External-Field Shifts of the [[blank].sup.199][Hg.sup.+] Optical Frequency Standard.Frequency shifts of thc [[blank].sup.199][Hg.sup+] [5d.sup.10]6s [[blank].sup.2][S.sub.1/2] (F = 0, [M.sub.F] = 0) to [5d.sup.9] [6s.sup.2] [[blank].sup.2][D.sub.5/2] (F = 2, [M.sub.F] = 0) elcctric-quadrupolc transition at 282 nm due to external fields are calculated, based on a combination of measured atomic parameters and ab initio [Latin, From the beginning; from the first act; from the inception.] An agreement is said to be "void ab initio" if it has at no time had any legal validity. calculations. This transition is under investigation as an optical frequency standard. The perturbations calculated are the quadratic quadratic, mathematical expression of the second degree in one or more unknowns (see polynomial). The general quadratic in one unknown has the form ax2+bx+c, where a, b, and c are constants and x is the variable. Zeeman shift, the scalar scalar, quantity or number possessing only sign and magnitude, e.g., the real numbers (see number), in contrast to vectors and tensors; scalars obey the rules of elementary algebra. Many physical quantities have scalar values, e.g. and tensor tensor, in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates). quadratic Stark shifts, and the interaction between an external electric field gradient Mathematically, the electric field gradient (EFG) is the hessian matrix (the matrix of the second derivatives) of the electrical potential V: Key words: atomic polarizabilities; electric quadrupole interaction; mercury ion; optical frequency standards; Stark shift; Zeeman shift. Accepted: November 1, 2000 Available online: http://www.nist.gov/jres 1. Introduction It has long been recognized that a frequency standard could be based on the 282 nm transition between the ground [5d.sup.10]6s [[blank].sup.2][D.sub.1/2] level and the metastable met·a·sta·ble adj. Of, relating to, or being an unstable and transient but relatively long-lived state of a chemical or physical system, as of a supersaturated solution or an excited atom. [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] level of [Hg.sup.+] [1]. The lifetime of the upper level is 86(3) ms [2], so the ratio of the natural linewidth [delta]v to the transition frequency [v.sub.0] is 2 X [10.sup.-15]. (Unless otherwise noted, all uncertainties given in this paper are standard uncertainties, i.e., one standard deviation In statistics, the average amount a number varies from the average number in a series of numbers. (statistics) standard deviation - (SD) A measure of the range of values in a set of numbers. estimates.) Doppler broadening In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect in which the thermal movement of atoms or molecules shifts the apparent frequency of each emitter. can be avoided if the transition is excited with two counter-propagating photons, as originally proposed by Bender et al. [1] and subsequently demonstrated by Bergquist et al. [3]. However, optical Stark shifts are greatly reduced if the transition is driven instead with a single photon by the electric-quadrupole interaction. In this case, Doppler broadening can be eliminated if the ion is confined con·fine v. con·fined, con·fin·ing, con·fines v.tr. 1. To keep within bounds; restrict: Please confine your remarks to the issues at hand. See Synonyms at limit. to dimensions much less than the optical wavelength, as was firs t demonstrated by Bergquist et al. [4]. Recently, the (F=0, [M.sub.F]=0) to (F=2, [M.sub.F]=0) hyperfine component of the [[blank].sup.199][Hg.sup.+] [5d.sup.10]6s [[blank].sup.2][D.sub.1/2] to [5d.sup.9] [6s.sup.2] [[blank].sup.2][D.sub.5/2] single-photon transition has been observed with a linewidth of only 6.7 Hz by Rafac et al. [5]. A laser servo-locked to this transition is an extremely stable and reproducible frequency reference. New developments in optical frequency metrology metrology Science of measurement. Measuring a quantity means establishing its ratio to another fixed quantity of the same kind, known as the unit of that kind of quantity. [6, 7] may soon make this system practical as an atomic frequency standard or clock. While the (F=0, [M.sub.F]=0) to (F=2, [M.sub.F]=0) hyperfine component has no linear Zeeman shift, it does have a quadratic Zeeman shift that must be accounted for. In addition, there is a second-order Stark shift and a shift due to the interaction between the electric-field gradient gradient In mathematics, a differential operator applied to a three-dimensional vector-valued function to yield a vector whose three components are the partial derivatives of the function with respect to its three variables. The symbol for gradient is ∇. and the atomic electric-quadrupole moment. None of these shifts has yet been measured accurately, so it is useful to have calculated values, even if they are not very precise. Also, it is useful to know the functional form of the perturbation perturbation (pŭr'tərbā`shən), in astronomy and physics, small force or other influence that modifies the otherwise simple motion of some object. The term is also used for the effect produced by the perturbation, e.g. , even if the magnitude is uncertain. For example, the quadrupole shift can be eliminated by averaging the transition frequency over three mutually orthogonal At right angles. The term is used to describe electronic signals that appear at 90 degree angles to each other. It is also widely used to describe conditions that are contradictory, or opposite, rather than in parallel or in sync with each other. magnetic-field orientations, independent of the orientation of the electric-field gradient. 