Guide Condensed Matter Theories: Volume 3

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Quantum electrodynamics gives us. When the quantized electromagnetic field is submitted to the boundary conditions imposed by a topological defect the correction obviously change. For an electron bound to a negative disclination it is found [29]. The well-known original Aharonov-Bohm effect involves a charged particle moving outside a region with magnetic flux.

If the flux is confined to a string, it can be described, in the Kaluza-Klein approach, as the five-dimensional metric. Now, compare this metric with the four-dimensional metric of a screw dislocation see Equation 2. Notice that the Burgers vector b plays the same role as the magnetic flux F making it clear that the Aharonov-Bohm effect should appear for a particle moving in the presence of a screw dislocation. Topological defects appear in nature as consequence of symmetry-breaking phase transitions.

An important issue in cosmology is whether the observed structure of the universe contains relics of topological defects formed as the early universe cooled. What is the defect density after a phase transition? In T. Kibble[31] proposed a statistical procedure for calculating the probability of string formation in the phase transitions in the early universe.

In Chuang and coworkers [32] and in Bowick et al. Since different experimental groups [3] have tested Kibble's mechanism in superfluid He 4. More interesting than superfluid He 4 are the superfluid phases of He 3 which are quantum liquids with interacting fermionic and bosonic fields. Its rich structure gives rise to a number of analogues of cosmological defects [33]: 1 the dysgiration, that simulates the extremely massive cosmic string; 2 the singular vortex, which is analogous to the rotating cosmic string, 3 the continous or ATC vortex, whose motion causes "momentogenesis", which is the analogue of baryogenesis in the early universe; 4 planar solitons, that have event horizons similar to rotating black and white holes; 5 symmetric vortices in a thin film , which admit the existence of closed timelike curves through which only superfluid clusters of anti-He 3 atoms can travel and violate causality; 6 moving domain walls that can generate Hawking radiation; and so on.

There are still the superconductors with their vortices; glasses with their curved space description that require disclinations and perhaps monopoles; magnets with their disclinations and domain walls Condensed Matter Physics has much benefited from tools and ideas from gravitation and cosmology and pays that back by offering a laboratory for testing some cosmological or gravitational hypotheses. My warmest thanks go to my numerous collaborators see references [] who made this work come out of the vacuum. D 44 , Bowick et al. Zurek, Phys. Katanaev and I. Volovich, Ann. NY , 1 Kleman, Points, Lignes, Parois: dans les fluides anisotropes et les solides cristallins Edition de Physique, France Balian et al.

North-Holland, Amsterdam, pp. Puntigam and Harald H. Soleng, Class. Quantum Grav. Vilenkin, Phys. D 23 , Gal'tsov and P. Letelier, Phys. D 47 , Tod, Class.

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A , Furtado et al. A , 90 A 13 , Bezerra, Phys. D 59 , A Bezerra de Mello, V. Bezerra, C. Furtado and F. Moraes, Phys. First however I need to introduce one of the main rules of quantum mechanics, The Pauli Exclusion Principle. The Pauli Exclusion Principle stated that no two electrons can occupy the same quantum mechanical state, and this basically means you can never have 2 electrons in identical situations in a quantum state, something has to be different between them. For, when you are building an atom from scratch you start off with the nucleus and then add an electron, now if you want to add a second electron it has to have something different to the first.

In this case it will be its spin. Electrons have a property called spin, which can be defined as being up or down, so the second electron you add to the atom will have the opposite spin to the first. Now what about if you want to add a third electron? There are only two possibilities for spin so something else will have to change, so the third electron creates the 2s shell to go into thus being in a different quantum state to the other two, and will more electrons the process continues, thus building up the periodic table.

So with this one principle we have explained most of chemistry. So the Pauli exclusion principle says that we can get 2 electrons on each of the energy levels, one of them spin up and one spin down. It is worth making a quick note here about the situation known as degeneracy. Degeneracy is when you have two or more separate energy levels which have the same energy.

For example if the value of were , , , this would give the exact same answer for the energy as , , , even though they are 2 physically different levels. Our equation for the energy of the states in an infinite 3D well is given by equation Our value for L in this equation will be of the order 1 meter as we are describing a sample of metal not an individual atom. If we put 1m into equation 26 then we get. In the photoelectric effect experiment the electrons were being emitted with energies of the order 1eV. To get these energies the n values have to be huge in order to make up 19 orders of magnitude.

We know that n changes by 1 from one level to the next, so the states must be packed in very closely on order to fit them all in. To help with all these closely packed states we introduce a concept know as the Density of States, D E , which is defined as the number of allowed states in a specific energy range. If we treat the , and values like vectors representing the distance from the lowest energy then we can use Pythagoras Theorem to combine them back into one resultant vector like so.

