DOCSLIB.ORG

Lecture 21: Hyperfine Structure of Spectral Lines

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 21: Hyperfine Structure of Spectral Lines Lecture 21: Hyperfine Structure of Spectral Lines: Page-1 In this lecture, we will go through the hyperfine structure of atoms. Various origins of the hyperfine structure are discussed The coupling of nuclear and electronic total angular momentum is explained. Page-2 When individual multiplet ( JJ→ transitions) components are examined with spectral apparatus of the highest possible resolution, it is found that in many atomic spectra each of these components is still further split into a number of components lying extremely close together. This splitting is called hyperfine structure. − The magnitude of the splitting is ~2cm 1 . Hyperfine structure is caused by properties of the atomic nucleus. Isotopic effect: Heavier isotopes present 1 in 5000 in ordinary hydrogen. Therefore, different isotopes of same element have slightly different spectral lines. Consider 1H (hydrogen) and 2H (deuterium): 1 71− RRH = ∞ =1.09677 × 10 meter 1+ m M H 1 71− RRD = ∞ =1.097074 × 10 meter 1+ m M D The wavelength difference is therefore: λD ∆=λλHDH − λ = λ1 − λH R =λ 1 − H H R Figure shows the Balmer D line of H & D This is known as isotope shift. observed calculated −−11 ()cm ()cm Hα 1.79 1.787 Hβ 1.33 1.323 Hγ 1.19 1.182 Hδ 1.12 1.117 Page-3 A quantitative explanation of the isotope effect is not simple, exception H atom. For the heavier elements the effect is traced back to the change of nucleus radius with mass. Sm150− Sm 152 is double that of Sm152− Sm 154 . Usual increase is not from Sm150→ Sm 152 . In many cases the isotope effect is not sufficient to explain the hyperfine structure. The number of hyperfine structure components is often considerably greater than the number of isotopes. In particular, elements which have only one isotope in appreciable amount also show hyperfine structure splitting. Likewise, the number of components of different lines is frequently quite different for one and the same element. These hyperfine structures can be quantitatively explained, when it is assumed that the “atomic nucleus possess an intrinsic angular momentum with which is associated a magnetic moment”. This angular momentum can have different magnitudes for different nuclei and also of course, for different isotope of the same element. This is known as Nuclear spin . Page-4 Magnetic moment & Angular momentum of the nucleus. Nucleus consists of Proton & Neutron. Proton: 1 (i) Possesses angular momentum I described by the spin quantum number ; this angular P 2 momentum obeys the general rules of quantization. 1 Component along oz-axis (I ) = ± P z 2 11 Magnitude IP = +1 22 (ii) Possesses a magnetic moment µP parallel and in the same sense as its angular momentum IP . e Magnetic momentum of proton µµ=( ) =2.79 = 2.79 µN PPz 2MK B em 1 µ N = nuclear magneton =≈=µµ B 2MKBB M 1836 µ So, µ N = B B 1836 Neutron: also possesses magnetic moment µµ= −1.913 N Neu B But µµNeu+≠ P total magnetic moment of the nucleus. Page-5 The structure of the nucleus is complex. Inter nuclear forces are non-central forces involving angles between the magnetic moments and the radius vector joining the nucleus. Further more, within the nucleus the nucleus possess an orbital angular momentum which can be zero for certain