ {2} \ hbar ^ {3}}} {\ sqrt {2m_ { e} E}} = {\ frac {3} {2}} {\ frac {n} {E_ {F}}} {\ sqrt {\ frac {E} {E_ {F}}}},}  ÂÂ$

where ${\ textstyle E \ geq 0}$ is the energy of the given electron. This formula takes into account the degeneration of the spin but does not consider a possible shift in energy due to the underside of the conduction band. For 2D, the state density is constant and for 1D is inversely proportional to the square root of the electron energy.

Chemical potential

In addition Fermi energy is used to determine the chemical potential of ${\ displaystyle \ mu}$ . Sommerfeld's expansion is a technique used to calculate chemical potential for higher energy, that is

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â?Â\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂTÂ\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          E                Â¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯                                    [             Â 1     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...                 ?                                   2     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,    ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,     Â 12        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                 ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                                                                            T     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂà   Ã...                      T                                         F              Â             Â                        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<                             2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...                 ?                          Â 4     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,    ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                 ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                                                                            T     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂà   Ã...                      T                                         F              Â             Â                        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ, <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<                   Â 4        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>      Â              ?                     ]                 ,               {\ displaystyle \ mu (T) = E_ {F} \ left [1 - {\ frac {\ pi ^ {2}} {12}} \ left ({\ frac {T} {T_ {F}}} \ right) ^ {2} - {\ frac {\ pi ^ {4}} {80}} \ left ({\ frac {T} {T_ { F}}} \ right) ^ {4} \ cdots \ right],}  Â}

di mana ${\ displaystyle T}  ÂÂ$ adalah suhu dan kami mendefinisikan ${\ displaystyle T_ {F} = E_ {F}/k_ {B}}  ÂÂ$ sebagai suhu Fermi ( ${\ displaystyle k_ {B}}  ÂÂ$ adalah konstanta Boltzmann). Pendekatan perturbatif dibenarkan karena suhu Fermi biasanya sekitar ${\ displaystyle 10 ^ {5}}  ÂÂ$ K untuk logam, maka pada suhu kamar atau menurunkan energi Fermi dan potensi bahan kimia secara praktis setara.

Kompresibilitas logam dan tekanan degenerasi

Energi total per satuan volume (pada ${\ textstyle T = 0}  ÂÂ$ ) juga dapat dihitung dengan mengintegrasikan ruang fase sistem, kami memperoleh

${\ displaystyle u (0) = {\ frac {3} {5}} nE_ {F},}  ÂÂ$

yang tidak bergantung pada suhu. Bandingkan dengan energi per elektron gas ideal: ${\ displaystyle {\ frac {3} {2}} k_ {B} T}  ÂÂ$ , yang nol pada suhu nol. Untuk gas ideal yang memiliki energi yang sama dengan gas elektron, suhu harus sesuai dengan suhu Fermi. Termodinamika, energi gas elektron ini sesuai dengan tekanan suhu nol yang diberikan oleh

$            {\displaystyle         P        =        -        {            \left(                          \frac{                  \mathrm{?}                  U                }{                  \mathrm{?}                  V                }                        \right)          }_{            T            ,            ?          }        =                  \frac{2}{3}                u        (        0        )        ,      }        \{\backslash \; displaystyle\; P\; =\; -\; \backslash \; left\; (\{\backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; V\}\}\; \backslash \; right)\; \_\; \{T,\; \backslash \; mu\}\; =\; \{\backslash \; frac\; \{\; 2\}\; \{3\}\}\; u\; (0),\}\;  ÂÂ$

di mana ${\ textstyle V}  ÂÂ$ adalah volume dan ${\ textstyle U (T) = u (T) V}  ÂÂ$ adalah total energi, turunan yang dilakukan pada suhu dan potensi kimia konstan. Tekanan ini disebut tekanan degenerasi elektron dan tidak berasal dari tolakan atau gerakan elektron tetapi dari pembatasan yang tidak lebih dari dua elektron (karena dua nilai putaran) dapat menempati tingkat energi yang sama. Tekanan ini menentukan kompresibilitas atau modulus bulk dari logam

${\ displaystyle B = -V \ kiri ({\ frac {\ partial P} {\ partial V}} \ right) _ {T, \ mu} = {\ frac {5} {3}} P = {\ frac {2} {3}} nE_ {F}.}  ÂÂ$

This expression gives the right order quantity for the bulk modulus for alkali metals and precious metals, which indicates that this pressure is just as important as other effects in the metal. For other metals the crystal structure must be taken into account.

