Yuki Takeuchi
== Reading assignment 1
Ch1.1
The spectrum correspond to waves which have various wavelength and frequency. λν=c Temperature and energy E=hν T=E/k
ch1.2
Flux = the measure of energy of all rays passing through a given area dAdt. Flux from isotropic source = assuming there are two spherical sources S and S' with radii r and r'. by conservation of energy, energy passing both elements are the same; F(r)*4πr^2 = F(r')*4πr'^2. => F= const/r^2 if S' is fixed.
ch1.3
radiation is the energy carried by individual rays, but we need to consider the energy carried by sets of rays rather than individual ray because single ray essentially does not carry energy.
Specific Intensity = describes the rate of radiative transfer of energy at a specific point P.
Net Flux = integration of flux over solid angle with direction n. F = ∫IcosθdΩ
Momentum flux = momentum flux along ray at angle θ is dF/c. and integrate it over solid angle.
Specific Energy Density = the energy per unit volume per unit frequency range. Total radiation density = integrating specific density over all frequency. Radiation Pressure P = 1/3 u which is also useful for thermodynamics of blackbody radiation.
Ch1.4
Emission => dI = jds Spontaneous emission coefficient: dE = jdVdΩdt Monochromatic emission coefficient: dE = jdVdΩdtdν (where j_ν = 1/4pi P)
- spontaneous emission can be defined as emissivity ε.
So, spontaneous emission can be dE = ερdVdtdν(dΩ/4π) then, compare those dE to get j = ερ/4pi
Absorption => dI = -αIds derivation is similar to emission above
Radiative transfer equation can be obtained by combining emission and absorption. => (dI/ds)=-αI+j Case.1) Emission only α=0 Case.2) Absorption Only j = 0
Also, transfer equation can be simpler by introducing optical depth τ =>(dI/dτ)=-I+S where S = j/α
Ch1.5
Kirchhoffs law for Thermal emission S = B(T) j = αB(T) If S>B, then I>B. If S<B, then I<B. Transfer equation becomes (dI/dτ)=-I+B(T) for blackbody radiation I = B for Thermal radiation S = B
Blackbody radiation adiabatic laws TV^(1/3) - const. pV^(4/3) = const.
Planck spectrum Derivation requires two parts. 1)density of photon state 2)average energy per photon state Also, planck law has 5 different properties. 1)hν<<kT:Rayleigh-Jean law (applied at low frequency) 2)hν>>kT:Wien Law 3)Monotonicity with temperature (On blackbody curve, one with higher temp. lies entirely above the other. ) 4)Wien Displacement Law (peak frequency of blackbody law shifts linearly with temperature.) 5)Relation of Radiation Constants to Fundamental Constants In addition Characteristic Temp. is related to Planck constant. a)Brightness Temp.I = B(T)used in radio astronomy and where Rayleigh-Jeans law is applicable. b)Color Temp.(By fitting the data to a blackbody curve without regard to vertical scale, a color temp. is obtained.) c)Effective temp. (Obtained by integrate flux over all frequency.
Reading assignment 2
ch1.6
spontaneous emission (transition prob. A_21) = when particle drop from energy level 2 to 1, the particle emit photon. And this occurs even if radiation field does not exist.
absorption (transition prob. B_12J) = this occur when a particle absorb photon and excited to from level 1 to 2. probability for this even is prop. to the density of photon, and the function is normalizeable.
A and B are both Einstein coefficient.
stimulated emission (transition prob. = B_21J) = when particle in level 2 is stimulated by other particle such as photon, the particle emit photon and drop from level 2 to 1.
Relation b/w Einstein coefficient by introducing n1, n2, and ,n3 being number density of atoms in level 1 and 2, the einstein coefficients are combined in one equation as in page 29 eq 1.69. Since J = B in thermodynamic equilibrium, the final relation of Einstein relation become g_1B_12 = g_2B_21 and A_21 = 2hv^3 / c^2 B_21 as eq 1.72 this einstein relation can include non-thermal emission in not TE.
LTE = this happen when the matter is in TE with itself. non-thermal emission = this occur when the atomic populations does not obey the Maxwellian velocity distribution law.
ch2.1 ch2.2
poynting theorem tells us that the rate of change of mechanical energy per unit volume plus the rate of change of field energy per unit volume equals minus the divergence of the field energy flux. Then after solving maxwell equation, we can get the time averaged poynting vector which satisfy <S> = <c/8pi)Re(EB*). Since E = B, we can obtain time averaged energy density equation from it.
ch2.3
the radiation spectrum a local spectrum is useful where the condition ⊿ω=1/T satisfy. Because electric field depends on time, the spectrum as determined by analyzing a portion of length T will depend on what portion is analyzed. Therefore, the concept of radiation spectrum or local spectrum depends on whether the changes of character of electric field occur on a long enough time scale.
ch2.4
Monochromatic wave and quasi-monochromatic wave monochromatic wave is completely polarized over short time with a definite state of elliptical polarization. However, in much larger time, the state of polarization changes completely. In this case, such a wave is called quasi-monochromatic wave. For this wave,⊿ω about the value ω can be estimated as ⊿ω>1/⊿t so that ⊿ω<<ω, where ⊿ω=bandwidth and ⊿t=coherence time.
ch2.5 ch2.6
so by manipulating maxwell equation, we can arrive at retarded potential equation and retarded time equation. the retarded time refers to conditions at the point r' that existed at a time earlier than t by just the time required for light to travel b/w r and r'.
