Einstein-Bose condensation

More detailed calculations show that BEC is reached if 

This gives the critical temperature T, where Bose-Einstein-condensation starts 

The critical temperature increases with the atomic density as. The minimum density for BEC depends on the atomic mass and on the temperature and decreases 

Example 9.12 Eor Na atoms at a temperature of 10 pK the critical density would be n = 6 X 10 /cm, which is at present not achievable. For experimentally realized densities of 10 2 the atoms have to be cooled below 100 nK in order to reach BEC. 

even at temperatures well above absolute zero, the electrons are essentially all in the lowest possible energy states. As a result, the electronic heat capacity at constant volume, which equals d tot/dr, is small at ordinary temperatures and approaches zero at low temperatures. 

The free-electron gas exerts a pressure on the walls of the infinite potential well in which it is contained. If the volume v of the gas is increased slightly by an amount ch , then the energy levels in equation (8.56) decrease 

Although we have explained Bose-Einstein condensation as a characteristic of an ideal or nearly ideal gas, i.e., a system of non-interacting or weakly interacting particles, systems of strongly interacting bosons also undergo similar transitions. Eiquid helium-4, as an example, has a phase transition at 2.18 K and below that temperature exhibits very unusual behavior. The properties of helium-4 at and near this phase transition correlate with those of an ideal Bose-Einstein gas at and near its condensation temperature. Although the actual behavior of helium-4 is due to a combination of the effects of quantum statistics and interparticle forces, its qualitative behavior is related to Bose-Einstein condensation. 

6 Consider two identical non-interacting particles, each of mass m, in a onedimensional box of length a. Suppose that they are in the same spin state so that spin may be ignored. 

7 Consider a crude approximation to the ground state of the lithium atom in which the electron-electron repulsions are neglected. Construct the ground-state wave function in terms of the hydrogen-like atomic orbitals. 

8 The atomic weight of silver is 107.9 gmol 1 and its density is 10.49 gem-3. Assuming that each silver atom has one conduction electron, calculate 

The minimum density for EEC depends on the temperature and decreases with 7 / [14.6,14.59]. 

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In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The fiinction N (T) is sketched in fignre A2.2.7. At zero temperature all the Bose particles occupy the ground state. This phenomenon is called the Bose-Einstein condensation and is the temperature at which the transition to the condensation occurs. 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Recent research (1995-) has produced at very low temperatures (nanokelvins) a Bose-Einstein condensation of magnetically trapped alkali metal atoms. Measurements [41] of the fraction of molecules in the ground... 

Ensher J R, Jin D S, Mathews M R, Weman C E and Cornell E A 1996 Bose-Einstein condensation in a dilute gas measurement of energy and ground-state occupation Phys. Rev. Lett. 77 4984-7... 

Pure magnetic traps have also been used to study cold collisions and tliey are critical for tire study of dilute gas-phase Bose-Einstein condensates (BECs) in which collisions figure importantly. We anticipate, tlierefore, tliat magnetic traps will play an increasingly important role in future collision studies in and near BEC conditions. 

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. 

Permutations of this type have to be considered in PIMC simulations if a full account of the quantum statistics is intended in the study and required by the physical effect under consideration, which means that additional permutation moves have to be done in the simulation. In this way quantum statistics has been included in a few PIMC simulations, in particular for the study of superfluidity in He [287] and in adsorbed H2 layers [92], for the Bose-Einstein condensation of hard spheres [269], and for the analysis of... 

All these multifarious activities took a lot of Einstein s energies but did not keep him from his physics research. In 1922 he published Ins first paper on unified field theoiy, an attempt at incorporating not only gravitation but also electromagnetism into a new world geometry, a subject that was his main concern until the end of his life. He tried many approaches none of them have worked out. In 1924 he published three papers on quantum statistical mechanics, which include his discoveiy of so-called Bose-Einstein condensation. This was his last contribution to physics that may be called seminal. He did continue to publish all through his later years, however. 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

Proceedings Bose-Einstein Condensates and Atom Lasers... 

Now — L is the Landau-Ginzburg free energy, where m2 = a(T — Tc) near the critical temperature, is a macroscopic many-particle wave function, introduced by Bardeen-Cooper-Schrieffer, according to which an attractive force between electrons is mediated by bosonic electron pairs. At low temperature these fall into the same quantum state (Bose-Einstein condensation), and because of this, a many-particle wave function (f> may be used to describe the macroscopic system. At T > Tc, m2 > 0 and the minimum free energy is at = 0. However, when T [Pg.173]

This topic is relevant to the physics of neutron stars (nuclei or quark bubbles embedded in a neutron gas), to dilute Bose-Einstein-condensate bubbles inside the background of a Fermi-Dirac condensate, to buckyballs in liquid mercury and to superconducting droplets in a Fermi liquid. 

