ˆd = 1 2π. d(t)e iωt dt. (1)
|
|
- Brittany Hall
- 6 years ago
- Views:
Transcription
1 Bremsstrahlung Initial questions: How does the hot gas in galaxy clusters cool? What should we see in their inner portions, where the density is high? As in the last lecture, we re going to take a more in-depth look at a particular radiative process, in this case bremsstrahlung. The term means braking radiation, and occurs when an electron is accelerated by passage near an ion, and hence radiates. There is also an inverse process, which you get by running the film backward in time : a photon is absorbed by an electron that is moving in the Coulomb field of an ion. This is free-free absorption. Bremsstrahlung and free-free absorption are basic radiative processes that show up in manycontexts. Givenourlimitedtime, wewillonlybeabletotouchonafewoftheconcepts; read Rybicki and Lightman or Shu if you want more details. Let s make things easier for ourselves by starting with nonrelativistic bremsstrahlung. From our discussion of acceleration radiation, this means we ll use a dipole approximation. In such a case, this means the radiation is proportional to the second derivative of the dipole moment e i r i. Note that for interactions of identical particles (all electrons, or all protons, for example), the dipole moment is proportional to m i r i, which is the center of mass. Ask class: what can we conclude about bremsstrahlung from identical particles? There is no dipole radiation (although there can be quadrupole radiation), because the center of mass is stationary. We thus consider electron-ion bremsstrahlung. In principle, both are moving, so we have to include acceleration radiation from both electrons and ions. Ask class: is there a simplification that can help us out? Since ions are so much more massive than electrons, to a good approximation we can treat them as fixed, while the electrons move in their static Coulomb field. That s an approximation that appears over and over again in plasma physics. As with Compton scattering, we ll simplify things further by considering single-speed electrons at the start. As with lots of these basic processes, one can simply grind through with brute force, but in this case there are several important obstacles to overcome, so we ll go into this in more detail. First, let s assume that the electron is only slightly deflected from its path; that means that instead of having to compute the full trajectory, we can assume a straight line path and determine the radiation emitted as a result. See Figure 1 for the geometry of bremsstrahlung. Suppose the electron has charge e, and the ion has charge Ze. The impact parameter is b; that means that if the path were a perfect straight line, the closest the electron would come to the ion would be a distance b. The dipole moment is d = er, where R is the instantaneous location of the electron, so the second derivative is d = e v, where v is the electron velocity. Ask class: schematically, how would they compute the total energy
2 radiated over the electron s trajectory? More generally, however, we would like to figure out the energy radiated as a function of frequency. Here s a case where Fourier transforms can add some insight. Recall that the Fourier transform of a quantity, in this case d, is ˆd = 1 2π d(t)e iωt dt. (1) Thus the Fourier transform of d at some frequency ω is ω 2ˆd(ω) (think of taking the derivative inside the integral, where it acts on exp(iωt)). This gives ω 2ˆd(ω) = (e/2π) ve iωt dt. (2) In general this still might be a mess to evaluate, but let s get some insight by looking at extreme cases: large and small ω. We know that for most of the trajectory of the electron it s far from the ion, but it is close to its minimum distance over a time called the collision time, τ = b/v. That means the integral above is only really important for times between τ/2 and τ/2, give or take. Now, if ωτ 1, then the exponential oscillates many times, so the net contribution is small. If instead ωτ 1, then the exponent is close to 0 and thus the exponential is near unity. Thus, in our two limits: ˆd(ω) (e/2πω 2 ) v, ωτ 1 0, ωτ 1. (3) Here v is the change of velocity during the collision. Now let s think about how we could get this to a form where we have the energy emitted per frequency interval. We know that the energy per area per time is given by the Poynting flux dtda = c 4π E2 (t). (4) Thus the energy per area is this, integrated over time: /da = (c/4π) E2 (t)dt. Parseval s theorem says that the integral over all times of a quantity is related to the integral over all frequencies of the quantity s Fourier transform via E 2 (t)dt = 2π Ê(ω) 2 dω. (5) Here the vertical bars mean the magnitude of the complex quantity tells us that Ê(ω) 2 = Ê( ω) 2, so this gives /da = c 0 Ê(ω) 2 dω /dadω = c Ê(ω) 2. Ê(ω). E is real, which (6)
