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V. TRANSPORT PROPERTIES OF DENSE MATTER

Properties of dense matter under non-equilibrium conditions, such as during the transfer of mass and conduction of heat and electricity, are of much physical interest. These are the transport properties, and knoweldge of them is important in understanding stellar structure and stellar evolution. We discuss here the following transport properties of dense matter: electrical conductivity, heat conductivity, and shear viscosity.

A. Electrical Conductivity

The electrical conductivity of a substance is given by the ratio of the induced electric current density to the applied electric field. In the case of a metal, the electric current is due to the flow of conduction electrons. If all conduction electrons are given an average drift velocity vd, then the current density je is given by

                                              (45)

where n is the number density of the conduction electrons and e the electric charge of the electrons. The electrons acquire the drift velocity as they are accelerated by the electric field. When an electron collides with either the lattice or another electron, its drift velocity is redirected randomly, and the subsequent velocity is averaged to zero statistically. Thus, after each collision the electron may be assumed to be restored to thermal equilibrium and begins anew with zero drift velocity. The average drift velocity is therefore determined by the mean time between collisions, t, during which an electron is accelerated to its drift velocity by the electric field E

                                                     (46)

t is also called the relaxation time. Putting these two expressions together gives the electrical conductivity:

                                                      (47)

The electrical conductivity is in units of inverse seconds, or sec-1, in the cgs system of units, and is related to the mks system of units by sec-1 = (9 x 109 W-meter)-1. When the electrons in the system are degenerate, the electron mass in Eq. (47) is replaced by pF/c, where pF is the Fermi momentum of the degenerate electron system. The relaxation time is the most crucial parameter in this investigation, which must be related to the electron density, the average electron speed, the number and types of scatterers in the system, and the scattering cross sections of the electrons with different types of scatterers.

When the temperature is below the melting temperature Tm given by Eq. (41), matter in the density range of 102-1014 g/cm3 is in a solid state possessing a crystalline structure. The constituent nuclei are organized into a lattice while the electrons are distributed more or less uniformly in the space between. In the neutron drip region, 1012-1014 g/cm3, neutrons are also outside the nuclei. The relaxation time of the electrons is determined by the frequency of scatterings with the lattice nuclei and with the other electrons and neutrons. When there are several scattering mechanisms present, the relaxation times due to different mechanisms are found by the inverse sum of their reciprocals

                                     (48)

Upon electron scattering, the lattice vibrates and the vibration propagates collectively like sound waves through the lattice. In quantized form the sound waves behave like particles, called phonons. Electron scattering from the lattice is usually described as electron-phonon scattering. Phonons increase rapidly in number with temperature. We therefore expect the electric conductivity due to electron-phonon scattering to decrease with increasing temperature.

Electron-electron scatterings have minor effects on the electric conductivity, because in dense matter all the electrons are not bound to the nuclei. Electrons lose energy only if they are scattered from bound electrons. Elastic scattering of electrons does not alter the current being transported. Hence this form of scattering does not affect electric conductivity and may be ignored.

Let us imagine that the dense matter in consideration is formed from a dynamical process, as in the formation of a neutron star, in which case the material substance is adjusting to reach the proper density while it is being cooled. If the solidification rate is faster than the nuclear equilibrium rate, there will be large admixtures of other nuclei with the equilibrium nuclei. The non-equilibrium nuclei act as impurities in the system. Also, there may be defects in the system due to rapid rates of cooling and rotation. Electrons and phonons are scattered by these impurities and defects which also limit the transport of electric charge by electrons.

There are estimates of the electric conductivity in dense matter that consider all the above discussed features. Typical results for a wide range of densities at a temperature of 108 K are shown in Fig. 8. Electric conductivities due to different scattering mechanisms are shown separately. The total conductivity should be determined by the lowest lying portions of the curves, since the total conductivity, like the relaxation time, is found by taking the inverse sum of the reciprocals of the conductivities due to different scattering mechanisms.

Electron-phonon scattering should be the dominant mechanism. The conductivity due to electron-phonon scattering has a temperature dependence of approximately T-1. Electron-impurities scattering depends on the impurity concentration and the square of the charge difference between the impurity charge and the equilibrium nuclide charge. Let us define a concentration factor:

                                         (49)

where the summation is over all impurity species designated by subscript i; ni and Zi are respectively the number density and charge of each impurity species. The conductivity curve due to impurity scattering simp is drawn in Fig. 8 with an assumed ximp = 0.01. It is independent of T.

The electron-phonon curve would not be correct at the low density end where the melting temperature is below 108 K. Above melting temperature, the lattice structure would not be there, and electrons would not be impeded by scattering from phonons but instead from the nuclei. Electric conductivity due to electron-nuclei scattering is shown in the possible melting region.