nuclei. The nuclear magnetic moment µN is related to nuclear angular momentum I, µµN= gI IB 1 = gNNµ I So, g= g N IB II1836 ggor N is called nuclear Lande’ factor. I I The general adopted sign for gI : The Lande’ factor is positive when the nuclear magnetic moment and angular momentum are in the same direction, and is considered negative when in the opposite direction. I I µN µN gII<<0,µ 0 gII>>0,µ 0 Page-6 Nuclear spin and Magnetic moment: (1) All isotopes having an even mass no. A and an even atomic number z , have zero nuclear spin and zero nuclear magnetic moment. 4 16 20 Example: 2He; 8 O ; 10 Ne ; ..... (2) All isotopes having an even mass number A and an odd atomic number z , have an integral nuclear spin. 2 6 10 135D, I= 1 ; Li , I = 1 ; B , I = 3 ; ..... (3) All isotopes having an odd mass number A , have a half integral nuclear spin. 1 113 39 3 1H,;,;, I= 2 He I= 19 K I = 222 1s 12 ss [ _4] s Page-7 Magnetic field due the orbital motion of electron: A point charge is qe= − is moving in a classical orbit with a velocity V . At a given instant, the field it creates at the nucleus, is µµid×− r r µ q B =00 =qV ×= 0m r × V 44ππr3 rr 33 4 πm µ q l = 0 4π m r3 r is directed from the nucleus towards the charge q . Hence the magnetic field due to the orbital motion of the electron is µµ00q 11 BllB= = − 2µ l 44ππm rr33 The interaction energy between the nuclear magnetic moment and the orbital motion of electron µ0 2 1 El=−⋅=µµ N B l2 g IB lI ⋅ 4π r3 Using special case of Wigner-Eckart theorem, ( Lande formula) Jm|| A⋅ J Jm Jm'| A | Jm = Jm'| J | Jm JJ( +1) jm|| l⋅ j jm jm'| l | jm = jm'| j | jm jj( +1) lj. So we can substitute, lj= and we get jj(+ 1) µµ0022 1 lj.1 El=2 g IBµµ l ⋅= I 2. gIB (Ij) 4ππrr334 jj(+ 1) ……………………..(21.1) Page-8 The interaction energy between two magnetic dipole moments µN and µS separated by r is given by µ µµ. 3(µµ⋅⋅rr)( ) =0 NS + NS ESpin 35 …………………………………(21.2) 4π rr Substituting the value of µN and µS in this equation, we get 2 µµgg Is. 3(Ir⋅⋅)( sr) E =0 ISB −+ ……………………………..(21.3) Spin 4π rr35 Is. Let us take the first term r3 jm|| s⋅ j jm Using the relation jm'| s | jm = jm'| j | jm jj( +1) sj. So we can substitute, sj= and we get jj(+ 1) 2 I. s js .1. ls+ s 1 = (.)Ij = (.)Ij r3 jj(++ 1) r 33 jj ( 1) r 3(Ir⋅⋅)( sr) Now let us take the second term r5 rj. we can substitute, rj= and we get jj(+ 1) 33(Ir⋅⋅)( sr) ( I ⋅⋅ j)( sr) rj. = r55r jj(+ 1) Page-9 jr.= ( l + s ). r = + Now, lr.. sr =0. + sr = sr. 2 33(Ir⋅⋅)( sr) ( I ⋅⋅ j)( sr) So, = r55 jj(+ 1) r Is. 3(Ir⋅⋅)( sr) Substituting the values in equation-21.3 of and , we get the interaction r3 r5 energy for spin 2 µµgg Is. 3(Ir⋅⋅)( sr) E =0 ISB −+ Spin 4π rr35 2 2 2 µµgg 3(I⋅⋅ j)( sr) ls.1+ s = 0 ISB− (.)Ij …………………..(21.4) 4π jj(++ 1) r53 jj ( 1) r 2 µµgg 2 (Ij⋅ ) 3(sr⋅ ) ls. + s2 = 0 ISB − π + 53 4 jj ( 1) r r Page-10 Now we will calculate the total interaction energy due to electron orbital (equation 21.1) and spin (equation 21.4) EHF= EE l + Spin 2 µµl.1 j 2g µ 2 (Ij⋅ ) 3(sr⋅ ) ls.+ s2 =+−002.gµ 2 Ij IB ππIB ++3( ) 53 4jj ( 1) r 4jj ( 1) r r Here we have substitutes gS = 2 for the electron. So 2 µ 2g µ 2 (Ij⋅ ) l.. j 3(sr⋅ ) ls+ s2 E =0 IB +− HF π + 35 3 4 jj ( 1) r r r 2 222 µ 2g µ (Ij⋅ ) l.. s++ l 3(sr⋅ ) ls s =0 IB +− π + 3 53 4 jj ( 1) r r r 2 µ 2g µ 2 (Ij⋅ ) ls223(sr⋅ ) =0 IB +− π + 3 53 4 jj ( 1) r r r 1 2 11 3 Substituting sr. = rand s= ss( += 1) + 1 =, we get 2 22 4 2 µ 2g µ 2 (Ij⋅ ) ll(+ 1)3(r) 3 E =0 IB +− HF π + 3 53 4 jj ( 1) r 4 r 4 r µ 2g µ 2 (Ij⋅ ) ll(+ 1) = 0 IB …………………….