Here I could not figure out how to get the eq 2.67.
Chapter 3 summary
The potentials in radiation from moving charge is called Lienard-Wiechart potential. Then, this potential has two differences in static electromagnetic theory. The first difference is that equation has factor k in it. This is k in only important when the speed of particle come close to the speed of light, where it tends to concentrate the potentials into a narrow come about the particle velocity. the second is the quantities are all evaluated at the retarded time. retarded t is derivative of t.
Larmor's formula = this is for emission from a single accelerated charge q. This formula has three significant things. 1)The power emitted is prop. to the square of the charge and acceleration. 2)no radiation is emitted along the direction of acceleration. 3)the instantaneous direction of E_rad is determined by acceleration and direction.
chapter 5 summary
bremsstrahlung = free-free emission = the radiation due to the acceleration of charge in Coulomb field of another charge. I did not understand how to derive small angle scatterings (eq. 5.6)
Thermal bremsstrahlung emission the probability that a particle has a thermal distribution of speeds. And, the prob. of it is given by as dp (eq.5.13). Then, to integrate that equation, we need to define limit over electron velocities. this is called photon discreteness effect. Then, we get eq 5.14 which contains velocity averaged Gaunt factor. however, I could not figure out what this factor actually mean...
thermal bremsstrahlung absorption the formula for this absorption come from the absorption of radiation and has very similar formula (5.16) After evaluate under Rayleigh-Jean regime, we get numerical formula for it.
Chapter 6 summary
the Particle accelerated by magnetic field radiate and this radiation is known as synchrotron radiation in extreme relativistic particles. The final equation for its total emitted power is given by combination of Thomson cross section and magnetic energy density. The feature of synchrotron radiation is a pulse confined to a time interval is much smaller than the gyration period. Then, the spectrum spread out much broader region than one of order ω/2π. Also, synchrotron radiation partially linearly polarized and direction of its power per unit frequency is perpendicular and parallel to projection of magnetic field on the plane of the sky.
Chapter 7 summary
Compton effect is the closely elastic scattering of photon and electron in long wavelength. After collision, the wave length become longer because of energy conservation. The differential cross section in the quantum effect is given by klein-Nishina formula.
Energy transfer moving electron has sufficient kinetic energy compared to the photon, net energy may be transferred from the electron to photon. In this case, the process is called inverse Compton. The spectrum of inverse Compton depends on the incident spectrum and the energy distribution of the electrons. Here what we have to consider is spectrum averaging over the actual distributions of photon and electron.
chapter 9 summary
Determining an approx. to atomic state. start from time independent Schrodinger eq.for the case of electron surrounding a nucleus charge Ze including relativistic and atomic effect.
One electron system n central field approx. each electron feels different potential, and then the potential is depend on its radius from nucleus.
Many electron system Form a products such as a(1)b(2)....k(N) where a,b,k represents the set of values (n,l,m,n) and 1,2,...N represent the space and spin coordinates for the 1st, 2nd and nth particle. Such particles are all identical, but we can not say that particle a is on orbital a and so on. What we have to use now is linear combination of the product. that is slater determinants so that it is clear that when two electron occupy the same orbital,tow row of this determinants are equal so vanishes. The particle which satisfy this is fermions.
Hartree-Fock Approximation = A method for choosing the orbitals used to construct the atomic state and it is based on the variational principle of expectation value of energy.
Hyper fine structure 1: Isotope effect: 2: Nuclear spin
Boltzmann population of level At the point of thermal equilibrium, the populations are completely determined by temperature. Then the population function is given by eq9.38 on text.
Saha equation This equation determines the distribution of an atomic species among its various stage. If the ration is large positive, that means a significant portion of atoms are ionized, and if the ration is tiny number, almost none or none of them are ionized.
Ch. 10 summary
semi-classical theory Atom is treated quantum mechanically and the radiation is treated classical mechanically. This theory predicts the induced radiation process and this process is described by Einstein coefficients. In the classical limit of radiation, the number of photons per per photon state is large , and then, the induced processes, which are prop. to the number of photons, dominate the spontaneous process, which is independent process in the number of photons. Einstein coefficients and oscillator strength from the relationship with einstein coefficient B, we can derive the relation between Einstein coefficient and absorption oscillator strength f. When upper state happens to lie in a continuum, it is meaningless to talk about the probability of a transition to a single state. However, we should rather define the probability per unit energy range. From this concept, we can come up with the continuum oscillator strength fc. This is the total oscillator strength to all continuum state. Line Broadening Mechanisms Doppler broadening Atom is in therma motion, so that the frequency of emissin or absorption in its own frame correxponds to a diff frequency for observers. Natural Broadening Lorentz profile Collisional Broadening Lorentz profile is applied to more certain type generally. When atom experience the collision with other particles while emitting, the phase of the emitted radiation can be altered suddenly. If the phase change is random at collision moments, the information about the emitted frequency is lost.