**3. The applicability of the Fourier heat equation for study of relativistic electron-solid interaction**

Taking into account the experimental data that were measured at the ALIN-10 linear acceler‐ ator from NILPRP [13] can be approximated a power distribution in the electron beam cross section as follows:

$$\mathbf{P}\{\quad\mathbf{x},\mathbf{y}\} \equiv P\_0 \cdot \mathbf{e}^{-\left(\mathbf{x}^{\frac{\gamma}{\epsilon} + \gamma^2/16\theta}\right)}.\tag{6}$$

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 ALIN-10 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:

$$\int\_{-2}^{2} \int\_{-8}^{8} e^{-(x^2 + y^2 / 16\theta)} d\mathbf{x} d\mathbf{y} = 13\pi \cdot E \text{rf} \{ 8/13 \} \cdot E \text{rf} \{ 2 \} \equiv 25.3 \tag{7}$$

**Figure 5.** The simulated intensity in the cross section of the beam delivered by ALIN-10.

**Figure 4.** The "geometrical" situation for **Figure 3**.

**electron-solid interaction**

section as follows:

418 Radiation Effects in Materials

and y coordinates.

The normalization condition is:

2 8

2 8


2 2

( /169)


**3. The applicability of the Fourier heat equation for study of relativistic**

Taking into account the experimental data that were measured at the ALIN-10 linear acceler‐ ator from NILPRP [13] can be approximated a power distribution in the electron beam cross

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 ALIN-10 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

> 13 [8 / 13] [2] 25.3 *x y e* p

=× × @ ò ò *dxdy Erf Erf* (7)

**P( x,y)**

2 2 ( /169) <sup>0</sup> . - + *x y*

@ × *P e* (6)

Taking into account that the average power in time of the electron beam is 62 W, we obtain *P*0 = 2.45 *W*.

**Figure 6.** The experimental spot on a plastic sample due to the electron beam after a 10 s exposure.

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.

**Figure 7.** Irradiation geometry for a small graphite sample.

The instantaneous energy loss of electrons passing through the material sample (∝ ∂E/ ∂z = f(E, M<sup>i</sup> )) 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]:

$$d\_{\rm max} \,\mathrm{[cm]} = (0.530 \cdot E\_{\rm max} \,\mathrm{[MeV]} - 0.106 \,\mathrm{[cm]}) / \,\rho \,\mathrm{[g/} \,\mathrm{cm}^3]. \tag{8}$$

Here, *d*max represents the maximum range of electron beam given in cm in a target of density *ρ* expressed in g/cm<sup>3</sup> . The energy *Emax* is introduced in expression of Eq. (8) in MeV and it refers to the maximum energy of the beam which determines the maximum range that the electrons of the beam can go through in a target. Based on this equation, a linear dependency of the energy absorbed in the material with the distance z is considered.

In our study it has used a graphite sample with *ρ* = 2.23 g/cm<sup>3</sup> giving *dmax* = 1.43 cm, below the length of our sample. This leads to the following absorption law:

$$E\_{\text{sub}}(\mathbf{z}) = \begin{cases} 6.23 - 4.36 \cdot \mathbf{z}, \text{ for } \mathbf{z} \le \mathbf{d}\_{\text{max}} \\ 0, \text{ for } \mathbf{z} > \mathbf{d}\_{\text{max}} \end{cases},\tag{9}$$

where: *Eabs* is in MeV and *z* is expressed in centimeters. Expression of Eq. (9) is the source term for the heat equation as it will be shown in the next section.

### **4. The Fourier heat equation**

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/

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

For electrons with energies greater than 2.5 MeV, their range in the target material is given by

Here, *d*max represents the maximum range of electron beam given in cm in a target of density

to the maximum energy of the beam which determines the maximum range that the electrons of the beam can go through in a target. Based on this equation, a linear dependency of the

> max 6.23 4.36 , ( ) , 0, *abs* <sup>ì</sup> -× £*<sup>z</sup>* <sup>=</sup> <sup>í</sup>

î > *z for d E z*

max max *d cm E MeV cm g cm* [ ] (0.530 [ ] 0.106 [ ]) / [ / ]. =× -

)) depends on material constants such as the mass density of the target, the atomic

3

(8)

giving *dmax* = 1.43 cm, below the

r

. The energy *Emax* is introduced in expression of Eq. (8) in MeV and it refers

max

*for z d* (9)

**Figure 7.** Irradiation geometry for a small graphite sample.

the formula put forward by Katz and Penfolds [18]:

energy absorbed in the material with the distance z is considered.

length of our sample. This leads to the following absorption law:

In our study it has used a graphite sample with *ρ* = 2.23 g/cm<sup>3</sup>

∂z = f(E, M<sup>i</sup>

420 Radiation Effects in Materials

[16, 17].

