Lecture 23. Degenerate Fermi Gas & Bose-Einstein condensation (Ch. 7)

1

exp 1BE

B

n

k T

1exp

1

Tk

n

B

FD

expMBB

nk T

-6 -4 -2 0 2 4 6

0

1

2

FD

MB BE

<

n >

()/kBT

n n

Today’s plan: high density limit n ~ nQ

1.Fermion: Degenerate Fermi Gas (7.3)

2.Boson: Bose-Einstein condensation (7.6)

QNV V

When will Maxwell-Boltzmann break down?

Answer: when 1 doesn't hold.QnV ln 0BQ

Vk T

NV

For a system with fixed density (n), is a function of T.

0 1 2 3-4

-2

0

2

MB

T/T*

32

2

2ln B

MB B

mk TVk T

N h

3

2

2

20 1Bmk TV

N h

Bk T

(fermion) degenerate fermi gas ( ) * ~

(Boson) Bose-Einstein condensation ( )F F

C C

T T TT

T T T

T*: the temperature (energy) scale where quantum (exchange) effect becomes pronounced.

2 2

3*2B

hk T n

m

Degenerate fermi gas (T=0 limit)

0, = , a step function.

0 : fermi energy

F

F

T n

1

exp 1FD

B

n

k T

3

3 3 230 0

42 1 2

3

FD DFD FN g n d g d S V m

h

22 33

02 2 1 4FD F

h N

m S V

B F Fk T

22 31 3

for electrons in metals: , and 300 K 2 8F F

e

h NS T

m V

At T=0, the total # of fermions in a degenerate fermi gas (spin S) is:

2 2

3*2B

hk T n

m

3

0

30

5D

FD FU g n d N

The total (internal) energy of a degenerate fermi gas is:

30

5 F

Uu

N

- a very appreciable zero-point energy!

Degenerate fermi gas (T=0 limit) (cont.)

, ,

3 3 2 2

5 5 3 3F F

N S N S

U UP N N

V V V V

The large (internal) energy U at 0 K is a consequence of Pauli exclusion principle. Recall in ideal gas, UIG=3/2 PV, we expect similar pressure generated by DFG. Indeed:

3

2U PV

This pressure is called degeneracy pressure. It is the physical mechanism that prevents white dwarf stars (electron) or neutron stars (neutron) from collapse by gravity. (Pr. 7.23, 7.24)

Bulk modules (compressibility-1) of electron in metal:T

PB V

V

5

32

2 5 10

3 3 9T

U P P UP V

V V V V

10

9

UB

V

Star Evolution

gas and radiation pressure supports stars in which thermonuclear reactions occur

pressure of a degenerate electron gas at high densities supports the objects with no fusion: dead stars (white dwarfs) and the cores of giant planets (Jupiter, Saturn)

pressure of a degenerate neutron gas at high densities support neutron stars

How stars can support themselves against gravity:

Nobel 1983

Chandrasekhar

Degenerate fermi gas (T≠0, but T<<TF)

= EF

2/1 g

To be quantitative, we need to apply Sommerfeld expansion. But the qualitative behavior can be captured by a back-of-envelope calculation. T>0

~ BF B

F F

k TNN g k T N

When T<<TF, the # of “excited” fermions is:

The extra thermal energy acquired by each fermion: Bu k T

2

BBB

F F

k Tk TU N u N k T N

2

BV

V F

kUC N T T

T

The characteristic behavior of electrons in metals.

21

~2

B

F F

k T

Slightly more complicate cal.:

22

112

B

F F

T k T

Chemical potential of fermi gas (with fixed n)

Sommerfeld expansion (T<<TF)

0 1 2

-2

-1

0

1

FD/ F

kBT /

F

Maxwell-Boltzmann (T>>TF):3

2

2

2ln B

B

mk TVk T

N h

The Fermi Gas of Nucleons in a NucleusLet’s apply these results to the system of nucleons in a large nucleus (both protons and neutrons are fermions). In heavy elements, the number of nucleons in the nucleus is large and statistical treatment is a reasonable approximation. We need to estimate the density of protons/neutrons in the nucleus. The radius of the nucleus that contains A nucleons:

3/115 m103.1 AR

Thus, the density of nucleons is:

3-m m

44

315

101103.1

34

A

An

For simplicity, we assume that the # of protons = the # of neutrons, hence their density is

-3m 44105.0 np nn

The Fermi energy

MeV 27J 104.3J 105.03

106.18

106.6 12

3/2

4427

234

FE

The average kinetic energy in a degenerate Fermi gas = 0.6 of the Fermi energy

MeV 61E - the nucleons are non-relativistic

EF >>> kBT – the system is strongly degenerate. The nucleons are very “cold” – they are all in their ground state!