2. Methods and Notation notation: see arithmetic and musical notation. How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system. The quadratic Zeeman shift can be calculated if the hyperfine constants and electronic and nuclear g-factors are known. Similarly, the quadratic Stark effect Stark effect The effect of an electric field on spectrum lines. The electric field may be externally applied; but in many cases it is an internal field caused by the presence of neighboring ions or atoms in a gas, liquid, or solid. Discovered in 1913 by J. can be calculated from a knowledge of the electric-dipole oscillator oscillator Mechanical or electronic device that produces a back-and-forth periodic motion. A pendulum is a simple mechanical oscillator that swings with a constant amplitude, requiring the addition of energy at each swing only to compensate for the energy lost because of air strengths. The quadrupole shift depends on the atomic wavefunctions. Some of these parameters have been measured, such as the hyperfine constants and some of the oscillator strengths. There are also published calculations for some of the oscillator strengths. Here, we estimate, by the use of the Cowan atomic-structure codes, values for parameters for which there are neither measured values nor published calculations. The Cowan codes are based on the Hartree-Fock approximation approximation /ap·prox·i·ma·tion/ (ah-prok?si-ma´shun) 1. the act or process of bringing into proximity or apposition. 2. a numerical value of limited accuracy. with some relativistic rel·a·tiv·is·tic adj. 1. Of or relating to relativism. 2. Physics a. Of, relating to, or resulting from speeds approaching the speed of light: relativistic increase in mass. corrections [8]. The odd-parity configurations included in the calculation were [5d.sup.10]np (n = 6,7,8,9), [5d.sup.10]5f, [5d.sup.9]6s6p, [5d.sup.9]6s7p, [5d.sup.9]6s5f, and [5d.sup.8][6s.sup.2]6p. The even-parity configurations were [5d.sup.10]ns (n = 6,7,8,9,10), [5d.sup.10]nd (n = 6,7,8,9), [5d.sup.9][6s.sup.2], [5d.sup.9]6s7s, [5d.sup.9]6s6d, and [5d.sup.9][6p.sup.2]. Recently, Sansonetti and Reader have made new measurements of the spectrum of [Hg.sup.+] and classified many new lines [9]. They also carried out a least-squares adjustment of the energy parameters that enter the Cowan-code calculations in order to match the observed energy levels. We use these adjusted parameters in our Cowan-code calculations. As one test of this method of calculation, we estimated the weakly weak·ly adj. weak·li·er, weak·li·est Delicate in constitution; frail or sickly. adv. 1. With little physical strength or force. 2. With little strength of character. allowed 10.7 [micro]m [5d.sup.10]6p [[blank].sup.2][P.sub.1/2] to [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.3/2] electric-dipole decay rate. This decay is allowed only because of configuration mixing, since it requires two electrons to change orbitals orbitals (ōrˑ·b  ·t . The calculation shows the decay to be due mostly to mixing between the [5d.sup.10]6p and [5d.sup.9]6s6p configurations. The calculated rate is 111 [s.sup.-1]; the measured rate is 52(16) [s.sup.-1] [2]. Another test is the electric-quadrupole decay rate of the [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] level to the ground level. The calculated rate is 12.6 [s.sup.-1], and the measured rate is 11.6(0.4) [s.sup.-1]. Similar calculations have been carried out by Wilson [10]. Let [H.sub.0] be the atomic Hamiltonian, exclusive of the hyperfine and external field effects, which are treated as perturbations. For convenience, we denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the eigenstates of [H.sub.0] corresponding to the electronic levels [5d.sup.10]6s [[blank].sup.2][S.sub.1/2] and [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] having [J.sub.Z] eigenvalue eigenvalue In mathematical analysis, one of a set of discrete values of a parameter, k, in an equation of the form Lx = kx. Such characteristic equations are particularly useful in solving differential equations, integral equations, and systems of [M.sub.J] by IS 1/2 [M.sub.J][greater than] and ID 5/2 [M.sub.J][greater than], respectively. The corresponding eigenvalues eigenvalues statistical term meaning latent root. of [H.sub.0] are denoted W(S, 1/2) and W(D, 5/2). An arbitrary eigenstate of [H.sub.0] with eigenvalue W([gamma], J) and electronic angular momentum angular momentum: see momentum. angular momentum Property that describes the rotary inertia of a system in motion about an axis. It is a vector quantity, having both magnitude and direction. J is denoted \[gamma] J [M.sub.J][greater than]. Since [[blank].sup.199][Hg.sup.+] has in addition a nuclear angular momentum I, where I = 1/2, the complete state designation is \[gamma][JFM JFM Journal of Fluid Mechanics JFM Just for Me JFM Japan Finance Corporation for Municipal Enterprises JFM Joint Forces Memorandum JFM Joint Frequency Management JFM Just Fine Magic (slang, polite form; explains unexplainable processes) .sub.F][greater than], where F is the total angular momentum, and [M.sub.F] is the eigenvalue of [F.sub.Z]. 