*ρ* expressed in g/cm<sup>3</sup>

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:

$$\frac{\varepsilon^2 \mathbf{T}}{\varepsilon \mathbf{x}^2} + \frac{\varepsilon^2 \mathbf{T}}{\varepsilon \mathbf{y}^2} + \frac{\varepsilon^2 \mathbf{T}}{\varepsilon \mathbf{x}^2} - \frac{1}{\gamma} \frac{\varepsilon \mathbf{T}}{\varepsilon \mathbf{y}} = -\frac{\mathcal{A}(\mathbf{X}\mathbf{Y}, \mathbf{Z}\mathbf{T})}{\mathbf{k}} \tag{10}$$

Here *T* represents the temperature variation relative to the initial sample temperature *T0*, which occurs during exposure to the electron beam, *A* is the energy deposited by electrons in the unit volume and unit time, *k* is the thermal conductivity and *γ* is the thermal diffusivity of the sample. We have *A*(*x, y, z, t*) *=Eabs* (*z*)*/* (*Vsample*× *t0*), where *Vsample=a . b . c* and *t0* is the irradiation time. The boundary conditions are:

$$
\left[\frac{\partial K\_x}{\partial X} + \frac{h}{k} \cdot K\_x\right]\_{x=-\frac{a}{2}} = 0, \quad \left[\frac{\partial K\_x}{\partial X} + \frac{h}{k} \cdot K\_x\right]\_{x=\frac{a}{2}} = 0,\tag{11a}
$$

$$
\left[\frac{\partial K\_{\boldsymbol{\gamma}}}{\partial \boldsymbol{Y}} + \frac{\boldsymbol{h}}{k} \cdot \boldsymbol{K}\_{\boldsymbol{\gamma}}\right]\_{\boldsymbol{y} = \frac{\boldsymbol{b}}{2}} = \mathbf{0}, \quad \left[\frac{\partial K\_{\boldsymbol{\gamma}}}{\partial \boldsymbol{Y}} + \frac{\boldsymbol{h}}{k} \cdot \boldsymbol{K}\_{\boldsymbol{\gamma}}\right]\_{\boldsymbol{y} = -\frac{\boldsymbol{b}}{2}} = \mathbf{0}, \tag{11b}
$$

$$
\left[\frac{\partial K\_{\varepsilon}}{\partial \mathcal{Z}} - \frac{h}{k} \cdot K\_{\varepsilon}\right]\_{\varepsilon=0} = 0, \quad \left[\frac{\partial K\_{\varepsilon}}{\partial \mathcal{Z}} + \frac{h}{k} \cdot K\_{\varepsilon}\right]\_{\varepsilon=\varepsilon} = 0,\tag{11c}
$$

where *a*, *b* and *c* are the geometrical lengths of the sample, along *X, Y* and *Z,* respectively, *h* is the heat transfer coefficient, *Kx*, *Ky*, *Kz* are the Eigen functions and *α<sup>i</sup>* , *β<sup>j</sup>*, *χ<sup>o</sup>* are their correspond‐ ing Eigen values, respectively.

