DOCSLIB.ORG

Optical Pumping of Rubidium

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Optical Pumping of Rubidium Optical pumping of rubidium Quinn Pratt, John Prior, Brennan Campbella) (Dated: 25 October 2015) The effects of a magnetic field incident on a sample of rubidium were examined both in the low-field Zeeman region as well as in the intermediate, quadratic region. In the Zeeman region, the Land´eg-factor (gF ) for Rubidium-85 and Rubidium-87 were calculated to be 0:3357 ± 0:010 and 0:5043 ± 0:023, respectively. These values were later used to calculate the nuclear spin for each of these isotopes. In turn, we calculate I = 1.513 and I = 2.518 compared to I = 3/2 and I = 5/2 for rubidium-87 and 85 respectively. Furthermore, in the quadratic region, we are able to resolve all of the magnetic sublevels for each isotope in accordance with quantum theory. I. INTRODUCTION A. Quantum Mechanics Optical pumping is the process of stimulating a sample There are a variety of quantum numbers which serve of atoms out of thermodynamic equilibrium through the to describe the various physical properties of a quantum resonant absorption of light.1In this experiment we use system. These quantum numbers and their significance optical pumping in conjunction with a magnetic field to are as follows,2 trap atoms in a specific quantum mechanical atomic sub- level. Then, we use radio frequency photons to stimulate • S: the S quantum number, or SZ corresponds to absorption. By varying the frequency of the incident rf- the z-component of the angular momentum of an photons, we can monitor the linear dependance of energy electron. with respect to the magnetic field. Furthermore, upon • L: the L quantum number corresponds to the an- increasing the magnetic field beyond the linear Zeeman gular momentum of the orbit of the electron. region, we can separate out the M-level spectrum of the atoms and account for every one of the magnetic sub- • J: the J quantum number is defined as: levels. Atomic spectroscopy is frequently the study of a se- J = L + S; (1) ries of quantum numbers. We will discuss how quantum mechanics explains effects such as Zeeman splitting and and it is known as the total angular momentum of optical pumping. Additionally we will describe the ex- the electron. perimental design including the optical and electronic as- • I: the I quantum number corresponds to the intrin- pects of the apparatus. Lastly we will discuss the results sic spin of the nucleus, this will be the subject of of both facets of this experiment: the exploration of the one of our experiments, it will be different for each linear Zeeman effect to derive the nuclear spin quantum isotope. number (I), as well as accounting for M-level absorption dips in the quadratic region. • F: this quantum number is best described as the We will begin with the theory of the experiments, then total atomic angular momentum and it is defined move onto the design of the apparatus, then the results of as: the experiments themselves, lastly we will discuss these results in the context of high resolution spectroscopy. F = J + I; (2) this is another one of the most important quantum numbers in our studies as it describes the coupling II. THEORY of the electron with the nucleus. The theoretical underpinnings of this set of experi- • M: this quantum number is different from the oth- ments can be broken down into two main categories. ers as it is dependent on the application on an ex- First, the theory surrounding the nuclear quantum me- ternal magnetic field, M is therefore the component chanics and its treatment of magnetic effects. Secondly of F along said magnetic field (B). the theory behind optical pumping and the other optical To further demonstrate the organization of these quan- aspects of this experiment. tum numbers one should consult the vector diagram in FIG.1. The most important quantum numbers for this exper- iment are F, M and I, the first two are important be- a)also at University of San Diego: Department of Physics & Bio- cause they serve to organize the energy levels based on physics. the applied magnetic field, as shown in FIG. 2, I is also 2 FIG. 1. This simple diagram illustrates the relationship be- tween these quantum numbers. The effect of the impinging magnetic field is what we will be studying in this lab. important as our experiment serves to calculate I for each isotope. The externally applied magnetic field acts as a pertur- bation on the atomic energy levels, hamiltonian which described this interaction is: µ µ FIG. 2. This is the principal energy diagram for this exper- H = haI · J − J J · B − I I · B (3) iment, it