3. Quadratic Zeeman Shift In order to calculate the energy shifts due to the hyperfine interaction and to an external magnetic field B [equivalent] BZ, we define effective Hamiltonian operators [H'.sub.S] and [H'.sub.D] that operate within the subspaces of hyperfine sublevels associated with the electronic levels [5d.sup.10]6s [[blank].sup.2][S.sub.1/2] and [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2], respectively: [H'.sub.S] = h[A.sub.S]I.J + [g.sub.J](S)[[micro].sub.B] J.B + [g'.sub.I][[micro].sub.B]I.B, (1) [H'.sub.D] = h[A.sub.D]I.J + [g.sub.J](D)[[micro].sub.B]J.B + [g'.sub.I][[micro].sub.B]I.B, (2) where [A.sub.S] and [A.sub.D] are the dipole hyperfine constants, [g.sub.J](S) and [g.sub.J](D) are the electronic g-factors, [g'.sub.I] is the nuclear g-factor, h is the Planck constant The Planck constant (denoted  ) is a physical constant that is used to describe the sizes of quanta. , and [[micro].sub.B] is the Bohr magneton Bohr magnetonSee under magneton. . All of the parameters entering [H'.sub.S] and [H'.sub.D] are known from experiments, although a more accurate measurement of [g.sub.J](D) would be useful. The ground-state hyperfine constant [A.sub.S] has been measured in a [[blank].sup.199][Hg.sup.+] microwave frequency standard to be 40 507.347 996 841 59 (43) MHz (MegaHertZ) One million cycles per second. It is used to measure the transmission speed of electronic devices, including channels, buses and the computer's internal clock. A one-megahertz clock (1 MHz) means some number of bits (16, 32, 64, etc. [11]. The excited-state hyperfine constant [A.sub.D] has been measured recently by an extension to the work described in Ref. (5], in which the difference in the frequencies of the IS 1/2 0 0[greater than] to ID 5/2 2 0[greater than] and the IS 1/2 0 0[greater than] to ID 5/2 3 0[greater than] transition frequencies was determined to be [3A.sub.D] = 2 958.57(12) MHz [12], in good agreement with an earlier, less precise measurement by Fabry-Perot spectroscopy spectroscopy Branch of analysis devoted to identifying elements and compounds and elucidating atomic and molecular structure by measuring the radiant energy absorbed or emitted by a substance at characteristic wavelengths of the electromagnetic spectrum (including gamma ray, [13]. The ground-state electronic g-factor [g.sub.J](S) was measured in [[blank].sup.198][Hg.sup.+] by rf-optical double resonance to be 2.003 174 5(74) [14]. The excited-state electronic g-factor [g.sub.J](D) was measured in [[blank].sup.198][Hg.sup.+] by conventional grating spectroscopy of the 398 nm [5d.sup.10]6p [[blank].sup.2][P.sub.3/2] to [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] line to be 1.198 0(7) [15]. The difference in [g.sub.J](S) or [g.sub.J](D) between [[blank].sup.198][Hg.sup.+] and [[blank].sup.199][Hg.sup.+] is estimated to be much less than the experimental uncertainties. The nuclear g-factor [g'.sub.I] is -5.422 967(9) X [10.sup.-4] [16]. The measurement was made with neutral ground-state [Hg.sup.199] atoms, so the diamagnetic di·a·mag·net·ic adj. Of or relating to a substance that is repelled by a magnet. di  a·mag shielding factor will be slightly different from that in the ion. However, this is effect is negligible  This article or section is written like a personal reflection or and may require .Please [ improve this article] by rewriting this article or section in an . , since the magnitude of [g'.sub.I] is so small compared to [g.sub.J](S) or [g.sub.J](D). The determination of [g.sub.J](D) could be improved by measuring the optical-frequency difference between two components of the 282 nm line and the frequency of a ground-state microwave transition at the same magnetic field. Since the uncertainty in the quadratic Zeeman shift is due mainly to the uncertainty in [g.sub.J](D), it is useful to see how accurately it can be estimated theoretically. The Lande g-factor for a [[blank].sup.2][D.sub.5/2] state, including the correction for the anomalous a·nom·a·lous adj. 1. Deviating from the normal or common order, form, or rule. 2. Equivocal, as in classification or nature. magnetic moment of the electron, is 1.200 464. The Cowan-code calculation shows that the configuration mixing does not change this value by more than about [10.sup.-6], i.e., 1 in the last place. There are several relativistic and diamagnetic corrections that modify [g.sub.J](D), one of which, called the Breit-Margenau correction by Abragam and Van Vleck Van Vleck , John Hasbrouck 1899-1980. American physicist. He shared a 1977 Nobel Prize for developments in computer memory. Noun 1. Van Vleck - United States physicist (1899-1980) John Hasbrouck Van Vleck, John Van Vleck [17], is proportional to the electron mean kinetic energy kinetic energy: see energy. kinetic energy Form of energy that an object has by reason of its motion. The kind of motion may be translation (motion along a path from one place to another), rotation about an axis, vibration, or any combination of . The other corrections are more difficult to calculate. The Cowan-code result for the mean kinetic energy of an electron in the 5d orbital orbital Mathematical expression, called a wave function, that describes properties characteristic of no more than two electrons near an atomic nucleus or molecule. An orbital can be considered a three-dimensional region in which there is a 95% probability of finding an of the [5d.sup.9][6s.sup.2] configuration is T = 19.32 hc[R.sub.