The solution for the heat equation is:

$$
\Delta T(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) = \sum\_{i=1}^{n} \sum\_{j=1}^{n} \sum\_{o=1}^{o} I\_1(\mathbf{a}\_i, \boldsymbol{\beta}\_j, \boldsymbol{\chi}\_o) I\_2(\mathbf{a}\_i, \boldsymbol{\beta}\_j, \boldsymbol{\chi}\_o, t) K\_x(\mathbf{a}\_i, \mathbf{x}) K\_y(\boldsymbol{\beta}\_j, \mathbf{y}) K\_z(\boldsymbol{\chi}\_o, \mathbf{z}), \tag{12}
$$

where

$$\begin{split} I\_{1}(\boldsymbol{\alpha}\_{\circ},\boldsymbol{\beta}\_{\circ},\boldsymbol{\chi}\_{\circ}) &= \frac{1}{C\_{\boldsymbol{\iota}}C\_{\boldsymbol{\iota}}C\_{\boldsymbol{\iota}}} \prod\_{\begin{subarray}{c}\frac{a}{2} \\ -\frac{a}{2} \end{subarray}}^{\frac{b}{2}} \prod\_{\begin{subarray}{c}\frac{a}{2} \\ \frac{a}{2} \end{subarray}}^{b} K(\boldsymbol{\alpha}\_{\circ},\boldsymbol{\chi}) K\left(\boldsymbol{\beta}\_{\circ},\boldsymbol{\chi}\right) \cdot P(\boldsymbol{\chi},\boldsymbol{\chi}) \cdot \operatorname{d\boldsymbol{x}} \operatorname{d\boldsymbol{y}} \Big| \operatorname{\boldsymbol{\Gamma}}\_{\circ}\left(\boldsymbol{\chi}\_{\circ},\boldsymbol{\varepsilon}\right) \Big(\boldsymbol{\delta}.23-4.36\cdot\boldsymbol{\varepsilon}\Big) \operatorname{d\boldsymbol{z}}, \\ I\_{2}(\boldsymbol{\alpha}\_{\circ},\boldsymbol{\beta}\_{\circ},\boldsymbol{\chi}\_{\circ},\boldsymbol{\epsilon}) &= \frac{1}{\boldsymbol{\alpha}\_{\circ}^{2}+\boldsymbol{\beta}\_{\circ}^{2}+\boldsymbol{\chi}\_{\circ}^{2}} [\operatorname{l} - e^{-\boldsymbol{\gamma}\_{\circ\circ}^{\boldsymbol{\beta}}\boldsymbol{\epsilon}} - (\operatorname{l} - e^{-\boldsymbol{\gamma}\_{\circ\circ}^{\boldsymbol{\beta}}\boldsymbol{\epsilon}\cdot\boldsymbol{\epsilon}\_{\circ}}) h(\boldsymbol{\epsilon} - \boldsymbol{t}\_{o})], \end{split} \tag{13}$$

with

$$
\chi^2\_{\rm jo} = \chi(\alpha\_i^2 + \beta\_j^2 + \chi\_o^2) \tag{14}
$$

Here, *Ci , Cj , Co* are normalization constants. The Eigen functions determined for the heat equation (10) with boundary conditions (11a–c) have the following explicit expressions:

$$K\_{\underline{x}}(\boldsymbol{\alpha}\_{\cdot}, \mathbf{x}) = \cos(\boldsymbol{\alpha}\_{\cdot} \cdot \mathbf{x}) + (h \,/ \, k \boldsymbol{\alpha}\_{\cdot}) \cdot \sin(\boldsymbol{\alpha}\_{\cdot} \cdot \mathbf{x}) \tag{15a}$$

$$K\_{\boldsymbol{\cdot}}(\boldsymbol{\beta}\_{/}, \boldsymbol{\text{y}}) = \cos^{2}\left(\boldsymbol{\beta}\_{/} \cdot \boldsymbol{\text{y}}\right) + (\boldsymbol{h} \,/ \, k \, \boldsymbol{\beta}\_{/}) \sin\left(\boldsymbol{\beta}\_{/} \cdot \boldsymbol{\text{y}}\right) \tag{15b}$$

$$K\_{\preceq}(\chi\_0, z) = \cos^{-1}\left(\chi\_0 \cdot z\right) + (h \,/\, k\,\chi\_0)\sin\left(\chi\_0 \cdot z\right) \tag{15c}$$

The Eigen values can be determined from the boundary equations:

$$\text{12\cot \ (a\_i a)} = \frac{a\_i k}{h} - \frac{h}{k \text{ } a\_i} \tag{16a}$$

$$2\cot\left(\beta\_{\rangle}b\right) = \frac{\beta\_{\rangle}k}{h} - \frac{h}{k\beta\_{\rangle}},\tag{16b}$$

$$\text{2\textbullet } \left( \chi\_0 c \right) = \frac{\chi\_0 k}{h} - \frac{h}{k \text{ } \chi\_0}. \tag{16c}$$