Bose-Einstein Condensation (Ch. 7 )

70 years after the Einstein prediction, the BEC in weakly interacting Bose systems has been experimentally demonstrated - by laser cooling of a system of weakly-interacting alkali atoms in a magnetic trap.

Nobel 2001

BEC and related phenomena

BEC of photons(lasers)

BEC in a strongly-interacting system

(superfluid 4He)

BEC in a weakly-interacting system

(atomic gases)

Townes Basov Prokhorov

Nobel 1964

LandauNobel 1962

KapitsaNobel 1978

Einstein described the phenomenon of condensation in an ideal gas of particles with nonzero mass in 1925. In the 1930’s Fritz London realized that superfluity 4He can be understood on terms of BEC. However, the analysis of superfluity 4He is complicated by the fact that the 4He atoms in liquid strongly interact with each other.

Two types of Bosons

Two types of bosons:

(a) Composite particles which contain an even number of fermions. These number of these particles is conserved if the energy does not exceed the dissociation energy (~ MeV in the case of the nucleus).

(b) Particles associated with a field, of which the most important example is the photon. These particles are not conserved: if the total energy of the field changes, particles appear and disappear. The chemical potential of such particles is zero in equilibrium, regardless of density.

Ideal Gas of Conserved Bosons

0 1 2 3-4

-2

0

2

MB

T/T*

2 2

3*2B

hk T n

mRecall:

An educated guess: something (BEC) will happen below T* for bosons with fixed density n. min 0

Q: Why bosons are special, i.e. forming BEC at low temperature?

A: The quantum nature, i.e. any # of bosons can occupy one quantum state (energy level) .

A layman’s definition: Bose-Einstein Condensation (BEC) is a special macrostate with macroscopic # of bosons occupying one quantum state (often, the ground state) of a bosonic system. As the system temperature is cooled below certain temperature TC, BEC spontaneously forms. It is a phase transition purely driven by quantum (exchange) effect.

BEC of Conserved Bosons

Let’s consider a simple but very special case: T = 0 K. What is the macrostate of a bosonic system?

All the bosons (a macroscopic #!) occupy the lowest energy level, i.e. the ground state, so that the system has lowest energy. 1

2

3

4

the multiplicity of the system: 1. i.e. ln 0.BS k

,

chemical potential 0 at =0.V U

ST T

N

On the other hand, 0 around T* according to Maxwell-Boltzmann distribution. Indeed, =0 right at T=TC and stay at 0 as T further decreases.

T

(n,T)

TC

Bose-Einstein Condensation (TC)

3/22

12612.22

s

n

mk

hT

BC

Critical temperature of BECCritical temperature of BEC

3/ 23 1/ 2

20 0

22 12

exp 1exp 1

DB

g m k TS xN d V dx

h x

1

exp 1BEn

Recall Bose-Einstein distribution

3/ 2

3 1/ 22 2

2 1 2

4D s m

g V

The total # of bosons: ii

N nUsing density of state approximation

Let’s perform the integration at TC, i.e. =0.3/ 2 1/ 2

20

22 12

1B C

x

mk TN S xn dx

V h e

1/ 2

0

1.3 2.315exp 1

xdx

x

2 2

3*2B

hk T n

mRecall Confirm: T*TC

Bose-Einstein Condensation (T<TC)

What happen at T<TC? is already 0 at TC. The right hand side decreases as T1.5. Something is wrong!

3/ 2 1/ 2

20

22 12

exp 1Bm k TS x

n dxh x

T>TC, the system can adjust (<0) to satisfy the constraint:

g

g()

=n()

Resolving the paradox: The problem is caused by the behavior of the 3D density of states and our use of the continuum approximation. Because g()=0 at =0, our calculations of n ignored all the particles in the ground (=0) state. At low energies, we have to take into account the discreteness of the quantum states.

g(

)n

()

Just excited states!

