Ideal Bose gas Thermodynamic behavior of an ideal

• Ideal Bose gas. • Thermodynamic behavior of an ideal Bose gas. • The temperature of condensation. • Elementary excitation in liquid helium II. • Thermodynamics of black-body radiation. • Planck’s formula for the distribution of energy over the black-body spectrum. • Stefan-Boltzmann law of black-body radiation. 1

Ideal Bose gas. We shall now study the properties of a perfect gas of bozons of non-zero mass. The Pauli principle does not apply in this case, and the lowtemperature properties of such a gas are very different from those of a fermion gas discussed in the last lecture. A B-E gas displays most remarkable quantal features. The properties of BE gas follow from the Bose-Einstein distribution (8. 1 ) We must always have nj 0 , as the number of particles in a state cannot be negative. This requires that (8. 2 ) 2

If the zero of energy is taken at the lowest energy state, state we must (8. 3) have or (8. 4) At absolute zero all the particles will be in the ground state, and we have for no (8. 5) This is satisfied by (8. 6) 3

We now consider the situation at finite temperatures. Let g( )d be the number of states in d at . We have (8. 8) (8. 9) The density of states g( ) can be presented as (8. 10) where (8. 11) 4

The power ½ in is coming from the following consideration: We must be cautious in substituting (8. 10) into (8. 8). At high temperatures there is no problem. But at low temperatures there may be a pile-up of particles in the ground state =0; then we will get an incorrect result for N. This is because g(0)=0 in the approximation we are using, whereas there is actually one state at =0. If this one state is going to be important we should write (8. 12) where ( ) is the Dirac delta function We have then, instead of (8. 8) 5

(8. 13) It is convenient to write (8. 14) where =e / and 0 1 , from (8. 3). If <<1, <<1 the classical Boltzmann distribution is a good approximation. If the distribution is degenerate and most of the particles will be in the ground state. In the treatment of BE gas we are going to need integrals of the form (8. 15) We have 6

(8. 16) The last integral is equal to (8. 17) where (x) is the gamma function. From (8. 16) (8. 18) We have 7

(8. 19) where (8. 20) Further, (8. 21) where (8. 22) Because 1 these series always converge. We note that (8. 23) 8

From (8. 13) (8. 24) or taking the spin to be zero (8. 25) and from (8. 9) (8. 26) 9

Here (8. 27) is the number in the ground state, and (8. 28) is the number of particles in excited states. At high temperatures <<1 we obtain the usual classical result for the energy: (8. 29) 10

Bose-Einstein Condensation Let us consider equation (8. 25) in the quantum region. For =1 we have (8. 30) where is the Riemann zeta function If N 0 is to be a large number (as at low temperatures), then must be very close to 1 and the number of particles in excited states will be given approximately by (8. 28) with F( )=F(1). (8. 28) 11

(8. 31) It should be pointed out that (8. 31) represents an upper limit to the number of particles in states other than the ground state, at the temperature for which is calculated. If N is appreciably greater than N , N 0 must be large and the number of particles in excited states must approach (8. 31) Let us define a temperature T 0 such that (8. 32) where 0 is thermal de Broglie wavelength at T 0. Then, from (8. 31) 12

(8. 33) The number of particles in excited states varies as temperature region for which T 3/2 for T< T 0, in the F( ) F(1)=2. 612. Further, the number of particles in the ground state is given approximately by (8. 34) Thus for T even a little less than T 0 a large number of particles are in the ground state, whereas for T>T 0 there are practically no particles in the ground state. We call T 0 the degeneracy temperature or the condensation temperature. It can be calculated easily from the relation 13

where VM is the molar volume in cm 3 and M is the molecular weight. For liquid helium VM=27. 6 cm 3; M=4, and T 0=3. 1 o. K. It is not correct to treat the atoms in liquid helium as non-interacting, but the approximation is not as bad in some respects as one might think. The rapid increase in population of the ground state below T 0 for a Bose gas is known as the Bose-Einstein condensation. It is illustrated in Figure 8. 1 a condensation in momentum space rather than a condensation in coordinate space such as occurs for liquid-gas phase transformation 14

Figure 8. 1 Comparison of the Einstein condensation of bosons in momentum space with the ordinary condensation of a liquid in coordinate space. 15

Normally, gases have no such transition because they all turn into liquids or solids under the conditions required for Bose condensation to occur. However, liquid helium (4 He) has two phases called He I and He II, and He II has anomalous thermal and mechanical properties. It is believed that the lambda-point transition observed in liquid helium at 2. 19 0 K is essentially an Einstein condensation. Remarkable physical properties described as superfluidity are exhibited by the low-temperature phase, which is known as liquid He II. It is generally believed that the superflow properties are related to the Einstein condensation in the ground state. 16

When a material does become a superfluid, superfluid it displays some very strange behaviour; • if it is placed in an open container it will rise up the sides and flow over the top • if the fluid's container is rotated from stationary, the fluid inside will never move, the viscosity of the liquid is zero, so any part of the liquid or it's container can be moving at any speed without affecting any of the surrounding fluid • if a light is shone into a beaker of superfluid and there is an exit at the top the fluid will form a fountain and shoot out of the top exit 17

