Bose—Einstein statistics

If particles are indistinguishable and there are no restrictions on how many particles can occupy a cell, which is the case for most atoms and molecules, these particles are said to obey Bose-Einstein (B-E) statistics. There are (gi + N ways of arranging the sequence of numbers and letters. Since particles are indistinguishable, only the number of particles in each cell, not which particles, is important. Therefore, we must divide by redundancies. Cells are distinguishable (each corresponds to a particular position and momenta), but the sequence in which they are taken is not important. That is, the above sequence could have been written as 3bd, Ic, 2, 4ae. without changing anything physically. Since there are cells, the sequence can be permuted gi ways that are redundant. 

Therefore, the number of ways of arranging Ni particles in celb corresponding to energy i is 

Now we must determine the most probable distribution of the N particles among the Ej energy states with the following constraints  

This last constraint can be eid orced by maximizing In Wj since ln(Wj) = In Wi. This corresponds to requiring the entropy, which is defined asS = fclnlV = fcJ lnWi, to be a maximum, which is the same as requiring the system to be in thermodynamic equilibrium. 

Taking the variation of Equation 15.2 with respect to Nj (same as taking the derivative with respect to Ni), 

In agreement with Pauli principle, for the particles with integer spin, there is no restriction to the number of particles that can be placed in an available sub-level of an energy level. However, the fundamental difference toward the Boltzmaim distribution, somehow similar in occupancy, is that the quantum one (the Bose-Einstein spin based ne) it can be obtained by arranging particles-imderstate also through the permutation of the walls between sub-states in the same way in which are permuted the particles id est, the particles and the walls that are separating them, can be 

Under these conditions, the thermodynamic Bose-Einstein probability that is by cumulating the entire permutation particles + walls (Af + g, -1) excluding the permutation of two particles TV and two walls (g, -l) , based on their identity so it can be written in an elementary maimer 

The standard construction requires that the thermodynamic function of the macrostate it will be written by combining the statistical information contained within the thermodynamic probability (1.157) with the Langrange constraints of particle and energy conservation 

TABLE 1.5 The Illustration of a Possible Mode (Arrangement) for the Quantum Distribution of Bosonic-Type Particles (with Int er Spin) on an Energetic Level withg Sub-Levels (Putz, 2010) 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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Bose-Einstein statistics Bose-Einstein systems Boson Bosons... 

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

Systems containing symmetric wave function components ate called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Ditac systems (130,131). Systems in which all components are at a single quantum state are called MaxweU-Boltzmaim systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Eermi-Ditac statistics (132). 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

Similar principles apply to ortho- and para-deuterium except that, as the nuclear spin quantum number of the deuteron is 1 rather than as for the proton, the system is described by Bose-Einstein statistics rather than the more familiar Eermi-Dirac statistics. Eor this reason, the stable low-temperature form is orriio-deuterium and at high temperatures the statistical weights are 6 ortho 3 para leading to an upper equilibrium concentration of 33.3% para-deuterium above about 190K as shown in Eig. 3.1. Tritium (spin 5) resembles H2 rather than D2. 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

The probability that an oscillator at a given temperature occupies a given energy state, en, is given by Bose-Einstein statistics (see e.g. C. Kittel and H. Kroemer, Further reading) and the mean value of n at a given temperature is given by... 

Because of the involvement of phonons in indirect transitions, one expects that the absorption spectrum of indirect-gap materials must be substantially influenced by temperature changes. In fact, the absorption coefficient must be also proportional to the probabihty of photon-phonon interactions. This probabihty is a function of the number of phonons present, t]b, which is given by the Bose-Einstein statistics ... 

Particles that obey Bose-Einstein statistics are called Bose particles or bosons. The probability density of bosons in their energy levels is represented by the Bose-Einstein function as shown in Eqn. 1-2 ... 

As mentioned above, we assume that the molecular energy does not depend on the nuclear spin state For the initial rovibronic state nuclear spin functions available, for which the product function 4 i) in equation (2) is an allowed complete internal state for the molecule in question, because it obeys Fermi-Dirac statistics by permutations of identical fermion nuclei, and Bose-Einstein statistics by permutations of identical boson nuclei (see Chapter 8 in Ref. [3]). By necessity [3], the same nuclear spin functions can be combined with the final rovibronic state form allowed complete... 

For a gas containing N molecules of the same chemical species, the molecules would all be indistinguishable from one another. The factor W has to be divided by Nl in this case. The proper explanation can only be understood through a detailed discussion of quantum mechanics and Bose-Einstein statistics. This explanation is beyond the realm of interest here, and we simply state the proper weighting for a collection of N indistinguishable molecules as... 

BOSONS. Those elementary particles for which there is symmetry under intra-pair production. They obey Bose-Einstein statistics. Included are photons, pi mesons, and nuclei with an even number of particles. (Those particles for which there is antisymmetry fermions.) See Mesons Particles (Subatomic) and Photon and Photonics. 

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

COVALENT BONDING involves a pair of electrons with opposite electron spin. The bond (or electron charge distribution) is essentially localized between nearest neighbor atoms that contribute electrons for the bonding. Since these electron pairs follow Bose-Einstein statistics, therefore they are known as boson. In this case the paired particles do not obey the Pauli Exclusion Principle and many electron pairs in the system may occupy the same energy level. 

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