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Ingolf V. Hertel and Claus-Peter Schulz Humboldt Universität zu Berlin and Max Born Institute Berlin-Adlershof Atoms, Molecules and Optical Physics January 13, 2014 Springer 7:57 h (UTC+1) Volume II Molecules and Photons – Spectroscopy and Collisions Preface The two textbooks on Atomic, Molecular and Optical (AMO) physics presented here aim at providing something like the canonical knowledge of modern atomic and molecular physics and to opening a first entry into optical physics and quantum optics. All of these topics constitute a vital area of active and highly productive research in physics. And in spite of, or perhaps even because of its remarkable history the field continues to constitute an indispensable basis for any more profound understanding of nearly all branches of modern physics, physical chemistry and partially even biological and material sciences. Specifically the latter appear to become more and more based on genuine molecular concepts. These textbooks address on the one hand advanced students of physics and physical chemistry, who have to study these topics within their respective curricula. At the same time, however, they should be useful to all those who discover in different contexts that they miss some essential basics from this field and who seek for suitable means to acquire that knowledge. Of course these books also address quite specifically the PhD-student or the young researcher who starts for the first time his or her own activities in the field – or just wants to know more about it. They will find here reliable knowledge and stimulating challenges for their own work. We thus have tried not only to provide the essential basics for working with these topics, but whenever possible also to inform the interested reader about the present state-of-the-art and to allow him a glimpse on today’s cutting edge research. The general remarks as well as details about formats, notation, units, and typography outlined in the preface to Vol. 1 (Hertel and Schulz, 2014) are equally valid for the present Vol. 2. In the following we just give a guide through the contents, as the the readers will find many topics and details far beyond the standard textbooks and routine teachings on AMO science. Chapter 11 resumes the discussion of light (comprising in the broadest sense the whole electromagnetic spectrum) and photons, key themes of these two volumes. The focus is here on lasers (one of the most important tools of modern AMO physics), Gaussian beams, polarization and nonlinear proi ii Preface cesses. In the following Chap. 12 the emphasis is on the properties of photons and coherence. In addition to discussing some basics of quantum optics and its applications, we also complete now the theory of photon induced transitions (Chap. 4) and introduce field quantization which allows us to treat spontaneous emission. After these preparations we are ready to enter into molecular physics and modern molecular spectroscopy. We start in Chap. 13 with diatomic molecules, the most simple prototypes, and enhance our view with ‘real’ polyatomic molecules in Chap. 14 – with a brief excursion into the subject of symmetries. While along this way we have already encountered various comparatively simple examples of molecular spectroscopy, Chapter 15 leads us deep into a variety of sophisticated modern methods of spectroscopy. The field is dominated today by laser based methods, but we also gain some insights e.g. into the possibilities of photoelectron spectroscopy. Three quite detailed Chapters 16-??coll3 are devoted to the present status in the physics of electronic, atomic, molecular and ionic collisions (including collisional ionization) – a topic of great practical importance and with demanding intellectual challenges, both from an experimental and theoretical view point. Chapter 19 gives a down to earth manual for using the density matrix. It also gives a brief look on the theory of measurement, including a powerful method to analyze radiation patterns from anisotropically populated mixtures of excited states. Finally, making use of these concepts, an introduction of the optical Bloch equations is given in Chap. 20, which again addresses many exciting facets of quantum optics with interesting examples. A possible extension of the two volumes published now is under consideration. Such a Vol. 3 would approach modern research even more closely and illuminate some particularly hot and rapidly developing areas such as ultra-cold matter and quantum gases, ultrafast dynamics, attosecond physics, cluster spectroscopy and similar themes. We wish all our readers an exciting and stimulating reading as well as efficient understanding and successful learning. In several readings, we have tried to produce text, mathematical formulas and figures as free from errors as possible. Clearly, this can only be an approximate process. Thus, we encourage the readers to kindly communicate any critical comments, errors or simply even typos which they may discover – and to make suggestions for improvements wherever it appears advisable. We shall correct such errors at the web-site http://staff.mbi-berlin.de/AMO/book-homepage/ if and as soon as they become known to us. As further reading we recommend