[infty], where [R.sub.[infty]] is the Rydberg constant Rydberg constant (rĭd`bərg), physical constant used in studies of the spectrum of a substance. Its value for hydrogen is 109,737.3 cm−1. . Using this value, we obtain a theoretical value of [g.sub.J](D), including the Breit-Margenau correction, of 1.199 85, which disagrees with the the experimental value by 1.85 X [10.sup.-3], which is 2.6 times the estimated experimental uncertainty of Ref. [15]. If we calculate [g.sub.J](D) for neutral gold, which is isoelectronic i·so·e·lec·tron·ic adj. Having equal numbers of electrons or the same electronic configuration. to [Hg.sup.+], by the same method, we obtain a value which differs from the accurately measured experimental one [18] by (7 [+ or -] 2) X [10.sup.-5]. Thus, the error in the calculated value for [g.sub.J](D) of [[blank].sup.99][Hg.sup.+] might be less than 1 X [10.sup.-4], but it is impossible to be certain of this, since there are uncalculated un·cal·cu·lat·ed adj. Not thought out in advance; spontaneous. terms. Measurements of the [[blank].sup.199][Hg.sup.+] optical clock frequency at different values of the magnetic field should result in a better experimental value for [g.sub.J]D) in the near future. For low magnetic fields magnetic fields, n.pl the spaces in which magnetic forces are detectable; created by magnetostrictive ultrasonic scalers to cause the tips of instruments such as ultrasonic scalers to vibrate. (B less than 1 mT), it is sufficient to calculate the energy levels to second order in B. To this order in B, the energies of the hyperfine-Zeeman sublevels for the ground electronic level are W(S, 1/2, 0, 0, B) = W(S, 1/2) - 3h[A.sub.s]/4 - [[g.sub.J](S) - [g'.sub.J]].sup.2][[[micro].sup.2].sub.B][B.sup.2]/4h[A.sub.s], (3) W(S, 1/2. 1, 0, B) = W(S, 1/2)+ h[A.sub.s]/4 + [[g.sub.J](S) - [g'.sub.I]].sup.2] [[[micro].sup.2].sub.B][B.sup.2]/4h[A.sub.s], (4) W(S, 1/2, 1, [+ or -]1, B) = W(S, 1/2)+ h[A.sub.s]/4 [+ or -] [[g.sub.J](S)+[g'.sub.I]][[micro].sub.B]B/2, (5) For the [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] level we have W(D, 5/2, 2, 0, B) = W(D, 5/2) - 7h[A.sub.D]/4 - [[[g.sub.J](D)- [g'.sub.I]].sup.2][[[micro].sup.2].sub.B][B.sup.2]/12h[A.sub.D], (6) W(D, 5/2, 2, [+ or -]1, B) = W(D, 5/2) - 7h[A.sub.D]/4 [+ or -] [7[g.sub.J](D) - [g.sub.I]][micro].sub.B]B/6 - 2[[[g.sub.J](D) - [g'.sub.I]].sup.2][[[micro].sup.2].sub.B][B.sup.2]/108h[A.sub.D], (7) W(D, 5/2, 2, [+ or -]2, B)= W(D, 5/2) 7h[A.sub.D]/4 [+ or -] [7[g.sub.J](D) - [g'.sub.I]][[micro].sub.B]B/3 - 5[[g.sub.J](D) - [g'.sub.I]].sup.2][[[micro].sup.2].sub.B][B.sup.2]/108h[A.sub.D], (8) W(D, 5/2, 3, 0, B) = W(D, 5/2) + 5h[A.sub.D]/4 + [[[g.sub.J](D) - [g'.sub.I]].sup.2][[[micro].sup.2].sub.B][B.sup.2]/12h[A.sub.D], (9) W(D, 5/2, 3, [+ or -]1, B) = W(D, 5/2)+ 5h[A.sub.D]/4 [+ or -] [5[g.sub.J](D) + [g'.sub.I][[micro].sub.B]B/6 + 2[[[g.sub.J](D) - g'.sub.I]].sup.2] [[[micro].sup.2].sub.B][B.sup.2]/27h[A.sub.D], (10) W(D, 5/2, 3, [+ or -]2, B) = W(D, 5/2) + 5h[A.sub.D]/4 [+ or -] [5[g.sub.J](D) + [g'.sub.I]][[micro].sub.B]B/3 + 5[[[g.sub.J](D) - [g'.sub.I]].sup.2][[[micro].sup.2].sub.B][B.sup.2]/108h[A.sub.D], (11) W(D, 5/2, 3, [+ or -]3, B) = W(D, 5/2)+ 5h[A.sub.D]/4 [+ or -] [5[g.sub.J](D)+[g'.sub.I]][[micro].sub.B]B/2. (12) Here, W([gamma], J, F, [M.sub.F], B) denotes the energy of the state \[gamma] JF[M.sub.F][greater than], including the effects of the hyperfine interaction and the magnetic field. At a value of B of 0.1 mT, the quadratic shift of the IS 1/2 0 0[greater than] to ID 5/2 2 0[greater than] transition (optical clock transition) is - 189.25(28) Hz, where the uncertainty stems mainly from the uncertainty in the experimental value of [g.sub.J](D). In practice, the error may be less than this if the magnetic field is determined from the Zeeman splittings within the ID 5/2 F [M.sub.F]) sublevels. The reason is that an error in [g.sub.J](D) leads to an error in the value of B inferred from the Zeeman splittings, which partly compensates for the [g.sub.J](D) error. If instead we use the calculated value of [g.sub.J](D), the quadratic shift for B = 0.1 mT is -189.98 Hz, where the uncertainty is difficult to estimate. 4. Quadratic Stark Shift The theory of the quadratic Stark shift in free atoms has been described in detail by Angel and Sandars [19]. The Stark Hamiltonian is [H.sub.E] = -[micro].E, (13) where [micro] is the electric-dipole moment operator, [micro] = -e[[sigma].sub.i][r.sub.i], (14) and E is the applied external electric field. In Eq. (14), [r.sub.i] is the position operator of the ith electron, measured relative to the nucleus, and the summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) is over all electrons. First consider an atom with zero nuclear spin, such as [[blank].sup.198][Hg.sup.