Thermal parameters of Cu.
Abstract
The investigation of the thermal field distribution in a material sample irradiated by a laser beam or an electron beam with the energy of a few MeV appears as a demand for all kinds of experiments that involve irradiation. When investigating the effects of accelerated electrons on a target, it is necessary to figure out the temperature rise in the target. Also during irradiation with laser beams, it is important to know the thermal behavior of the target. A parallel between laser and electron beam irradiation is also made. The results are very interesting. Also, a very interesting case of cluster nanoparticles (20–100 nm; inserted in a Cu surface) heated with a laser beam is tacking in to account.
Keywords
 laser
 electron (beam)
 irradiation
 nanoparticle
 interaction
 W and C
1. Introduction
There are many methods for evaluating the thermal fields in radiationmatter interaction, but most of them require a complex mathematical handling [1–4]. This chapter presents a direct and powerful mathematical approach to compute the thermal field for electron beammaterial and lasersample interaction. The solving procedure is based on applying the integral transform technique which was developed in the 1960s, by the Russian School of Theoretical Physics [5]. As an example, the integral transform technique is used in [3] to solve the heat equation for a sample exposed to an infrared laser beam in order to find the solution for the absorption coefficient, which is then checked experimentally. It should be pointed out straightforwardly that the heat equation has the same form in the case of irradiation with a laser beam or an electron beam, at sufficiently large beam intensities [6, 7]. There is, however, a disadvantage in this model as it cannot take into account simultaneously the variation with temperature of several thermal parameters involved in the interaction like, for example, the thermal conductivity or thermal diffusivity. In consequence, the model should be regarded as a first approximation of the thermal field. The main advantage is that the solution is a series which converges rapidly. It is important to note that the integral transform technique, as it will be shown in the next sections, belongs to the “family” of Eigen functions and Eigen valuesbased methods.
2. The applicability of the Fourier heat equation for study of lasernano particles clusters interaction
Light has always played a central role in the study of physics, chemistry and biology. In the past century, a new form of light, laser light, has provided important contributions to medicine, industrial material processing, data storage, printing and defense [8] applications. In all these areas of applications, the lasersolid interaction played a crucial role. The theory of heat conduction was naturally applied to explain this interaction since it was well studied for a long time [9]. For describing this interaction, the classical heat equation was used in a lot of applications. Apart of some criticism [10], the heat equation still remains one of the most powerful tools in describing most thermal effects in lasersolid interactions [11]. In particular, the heat equation can be used for describing both of light interaction with homogeneous and inhomogeneous solids. In the literature, thus a special attention was given to cases of light interaction with multilayered samples and thin films.
It is undertaken in the following treatment that it has a solid consisting of a layer of a metal such as Au, Ag, Al or Cu, respectively. Assuming that only a photothermal interaction takes place, and that all the absorbed energy is transformed into heat, the linear heat flow in the solid is fully described by the heat partial differential equation, Eq. (1):
where: T(
where:
with
Here,
The coefficients
It has used the thermal parameters of the Cu sample as given in Table 1.





Cu  3.95  1.14  7.7·10^{5} 
It has used the heat equation for a configuration where the layers are assumed to have a thickness of 1 mm onto which are included clusters of nanospheres. The heat term for such a system can be represented by the following equation:
where,
For the simulation, we have to consider:
Inserting groups or clusters of nanoparticlesclusters on top of a layer exposed to irradiation gives a detectable increase of temperature in comparison with the bulk material in pure form. This result can be seen in the following simulations.
For
The present chapter continues the numerous ideas developed in the past few years with the integral transform technique applied to classical Fourier heat equation [1, 2].
From practical point of view, consider the formula (2), that
In conclusion, it is considered that the method of integral transform technique is a serious candidate in competition with: Born approximation, Green function method or numerical methods. In Figure 44 is represented the “geometrical” situation for Figure 3.
The nanoparticlesclusters should be of the order of magnitude of 20 nm, which is the limit of availability of Fourier model [12].
3. The applicability of the Fourier heat equation for study of relativistic electronsolid interaction
Taking into account the experimental data that were measured at the ALIN10 linear accelerator from NILPRP [13] can be approximated a power distribution in the electron beam cross section as follows:
Therefore, it is supposing that the irradiation source emits relativistic electrons with an asymmetric Gaussian distribution [14, 15]. This intensity distribution of the accelerated electron beam is represented in Figure 5 and is obtained using the experimental data shown in Figure 6. The approximation of the ratio between the two planar coordinates of the electron beam spot is obtained from Figure 6. This shows the experimental transverse profile of the beam at the exit of the ALIN10 accelerator. The average beam power is 62 W for a beam current of 10 μA. The measured beam dimensions in the transverse plane are 14 and 2 mm on the x and y coordinates.
