62

3

= n 5 eF(g 1)

						CHAPTER 3. FERMIONS
	5		11	2 ln 2
		l +		( )	l		(3.129)
	9			21

Here l is the Gas Parameter

l =	2apF	=	mpF 4p~2a			= nj(eF)V0 1	(3.130)
	p~		2p2~3		m

where nj is the density of states.

3.3.2 Decay of excitations

We want to discuss the life time of excitations, i.e. the inverse scattering time t. We discuss the regime of T eF.

Figure 3.4: Schematic plot of excitations and scattering around the FERMI sphere

The excitations 1 and 2 can "collide", i.e. interaction occurs. The lifetime of the excitation 1 is according to FERMIs Golden Rule (appendix C)

1		a2 Z d3 p2d p130d p230				d e1 + e2 e10 e20		d ~p1 +~p2 ~p10	~p20
	t
			n p		n p		n p
			(~2)(1		(~10))(1		(~20))		(3.131)

The first line is the classical value while the second line takes the quantum statistics into account.

e	10	eF <0	e1 . eF + T	) eF T . e2 < eF;	(3.132)
		+ e2	= e1 + e2 > 2eF

3.3. WEAKLY INTERACTING FERMI GAS

63

i.e. the differences e1 eF and eF e2 are positive and of the order of temperature T . First we expand the energy momentum relation near the FERMI energy:

e(p) eF + vF(p pF) for p pF			(3.133)
de T ) d p vF T ) d p	T		(3.134)
	vF

We can now calculated the fraction of particles involved in the collisions by di-

viding the number of particles in the shell in momentum space of width T by the

vF

total number of particles:

pF2 d p		d p			T		T
		=				=		1	(3.135)
pF3			pF		vF pF		eF

If we now look at the final states we have

2eF	<	e10 + e20	. 2eF + T with (3.132)	(3.136)
eF	<	e10	. eF + T:	(3.137)

Without the PAULI principle the outgoing particles could have any energy allowed, i.e. between 0 and 2eF + T . But with FERMI statistics being taken into account

we have only a fraction T 1 of final states available. Thus (3.131) becomes 2

eF

1	na2vF	T		2
				T 0
				!!	0:	(3.138)
t		eF

The first three terms are the classical value, while the fraction is caused by the PAULI principle. The square originates in the fact that FERMI statistics imposes restrictions on the possible momenta of the incoming and outgoing particles.

At T = 0 calculation shows

	1	na2vF	e	:			(3.139)
	t(e)		eF
The energy of the excited particles obeys
1
eex. T eex.						:	(3.140)
					t

Therefore the imaginary part of the excitation spectrum is much less than the real part and hence the excitations are well-defined.

2The exact and rather lengthy derivation is not included here