for comparison or for a deeper look into some specialties the textbooks listed below. . Berlin Adlershof, January 2014 Ingolf V. Hertel and Claus-Peter Schulz References iii Acronyms and terminology AMO: ‘Atomic, molecular and optical’, physics. References Atkins, P. W. and R. S. Friedman: 2010. Molecular Quantum Mechanics. Oxford: Oxford University Press, 2nd edn. Bergmann, L. and C. Schaefer: 1997. Constituents of Matter - Atoms, Molecules, Nuclei and Particles. Berlin, New York: Walter der Gruyter, 902 pages. Blum, K.: 2012. Density Matrix Theory and Applications. Atomic, Optical, and Plasma Physics 64. Berlin, Heidelberg: Springer Verlag, 3rd edn., 343 pages. Born, M. and E. Wolf: 2006. Principles of Optics. Cambridge University Press, 7th (expanded) edn. Bransden, B. H. and C. J. Joachain: 2003. The Physics of Atoms and Molecules. Prentice Hall Professional. Brink, D. M. and G. R. Satchler: 1994. Angular Momentum. Oxford: Oxford University Press, 3 edn., 182 pages. Demtröder, W.: 2010. Atoms, Molecules and Photons. Berlin, Heidelberg, New York: Springer, 2nd edn. Demtröder, W.: 2008a. Laser Spectroscopy, vol. 1: Basic Principles. Berlin, New York: Springer, 4 edn., 457 pages. Demtröder, W.: 2008b. Laser Spectroscopy, vol. 2: Experimental Techniques. Berlin, New York: Springer, 4 edn., 697 pages. Drake, G. W. F., ed.: 2006. Handbook of Atomic, Molecular and Optical Physics. Heidelberg, New York: Springer. Edmonds, A. R.: 1996. Angular Momentum in Quantum Mechanics. Princeton, NJ, USA: Princeton University Press, 154 pages. Hertel, I. V. and C. P. Schulz: 2014. Atoms, Molecules and Optical Physics 1; Atoms and Spectroscopy, vol. 1 of Springer-Textbook. Berlin Heidelberg: Springer, 1st edn. Loudon, R.: 2000. Quantum Theory of Light. Oxford, New York: Oxford University Press, 3rd edn. Mukamel, S.: 1999. Principles of Nonlinear Optical Spectroscopy. Oxford: Oxford University Press, 576 pages. Steinfeld, J. I.: 2005. Molecules and Radiation - 2nd Edition, An Introduction to Modern Molecular Spectroscopy. Mineola, NY: Dover Edition. Weissbluth, M.: 1978. Atoms and Molecules. Student Edition. New York, London, Toronto, Syndey, San Francisco: Academic Press, 713 pages. Acknowledgements Over the past years, many colleagues have encouraged and stimulated us to move forward with this work, and helped with many critical hints and suggestions. Most importantly, we have received a lot of helpful material and state of the art data for inclusion in these textbooks. We would like to thank all those who have in one or the other way contributed to close a certain gap in the standard textbook literature in this area – that is at least what we hope to have achieved. Specifically we mention Robert Bittl, Wolfgang Demtroeder, Melanie Dornhaus, Kai Godehusen, Uwe Griebner, Hartmut Hotop, Marsha Lester, John P. Maier, Reinhardt Morgenstern, Hans-Hermann Ritze, Horst Schmidt-Boecking, Ernst J. Schumacher, Guenter Steinmeyer, Joachim Ullrich, Marc Vrakking und Roland Wester ; their contributions are specifically noted in the respective lists of references. Of course, there all other sources are documented which we have used for informations and which have provided data which we have used to generate the figures in these books. One of us (IVH) is particularly grateful to the Max-Born-Institute for providing the necessary resources (including computer facilities, library access, and office space etc.) for continuing the work on this book after official retirement. iv Contents xxvii Volume II Molecules and Photons – Spectroscopy and Collisions Preface Vol. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iii 11 Lasers, light beams and light pulses . . . . . . . . . . . . . . . . . . . . . . . 11.1 Lasers – a brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Basic principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 FABRY-PÉROT resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Stable, transverse modes and diffraction losses . . . . . . . 11.1.4 The amplifying medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.5 Threshold condition and stationary state . . . . . . . . . . . . 11.1.6 Laser rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.7 Line profiles and hole burning . . . . . . . . . . . . . . . . . . . . . . 11.2 Gaussian beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Diffraction limited profile of a laser beam . . . . . . . . . . . . 11.2.2 FAUNHOFER diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Ray transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Focussing a Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . 11.2.5 Measuring beam profiles with a razor blade . . . . . . . . . . 11.2.6 The M2 factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Polarization and time dependent intensity . . . . . . . . . . . 11.3.2 Lambda-quarter and half-wave plates . . . . . . . . . . . . . . . 11.3.3 STOKES parameters, partially polarized light . . . . . . . . . 11.4 Wave-packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Description of laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Spatial and temporal intensity distribution . . . . . . . . . . 11.4.3 Frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Measuring durations of short laser pulses . . . . . . . . . . . . . . . . . 