(21.5) 3 4π jj (+ 1) r µ 2g µ 2 ll(+ 1) 1 = 0 IB ⋅ 3 (Ij) 4π jj (+ 1) r Page-11 So the Hamiltonian including the hyperfine interaction for one electron system is HH=++0 HSpin− Orbit H Hyperfine Here the hyperfine interaction is coupling the total angular momentum of the electron j and the nuclear angular momentum I. So we need the new angular momentum F which will be the good quantum number for the total Hamiltonian. So we define, F= jI + and the eigenfunction is FmF which will be the coupled state arising from the uncoupled states of jmjI Im The interaction energy EHF= A HF′ Ij ⋅ AHF′ → constant, characteristic of the level jl and . Note that the value of AHF = 0 for l = 0, i.e. for the s-states. 2 However, experimentally splitting is observed for the S 1 state of hydrogen. This can not be 2 explained by this classical explanation. However, starting from the Dirac equation, if one evaluates the Hamiltonian for the hyperfine interaction including the vector potential (Reference: Atoms and Molecules by M. Weissbluth), it becomes µ 2g µ 2 ( +1) 18 π = 0 IB ⋅+δ ⋅ Hh 3 I j( r)( Is) 4π jj( + 13) r ( +1) =A I⋅+ j AIs ⋅= A′ I ⋅+ j AIs ⋅……………………(21.6) HF jj( +1) F HF F The first term is the dipole-dipole interaction with corresponding to classical expression as we derived earlier. The last term is known as Fermi Contact Interaction term, it has no classical analog and contributes only for s-states. Since ψ (0) at r = 0 for non-s states is zero, Fermi contact term goes to zero for non-s states.
Load more

Recommended publications
  • Electric Quadrupole Interaction in Cubic BCC Α-Fe
    Electric quadrupole interaction in cubic BCC α-Fe A. Błachowski 1, K. Kom ędera 1, K. Ruebenbauer 1* , G. Cios 2, J. Żukrowski 2, and R. Górnicki 3 1Mössbauer Spectroscopy Division, Institute of Physics, Pedagogical University ul. Podchor ąż ych 2, PL-30-084 Kraków, Poland 2AGH University of Science and Technology, Academic Center for Materials and Nanotechnology Av. A. Mickiewicza 30, PL-30-059 Kraków, Poland 3RENON ul. Gliniana 15/15, PL-30-732 Kraków, Poland *Corresponding author: [email protected] PACS: 76.80.+y, 75.50.Bb, 75.70.Tj Keywords: cubic ferromagnet, electric field gradient (EFG), Mössbauer spectroscopy Short title: EFG in α-Fe Abstract Mössbauer transmission spectra for the 14.41-keV resonant line in 57 Fe have been collected at room temperature by using 57 Co(Rh) commercial source and α-Fe strain-free single crystal as an absorber. The absorber was magnetized to saturation in the absorber plane perpendicular to the γ-ray beam axis applying small external magnetic field. Spectra were collected for various orientations of the magnetizing field, the latter lying close to the [110] crystal plane. A positive electric quadrupole coupling constant was found practically independent on += 19 -2 the field orientation. One obtains the following value Vzz )4(61.1 x 10 Vm for the (average) principal component of the electric field gradient (EFG) tensor under assumption that the EFG tensor is axially symmetric and the principal axis is aligned with the magnetic hyperfine field acting on the 57 Fe nucleus. The nuclear spectroscopic electric quadrupole moment for the first excited state of the 57 Fe nucleus was adopted as +0.17 b.