3 Back when we discussed acceleration and radiation, we found that E(t) = d(t) sinθ c 2 R 0 (7) for an angle Θ relative to the polarization direction and an average distance R 0 from the charges. Integrating over solid angle, and realizing that da = R 2 0dΩ, we get dω = 8πω4 3c 3 ˆd(ω) 2. (8) Plugging this into our bremsstrahlung formula, this means that the energy emitted during the whole encounter in a frequency interval dω is /dω (2e 2 /3πc 3 ) v 2, ωτ 1 0, ωτ 1. But how do we estimate v? Most of the change will be deflecting the electron, rather than speeding it up or slowing it down. That means we can consider just the change in velocity normal to the path: v = Ze2 m (9) bdt 2Ze2 = (b 2 +v 2 t 2 ) 3/2 mbv. (10) Putting it together, this means the energy radiated in a frequency interval dω, for electrons with impact parameter b, is (b)/dω = [8Z 2 e 6 /(3πc 3 m 2 v 2 b 2 )], = 0, b v/ω. b v/ω Here b v/ω is a restatement of ωτ 1. Now wait a second: isn t there a problem? This says that the power per frequency emitted for extremely small b diverges. Hm. Ask class: any ideas at this point what we can do? If not, let s plow on anyway. If we have lots of electrons (say, number density n e ) moving with a fixed speed v, then the flux of electrons (number per area per time) is n e v. The number per time that have an impact parameter between b and b + db is this flux times the area element, 2πbdb. If the number density of ions is n i, this means that the energy per frequency per volume per time is dωdvdt = n en i 2πv 0 (11) (b) bdb. (12) dω Yikes! Danger, Will Robinson! The integral in question is of 1/b, so we get a logarithm that diverges at both large b and small b. Ask class: what do we do? This is one manifestation of what we discussed last time: physically, power laws must be cut off at some limits. If, for example, we decide that the integral will really extend from b min to b max, we get dωdvdt = 16e6 3c 3 m 2 v n en i Z 2 ln(b max /b min ). (13)
4 You can do the usual checks: e = 0 implies no radiation, large m implies less acceleration and hence less radiation, and so on. You are in the presence of the famed Coulomb logarithm much spoken of in legend. This occurs a lot when Coulomb forces are involved, since the forcescalesas1/r 2 andtheareaelementscalesasr. Thefirstthingtonoteisthatlogarithms are blessedly insensitive to precise values; ln(10 16 ) is only twice ln(10 8 ), for example. That means that we don t have to be too precise in defining b max and b min to get a decent answer. But what is it that produces upper and lower cutoffs? Well, our approximation is invalid if b v/ω, so we can try b max = v/ω. What about the other limit? In our derivation we assumed that the electron was basically going in a straight line path. This will clearly be invalid when v v, which occurs when b = b min = 4Ze 2 /(πmv 2 ). Or, we could adopt a quantum approach: the classical calculation we ve done is invalid when b < h/mv, since that would imply x p <, where x b and p mv. Thus, we could also try b min = h/mv. Again, the insensitivity of the logarithm means we have latitude here. We will, of course, take the larger of the two possible values of b min. One reason we have to use these approximations is that we re trying to do a classical calculation of a quantum process. We can sort of adjust for that by writing the exact results in terms of a Gaunt factor g ff (v,ω): where dωdvdt = 16πe6 3 3c 3 m 2 v n en i Z 2 g ff (v,ω), (14) g ff (v,ω) = ( 3/π)ln(b max /b min ). (15) The Gaunt factor is normalized in this way so that typically its value is near unity. Note, by the way, that the expression has a 1/v in it. Ask class: does that mean that for a given frequency ω the power per volume diverges for electrons of arbitrarily low velocity? No! We have to realize that the energy of the photons comes from the kinetic energy of the electrons, so if 1 2 mv2 < hν then a photon of energy hν can t be created. There is thus another cutoff if you want the power at a particular frequency: only electrons in a certain velocity range contribute. This is called a photon discreteness effect. Now that we have an answer for a particular velocity, we can integrate over an electron velocity distribution to get bulk emission from a region. Consider a thermal distribution, meaning that electron velocities are apportioned according to a Maxwell-Boltzmann distribution. Rybicki and Lightman get the integrated emission for an electron temperature T, in erg s 1 cm 3 Hz 1 : ǫ ff ν = dvdtdν = Z 2 n e n i T 1/2 e hν/kt ḡ ff, (16) where ḡ ff is the velocity-averaged Gaunt factor at temperature T.