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Additional predictions

Heat Capacity

An open problem in solid state physics before the arrival of a free electron model is associated with low heat capacity of the metal. Even when the Drude model is a good approach to Lorenz's number from Wiedemann-Franz's law, the classic argument is based on the idea that volumetric heat capacity of ideal gas is

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âc_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂVÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}^{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\text{Drude}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{3}{2}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; ÂnÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Âk_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂBÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; ÂAnnotation\; encoding\; =\; "application/x-tex">\; \{\backslash \; displaystyle\; c\_\; \{V\}\; ^\; \{\backslash \; text\; \{Drude\}\}\; =\; \{\backslash \; frac\; \{3\}\; \{2\}\}\; nk\_\; \{B\}\}\; Â\; Â$ .

If that is the case, the metal heat capacity could be much higher because of this electronic contribution. However, large heat capacity is never measured, raising suspicions about the argument. By using Sommerfeld's expansion, one can obtain the energy density correction at a temperature up to and obtain the volumetric heat capacity of the electron gas, provided by:

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â » Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂ ( Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...
                ?                  u        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,
                ?                 T        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                       Â )                           Â ·                           =                      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,               ?                             2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>     Â 2                                          Â¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯      Â              T                       Â         ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,          Â                           n            Â                Â  <Â> B                                {\ displaystyle c_ {V} = \ left {{\ frac {\ partial u} {\ partial T}} \ right) _ {n} = { \ frac {\ pi ^ {2}} {2}} {\ frac {T} {T_ {F}}} nk_ {B}}  Â} ,

di mana prefaktor untuk ${\ displaystyle nk_ {B}}  ÂÂ$ jauh lebih kecil daripada 3/2 yang ditemukan di ${\ displaystyle c_ {v} ^ {\ text {Drude}}}  ÂÂ$ , sekitar 100 kali lebih kecil pada suhu kamar dan jauh lebih kecil di bawah ${\ textstyle T}  ÂÂ$ . Perkiraan yang baik dari nomor Lorenz dalam model Drude adalah hasil dari kecepatan rata-rata elektron klasik yang sekitar 100 lebih besar dari versi kuantum, mengkompensasi nilai besar dari kapasitas panas klasik. Perhitungan model elektron bebas dari faktor Lorenz kira-kira dua kali lipat dari nilai Drude dan lebih dekat ke nilai eksperimental. Dengan kapasitas panas ini model elektron bebas juga mampu memprediksi urutan besaran dan ketergantungan suhu yang tepat pada rendah T untuk koefisien Seebeck dari efek thermoelectric.

Evidently, the electronic contribution alone does not predict Dulong-Petit's law, ie the observation that the heat capacity of the metal is constant at high temperatures. The free electron model can be improved in this sense by adding a lattice vibration contribution. Two well-known schemes to include lattice into the problem are Einstein's solid model and Debye's model. With the later addition, the metal volumetric heat capacity at low temperatures can be more precisely written in the form,

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{Â Â Â Â » Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â ? Â Â Â Â Â Â Â Â Â T Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â A Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â T Â Â Â Â Â Â Â Â Â Â Â Â Â ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ 3 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle c_ {V} \ approximately \ gamma T AT ^ {3}} Â Â} ,

di mana ${\ displaystyle \ gamma}  ÂÂ$ dan ${\ displaystyle A}  ÂÂ$ adalah konstanta yang terkait dengan materi. Istilah linier berasal dari kontribusi elektronik sedangkan istilah kubik berasal dari model Debye. Pada suhu tinggi, ekspresi ini tidak lagi benar, kapasitas panas elektronik dapat diabaikan, dan kapasitas panas total logam cenderung konstan.

