Caterina Michelagnoli

The performance of AGATA: From the LNL demonstrator to the GANIL setup

The performance of the Advanced-GAmma-Tracking-Array AGATA is demonstrated through selected examples from the different recent experimental campaigns.

Introduction

In order to perform nuclear structure studies in extreme conditions of neutron/proton asymmetry and angular momentum, in the last decade, the challenge of the realization of the gamma-ray spectroscopy dream has been taken on by two international collaborations, AGATA (Advanced-GAmma-Tracking-Array) [1] in Europe and GRETA (Gamma-Ray Energy Tracking Array) in the US [2]. The dream consists in covering the whole 4 π solid angle by germanium material and track the path of the gamma rays inside the detector medium, corresponding to maximum efficiency and "infinite" position resolution, in addition to an excellent energy resolution.

On this purpose, the digitally recorded waveforms from highlysegmented HPGe detectors are treated with Pulse-Shape-Analysis (PSA) techniques in order to extract the position of the interaction points in the detector with a 4 mm position resolution (at 1 MeV) [3,4]. These interaction points (hits) are grouped in events on the basis of the timestamp. Different gamma rays in the same event and their ordered path are reconstructed via tracking algorithms. An efficiency four times larger than standard arrays (also for high gamma-ray multiplicities) is expected, together with the capability to stand high count-rate (up to hundreds of kHz) thanks to the use of digital electronics. For the first time an almost continuous angular distribution of gamma interaction points will be available, allowing for a "perfect" Doppler correction and new degree of sensitivity in the determination of nuclear structure observables.

This contribution focuses on the AGATA array. It run its Demonstrator phase in 2009-2010 [5] and first physics campaign with stable beams at the Legnaro National Laboratories, consisting mainly in the spectroscopy and lifetime measurements in neutron-rich nuclei produced in deep-inelastic reactions at the PRISMA magnetic spectrometer. It then moved to GSI for the PreSpec campaign at the FRS separator and used for the first measurements with fast moving radioactive ion beams (beta up to 70%). It is now running its first experimental campaign at GANIL (France) coupled to the large acceptance magnetic spectrometer VAMOS [6]. Details on the AGATA project can be found in the reference paper [1]. It the following, the focus will be on particular aspects, that bring to new physics opportunities already with a sub-system of the whole array.

High count-rate capabilities

The use of digital electronics to preprocess the detector preamplifier signals allow to preserve a good efficiency and energy resolution even for count-rates larger than 10 kHz (typical limitation when using analogue electronics). In Fig. 1 an example of the output of the detector preamplifier on the core contact of an AGATA detector in response to a count rate of 30 kHz is shown (in red), together with the corresponding amplitudes reconstructed via the trapezoidal filter, corresponding to two different risetimes of the trapezoid (in blue and black). The proper choice of the risetime (and of the filter parameters in general), together with the baseline restoration, allow for extracting the amplitudes from the signals and avoiding pileup. A detailed study of the efficiency and resolution of an AGATA detector as a function of the count rate is reported in [7].

Correction for neutron damage

The operation of the germanium detectors in position-sensitive mode allows for the correction for the effects of the damage of the germanium crystalline structure due to the interaction of fast neutrons. Such an effect results in an incomplete charge collection causing a worsening of the energy resolution.

Since the charge loss due to neutron damage is proportional to the length of the path to the electrodes, the interaction points resulting from PSA can be used together with a modeling of charge trapping to correct for such an effect [8]. In Fig. 2 the energy spectra corresponding to the 1.3 MeV line ( 60 Co calibration source) before and after the correction are compared. This procedure is available and used for the detectors that suffers from neutron damage and applied in the data flow (or offline processing) after the PSA, before the data are passed to the global level stage, where event building and tracking are performed.

Some benefits from position resolution

The position resolution of AGATA can be used for the determination of nuclear structure observables with a new degree of sensitivity, giving access to new physics opportunities. Few selected examples from the Legnaro and GANIL campaigns will be given in the following. The left tail consequent to inefficiency in charge collection due to neutron damage is evident in the spectra in red (no correction applied) and it is significantly reduced in the spectra in blue, obtained after correction for neutron damage (resolution for the sum of all segments from 5.9 keV to 2.9 keV after correction. The data are taken during the Legnaro campaign (picture taken from [9]).