5 Notethatthisisarather flat spectrumforhν < kt,andthatitrollsoverexponentially for higher energies (see Figure 2). That high-energy cutoff comes from photon discreteness and the lack of too many high-energy electrons. Ask class: given this spectrum, what is the rough frequency range where most of the energy will come out? Somewhere in the kt range. Note that if the electron distribution is nonthermal, one has to do different integrals. If we integrate this emission over all frequencies, we get a total emission dvdt = T 1/2 n e n i Z 2 ḡ B (17) (in cgs units), where ḡ B is the frequency and velocity averaged Gaunt factor, usually in the range 1.1 to 1.5. Now, we ve discussed this in terms of an emission process. But there must therefore be a corresponding absorption process; in this case, free-free absorption. We can get a free-free absorption coefficient from Kirchoff s law, and we find that α ff ν = cm 1 T 1/2 Z 2 n e n i ν 3( 1 e hν/kt) ḡ ff. (18) Ask class: why is there a 1 e hν/kt factor in there? It corrects for stimulated emission. The Rosseland mean of the free-free absorption coefficient is α ff R = T 7/2 Z 2 n e n i ḡ R, (19) where quantities are measured in cgs units, and ḡ R is weighted as in the Rosseland mean and is of order unity. The particular form T 7/2 for the Rosseland mean, or ν 3 for the frequency dependence, is called a Kramers opacity and occurs for bound-free as well as free-free. The preceding was all done for nonrelativistic electrons. In certain very high-energy situations, though, the electron speeds can be relativistic. What is to be done? A cool way of approaching this problem is through the method of virtual quanta. First, transform into a frame in which the electron is stationary and the ion is moving. The moving ion produces a pulse of electric field, which Compton scatters off of the electron. The radiation that emerges is the bremsstrahlung radiation as seen in this new frame. You can then Lorentz transform back into the lab frame where the electron is moving to get the radiation rate. Stuff like this is common in quantum electrodynamics. One can often find symmetries between apparently unrelated processes. For example, consider Compton scattering. A photon hits an electron, and bounces off; this means that if one plots the interaction on a time axis, then a photon and electron converge, then diverge. It happens, though, that a positron can be treated as an electron moving backward in time (that s Feynman s insight). This means that if you look at Compton scattering sideways, two photons converge and produce an electron-positron pair, or the pair annihilates to produce two photons. Thus, there is an essential relation between Compton scattering and pair annihilation/production.
6 Enough of this diversion. When one puts in the relativistic effects, a reasonable fitting formula in cgs units is dvdt = T 1/2 Z 2 n e n i ḡ B ( T). (20) One final set of comments. We have now discussed in some more detail two processes: Compton scattering and bremsstrahlung. As observers, we are often presented with a spectrum and we d like to know the fundamental process that created it. In principle it may not sound too bad: look at the characteristics of the spectrum, then just identify it! In reality, though, if only continuum processes are operating (these vary only slowly with frequency), it can be difficult to do unique identification. Is your spectrum due to Comptonization, or is it the sum of many blackbodies of different temperatures? Be careful in these circumstances. Many, many people have a tendency to use the following approach: (1) assume some form or mechanism (e.g., bremsstrahlung), (2) derive parameters from the spectral fit (e.g., temperature or density), (3) assign great meaning to those derived parameters. The fact is that the derived parameters can vary significantly given different fits. A good fit does not guarantee physical meaning! If one has lines or edges it s usually easier, but beware of continuum fits. Recommended Rybicki and Lightman problem: 5.1
7 Fig. 1. Geometry of bremsstrahlung, from Wikipedia. An electron of initial energy E 1 and speed v 1 as seen in the reference frame of the positive ion is deflected by the ion, radiates a photon, and ends up with energy E 2 < E 1 and speed v 2 < v 1 as seen in the original frame. The inverse process, in which an electron absorbs a photon as it goes by an ion, is free-free absorption.
8 Fig. 2. Representation of the spectrum of bremsstrahlung, from cflynn/astroii/brems.gif. The energy per volume per time per frequency is nearly constant until hν > kt, after which it rolls over exponentially.
Electrodynamics of Radiation Processes
Electrodynamics of Radiation Processes 7. Emission from relativistic particles (contd) & Bremsstrahlung http://www.astro.rug.nl/~etolstoy/radproc/ Chapter 4: Rybicki&Lightman Sections 4.8, 4.9 Chapter
More informationBremsstrahlung. Rybicki & Lightman Chapter 5. Free-free Emission Braking Radiation
Bremsstrahlung Rybicki & Lightman Chapter 5 Bremsstrahlung Free-free Emission Braking Radiation Radiation due to acceleration of charged particle by the Coulomb field of another charge. Relevant for (i)
More informationBremsstrahlung Radiation
Bremsstrahlung Radiation Wise (IR) An Example in Everyday Life X-Rays used in medicine (radiographics) are generated via Bremsstrahlung process. In a nutshell: Bremsstrahlung radiation is emitted when
More information1 Monday, October 31: Relativistic Charged Particles
1 Monday, October 31: Relativistic Charged Particles As I was saying, before the midterm exam intervened, in an inertial frame of reference K there exists an electric field E and a magnetic field B at