shows the low-field Zeeman splitting effect wherein J I the energy level separation between M-levels varies linearly µ is the total electronic magnetic dipole moment, µ with increasing magnetic field (B). It is important to know J I that this diagram is for rubidium-87, not rubidium-85. The is the nuclear magnetic dipole moment.3As we can see F-levels correspond to the hyperfine structure, whereas the this perturbation is linked to the nuclear spin as well M-levels correspond to Zeeman splitting. as the total electronic spin. The ultimate result of this perturbation is shown in FIG. 2. Upon an increasing B-field each F state gives rise to where it is clear that W scales linearly with B. 2F + 1 sublevels. Although these energy levels appear to Furthermore, split linearly with increasing B, their true nature is given by the Breit-Rabi equation: F(F + 1) + J(J + 1) − I(I + 1) g = g (7) F J 2F(F + 1) −∆W µ ∆W 4M W (F; M) = − I BM± [1+ x+x2]1=2; This relationship will be used later to calculate the nu- 2(2I + 1 I 2 2I + 1 clear spin (I) for each isotope. (4) We will also collect data in the intermediate, quadratic where, region where a simplified linear approximation cannot be made. µoB µI x = (gJ − gI ) and gI = − : (5) ∆W IµI B. Optical Pumping In terms of the dimensionless number x, three principal regions of interest exist. Next, we must address the other theoretical aspects of 1. Zeeman Region: x = 0, the energy levels split lin- the experiment regarding optical pumping and the stim- early with B. ulation of the rubidium atoms by RF-photons and light. The rubidium cell is continuously being exposed to fo- 2. Intermediate Region: 0 < x < 2, energy levels split cused light of wavelength λ = 795nm, which is consistent quadratically with B. 1=2 1=2 with the resonant wavelength for the S1=2 ! P1=2 transition. This light is also circularly polarized to cause 3. Paschen-Bach Region: 2 < x, energy levels are lin- the only allowed transitions to be ∆M = +1. early separated where I and J have been decoupled. The following diagram serves best to illustrate the al- We will be investigating effects in the low-field Zeeman lowed transitions under the optical pumping conditions, region where the energy level splitting is given approxi- Note that there is no transition out of the M = 2 state. mately as: This means that the rubidium atoms will transition out of the ground state, and naturally cascade back down W = gF µoBM; (6) and transition up through the Zeeman Split states until 3 A. Investigations in Zeeman Splitting The procedure for this experiment is as follows, 1. We begin by aligning the apparatus itself as to best cancel-out the local components of Earth's mag- netic field. Other setup procedures include setting the temperature in the Rubidium cell to 50oC, and we must ensure the horizontal coils are set to 0. 2. Next, we monitor the current being delivered to the aligned-Helmholtz coils on the oscilloscope ver- FIG. 3. This diagram, although similar to FIG. 2 has an im- sus the absorption from the optical detector. We portant difference. Here we can see the allowed transitions adjust the range and speed of the sweep until we between energy levels. It is critical to note that there is no have a clear image of the zero-field dip. This dip transition out of the M = 2 ground state. This is the key to corresponds to the whatever magnetic field is neces- optical pumping, while an external B field is applied, the ru- sary to cancel out any residual effects from Earth's bidium atoms will be pumped into this state. It is important magnetic field. to note that this diagram only applies to rubidium-87. 3. After obtaining a clear image of the zero-field dip we are prepared to generate RF-photons to stim- population largely inhabits the M = 2 state. This is the ulate the sample. The transition energy of these basis for optical pumping. photons is given by, E = hν: (8) These photons are used to stimulate the atoms III. EXPERIMENT DESIGN trapped in the pumped state to a more absorbent state. This de-pumping causes two more dips to appear on either side of the central zero-field dip. Using the apparatus displayed in FIG. 4, along with There are two because there is one for each isotope. other instrumentation to be explained later, we will con- duct two experiments, The first serves to investigate the 4. Starting at a frequency of about ν = 20kHz and linear Zeeman splitting at low B field values. We will use incrementing up to values near ν = 150kHz collect the data collected from the first experiment to calculate the relevant data to see the de-pumping dips drift the nuclear spin values I for each isotope.