In the neutron matter region, 1014-1015 g/cm3, the system is composed basically of three types of particles, neutrons, protons, and electrons, of which both protons and electrons act as carriers for electrical conduction. The main difficulty in dealing with this situation is to take into full account the interactions among the particles. With the presence of a strong attractive interaction among the nucleons, it is quite likely that the protons will be paired to turn the system into a superconducting state when the temperature falls below the critical temperature, in which case the electric conductivity becomes infinite. The critical temperature Tc is estimated to be as high as 109-1010 K. Such a temperature, though it appears high, corresponds actually to a thermal energy kBT that is small compared with the Fermi energies of the particles present.

If the system is not in a superconducting state, or in other words it is in a normal state, its electrical conductivity is due mainly to electrons as carriers. The electron relaxation time is determined by scattering by protons and has been evaluated. The electrical conductivity of neutron matter in the normal state is plotted in Fig. 9 in the density range 1014-1015 g/cm3.

B. Thermal Conductivity

The thermal conductivity of a substance is given by the ratio of the amount of heat transferred per unit area per unit time to the temperature gradient. Kinetic theory of dilute gas yields the following expression for thermal conductivity:

                                                 (50)

where n is the carrier number density, cs its specific heat, v the average thermal velocity, and t the relaxation time of the carriers. The thermal conductivity is expressed in units of erg/cm-K-sec. When there are several types of carriers participating in the transport of heat, the final thermal conductivity is the sum of individual conductivities due to different types of carriers. In the case of a solid, the important carriers are the electrons and phonons, but in general the phonon contribution to thermal conduction is negligible compared with the electron contribution.

In thermal conduction by electrons, no electric current is generated while energy is being transported. The type of electron-phonon scattering important for thermal conduction is different from that of electrical conduction. At each point in the system, the electron number density obeys the Fermi-Dirac distribution of Eq. (32), which assigns a higher probability of occupation of high-energy states when the temperature is high than when the temperature is low. When a thermal gradient exists in the system, neighboring points have different electron distributions. Electrons moving from a high-temperature point to a low-temperature point must lose some of their energy to satisfy the distribution requirement. If this can be accomplished over a short distance, the thermal resistivity of the substance is high, or its thermal conductivity is low. The most important mechanism of energy loss is through inelastic scatterings of electrons by phonons at small angles. Such scatterings constitute the major source of thermal resistivity. On the other hand, elastic scatterings of electrons do not lead to energy loss and aid in thermal conduction. The frequency of inelastic scattering to that of elastic scattering depends on the thermal distribution of phonons. The temperature dependence of thermal conductivity due to electron-phonon scatterings is therefore complicated. For matter density below 108 g/cm3, it is relatively independent of temperature, but above that it shows a decrease with increasing temperature.

Although electron-electron scatterings do not contribute to the electric conductivity, they contribute to the thermal conductivity by redistributing the electron energies. The thermal conductivity due to impurity scattering kimp is directly related to the electric conductivity due to impurity scattering simp by the Wiedemann-Franz rule:

                                 (51)

The thermal conductivities due to different mechanisms at 108 K are shown in Fig. 9. The thermal conductivity due to impurity scattering is drawn with ximp =0.01, and it has a temperature dependence linear in T. The thermal conductivity due to electron-electron scattering is expected to have a temperature dependence of T-1. The thermal conductivity due to electron-ion scattering is shown for the region where 108 K is expected to be above the melting temperature. The total thermal conductivity of dense matter is determined by the mechanism that yields the lowest thermal conductivity at that density, since the total thermal conductivity, like the relaxation time, is found from the inverse sum of the reciprocals of the conductivities due to different mechanisms.

In the neutron matter region, 1014-1015 g/cm3, all three types of particles, neutrons, protons, and electrons, contribute to the thermal conductivity

where the subscripts e, n, and p denote contributions to the thermal conductivity by electrons, neutrons and protons, respectively. When the particles are in a normal state (i.e., not in a superfluid or superconducting state), the thermal conductivity is determined primarily by the highly mobile electrons, whose motion is impeded largely by scatterings with the protons and other electrons, and much less by scatterings with the neutrons. The neutron contribution to the thermal conductivity is substantial because of its high number density. Neutrons will encounter neutron-proton scattering and neutron-neutron scattering in the process. The proton contribution to the thermal conductivity is small because the proton number density is low, but otherwise the protons contribute in a manner similar to the neutrons. The thermal conductivities due to these three components for a system in the normal state are shown in short dashed lines in Fig. 9. The total conductivity is drawn as a solid line there.

