 A three-component Hydrogen (H2) plasma at a background gas pressure of 10−2 Pa and gas
temperature 𝑇𝑔=600K has an electron number density of 𝑛𝑒=1011 𝑐𝑚−3
and electron temperature of
𝑇𝑒=4 eV. Find in the lecture notes (Section 3: Plasma Fundamentals) and/or calculate the following
properties of this plasma:
A. Debye radius
B. Gas number density (use ideal gas equation of state for the background gas)
C. Elastic collisional cross-section between the electrons and the gas atoms
D. Mean free path of the electron-gas elastic collisions

Let's go through the required calculations step-by-step for each part:

### Given Data:
- **Background gas pressure**, \( P = 10^{-2} \, \text{Pa} \)
- **Gas temperature**, \( T_g = 600 \, \text{K} \)
- **Electron number density**, \( n_e = 10^{11} \, \text{cm}^{-3} \) = \( 10^{17} \, \text{m}^{-3} \)
- **Electron temperature**, \( T_e = 4 \, \text{eV} \)

#### **A. Debye Radius**

The Debye radius (\( \lambda_D \)) is given by:

\[
\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}
\]

Where:
- \( \varepsilon_0 \) is the permittivity of free space, \( \varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \),
- \( k_B \) is the Boltzmann constant, \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \),
- \( T_e \) is the electron temperature in Kelvin. To convert \( T_e \) from eV to K, use \( T_e (\text{K}) = T_e (\text{eV}) \times 11604 \),
- \( e \) is the elementary charge, \( e = 1.602 \times 10^{-19} \, \text{C} \),
- \( n_e \) is the electron number density in \( \text{m}^{-3} \).

First, convert the electron temperature to Kelvin:
\[
T_e = 4 \, \text{eV} \times 11604 \, \text{K/eV} = 46416 \, \text{K}
\]

Now we can calculate \( \lambda_D \).

#### **B. Gas Number Density**

Use the ideal gas law to find the number density (\( n_g \)):

\[
n_g = \frac{P}{k_B T_g}
\]

Where:
- \( P = 10^{-2} \, \text{Pa} \),
- \( T_g = 600 \, \text{K} \),
- \( k_B = 1.38 \times 10^{-23} \, \text{J/K} \).

#### **C. Elastic Collisional Cross-Section between Electrons and Gas Atoms**

The elastic collisional cross-section (\( \sigma \)) between electrons and hydrogen atoms can be approximated by:

\[
\sigma \approx 10^{-20} \, \text{m}^2
\]

This is a typical value for electron-hydrogen collisions. For more precise calculations, detailed quantum mechanical computations would be required.

#### **D. Mean Free Path of Electron-Gas Elastic Collisions**

The mean free path (\( \lambda \)) is given by:

\[
\lambda = \frac{1}{n_g \sigma}
\]

Where:
- \( n_g \) is the gas number density,
- \( \sigma \) is the collisional cross-section.

### Let's perform these calculations now.Here are the results for each part of the problem:

#### **A. Debye Radius**
The Debye radius (\( \lambda_D \)) is approximately \( 4.70 \times 10^{-5} \, \text{m} \) or \( 47 \, \mu\text{m} \).

#### **B. Gas Number Density**
The gas number density (\( n_g \)) is approximately \( 1.21 \times 10^{18} \, \text{m}^{-3} \).

#### **C. Elastic Collisional Cross-Section**
The cross-section is given as \( \sigma \approx 10^{-20} \, \text{m}^2 \) (approximated value for electron-hydrogen collisions).

#### **D. Mean Free Path of Electron-Gas Elastic Collisions**
The mean free path (\( \lambda \)) is approximately \( 82.8 \, \text{m} \).

If you have further questions or need more detailed explanations, let me know!

### Here are 5 related questions to expand your understanding:
1. How does the electron temperature affect the Debye radius in a plasma?
2. What happens to the mean free path if the gas pressure increases?
3. How would changing the gas species from hydrogen to helium affect the collisional cross-section?
4. Why is the Debye radius an important parameter in plasma physics?
5. What role does the mean free path play in determining the behavior of plasma?

**Tip:** The mean free path is inversely proportional to the gas number density; as the gas density increases, the mean free path decreases.

A three-component Hydrogen (H2) plasma at a background gas pressure of 10−2 Pa and gas temperature 𝑇𝑔=600K has an electron number density of 𝑛𝑒=1011 𝑐𝑚−3 and electron temperature of 𝑇𝑒=4 eV. Find in the lecture notes (Section 3: Plasma Fundamentals) and/or calculate the following properties of this plasma: A. Debye radius B. Gas number density (use ideal gas equation of state for the background gas) C. Elastic collisional cross-section between the electrons and the gas atoms D. Mean free path of the electron-gas elastic collisions.

This problem focuses on calculating key plasma properties, including the Debye radius, gas number density using the ideal gas law, elastic collisional cross-section, and the mean free path of electron-gas collisions. Suitable for university-level students studying plasma physics.

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