The normalization condition is:
Taking into account that the average power in time of the electron beam is 62 W, we obtain
The geometry of the simulation is shown in Figure 7 where the electron beam propagates along the z axis and is incident on a graphite sample with dimensions 10 × 10 × 15 mm. The geometry is described in Cartesian coordinates and the transversal plane of the beam is the xy plane.
The instantaneous energy loss of electrons passing through the material sample (∝ ∂E/∂z = f(E, M_{i})) depends on material constants such as the mass density of the target, the atomic number, the classical radius of the electron and some empirical numerical constants, and does not depend explicitly on the distance travelled, in our case the z direction. When calculating the stopping power, three physical phenomena are taken into account: the secondary electron emission, the polarization of the target and the effect of magnetic field on the incident beam [16, 17].
For electrons with energies greater than 2.5 MeV, their range in the target material is given by the formula put forward by Katz and Penfolds [18]:
Here,
In our study it has used a graphite sample with
where:
4. The Fourier heat equation
The goal to establish the thermal field during electron beam irradiation is not a new issue. For achieving it in the irradiation geometry described in Figure 7, the heat equation in Cartesian coordinates is the starting point:
Here
where
The solution for the heat equation is:
where
with
Here,
The Eigen values can be determined from the boundary equations:
5. Experiment and simulations
For small samples the thermal field distribution is determined by two important factors: the energy denoted by the term
The temperature of a rectangular graphite sample was measured using two thermocouples attached on the lateral and back sides of the sample, respectively. No thermocouple was mounted on the face directly exposed to the incident electron beam to prevent the obstruction of the beam and to protect the sensor. The beam was incident on the square face of the sample with a cross section of 10 × 10 mm and propagated along its length of 15 mm. Each thermocouple consisted in a small size junction with a rounded head of about 1 mm in diameter and was connected to a FLUKA unit which displayed in real time the measured temperature during irradiation. The acquisition of temperature time series was done simultaneously with the two thermocouples. The electron beam exited the vacuum structure of a lowpower LINAC through an aluminum window and was incident on the sample placed in air, at normal pressure and temperature. The irradiation time was limited to a few tens of seconds such that no damages would be induced in the sample, its support and the thermocouples. Longer irradiation times of over 50 s could easily induce temperatures well above 500°C.
Figures 8 and 9 present the evolution in time of the temperature on the surface of the graphite sample at two locations (x, y, z) given by (5, 0, 7.5 mm) and (0, 0,15 mm), respectively. Both locations were conveniently chosen to be at the center of the sample faces and coincided with the position of the sensors. In the figures
In Figure 8 a slightly higher peak in the temperature with about 30 degrees is observed compared to Figure 9. The reason is that the position on the sample surface at which data presented in Figure 8 has been recorded was closer to the heating source. Figures 10 and 11 present the comparison of experimental data (dotted line) with our simulations (continuous line) according to integral transform technique. The agreement is quite well, an improving of the future simulations being the consideration of nonFourier models.
6. Laser versus electron interaction in w bulk target processing
As it is known, in the case of a Gaussian laser beam having a waist of
Here,
We assumed that one is in the case:
For electron irradiation, one should apply, in the particular case of
The total power of the laser and electron beams is around of 200 W.
The maximum propagation length of an electron beam in cm in targets with high
Here,
The constants
1.  0.2335 
2.  1.209 
3.  1.78 × 10^{−4} 
4.  0.9891 
5.  3.01 × 10^{−4} 
6.  1.468 
7.  1.180 × 10^{−2} 
8.  1.232 
9.  0.109 
Following the formalism from our previous paper, one may write:
where
In Figures 12 and 13, we present the variation of thermal fields under laser and electron irradiation at the same continuous power (200 W) after an exposure time of 25 s. It can be observed that the thermal fields are almost identical, despite the fact that Lambert Beer law (Eq. (17) with: (
In Figures 14 and 15 we present the variation of the thermal fields under laser and electron irradiation at the same continuous power (250 W) after an exposure time of 20 s. Noticeably, the thermal fields are almost identical, also despite the fact that Lambert Beer law (
Acknowledgments
This work was supported by the NUCLEU project (M. Oane) and MERANET MAGPHOGLAS/20132015—(R.V. Medianu).
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