ITER

FESTIM implements these models

FESTIM

Finite element

H transport

Heat transfer

Complex geometries

The finite element method

Steady state weak formulation:

Transient weak formulation:

In FESTIM

Type of finite elements:

• P1 (CG1) for \( c_\mathrm{m} \)
• P1 or DG1 for \( c_\mathrm{t, i} \)

FESTIM: a code verified & validated

Experimental validation

Parametric optimisation technique

Cross-check with TMAP7

FESTIM: an open-source code

✅ More transparency

✅ More collaborations

✅ More flexibility

FESTIM workshop

A FESTIM course to learn how to run H transport simulations

FESTIM model

Boundary conditions

Convective flux: \( -\lambda \nabla T \cdot n = h \ (T-T_\mathrm{coolant}) \)

Recombination flux: \( -D \nabla c_\mathrm{m} \cdot n = K_r \ c_\mathrm{m}^2 \)

Incident heat flux: \( -\lambda \nabla T \cdot n = 10 \ \mathrm{MW \ m^{-2}} \)

Heat transfer coefficient: \( h = 70,000 \ \mathrm{W \ m^{-2} \ K^{-1}} \)

Coolant temperature: \( T_\mathrm{coolant} = 323 \ \mathrm{K} \)

Recombination coefficient: \( K_r =2.9 \times 10^{-14} \ \exp{(-1.92/k_B T)}\)  (\(\mathrm{m^4 \ s^{-1} }\) )

FESTIM model

Materials properties

Simulation parameters

Thermal properties

W

Cu

CuCrZr

• Maximum 77% difference

• Equivalent trap:
  • $$n=n_1 + n_2$$
  • $$E_p = \frac{E_{p,1}\ n_1 + E_{p,2} \ n_2}{n} $$

Approximations

Neglecting Trap 2

Assuming a continuous exposure allows to have a bigger stepsize

H inventory (\(\mathrm{m^{-2}}\) )

Low flux:

\( 5.0 \ \mathrm{MW \ m^{-2}} \)

\( 5.0 \times 10^{21}\ \mathrm{m^{-2} \ s^{-1}} \)

High flux:

\( 13 \ \mathrm{MW \ m^{-2}} \)

\( 1.6 \times 10^{22}\ \mathrm{m^{-2} \ s^{-1}} \)

Similar results were obtained with MHIMS

Monoblock thermal response

• High temperature gradient
• 2D temperature field

H concentration

• Penetration front

• High retention zone in colder regions

Permeation to coolant

DEMO monoblock

Influence of non-instantaneous recombination

wo gap and wo recombination, independent of thickness (2D)

Monoblock baking

To be published

Parametric study

Gaussian Process Regression (GPR)

\( T_\mathrm{surface} \) (K)

\( c_\mathrm{surface} \) (\(\mathrm{m}^{-3}\))

Gaussian process regression

https://juanitorduz.github.io/gaussian_process_reg/

Parametric study

Monoblock behaviour law

Rapid assessment of monoblock inventories

\( c_\mathrm{surface} \) (\(\mathrm{m}^{-3}\))

\( T_\mathrm{surface} \) (K)

+

+

+

+

+

+

+

+

+

+

A better behaviour law

Instantaneous recombination

✔️Non homogeneous surface temperature

✔️Non homogeneous surface concentration

❌Only works for instantaneous recombination

+

+

+

+

+

+

+

+

+

+

A better behaviour law

Non-instantaneous recombination

✔️Non homogeneous surface temperature

✔️Non homogeneous surface concentration

✔️Non-instantaneous recombination

❌3 independent variables

+

A better behaviour law

Automatic sampling

Converting plasma inputs...

Heat flux from SOLPS

Shot #2399

Plasma codes

Mass transport

Momentum

Ions energy \(T_\mathrm{i}\)

Electrons energy \(T_\mathrm{e}\)

\( n \): density

\( u \): velocity

Converting plasma inputs...

Implantation range and reflection coeff. are obtained from SRIM

Use of SRIM is questionable here...

The flux of He is \( \approx \) 1 % of that of D 

Safety limit \( = \) 700 g of tritium

\( \max \approx \) 2 % limit ✅

Assuming 50% T

\( \approx \) 1% limit ✅

ITER divertor inventory

Divertor H inventory (g) at \( t = 10^7 \, \mathrm{s}\)

Neglecting trap 2

Sources of helium

Neutronics monoblock simulations with OpenMC

500 MW of fusion power

\(\approx 10^{20} \mathrm{n/s} \)

500 MW of fusion power

\(\approx 10^{20} \mathrm{n/s} \)

Energy released by 1 DT reaction:

\(17.58 \ \mathrm{MeV} = 2.81 \times 10^{-12} \ \mathrm{J}\)