#### **5. Experiment and simulations**

For small samples the thermal field distribution is determined by two important factors: the energy denoted by the term *A* in Eq. (10) released by the electrons during the time and volume unit within the target, and the heat transfer constant *h*, which shows how fast the target loses its heat to the surrounding environment depending on the material of the target, pressure of the surrounding gas and magnitude of the contact surface between the target and the envi‐ ronment. The temperature increases with the absorbed energy *A*, and with the decrease of *h.* There are in general three types of heat transfer by: (i) radiation, (ii) convection and (iii) conduction. In the present case, the heat lost by conduction is neglected as the sample is fixed on two Teflon claws. The heat rate lost by radiation may be written as *<sup>σ</sup>* <sup>⋅</sup>*<sup>E</sup>* <sup>⋅</sup> (*<sup>T</sup>* <sup>4</sup> <sup>−</sup>*T*<sup>0</sup> 4 ), which in linear approximation is given by 4*σ* ⋅*T*<sup>0</sup> <sup>3</sup> <sup>⋅</sup>*<sup>E</sup>* <sup>⋅</sup> (*<sup>T</sup>* <sup>−</sup>*T*0)≡*hrad* <sup>⋅</sup> (*<sup>T</sup>* <sup>−</sup>*T*0). Here, *hrad* =4⋅*<sup>σ</sup>* <sup>⋅</sup>*T*<sup>0</sup> <sup>3</sup> ⋅*E*, where *T0 =* 298 K, *σ* = 5.6 × 10−8 Wm−2K−4 is the Stephan Boltzmann constant*,* and *E* is the thermal emissivity which for polished metallic surfaces can be taken as 0.05. We obtain *hrad* =3⋅10−<sup>7</sup> *W mm*−<sup>2</sup> *K* <sup>−</sup><sup>1</sup> . The heat rate loss by convection when the sample is in air obeys a power law given by: 20⋅10−<sup>9</sup> (*<sup>T</sup>* <sup>−</sup>*T*0)5/4 *W mm*−<sup>2</sup> . This expression can be further made linear: 20⋅10−<sup>9</sup> (*<sup>T</sup>* <sup>−</sup>*T*0)1/4(*<sup>T</sup>* <sup>−</sup>*T*0) *W mm*−<sup>2</sup> <sup>=</sup> *hconv* <sup>⋅</sup> (*<sup>T</sup>* <sup>−</sup>*T*0) *W mm*−<sup>2</sup> . In consequence we can conclude: *hconv* <sup>≅</sup>0.8⋅10−<sup>7</sup> *W mm*−<sup>2</sup> *K* <sup>−</sup><sup>1</sup> , where we have considered: *T* − *T*0 = 300*K*. The total heat transfer coefficient is: *htotal* <sup>=</sup>*hrad* <sup>+</sup> *hconv* <sup>≅</sup>3.8⋅10−<sup>7</sup> *W mm*−<sup>2</sup> *K* <sup>−</sup><sup>1</sup> , which corresponds to the sample sur‐ rounded by air. For a sample in vacuum we neglect cooling by convection *hconv* = 0, and therefore: *htotal* <sup>=</sup>*hrad* <sup>≅</sup>3⋅10−<sup>7</sup> *W mm*−<sup>2</sup> *K* <sup>−</sup><sup>1</sup> .

( ) ( ) ( ) ( ) ( )( )

ò ò ò

*I K x K y P x y dxdy K z z dz*

<sup>1</sup> , , , ,, , 6.23 4.36 ,

= × × - ×

 g 0

c

= ++ (14)

= ×+ × × *<sup>i</sup> i i* (15a)

= ×+ ( ) *<sup>j</sup>* ( / )sin *j j* ( ) × (15b)

= - (16a)

= - (16b)

= - (16c)

0 0 00 = ×+ ( ) ( / )sin( ) × (15c)

(13)