+]. To second order in the electric field, the Stark shifts of the set of sublevels \[gamma] [JM.sub.J][greater than] depend on two parameters, [[alpha].sub.scalar] ([gamma], J) and [[alpha].sub.tensor] ([gamma], J), called the scalar and tensor polarizabilities. In principle, when both magnetic and electric fields are present but are not parallel, the energy levels are obtained by simultaneously diagonalizing the hyperfine, Zeeman, and Stark Hamiltonians. In practice, the Zeeman shifts are normally much larger than the Stark shifts, so that HE does not affect the diagonalization. In that case, the energy shift of the state \[gamma] [JM.sub.J][greater than] due to [H.sub.E] is [delta][W.sub.E]([gamma], J, [M.sub.J], E) = - 1/2 [[alpha].sub.scalar]([gamma], J) [E.sup.2] -1/4 [[alpha].sub.tensor]([gamma], J) \3[[M.sup.2].sub.J] - J(J + 1)]/J (2J - 1) (3[[M.sup.2].sub.z] - [E.sup.2]). (15) Treating [H.sub.E] by second-order perturbation theory perturbation theory A set of mathematical methods for obtaining approximate solutions to complex equations for which no exact solution is possible or known, generally involving an iterative algorithm in which each new term contributing to the solution has leads to the following expressions for the polarizabilities [19]: [[alpha].sub.sealar]([gamma], J) = 8[pi][[epsilon].sub.0]/3(2J + 1) [[sigma].sub.[gamma]J'] [\([gamma] J [parallel to][[micro].sup.(1)][parallel to] [gamma]' J')\].sup.2]/W ([gamma]', J') - W([gamma], J), (16) [[alpha].sub.tensor]([gamma], J) = 8[pi][[epsilon].sub.0] [[10J(2J - 1)/3(2J + 3)(J + 1)(2J + 1)].sup.1/2] [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (17) The summations are over all levels other than \[gamma]J[greater than]. Equations (16) and (17) can be rewritten in terms of the oscillator strengths [f.sub.[gamma]J,[gamma]J']: [[alpha].sub.scalar]([gamma], J) = 4[pi][[epsilon].sub.0][e.sup.2][h.sup.2]/[m.sub.c] [[sigma].sub.[gamma]'J'] [f.sub.[gamma]J,[gamma]'J']/[[W([gamma]', J') - W([gamma], J)].sup.2], (18) [[alpha].sub.tensor]([gamma], J) = 4[pi][[epsilon].sub.0][e.sup.2][h.sup.2]/[m.sub.c] [[30 J(2J - 1)(2J + 1)/(2J + 3)(J + 1)].sup.1/2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19) where [m.sub.c] is the electron mass. The tensor polarizability is zero for levels with J [less than] 1, such as the [Hg.sup.+] [5d.sub.10][6s.sub.2][S.sub.1/2] level. For an atom with nonzero non·ze·ro adj. Not equal to zero. nonzero Not equal to zero. nuclear spin I, the quadratic Stark shift of the state \[gamma] [JFM.sub.F][greater than] is [delta][W.sub.E]([gamma], J, [M.sub.F], E) = -1/2 [[alpha].sub.sealer sealer, n a substance used to fill the space around silver or gutta-percha points in a pulp canal. Most contain some combination of zinc, barium, and bismuth salts and eugenol, Canadian balsam, and eucalyptol. ]([gamma], J, F) [E.sup.2] - 1/4 [[alpha].tensor]([gamma], J, F) [3[[M.sup.2].sub.F] - F(F + 1)]/F(2F - 1) (3[[E.sup.2].sub.z] - [E.sup.2]). (20) We make the approximation that hyperfine interaction does not modify the electronic part of the atomic wave-functions (the I J-coupling approximation of Angel and Sandars [19]). This approximation is adequate for the present purpose, which is to evaluate the Stark shift of the [[blank].sup.199][Hg.sup.+] optical clock transition. Obtaining the differential Stark shift between the hyperfine levels of the ground state, which is significant for the [[blank].sup.199][Hg.sup.+] microwave frequency standard [11], requires going to a higher order of perturbation theory [20]. In the I 1-coupling approximation [19], [[alpha].sub.scalar]([gamma], J, F) = [[alpha].sub.scalar]([gamma], J), (21) [[alpha].sub.tensor]([gamma], J, F) = [(-1).sup.I+J+F] [[F(2F - 1)(2F + 1)(2J + 3)(2J + 1)(J + 1)/(2F + 3)(F + 1)J(2J - 1)].sup.1/2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22) Equations (18) and (19) were used to evaluate the polarizabilities for the [Hg.sup.+] [5d.sup.10]6s [[blank].sup.2][S.sub.1/2] and [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] levels. For the calculation of [[alpha].sub.scalar] (S, 1/2), the oscillator strengths for all electric-dipole transitions connecting the [5d.sup.10]6s configuration to the [5d.sup.10]np (n = 6,7,8) and [5d.sup.9]6s6p configurations were included. These were taken from the theoretical work of Brage et al. [21]. The final result is [[alpha].sub.scalar] (S, 1/2)/(4[phi][[epsilon].sub.0]) = 2.41 X [10.sup.-24] [cm.sup.3], which compares very well with the value of 2.22 x [10.sup.-24] [cm.sup.3] obtained by Henderson et al. from a combination of experimental and calculated oscillator strengths [22]. For the calculations of [[alpha].sub.scalar(D, 5/2) and [[alpha].sub.tensor](D, 5/2), thc oscillator strengths for electric-dipole transitions to the [5d.sup.10]np (n = 6,7,8), [5d.sup.10]5f, and [5d.sup.9]6s6p configurations were taken from Brage et al. [21]. The oscillator strengths for electric-dipole transitions to the [5d.sup.9]6s7p and [5d.sup.8][6s.sup.2]6p configurations were taken from the Cowan-code calculations. The results were [[alpha].sub.scalor](D, 5/2)/(4[pi][[epsilon].sub.0]) = 3.77 X [10.sup.-24] [cm.sup.3] and [[alpha].sub.tensor](D, 5/2)/(4[pi][[epsilon].sub.0]) = -0.263 x [10.sup.