The number of states with energy will just be the number of states inside the sphere of radius n.

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We can also make the length term more useful. We know that so so , which we can put in and then take inside the brackets along with the and get. This is called the Fermi Energy, and is the highest occupied state after all the available electrons have been piled in from the lowest states. Think of it as the surface of the electron sea in the bathtub model.

From equation 28 you can see that most of the equation is made of constants and the energy only depends on the and terms, so we can write. But the number of electrons per volume is just the density, D, so we can write it as. What we want to know however is how D varies with E, so we just differentiate with respect to E and get.

Which bears a striking resemblance to the distribution we are looking for. The actual equation is given by. Originally we assumed that each of the atoms gave one electron which gave us an electron density of about — electrons per square meter. If we put this value into equation 28 we get. Which agrees with the results found in the photoelectric effect experiment. The distribution we have just found and equation 28 and 29 apply when all the states are filled in order up to the Fermi energy, and this only applies at absolute 0 i.

The distribution at higher temperatures depends on the probability of a state being occupied at a given temperature. This probability is given by the Fermi-Dirac distribution. For the probability is close to 1, you are very likely to find electrons below the Fermi energy at most temps. At is is exactly a half as you can see from the equation. Because of the exponential term, P E is usually 1 or 0 except for a small region around the fermi energy. So now if we include this probability into our equation then our distribution is given by.

So now we have a working theory and set of equations that describe the distribution of electrons and their energies. Our P E D E equation becomes. The first term is equation 30 and the second term is equation The constants from equation 29 have been combined into one constant, A, for the sake of simplicity. Which is the Maxwell Distribution!!! The electrons can only be excited to higher states if there is a free state for them to go in to. The number of electrons that can be excited is equal to the shaded area. This area is roughly equal to the density of electrons at the fermi energy times the width kT, so D Ef kT.

This means that the ratio of electrons excited is. So the total number of electrons excited will just be this fraction times the number of electrons N. This gives. This difference is usually of the order 0. The equation we obtained for the specific heat uses some very rough estimations.

If you use more detailed calculations you get. Just like before we are using the idea that conductivity arises from the electrons gaining a drift velocity from an eternal field, however this time we have to take into account the pauli exclusion principle. The main difference between the classical and quantum descriptions comes from the mean free path. For an electron to scatter it must have a free state to scatter into, so only the electrons around the fermi energy can do this. Their velocity will correspond to the the fermi energy so we have. The fermi velocity is independent of temperature and varies with T , so which means that resistivity is proportional to , which was found here.

We will use the 6 streams method as we did before so we get the thermal conductivity as. As we just stated, only electrons with at the fermi energy will be able to move around so we can replace the average velocity, , with the fermi velocity. We can also replace the specific heat with the new one calculated in equation 31 to get. We know that at high temperatures , so is independent of , which is now correct. And at low temperatures is virtually independent of temperature so , which is again correct.

Now that we have 2 new values for and we can now get a new value for the Wiedemann-Franz law, which is. This number matches the one found experimentally a lot better than before. As well as electric fields the electrons inside a metal will also react to an applied magnetic field.

Electrons have a magnetic moment due to their spin Remember that a magnetic field is created by a moving charge, which includes a spinning electron. The magnetic moment of an electron, , is given by. In a magnetic field the electrons will gain an extra of magnetic energy depending on the direction of their spin, which will separate the up and down electrons.

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Half the electrons will be spin up and half will be spin down 1 In the diagram below , so half will get an increase in energy and half will get a decrease 2 in the diagram below. The electrons that gained energy will want to return to the lowest energy state, the fermi energy. To do this they will undergo a process known as spin flipping.

This causes a net magnetic moment in the metal as there are now more electrons of one spin than the other. The net magnetic moment caused by the imbalance of electrons is equal to. Using the distribution at the fermi energy we can rewrite the susceptibility as.

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The value of is roughly and is in good agreement with experimental results. What happens if you have an electric field and a magnetic field? To test this we take a thin rectangular slap of metal and apply a voltage across it and a magnetic field through it like so. So we have an E field in the x direction and a B field in the y direction. To work out what happens we have to use the Lorentz Force Law. Due to the cross product between the velocity of the electron and the magnetic field the electrons move in a circular path like so.

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This creates an imbalance of charge in the metal that creates a third field, an E field in the z direction. When the system gets to equilibrium this E field cancels the effect of the B field and a normal current can flow. Variational Theory of Impurities in Liquid 4 He. Ristig, J. Spin-Polarized Deuterium. Variational Density Matrix Theory I. Campbell, K.


Senger, M. Ristig, G. Senger, K. Thermal Response of Hot Nuclei. Vary, G. Bozzolo, H. Miller, R. Convergence Properties of an Exact Band Theory. Correlations in Fractional Hall Effect. Kallio, P.