11.5.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Interferometric measurement . . . . . . . . . . . . . . . . . . . . . . . 11.5.4 Experimental examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Nonlinear processes in Gaussian laser beams . . . . . . . . . . . . . . . 11.6.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Cylindrical geometry (2D geometry) . . . . . . . . . . . . . . . . 11.6.3 Conical geometry (3D geometry) . . . . . . . . . . . . . . . . . . . 11.6.4 Spatially resolved measurements . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 685 687 688 690 693 695 696 698 701 701 707 710 713 718 718 719 720 721 725 729 729 732 733 736 736 737 738 743 745 745 747 748 750 752 753 xxviii Contents 12 Coherence and photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Some basics for quantum optics . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 First-order degree of coherence . . . . . . . . . . . . . . . . . . . . 12.1.3 Quasi-monochromatic light . . . . . . . . . . . . . . . . . . . . . . . . 12.1.4 Temporal or longitudinal coherence . . . . . . . . . . . . . . . . . 12.1.5 Higher-order degree of coherence . . . . . . . . . . . . . . . . . . . 12.1.6 Photon “bunching” experiments . . . . . . . . . . . . . . . . . . . . 12.1.7 Spatial or lateral coherence . . . . . . . . . . . . . . . . . . . . . . . . 12.1.8 Astronomical interferometry . . . . . . . . . . . . . . . . . . . . . . . 12.1.9 HANBURY BROWN-TWISS stellar interferometer . . . . . . 12.1.10 Bunching and anti-bunching . . . . . . . . . . . . . . . . . . . . . . 12.2 Photons, photon states, and radiation modes . . . . . . . . . . . . . . . 12.2.1 Towards quantization of the radiation field . . . . . . . . . . 12.2.2 Modes of the radiation field . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Density of states and black body radiation . . . . . . . . . . . 12.2.4 Number of photons per mode . . . . . . . . . . . . . . . . . . . . . . 12.2.5 The multi-mode field and energy . . . . . . . . . . . . . . . . . . . 12.3 Field quantization and optical transitions . . . . . . . . . . . . . . . . . . 12.3.1 Second quantization and photon number states . . . . . . . 12.3.2 The electric field operator . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 GLAUBER states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Addendum for multi-mode states . . . . . . . . . . . . . . . . . . . 12.3.5 Interaction Hamiltonian for dipole transitions . . . . . . . . 12.3.6 Perturbation theory and spontaneous emission . . . . . . 12.3.7 Spontaneous emission in a cavity . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 755 755 756 758 762 766 768 770 775 779 781 784 785 786 789 790 793 794 794 797 798 801 802 805 811 815 817 13 Diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Characteristic energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Electronic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 Vibrational energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 Rotational energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Molecular potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 General form of molecular potentials . . . . . . . . . . . . . . . . 13.2.3 Nuclear wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 Harmonic potential and harmonic oscillator . . . . . . . . . . 13.2.5 Morse potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.6 VAN DER WAALS molecules . . . . . . . . . . . . . . . . . . . . . . . 13.3 Nuclear motion: rotation and vibration . . . . . . . . . . . . . . . . . . . . 13.3.1 SCHRÖDINGER equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Rigid rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 820 820 822 822 822 823 823 824 826 827 829 831 834 834 836 Contents xxix 13.3.3 Population of rotational levels and nuclear spin . . . . . . 13.3.4 Specific heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.6 Non-rigid rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.7 DUNHAM coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Dipole transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Rotational transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Centrifugal distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.3 STARK effect: polar molecules in an electric field . . . . . . 13.4.4 Vibrational transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 Vibration-rotation spectra . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.6 RYDBERG-KLEIN-REES method . . . . . . . . . . . . . . . . . . . . 13.5 Molecular orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Variational method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Specialization for H+ 2 ............................. 13.5.3 Charge exchange in the H+ 2 system . . . . . . . . . . . . . . . . . 13.5.4 MOs for homonuclear molecules . . . . . . . . . . . . . . . . . . . . 13.6 Construction of total angular momentum states . . . . . . . . . . . . 13.6.1 Total orbital angular momentum . . . . . . . . . . . . . . . . . . . 13.6.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.3 Total angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.4 HUND’s coupling cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.5 Reflection symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.6 Lambda-type doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.7 Example H2 – MO ansatz . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.8 Valence bond theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.9 Nitrogen and oxygen molecule . . . . . . . . . . . . . . . . . . . . . 13.7 Heteronuclear molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.1 Energy terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.2 Filling the orbitals with electrons . . . . . . . . . . . . . . . . . . . 13.7.3 Lithiumhydrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.4 Alkali halides: ionic bonding . . . . . . . . . . . . . . . . . . . . . . . 13.7.5 Nitrogen monoxide, NO . . . . . . . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 841 843 846 848 849 850 852 853 855 856 860 861 861 863 867 872 879 879 880 881 882 884 888 888 893 896 897 897 899 900 903 907 909 910 14 Polyatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Rotation of polyatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Spherical rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 Symmetric rigid rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.4 Asymmetric rigid rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Vibrational modes of polyatomic molecules . . . . . . . . . . . . . . . . 14.2.1 Normal modes of vibration . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Energies and transitions of normal modes . . . . . . . . . . . 913 913 913 916 916 918 920 921 923 xxx Contents 14.2.3 Linear, triatomic molecules AB2 . . . . . . . . . . . . . . . . . . . . 14.2.4 Nonlinear triatomic molecules AB2 . . . . . . . . . . . . . . . . . 14.2.5 Inversion vibration in ammonia . . . . . . . . . . . . . . . . . . . . 14.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Symmetry operations and elements . . . . . . . . . . . . . . . . . 14.3.2 Point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Eigenstates of polyatomic molecules . . . . . . . . . . . . . . . . 14.3.4 JAHN-TELLER effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Electronic states of some polyatomic molecules . . . . . . . . . . . . . 14.4.1 A first example: H2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Hybridization – sp3 orbitals . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Electronic states of NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 sp2 hybrid orbitals forming double bonds from . . . . . . 14.4.5 Triple bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Conjugated molecules and the HÜCKEL method . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Molecular spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Microwave spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Infrared spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 FOURIER transform infrared spectroscopy . . . . . . . . . . . 15.3.3 Infrared action spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 15.4 Electronic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 FRANCK-CONDON factors . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Selection rules for electronic transitions . . . . . . . . . . . . . 15.4.3 Radiationless transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.4 Rotational excitation in electronic transitions . . . . . . . . 15.4.5 Classical emission and absorption spectroscopy . . . . . . . 15.5 Laser spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Laser induced fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 REMPI for a ‘simple’ triatomic molecule . . . . . . . . . . . . 15.5.3 Cavity ring down spectroscopy . . . . . . . . . . . . . . . . . . . . . 15.5.4 Spectroscopy of small free biomolecules . . . . . . . . . . . . . 15.5.5 Other important methods . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 RAMAN spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.2 Classical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.3 Quantum mechanical theory . . . . . . . . . . . . . . . . . . . . . . . 15.6.4 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.5 Examples of RAMAN spectra . . . . . . . . . . . . . . . . . . . . . . . 15.6.6 Nuclear spin statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Nonlinear spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924 927 929 934 935 935 939 944 948 948 952 956 957 958 959 967 968 971 971 974 978 978 980 984 987 987 991 993 995 996 1000 1000 1002 1010 1013 1017 1018 1018 1020 1021 1026 1027 1029 1032 Contents xxxi 15.7.1 Some basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Photoelectron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.1 Experimental basis and the principle of PES . . . . . . . . . 15.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.3 TPES, PFI, ZEKE, KETOF, MATI . . . . . . . . . . . . . . . . 15.8.4 PES for negative ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.5 PEPICO, TPEPICO and variations . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 1037 1039 1039 1042 1047 1049 1050 1056 1061 16 Basics of atomic collision physics: elastic processes . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Integral and total cross sections . . . . . . . . . . . . . . . . . . . . 16.1.2 Principle of detailed balance . . . . . . . . . . . . . . . . . . . . . . . 