Sub-femstosecond lifetime of interest in nuclear astrophysics

The position resolution of AGATA can be used to measure nuclear level lifetimes via Doppler Shift techniques exploiting a continuous distribution of angles. It is well known that especially in the case of the Doppler Shift Attenuation Method (DSAM) the contribution of the detector response function, namely the finite angular coverage of the detectors, is one of the main factors that limits the sensitivity. A new sub-femtosecond upper limit has been recently obtained for the 6.79 MeV state in 15 O [10]. The experiment was performed during the Demonstrator campaign in Legnaro. The reaction of 14 N beam on a 2 H target has been used to populate the state of interest.

The effect of fs lifetimes can be observed by analysing the line-shape of the gamma rays, emitted by the reaction products slowing down in a gold layer, as a function of the observation angle. In Fig. 3 the energy of the reconstructed gamma rays sorted according to the angle of the first interaction point is reported with an unprecedented angular resolution of 2 degrees (see figure caption for details).

AGATA as a Compton polarimeter

The second point of interaction of a gamma ray in its attenuation path in the germanium medium can be used to determine the linear polarization of gamma rays emitted by aligned nuclear states. The asymmetry in the distribution of the Compton scattered gamma rays as a function of the azimuthal angle can be used to determine the linear polarization. A test of AGATA modules as Compton polarimeters has been performed in the Legnaro campaign [11], using the Coulomb excitation of the first excited states in 104,106 Pd targets by a 12 C beam. As evident from Fig. 4 (taken from [11]), with respect to standard clover detection system, this asymmetry can be analysed as a continuous function of the angle. This results in a larger analysing power, that allows for the measurement of gamma linear polarization in cases of weaker nuclear alignments or, in general, for rarer phenomena. This is the case of the study of the photon entanglement through the measurement of the relative polarization of the 511 keV gamma rays emitted by the singlet positronium [12].

Doppler correction capabilities

In Fig. 5 the advantages of using PSA and tracking techniques for the detection of gamma rays with respect to traditional arrays are shown, by using the comparison of spectra obtained in the same experimental conditions, exception made for the gamma array. The 98 Zr nucleus was populated in the fission process in inverse kinematics of 238 U beam on a 9 Be target at 6.6 MeV/u. The same reaction has been recently used in GANIL for the study of isotopically-identified neutron-rich fission fragments at the VA-MOS++ magnetic spectrometer [13]. Here a spectrum obtained from those data is used for comparison. The gain of almost a factor of two in energy resolution is evident.

The AGATA+VAMOS campaign at GANIL

The commissioning runs in GANIL took place in November-December 2014 and the Doppler correction capabilities and physics correlations between the gamma and magnetic spectrometers were verified and tested for different range of velocities and A and Z of the emitting nuclei [14]. Nowadays 8 triple clusters of the AGATA array are available, Fig. 6. The AGATA array positioned at backward angles with respect to the beam line coupled to the magnetic spectrometer VAMOS++ were used in the first campaign with stable beams at GANIL, in March-July 2015. The sensitivity of this unique combination was used to measure lifetimes and other spectroscopic properties of neutron rich nuclei produced by fission and deepinelastic reactions in different region of the nuclear chart.

2 Figure 1 :

 21 Figure 1: Response of the core preamplifier of an AGATA HPGe detector to an average 30 kHz count rate, during an in-beam experiment (in red). The amplitudes extracted from the signals when using a trapezoidal filter with two different risetimes are reported in black and blue. In the right panel, a zoom of the region indicated by the dashed rectangle is reported. The shorter risetime acts as a pile-up rejector.

3 Figure 2 :

 32 Figure 2: Segment spectra corresponding to the 1.3 MeV line ( 60 Co calibration source) -see schematic representation of an AGATA detector in the inset for segment labeling.The left tail consequent to inefficiency in charge collection due to neutron damage is evident in the spectra in red (no correction applied) and it is significantly reduced in the spectra in blue, obtained after correction for neutron damage (resolution for the sum of all segments from 5.9 keV to 2.9 keV after correction. The data are taken during the Legnaro campaign (picture taken from[9]).

Figure 3 :Figure 4 :

 34 Figure 3: Tracked gamma-ray energy sorted according to the angle of the reconstructed first interaction point, from the experiment for the measurement of the lifetime of the 6.79 MeV state in 15 O. The "straight" lines correspond to the emission at rest by a radioactive source (present while beam-on-target for monitoring of gain instabilities). The tilted broad lines correspond to gamma rays emitted from short (≈ fs) lived states. The lifetimes are obtained with Doppler Shift Attenuation technique over an almost continuous distribution of angles.