    [Show full text]
  • Fundamental Nature of the Fine-Structure Constant Michael Sherbon
    Fundamental Nature of the Fine-Structure Constant Michael Sherbon To cite this version: Michael Sherbon. Fundamental Nature of the Fine-Structure Constant. International Journal of Physical Research , Science Publishing Corporation 2014, 2 (1), pp.1-9. 10.14419/ijpr.v2i1.1817. hal-01304522 HAL Id: hal-01304522 https://hal.archives-ouvertes.fr/hal-01304522 Submitted on 19 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Fundamental Nature of the Fine-Structure Constant Michael A. Sherbon Case Western Reserve University Alumnus E-mail: michael:sherbon@case:edu January 17, 2014 Abstract Arnold Sommerfeld introduced the fine-structure constant that determines the strength of the electromagnetic interaction. Following Sommerfeld, Wolfgang Pauli left several clues to calculating the fine-structure constant with his research on Johannes Kepler’s view of nature and Pythagorean geometry. The Laplace limit of Kepler’s equation in classical mechanics, the Bohr-Sommerfeld model of the hydrogen atom and Julian Schwinger’s research enable a calculation of the electron magnetic moment anomaly. Considerations of fundamental lengths such as the charge radius of the proton and mass ratios suggest some further foundational interpretations of quantum electrodynamics.
    [Show full text]
  • ß-NM R Study of the Electric Field Gradient in the Metallic Intercalation
    ß-NMR Study of the Electric Field Gradient in the Metallic Intercalation Compound LiC6 P. Freiländer, P. Heitjans, H. Ackermann, B. Bader, G. Kiese, A. Schirmer and H.-J. Stockmann Fachbereich Physik, Universität Marburg, West Germany and Institut Laue-Langevin, F-38042 Grenoble Cedex, France Z. Naturforsch. 41 a, 109 -1 1 2 (1986); received July 22, 1985 In the layered compound LiC 6 polarized /?-active 8Li probe nuclei were produced by capture of polarized thermal neutrons. Nuclear magnetic resonance spectra of 8Li were recorded via the /5-radiation asymmetry. The 8 Li quadrupole coupling constant e2qQ/h, measured in the temperature range 7 = 5... 500 K decreases with increasing 7 from 45.2(8) to 22.8(4) kHz. Anomalies in the overall temperature dependence are discussed in terms of phase transitions proposed for LiC6. 1. Introduction graphite the carbon layers are stacked in direct registry with respect to each other. The lithium A characteristic feature of graphite intercalation atoms, being sandwiched between two carbon layers compounds (GICs) is the variety of ordered struc­ such that one-third of the C hexagons have Li tures and phase transitions observed. A GIC of atoms directly above and below, form a 2D hexa­ stage n is formed by a periodic arrangement of one gonal superlattice (j/3 x |/3 30° structure: the Li- layer of intercalant followed by n layers of graphite. superlattice vectors are measured in units of the Both intercalant in-plane order and layer sequence graphite lattice, their direction is obtained by 30° are known to undergo various phase transitions for rotation of the graphite lattice vectors).
    [Show full text]
  • Two Photon Processes Within the Hyperfine Structure of Hydrogen
    University of Windsor Scholarship at UWindsor Electronic Theses and Dissertations Theses, Dissertations, and Major Papers 12-10-2018 Two Photon Processes within the Hyperfine Structure of Hydrogen Spencer Dylan Percy University of Windsor Follow this and additional works at: https://scholar.uwindsor.ca/etd Recommended Citation Percy, Spencer Dylan, "Two Photon Processes within the Hyperfine Structure of Hydrogen" (2018). Electronic Theses and Dissertations. 7622. https://scholar.uwindsor.ca/etd/7622 This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email ([email protected]) or by telephone at 519-253-3000ext. 3208. Two Photon Processes within the Hyperfine Structure of Hydrogen by Spencer Percy A Thesis Submitted to the Faculty of Graduate Studies through the Department of Physics in Partial Fulfillment of the Requirements for the Degree of Master of Science at the University of Windsor Windsor, Ontario, Canada c 2018 Spencer Percy Two Photon Processes in the Hyperfine Structure of Hydrogen by Spencer Percy APPROVED BY: M.
