**2. Experimental**

2 2

For the intermediate η values, *L* exhibits a strong dependence on both Fermi energy and the

Since Cu is highly conductive metal, the value of η at room temperature may be calculated using eq. 8 and the physical properties of Cu (effective mass, *m*\*=1.01*m*<sup>o</sup> [7], and carrier concentration, *n*=8.4x1022 cm-3 [8]). The estimated η value is equal to 270 and corresponds to

2

*h* p

*m kT*

Nevertheless, the Lorenz number differs slightly from the Sommerfeld value due to the

The simple consideration presented above shows that for the applying the Weidemann-Franz relation between the transport properties of materials, the specific Lorenz number has to be

It was shown [9], for instance, that Lorenz number of copper alloys depends on their purity and thermo-mechanical treatments. Moreover, in [10] it was established that the Lorenz number for copper films depends on its thickness. This effect was partly attributed to scattering of electrons at the films surfaces and partly to scattering by frozen-in structural defects.

This explanation is based on a distinction between the free paths of electrons for the electrical conductivity process and those for the thermal conductivity process. Being more specific, in a metal, an electric field or a temperature gradient causes an electron drift, which is restricted only by the collisions of the electrons with lattice imperfections (static defects or lattice vibrations). When the electron distribution function is disturbed from its equilibrium value, the rate of return to equilibrium may be expressed by collision processes, which are usually expressed in terms of a relaxation time. Only in case that the relaxation time is the same for both electrical and thermal transport, eq. 7 can be used. This equation is based on the assump‐ tion that *L* is a constant independent of the band structure or the relaxation time. Regarding relaxation, it was pointed out in [11] that equilibrium can be reached in two ways: either by processes changing the direction of motion of an electron but not changing its energy signifi‐ cantly, or by processes changing its energy but not the direction – the so-called "horizontal" or "vertical" movements on the Fermi surface. Since the "vertical" movement was found as ineffective in producing electrical resistance, the relaxation times for electrical and thermal conduction are equal only in case that the "vertical" movement is absent. The effective scattering by phonons at high temperatures and by impurities at low temperatures is elastic,

8 2

\* æ ö <sup>=</sup> ç ÷ è ø

p

3

3 <sup>2</sup> <sup>3</sup> 2

h

(7)

(8)

3 *<sup>k</sup> <sup>L</sup> e*

the energetic range where the constant Sommerfeld value valids.

*n*

scattering mechanism.

158 Sintering Techniques of Materials

inelastic electrons scattering.

determined.

p æ ö <sup>=</sup> ç ÷ è ø

> Pure (99.9%) copper powder with a nearly spherical particles shape and an average particle size of about 8μm was consolidated by SPS apparatus (FCT Systems GmbH, Germany) under argon atmosphere with heating and cooling rates of 50°C/min. The specific parameters of the SPS process are presented in Table 1. The porosity of the sintered specimens, (measured by the Archimedes method) was varied in the 0-30 vol.% range.

> The microstructure was characterized by optical microscope *Zeiss* (Germany). Electrical resistivity was measured at room temperature by a four-probe method using 1V/50 Hz alternating power source and *Keithley 2182A Nanovoltmeter*. The thermal conductivity was tested at room temperature using the flash diffusivity method (*LFA 457, Netzsch*). Thermal conductivity (κeff) values were calculated using the equation κeff=α ρ *C*p where, α is the thermal diffusivity, *C*<sup>p</sup> is the specific heat (measured using differential scanning calorimetry, *STA 449- Netzsch*), and ρ is the bulk density of the sample.


**Table 1.** Spark Plasma Sintering conditions and porosity of the specimens.

The thermal conductivity measurements were conducted under ~1atm argon, similarly to the SPS conditions.