Tally (n,Xa) reaction

DT neutron source energy spectrum

He generation distribution

Standard deviation

Neutronic heating

Todo: assess influence of this on the thermals

Sources of helium

Tritium decay simulations with FESTIM

Neutronics monoblock simulations with OpenMC

Tritium decay

Decay constant \( \lambda = 1.77 \times 10^{-9} \ \mathrm{s}^{-1}\)

Decay source for Helium:    \( \lambda \sum  c_i \)

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{6}k_{1,i}^+ c_1 c_i - \langle k_b^+\rangle c_1 c_b \)

$$ \vdots $$

\( \frac{\partial c_6}{\partial t} = \nabla \cdot (D_6 \nabla c_6) - k^+_{1, 6} c_1 c_6 + k_{1,5}^+ c_1 c_{5} \)

\( \frac{\partial c_b}{\partial t} = k_{1,6}^+ c_1 c_{6} \)

\(\frac{\partial \langle i_b \rangle c_b}{\partial t} = 7 k_{1, 6}^+ c_1 c_6 + \langle k_b^+ \rangle c_1 c_b \)

Our clustering scheme

average clustering rate in bubbles

\( \langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)\)

average bubble radius

\( \langle r_b \rangle = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{\langle i_b \rangle}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}\)

Only 8 equations:

Let's consider the following reactions

Helium clustering (or emission)

Trap mutation

For each reaction

3-species model:

Rate constants:

N-species model:

Diffusion coefficients

Capture radii

Diffusion

Production

Reaction

Binding energy

Derivation of the forward reaction rate

\(n_i\): local concentration

\(c_i \): mean concentration

\( r_{i,j} = r_i + r_j \): coordinate where the reaction happens (capture radii)

$$c_i$$

in spherical coordinates

2 diffusive species from superpostion principle:

$$k_{i,j}^+ = 4\pi \ r_{i,j} \ (D_i + D_j) $$

Issue #1: Too many equations

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \frac{\partial c_{N}}{\partial t} =\) \(- k^+_{1, N} c_1 c_{N}\) \( + k_{1,N-1}^+ c_1 c_{N-1} \)

\( \frac{\partial c_{N+1}}{\partial t} =\) \(- k^+_{1, N+1} c_1 c_{N+1}\) \( + k_{1,N}^+ c_1 c_{N} \)

\( \frac{\partial c_{N+2}}{\partial t} =\) \(- k^+_{1, N+2} c_1 c_{N+2}\) \( + k_{1,N+1}^+ c_1 c_{N+1} \)

$$ \vdots $$

💪

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \frac{\partial c_{N+1}}{\partial t} =\) \(- k^+_{1, N+1} c_1 c_{N+1}\) \( + k_{1,N}^+ c_1 c_{N} \)

\( \frac{\partial c_{N+2}}{\partial t} =\) \(- k^+_{1, N+2} c_1 c_{N+2}\) \( + k_{1,N+1}^+ c_1 c_{N+1} \)

\( \frac{\partial c_{N+3}}{\partial t} =\) \(- k^+_{1, N+3} c_1 c_{N+3}\) \( + k_{1,N+2}^+ c_1 c_{N+2} \)

$$ \vdots $$

💪

Issue #1: Too many equations

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \sum\limits_{i=N+1}^{\infty} \frac{\partial c_i}{\partial t} = k_{1,N}^+ c_1 c_{N} \)

💪

Issue #1: Too many equations

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^\infty k_{1,i}^+ c_1 c_i \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N} \)

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

💪

Issue #1: Too many equations

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{N}k_{1,i}^+ c_1 c_i - \sum\limits_{i=N+1}^\infty k_{i, 1}^+c_i c_1 \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N} \)

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

💪

Issue #1: Too many equations

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{N}k_{1,i}^+ c_1 c_i - \langle k_b^+\rangle c_1 c_b \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N} \)

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = \left( \sum\limits_{i=N+1}^{\infty} k_ {i, 1}^+ c_i \right) / c_b\) : average clustering rate in bubbles

💪

Issue #1: Too many equations

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = \left( \sum\limits_{i=N+1}^{\infty} k_ {i, 1}^+ c_i \right) / c_b\)

\( = \left(\sum\limits_{i=N+1}^{\infty} 4 \pi D_1 (r_1 + r_i) c_i\right) / c_b \)

💪

average clustering rate in bubbles

Issue #2: what's the value of \( \langle k_b^+ \rangle \)?

Issue #2: what's the value of \( \langle k_b^+ \rangle \)?

