(a) Suppose the temperature of the electron gas is much higher than the

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Fermi temperature. Then the electron gas may be considered as an ideal Boltzmann gas, with kT 3kT M

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Po = - = - - 3

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87Tm p R

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STATISTICAL MECHANICS

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Substitution of this into (11.48) yields the linear relation

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P R=2 a M- 3 kT

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(11.49)

This case, however, is never applicable for a white dwarf star.

(b) Suppose the electron gas is at such a low density that nonrelativistic dynamics may be used (x F 1). Then Po is given by (11.42), and

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(11.48) leads to the equilibrium condition

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5/ 3

. K---=- =

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where

(11.50) 4'7T 9'7T h Thus the radius of the star decreases as the mass of the star increases: __ 4 K M 1/ 3R = - (11.51) 5 K' This condition is valid when the density is low. Hence it is valid for small M and large R. (c) Suppose the electron gas is at such a high density that relativistic effects are important (x F 1). Then Po is given by (11.43). The equilibrium condition becomes (11.52)

K =-y -

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(11.53)

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where

(~)3/2 =

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)3/2(~)3/2

(11.54)

Numerically,

__ ::::: 10 39

(11.55)

This interesting pure number is the rest energy of X divided by the gravitational attraction of two protons separated by the Compton wavelength of X, where X is anything. The mass M o corresponding to the reduced quantity M o is (taking a ::::: 1): 8 _ M o = 9'7T mpMo ::::: 10 g::::: M 0 (11.56)

the mass of the sun. The formula (11.53) is valid for high densities or for R

FERMI SYSTEMS

' - - - - - - - - -.........-+M

FIg. 11.6

Radius-mass relationship of a white dwarf

star.

Hence it is valid for M near Mo. Our model yields the remarkable prediction that no white dwarf star can have a mass larger than M o, because otherwise (11.53) would give an imaginary radius. The physical reason underlying this result is that if the mass is greater than a certain amount, the pressure coming from the Pauli exclusion principle is not sufficient to support the gas against gravitational collapse. The radius-mass relationship of a white dwarf star, according to our model, has the form shown in Fig. 11.6, where the solid lines indicate the regions covered by formulas (11.51) and (11.53). We have not been able to calculate a, so that an exact value of M o cannot be obtained. More refined considerations* give the result M o = 1.4M0 (11.57) This mass is known as the Chandrasekhar limit. Thus according to our model no star can become a white dwarf unless its mass is less than 1.4M0 . This conclusion has so far been verified by astronomical observations. If the mass of a star is greater than the Chandrasekhar limit then it will eventually collapse under its own gravitational attraction. When the density becomes so high that new interactions, dormant thus far, are awakened, a new regime takes over. For example, the star could explode as a supernova.

11.3 LANDAU DIAMAGNETISM Van Leeuwen's theoremt states that the phenomenon of diamagnetism is absent in classical statistical mechanics. Landau first showed how diamagnetism arises from the quantization of the orbits of charged particles in a magnetic field. The magnetic susceptibility per unit volume of a system is defined to be 8v1t X == (11.58)

S. Chandrasekhar, Stellar Structure (Dover, New York, 1957), XI. tSee Problem 8.7. *L. Landau, Z. Phys. 64,629 (1930).

STATISTICAL MECHANICS

where Jt is the average induced magnetic moment per unit volume of the system along the direction of an external magnetic field H:

(11.59)

where Yl' is the Hamiltonian of the system in the presence of an external magnetic field H. For weak. fields the Hamiltonian Yl'depends on H linearly. In the canonical ensemble we have

Jt=kT--JH V

a 10gQN

(11.60)

and in the grand canonical ensemble we have

Jt= kT!-(log2)

JH V

T,V,z

(11.61)

where z is to be eliminated in terms of N by the usual procedure. A system is said to be diamagnetic if X < 0; paramagnetic if X> O. To understand diamagnetism in the simplest possible terms, we construct an idealized model of a physical substance that exhibits diamagnetism. The magnetic properties of a physical substance are mainly due to the electrons in the substance. These electrons are either bound to atoms or nearly free. In the presence of an external magnetic field two effects are important for the magnetic properties of the substance: (a) The electrons, free or bound, move in quantized orbits in the magnetic field. (b) The spins of the electrons tend to be aligned parallel to the magnetic field. The atomic nuclei contribute little to the magnetic properties except through their influence on the wave functions of the electrons. They arc too massive to have significant orbital magnetic moments, and their intrinsic magnetic moments are about 10- 3 times smaller than the electron's. The alignment of the electron spin with the external magnetic field gives rise to paramagnetism, whereas the orbital motions of the electrons give rise to diamagnetism. In a physical substance these two effects compete. We completely ignore paramagnetism for the present, however. The effect of atomic binding on the electrons is also ignored. Thus we consider the idealized problem of a free spinless electron gas in an external magnetic field. Landau Levels The Hamiltonian of a nonrelativistic electron in an external magnetic field is

Yl'=