5 Figure 5 :

 55 Figure5: Gamma-ray spectra corresponding to the 98 Zr nucleus produced in the fission of a 238 U beam on a 9 Be target at 6.6 MeV/u and identified in the magnetic spectrometer VAMOS. The comparison of performance in gamma-ray detection between the EXOGAM (lower panel, data taken from[13]) and AGATA (upper panel, data taken from December 2014 commissioning runs) arrays is shown. The FWHM at 1.2 MeV is reported.

  Hit pattern from 24 crystals.

Figure 6 :

 6 Figure 6: The AGATA array installed for the first campaign at the VAMOS spectrometer at GANIL (spring/summer 2015). Eight triple clusters are mounted at backward angles with respect to the axis of the VAMOS spectrometer. The corresponding hits reconstructed via PSA are also shown (in the XY plane) together with the simulation of the 1 π array foreseen for the forthcoming experimental campaign at GANIL.


Amelia Carolina Sparavigna

On the Boltzmann Equation of Thermal Transport for Interacting Phonons and Electrons

Keywords: Boltzmann Equation, Thermal Conductivity, Phonons and Electrons

The thermal transport in a solid can be determined by means of the Boltzmann equations regarding its distributions of phonons and electrons, when the solid is subjected to a thermal gradient. After solving the coupled equations, the related thermal conductivities can be obtained. Here we show how to determine the coupled equations for phonons and electrons.

Introduction.

. Phonons are the quantization of the vibrations of the ions in crystals. They are the quanta which are carrying the thermal energy in the dielectric solids. Since the phonons are interacting among them and with the crystal defects and boundaries, a finite thermal conductivity is observed. Here, we will consider the presence in the solid of both phonons and electrons, subjected to a thermal gradient. The electrons we consider are those which can freely move in the crystal. In fact, these electrons can give rise to several phenomena connected to charge transport: these phenomena will be the subject of a future paper. Here we concentrate just on the effect of a thermal gradient and on the interaction of electrons with phonons and how it can be described by the Boltzmann equation, aiming to introduce the subject to students of engineering colleges.

As shown in [START_REF] Srivastava | The physics of phonons[END_REF][START_REF] Kittel | Quantum theory of solids[END_REF][START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF], the interaction among phonons, in three phonons scattering processes for instance, is coming from the Lagrangian of the lattice. In the case that in the solid there are electrons which can support an energy/charge conductivity, these particles, when moving in the solid, are interacting with the ions of the lattice. The use of a pseudopotential [START_REF]Condensed Matter -Disordered Solids[END_REF] allows us to consider the conducing electrons as electrons described by means of a plane waves basis. In this manner we can analyse the transport with an approach based on a perturbative interaction with phonons, neglecting electron-electron interaction and the scattering of electrons with lattice defects. As we have seen for phonons, we can solve the Boltzmann equation for these electrons and determine their contribution to the thermal conductivity. However, the electron-phonon interaction is also influencing the phonon transport, and therefore the phonon thermal conductivity is changed too.

The Boltzmann equation for phonons in a thermal gradient. A thermal gradient in a solid is exciting electrons and phonons which are present in the crystal; they are gaining energy which are transferring, that is conducting, from one end of the solid to the other, moving from the hotter to the coldest side. In metals, both electrons and phonons are playing a relevant role in the thermal conduction, whereas in dielectrics, only phonons are involved. To find the thermal resistance and then the thermal conductivity of the solid, we need to describe several interactions among phonons, electrons, those of phonons with electrons and of phonons and electrons with defects and impurities of the crystals. Moreover the scattering with the boundaries of the crystal have to be considered too.