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b\) : average bubble radius

\( \langle k_b^+ \rangle \)\( = \left( \sum\limits_{i=N+1}^{\infty} k_ {i, 1}^+ c_i \right) / c_b\)

\( = \left(\sum\limits_{i=N+1}^{\infty} 4 \pi D_1 (r_1 + r_i) c_i\right) / c_b \)

\( =4 \pi D_1 (r_1 + \langle r_b \rangle)\)

💪

average clustering rate in bubbles

\( r_i = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} n_{\mathrm{V},i} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}\)

💪

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)\) : average clustering rate in bubbles

\( \langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b\) : average bubble radius

Issue #3: what's the value of \( \langle r_b \rangle \)?

\( r_i = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{i}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}\)

\( n_{\mathrm{V},i} = i/4 \) : 4 He per vacancy

💪

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)\) : average clustering rate in bubbles

\( \langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b\) : average bubble radius

Issue #3: what's the value of \( \langle r_b \rangle \)?

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)\) : average clustering rate in bubbles

\( \langle r_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} r_ i c_i \right) / c_b\) : average bubble radius

\( \langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b\) : average He content in bubbles

\( n_{\mathrm{V},i} = i/4 \) : 4 He per vacancy

💪

\( \langle r_b \rangle = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{\langle i_b \rangle}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}\)

\( r_i = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{i}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}\)

Issue #3: what's the value of \( \langle r_b \rangle \)?

\( \langle r_b \rangle = r_{\mathrm{He}_0\mathrm{V}_1} + \left( \frac{3}{4\pi} \frac{a_0^3}{2} \frac{\langle i_b \rangle}{4} \right)^{1/3} - \left( \frac{3}{4\pi} \frac{a_0^3}{2} \right)^{1/3}\)

\(  \frac{\sum\limits_{i=N+1}^{\infty} i^{1/3} c_i } { c_b}  \approx \left( \sum\limits_{i=N+1}^{\infty} i c_i  / c_b \right)^{1/3} = \langle i_b \rangle ^{1/3}\)

When \( c_i \) has a narrow gaussian distribution (ie. \(\sigma / \mu \ll 1\) )

Sum approximation

\( \langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b\)

\( \langle i_b \rangle c_b=  \sum\limits_{i=N+1}^{\infty} i c_i \)

\(\frac{\partial \langle i_b \rangle c_b}{\partial t} = \sum\limits_{i=N+1}^{\infty} i \frac{\partial c_i}{\partial t}\)

💪

\(\frac{\partial \langle i_b \rangle c_b}{\partial t} = (N+1) k_{1, N}^+ c_1 c_N + \langle k_b^+ \rangle c_1 c_b \)

↓ trust me

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)\) : average clustering rate in bubbles

\( \langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b\) : average He content in bubbles

Issue #4: what's the value of \( \langle i_b \rangle \)?

\( \frac{\partial c_1}{\partial t} = \nabla \cdot (D_1 \nabla c_1) + P_1 - 2 k^+_{1, 1} c_1^2 - \sum\limits_{i=2}^{N-1}k_{1,i}^+ c_1 c_i - \langle k_b^+\rangle c_1 c_b \)

$$ \vdots $$

\( \frac{\partial c_i}{\partial t} = \nabla \cdot (D_i \nabla c_i) - k^+_{1, i} c_1 c_i + k_{1,i-1}^+ c_1 c_{i-1} \)

$$ \vdots $$

\( \frac{\partial c_b}{\partial t} = k_{1,N}^+ c_1 c_{N} \)

💪

\(\frac{\partial \langle i_b \rangle c_b}{\partial t} = (N+1) k_{1, N}^+ c_1 c_N + \langle k_b^+ \rangle c_1 c_b \)

\( c_b = \sum\limits_{i=N+1}^{\infty} c_i \) : bubble concentration

\( \langle k_b^+ \rangle = 4 \pi D_1 (r_1 + \langle r_b \rangle)\) : average clustering rate in bubbles

\( \langle i_b \rangle = \left( \sum\limits_{i=N+1}^{\infty} i c_i \right) / c_b\) : average He content in bubbles

Issue #4: what's the value of \( \langle i_b \rangle \)?

Assumption time!

• Only pure He clusters are mobile
• Trap-mutation from 7 He
• Grouping starts at \( i > 6 \)
• He/V ratio: 4
• No He emission from bubbles
• Pre-existing vacancies are neglected
• We don't solve for W self-interstitials

• Diffusion coefficients from Faney et al. Nucl. Fusion (2015)

• Dissociation energies from Becquart et al. J. Nucl. Mater. (2010)

Varying the grouping threshold

Results on a half-slab case

• Helium first diffuses rapidly into the bulk

• Mobile helium is consumed to form bubbles

\(c_i = 0 \)