-24] [cm.sup.3]. Evaluating Eq. (22) for F = 2 and F = 3 in the [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] level, we obtain [[alpha].sub.tensor](D, 5/2 2) = 4/5 [[alpha].sub.tensor](D, 5/2) and [[alpha].sub.tensor](D, 5/2, 3) = [[alpha].sub.tensor](D, 5/2). The tensor polarizability is much smaller than the scalar polarizabilities and in any case does not contribute if the external electric field is isotropic Refers to properties that do not differ no matter which direction is measured. For example, an isotropic antenna radiates almost the same power in all directions. In practice, antennas cannot be 100% isotropic. , as is the case for the blackbody radiation blackbody radiation The electromagnetic radiation that a perfect blackbody would give off at a given temperature. A warm blackbody would emit radiation with a higher average frequency than a cooler one. Noun 1. field. The net shift of the optical clock transition due to the scalar polarizabilities is 1/2[[[alpha].sub.scalar](S, 1/2) - [[alpha].sub.scalar](D, 5/2)]][E.sup.2]. In frequency units, the shift is - 1.14 X [10.sup.-3] [E.sup.2] Hz, where E is expressed in V/cm. The error in the coefficient coefficient /co·ef·fi·cient/ (ko?ah-fish´int) 1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities. 2. is difficult to estimate, particularly since it is a difference of two quantities of about the same size. However, the total shifts are small for typical experimental conditions. If the electric field is time-dependent, as for the blackbody blackbody Theoretical surface that absorbs all radiant energy that falls on it, and radiates electromagnetic energy at all frequencies, from radio waves to gamma rays, with an intensity distribution dependent on its temperature. field, the mean-squared value [less than][E.sup.2][greater than] is taken. At a temperature of 300 K, the shift of the optical clock transition due to the blackbody electric field is -0.079 Hz. The mean-squared blackbody field is proportional to the fourth power of the temperature. For a single, laser-cooled ion in a Paul trap, the mean-squared trapping trapping, most broadly, the use of mechanical or deceptive devices to capture, kill, or injure animals. It may be applied to the practice of using birdlime to capture birds, lobster pots to trap lobsters, and seines to catch fish. electric fields can be made small enough that the Stark shifts are not likely to be observable ob·serv·a·ble adj. 1. Possible to observe: observable phenomena; an observable change in demeanor. See Synonyms at noticeable. 2. [23]. 5. Electric Quadrupole Shift The atomic quadrupole moment is due to a departure of the electronic charge distribution of an atom from spherical spher·i·cal adj. Having the shape of or approximating a sphere; globular. symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. . Atomic quadrupole moments were first measured by the shift in energy levels due to an applied electric-field gradient in atomic-beam resonance experiments [24, 25]. The interaction of the atomic quadrupole moment with external electric-field gradients, for example those generated by the electrodes Electrodes Tiny wires in adhesive pads that are applied to the body for ECG measurement. Mentioned in: Electrocardiography of an ion trap ion trap n. A device, such as a magnet, used to prevent ions in an electron beam from striking other apparatus. ion trap , is analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development. a·nal·o·gous adj. to the interaction of a nuclear quadrupole moment with the electric field gradients due to the atomic electrons. Hence, we can adapt the treatment used for the electric-quadrupole hyperfine interaction of an atom [26]. The Hamiltonian describing the interaction of external electric-field gradients with the atomic quadrupole moment is [H.sub.Q] = [nabla][E.sup.(2)]*[[theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ].sup.(2)] = [[[sigma].sup.2].sub.q=2][(-1).sup.q][nabla][[E.sup.(2)].sub.q][[[the ta].sup.(2)].sub.-q], (23) where [nabla][E.sup.(2)] is a tensor describing the gradients of the external electric field at the position of the atom, and [[theta].sup.(2)] is the electric-quadrupole operator for the atom. Following Ref. [26], we define the components of as [nabla][E.sup.(2)] as [nabla][[E.sup.(2)].sub.0] = - 1/2 [partial][E.sub.z]/[partial]z, (24) [nabla][[E.sup.(2)].sub.[+ or -]1] = [+ or -] [square root]6/6 [partial][E.sub.[+ or -]]/[partial]z = [+ or -] [square root]6/6 [partial][+ or -] [E.sub.z], (25) [nabla][[E.sup.(2)].sub.[+ or -]2] = - [square root]6/12 [[partial].sub.[+ or -] [E.sub.[+ or -]], (26) where [E.sub.[+ or -]] [equivalent] [E.sub.x] [+ or -] [iE.sub.y] and [[partial].sub.[+ or -]] [equivalent] [partial]/[partial]x [+ or -] i[partial]/[[partial].sub.y]. The operator components [[theta].sup.(2)].sub.q] are defined in terms of the electronic coordinate operators as [[[theta].sup.(2)].sub.0] = - e/2 [[sigma].sub.j] (3[[z.sup.2].sub.j] - [[r.sup.2].sub.j]), (27) [[[theta].sup.(2)].sub.[+ or -]1] = - e[square root]3/2 [[sigma].sub.j] [z.sub.j]([x.sub.j] [+ or -] [iy.sub.j]), (28) [[[theta].sup.(2)].sub.[+ or -]2 = - e[square root]3/8 [[sigma].sub.j] ([x.sub.j] [+ or -] [iy.sub.j]).sup.2], (29) where the sums are taken over all the electrons. The quadrupole moment [theta]([gamma], J) of an atomic level \[gamma]J[greater than] is defined by the diagonal matrix Noun 1. diagonal matrix - a square matrix with all elements not on the main diagonal equal to zero square matrix - a matrix with the same number of rows and columns scalar matrix - a diagonal matrix in which all of the diagonal elements are equal element in the state with maximum [M.sub.j]: [theta]([gamma], J) = [less than][gamma] JJ\[[[theta].sup.