More information1. Why photons? 2. Photons in a vacuum
Photons and Other Messengers 1. Why photons? Ask class: most of our information about the universe comes from photons. What are the reasons for this? Let s compare them with other possible messengers,
More informationChapter 2 Bremsstrahlung and Black Body
Chapter 2 Bremsstrahlung and Black Body 2.1 Bremsstrahlung We will follow an approximate derivation. For a more complete treatment see [2] and [1]. We will consider an electron proton plasma. Definitions:
More informationBasics of Radiation Fields
Basics of Radiation Fields Initial questions: How could you estimate the distance to a radio source in our galaxy if you don t have a parallax? We are now going to shift gears a bit. In order to understand
More informationCompton Scattering. hω 1 = hω 0 / [ 1 + ( hω 0 /mc 2 )(1 cos θ) ]. (1) In terms of wavelength it s even easier: λ 1 λ 0 = λ c (1 cos θ) (2)
Compton Scattering Last time we talked about scattering in the limit where the photon energy is much smaller than the mass-energy of an electron. However, when X-rays and gamma-rays are considered, this
More informationThermal Bremsstrahlung
Thermal Bremsstrahlung ''Radiation due to the acceleration of a charge in the Coulomb field of another charge is called bremsstrahlung or free-free emission A full understanding of the process requires
More informationde = j ν dvdωdtdν. (1)
Transfer Equation and Blackbodies Initial questions: There are sources in the centers of some galaxies that are extraordinarily bright in microwaves. What s going on? The brightest galaxies in the universe
More informationNeutrinos, nonzero rest mass particles, and production of high energy photons Particle interactions
Neutrinos, nonzero rest mass particles, and production of high energy photons Particle interactions Previously we considered interactions from the standpoint of photons: a photon travels along, what happens
More informationThe Stellar Opacity. F ν = D U = 1 3 vl n = 1 3. and that, when integrated over all energies,
The Stellar Opacity The mean absorption coefficient, κ, is not a constant; it is dependent on frequency, and is therefore frequently written as κ ν. Inside a star, several different sources of opacity
More informationAstrophysical Radiation Processes
PHY3145 Topics in Theoretical Physics Astrophysical Radiation Processes 5:Synchrotron and Bremsstrahlung spectra Dr. J. Hatchell, Physics 406, J.Hatchell@exeter.ac.uk Course structure 1. Radiation basics.
More informationCHAPTER 27. Continuum Emission Mechanisms
CHAPTER 27 Continuum Emission Mechanisms Continuum radiation is any radiation that forms a continuous spectrum and is not restricted to a narrow frequency range. In what follows we briefly describe five
More informationThe next three lectures will address interactions of charged particles with matter. In today s lecture, we will talk about energy transfer through
The next three lectures will address interactions of charged particles with matter. In today s lecture, we will talk about energy transfer through the property known as stopping power. In the second lecture,
More informationφ(ν)dν = 1. (1) We can define an average intensity over this profile, J =
Ask about final Saturday, December 14 (avoids day of ASTR 100 final, Andy Harris final). Decided: final is 1 PM, Dec 14. Rate Equations and Detailed Balance Blackbodies arise if the optical depth is big
More information2. NOTES ON RADIATIVE TRANSFER The specific intensity I ν
1 2. NOTES ON RADIATIVE TRANSFER 2.1. The specific intensity I ν Let f(x, p) be the photon distribution function in phase space, summed over the two polarization states. Then fdxdp is the number of photons
More informationRetarded Potentials and Radiation
Retarded Potentials and Radiation No, this isn t about potentials that were held back a grade :). Retarded potentials are needed because at a given location in space, a particle feels the fields or potentials
More informationRecap Lecture + Thomson Scattering. Thermal radiation Blackbody radiation Bremsstrahlung radiation
Recap Lecture + Thomson Scattering Thermal radiation Blackbody radiation Bremsstrahlung radiation LECTURE 1: Constancy of Brightness in Free Space We use now energy conservation: de=i ν 1 da1 d Ω1 dt d
More informationAddition of Opacities and Absorption
Addition of Opacities and Absorption If the only way photons could interact was via simple scattering, there would be no blackbodies. We ll go into that in much more detail in the next lecture, but the
More information1. Why photons? 2. Photons in a vacuum
1 Photons 1. Why photons? Ask class: most of our information about the universe comes from photons. What are the reasons for this? Let s compare them with other possible messengers, specifically massive
More informationThe Larmor Formula (Chapters 18-19)
2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 1 The Larmor Formula (Chapters 18-19) T. Johnson Outline Brief repetition of emission formula The emission from a single free particle - the Larmor
More informationOutline. Today we will learn what is thermal radiation
Thermal Radiation & Outline Today we will learn what is thermal radiation Laws Laws of of themodynamics themodynamics Radiative Radiative Diffusion Diffusion Equation Equation Thermal Thermal Equilibrium
More informationLet s consider nonrelativistic electrons. A given electron follows Newton s law. m v = ee. (2)
Plasma Processes Initial questions: We see all objects through a medium, which could be interplanetary, interstellar, or intergalactic. How does this medium affect photons? What information can we obtain?