The problem of evaluating the thermal conductivity in solids is therefore quite complex. It can be solved, from a microscopic point of view [START_REF] Peierls | Quelques propriétés typiques des corps solides[END_REF], using the relaxation time approximation [START_REF] Sparavigna | The Boltzmann equation of phonon thermal transport solved the relaxation time approximation -I -Theory[END_REF][START_REF] Sparavigna | The Boltzmann equation of phonon thermal transport solved in the relaxation time approximation -II -Data analysis[END_REF][START_REF] Peierls | Quelques propriétés typiques des corps solides[END_REF][START_REF] Klemens | The thermal conductivity of dielectric solids at low temperatures (theoretical)[END_REF][START_REF] Klemens | Thermal conductivity and lattice vibrational modes[END_REF][START_REF] Callaway | Model for lattice thermal conductivity at low temperatures[END_REF][START_REF] Krumhansl | Thermal conductivity of insulating crystals in the presence of normal processes[END_REF][START_REF] Srivastava | Calculation of lattice thermal conductivity of Ge from 4 to 900 K[END_REF][START_REF] Srivastava | Phonon conductivity of insulators and semiconductors[END_REF][START_REF] Parrott | Generalization of the Callaway thermal conductivity equation[END_REF], using some variational [START_REF] Guyer | Solution of the linearized phonon Boltzmann equation[END_REF][START_REF] Srivastava | Phonon conductivity of insulators and semiconductors[END_REF][START_REF] Omini | Thermal conductivity of rare gas crystals the role of three-phonon processes[END_REF] or iterative methods [1-4], or the Green functions [START_REF] Jancel | Foundations of classical and quantum statistical mechanics[END_REF][START_REF] Gurevich | Transport in phonon systems[END_REF]. The methods based on relaxation time approximations and on variational and iterative methods are based on the Boltzmann equation, whereas the approach with the Green functions is based on quantum statistics. Here we consider the approach based on the Boltzmann equation.

Let us start from phonons. To derive the Boltzmann equation we need that a distribution function ) , ( t n p r q exists concerning the phonon state ) , ( p q about position r, at time t. The state is described by the wave vector q and by its polarization p . If a thermal gradient ∇T exists, which is changing the temperature T = T (r) in the sample, we will have a diffusion process in the distribution. In this case we can write:

T n T T t n p p p ∂ ∂ ∇ ⋅ - = ∂ ∂ q q q v ) ( k B diff (1) In (1), ) (q v p
is the group velocity of phonon (q, p). Let us observe that the phonon distribution p n q is depending on space through the temperature T = T (r).

The scattering processes will give a change of the phonon distribution. In the case of a stationary condition, we must have:

0 ) ( k scatt B = ∂ ∂ + ∂ ∂ ∇ ⋅ - t n T n T T p p p q q q v
(2) This is the general form of the Boltzmann equation [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF][START_REF] Mizutani | Introduction to the Electron Theory of Metals[END_REF]. It is an integral-differential equation. A way to solve this equation is in its linearization, considering only the deviation from equilibrium 0 p p n n q q -, being 0 p n q the equilibrium distribution.

The simplest case of scattering we can discuss is that of an elastic scattering. The phonon process is ) , ( ) , ( p p ′ ′ → q q , conserving the energy. In this process, we have a transition from the phonon state ) , ( p q to the state ) , ( p′ ′ q

. The relative probability of a transition in the unit of time is:

p p p p p p Z n n P ′ ′ ′ ′ ′ ′ + = q q q q q q ) 1 ( (3)
In (3),

p p Z ′ ′ q q
is the intrinsic probability, that is the probability not depending on the occupation of phonon states ) , ( p q and ) , ( p′ ′ q . Therefore, the probability of the transition from ) , ( p′ ′ q to ) , ( p q is:

p p p p p p Z n n P q q q q q q ′ ′ ′ ′ ′ ′ + = ) 1 ( (4)
where

p p p p Z Z ′ ′ ′ ′ = q q q q (5)
Because the variation of the phonon distribution is equal to the probability of having a scattering per unit of time, we have:

( ) [ ] p p p p p p p p p p p p p Z n n n n P P t n ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + - + = - = ∂ ∂ ∑ ∑
q q q q q q q q q q q q q ) 1 ( ) 1 ( scatt [START_REF] Sparavigna | The Boltzmann equation of phonon thermal transport solved in the relaxation time approximation -II -Data analysis[END_REF] After simplifying and linearizing we find:

[ ] p p p p p p p p Z n n n n t n ′ ′ ′ ′ ′ ′ ′ ′ - - - = ∂ ∂ ∑ q q q q q q q q ) ( ) ( 0 0 scatt (7) In (7), 0 0 p p n n ′ ′ = q q
, because 0 p n q is depending just on energy and that in an elastic process:

) , ( ) , ( p p ′ ′ = q q ω ω h h (8)
The linearized Boltzmann equation turns out to be:

[ ] p p p p p p p p p Z n n n n T n T T ′ ′ ′ ′ ′ ′ ′ ′ - - - = ∂ ∂ ∇ ⋅ - ∑ q q q q q q q q q v ) ( ) ( ) ( k 0 0 0 B (9)
In [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF], to evaluate

T n p ∂ ∂ / q
we used only the equilibrium distribution 0 p n q .

