Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 1 2. Basic properties of plasma. Conditions affecting basic properties of plasma • Quantum degeneracy • Electrostatic coupling • Quasi-neutrality • Debye shielding • Collisionality Quantum degeneracy Quantum effects become significant when the plasma density is high. Plasma with relatively low density can be described by the classical physics and is called classical plasma. Consider conditions for quantum and classical plasmas. If the distance between particles d is much less than the quantum (De Broglie) wavelength, λ D , then the quantum effects are important.
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Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 1
2. Basic properties of plasma.
Conditions affecting basic properties of plasma
• Quantum degeneracy
• Electrostatic coupling
• Quasi-neutrality
• Debye shielding
• Collisionality
Quantum degeneracy
Quantum effects become significant when the plasma density is high.
Plasma with relatively low density can be described by the classical physics
and is called classical plasma.
Consider conditions for quantum and classical plasmas. If the distance
between particles d is much less than the quantum (De Broglie) wavelength,
λD, then the quantum effects are important.
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 2
Conditions for quantum and classical plasma:
d ≪ λD - quantum plasma
d ≫ λD - classical plasma
A characteristic distance between particles can be estimated as: d ≃ n−1/3.
The De Broglie wavelength is: λB ≃ h
p, where h is the Planck constant;
and p is the particle momentum: p =√2mE =
√2mT ,
temperature T in this formula is measured in the energy units, eV.
Then, in term of temperature the De Broglie wavelength is: λD ≃ h√2mT
.
If n−1/3 ≫ h√2mT
or T ≫ h2n2/3
2m
then the plasma is classical.
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 3
The quantum effects first become significant for electrons.
Electron temperature Te =12me〈v2e〉,
Ion temperature Ti =12mi〈v2i 〉.
Electron and ion velocities: ve =
√
2Te
meand vi =
√
2Ti
mi.
If a one-temperature plasma Te = Ti = T and vi =
√
me
mive.
Ions move slower than electrons.
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 4
Problem 2. Plasma temperature is often measured in eV:
1eV=1.6 · 10−12 erg=11600 K, and plasma density is measured in cm−3
(the number of particles in cubic cm).
a) Derive the practical formula for the condition of classical plasma:
TeV ≫ 3.5 · 10−16n2/3cm−3 .
b) Estimate significance of quantum effects in the Sun’s core
(T = 1.5× 107K, ρ = 150 g/cm3).
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 5
Electrostatic coupling
Compare the kinetic and electrostatic energies of particles in a fully
ionized plasma.
Kinetic energy per particle is: EK ≃ T .
Electrostatic energy per particle is:
EE ≃ e2
2d≃ e2
2n−1/3
(
SI :EE ≃ e2
8πǫ0n−1/3
)
When EK ≫ EE the plasma is weakly coupled or ideal.
The condition for the ideal plasma is:
T ≫ e2n1/3
2
(
SI :T ≫ e2n1/3
8πǫ0
)
Practical formula:
TeV ≫ 0.72 · 10−7n1/3cm−3
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 6
If EK ≪ EE , then the electrostatic interactions dominate, and such
plasma is called strongly coupled plasma.
In quantum plasma the kinetic energy is not equal to temperature.
When the density increases, the plasma becomes degenerate, so
that there cannot be more than two electrons at the same point
with the spins ”up” and ”down”.
If the distance between particles d ∼ n−1/3 then each electron is
confined in a box of size ∆x ∼ d ∼ n−1/3.
According to the uncertainty principle a particle confined in a box
of size ∆x has a momentum p ∼ h
∆x, and the kinetic energy is:
EK = p2/m. Then,
EK ≃ h2n2/3
2me
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 7
This energy is called ⁀the Fermi energy of degenerate electron gas.
This is the lowest plasma energy.
In the degenerate case, the condition for the ideal weakly coupled
plasma is
EK ≫ e2n1/3
2
(
SI :e2n1/3
8πǫ0
)
h2n2/3
2m≫ e2n1/3
2
(
SI :e2n1/3
8πǫ0
)
n ≫ n∗ ≡(
me2
h2
)3
≡ a−3B = 6.75 · 1024cm−3
Here aB is the Bohr radius of electrons.
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 8
If n ≪ n∗ and T ≫ T∗ ≡ e2n1/3∗ the plasma is classical weakly
coupled.
If n ≪ n∗ and T ≪ T∗ then the plasma is quantum strongly coupled.
T∗ =1
2e2n
1/3∗ =
me4
2h2= Ry = 13.6eV
Ry (Rydberg) is the energy of the lowest state of hydrogen atom.
Phys780: Plasma Physics Lecture 2: Basic properties of plasma. 9
Problem 3. Draw the T-n diagram using a computer and indicate
the location of your favorite plasma (e.g. in the solar
chromosphere, corona, magnetosphere, white dwarf).