More informationThermal Equilibrium in Nebulae 1. For an ionized nebula under steady conditions, heating and cooling processes that in
Thermal Equilibrium in Nebulae 1 For an ionized nebula under steady conditions, heating and cooling processes that in isolation would change the thermal energy content of the gas are in balance, such that
More informationSpecial relativity and light RL 4.1, 4.9, 5.4, (6.7)
Special relativity and light RL 4.1, 4.9, 5.4, (6.7) First: Bremsstrahlung recap Braking radiation, free-free emission Important in hot plasma (e.g. coronae) Most relevant: thermal Bremsstrahlung What
More informationLecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization
22.101 Applied Nuclear Physics (Fall 2006) Lecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization References: R. D. Evans, The Atomic Nucleus (McGraw-Hill, New York,
More information1 Monday, November 7: Synchrotron Radiation for Beginners
1 Monday, November 7: Synchrotron Radiation for Beginners An accelerated electron emits electromagnetic radiation. The most effective way to accelerate an electron is to use electromagnetic forces. Since
More informationCompton Scattering I. 1 Introduction
1 Introduction Compton Scattering I Compton scattering is the process whereby photons gain or lose energy from collisions with electrons. It is an important source of radiation at high energies, particularly
More informationParticle acceleration and generation of high-energy photons
Particle acceleration and generation of high-energy photons For acceleration, see Chapter 21 of Longair Ask class: suppose we observe a photon with an energy of 1 TeV. How could it have been produced?
More informationLine Broadening. φ(ν) = Γ/4π 2 (ν ν 0 ) 2 + (Γ/4π) 2, (3) where now Γ = γ +2ν col includes contributions from both natural broadening and collisions.
Line Broadening Spectral lines are not arbitrarily sharp. There are a variety of mechanisms that give them finite width, and some of those mechanisms contain significant information. We ll consider a few
More informationRadiative Processes in Flares I: Bremsstrahlung
Hale COLLAGE 2017 Lecture 20 Radiative Processes in Flares I: Bremsstrahlung Bin Chen (New Jersey Institute of Technology) The standard flare model e - magnetic reconnection 1) Magnetic reconnection and
More informationRadiation processes and mechanisms in astrophysics I. R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 2009
Radiation processes and mechanisms in astrophysics I R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 009 Light of the night sky We learn of the universe around us from EM radiation, neutrinos,
More informationStatistical Equilibria: Saha Equation
Statistical Equilibria: Saha Equation We are now going to consider the statistics of equilibrium. Specifically, suppose we let a system stand still for an extremely long time, so that all processes can
More informationRadiation from Charged Particle Interaction with Matter
Chapter 7 Radiation from Charged Particle Interaction with Matter 7.1 Bremsstrahlung When charged particles collide, they accelerate in each other s electric field. As a result, they radiate electromagnetic
More informationElectromagnetic Spectra. AST443, Lecture 13 Stanimir Metchev
Electromagnetic Spectra AST443, Lecture 13 Stanimir Metchev Administrative Homework 2: problem 5.4 extension: until Mon, Nov 2 Reading: Bradt, chapter 11 Howell, chapter 6 Tenagra data: see bottom of Assignments
More informationChapter 2 Radiation of an Accelerated Charge
Chapter 2 Radiation of an Accelerated Charge Whatever the energy source and whatever the object, (but with the notable exception of neutrino emission that we will not consider further, and that of gravitational
More informationBethe-Block. Stopping power of positive muons in copper vs βγ = p/mc. The slight dependence on M at highest energies through T max
Bethe-Block Stopping power of positive muons in copper vs βγ = p/mc. The slight dependence on M at highest energies through T max can be used for PID but typically de/dx depend only on β (given a particle
More informationRelativistic corrections of energy terms
Lectures 2-3 Hydrogen atom. Relativistic corrections of energy terms: relativistic mass correction, Darwin term, and spin-orbit term. Fine structure. Lamb shift. Hyperfine structure. Energy levels of the
More information1 Fundamentals. 1.1 Overview. 1.2 Units: Physics 704 Spring 2018
Physics 704 Spring 2018 1 Fundamentals 1.1 Overview The objective of this course is: to determine and fields in various physical systems and the forces and/or torques resulting from them. The domain of
More informationIn today s lecture, we want to see what happens when we hit the target.
In the previous lecture, we identified three requirements for the production of x- rays. We need a source of electrons, we need something to accelerate electrons, and we need something to slow the electrons
More information1 Monday, November 21: Inverse Compton Scattering
1 Monday, November 21: Inverse Compton Scattering When I did the calculations for the scattering of photons from electrons, I chose (for the sake of simplicity) the inertial frame of reference in which
More informationAy Fall 2004 Lecture 6 (given by Tony Travouillon)
Ay 122 - Fall 2004 Lecture 6 (given by Tony Travouillon) Stellar atmospheres, classification of stellar spectra (Many slides c/o Phil Armitage) Formation of spectral lines: 1.excitation Two key questions:
More informationAtomic cross sections
Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified
More informationPhysics Oct A Quantum Harmonic Oscillator
Physics 301 5-Oct-2005 9-1 A Quantum Harmonic Oscillator The quantum harmonic oscillator (the only kind there is, really) has energy levels given by E n = (n + 1/2) hω, where n 0 is an integer and the
More informationInstability and different burning regimes
1 X-ray bursts Last time we talked about one of the major differences between NS and BH: NS have strong magnetic fields. That means that hot spots can be produced near the magnetic poles, leading to pulsations
More informationPassage of particles through matter
Passage of particles through matter Alexander Khanov PHYS6260: Experimental Methods is HEP Oklahoma State University September 11, 2017 Delta rays During ionization, the energy is transferred to electrons
More informationHigh-Energy Astrophysics
M.Phys. & M.Math.Phys. High-Energy Astrophysics Garret Cotter garret.cotter@physics.ox.ac.uk High-Energy Astrophysics MT 2016 Lecture 2 High-Energy Astrophysics: Synopsis 1) Supernova blast waves; shocks.