The system may also become superfluid when its temperature falls below the critical temperature of Tc @ 109-1010 K. The critical temperature of the protons is in general different from that of the neutrons, and therefore it is possible that while one turns superfluid, the other remains normal. Also, when the temperature falls below the critical temperature of a certain type of particles, say the neutrons, there remains a normal component of neutrons in the system. This situation is usually described by a two-fluid model that consists of both the superfluid and normal fluid components. In general, scattering of particles off the superfluid component is negligible for transport purposes.

If only the protons turn superfluid (and superconducting) while the neutrons remain normal, the thermal conductivity due to the superfluid protons vanishes. The thermal conductivity found for the system in the normal state is basically unaltered, because the protons give a very small contribution to the thermal conductivity, as shown in Fig. 9.

If the neutrons turn superfluid while the protons remain normal, the superfluid component of the neutrons gives vanishing thermal conductivity, and the contribution by the normal component of the neutrons to the thermal conductivity diminishes rapidly with decreasing temperature below the critical temperature, because the number density of the normal neutrons decreases rapidly with decresing temperature. The thermal conductivity in this case is therefore determined entirely by the electron contribution to the thermal conductivity, which is modified slightly from the normal matter case due to the absence of scattering by superfluid. The net result is that the thermal conductivity is only slightly reduced from the normal case shown in Fig. 9. However, if both the neutrons and protons turn superfluid, the thermal conductivity is due entirely to electron-electron scattering, and the general result is indicated by the extension of the electron-electron curve for densities below 1014 g/cm3.

C. Shear Viscosity

When a velocity gradient exists in a fluid, a shearing stress is developed between two layers of fluid with differential velocities. The shear viscosity is given by the ratio of the shearing stress to the transverse velocity gradient. Elementary kinetic theory suggests that the shear viscosity of a dilute gas is given by

                                         (52)

where n is the molecular density, m the mass of each molecule, v the average thermal velocity of the molecules, and t the relaxation time. Viscosity is expressed in units of g/cm-sec, which is also called poise. In appearance it is similar to the expression for thermal conductivity with the exception that the specific heat per particle in the thermal conductivity is being replaced by the particle mass. Consequently, we may suspect that the electron component of the viscosity should behave similarly to the thermal conductivity. This, however, is not true due to the fact that the relaxation time involved relates to different aspects of the scattering mechanism. Also, there is an additional component to the viscosity. The solid lattice can make a great contribution to the total viscosity. Unfortunately, the determination of the lattice viscosity is very difficult, and no adequate work has been performed to determine its properties at the present time. The following discussion relates only to the electron viscosity.

The electron relaxation time is determined by electron scatterings by phonons, impurities, electrons, and nuclei. Shearing stress is developed when electrons belonging to fluid layers of different velocities are exchanged. Thus, viscosity is related to mass transfer or the transfer of electrons. This is similar to electric conduction where the transfer of electrons gives rise to charge transfer and is different from heat conduction which involves the adjustment of electron energy distributions. The evaluation of the relaxation times for the viscosity due to different scattering mechanisms is similar to that for the electrical conductivity.

The viscosity of dense matter at 108 K due to different scattering mechanisms is shown in the Fig. 10. The temperature dependence of the viscosity due to electron-phonon scattering is approximately T-1, as in the case of the electrical conductivity. This is also true of the viscosity due to electron-impurities scattering, which is independent of temperature as in the case of the electrical conductivity. While electron-electron scatterings do not contribute to the electrical conductivity, they play a role in viscosity giving rise to a temperature dependence of T-2.

In the neutron matter region, 1014-1015 g/cm3, all three types of particles, neutrons, protons, and electrons, contribute to the viscosity. The contributions from neutrons, protons, and electrons to the viscosity are shown separately in Fig. 10 in this density range by dashed lines. The total viscosity is drawn as a solid line. They all have a temperature dependence of T-2.

When the temperature drops below the critical temperature Tc, superfluid proton and/or neutron components appear. The behavior of the viscosity in the superfluid state is very similar to the thermal conductivity. If the protons turn superfluid, the viscosity is basically unaltered from the normal viscosity, because the proton contribution is small. If the neutrons turn superfluid, the superfluid component of the neutrons has vanishing viscosity. The viscosity is dominated by the electron contribution which is determined by the electron-electron scattering and electron-proton scattering mechanisms. When both protons and neutrons turn superfluid, then the viscosity is detemined entirely by electron-electron scattering, and the general result is indicated by the extension of the electron-electron curve for densities below 1014 g/cm3.