The Lambda Point There are other interesting facts about superfluids, the point at which a liquid becomes a superfluid is named the lambda point. This is because at around this area the graph of specific heat capacity against temperature is shaped like the Greek letter . 18

It took 70 years to realize Einstein's concept of Bose-Einstein condensation in a gas. It was first accomplished by Eric Cornell and Carl Wieman in Boulder, Colorado in 1995. They did it by cooling atoms to a much lower temperature than had been previously achieved. Their technique used laser light to first cool and hold the atoms, and then these atoms were further cooled by something called evaporative cooling. 19

Black body radiation and the Plank radiation law The term "black body" was introduced by Gustav Kirchhoff in 1862. The light emitted by a black body is called black-body radiation We now consider photons in thermal equilibrium with matter Among the important properties of photons are: They are Bose particles, with spin 1, having two modes of propagation. The two modes may be taken as clockwise and counter-clockwise circular polarization We are therefore to replace the factor (2 I+1) in the density states by 2. A particle traveling with the velocity of light must look the same in any frame of reference in uniform motion. 20

Because photons are bosons we may excite as many photons into a given state as we like: the electric and magnetic field intensities may be made as large as we like. It is worth remarking that all fields which are macroscopically observable arise from bosons; the field amplitude of a fermion state is restricted severely by the population rule 0 or 1 and so cannot be measured. Boson fields include photons, phonons (elastic waves), and magnons (spin waves in ferromagnets). Photons have zero rest mass. This suggests, recalling the definition (8. 35) that the degeneracy temperature is infinite for a photon gas. We can consider photons as the uncondensed portion of a B-E gas below T 0. 21

We take =0 ( =1) in the distribution law as there is no requirement that the total number of photons in the system be conserved. Thus the distribution function (8. 1) becomes (8. 36) We can say it in another way. We recall from the Grand Canonical ensemble lecture that appears in the distribution law for the grand canonical ensemble as giving the rate of change of the entropy of the heat reservoir with a change in the number of particles in the subsystem. For photons a change in the number of photons in the subsystem (without change of energy of the subsystem) will cause no change in the entropy of the reservoir. Thus we have to put equal to zero if N refers to the number of photons: this is true for the grand canonical ensemble and so for all results derived from it. 22

The number of states having wave vector k is (8. 37) where k=2 / is a wave vector. The de Broglie relation = /p may be written as p=k. Now for photons (8. 38) The zero of energy is taken at the ground state, so that the usual zero-point energy does not appear below. Defining (8. 39) we have from (8. 37) (8. 40) (8. 41) 23

Plank radiation law Thus the number of photons s( )d in d at in thermal equilibrium is (8. 42) where n( ) is given by (8. 1). The energy per unit volume ( , T)d in d at is V-1 h s( )d , so that (8. 43) This is the Plank radiation law for the energy density of radiation in thermal equilibrium with material temperature T. 24

The total energy density is (8. 44) By (8. 18) the last integral on the right is equal to (8. 45) where is the gamma function and the Riemann zeta function. We have for the radiant energy per unit volume where the constant (which is not the entropy) is given by Stefan constant This is the Stefan-Boltzmann law. (8. 46) (8. 47) 25

BLACK-BODY RADIATION Let us consider another approach to the black-body radiation. We consider a radiation cavity of volume V at temperature T. Historically, this system has been looked upon from two, practically identical but conceptually different, points of view: • as an assembly of harmonic oscillators with quantized energies where ns=0, 1, 2, …. , and s is the angular frequency an oscillator, or • as a gas of identical and indistinguishable quanta - the so-called photons the energy of a photon (corresponding to the angular frequency s of the radiation mode) being . 26

The first point of view is essentially the same as adopted by Plank (1900), except that we have also included here the zero-point energy of the oscillator; for thermodynamics of the radiation, this energy is of no great consequence and may be dropped altogether. The oscillators, being distinguishable from one another (by the very values of s ), would obey Maxwell-Boltzmann statistics; however, the expression for single-oscillator partition function Z 1(V, T) would be different from the classical expression because now the energies accessible to the oscillator are discrete, rather than continuous; see (4. 50) and (4. 62). 27

The expectation value of a energy of a Planck oscillator of frequency s is then given by eqn. (4. 68), excluding the zero-point term : (8. 48) Now the number of normal modes of vibration per unit volume of the enclosure in the frequency range ( , +d ) is given by Rayleigh expression (8. 49) where the factor 2 has been included to take into account the duplicity of the transverse modes; c here denotes the velocity of light 28

Planck’s formula (8. 50) which is the Planck’s formula for the distribution of energy over the black-body spectrum. Integrating (8. 50) over all values of , we obtain an expression for the total energy density in the radiation cavity. 29

Radiation Curves 30

Radiation Curves 31

Somewhere in the range 900 K to 1000 K, the blackbody spectrum encroaches enough in the visible to be seen as a dull red glow. Most of the radiated energy is in the infrared. 32

Essentially all of the radiation from the human body and its ordinary surroundings is in the infrared portion of the electromagnetic spectrum, which ranges from about 1000 to 1, 000 on this scale. 33

3 K Background Radiation A uniform background radiation in the microwave region of the spectrum is observed in all directions in the sky. It shows the wavelength dependence of a "blackbody" radiator at about 3 Kelvins temperature. 34