More informationPhysics Dec The Maxwell Velocity Distribution
Physics 301 7-Dec-2005 29-1 The Maxwell Velocity Distribution The beginning of chapter 14 covers some things we ve already discussed. Way back in lecture 6, we calculated the pressure for an ideal gas
More information_ int (x) = e ψ (x) γμ ψ(x) Aμ(x)
QED; and the Standard Model We have calculated cross sections in lowest order perturbation theory. Terminology: Born approximation; tree diagrams. At this order of approximation QED (and the standard model)
More informationQuantum Statistics (2)
Quantum Statistics Our final application of quantum mechanics deals with statistical physics in the quantum domain. We ll start by considering the combined wavefunction of two identical particles, 1 and
More informationNotes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
More informationRadiative processes from energetic particles II: Gyromagnetic radiation
Hale COLLAGE 2017 Lecture 21 Radiative processes from energetic particles II: Gyromagnetic radiation Bin Chen (New Jersey Institute of Technology) e - Shibata et al. 1995 e - magnetic reconnection Previous
More informationAtomic Structure and Processes
Chapter 5 Atomic Structure and Processes 5.1 Elementary atomic structure Bohr Orbits correspond to principal quantum number n. Hydrogen atom energy levels where the Rydberg energy is R y = m e ( e E n
More informationGEORGE B. RYBICKI, ALAN P. LIGHTMAN. Copyright W Y-VCH Verlag GmbH L Co. KCaA
RADIATIVE PROCESSE S IN ASTROPHYSICS GEORGE B. RYBICKI, ALAN P. LIGHTMAN Copyright 0 2004 W Y-VCH Verlag GmbH L Co. KCaA 5 BREMSSTRAHLUNG Radiation due to the acceleration of a charge in the Coulomb field
More informationIf light travels past a system faster than the time scale for which the system evolves then t I ν = 0 and we have then
6 LECTURE 2 Equation of Radiative Transfer Condition that I ν is constant along rays means that di ν /dt = 0 = t I ν + ck I ν, (29) where ck = di ν /ds is the ray-path derivative. This is equation is the
More informationInterstellar Medium Physics
Physics of gas in galaxies. Two main parts: atomic processes & hydrodynamic processes. Atomic processes deal mainly with radiation Hydrodynamics is large scale dynamics of gas. Start small Radiative transfer
More informationFundamental Interactions (Forces) of Nature
Chapter 14 Fundamental Interactions (Forces) of Nature Interaction Gauge Boson Gauge Boson Mass Interaction Range (Force carrier) Strong Gluon 0 short-range (a few fm) Weak W ±, Z M W = 80.4 GeV/c 2 short-range
More informationRadiative Processes in Astrophysics
Radiative Processes in Astrophysics 11. Synchrotron Radiation & Compton Scattering Eline Tolstoy http://www.astro.rug.nl/~etolstoy/astroa07/ Synchrotron Self-Absorption synchrotron emission is accompanied
More informationApplied Nuclear Physics (Fall 2006) Lecture 19 (11/22/06) Gamma Interactions: Compton Scattering
.101 Applied Nuclear Physics (Fall 006) Lecture 19 (11//06) Gamma Interactions: Compton Scattering References: R. D. Evans, Atomic Nucleus (McGraw-Hill New York, 1955), Chaps 3 5.. W. E. Meyerhof, Elements
More informationPhysics Nov Bose-Einstein Gases
Physics 3 3-Nov-24 8- Bose-Einstein Gases An amazing thing happens if we consider a gas of non-interacting bosons. For sufficiently low temperatures, essentially all the particles are in the same state
More information3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell. Multiwavelength Milky Way
Multiwavelength Milky Way PHY3145 Topics in Theoretical Physics Astrophysical Radiation Processes Dr. J. Hatchell, Physics 406, J.Hatchell@exeter.ac.uk Textbooks Main texts Rybicki & Lightman Radiative
More informationRadiation Processes. Black Body Radiation. Heino Falcke Radboud Universiteit Nijmegen. Contents:
Radiation Processes Black Body Radiation Heino Falcke Radboud Universiteit Nijmegen Contents: Planck Spectrum Kirchoff & Stefan-Boltzmann Rayleigh-Jeans & Wien Einstein Coefficients Literature: Based heavily
More informationProperties of Electromagnetic Radiation Chapter 5. What is light? What is a wave? Radiation carries information
Concepts: Properties of Electromagnetic Radiation Chapter 5 Electromagnetic waves Types of spectra Temperature Blackbody radiation Dual nature of radiation Atomic structure Interaction of light and matter
More information1 Radiative transfer etc
Radiative transfer etc Last time we derived the transfer equation dτ ν = S ν I v where I ν is the intensity, S ν = j ν /α ν is the source function and τ ν = R α ν dl is the optical depth. The formal solution
More informationChapter 1: Useful definitions
Chapter 1: Useful definitions 1.1. Cross-sections (review) The Nuclear and Radiochemistry class listed as a prerequisite is a good place to start. The understanding of a cross-section being fundamentai
More informationParticle Interactions in Detectors
Particle Interactions in Detectors Dr Peter R Hobson C.Phys M.Inst.P. Department of Electronic and Computer Engineering Brunel University, Uxbridge Peter.Hobson@brunel.ac.uk http://www.brunel.ac.uk/~eestprh/