Load more

Recommended publications
  • H20 Maser Observations in W3 (OH): a Comparison of Two Epochs
    DePaul University Via Sapientiae College of Science and Health Theses and Dissertations College of Science and Health Summer 7-13-2012 H20 Maser Observations in W3 (OH): A comparison of Two Epochs Steven Merriman DePaul University, [email protected] Follow this and additional works at: https://via.library.depaul.edu/csh_etd Part of the Physics Commons Recommended Citation Merriman, Steven, "H20 Maser Observations in W3 (OH): A comparison of Two Epochs" (2012). College of Science and Health Theses and Dissertations. 31. https://via.library.depaul.edu/csh_etd/31 This Thesis is brought to you for free and open access by the College of Science and Health at Via Sapientiae. It has been accepted for inclusion in College of Science and Health Theses and Dissertations by an authorized administrator of Via Sapientiae. For more information, please contact [email protected]. DEPAUL UNIVERSITY H2O Maser Observations in W3(OH): A Comparison of Two Epochs by Steven Merriman A thesis submitted in partial fulfillment for the degree of Master of Science in the Department of Physics College of Science and Health July 2012 Declaration of Authorship I, Steven Merriman, declare that this thesis titled, `H2O Maser Observations in W3OH: A Comparison of Two Epochs' and the work presented in it are my own. I confirm that: This work was done wholly while in candidature for a masters degree at Depaul University. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.
    [Show full text]
  • Quantum Mechanics of Atoms and Molecules Lectures, the University of Manchester 2005
    Quantum Mechanics of Atoms and Molecules Lectures, The University of Manchester 2005 Dr. T. Brandes April 29, 2005 CONTENTS I. Short Historical Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 I.1 Atoms and Molecules as a Concept . 1 I.2 Discovery of Atoms . 1 I.3 Theory of Atoms: Quantum Mechanics . 2 II. Some Revision, Fine-Structure of Atomic Spectra : : : : : : : : : : : : : : : : : : : : : 3 II.1 Hydrogen Atom (non-relativistic) . 3 II.2 A `Mini-Molecule': Perturbation Theory vs Non-Perturbative Bonding . 6 II.3 Hydrogen Atom: Fine Structure . 9 III. Introduction into Many-Particle Systems : : : : : : : : : : : : : : : : : : : : : : : : : 14 III.1 Indistinguishable Particles . 14 III.2 2-Fermion Systems . 18 III.3 Two-electron Atoms and Ions . 23 IV. The Hartree-Fock Method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 IV.1 The Hartree Equations, Atoms, and the Periodic Table . 24 IV.2 Hamiltonian for N Fermions . 26 IV.3 Hartree-Fock Equations . 28 V. Molecules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 V.1 Introduction . 35 V.2 The Born-Oppenheimer Approximation . 36 + V.3 The Hydrogen Molecule Ion H2 . 39 V.4 Hartree-Fock for Molecules . 45 VI. Time-Dependent Fields : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48 VI.1 Time-Dependence in Quantum Mechanics . 48 VI.2 Time-dependent Hamiltonians . 50 VI.3 Time-Dependent Perturbation Theory . 53 VII.Interaction with Light : : : : : : : : : : : : : : : : : : : : : : :
    [Show full text]
  • Isotopic Composition of Some Metals in the Sun
    SNSTITUTE OF THEORETICAL ASTROPHYSICS BLINDERN - OSLO REPORT .No. 35 ISOTOPIC COMPOSITION OF SOME METALS IN THE SUN by ØIVIND HAUGE y UNIVERSITETSFORLAqET • OSLO 1972 Universitetsfc lagets trykningssentral, Oslo INSTITUTE OF THEORETICAL ASTROPHYSICS BLINDERN - OSLO REPORT No. 35 ISOTOPIC COMPOSITION OF SOME METALS IN THE SUN by ØIVIND HAUGE UNIVERSITETSFORLAGET • OSLO 1972 Universitetsforlagets tryknlngssentral, Oslo CONTENTS Abstract 1 1. Introduction 2 2. Fine structure in spectral lines from atoms 5 1. Isotope shift 5 2. Hyperfine structure 6 3. Applications to atomic lines in photospheric spectrum .... 8 1. Elements with one odd isotope , 9 2. Elements with two odd isotopes 9 3. Elements with one odd and several even isotopes 11 k. Elements with several odd and even isotopes 11 h. Studies of elements in the Sun with two odd isotopes 1. Isotopes of rubidium 12 A. Observations lk B. Calculations 16 C. The Rb I line at 78OO Å 1. The continuum level 16 2. Line profiles and turbulent velocities 18 3. The asymmetry of the Si I line 19 h. Isotope investigations 21 P. The Rb I line at 79^7 A 28 E. The isotope ratio of rubidium 31 F. The abundance of rubidium 3k 2. Isotopes of antimony 35 A. Spectroscopic data 35 B. The Sb I lines at 3267 and 3722 A 37 3* Isotopes of europium 1*0 A. Observations and methods of analysis ^1 B. Spectroscopic data 1*1 C. Spectral line investigations 1. Investigations of four Eu II lines **3 2. The Eu II lines at Ul29 and U205 k ^6 D. The isotope ratio of europium 50 E.