16.1.3 Integral elastic cross sections . . . . . . . . . . . . . . . . . . . . . . . 16.2 Differential cross sections and kinematics . . . . . . . . . . . . . . . . . . 16.2.1 Experimental considerations . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Collision kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Mass selection of atomic clusters . . . . . . . . . . . . . . . . . . . 16.3 Elastic scattering and classical theory . . . . . . . . . . . . . . . . . . . . . 16.3.1 The differential cross section . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 The optical rainbow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.3 The classical deflection function . . . . . . . . . . . . . . . . . . . . 16.3.4 Rainbows and other remarkable oscillations . . . . . . . . . . 16.4 Quantum theory of elastic scattering . . . . . . . . . . . . . . . . . . . . . . 16.4.1 General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Angular momentum and impact parameter . . . . . . . . . . 16.4.3 Partial wave expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.4 Semiclassical approximation . . . . . . . . . . . . . . . . . . . . . . . 16.4.5 Scattering phase shifts at low kinetic energies . . . . . . . 16.4.6 Scattering matrices for pedestrians . . . . . . . . . . . . . . . . . 16.5 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 Types and phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.3 An example: electron helium scattering . . . . . . . . . . . . . . 16.6 BORN approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 1069 1071 1073 1075 1079 1079 1082 1086 1088 1089 1089 1092 1095 1104 1105 1107 1109 1111 1114 1118 1122 1122 1124 1127 1131 1134 1135 17 Inelastic collisions – a first overview . . . . . . . . . . . . . . . . . . . . . . 17.1 Simple models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Reactions without threshold energy . . . . . . . . . . . . . . . . . 17.1.2 The absorbing sphere model . . . . . . . . . . . . . . . . . . . . . . . 17.1.3 An example: charge exchange . . . . . . . . . . . . . . . . . . . . . . 1139 1139 1139 1141 1142 xxxii Contents 17.1.4 MASSEY criterium for inelastic collisions . . . . . . . . . . . . 17.2 Excitation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Impact excitation by electrons and protons . . . . . . . . . . 17.2.2 Electron impact excitation of He . . . . . . . . . . . . . . . . . . . 17.2.3 Finer details in e− + He impact excitation . . . . . . . . . . . 17.2.4 Electron collisions with rare gases . . . . . . . . . . . . . . . . . . 17.2.5 Electron impact at atomic mercury – the FRANCK-HERTZ experiment . . . . . . . . . . . . . . . . . . . . . . . 17.2.6 Molecular excitation by electron impact . . . . . . . . . . . . . 17.2.7 Threshold laws for excitation and ionization . . . . . . . . . 17.3 Scattering theory for the multichannel problem . . . . . . . . . . . . . 17.3.1 General formulation of the problem . . . . . . . . . . . . . . . . . 17.3.2 Potential matrix and coupling elements . . . . . . . . . . . . . 17.3.3 The adiabatic representation . . . . . . . . . . . . . . . . . . . . . . . 17.3.4 The diabatic representation . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Semiclassical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Time dependent SCHRÖDINGER equation . . . . . . . . . . . . 17.4.2 Coupling elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 Solution of the coupled differential equations . . . . . . . . . 17.4.4 LANDAU-ZENER formula . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.5 A simple example: Na+ +Na(3p) . . . . . . . . . . . . . . . . . . . . 17.4.6 STÜCKELBERG oscillations . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Collision processes with highly charged ions (HCI) . . . . . . . . . . 17.5.1 Above-barrier model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.2 An experiment on electron exchange . . . . . . . . . . . . . . . . 17.5.3 HCI collisions and ultrafast dynamics . . . . . . . . . . . . . . . 17.6 Surface hopping, conical intersections and reactions . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143 1146 1146 1147 1150 1151 1153 1155 1156 1159 1159 1163 1164 1167 1170 1170 1172 1174 1175 1178 1182 1186 1187 1190 1192 1193 1196 1197 18 Electron impact excitation and ionization . . . . . . . . . . . . . . . . . 18.1 Formal scattering theory and applications . . . . . . . . . . . . . . . . . 18.1.1 Close-coupling equations . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 Theoretical methods and experimental examples . . . . . 18.2 BORN approximation for inelastic collisions . . . . . . . . . . . . . . . . 18.2.1 FBA scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.3 BORN approximation and RUTHERFORD scattering . . . 18.2.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Generalized oscillator strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Expansion for small momentum transfer . . . . . . . . . . . . . 18.3.3 Explicit evaluation of GOS for an example . . . . . . . . . . 18.3.4 Integral inelastic cross sections . . . . . . . . . . . . . . . . . . . . . 