optical properties of plasma
TRANSCRIPT
Optical Properties of Plasma
Course: B.Sc. Physical Sciences
Semester: VI, Section C
Paper: Solid State Physics
Instructor: Manish K. Shukla
Plasma β’ Plasma is a gas of charge particles.
β’ The plasma is overall neutral, i.e., the number density of the electrons and ions are the same.
β’ Under normal conditions, there are always equal numbers positive ions and electrons in any volume of the plasma, so the charge density π = 0, and there is no large scale electric field in the plasma.
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Plasma Oscillation
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Now imagine that all of the electrons are displaced to the right by a small amount x, while the positive
ions are held fixed, as shown on the right side of the figure above.
The displacement of the electrons to the right leaves an excess of positive charge on the left side of the
plasma slab and an excess of negative charge on the right side, as indicated by the dashed rectangular
boxes.
The positive slab on the left and the negative slab on the right produce an electric field pointing
toward the right that pulls the electrons back toward their original locations.
However, the electric force on the electrons causes them to accelerate and gain kinetic energy, so they
will overshoot their original positions.
This situation is similar to a mass on a horizontal frictionless surface connected to a horizontal spring.
In the present problem,. the electrons execute simple harmonic motion at a frequency that is called the
βplasma frequencyβ.
Plasma Oscillation contd.
Derivation of Plasma frequency
Derivation of Plasma frequency contd.
Derivation of Plasma frequency
π0= 8.98 π
Plasmon
o A plasmon is a quantum of plasma oscillation.
o Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of
plasmons.
o The plasmon can be considered as a quasiparticle since it arises from the quantization of plasma
oscillations, just like phonons are quantizations of mechanical vibrations.
o Thus, plasmons are collective (a discrete number) oscillations of the free electron gas density.
o At optical frequencies, plasmons can couple with a photon to create another quasiparticle called
a plasmon polariton.
o The plasmon energy can often be estimated in the free electron model as,
πΈ = βπ0(= βπ0) where π0, π0 are plasma angular frequency and plasma frequency.
Dispersion Relation for plasma o Recall the Lorentz Dispersion relation for gaseous medium
ππ ππ π2 = 1 +
ππ2
π0π
π
ππ
ππ2 βπ2 β π πΎπ
In plasma
i. The electrons are free and not bound to ions by any kind of spring like force, so π½ = 0 βππ = 0
ii. All electrons experience same force, so no summation required,
iii. Also, q=e , and there is negligible damping in plasma: so, πΎ = 0.Thus, we have :
ππ = π2 = 1 β
ππ2
π0π
1
βπ2. We know π0
2 = ππ2/ππ0,
Thus,ππ = π
2 = 1 βπ02
π2
β’ Expression of dielectric constant and refractive index of
plasma.
β’ This expression is also called the dispersion relation of
plasma.
β’ It is clear that refractive index of plasma n depends on
the frequency (π) of the wave passing through it.
Surprise in Dispersion relation of plasma
ππ = π2 = 1 β
π02
π2
In this relation two surprising elements can be seen
1. π2 < 1πππππ π/π£ < 1 βΉ π£ > π , Surprising !!!! Is it violation of Einsteinβs
special relativity??
2. For π < π0, dielectric constant ππ is negative and refractive index n is pure
imaginary.
β’ In the definition of n=c/v, v is in fact phase velocity of EM wave in a given medium. In above expression, putting n=c/v,
we get π£ =π
1βπ02
π2
β’ As, π£ = π£π = π/π, substitution of v gives,
π2π2
π2= 1 β
π02
π2β
β’ Above expression is an alternate form of dispersion relation (DR) written in terms of wave vector k and angular freq. π.β’ Group velocity is the physical velocity which gives the rate of energy/signal transfer in a medium.
β’ π£π =ππ
ππ: differentiation of DR w.r.t. k gives
2ππ2ππ = 2πππ βΉππ
ππ=ππ2
π=π2
π£π
βΉ π£π = π 1 βπ02
π2, now, π£π < π which is cosistant with Relativity
π2π2 = π2 β π02
Imaginary Refractive index & Negative Dielectric constant: Physical meaning
π2π2 = π2 β π02 β ππ = (π2 β π0
2)
β’ Wave Equation : πΈ = πΈ0 ππ(ππ§βππ‘),
β’ Waves can propagate through a medium only if k has a real part.
β’ If π < π0, dielectric constant ππ is negative and refractive index n is pure imaginary. Also π = π π i.e. k is also a pure
imaginary number.
β’ Hence wave equation becomes, πΈ = πΈ0ππ(ππ§βππ‘) = πΈ0π
β|π|π§πβπππ‘.
β’ Here the first term πβ|π|π§ is exponentially decaying with space while the second term πβπππ‘ gives oscillation with time.
Thus for π < π0, wave take the form of decaying standing waves.
β’ In fact, for π < π0, the EM wave incident on plasma does not propagate in plasma , instead it will be totally reflected.
(How ??) Answer comes from the EM theory.
β’ Energy flux of EM wave π’ =πΈ 2
π0πRe(k). For π < π0, i.e. k is pure imaginary, so, π’ = 0 which means NO energy is
transferred in the plasma. Thus wave is completely reflected.
ππ = π2 = 1 β
π02
π2
o Plasma frequency sets a lower cutoff for the frequencies of electromagnetic radiation.
o Waves of frequencies lower than plasma frequency (π < π0), are reflected back by the plasma.
o Only those radiations for which frequency is greater than plasma frequency (π > π0), can pass
through a plasma.
o For very high frequencies i.e. π β« π0 plasma behaves as a transparent non-dispersive medium.
How??
o In the limit π β« π0, ππβΆ 1, π βΆ 1. DR becomes π2π2 = π2, where π£π = π£π = π. Since,
dielectric constant, refractive index and phase velocity is independent of frequency, there is no
dispersion of wave.
Conclusion
ππ = π2 = 1 β
π02
π2π2π2 = π2 β π0
2orDispersion Relation (DR)
Application of Plasma : Ionosphere
Application of plasma II : Metalsβ’ The metals shine by reflecting most of light in visible range.
β’ The concept of electron plasma oscillations can also be applied to the
free electrons in a conductor.
β’ The visible light can't pass through the metal because the plasma
frequency of electrons in metal falls in ultraviolet region.
β’ For frequencies in UV, metals are transparent.
β’ For example, the free electron density in Cu is 8.4 Γ 1028 mβ3. (By
way of comparison, the density of air molecules at a pressure of one
atmosphere and T = 300K is 2.4 Γ 1025 mβ3.)
β’ In Cu, then plasma frequency, fo β 2.6 Γ 1015 Hz.
β’ This is higher than the frequencies of visible light, and explains why
metals are opaque to visible light and transparent to UV light.