    [Show full text]
  • Optical Pumping 1
    Optical Pumping 1 OPTICAL PUMPING OF RUBIDIUM VAPOR Introduction The process of optical pumping is a beautiful example of the interaction between light and matter. In the Advanced Lab experiment, you use circularly polarized light to pump a particular level in rubidium vapor. Then, using magnetic fields and radio-frequency excitations, you manipulate the population of the pumped state in a manner similar to that used in the Spin Echo experiment. You will determine the energy separation between the magnetic substates (Zeeman levels) in rubidium as well as determine the Bohr magneton and observe two-photon transitions. Although the experiment is relatively simple to perform, you will need to understand a fair amount of atomic physics and experimental technique to appreciate the signals you witness. A simple example of optical pumping Let’s imagine a nearly trivial atom: no nuclear spin and only one electron. For concreteness, you can think of the 4He+ ion, which is similar to a Hydrogen atom, but without the nuclear spin of the proton. Its ground state is 1S1=2 (n = 1,S = 1=2,L = 0, J = 1=2). Photon absorption can excite it to the 2P1=2 (n = 2,S = 1=2,L = 1, J = 1=2) state. If you place it in a magnetic field, the energy levels become split as indicated in Figure 1. In effect, each original level really consists of two levels with the same energy; when you apply a field, the “spin up” state becomes higher in energy, the “spin down” lower. The spin energy splitting is exaggerated on the figure.
    [Show full text]
  • ILPAC Unit S2: Atomic Structure
    UNIT INDEPENDENT LEARNING PROJECT FOR ADVANCED CHEMISTRY Periodic Table of the Elements o 2 I He II ill] III IV V VI VIII 4.0 3 4 5 6 7 8 9 10 Li Be B C N 0 F Ne 6.9 9.0 10.8 12.0 14.0 16.0 19.0 20.2 11 12 13 14 15 16 17 18 Na Mg Al Si P S CI Ar 23.0 24.3 27.0 28.1 31.0 32.1 35.5 39.9 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 3! 36 K Ca Sc Ti V Cr Mn Fe Co Ni eu Zn Ga Ge As Se BrlKr 39.1 40.1 45.0 47.9 50.9 52.0 54.9 55.9 58.9 58.7 63.5 65.4 69.7 72.6 74.9 79.0 79 83.8 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 S4 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 85.5 87.6 88.9 91.2 92.9 95.9 99.0 101.1 102.9 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.91 ' 3 ' . 3 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La 4 Hf Ta W Re Os If Pt Au Hg Tl Pb Bi Po AtlRn 132.9 137.3 138.9 178.5 181.0 183.9 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 210.0 210.01222.0 87 88 89 Fr Ra Ac~ 223.0 226.0 227.0 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 140.1 140.9 144.2 (147) 150.4 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 90 91 92 93 94 95 96 97 98 99 100 101 102 103 "--- Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lw 232.0 231.0 238.1 (237) 239.1 (243) (241) (247) (251 ) (254) (253) (256) ( 254) (257) A value in brackets denotes the mass number of the most stable isotope.