He implantation

Mobile helium

Bubbles

Retention

$$\mathrm{m}^{-3}$$

Half-slab case

• "Semi-infinite" (0.6 mm)
• Helium source
  • ​100 eV
  • Gaussian distribution
    • ​ \( \mu =1.5 \mathrm{nm} \,      \sigma = 1.5 \mathrm{nm} \)
  • Flux \( 10^{22} \, \mathrm{m^{-2} \, s^{-1} }\)
• Temperature 1000 K

\(c_i = 0 \)

0.6 mm

Code comparison

Faney et al. Nucl. Fusion 2014

• 100 eV He
• Flux \( 10^{22} \mathrm{\, m^{-2} \, s^{-1}} \)
• Fluence \(5 \times 10^{25} \mathrm{\ m^{-2}} \)

30 nm

\( c_i = 0 \)

\( c_i = 0 \)

Discrepancies at high T due to different sets of dissociation energies

Dissociation energy sensitivity

Solid: +0

Dashed: + 0.5 eV

Dash-point: - 0.5 eV

Varying flux and temperature

\( c_\mathrm{He_1 \, ideal} = \frac{\varphi_\mathrm{imp} R_p}{D_1(T)} \)

Varying temperature and particle flux

Parametric study

1 h

\( \int c_b \langle i_b \rangle dx\)

He inventory in bubbles

Parametric study

1 h

\( \bar{\langle i_b \rangle} = \frac{\int c_b \langle i_b \rangle dx}{\int c_b dx} = \frac{\mathrm{inventory}}{\mathrm{total \, bubbles}}\)

Mean helium content in bubbles

Parametric study

 \(\int c_b dx\)

Total bubbles

1 h  efef

Two regimes can be identified

Nucleation

🡺Self trapping

🡺\( c_b \) increases

Growth

🡺\( \langle i_b \rangle \) increases

🡺\( \langle k_b^+ \rangle \) increases

\( \langle i_b \rangle \) is low

\( \langle k_b^+ \rangle \) is low

When \( c_b \) is large enough

Two regimes can be identified

\(\frac{\partial c_b}{\partial t} = k_{1, N}^+ c_1 c_N\)

\(\frac{\partial \langle i_b \rangle c_b}{\partial t} = (N+1) k_{1, N}^+ c_1 c_N + \langle k_b^+ \rangle c_1 c_b \)

\( \langle i_b \rangle \approx 7 \Leftrightarrow \langle k_b^+ \rangle \approx 0\)

\( \Leftrightarrow \frac{\partial \langle i_b \rangle c_b}{\partial t} \approx (N+1) k_{1, N}^+ c_1 c_N \)

\( \Leftrightarrow \langle i_b \rangle \frac{\partial c_b}{\partial t} + c_b \frac{\partial \langle i_b \rangle}{\partial t} \approx (N+1) k_{1, N}^+ c_1 c_N  \)

\( \Leftrightarrow \frac{\partial \langle i_b \rangle}{\partial t} \propto N +1 - \langle i_b \rangle \approx 0\)

\( c_b \gg c_N \)

\(\Leftrightarrow c_N \approx 0 \)

\(\Leftrightarrow \frac{\partial c_b}{\partial t} \approx 0 \)

\( \Leftrightarrow \frac{\partial \langle i_b \rangle c_b}{\partial t} \approx \langle k_b^+ \rangle c_1 c_b \)

\( \Leftrightarrow \frac{\partial \langle i_b \rangle}{\partial t} \approx \langle k_b^+ \rangle c_1\)

Nucleation regime

Growth regime

N cannot be lower than 6

Regime where intermediate clusters don't matter anymore

Ialovega's experiment

He exposure

W

Experimental conditions

Sample: \(100 \ \mathrm{\mu m} \) W

Pre-damage: \(75 \ \mathrm{eV}\) He at \( 2.3 \times 10^ {22} \ \mathrm{m^{-2} \ s^{-1}}\) and \(1053 \ \mathrm{K} \) for \(13 \ \mathrm{s}\)

Initial cleaning TDS after He implantation up to 870 K

D exposure: 250 eV at room temperature

Flux: \( 1.7 \times 10^ {16} \ \mathrm{m^{-2} \ s^{-1}}\)

Fluence: \( 4.5 \times 10^ {19} \ \mathrm{m^{-2}}\)

TDS temperature ramp: 1 K/s

Ialovega's experiment

Lack of control experiment

❌No initial TDS

❌No cycling

❌Not performed at the time of the experiment

→ Can only be compared qualitatively!

Ialovega's experiment

Ialovega's experiment

Correlation He release/D release

The model reproduced the experiment

Bubble-induced trap

• 4 traps including a bubble-induced trap

• TDS 1: density of available pre-existing traps is zero

• TDS 2-5: density of pre-existing traps  \( \approx 2\times10^{-3}\ \mathrm{at.fr.}\)

• \( f \) is unchanged \( 3 \times 10^{18} \ \mathrm{m^{-2}}\)