More informationPlasma Processes. m v = ee. (2)
Plasma Processes In the preceding few lectures, we ve focused on specific microphysical processes. In doing so, we have ignored the effect of other matter. In fact, we ve implicitly or explicitly assumed
More informationParticles and Waves Particles Waves
Particles and Waves Particles Discrete and occupy space Exist in only one location at a time Position and velocity can be determined with infinite accuracy Interact by collisions, scattering. Waves Extended,
More informationInteraction of Ionizing Radiation with Matter
Type of radiation charged particles photonen neutronen Uncharged particles Charged particles electrons (β - ) He 2+ (α), H + (p) D + (d) Recoil nuclides Fission fragments Interaction of ionizing radiation
More information3 Some Radiation Basics
12 Physics 426 Notes Spring 29 3 Some Radiation Basics In this chapter I ll store some basic tools we need for working with radiation astrophysically. This material comes directly from Rybicki & Lightman
More informationUniversity of Michigan Physics : Advanced Laboratory Notes on RADIOACTIVITY January 2007
University of Michigan Physics 441-442: Advanced Laboratory Notes on RADIOACTIVITY January 2007 1. As usual in the lab, you are forced to learn in several categories at the same time. Your goals in this
More informationParticle nature of light & Quantization
Particle nature of light & Quantization A quantity is quantized if its possible values are limited to a discrete set. An example from classical physics is the allowed frequencies of standing waves on a
More informationThe Strange Physics of Nonabelian Plasmas
The Strange Physics of Nonabelian Plasmas Guy Moore, McGill University Review: What does Nonabelian mean? Instability of a Uniform magnetic field Radiation and the LPM effect Plasma instabilities My units:
More informationCHAPTER 26. Radiative Transfer
CHAPTER 26 Radiative Transfer Consider an incoming signal of specific intensity I ν,0 passing through a cloud (i.e., any gaseous region). As the radiation transits a small path length dr through the cloud,
More informationPHYS 352. Charged Particle Interactions with Matter. Intro: Cross Section. dn s. = F dω
PHYS 352 Charged Particle Interactions with Matter Intro: Cross Section cross section σ describes the probability for an interaction as an area flux F number of particles per unit area per unit time dσ
More informationThe Nature of Light I: Electromagnetic Waves Spectra Kirchoff s Laws Temperature Blackbody radiation
The Nature of Light I: Electromagnetic Waves Spectra Kirchoff s Laws Temperature Blackbody radiation Electromagnetic Radiation (How we get most of our information about the cosmos) Examples of electromagnetic
More informationModal Analysis: What it is and is not Gerrit Visser
Modal Analysis: What it is and is not Gerrit Visser What is a Modal Analysis? What answers do we get out of it? How is it useful? What does it not tell us? In this article, we ll discuss where a modal
More informationFinal Exam - Solutions PHYS/ECE Fall 2011
Final Exam - Solutions PHYS/ECE 34 - Fall 211 Problem 1 Cosmic Rays The telescope array project in Millard County, UT can detect cosmic rays with energies up to E 1 2 ev. The cosmic rays are of unknown
More informationLecture: Scattering theory
Lecture: Scattering theory 30.05.2012 SS2012: Introduction to Nuclear and Particle Physics, Part 2 2 1 Part I: Scattering theory: Classical trajectoriest and cross-sections Quantum Scattering 2 I. Scattering
More informationInteraction of Particles and Matter
MORE CHAPTER 11, #7 Interaction of Particles and Matter In this More section we will discuss briefly the main interactions of charged particles, neutrons, and photons with matter. Understanding these interactions
More informationPHY-105: Nuclear Reactions in Stars (continued)
PHY-105: Nuclear Reactions in Stars (continued) Recall from last lecture that the nuclear energy generation rate for the PP reactions (that main reaction chains that convert hyogen to helium in stars similar
More informationPHYSICS OF HOT DENSE PLASMAS
Chapter 6 PHYSICS OF HOT DENSE PLASMAS 10 26 10 24 Solar Center Electron density (e/cm 3 ) 10 22 10 20 10 18 10 16 10 14 10 12 High pressure arcs Chromosphere Discharge plasmas Solar interior Nd (nω) laserproduced
More informationI. INTRODUCTION AND HISTORICAL PERSPECTIVE
I. INTRODUCTION AND HISTORICAL PERSPECTIVE A. Failures of Classical Physics At the end of the 19th century, physics was described via two main approaches. Matter was described by Newton s laws while radiation
More informationAstrophysics Assignment; Kramers Opacity Law
Astrophysics Assignment; Kramers Opacity Law Alenka Bajec August 26, 2005 CONTENTS Contents Transport of Energy 2. Radiative Transport of Energy................................. 2.. Basic Estimates......................................