In the general case, to obtain the linearized Boltzmann equation we use the deviation from equilibrium p q Ψ in the following manner:

) ( 0 0 p p p p p n n n q q q q q ω h ∂ ∂ Ψ - = (10') Let us assume T k x B p q ω h =
, where B k is the Boltzmann constant. From (10') we have:

( ) 1 k 1 1 1 k 1 ) ( ) ( 0 0 + =       - ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ p p B x B p o p p o p n n T e x T x x n n q q q q q q ω ω h h (10'') Therefore: ) 1 ( k ) ( 0 0 B p 0 0 0 + Ψ + = ∂ ∂ Ψ - = p p p p p p p p n n T n n n n q q q q q q q q q ω h (10''')
For the elastic scattering:

[ ] p p p p p p p p p p p Z n n n n T n T T ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ + Ψ - + Ψ = ∂ ∂ ∇ ⋅ - ∑ q q q q q q q q q q q v ) 1 ( ) 1 ( ) ( k 0 0 0 0 0 B (11) That is: [ ] p p p p p p p Q T n T T ′ ′ ′ ′ ′ ′ Ψ - Ψ = ∂ ∂ ∇ ⋅ ∑ q q q q q q q v ) ( k 0 B ( 12
) p p p p p p Z n n Q ′ ′ ′ ′ + = q q q q q q
) 1 ( 0 0 [START_REF] Klemens | Thermal conductivity and lattice vibrational modes[END_REF] With this formalism, in the case that we have three-phonon processes, the linearized Boltzmann equation is:

[ ] [ ] p p p p p p p p p p p p p p p p p p Q Q T n T T q q q
q q q q q q q q q q q q q q q v Ψ -

Ψ + Ψ + Ψ - Ψ - Ψ = ∂ ∂ ∇ ⋅ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ∑ ∑ ∑ ∑ , , 0 B 2 1 ) ( k (14)
In the linearized Boltzmann equation, we have the additions on phonons ′ q and ′′ q of the Brillouin Zone. However, sums can be converted into integrals:

∫ ∫ ∫ ∫ ∑ ′ ′ ′ ′ - ′ ′ ′ ′ ′ ′         Ω = ′ Ω → ) , ( ) , ( ) ( 0 2 0 1 3 3 3 d d d 2 2 ) 2 ( d ) 2 ( η θ η θ θ π ζ η η θ π π π M M H h q q (15)
In [START_REF] Krumhansl | Thermal conductivity of insulating crystals in the presence of normal processes[END_REF],

NV = Ω
is the volume of the crystal, made of N lattice cells each with volume V. H and M are the functions describing the boundary of the Brillouin Zone, and 1 h is the nearest neighbor site distance [1].

Once obtained the deviation of the distribution function, the Boltzmann equation is solved and the thermal current U can be calculated as:

p p p p p p p p p p n n q q q q q q q q q q v v U Ψ ∂ ∂ Ω - = Ω = ∑ ∑ ) ( 1 1 0 ω ω ω h h h (16)
In a Cartesian frame of reference having unit vercots u i , current U is represented by:

i i ji j x T U ∂ ∂ - = ∑ κ (17)
In [START_REF] Srivastava | Phonon conductivity of insulators and semiconductors[END_REF] κ ij is a term of the thermal conductivity tensor, which is diagonal (κ xx = κ yy = κ zz = κ).

Besides the phonon-phonon interaction, a phonon can be subjected to a scattering coming from the lattice defects and from electrons [1-6].

The Boltzmann Equation for electrons Let us use symbol Q when considering a phonon having wave-vector q, polarization p and frequency p q ω . And also use simply k for an electron having wave-vector k, and energy ε k and spin σ . Supposing a thermal gradient disturbing the distribution of electrons from equilibrium 0 k f to a distribution k f , let us assume, as we did for phonons:

T f f f f k k k k k B 0 0 0 k ) 1 ( - Φ = - (18)
The equilibrium distribution 0 k f is that of Fermi-Dirac, where each energy level is occupied by an electron. In it, we have the Fermi energy ε F . This threshold is obtained by the distribution 0 k f which becomes a step function of energy when the Kelvin temperature is null. For non-null temperature:

T k f B k ) F ( 0 e 1 1 ε ε - + = (19)
Let us have free electrons described by k . At null temperature, the energy levels would be occupied till the radius k F so that:

2 F e 2 F m 2 k h = ε (20)
In [START_REF] Srivastava | Phonon conductivity of insulators and semiconductors[END_REF], e m is the mass of the electron. For a non-null temperature, some electrons have

F k > | | k
and them they can be above the Fermi energy.