    [Show full text]
  • Quantum State Detection of a Superconducting Flux Qubit Using A
    PHYSICAL REVIEW B 71, 184506 ͑2005͒ Quantum state detection of a superconducting flux qubit using a dc-SQUID in the inductive mode A. Lupașcu, C. J. P. M. Harmans, and J. E. Mooij Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands ͑Received 27 October 2004; revised manuscript received 14 February 2005; published 13 May 2005͒ We present a readout method for superconducting flux qubits. The qubit quantum flux state can be measured by determining the Josephson inductance of an inductively coupled dc superconducting quantum interference device ͑dc-SQUID͒. We determine the response function of the dc-SQUID and its back-action on the qubit during measurement. Due to driving, the qubit energy relaxation rate depends on the spectral density of the measurement circuit noise at sum and difference frequencies of the qubit Larmor frequency and SQUID driving frequency. The qubit dephasing rate is proportional to the spectral density of circuit noise at the SQUID driving frequency. These features of the back-action are qualitatively different from the case when the SQUID is used in the usual switching mode. For a particular type of readout circuit with feasible parameters we find that single shot readout of a superconducting flux qubit is possible. DOI: 10.1103/PhysRevB.71.184506 PACS number͑s͒: 03.67.Lx, 03.65.Yz, 85.25.Cp, 85.25.Dq I. INTRODUCTION A natural candidate for the measurement of the state of a flux qubit is a dc superconducting quantum interference de- An information processor based on a quantum mechanical ͑ ͒ system can be used to solve certain problems significantly vice dc-SQUID .
    [Show full text]
  • Quantum Mechanics Department of Physics and Astronomy University of New Mexico
    Preliminary Examination: Quantum Mechanics Department of Physics and Astronomy University of New Mexico Fall 2004 Instructions: • The exam consists two parts: 5 short answers (6 points each) and your pick of 2 out 3 long answer problems (35 points each). • Where possible, show all work, partial credit will be given. • Personal notes on two sides of a 8X11 page are allowed. • Total time: 3 hours Good luck! Short Answers: S1. Consider a free particle moving in 1D. Shown are two different wave packets in position space whose wave functions are real. Which corresponds to a higher average energy? Explain your answer. (a) (b) ψ(x) ψ(x) ψ(x) ψ(x) ψ = 0 ψ(x) S2. Consider an atom consisting of a muon (heavy electron with mass m ≈ 200m ) µ e bound to a proton. Ignore the spin of these particles. What are the bound state energy eigenvalues? What are the quantum numbers that are necessary to completely specify an energy eigenstate? Write out the values of these quantum numbers for the first excited state. sr sr H asr sr S3. Two particles of spins 1 and 2 interact via a potential ′ = 1 ⋅ 2 . a) Which of the following quantities are conserved: sr sr sr 2 sr 2 sr sr sr sr 2 1 , 2 , 1 , 2 , = 1 + 2 , ? b) If s1 = 5 and s2 = 1, what are the possible values of the total angular momentum quantum number s? S4. The energy levels En of a symmetric potential well V(x) are denoted below. V(x) E4 E3 E2 E1 E0 x=-b x=-a x=a x=b (a) How many bound states are there? (b) Sketch the wave functions for the first three levels (n=0,1,2).
    [Show full text]
  • The Elements.Pdf
    A Periodic Table of the Elements at Los Alamos National Laboratory Los Alamos National Laboratory's Chemistry Division Presents Periodic Table of the Elements A Resource for Elementary, Middle School, and High School Students Click an element for more information: Group** Period 1 18 IA VIIIA 1A 8A 1 2 13 14 15 16 17 2 1 H IIA IIIA IVA VA VIAVIIA He 1.008 2A 3A 4A 5A 6A 7A 4.003 3 4 5 6 7 8 9 10 2 Li Be B C N O F Ne 6.941 9.012 10.81 12.01 14.01 16.00 19.00 20.18 11 12 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 3 Na Mg IIIB IVB VB VIB VIIB ------- VIII IB IIB Al Si P S Cl Ar 22.99 24.31 3B 4B 5B 6B 7B ------- 1B 2B 26.98 28.09 30.97 32.07 35.45 39.95 ------- 8 ------- 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 39.10 40.08 44.96 47.88 50.94 52.00 54.94 55.85 58.47 58.69 63.55 65.39 69.72 72.59 74.92 78.96 79.90 83.80 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 5 Rb Sr Y Zr NbMo Tc Ru Rh PdAgCd In Sn Sb Te I Xe 85.47 87.62 88.91 91.22 92.91 95.94 (98) 101.1 102.9 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.9 131.3 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 6 Cs Ba La* Hf Ta W Re Os Ir Pt AuHg Tl Pb Bi Po At Rn 132.9 137.3 138.9 178.5 180.9 183.9 186.2 190.2 190.2 195.1 197.0 200.5 204.4 207.2 209.0 (210) (210) (222) 87 88 89 104 105 106 107 108 109 110 111 112 114 116 118 7 Fr Ra Ac~RfDb Sg Bh Hs Mt --- --- --- --- --- --- (223) (226) (227) (257) (260) (263) (262) (265) (266) () () () () () () http://pearl1.lanl.gov/periodic/ (1 of 3) [5/17/2001 4:06:20 PM] A Periodic Table of the Elements at Los Alamos National Laboratory 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Lanthanide Series* Ce Pr NdPmSm Eu Gd TbDyHo Er TmYbLu 140.1 140.9 144.2 (147) 150.4 152.0 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Actinide Series~ Th Pa U Np Pu AmCmBk Cf Es FmMdNo Lr 232.0 (231) (238) (237) (242) (243) (247) (247) (249) (254) (253) (256) (254) (257) ** Groups are noted by 3 notation conventions.
