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Using the fact that the angular velocity is a vector as infinitesimal rotation commute.

Then the relative angular velocity of body 1 w.r.t 2 is given by

w₁₂ = w₁ -w₂     (as for relative linear velocity)

The relative acceleration of 1 w.r.t 2 is

(dw₁ / dt)_S'

Where S' is a frame rotating with the second body and S is a space fixed frame with origin coinciding with the point of intersection of the two axes,

but (dw₁ / dt)_S = (dw₁ / dt)_S' + w₁ X w₂

Since S' rotates with angular velocity w₂. However (dw₁ / dt)_S = 0 as the first body rotates with constant angular velocity in space, thus

Beta₁₂ = w₁ X w₂

Note that for any vector b, the relation in space forced frame (k) and a frame (k') rotating with angular velocity w is

db/dt I_K = db/dt I_{K '} + w X b

Bose-Einstein Condensation

In 1924 Einstein pointed out that bosons could "condense" in unlimited numbers into a single ground state since they are governed by Bose-Einstein statistics and not constrained by the Pauli exclusion principle. Little notice was taken of this curious possibility until the anomalous behavior of liquid helium at low temperatures was studied carefully.

When helium is cooled to a critical temperature of 2.17 K, a remarkable discontinuity in heat capacity occurs, the liquid density drops, and a fraction of the liquid becomes a zero viscosity "superfluid". Superfluidity arises from the fraction of helium atoms which has condensed to the lowest possible energy.

A condensation effect is also credited with producing superconductivity. In the BCS Theory, pairs of electrons are coupled by lattice interactions, and the pairs (called Cooper pairs) act like bosons and can condense into a state of zero electrical resistance.

The conditions for achieving a Bose-Einstein condensate are quite extreme. The participating particles must be considered to be identical, and this is a condition that is difficult to achieve for whole atoms. The condition of indistinguishability requires that the deBroglie wavelengths of the particles overlap significantly. This requires extremely low temperatures so that the deBroglie wavelengths will be long, but also requires a fairly high particle density to narrow the gap between the particles.

 

Since the 1990s there has been a surge of research into Bose-Einstein condensation since it was discovered that Bose-Einstein condensates could be formed with ultra-cold atoms. The use of laser cooling and the trapping of ultra-cold atoms with magnetic traps has produced temperatures in the nanokelvin range. Cornell and Wieman along with Ketterle of MIT received the 2001 Nobel Prize in Physics "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates". Cornell and Wieman led an active group at the University of Colorado, Boulder which has produced Bose-Einstein condensates with rubidium atoms. Other groups at MIT, Harvard and Rice have been very active in this rapidly advancing field.

 

Chilling the atoms

Bose-Einstein condensation has been cited as an important phenomenon in many areas of physics, but until recently the only evidence for condensation came from studies of superfluid liquid helium and excitons in semiconductors. In the case of liquid helium, however, the strong interactions that exist in a liquid qualitatively alter the nature of the transition. For this reason a long-standing goal in atomic physics has been to achieve BEC in a dilute atomic gas. The challenge was to cool the gases to temperatures around or below one microkelvin, while preventing the atoms from condensing into a solid or a liquid.

Efforts to Bose condense atoms began with hydrogen more than 15 years ago. In these experiments hydrogen atoms are first cooled in a dilution refrigerator, then trapped by a magnetic field and further cooled by evaporation (see below). This approach has come very close to observing BEC, but is limited by the recombination of individual atoms to form molecules and by the detection efficiency.

In the 1980s laser-based techniques such as Doppler cooling, polarization-gradient cooling and magneto-optical trapping were developed to cool and trap atoms. These techniques profoundly changed the nature of atomic physics and provided a new route to ultracold temperatures that does not involve cryogenics. Atoms at sub-millikelvin temperatures are now routinely used in a variety of experiments. Alkali atoms are well suited to laser-based methods because their optical transitions can be excited by available lasers and because they have a favourable internal energy-level structure for cooling to low temperatures.