Bose-Einstein condensation, 17 352 Bosons, 17 352 Boswellic acids, 24 557 Botox (Clostridium botulinum toxin type A), 2 816 Bottle centrifuge operation, 5 528-529 theory of performance, 5 507-508 Bottle polymerizatioi, 20 376 Bottles... 

One of the most amazing phenomena in quantum many-particle systems is the formation of quantum condensates. Of particular interest are strongly coupled fermion systems where bound states arise. In the low-density limit, where even-number fermionic bound states can be considered as bosons, Bose-Einstein condensation is expected to occur at low temperatures. The solution of Eq. (6) with = 2/j, gives the onset of pairing, the solution of Eq. (7) with EinP = 4/i the onset of quartetting in (symmetric) nuclear matter. At present, condensates are investigated in systems where the cross-over from Bardeen-Cooper-Schrieffer (BCS) pairing to Bose-Einstein condensation (BEC) can be observed, see [11,12], In these papers, a two-particle state is treated in an uncorrelated medium. Some attempts have been made to include the interaction between correlated states, see [7,13]. 

In a simplistic and conservative picture the core of a neutron star is modeled as a uniform fluid of neutron rich nuclear matter in equilibrium with respect to the weak interaction (/3-stable nuclear matter). However, due to the large value of the stellar central density and to the rapid increase of the nucleon chemical potentials with density, hyperons (A, E, E°, E+, E and E° particles) are expected to appear in the inner core of the star. Other exotic phases of hadronic matter such as a Bose-Einstein condensate of negative pion (7r ) or negative kaon (K ) could be present in the inner part of the star. 

Fig. 1-1. Probability density functions of partiele energy distribution (a) Fermi function, (b) Bose-Einstein function, e = particle energy f(i) - probability density function cp = Fermi level sb - Bose-Einstein condensation level.

A newly discovered, highly organized state of matter in which clusters of 20-30 component atoms are magnetically contained and adiabatically cooled to within 2-3 X 10 K of absolute zero. At this point, the motions of the contained atoms are overcome by very weak cohesive forces of the Bose-Einstein condensate. While of no apparent relevance to biochemical kinetics, the Bose-Einstein condensate represents one of the most perfect forms of self-assembly, inasmuch as aU atoms within the condensate share identical Schrodinger wave equations. 

BORN-OPPENHEIMER APPROXiMATION ADIABATIC PHOTOREACTION DIABATIC PHOTOREACTION BORNYL PYROPHOSPHATE SYNTHASE BOROHYDRIDE REDUCTION ACYL-PHOSPHATE INTERMEDIATE BOSE-EINSTEIN CONDENSATE Bound water,... 

The previous chapter dealt with chemical bonding and the forces present between the atoms in molecules. Forces between atoms within a molecule are termed intramolecular forces and are responsible for chemical bonding. The interaction of valence electrons between atoms creates intramolecular forces, and this interaction dictates the chemical behavior of substances. Forces also exist between the molecules themselves, and these are collectively referred to as intermolecular forces. Intermolecular forces are mainly responsible for the physical characteristics of substances. One of the most obvious physical characteristics related to intermolecular force is the phase or physical state of matter. Solid, liquid, and gas are the three common states of matter. In addition to these three, two other states of matter exist—plasma and Bose-Einstein condensate. 

Bose-Einstein Condensate phase of matter that is created just above absolute zero when atoms lose their individual identity Boyle s Law law that states volume of a gas is inversely related to its pressure Breeder Reactor type of nuclear reactor that creates or breeds fissionable plutonium from nonfissionable U-238 Buckministerfullerene Cg, allotrope of carbon consisting of spherical arrangement of carbon, named after architect Buckmin-ister Fuller, Invertor of geodesic dome Buffer a solution that resists a change in pH... 

E. Torrontegui, X. Chen, M. Modugno, S. Schmidt, A. Ruschhaupt, and I. G. Muga. Fast transport of Bose-Einstein condensates. Afew T. Phys., 14 013031-013041(2012). 

D. Stefanatos and J. S. Li. Frictionless decompression in minimum time of Bose-Einstein condensates in the Thomas-Fermi regime. Phys. Rev. A, 86(6) 063602—063608(2012). 

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