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Note that without the relaxation time approach, there is no reason for the electrons to divert their movement, because there is no interaction, so the free path means to be unlimited. The Drude model considers the free electron path to mean close to the distance between the ions in the material, implying the earlier conclusion that the diffusive motion of the electrons is due to collisions with ions. The average free path in the free electron model is given by $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂlÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{    Â     v                ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯                          ?               {\ displaystyle \ ell = v_ {F} \ tau}  Â} (where $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{    Â     v                ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯                          =                       Â 2      Â             E                       Â         ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,          Â                      Â /                  Â    Â ï <½Â                             e        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,          Â                                {\ displaystyle v_ {F} = {\ sqrt {2E_ {F}/m_ {e}}}}  Â} is Fermi speed) and is in the order of hundreds of  ¥ ngstrÃÆ'¶ms, at least one larger order larger than the classical calculation possible. Free path means is not due to electron-ion collision but is related to imperfections in material, either because of defects and impurities in metal, or due to thermal fluctuations.

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Inaccuracy and extensions

The free electron model presents several deficiencies that conflict with the experimental observations. We list some of the inaccuracies below:

Temperature dependency

The free electron model presents some physical quantities that have incorrect temperature dependence, or no dependence at all like electrical conductivity. Thermal conductivity and specific heat are well predicted for alkali metals at low temperatures, but fail to predict high temperature behavior derived from ion motion and phonon scattering.

Hall effect and magnetoresistance

Hall coefficient has constant value $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{         R                Â Â¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ <¯¯¯¯¯ <¯¯¯ <¯¯¯ <¯¯¯ <¯ H¯ H¯ H¯ H¯                          =         -        (          n         e    ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂï mi½ <Â>                  )                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -   Â 1                                {\ displaystyle R_ {H} = - (nec) ^ {- 1}}  Â} in the free electron model. This value is independent of the temperature and strength of the magnetic field. Hall coefficients actually depend on the band structure and differences with the model can be very dramatic when studying elements such as magnesium and aluminum that have strong magnetic field dependence. The free electron model also predicts that the passing magnetoresistance, the resistance in the current direction, is independent of the field strength. In almost all cases.

Directional

The conductivity of some metals may depend on the orientation of the sample with respect to the electric field. Sometimes even the electric current is not parallel to the terrain. This possibility is not explained because the model does not integrate metal crystallinity, ie the presence of a periodic ionic lattice.

Diversity in conductivity

Not all metals are electrical conductors, some do not conduct electricity well (isolators), some can do when dirt is added like a semiconductor. Semimetal, with narrow conduction bands also exist. This diversity is not predicted by the model and can only be explained by analyzing valence bands and conduction. In addition, electrons are not the only carriers in metal, electron voids or holes can be seen as quasiparticles carrying a positive electrical charge. The hole conduction leads to the opposite sign for the Hall and Seebeck coefficients predicted by the model.

Other inadequacies are present in the Wiedemann-Franz law at intermediate temperatures and metal frequency dependence in the optical spectrum.

More precise values ​​for electrical conductivity and Wiedemann-Franz law can be obtained by softening the relaxation-time approach using Boltzmann's transportation equation or Kubo formula.

Spin is largely ignored in the free electron model and consequently can cause emerging magnetic phenomena such as paramagnetism and Pauli ferromagnetism.

A direct progression to the free electron model can be obtained by assuming an empty lattice estimate, which forms the basis of the band's structure model known as the nearly free electron model.

Interestingly, adding a disgusting interaction between electrons does not change the picture much presented here. Lev Landau suggests that Fermi gases beneath disgusting interactions, can be seen as equivalent quasiparticles gas that slightly modifies the properties of metals. The Landau model is now known as the Fermi liquid theory. More exotic phenomena such as superconductivity, where interactions can be interesting, require a more su

Source of the article : Wikipedia