    [Show full text]
  • Hyperfine Structure Constants on the Relativistic Coupled Cluster Level With
    Hyperfine structure constants on the relativistic coupled cluster level with associated uncertainties Pi A. B. Haase,∗,y Ephraim Eliav,z Miroslav Iliaˇs,{ and Anastasia Borschevskyy yVan Swinderen Institute, University of Groningen, 9747 Groningen, The Netherlands zSchool of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel {Department of Chemistry, Faculty of Natural Sciences, Matej Bel University,Tajovsk`eho 40 , SK-97400 Banska Bystrica, Slovakia E-mail: [email protected] arXiv:2002.00887v1 [physics.atom-ph] 3 Feb 2020 1 Abstract Accurate predictions of hyperfine structure (HFS) constants are important in many areas of chemistry and physics, from the determination of nuclear electric and magnetic moments to benchmarking of new theoretical methods. We present a detailed investigation of the performance of the relativistic coupled cluster method for calculating HFS constants withing the finite-field scheme. The two selected test systems are 133Cs and 137BaF. Special attention has been paid to construct a theoretical uncertainty estimate based on investigations on basis set, electron correlation and relativistic effects. The largest contribution to the uncertainty estimate comes from higher order correlation contributions. Our conservative uncertainty estimate for the calculated HFS constants is ∼ 5.5%, while the actual deviation of our results from experimental values was < 1% in all cases. Introduction The hyperfine structure (HFS) constants parametrize the interaction between the electronic and the nuclear electromagnetic moments. The HFS consequently provides important infor- mation about the nuclear as well as the electronic structure of atoms and molecules and can serve as a fingerprint of, for example, transition metal complexes, probed by electron para- magnetic resonance (EPR) spectroscopy,1 or of atoms, ions, and small molecules in the field of atomic and molecular physics, investigated by optical or microwave spectroscopy.
    [Show full text]
  • Investigation on Radiative Corrections Related to the Fine Structure and Hyperfine Structure
    UNIVERSITY OF NAIROBI INVESTIGATION ON RADIATIVE CORRECTIONS RELATED TO THE FINE STRUCTURE AND HYPERFINE STRUCTURE By OBEGI, ISAAC SANI I56/68841/2013 A research project submitted in partial fulfillment for the requirements for the award of a Master of Science degree in Physics of the University of Nairobi. APRIL, 2015 i DECLARATION AND CERTIFICATION This research project is my own work and has never been submitted for examination in any other University or Research Institution for work leading to award of a degree Mr. Obegi Sani Isaac Department of Physics, University of Nairobi Signed……………………………………….. Dated………………………………………… This work has been submitted with our authority as University supervisors DR. SILAS MURERAMANZI Department of Physics, University of Nairobi Signed…………………….. Dated……………………… DR. N. M. MONYONKO DR. GEORGE MAUMBA Department of Physics, Department of Physics, University of Nairobi University of Nairobi Signed…………………….. Signed…………………… Dated……………………… Dated……………………… ii Abstract Radiative corrections related to the fine structure, Lamb shift and hyperfine structure of atomic energy levels are studied. Since the perturbation Hamiltonian is very small compared to zero order Hamiltonian first order perturbation is used theory to find the respective corrections. The expressions obtained for the fine structure of hydrogen atom using the relativistic approach coincides with those obtained using the Dirac equation and the fine structure correction is 2 표 proportional to 훼 퐸푛 , which is a small fraction of the Bohr energy. The Lamb shift correction is 3 표 -3 2 표 proportional to 훼 퐸푛 . While the hyperfine structure correction is proportional to 10 훼 퐸푛 . The implications of fine structure and hyperfine structure are seen as an outcome of very small corrections.
    [Show full text]
  • Fine and Hyperfine Structure of the Hydrogen Atom
    UNIT 4 Fine and hyperfine structure of the hydrogen atom Notes by N. Sirica and R. Van Wesep Previously, we solved the time-independent Sch¨odingerequation for the Hydro- gen atom, described by the Hamiltonian p2 e2 H = − (1) 2m r Where e is the electron’s charge in your favorite units. However, this is not really the Hamiltonian for the Hydrogen atom. It is non-relativistic and it does not contain spin. In order to completely describe the Hydrogen we would need to use the Dirac equation. We will not introduce that equation here, but we will say a few words about the most important energy level of the relativistic Hydrogen atom, namely the rest mass energy. E = mc2 ≈ 0.5MeV (2) We can compare this to the ground state (ionization) energy we found for the Hamiltonian in 1. 2 E = −Eion ≈ 13.6eV ¿ mc (3) Even though the rest mass energy is so much larger, it appears constant in the non-relativistic regime. Since differences in energy are important, we could ignore it before. The rest mass energy may be large, but it does not enter the world of everyday experience. Regardless, it is fruitful to investigate the relative size of the ionization energy to the rest mass energy. E me4 1 e4 α2 ion = = = (4) mc2 2~2 mc2 2c2~2 2 86 UNIT 4: Fine and hyperfine structure of the hydrogen atom e2 1 Where α = ~c = 137 is a fundamental constant of nature. No one understands it, but it is important that it is small.