*c*

2 2

*a b*

2 2


*i jo*

abc

*i jo a b*

*CC C*

<sup>1</sup> ( , , ,) [1 (1 ) ( )], *ijo ijo o*

*i jo o*

g ga

*I t e e ht t*

a

b

c

**5. Experiment and simulations**

a

*i jo i j o o*

 b

g

<sup>=</sup> - -- - + +

2 222

1

with

Here, *Ci*

*, Cj*

ab c

422 Radiation Effects in Materials

ab c 2 2


*t t t*

2 222 ( ) *ijo i j o*

equation (10) with boundary conditions (11a–c) have the following explicit expressions:

( , ) cos( ) ( / ) sin( ) *K x x hk x x i*

*K y y hk y y j* ( , ) cos

*K z z hk z <sup>z</sup>* ( , ) cos

*k h <sup>a</sup> h k* a

*<sup>k</sup> <sup>h</sup> <sup>b</sup> h k* b

For small samples the thermal field distribution is determined by two important factors: the energy denoted by the term *A* in Eq. (10) released by the electrons during the time and volume

2cot ( ) , *<sup>j</sup> j*

> ( ) <sup>0</sup> 0

2cot . *k h <sup>c</sup> h k* c

 a

> b

 c

2cot ( ) *<sup>i</sup> i*

a

b

c

The Eigen values can be determined from the boundary equations:

 b c

( )

*, Co* are normalization constants. The Eigen functions determined for the heat

 aa

> bb

 cc

*i*

a

*j*

0

c

b

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 thermo‐ couple 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 low-power 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 *Tsample* is the temperature of the sample in Celsius. The irradiation time was 36 s, during which *Tsample* increased continuously. When the irradiation stopped, the temperature started to decrease and the sample cooled down. One can observe that *Tsample* dropped relatively fast during the first 20 to 30 s of cooling down process and at a much slower rate after about 100 s.

**Figure 8.** The experimental results for temperature variation in time at the point (*x*, *y*, *z*) = (5, 0, 7.5 mm).

**Figure 9.** The experimental results for temperature variation in time at the point (*x*, *y*, *z*) = (0, 0, 15 mm).

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 non-Fourier models.

**Figure 10.** The experimental (dotted line) and simulation (continuous line) results for temperature versus time at the point: (*x*, *y*, *z*) = (5, 0, 7.5 mm), during 36 s irradiation time.

**Figure 8.** The experimental results for temperature variation in time at the point (*x*, *y*, *z*) = (5, 0, 7.5 mm).

424 Radiation Effects in Materials

**Figure 9.** The experimental results for temperature variation in time at the point (*x*, *y*, *z*) = (0, 0, 15 mm).

the future simulations being the consideration of non-Fourier models.

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

**Figure 11.** The experimental (dotted line) and simulation (continuous line) results for temperature versus time at the point: (*x*, *y*, *z*) = (0, 0, 15 mm), during 36 s irradiation time.

#### **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 *w* = 1 mm, the Lambert Beer absorption law reads:

$$I\_{nm}(\mathbf{x}, \mathbf{y}) = I\_{0mn}(\mathbf{x}, \mathbf{y}) \times \text{Exp}[-az\mathbf{y}] \tag{17}$$

$$I\_{nn}(\mathbf{x}, \mathbf{y}) = I\_{0nn} \left[ H\_n(\frac{\mathfrak{S}\_{\mathbf{x}}}{\mathfrak{s}}) H\_n(\frac{\mathfrak{S}\_{\mathbf{y}}}{\mathfrak{s}}) \times \exp\left[ -\left( \frac{\mathfrak{s}^2 \ast \mathfrak{y}^2}{\mathfrak{s}^2} \right) \right] \right]^2 \tag{18}$$

Here, *Imn*, *I0mn*, *Hm*, *Hn* are lasers intensity in the mode {*m*,*n*}, maximum laser intensity in the mode {*m*,*n*}, the Hermite polynomial of order *m*, respectively of order *n*. We are dealing with CO2 lasers in *cw* mode.

We assumed that one is in the case: *m* = 0 and *n* = 0 in Eq. (18).

For electron irradiation, one should apply, in the particular case of *W*, the empirical absorption Tabata-Ito-Okabe law [19]; it is also considered a Gaussian profile of the electron beam.