More informationThermal radiation (a.k.a blackbody radiation) is the answer to the following simple question:
Thermal radiation (a.k.a blackbody radiation) is the answer to the following simple question: What is the state of the electromagnetic (EM) field in equilibrium with its surroundings at temperature T?
More informationSome fundamentals. Statistical mechanics. The non-equilibrium ISM. = g u
Some fundamentals Statistical mechanics We have seen that the collision timescale for gas in this room is very small relative to radiative timesscales such as spontaneous emission. The frequent collisions
More informationOpacity and Optical Depth
Opacity and Optical Depth Absorption dominated intensity change can be written as di λ = κ λ ρ I λ ds with κ λ the absorption coefficient, or opacity The initial intensity I λ 0 of a light beam will be
More informationWeek 8: Stellar winds So far, we have been discussing stars as though they have constant masses throughout their lifetimes. On the other hand, toward
Week 8: Stellar winds So far, we have been discussing stars as though they have constant masses throughout their lifetimes. On the other hand, toward the end of the discussion of what happens for post-main
More informationThe Cosmic Microwave Background
The Cosmic Microwave Background Class 22 Prof J. Kenney June 26, 2018 The Cosmic Microwave Background Class 22 Prof J. Kenney November 28, 2016 Cosmic star formation history inf 10 4 3 2 1 0 z Peak of
More informationQuantum physics. Anyone who is not shocked by the quantum theory has not understood it. Niels Bohr, Nobel Price in 1922 ( )
Quantum physics Anyone who is not shocked by the quantum theory has not understood it. Niels Bohr, Nobel Price in 1922 (1885-1962) I can safely say that nobody understand quantum physics Richard Feynman
More informationUnits, limits, and symmetries
Units, limits, and symmetries When solving physics problems it s easy to get overwhelmed by the complexity of some of the concepts and equations. It s important to have ways to navigate through these complexities
More informationPhysics 11b Lecture #24. Quantum Mechanics
Physics 11b Lecture #4 Quantum Mechanics What We Did Last Time Theory of special relativity is based on two postulates: Laws of physics is the same in all reference frames Speed of light is the same in
More informationPHYS 231 Lecture Notes Week 3
PHYS 231 Lecture Notes Week 3 Reading from Maoz (2 nd edition): Chapter 2, Sec. 3.1, 3.2 A lot of the material presented in class this week is well covered in Maoz, and we simply reference the book, with
More informationMITOCW watch?v=wr88_vzfcx4
MITOCW watch?v=wr88_vzfcx4 PROFESSOR: So we're building this story. We had the photoelectric effect. But at this moment, Einstein, in the same year that he was talking about general relativity, he came
More informationLecture 15 Notes: 07 / 26. The photoelectric effect and the particle nature of light
Lecture 15 Notes: 07 / 26 The photoelectric effect and the particle nature of light When diffraction of light was discovered, it was assumed that light was purely a wave phenomenon, since waves, but not
More informationRadiative Processes in Astrophysics
Radiative Processes in Astrophysics 9. Synchrotron Radiation Eline Tolstoy http://www.astro.rug.nl/~etolstoy/astroa07/ Useful reminders relativistic terms, and simplifications for very high velocities
More informationModern Physics notes Spring 2005 Paul Fendley Lecture 38
Modern Physics notes Spring 2005 Paul Fendley fendley@virginia.edu Lecture 38 Dark matter and energy Cosmic Microwave Background Weinberg, chapters II and III cosmological parameters: Tegmark et al, http://arxiv.org/abs/astro-ph/0310723
More information2. Passage of Radiation Through Matter
2. Passage of Radiation Through Matter Passage of Radiation Through Matter: Contents Energy Loss of Heavy Charged Particles by Atomic Collision (addendum) Cherenkov Radiation Energy loss of Electrons and
More information