Liquid helium-3: a fermi liquid 161However, since most measuredpropertiesinvolve rather smoothfunctions ofangle
for the relevantδδf(k)variations, it is no real surprise that most of the physics is found
inthefirst twoorthree oftheseFsnumbers. The numbers are often calledLandau
parameters.
Actually, we have over-simplified the situation in one important respect.As stressed
inChapter 4, ‘quantum states arek-states plus’. Theinteractionfunction must also
specifythespins ofstatesk′andk.Hence,besides thespace-dependentFsparameters
there is also a spin-dependent set of correspondingFaparameters,FFF 0 a,F 1 aetc. These
will beimportant whenamagneticfield is applied,togiveaspin-dependentδδf(k).
The usual superscripts stand for symmetricsand antisymmetrica.
14.2.2 Application to liquid^3 He
When weinterpret experimentalresults using Landau theory, wefindthat (i)
the corrections toidealgas theoryare really large, but (ii)thetheoryworks
remarkably well.
Most properties are explainedusing onlythethree parametersFFF 0 s,F 1 sandFFF 0 a.
Forinstance, theeffective mass turns out todependonF 1 s(thecosθtreatsnorth and
south differently, as required forAto B motion), and calculation shows thatM∗/M=
1 +F 1 s/3. Theeffective massis experimentally accessiblethroughthedensity ofstates,
i.e.from theheat capacity,whichis proportionaltoM∗.As an example, considerpure
liquid^3 He at low temperatures and at zero pressure – remember it can be studied at
pressures up to 35 bar before it solidifies. As noted in Chapter 8, the heat capacityis
found to be linear in the millikelvin range, characteristic of an ideal FDgas. However,
the magnitude ofCat zero pressure is about 2.76 times larger than expected from
idealgas theorywiththe normal^3 He mass. Thisis explainedin Landau theoryby
setting2.7 6 =M∗/M = 1 +F 1 s/3,and henceF 1 s= 5 .3. As noted above,F= 0
would imply no interactions; this is a big effect, and it gets even bigger as pressure is
applied, reachingF 1 s=14.6 at the meltingpressure (35 bar).
The first Landau parameterFFF 0 sturns out to be even larger(FFF 0 s= 9 .2)for^3 Heat
zero pressure. Thisinteraction termisuniform over allstates, soit comesinto play
when the densitychanges as in a sound wave. The value of 9.2 derives from the
experimental sound velocity of around 180 ms−^1.
Thevalue ofFFF 0 aisobtainedfrom measurements ofthemagnetic susceptibility
(see (8.13)). This depends on the density of states (as does the heat capacity) but
also on the tendencyfor alignment ofthespins; thediscussion offerromagnetismin
section 11.2isrelevant. Infact the susceptibilityisabout afactor of 3 higher than
would be expected from non-interacting spin-^12 fermions in a gas with the density of
statesdeterminedfrom theheat capacity. Spin-dependentinteractions enter through
a factor( 1 +FFF 0 a)−^1 inthe susceptibility,andthe experimentally determinedvalue
ofFFF 0 ais – 0.7. This is a very strong coupling, with the negative sign implying that
interactions tendto align thespins. Itis worthyofnote thatavalueforFFF 0 aof –1
would implythat theliquidwasferromagnetic!