However, the lowest temperature that these laser cooling techniques can reach is limited by the energy of a single photon. As a result, the "phase-space density" - the number of atoms within a volume lambda_dB³ - is typically about a million times lower than is needed for BEC.

The successful route to BEC turned out to be a marriage of the cooling techniques developed for hydrogen and those for the alkalis: an alkali vapour is first laser cooled and then evaporatively cooled. In evaporative cooling, high-energy atoms are allowed to escape from the sample so that the average energy of the remaining atoms is reduced. Elastic collisions redistribute the energy among the atoms such that the velocity distribution reassumes a Maxwell-Boltzmann form, but at a lower temperature. This is the same evaporation process that happens when tea cools, but the extra trick for trapped atoms is that the threshold energy can be gradually lowered. This allows the atomic sample to be cooled by many orders of magnitude, with the only drawback being that the number of trapped atoms is reduced.

The challenge in combining these two cooling schemes for alkalis was a question of atomic density. Optical methods work best at low densities, where the laser light is not completely absorbed by the sample. Evaporation, on the other hand, requires high atomic densities to ensure rapid rethermalization and cooling. This changed the emphasis for optical methods: while they had previously been used to produce low temperatures and high phase-space density simultaneously, they now needed to produce high elastic collision rates. Furthermore, this had to be achieved in an ultrahigh vacuum chamber to prolong the lifetime of the trapped gas. Thus no new concept was needed to achieve BEC, but rather it was an experimental challenge to improve and optimize existing techniques. These developments were pursued mainly at MIT and Boulder from the early 1990s.

Improved techniques in magnetic trapping

For evaporative cooling to work, the atoms must be thermally isolated from their surroundings. This must be done with electromagnetic fields, since at ultracold temperatures atoms stick to all surfaces. The best method for alkalis is magnetic confinement, which takes advantage of the magnetic moment of alkali atoms. After the atoms are trapped and cooled with lasers, all light is extinguished and a potential is built up around the atoms with an inhomogeneous magnetic field. This confines the atoms to a small region of space.

Atoms can only be cooled by evaporation if the time needed for rethermalization is much shorter than the lifetime of an atom in the trap. This requires a trap with tight confinement, since this allows high densities and hence fast rethermalization times. For this reason, the first experiments that observed BEC used so-called linear quadrupole traps, which have the steepest possible magnetic fields.

These techniques do indeed produce high densities and fast evaporation, but with one major problem: the magnetic field is zero at the centre, which causes an atom to become "disorientated" and lose the alignment of its magnetic moment. Since a magnetic field can only confine atoms with magnetic moments that are antiparallel to the field, these "spin flips" result in a disastrous loss of atoms from the trap. Both the Boulder and MIT groups found ways to circumvent this problem. The Boulder group added a rotating magnetic field to keep the atoms away from the "hole", while we "plugged" the hole with the repulsive force from a focused laser beam.

The set-up for evaporative cooling from the cloverleaf magnetic trap at MIT. The central (curvature) coils provide axial confinement while the outer coils (the "cloverleaves" or gradient coils) give tight radial confinement. The resulting anisotropic potential gives rise to cigar-shaped clouds.

Commercial preparationog ZnO is obtained by burning Zn-vapours in air. ZnO is collected as white wooly flocks and hence is called as philosopher's wool.

Properties: It is a white amorphous powder which becomes sulpur-yellow on heating but again white on cooling. It is insoluble in water. It is an amphoteric oxide since, it dissolves radily in acids forming the zinc salt and in alkalies forming zincates.

Laser Cooling

Starting in about 1985 with the work of Steven Chu and others, the use of lasers to achieve extremely low temperatures has advanced to the point that temperatures of 10^-9 K have been reached. If an atom is traveling toward a laser beam and absorbs a photon from the laser, it will be slowed by the fact that the photon has momentum p = E/c = h/λ.