(2)].sub.0]\[gamma] JJ[greater than]. (30) This is the definition used by Angel et al. [24]. In order to simplify the form of [nabla][E.sup.(2)], we make a principal-axis transformation as in Ref. [27]. That is, we express the electric potential in the neighborhood of the atom as [phi](x', y', z') = A [([x.sup.'2] + [y.sup.'2] - [2z.sup.'2]) + [epsilon]([x.sup.'2] - [y.sup.'2])]. (31) The principal-axis (primed) frame (x', y', z') is the one in which [phi] has the simple form of Eq. (31), while the laboratory (unprimed) frame (x, y, z) is the one in which the magnetic field is oriented o·ri·ent n. 1. Orient The countries of Asia, especially of eastern Asia. 2. a. The luster characteristic of a pearl of high quality. b. A pearl having exceptional luster. 3. along the z axis. The tensor components of [nabla][E.sup.(2)] in the principal-axis frame are obtained by taking derivatives of [phi](x', y', z'): [nabla][[E.sup.(2)'].sub.0] = -2A, (32) [nabla][[E.sup.(2)'].sub.[+ or -]1]= 0, (33) [nabla][[E.sup.(2)'].sub.[+ or -]2] = [square root]2/3[epsilon]A. (34) In the principal-axis frame, [H.sub.Q] has the simple form [H.sub.Q]= -2A [[[theta].sup.(2)'].sub.0] + [square root]2/3 [epsilon]A ([[[theta].sup.(2)'].sub.2] + [[[theta].sup.(2)'].sub.-2]). (35) As long as the energy shifts due to [H.sub.Q] are small relative to the Zeeman shifts, which is the usual case in practice, [H.sub.Q] can be treated as a perturbation. In that case, it is necessary only to evaluate the matrix elements of [H.sub.Q] that are diagonal in the basis of states \[gamma][JFM.sub.F]), where F is the total atomic angular momentum, including nuclear spin I, and [M.sub.F] is the eigenvalue of [F.sub.z] with respect to the laboratory (not principal-axis) frame. Let [omega] denote the set of Euler angles ([alpha], [beta], [gamma]} that takes the principal-axis frame to the laboratory frame. To be explicit, starting from the principal-axis frame, we rotate the coordinate system coordinate system Arrangement of reference lines or curves used to identify the location of points in space. In two dimensions, the most common system is the Cartesian (after René Descartes) system. coincides about the z axis by [alpha], then about the new y axis Y axis, n See axis, Y. by [beta], and then about the new z axis by [gamma] so that the rotated rotated turned around; pivoted. rotated tibia see rotated tibia. coordinate system coincides with the laboratory coordinate system. We can set [gamma] = 0, since the final rotation about the laboratory z axis, which is parallel to B, has no effe ct. The states \[gamma]JF[micro])' defined in the principal-axis frame and the states /[gamma]JF[micro]) defined in the laboratory frame are related by \[gamma]JFm' = [[sigma].sub.[micro]] [D.sup.(F).sub.[micro]m]([omega])/[gamma]JF[micro][greater than], (36) where [[D.sup.(F)].sub.[micro]m]([omega]) is a rotation matrix In matrix theory, a rotation matrix is a real square matrix whose transpose is its inverse and whose determinant is +1. /[gamma]JF[micro][greater than] = [[sigma].sub.m] [[D.sup.(F)*].sub.[micro]m] ([omega]/[gamma] JFm[greater than]'. (37) In order to evaluate the diagonal matrix elements of [H.sub.Q] in the laboratory frame, it is necessary to evaluate matrix elements of the operators [[[theta].sup.(2)'].sub.q], defined in the principal-axis frame. These matrix elements are of the form [lesser than][gamma]JF[micro]/[[[theta].sup.(2)'].sub.q]/[gamma]JF[micro][gre ater than] = [sigma].sub.m'm][[D.sup.(F)].sub.[micro]m'].([omega])[[D.sup.(F)*].su b.[micro]m] ([omega]) '([gamma]JFm'/[[[theta].sup.(2)'].sub.q]/[gamma]JFm)', (38) = ([gamma]JF//[[theta].sup.(2)]//[gamma]JF)[[sigma].sub.m'm][(-1).sup.F -m'] X[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][[D.sup.{F}].sub.[micro]m']([omega][[D.sup.(F)*].sub.[micro]m] ((omega]), (39) = [(-1).sup.F-[micro]-q]([gamma]JF//[[theta].sup.(2)]//[gamma]JF) x [[sigma].sub.m'm][MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][[D.sup.(F)].sub.-[micro]-m([omega]), (40) = [(-1).sup.F-[micro]-q]([gamma]JF//[[theta].sup.(2)]//[gamma]JF) [[sigma].sub.Kmm'nn'] (2K+1) X[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][[D.sup.(K)*].sub.n'n] ([omega]), (41) = [(-1).sup.F-[micro]-q]([gamma]JF//[[theta].sup.2]//[gamma]JF)[MATHEMA TICAL Ti`cal´ n. 1. A bean-shaped coin of Siam, worth about sixty cents; also, a weight equal to 236 grains troy. 2. A money of account in China, reckoning at about $1.60; also, a weight of about four ounces avoirdupois. EXPRESSION NOT REPRODUCIBLE IN ASCII][[D.sup.(2)*].sub.0-q] ([omega]), (42) where Eq. (39) follows from the Wigner-Eckart theorem The Wigner-Eckart theorem is a theorem of representation theory and quantum mechanics allowing operators to be transformed from one basis to another. These transformations involve the use of Clebsch-Gordan coefficients. , and Eqs. (40), (41), and (42) follow from Eqs. (4.2.7), (4.3.2), and (3.7.8) of Ref. [28], respectively. The required rotation matrix elements are, from Eq. (4.1.25) of Ref. [281 (with correction of a typographical error typographical error - (typo) An error while inputting text via keyboard, made despite the fact that the user knows exactly what to type in. This usually results from the operator's inexperience at keyboarding, rushing, not paying attention, or carelessness. Compare: mouso, thinko. ), [[[D.sup.