    [Show full text]
  • High-Precision QED Calculations of the Hyperfine Structure in Hydrogen
    Institut f¨ur Theoretische Physik Fakult¨at Mathematik und Naturwissenschaften Technische Universit¨at Dresden High-precision QED calculations of the hyperfine structure in hydrogen and transition rates in multicharged ions Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium vorgelegt von Andrey V. Volotka geboren am 23. September 1979 in Murmansk, Rußland Dresden 2006 Ä Eingereicht am 14.06.2006 1. Gutachter: Prof. Dr. R¨udiger Schmidt 2. Gutachter: Priv. Doz. Dr. G¨unter Plunien 3. Gutachter: Prof. Dr. Vladimir M. Shabaev Verteidigt am Contents Kurzfassung........................................ .......................................... 5 Abstract........................................... ............................................ 7 1 Introduction...................................... ............................................. 9 1.1 Theoryandexperiment . .... .... .... .... ... .... .... ........... 9 1.2 Overview ........................................ ....... 14 1.3 Notationsandconventions . ............. 15 2 Protonstructure................................... ........................................... 17 2.1 Hyperfinestructureinhydrogen . .............. 17 2.2 Zemachandmagneticradiioftheproton. ................ 20 2.3 Resultsanddiscussion . ............ 23 3 QED theory of the transition rates..................... ....................................... 27 3.1 Bound-stateQED .................................. ......... 27 3.2 Photonemissionbyanion . ........... 31 3.3 Thetransitionprobabilityinone-electronions
    [Show full text]
  • Positronium Hyperfine Splitting Corrections Using Non-Relativistic QED
    Positronium Hyperfine Splitting Corrections Using Non-Relativistic QED Seyyed Moharnmad Zebarjad Centre for High Energy Physics Department of Physics, McGil1 University Montréal, Québec Novernber 1997 A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulWment of the requirements of the degree of Doctor of Philosophy @ Seyyed Moh~unmadZebarjad, 1997 National Library Bibliothèque nationale of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Weiiington Street 395. nie Wellington OttawaON K1A ON4 Ottawa ON K 1A ON4 Canaâa Canada Your fïb Votre reiemnw Our fî& Notre reIPrmce The author has granted a non- L'auteur a accordé une Licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of ths thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantiai extracts fiom it Ni la thèse ni des extraits substantiels may be prïnted or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son pemksion. autorisation. Contents Abstract vii Résumé viii Acknowledgement s x Statement of Original Contributions xi 1 Motivation and Outline of this Thesis 1 2 Introduction to NRQED 7 2.1 NRQED Lagrangian. 7 2.2 Matching at Leading Order.
    [Show full text]
  • Determination of the Hyperfine Coupling Constant of the Cesium 7S1/2 State
    Determination of the hyperfine coupling constant of the cesium 7S1/2 state Guang Yang 1, 2, Jie Wang 1, 2, Baodong Yang 1, 2, and Junmin Wang 1, 2, 3, * 1 Institute of Opto-Electronics, Shanxi University, Tai Yuan 030006, Shan Xi, People’s Republic of China 2 State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Tai Yuan 030006, Shan Xi, People’s Republic of China 3 Collaborative Innovation Center of Extreme Optics, Shanxi University, Tai Yuan 030006, Shan Xi, People’s Republic of China * E-mail: [email protected] Abstract We report the hyperfine splitting (HFS) measurement of the cesium (Cs) 7S1/2 state by optical-optical double-resonance spectroscopy with the Cs 6S1/2-6P3/2-7S1/2 (852 nm + 1470 nm) ladder-type system. The HFS frequency calibration is performed by employing a phase-type waveguide electro-optic modulator together with a stable confocal Fabry-Perot cavity. From the measured HFS between the F’’ = 3 and F’’ = 4 manifolds of the Cs 7S1/2 state (HFS = 2183.273 0.062 MHz), we have determined the magnetic dipole hyperfine coupling constant (A = 545.818 0.016 MHz), which is in good agreement with the previous work but much more precise. Keywords: hyperfine splitting (HFS), hyperfine coupling constant (HCC), cesium atoms, optical-optical double resonance (OODR), phase-type electro-optic modulator (EOM) PACS: 32.10.Fn, 32.30.−r 1. Introduction Precise measurement of the atomic hyperfine structure attracts more and more attentions for the reason that it can test the accuracy of fundamental physics.