    [Show full text]
  • Molecular Column Density Calculation
    Molecular Column Density Calculation Jeffrey G. Mangum National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA [email protected] and Yancy L. Shirley Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA [email protected] May 29, 2013 Received ; accepted – 2 – ABSTRACT We tell you how to calculate molecular column density. Subject headings: ISM: molecules – 3 – Contents 1 Introduction 5 2 Radiative and Collisional Excitation of Molecules 5 3 Radiative Transfer 8 3.1 The Physical Meaning of Excitation Temperature . 11 4 Column Density 11 5 Degeneracies 14 5.1 Rotational Degeneracy (gJ)........................... 14 5.2 K Degeneracy (gK)................................ 14 5.3 Nuclear Spin Degeneracy (gI ).......................... 15 5.3.1 H2CO and C3H2 ............................. 16 5.3.2 NH3 and CH3CN............................. 16 5.3.3 c–C3H and SO2 .............................. 18 6 Rotational Partition Functions (Qrot) 18 6.1 Linear Molecule Rotational Partition Function . 19 6.2 Symmetric and Asymmetric Rotor Molecule Rotational Partition Function . 21 2 7 Dipole Moment Matrix Elements (|µjk| ) and Line Strengths (S) 24 – 4 – 8 Linear and Symmetric Rotor Line Strengths 26 8.1 (J, K) → (J − 1, K) Transitions . 28 8.2 (J, K) → (J, K) Transitions . 29 9 Symmetry Considerations for Asymmetric Rotor Molecules 29 10 Hyperfine Structure and Relative Intensities 30 11 Approximations to the Column Density Equation 32 11.1 Rayleigh-Jeans Approximation . 32 11.2 Optically Thin Approximation . 34 11.3 Optically Thick Approximation . 34 12 Molecular Column Density Calculation Examples 35 12.1 C18O........................................ 35 12.2 C17O........................................ 37 + 12.3 N2H ....................................... 42 12.4 NH3 .......................................
    [Show full text]
  • Introduction to Biomolecular Electron Paramagnetic Resonance Theory
    Introduction to Biomolecular Electron Paramagnetic Resonance Theory E. C. Duin Content Chapter 1 - Basic EPR Theory 1.1 Introduction 1-1 1.2 The Zeeman Effect 1-1 1.3 Spin-Orbit Interaction 1-3 1.4 g-Factor 1-4 1.5 Line Shape 1-6 1.6 Quantum Mechanical Description 1-12 1.7 Hyperfine and Superhyperfine Interaction, the Effect of Nuclear Spin 1-13 1.8 Spin Multiplicity and Kramers’ Systems 1-24 1.9 Non-Kramers’ Systems 1-32 1.10 Characterization of Metalloproteins 1-33 1.11 Spin-Spin Interaction 1-35 1.12 High-Frequency EPR Spectroscopy 1-40 1.13 g-Strain 1-42 1.14 ENDOR, ESEEM, and HYSCORE 1-43 1.15 Selected Reading 1-51 Chapter 2 - Practical Aspects 2.1 The EPR Spectrometer 2-1 2.2 Important EPR Spectrometer Parameters 2-3 2.3 Sample Temperature and Microwave Power 2-9 2.4 Integration of Signals and Determination of the Signal Intensity 2-15 2.5 Redox Titrations 2-19 2.6 Freeze-quench Experiments 2-22 2.7 EPR of Whole Cells and Organelles 2-25 2.8 Selected Reading 2-28 Chapter 3 - Simulations of EPR Spectra 3.1 Simulation Software 3-1 3.2 Simulations 3-3 3.3 Work Sheets 3-17 i Chapter 4: Selected samples 4.1 Organic Radicals in Solution 4-1 4.2 Single Metal Ions in Proteins 4-2 4.3 Multi-Metal Systems in Proteins 4-7 4.4 Iron-Sulfur Clusters 4-8 4.5 Inorganic Complexes 4-17 4.6 Solid Particles 4-18 Appendix A: Rhombograms Appendix B: Metalloenzymes Found in Methanogens Appendix C: Solutions for Chapter 3 ii iii 1.
