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Calculate α ν and j ν from the Einstein coefficients PowerPoint PPT Presentation


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Jan. 31, 2011 Einstein Coefficients Scattering E&M Review: units Coulomb Force Poynting vector Maxwell’s Equations Plane Waves Polarization. Calculate α ν and j ν from the Einstein coefficients. (1). Consider emission: the emitted energy is. where.

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Calculate α ν and j ν from the Einstein coefficients

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Calculate and j from the einstein coefficients

Jan. 31, 2011Einstein CoefficientsScatteringE&M Review: unitsCoulomb ForcePoynting vectorMaxwell’s EquationsPlane WavesPolarization


Calculate and j from the einstein coefficients

Calculate αν and jν from the Einstein coefficients

(1)

Consider emission: the emitted energy is

where


Calculate and j from the einstein coefficients

Each emission event produces energy hν0

spread over 4π steradians

so

(2)

From (1) and (2):

Emission coefficient


Absorption coefficient

Absorption coefficient:

Total energy absorbed in

In volume dV is

or

Let

Recall:

so


Calculate and j from the einstein coefficients

Stimulated Emission

Repeat as above for absorption, but change sign,

and level 1 for level 2

So the “total absorption coefficient” or

the “absorption coefficient corrected for simulated emission” is


Calculate and j from the einstein coefficients

Equation of Radiative Transfer

The Source

Function


Calculate and j from the einstein coefficients

Recall the Einstein relations,


Calculate and j from the einstein coefficients

In Thermodynamic Equilibrium:


Calculate and j from the einstein coefficients

And

The SOURCE FUNCTION is the PLANCK FUNCTION

in thermodynamic equilibrium


Calculate and j from the einstein coefficients

LASERS and MASERS

When

the populations are inverted

Since

increases along ray, exponentially

HUGE amplifications


Scattering term in equation of radiative transfer rybicki lightman section 1 7

Scattering termin equation of radiativetransferRybicki & Lightman, Section 1.7

Consider the contribution to the emission coefficient from scattered photons

  • Assume:

  • Isotropy: scattered radiation is emitted equally in all angles

  •  jνis independent of direction

  • 2. Coherent (elastic) scattering: photons don’t change energy

  • ν(scattered) = ν(incident)

  • 3. Define scattering coefficient:


Calculate and j from the einstein coefficients

 Scattering source function

An integro-differential equation: Hard to solve.

You need to know Iν to derive Jν to get dIν/ds


Calculate and j from the einstein coefficients

Review of E&M

Rybicki & Lightman, Chapter 2


Qualitative picture the laws of electromagnetism

Qualitative Picture:The Laws of Electromagnetism

  • Electric charges act as sources for generating electric fields. In turn, electric fields exert forces that accelerate electric charges

  • Moving electric charges constitute electric currents. Electric currents act as sources for generating magnetic fields. In turn, magnetic fields exert forces that deflect moving electric charges.

  • Time-varying electric fields can induce magnetic fields; similarly time-varying magnetic fields can induce electric fields. Light consists of time-varying electric and magnetic fields that propagate as a wave with a constant speed in a vacuum.

  • Light interacts with matter by accelerating charged particles. In turn, accelerated charged particles, whatever the cause of the acceleration, emit electro-magnetic radiation

After Shu


Lorentz force

Lorentz Force

A particle of charge q at position

With velocity

Experiences a FORCE

= electric field at the location of the charge

= magnetic field at the location of the charge


Calculate and j from the einstein coefficients

Law #3: Time varying E  B

Time varying B  E


Calculate and j from the einstein coefficients

Lorenz

Force


Calculate and j from the einstein coefficients

More generally, let

current

density

charge

density

Force per

Unit volume


Review vector arithmetic

Review Vector Arithmetic


Calculate and j from the einstein coefficients

Cross product

Is a vector


Calculate and j from the einstein coefficients

Direction of cross product: Use RIGHT HAND RULE


Calculate and j from the einstein coefficients

NOTE:


Calculate and j from the einstein coefficients

Gradient of scalar field T

is a vector with components


Calculate and j from the einstein coefficients

scalar


Calculate and j from the einstein coefficients

A vector with components


Calculate and j from the einstein coefficients

THEOREM:


Calculate and j from the einstein coefficients

THEOREM


Calculate and j from the einstein coefficients

Laplacian

Operator:

T is a scalar field

Can also operate on a vector,

Resulting in a vector:


Units

UNITS

  • R&L use Gaussian Units

    convenient for treating radiation

  • Engineers (and the physics GRE) use

    MKSA (coulombs, volts, amperes,etc)

  • Mixed CGS

    electrostatic quantities: esu

    electromagnetic quantities: emu


Calculate and j from the einstein coefficients

Units in E&M

We are used to units for e.g. mass, length, time

which are basic: i.e. they are based on the standard Kg in Paris, etc.

In E&M, charge can be defined in different ways, based on different experiments

ELECTROSTATIC: ESU

Define charge by Coulomb’s Law:

Then the electric field

is defined by


Calculate and j from the einstein coefficients

So the units of charge in ESU can be written in terms of M, L, T:

[eESU]  M1/2 L-3/2 T-1

And the electric field has units of [E]  M1/2 L-3/2 T-1

The charge of the electron is 4.803x10-10 ESU


Calculate and j from the einstein coefficients

In the ELECTROMAGNETIC SYSTEM (or EMU) charge is defined in

terms of the force between two current carrying wires:

Two wires of 1 cm length, each carrying 1 EMU of current

exert a force of 1 DYNE when separated by 1 cm.

Currents produce magnetic field B:


Calculate and j from the einstein coefficients

Units of JEMU (current density):

Since

[jEMU] = M1/2 L1/2 T -1 current

[JEMU] = [jEMU] L-2 = M1/2 L-3/2 T-1

So [B]  M1/2 L-1/2 T-1

Recall [E]  M1/2 L-1/2 T-1

So E and B have the same units


Emu vs esu

EMU vs. ESU

Current density = charge volume density * velocity

So the units of CHARGE in EMU are:

[eEMU] = M1/2 L1/2

Since M1/2 L-3/2 T-1 = [eEMU]/L3 * L/T

Thus,

Experimentally,


Maxwell s equations wave equations

MAXWELL’S EQUATIONSWave Equations


Calculate and j from the einstein coefficients

Maxwell’s Equations

Let

Charge density

Current density


Maxwell s equations

Maxwell’s Equations

Gauss’ Law

No magnetic monopoles

Faraday’s Law


Calculate and j from the einstein coefficients

We will be mostly concerned with Maxwell’s equations

In a vacuum, i.e.


Calculate and j from the einstein coefficients

Dielectric Media: E-field aligns polar molecules,

Or polarizes and aligns symmetric molecules


Calculate and j from the einstein coefficients

Diamagnetic: μ < 1 alignment weak, opposed to external

field so B decreases

Paramagnetic μ > 1 alignment weak, in direction of field

Ferromagnetic μ >> 1 alignment strong, in direction of external

field


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