    [Show full text]
  • Accelerator Glossary of Terms
    Accelerator Glossary of Terms Table of Contents The Obligatory Disclaimer 2 -M- 75 -Numbers- 3 -N- 83 -A- 5 -O- 87 -B- 13 -P- 89 -C- 23 -Q- 99 -D- 37 -R- 101 -E- 45 -S- 107 -F- 49 -T- 121 -G- 53 -U- 130 -H- 55 -V- 132 -I- 61 -W- 134 -J- 65 -X- 136 -K- 67 -Y- 136 -L- 69 -Z- 136 Glossary v3.6 Last Revised: 6/8/11 Printed 6/8/11 Accelerator Glossary of Terms The Obligatory Disclaimer This is a glossary of accelerator related terms that are important to the Beams Division, Operations Department, Fermilab National Accelerator Laboratory. In the course of the day to day working with the accelerator there is a bewildering array of "Accelerator Technospeak" used in the Operations Group and other areas of the Beams Division. This glossary is meant to be an attempt at helping the uninitiated understand some of the jargon, slang, and buzz words that are commonly used in everyday communication within the Beams Division and the lab in general. Since this glossary was created primarily for Operations Department personnel there are some terms included in this glossary that might be confusing or unfamiliar to those outside of the Operations Department. By the same token there are many commonly understood terms used by other divisions and groups in the lab that are not well known or non existent within the Operations Department. Many terms will not be found within this document. If anybody has any contributions of terms, definitions or sources of other glossaries to be included please feel free to pass them on, even if they are not related to the Beams Division.
    [Show full text]
  • Lecture 12 Atomic Structure Atomic Structure: Background
    Lecture 12 Atomic structure Atomic structure: background Our studies of hydrogen-like atoms revealed that the spectrum of the Hamiltonian, pˆ2 1 Ze2 Hˆ0 = 2m − 4π"0 r is characterized by large n2-fold degeneracy. However, although the non-relativistic Schr¨odingerHamiltonian provides a useful platform, the formulation is a little too na¨ıve. The Hamiltonian is subject to several classes of “corrections”, which lead to important physical ramifications (which reach beyond the realm of atomic physics). In this lecture, we outline these effects, before moving on to discuss multi-electron atoms in the next. Atomic structure: hydrogen atom revisited As with any centrally symmetric potential, stationary solutions of Hˆ0 index by quantum numbers n#m, ψn!m(r)=Rn!(r)Y!m(θ,φ). For atomic hydrogen, n2-degenerate energy levels set by 2 1 e2 m e2 1 E = Ry , Ry = = n − n2 4π" 2 2 4π" 2a ! 0 " ! 0 0 2 4π#0 ! where m is reduced mass (ca. electron mass), and a0 = e2 m . For higher single-electron ions (He+, Li2+, etc.), E = Z 2 Ry . n − n2 Allowed combinations of quantum numbers: n # Subshell(s) 1 0 1s 20, 12s 2p 30, 1, 23s 3p 3d n 0 (n 1) ns ··· − ··· Atomic structure: hydrogen atom revisited However, treatment of hydrogen atom inherently non-relativistic: pˆ2 1 Ze2 Hˆ0 = 2m − 4π"0 r is only the leading term in relativistic treatment (Dirac theory). Such relativistic corrections begin to impact when the electron becomes relativistic, i.e. v c. ∼ Since, for Coulomb potential, 2 k.e. = p.e.
    [Show full text]