    [Show full text]
  • Optical Pumping on Rubidium
    Optical Pumping on Rubidium E. Lance (Dated: March 27, 2011) By means of optical pumping the values of gf and nuclear spin for Rubidium were determined. 85 87 85 For Rb gf was found to be .35. For Rb gf was found to be .50. The nuclear spin for Rb was found to be 2.35 with a theoretical value of 2.5, for Rb87 the nuclear spin was found to be 1.5 with a theoretical value of 1.5. Quadratic Zeeman effect resonances were observed around magnetic fields of 6.98 gauss and RF of 4.064 MHz. I. INTRODUCTION Rabi equation: Optical pumping refers to the redistribution of oc- cupied energy states, in thermal equilibrium, inside an ∆W µ ∆W 4M 1=2 W (F; M) = − − j BM± 1+ x+x2 : atom. The redistribution is achieved by polarizing inci- 2(2I + 1) I 2 2I + 1 dent light, therefor, pumping electrons to less absorbent (4) energy states. By confining electrons in less absorbent A plot of Breit-Rabi equation is shown in figure 2, this energy states we can then study the relationship between plot shows how when the magnetic field becomes large, magnetic fields and Zeeman splitting. the energy splittings stop behaving in a linear fashion and a quadratic Zeeman effect can be seen. II. THEORY We will be dealing with Rubidium atoms which can, essentially, be conceived as a hydrogen like single elec- tron atoms. The single valence electron in Rubidium can be described by a total angular momentum vector J which couples the orbital angular momentum L with the spin orbital angular momentum S.
    [Show full text]
  • Measuring Optical Pumping of Rubidium Vapor
    Measuring Optical Pumping of Rubidium Vapor Shawn Westerdale∗ MIT Department of Physics (Dated: April 4, 2010) Optical pumping is a process that stochastically increases the energy levels of atoms in a magnetic field to their highest Zeeman energy state. This technique is important for its applications in applied atomic and molecular physics, particularly in the construction of lasers. In this paper, we will construct a model of optical pumping for the case of rubidium atoms and will use experimental results to determine the Land´eg-factors for the two stable isotopes of rubidium and measure the relaxation time for pumped rubidium. We will then show how the relaxation time constant varies with temperature and pumping light intensity. Lastly, we will use our model and observations to determine the mean free path of 7948 Aphotons˚ through rubidium vapor at 41.8◦C. 1. INTRODUCTION vector F~ = I~+ J~, we arrive at the conclusion that ∆E = gf mf µBB, where gf is given by, In the 1950s A. Kastler developed the process known f(f + 1) + j(j + 1) − i(i + 1) as optical pumping, for which he later received the Nobel gf ≈ gJ (2) prize. Optical pumping is the excitation of atoms with 2f(f + 1) a resonant light source and the stochastically biasing of the transitions in order to migrate the atoms into the Note that f, j, and i are the eigenvalues of F , J, and I, highest energy Zeeman state. Rubidium vapor was used respectively. in this experiment due to its well understood hyperfine levels. The use of rubidium also simplifies the physics, since there are only two stable isotopes of rubidium, 85Rb 1.1.1.
    [Show full text]