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 *Z* of density *ρ* expressed in g/cm3 is:

$$d\_{\max} = a\_1 \left[ \begin{pmatrix} 1/a\_2 \\ \end{pmatrix} \text{ ln } \begin{pmatrix} 1+a\_2\tau \\ \end{pmatrix} \begin{pmatrix} -a\_3\tau \\ \end{pmatrix} \begin{pmatrix} 1+a\_4\tau^{\*\beta} \\ \end{pmatrix} \right] / \rho \tag{19}$$

Here, *τ* is a unitless ratio between the kinetic energy of the electron beam (express in MeV) and the electron rest mass energy.

$$a\_1 = b\_{\,1}A \wedge Z^{\nu 2}, \, a\_2 = b\_{\,2}Z, \, a\_3 = b\_{\,4} - b\_{\,3}Z, \, a\_4 = b\_{\,6} - b\_{\,7}Z, \, a\_5 = b\_{\,8} \, | \, Z^{\nu 9} \tag{20}$$

The constants *bi* are given in Table 2:


**Table 2.** The constants *bi* in Tabata-Ito-Okabe formula.

Following the formalism from our previous paper, one may write:

$$E\_{\text{abs}}(z) = \begin{cases} 6.23 - 49.84 \cdot z, for \, z \le 0.125 \text{ } cm \\ 0, for \, z > 0.125 \text{ } cm \end{cases} \tag{21}$$

where *z,* E are expressed in cm and MeV, respectively.

Here, *Imn*, *I0mn*, *Hm*, *Hn* are lasers intensity in the mode {*m*,*n*}, maximum laser intensity in the mode {*m*,*n*}, the Hermite polynomial of order *m*, respectively of order *n*. We are dealing with

For electron irradiation, one should apply, in the particular case of *W*, the empirical absorption Tabata-Ito-Okabe law [19]; it is also considered a Gaussian profile of the electron beam.

The maximum propagation length of an electron beam in cm in targets with high *Z* of density

( )( ) ( ) <sup>5</sup> max 1 2 2 3 <sup>4</sup> 1 / ln 1 / 1 / *<sup>a</sup> da a a a a* = +- + é ù t

Here, *τ* is a unitless ratio between the kinetic energy of the electron beam (express in MeV)

2 9 1 1 2 3 3 45 4 67 5 8 / , , , , / *<sup>b</sup> <sup>b</sup> a bA Z a bZ a b bZ a b bZ a b Z* = = =- =- = (20)

 t

 trë û (19)

We assumed that one is in the case: *m* = 0 and *n* = 0 in Eq. (18).

The total power of the laser and electron beams is around of 200 W.

CO2 lasers in *cw* mode.

426 Radiation Effects in Materials

*ρ* expressed in g/cm3

The constants *bi*

*i bi* 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

**Table 2.** The constants *bi*

is:

are given in Table 2:

in Tabata-Ito-Okabe formula.

Following the formalism from our previous paper, one may write:

6.23 49.84 , 0.125 ( ) 0, 0.125 *abs z for z cm E z for z cm* <sup>ì</sup> -× £ <sup>=</sup> <sup>í</sup>

<sup>î</sup> <sup>&</sup>gt; (21)

and the electron rest mass energy.

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: (*α* = 8 × 10−2cm−1)) and Tabata-Ito-Okabe (Eq. (21)) law are quite different.

**Figure 12.** Temperature field on *W* surface after 25 s irradiation with an IR laser beam of 200 W power. In the integral transform technique *T* is the variation temperature rather than the absolute temperature.

**Figure 13.** Temperature field on *W* surface after 25 s electron beam irradiation at 200 W power and 6.5 MeV energy.

**Figure 14.** Temperature field on graphite surface after 20 s irradiation with an IR laser beam of 250 W power. In the integral transform technique *T* is the variation temperature rather than the absolute temperature.

**Figure 15.** Temperature field on graphite surface after 20 s electron beam irradiation at 250 W power and 6.5 MeV en‐ ergy.

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 (*α* = 10−1cm−1) and Katz and Penfolds (Eq. (9)) law are quite different. This implies that initial supposition regarding the similarity between laser and electron irradiation at relatively high power is fully justified. For the near future shall be developed non-Fourier models in order to be more accurate in the tentative to explain the experimental data [20, 21].
