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The Hall resistivity exhibits plateaus at p = 1, ~, t, with values equal to 1jP, in units of hje 2 At the same time, the conventional resistivity Pxx drops to very low values. This indicates that in the neighborhood of these special filling fractions the two-dimensional electron fluid flows with almost no resistance. The value at p = 1, called the integer quantized Hall effect, was first observed in a MOSFIT (metal-oxide semiconductor field-effect transistor) at T = 1.5 K. The Hall resis-

The Hall effect. A current I flows in a direction orthogonal to crossed electric and magnetic fields. The Hall resistivity is defined as Pxy = V/1. The conventional resistivity Pxx can be obtained by measuring the voltage drop along the direction of the current.

tlVlty was found to be quantized with a preClSlon of one part in 10 5.* The fractional values were found soon after. t The integer effect is easy to understand on a naive basis. Since at p = 1 the lowest Landau level is completely filled, there is an energy gap above the Fermi level. Low-energy excitations are therefore impossible, and so the centers of the electron orbits flow like a free gas. Using (11.94) with n = eH/hc, the Landau degeneracy per unit area, we immediately obtain the desired result.

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The equation of state of He 4 is shown in Fig. 13.2. In the liquid phase there is a second-order (i.e., non-first-order) phase transition dividing the liquid further into He I and He II. The former is a normal-type liquid, but He II exhibits superfluid behavior, and will be the focus of our attention. The transition from He I to He II is called the A transition. Along the vapor pressure curve it occurs at the temperature Tx and the specific volume Vx, given by T x = 2.18 K (13.3) Vx = 46.2 N /atom The specific heat along the vapor curve becomes logarithmically infinite as the point of A transition is approached from either side, * as shown in Fig. 13.3. The shape of the specific-heat curve near T x gives rise to the name A transition. It is significant that He 4 exhibits the A transition and He 3 does not. Apart from a difference in atomic masses, the only difference between these two

OW. M. Fairbank, M. J. Buckingham, and C. F. Kellers, in Low Temperature Physics and Chemistry, edited by 1. R. Dillinger (University of Wisconsin Press, Madison, 1958), p. 50.

substances is that He 4 atoms are bosons, whereas He 3 atoms are fermions. Hence it is natural to assume that the A transition is the Bose-Einstein condensation. modified, of course, by molecular interactions. In fact, an ideal Bose gas with the same mass and density as liquid He 4 would undergo the Bose-Einstein condensation at 3.14 K, which is of the same order of magnitude as T A The isotope He 3 undergoes a phase transition to a superfluid phase at much lower temperature (about 10- 3 K.) The physical mechanism is similar to that in superconductivity, in which bound fermion pairs form a condensate. We shall not discuss this case, and will consider He 4 exclusively from now on.

PARABOLIC STOP AND REVERSAL (PSAR)

The lowest excitations of liquid He 4 can be found experimentally by neutron scattering. * If the temperature is sufficiently low, the scattering proceeds predominantly through the creation of an elementary excitation whose energy and momentum is equal to the energy and momentum transferred to the system by the neutron. In this manner, one finds the spectrum of elementary excitations Wi. shown schematically in Fig. 13.4. The line..ar portion near k = 0 represents phonons. The portion near k = k o corresponds to "rotons," which behaves like a

*Palevsky, Otnes, and Larsson, Phys. Rev. 112, 11 (1958); Yarnell, Arnold, Bendt, and Kerr. Phys. Rev. 113,1379 (1959). The measurements are made at about 1 K.

You can remove a field from the report in the Wizard by clicking the Back button until you reach the wizard s first screen.Then click the field you want to remove in the Selected Fields list and click the Remove button ( ).To remove all the fields, click the Remove All button ( ).

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