If we take a sodium atom as an example, and assume that a number of sodium atoms are freely moving in a vacuum chamber at 300K, the rms velocity of a sodium atom from the Maxwell speed distribution would be about 570 m/s. Then if a laser is tuned just below one of the sodium d-lines (589.0 and 589.6 nm, about 2.1 eV), a sodium atom traveling toward the laser and absorbing a laser photon would have its momentum reduced by the amount of the momentum of the photon. It would take a large number of such absorptions to cool the sodium atoms to near 0K since one absorption would slow a sodium atom by only about 3 cm/s out of a speed of 570 m/s. A straight projection requires almost 20,000 photons to reduce the sodium atom momentum to zero. The change in speed from the absorption of one photon can be calculated from

That seems like a lot of photons, but according to Chu, a laser can induce on the order of 10⁷ absorptions per second so that an atom could be stopped in a matter of milliseconds.

A conceptual problem is that an absorption can also speed up an atom if it catches it from behind, so it is necessary to have more absorptions from head-on photons if your goal is to slow down the atoms. This is accomplished in practice by tuning the laser slightly below the resonance absorption of a stationary sodium atom. From the atom's perspective, the headon photon is seen as Doppler shifted upward toward its resonant frequency and it therefore more strongly absorbed than a photon traveling in the opposite direction which is Doppler shifted away from the resonance. In the case of our room temperature sodium atom above, the incoming photon would be Doppler shifted up 0.97 GHz, so to get the head on photon to match the resonant frequency would require that the laser be tuned below the resonant peak by that amount.

This method of cooling sodium atoms was proposed by Theodore Hansch and Arthur Schawlow at Stanford University in 1975 and achieved by Chu at AT&T Bell Labs in 1985. Sodium atoms were cooled from a thermal beam at 500K to about 240 mK. The experimental technique involved directing laser beams from opposite directions upon the sample, linearly polarized at 90° with respect to each other. Six lasers could then provide a pair of beams along each coordinate axis. The effectively "viscous" effect of the laser beams in slowing down the atoms was dubbed "optical molasses" by Chu.

Continuing to cool the sodium atoms by this method requires the tuning of the laser upward in frequency toward the atomic resonant frequency because the Doppler shift will be smaller. This places a practical limit on how much cooling can be achieved, because the differential cooling rate is reduced and at a certain point the cooling mechanism is foiled by heating due to the random absorption and reemission of photons. This practical limit is characterized by 2kT = E_{resonant photon}, which at the low temperature of 240 mK would correspond to photon energies around 4 x 10^-8 eV. Such energies can characterize the Zeeman split energy levels of the atoms in the magnetic fields produced by the laser photons.

It was found that the splittings which limited the original laser cooling processes could be exploited to lower the ultimate temperatures below these limits. With the opposing laser beams with perpendicular linear polarization, atoms could be selectively driven or "optically pumped" into the lower energy levels. These lasers create a small region of space about a quarter wavelength in extent where the atoms can drift to a region where their energy is relatively higher, only to be pumped downward to a lower energy again. This was dubbed "Sisyphus cooling" after the legendary tormented man who was condemned to perpetually roll a rock up a hill, only to have it roll down again. With the "optical molasses" and the polarization gradient in the region of opposing laser beams, temperatures as low as 35 mK for sodium and 3 mK for cesium were obtained.

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As electrons are fermions thus follow Pauli's exclusion priciple.



 

Pauli Exclusion Principle

No two electrons in an atom can have identical quantum numbers. This is an example of a general principle which applies not only to electrons but also to other particles of half-integer spin (fermions). It does not apply to particles of integer spin (bosons).

The nature of the Pauli exclusion principle can be illustrated by supposing that electrons 1 and 2 are in states a and b respectively. The wavefunction for the two electron system would be but this wavefunction is unacceptable because the electrons are identical and indistinguishable. To account for this we must use a linear combination of the two possibilities since the determination of which electron is in which state is not possible to determine.

The wavefunction for the state in which both states "a" and "b" are occupied by the electrons can be written

The Pauli exclusion principle is part of one of our most basic observations of nature: particles of half-integer spin must have antisymmetric wavefunctions, and particles of integer spin must have symmetric wavefunctions. The minus sign in the above relationship forces the wavefunction to vanish identically if both states are "a" or "b", implying that it is impossible for both electrons to occupy the same state.