Atomic Transitions II & Molecular Structure

Transcription

1 Atomic Transitions II & Molecular Structure

2 Atomic Transitions II Transition Probability Dipole Approximation Line Broadening

3 Transition Probability: The Hamiltonian To calculate explicitly the transition probabilities in an atom, we start from the Hamiltonian and we split it in a time independent (H0) and time dependent (H1) component: The atomic Hamiltonian has the form (in the non-relativistic limit): Here p is the momentum operator and A the vector potential. The terms containing A give a very different contribution since their ratio is usually very large:

4 Transition Probability: The Hamiltonian This term is usually >>1 (unless the photon density is >> 10^25 cm^-3, i.e., 8 order of mangitude larger than on the surface of the Sun). So we will ignore the A^2 terms. These A^2 terms represent processes that occur when an atom simultaneously absorbs two or more photons. Such multi-photon processes can lead to ionization of atoms or dissociation of molecules in intense laser fields, and phenomena such as free-to-free transitions in which an electron absorbs successive quanta as it flies out from the field of an atom. Our Hamiltonian can now be written: The time-dependent part (H1) will thus be perturbation. which can be considered a The Schroedinger equation can be written as: where the E_k energies are the solutions (eigenvalues) of the time-independent part:

5 Transition Probability If one solves now the time independent part one can find the probability per unit time for a transition from state i to state f: Here: and The interval 0 T is the time interval during which the perturbation (H1) is active. In several cases, the perturbation can be thought of as the action of an external field (e.g., a photon) incident upon the atom. If one interchanges the i and f symbols, it s easy to show that: which is the principle of detailed balance (see Lecture 3).

6 (Electric) Dipole Approximation If we take the potential vecotor A and we define it in the usual way then we can get rid of the exponential by Taylor expanding it: This is something allowed since If we take only the leading order of the expansion ( = 1) then we are in the so-called electric dipole approximation. This is an important point because the transition rates depend on A and thus, if we Taylor expand, on the order at which we truncate the exponential. The transition rates which are not zero in the dipole approximation are the strongest. If the transition rate is equal to zero then one needs to select higher orders in the Taylor expansion and re-calculate the transition rates. The selection rules we discussed briefly in Lecture 11 refer to this approximation. The forbidden transitions we discussed are those which occur outside the electric dipole approximation, i.e., they have a very small probability to occur and this probability is exactly zero if =1

7 Line Broadening I : Doppler Broadening Δ ν=ν0 vz c If the atom is in a thermal distribution, then the Maxwell Boltzmann distribution associates a certain velocity v to each atom. The number of atoms with velocity between and will be proportional to: The profile function will be: where

8 Line Broadening II: Natural Broadening The uncertainty relation means that the emission/absorption line will always have a certain natural width due to the fact that the emission/absorption process is not instantaneous but occurs on a timescale (See for example Lecture 7, slides 34-39). Γ= l A ul

9 Line Broadening III: Collisional Broadening Collisional broadening occurs become atoms collide and the phas of the emitted radiation can be suddenly altered. If he atom collides with frequency (i.e., collisions per unit time) then the profile is still Lorentizian:

10 Molecular Structure & Molecular Transitions

11 Very complex molecules are found in star forming regions and throughout the Universe!

12 amino acid, glycine (NH2CH2COOH) Large Molecule Heimat

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31 Occupation of Energy Levels

32

33

34 Summary of Radiation Properties Optically thick Thermal Blackbody Bremsstrahlung Synchrotron Inverse Compton YES NO Maxwellian distribution of velocities YES YES NO Relativistic speeds YES YES Matter AND radiation in thermal equilibrium Radiation emitted by Radiation emitted accelerating particles by accelerated Main Properties Matter in thermal equilibrium particles in B field. Relativistic electron/photon collisions Rules of thumb: 1. Blackbody is always thermal, but thermal radiation is not always blackbody (e.g., thermal Bremsstrahlung) 2. Bremsstrahlung can be thermal or non-thermal. 3. Bremsstrahlung becomes blackbody when optical depth >>1. 4. Synchrotron emission exists only for non-thermal distribution of particles.

35

36 Rules for the Final Exam (16 Dec. 2016) 1. You will get 3-4 exercises with a level of difficulty comparable to the exercises given during the werkcolleges. This does not mean that the exercises will be those of the werkcolleges. 2. You pass the exam if you score a minimum of 5.5 points. Below 5.5 you need to do the retake on January. 3. If you have done the tests, then I will calculate an average of your scores on your best 9 scores. This means that if you have done 8 or less test I will count the missing ones as zero. 4. The final score of the exam will be: A. Score test > Score written exam 30% score tests + 70% score written exam B. Score test < Score written exam 100% score exam. 5. If you fail the exam, then at the retake the final score is calculated as 100% score retake (i.e., the test score does not count).

37 How to prepare for the exam 1. You have two examples of past exams, so you can check at home if you are comfortable with the level of the exercises. If you are not please contact me and take an appointment with me (or the TAs) so we can go together through the unclear points. I am here to help you, don t be shy, it s your exam! 2. In the exam there is a List of Equations that you will be using during the exam. The exam is a close-book exam, so you re not allowed to bring any material beside a calculator and your exam papers. 3. Do not use a pencil, please use a pen, whatever color you prefer. 4. You are expected to know the spectral shapes of each radiative process. 5. You are expected to know the summary table on radiative processes and on Compton scattering. 6. You re not expected to remember values of constants, emissivity, absorption coefficients etc., those will all be given in the List of Equations 7. You are allowed to ask questions/clarifications during the exam, so please do so. Better to ask than to miss the opportunity of receiving some help and do a good exam.

38 Some extra suggestions During the course we discussed many astrophysical examples of radiative processes. Several of these were then discussed in more detail during the Werkcolleges. Sometimes extra examples were given in class together with some quantitative calculations: Go through these examples given during the lectures, especially the following: - Lecture 4: superluminal motion in the M87 jet - Lecture 5: Cooling time of HII regions & ICM - Lecture 6: Source function and ALMA, Orion Nebula - Lecture 7: LHC - Lecture 9: Inverse Compton for an electron - Lecture 11: Iron Kalpha line