18.4 Electron impact ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 1203 1203 1207 1213 1213 1215 1216 1217 1218 1218 1219 1221 1222 1222 Contents xxxiii 18.4.1 18.4.2 18.4.3 18.4.4 Integral cross sections and the LOTZ formula . . . . . . . . . SDCS: Energy partitioning between the electrons . . . . Behaviour at the ionization threshold . . . . . . . . . . . . . . . DDCS: Double-differential cross section and the BORN-BETHE approximation . . . . . . . . . . . . . . . 18.4.5 TDCS: Triple-differential cross sections . . . . . . . . . . . . . . 18.4.6 Electron momentum spectroscopy (EMS) . . . . . . . . . . . . 18.5 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225 1227 1229 1232 1237 1246 1251 1255 1257 19 The density matrix – a first approach . . . . . . . . . . . . . . . . . . . . . 19.1 Some terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1 Pure and mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Density operator and density matrix . . . . . . . . . . . . . . . . 19.1.3 Matrix representation for selected examples . . . . . . . . . . 19.1.4 Coherence and degree of polarization . . . . . . . . . . . . . . . 19.2 Theory of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 State selector and analyzer . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 Interaction experiment with state selection . . . . . . . . . . 19.3 Selected examples of the density matrix . . . . . . . . . . . . . . . . . . . 19.3.1 Polarization matrix and STOKES parameters . . . . . . . . . 19.3.2 Atom in an isolated 1 P state . . . . . . . . . . . . . . . . . . . . . . . 19.4 Angular distribution and polarization of radiation . . . . . . . . . . 19.4.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . 19.4.2 General discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.3 Details of the evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263 1265 1265 1271 1272 1275 1278 1278 1280 1286 1286 1292 1301 1301 1306 1309 1313 1313 20 Optical BLOCH equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Two level system in quasi-monochromatic light . . . . . . . . . . . . 20.2.1 Dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.2 RABI frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.3 Rotating wave approximation . . . . . . . . . . . . . . . . . . . . . . 20.2.4 The coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 MOLLOW triplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 AUTLER-TOWNES effect . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Quantum systems in strong electromagnetic fields . . . . . . . . . . 20.4.1 Temporal evolution of the density matrix . . . . . . . . . . . . 20.4.2 Optical BLOCH equations for a two state system . . . . . 20.5 Excitation with continuous wave (cw) light . . . . . . . . . . . . . . . . 20.5.1 Relaxed steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315 1315 1319 1319 1320 1321 1322 1324 1324 1326 1328 1328 1329 1332 1332 xxxiv Contents 20.5.2 Saturation broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.3 Broad band and narrow band excitation . . . . . . . . . . . . . 20.5.4 Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.5 Continuous excitation without relaxation . . . . . . . . . . . . 20.5.6 Continuous excitation with relaxation . . . . . . . . . . . . . . 20.6 BLOCH equations and short pulse spectroscopy . . . . . . . . . . . . . 20.6.1 Excitation with ultrafast laser pulses . . . . . . . . . . . . . . . . 20.6.2 Ultrafast spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6.3 Rate equations and optical BLOCH equations . . . . . . . . 20.7 STIRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7.1 Three level system in two laser fields . . . . . . . . . . . . . . . . 20.7.2 Energy splitting and state evolution . . . . . . . . . . . . . . . . 20.7.3 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332 1334 1335 1336 1338 1339 1339 1341 1342 1347 1347 1349 1351 1355 1355 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359 K First BORN approximation for e+Na(3s)→e+Na(3p) . . . . . . K.1 Evaluation of the generalized oscillator strength . . . . . . . . . . . . K.2 Integration of the differential cross section . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362 1362 1365 1365 1366 L Guiding, detecting and energy analysis of electrons and ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L.1 SEM, channeltron, microchannel plate . . . . . . . . . . . . . . . . . . . . L.2 Index of refraction, lenses and directional intensity . . . . . . . . . . L.3 Hemispherical energy selector . . . . . . . . . . . . . . . . . . . . . . . . . . . . L.4 Magnetic bottle and other time of flight methods . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367 1367 1372 1374 1377 1379 1380 M Statistical tensor and state multipoles . . . . . . . . . . . . . . . . . . . M.1 Multipole expansion of the density matrix . . . . . . . . . . . . . . . . . M.2 State multipoles and expectation values of multipole tensor operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.3 Recoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382 1382 N 1390 1390 1393 1396 1397 1398 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N.1 A standard case: Na(3 2 S1/2 ↔ 3 2 P3/2 ) . . . . . . . . . . . . . . . . . . . . N.2 Multipole moments and their experimental detection . . . . . . . . N.3 Optical pumping with two frequencies . . . . . . . . . . . . . . . . . . . . . Acronyms and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385 1387 1389