In [START_REF] Parrott | Generalization of the Callaway thermal conductivity equation[END_REF], the deviation from equilibrium 0 k k f f -, coming from the presence of a thermal gradient, is considered with the same approach we followed to solve the Boltzmann equation for phonons (see Eqs. (10')-(10''').

Let

q q k k k k ′ ′ Π Π ,
be the intrinsic probability of an electron-phonon scattering of an electron k interacting with a phonon q as shown in the following scheme (Figure 1). After the scattering, the electron has momentum ' k , because it had gained a phonon or lost it [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF]. Of course, as in the case of phonons, we can have processes involving the g vectors of the reciprocal lattice. The electron-phonon scatterings are adding new contributions in the scattering term of the Boltzmann equation for phonons. For instance, if we consider the case when just threephonon scattering processes are present, to them we have to add the scattering from electrons:

( ) ( ) ( ) ( ) ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2 1 Q Q Q Q Q Q QQ Q Q Q Q Q Q Q Q k Q k Q k k k Q k k kQ scatt Q Q Q P P t n Ψ - Ψ - Ψ + Ψ + Ψ - Ψ + Φ - Ψ + Φ + Φ - Ψ - Φ =         ∂ ∂ ∑ ∑ ∑ ∑ ∑ ∑ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ σ σ σ σ σ σ σ σ δ δ k k k k (21')
Then, the linearized Boltzmann equation is:

( ) ( ) ( ) ( ) ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 0 B 2 1 ) ( k Q Q Q Q Q Q QQ Q Q Q Q Q Q Q Q k Q k Q k k k Q k k kQ p p Q Q P P T n T T Ψ - Ψ - Ψ + Ψ + Ψ - Ψ + Φ - Ψ + Φ + Φ - Ψ - Φ = ∂ ∂ ∇ ⋅ ∑ ∑ ∑ ∑ ∑ ∑ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ σ σ σ σ σ σ σ σ δ δ k k k k q q v (21'')
Symbol σ is indicating electronic spin. Delta σ σ δ ′ is considering the conservation of electronic spin. Probabilities are given by the following expressions:

( ) 0 0 0 1 k k k k k k f n f P ′ ′ ′ - Π = q q q (22) ( )( ) 0 0 0 1 1 k k k k k k f n f P ′ ′ ′ - + Π = q q q (23)
In metals, electrons too are contributing to the thermal transport. In the Boltzmann equation for electrons, we find the scattering with phonons [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF]:

( ) ( ) k Q k Q Q k Q k Q Q scatt k P P t f ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Φ - Ψ - Φ - Φ - Ψ + Φ - =       ∂ ∂ ∑ ∑ σ σ σ σ σ σ δ δ k k k k k k (24')
Then, the linearize Boltzmann equation for electrons is:

( ) ( ) k Q k Q Q k Q k Q Q k k P P T f T T ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Φ - Ψ - Φ - Φ - Ψ + Φ - = ∂ ∂ ∇ ⋅ ∑ ∑ σ σ σ σ σ σ δ δ k k k k k k v 0 B k (24'')
From the theory of electron-phonon scattering [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF], the probability k k ′ Q P turns out to be:

( ) ( ) [ ] ( ) ( ) k q k k q k q q k k k k e k k ′ ′ ′ - + - - ′ ⋅ - ′ ℜ = ε ω ε δ ω p p p Q f n f P h 0 0 0 2 1 (25) ( ) ( ) [ ] ( )( ) ( ) k q k k q k q q k k k k e k k ′ ′ ′ ′ - - - + - ′ ⋅ - ′ ℜ = ε ω ε δ ω h 0 0 0 2 1 1 f n f P p p Q (26)
In ( 25) and ( 26) we find the polarization vectors e of phonons. The interaction between electron and lattice is described by the function ℜ(k). It can be described by means of a pseudopotential for instance.

Let us note that we have to consider the conservation of energy and of the generalized conservation of momentum given by the following expression:

g k q k + ′ = + ; g k q k + ′ = -
The coupled Boltzmann linearized equations for phonons and electrons can be solved in an iterative approach for instance. Another question we have to discuss before solving the equation is the description of interaction. This will be shortly analysed in the next section.