Bose-Einstein Condensation (T<TC) (cont.)

The eq. n(T) with = 0 still works at T<TC for calculating the number of particles not in the ground state (ignore spin):

The density of particles in the ground state:

T

n

TC

n>0

n0

a tremendous number of particles all sitting in the very lowest available energy state

g(

)n

()

T < TC

3/ 23/ 2

0 2

22.612 B

CC

mk T Tn n T T

h T

3/ 2

0 0 1 CC

Tn n n n T T

T

Bose-Einstein Condensation (Summary)We can discuss the ideal Bose gas in the same terms of a phase transition. That is, at a critical value of temperature, TC, (n,T) reaches the limit of = 0 and stops increasing. Beyond this point, the relation

is no longer able to keep track of all the particles – we miss the particles in the ground state. Below TC, bosons begin to condense into the ground state. The abrupt accumulation of bosons in the ground state is called Bose-Einstein condensation.

3/22

2

527.00 n

m

h

kST

BC

3/22

126.22

s

n

mkT

BC

T

(n,T)

TC

3/ 2

20

2 1 2

exp 1

S m dn

h

Realization of BEC in a Dilute 87Rb Vapor

K 108

107.18721038.1

101106.653.0

2

53.0 82723

3/2192343/22

V

N

m

h

kT

BC

First observation of the BEC with weakly-interacting gases was observed with relatively heavy atoms of 87Rb. 10,000 rubidium-87 atoms were confined within a “box” with dimensions ~ 10 m (the density ~ 1019 m-3). The spacing between the energy levels:

K 108J 101.110107.187

106.6

8

3

8

3~ 1032

2527

234

2

2

1

mL

h

The transition was observed at ~ 0.1 K. This is in line with the estimate:

100CBTk ! - again, it is worth emphasizing that the BEC occurs at kBT >> :