Scattering electron-phonon Let us consider the following scheme (Figure 2): In this process we have a phonon having momentum q and polarization p added to the population Q n so that population becomes 1 + Q n . Then, the initial quantum state is given by

Q k n i ψ = and the final quantum state is 1 + = Q k n j ψ
. If we use a pseudopotential approach [START_REF] Mizutani | Introduction to the Electron Theory of Metals[END_REF], the electron can be described by plane waves [START_REF] Barbero | Vacancy formation volume of sodium, aluminium and argon[END_REF][START_REF] Omini | Phonon-phonon interaction and phonon drag in potassium at low temperatures[END_REF][START_REF] Sparavigna | Appunti di un seminario del Professor Omini sullo pseudopotenziale[END_REF], so that:

r k r k ⋅ ′ ′ ⋅ Ω = Ω = i i e 1 ; e 1 k k ψ ψ (27)
Let us remember that a pseudopotential is an effective potential used as an approximation for the simplified description of complex systems. It is an attempt to consider the effects of the motion of the core, that, is of the non-valence electrons of an atom and of the nucleus with an effective potential, so that the Schrödinger equation contains such effective potential instead of the Coulomb terms for the core electrons. In this manner the valence electrons are described by pseudo-wave functions, in which a plane-wave basis is used.

We can write the scattering as the passage from the initial state | i 〉 to the final state | f 〉 through pseudopotential W [START_REF] Barbero | Vacancy formation volume of sodium, aluminium and argon[END_REF][START_REF] Omini | Phonon-phonon interaction and phonon drag in potassium at low temperatures[END_REF]:

Q Q Q Q Q k k Q k Q Q k n W n n x W n n x W n x n W n i W f k k r k k r r r r r r r - ′ ⋅ - ′ - ′ ′ + = Ω + = Ψ Ψ + = Ψ + Ψ = ∫ ∫ ∫ 1 d ) ( e 1 1 d ) ( ) ( ) ( 1 d ) ( ) ( 1 ) ( 3 ) i( 3 * 3 * (28)
Moreover, we can write [START_REF] Barbero | Vacancy formation volume of sodium, aluminium and argon[END_REF][START_REF] Omini | Phonon-phonon interaction and phonon drag in potassium at low temperatures[END_REF][START_REF] Sparavigna | Appunti di un seminario del Professor Omini sullo pseudopotenziale[END_REF]:

) ( ) ( ) ( k k k k k k k k - ′ - ′ - ′ = - ′ ε w NS W (29)
In [START_REF] Ponasso | Determinazione del trasporto termico nei metalli con approccio iterativo all'equazione di Boltzmann[END_REF], w is the pseudopotential generated by a single ion which is acting on the electron. N is the number of lattice sites. We find also in [START_REF] Ponasso | Determinazione del trasporto termico nei metalli con approccio iterativo all'equazione di Boltzmann[END_REF] the static dielectric constant ε, which can be written as the following function [START_REF] Barbero | Vacancy formation volume of sodium, aluminium and argon[END_REF][START_REF] Omini | Phonon-phonon interaction and phonon drag in potassium at low temperatures[END_REF][START_REF] Sparavigna | Appunti di un seminario del Professor Omini sullo pseudopotenziale[END_REF] (e is the electron charge):

( ) ( ) ( ) ( )         + - + - + = 1 2 / 1 2 / 1 ln 2 / 2 / 1 2 m e 4 1 2 2 2 F e 2 2 F F F F Q k Q k Q k Q k Q k Q h π π ε (30)
Moreover (l is giving the lattice site described by the position vector l):

...] ) i( 1 [ e e e e ) ( ) i( ) ( ) i( ) ( ) i( + ⋅ - ′ - = = = - ′ ∑ ∑ ∑ ⋅ - ′ - ⋅ - ′ - ⋅ - ′ - + ⋅ - ′ - l l l l i l l NS u k k k k l k k u k k l k k u l k k
As usual, the phononic function of the displacement of the lattice site given by the position vector l is [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF][START_REF]Condensed Matter -Disordered Solids[END_REF]:

[ ] ∑ + ⋅ ⋅ - - = p p p p p p l a a N q q l q q q l q q q e e u i i * e e 1 2m i ω h (31)
In this manner we can write the matrix:

Q l l Q n n w i W f ] ) i( 1 [ e 1 ) ( ) ( ) i( u k k k k k k l k k ⋅ - ′ - + - ′ - ′ = ∑ ⋅ - ′ - ε (32)
Then, we can obtain:

[ ]      - - ′ + + +      - ′ - ′ = ∑ ∑ ∑ + ⋅ ⋅ - ⋅ - ′ - ⋅ - ′ - Q p p p p p p l i Q l Q n a a N n n w i W f q q l q q q l q q q l k k l k k e e k k k k k k i i * ) ( Q ) i( e e 1 e 2m ) ( 1 n e 1 ) ( ) ( ω ε h (33)
Of the matrix, just the element having + Q a , which is creating phonon p , q survives, and then:

Q p Q l p p n a n N w i W f + ⋅ ⋅ - ′ - + - ′ ⋅ - ′ - ′ - = ∑ q l q l k k q q k k e k k k k 1 e e ) ( 2m ) ( ) ( i ) i( ω ε h (34) Since g q k k l q k k , ) i( e + - ′ ⋅ + - ′ - = ∑ δ N l and 1 1 + = + + Q Q p Q n n a n q , we have: 1 ) ( 2mN ) ( ) ( + - ′ ⋅ - ′ - ′ - = Q p p n N w i W f q q k k e k k k k ω ε h (35)
Of course, we have the generalized conservation of momentum ′ -+ = k k q g. Therefore, the probability for the unit of time is:

[ ] ) ( ) 1 ( ) ( 2m ) ( ) ( 2
) ( 2 2 2 i f 2 k p k Q p p n N w E E i W f ε ω ε δ ω ε π δ π - + + - ′ ⋅       - ′ - ′ = - ′ q q q k k e k k k k h h h h ( 36
)
The conservation of energy is represented by . Therefore we have:

) ( i f E E - δ . Eq.
[ ] ) ( ) 1 )( 1 ( ) ( 2m ) ( ) ( 2 2 2 k Q k Q k k p p Q k k n f f N w P ε ω ε δ ω ε π - + + - - ′ ⋅       - ′ - ′ = ′ ′ ′ h h h q q k k e k k k k (37)
This is in agreement to (25) if we have:

) ( ) ( m ) (       - ′ - ′ = - ′ ℜ k k k k k k ε π w N ( 38
)
In the Boltzmann equation we have also to consider the other scattering mechanism shown in the Figure 3. In this case we have to consider

Q Q Q n a n 1
and generalized conservation k q k g + = ′ + . In this manner, it is straightforward to obtain k kQ P ′ . Thermal conductivity As given in [START_REF] Ziman | I principi della teoria dei solidi[END_REF], the thermal current of electrons is the following:

( ) ∑ - Ω = k F k k k f ε ε v U 2 (39)
In (39), we find the velocity and energy of electrons. We have also the chemical potential, that we will consider as F ε . We can follow the same approach used for the phonons so that:

( ) ∑ Φ ∂ ∂ - Ω = k k k k F k k f ε ε ε 0 2 v U (40)
In the approach we have shown here, the equilibrium distribution of electrons 0 f is a spherical symmetric function. Then, for the evaluation of the thermal electronic conductivity we can use spherical coordinates k, α, θ. The electronic thermal conductivity is given by [START_REF] Ziman | Electrons and phonons: the theory of transport phenomena in solids[END_REF][START_REF] Ziman | I principi della teoria dei solidi[END_REF] In a future paper we will explain how to solve iteratively the coupled equations (21'') and (24''). Once the deviation functions of phononic and electronic distributions have been obtained the thermal phononic and the electronic thermal conductivities can be calculated from ( 16) and (41).

Figure 1 :

 1 Figure 1: An electron changes its momentum after it has gained or lost a phonon.

Figure 2 :

 2 Figure 2: A scattering electron-phonon

2

 2

Figure 3 :

 3 Figure 3: The other scattering electron-phonon process.

  :

  ), a reduced momentum k and the dimensionless deviation function φ had been used so that the dimensional factor turns out to be[START_REF] Ponasso | Determinazione del trasporto termico nei metalli con approccio iterativo all'equazione di Boltzmann[END_REF], parameter o U is linked to the potential used for calculating the lattice thermal conductivity [1,2].

  36 is the intrinsic probability that an electron can move from k to ′ k . This probability is conditioned by the fact that k state is occupied and ′ k is empty, due to the Pauli exclusion principle. The true probability is then (36)

	multiplied by	f	k	1 (	f	k	)

′ -