3/2

2

3/22

3/22

~~~ NL

NL

V

N

m

hTk CB2

2

1 ~~mL

h

- the greater the total number of particles in the system, the greater this difference.

In principle, the lighter the bosons, the greater TC. For example, the BE condensation of excitons (light-induced electron-hole pairs) in semiconductors has been observed before the BE condensation in dilute gases (electron is a fermion, but an electron-hole pair has an integer spin).

Realization of BEC in Dilute Vapor (cont.)

At T=0.9TC, the number of atoms in the ground state: 500,112/3

0

CT

TNN

For comparison, in the first excited state: 300101.01

3

1/exp

31

TkN

B

degeneracy of the 1st excited state in a cube

The ratio N0/N1, which is ~ 5 for N = 104, rapidly increases with N at a fixed T/TC (it becomes ~ 25 for N = 106).

The atoms are not very close to each other in the classic sense - in fact, the average density of this condensate is very low—one billionth the density of normal solids or liquids. But at this temperature, the quantum volume becomes comparable to the average volume per atom:

319

2/327823

334

2/3

3

m 106107.18721051038.1

106.6

2

Tmk

hV

B

Q

T

n

TC

n>0

n0

Problem(a) Calculate the critical temperature for BE condensation of diatomic hydrogen H2

if the density of liquid hydrogen is 60 kg/m3. Would you expect superfluidity in liquid hydrogen as well? Hydrogen liquefies around 20K and solidifies at 14K.

(b) Above TC, the pressure in a degenerate Bose gas is proportional to T. Do you expect the temperature dependence of pressure to be stronger or weaker at T <TC ? Explain and draw a qualitative graph of the temperature dependence of pressure over the temperature range 0 < T < 2 TC.

(a) Liquid hydrogen:

Since hydrogen solidifies at 14K, we do not expect to observe superfluidity in liquid hydrogen.

(b) The atoms in the ground state do not contribute to pressure. At T < TC, two factors contribute to the fast increase of P with temperature: (i) an increase of the number of atoms in the excited states, and (b) an increase of the average speed of atoms with temperature. As the result, the rate of the pressure increase with temperature is greater at T < TC than that at T > TC (in fact , P~T5/2 at T < TC) .

P(T)

TTC

2342 / 3 2 / 32

23 27 27

6.626 100.527 0.527 605.48 K

2 1.38 10 2 2 1.67 10 2 1.67 10CB

h NT

k m V

Laser Cooling-works for a dilute gas of neutral atoms

(cannot be applied to cool solids) E1

E2 For photon absorption or emission, the photon energy h must be equal to E2-E1

If the laser frequency is tuned slightly below E2-E1, an atom scatters (absorbs and re-emits) photons only is it moves towards the laser (Doppler effect). Atom at rest or moving in the opposite direction doesn’t scatter.

If a resonant photon is absorbed, the atom acquires momentum:c

Mv

The corresponding energy:

eVeV

eV

cMM

pK 10

9

2

2

22

1010232

2

22

Na, A=23

Na)for (1/1010 4101 KeVKeVT

An apparent limit on T achieved by laser cooling is reached when an atom’s recoil energy from absorbing or emitting a single photon is comparable to its total K. The single-photon recoil temperature limit (for Na):

How to cool the gas of Rb atoms down to ~0.1 K? The first stage – laser cooling, the second stage – evaporative cooling.

Laser Cooling and Trapping

By laser cooling, T ~ 10 K can be reached. At this temperature, the atom’s speed is a few cm/s. These slow-moving atoms are relatively easy to confine in a non-uniform magnetic trap. The magnetic field has a minimum value in the center of the “magnetic bowl”. An atom with spin parallel to the magnetic field (i.e., atomic magnetic moment anti-parallel to the magnetic field), is attracted to the minimum; for spin anti-parallel to the field, the atom is repelled from the minimum.

Laser cooling has been used in the experiments on BEC observation for pre-cooling of the gas of alkali atoms. However, to observe this phenomenon, even lower T are necessary. Further reduction of T by ~3 orders of magnitude (below 0.1 K) is required for the exp. vapor densities ~ 1017 m-3. This is achieved by the evaporation cooling after the lasers are turned off.

Magneto-Optical Trap

Evaporative Cooling

Radio-frequency forced evaporative cooling.

The resonance excitation flips the spins and those atoms are ejected (evaporated). Reducing fr frequency evaporates lower energy atoms.

Metastability is the Key

The experiments with Rb vapor were aimed at realization of BEC in a weakly-interacting system.

Though the interactions are weak in the vapor of Rb atoms, they are sufficiently strong for the phase transition vapor-solid at ultra-low temperatures. In conditions of thermal equilibrium, one cannot get below the blue line without phase separation. constnTC ln

3

2ln

How to cheat the Nature? The key is metastability. If the process of cooling is slow and “gentle” enough, one can realize a “super-saturated” vapor below the coexistence line without a condensed phase ever forming. For this, not only the interaction with walls must be excluded, but also the three-particle collisions that assist forming molecules and, eventually, condensed-matter phase – hence, very low densities.

nln

Tln

BECliquid

He

vapor

cond. matter

vapor-solid/liquidphase boundary

~ 1010

Observation of BEC

To observe the distribution of velocities of atoms in the system, the magnetic trap is turned off. The atoms find themselves in free space, and, because they have some residual velocity, they just fly apart.

The picture shows the velocity distribution of atoms in the cloud at the time of its release, instead of the spatial distribution.

For T > TC, atoms are distributed among many energy levels of the system, and have a Gaussian distribution of velocities. With cooling of the cloud, a spike appears right in its middle. It corresponds to atoms which are hardly moving at all: for T < TC, the concentration of atoms in the lowest state gives rise to a pronounced peak in the distribution at low velocities (condensation in the momentum space).

After they have flown apart for some time, the cloud is much bigger, and it is easier to take a snapshot of the atomic cloud (to make a snapshot, a laser beam is scattered by the cloud).

Each frame corresponds to the distance the atoms have moved in about 1/20 s after turning off the trap.This two-component cloud resembles the

situation in superfluid helium, where two components coexist: normal and superfluid .

Thermodynamic Functions of a Degenerate Bose Gas

By integrating the heat capacity at constant volume, we can get the entropy:

T

U

T

UCV 2

5

T

U

T

dTTCS

TV

3

5

0*

**

The Helmholtz free energy ( = 0): UTSUF3

2

The pressure exerted by a degenerate Bose gas: 2/5TV

FP

T

does not depend on volume!

This is due to the fact that, when compressing a degenerate Bose gas, we just force more particles to occupy the ground state. The particles in the ground state do not contribute to pressure – except of the zero-motion oscillations, they are at rest.