(2)*].sub.00]([omega]) = 1/2(3 [cos.sup.2] [beta] - 1), (43) [[[D.sup.(2)*].sub.0][+ or -]2]([omega]) = [square root]3/8 [sin.sup.2] [beta](cos 2[alpha] [- or +] i sin 2[alpha]). (44) The 3-j symbol in Eq. (42) is [MATHEMATICAL Expression not reproducible in ASCII] = [(-1).sup.F-[micro]] 2[3[[micro].sup.2] - F(F+1)]/[[(2F + 3)(2F + 2)(2F + 1)2F(2F - 1)].sup.1/2]. (45) The diagonal matrix elements of [H.sub.Q] in the laboratory frame are ([gamma][JFM.sub.F][absolute val. of [H.sub.Q]][gamma][JFM.sub.F]) = -2[[[3M.sup.2].sub.F] - F(F + 1)]A([gamma] JF[parallel][[theta].sup.(2)][parallel][gamma] JF)/[[(2F + 3)(2F + 2)(2F + 1)2F(2F - 1)].sup.1/2] X [(3 [cos.sup.2] [beta] - 1) - [epsilon] [sin.sup.2] [beta]([cos.sup.2] [alpha] - [sin.sup.2] [alpha])]. (46) It is simple to show, by directly integrating the angular angular /an·gu·lar/ (ang´gu-lar) sharply bent; having corners or angles. factor in square brackets square bracket n. One of a pair of marks, [ ], used to enclose written or printed material or to indicate a mathematical expression considered in some sense a single quantity. in Eq. (46), that the average value of the diagonal matrix elements of [H.sub.Q], taken over all possible orientations of the laboratory frame with respect to the principal-axis frame, is zero. This also follows directly from the fact that the quantity in square brackets is a linear combination of spherical harmonics In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates. Spherical harmonics are important in many theoretical and practical applications, particularly in the . It is less obvious that the average, taken over any three mutually perpendicular orientations of the laboratory z quantization (1) The division of a range of values into a single number, code or classification. For example, class A is 0 to 999, class B is 1000 to 9999 and class C is 10000 and above. (2) In analog to digital conversion, the assignment of a number to the amplitude of a wave. axis, is also zero. This result is proven in Appendix A. This provides a method for eliminating the quadrupole shift from the observed transition frequency. The magnetic field must be oriented in three mutually perpendicular directions with respect to the trap electrodes, which are the source of the external quadrupole field, but with the same magnitude of the magnetic field. The average of the transition frequencies taken under these three conditions does not contain the quadrupole shift. The reduced matrix element in Eq. (46) is, in the IJ-coupling approximation, [MATHEMATICAL EXPRESSION IS NOT REPRODUCIBLE IN ASCII] (47) where I is included in the state notation in order to specify the order of coupling of land J. For the particular case of the [[blank].sup.199][Hg.sup.+] [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] level, the reduced matrix elements are (D 5/2 2[parallel][[theta].sup.(2)][parallel]D 5/2 2) = 2 [square root]14/5 [theta](D, 5/2), (48) (D 5/2 3[parallel][[theta].sup.(2)][parallel]D 5/2 3) = 2 [square root]21/5 [theta](D, 5/2), (49) Since the Cowan-code calculation shows that there is very little configuration mixing in the [[blank].sup.199][Hg.sup.+] [5d.sup.9][6s.sup.2] [[blank].sup.2][D.sub.5/2] level, [theta](D, 5/2) can be reduced to a matrix element involving only the 5d orbital: [theta](D, 5/2) = e/2 ([5d.sup.2][d.sub.5/2], [m.sub.j] = 5/2[absolute val. of [3z.sup.2] - [r.sup.2]][5d.sup.2][d.sub.5/2], [m.sub.j] = 5/2), (50) = e/2 (5d, [m.sub.l] = 2[absolute val. of [3z.sup.2] - [r.sup.2]]5d, [m.sub.l] = 2), (51) = e [square root]4[pi]/5 (5d, [m.sub.l] = 2[absolute val. of [Y.sub.2,0]([theta, [phi])]5d, [m.sub.l] = 2), (52) = e [square root]4[pi]/5 (5d[absolute val. of [r.sup.2]]5d) X [[[integral].sup.2[pi]].sub.0] [[[integral].sup.[pi]].sub.0] [[Y.sup.*].sub.2,2]([theta], [phi])[Y.sub.2,0]([theta], [phi])[Y.sub.2,2]([theta], [phi])sin [theta]d[theta]d[phi], (53) [MATHEMATICAL EXPRESSION IS NOT REPRODUCIBLE IN ASCII] (54) = -2e/7 (5d[absolute val. of [r.sup.2]]5d). (55) The apparent sign reversal in Eq. (50) relative to Eqs. (27) and (30) is due to the fact that the quadrupole moment is due to a single hole in the otherwise filled 5d shell rather than to a single electron. According to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. the Cowan-code calculation, (5d[absolute val. of [r.sup.2]]5d) = 2.324 [[a.sup.2].sub.0] = 6.509 X [10.sup.-17] [cm.sup.2], (56) where [a.sub.0] is the Bohr radius In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy. . Since the quadrupole shifts are zero in the [5d.sup.10]6s [[blank].sup.2][S.sub.1/2] level, the quadrupole shift of the [[blank].sup.199][Hg.sup.+] optical clock transition is due entirely to the shift of the ID 5/2 2 0) state, and is given by (D 5/2 2 0[absolute val. of [H.sub.Q]]D 5/2 2 0) = 4/5 A [theta](D, 5/2)[(3 [cos.sup.2] [beta] - 1) - [epsilon] [sin.sup.2] [beta]([cos.sup.2] [alpha] - [sin.sup.2] [alpha])], (57) = -8/35 Ae (5d[absolute val. of [r.sup.2]]5d)[(3 [cos.sup.2] [beta] - 1) - [epsilon] [sin.sup.2] [beta]([cos.sup.2] [alpha] - [sin.sup.2] [alpha])], (58) [approximate] - 3.6 X [10.sup.-3]hA[(3 [cos.sup.2] [beta] - 1) - [epsilon] [sin.sup.2] [beta]([cos.sup.2] [alpha] - [sin.sup.2] [alpha])]Hz, (59) where A is expressed in units of V/[cm.sup.2]. Thus, for typical values A [approximate] [10.sup.3]V/[cm.sup.2] and [absolute val. of [epsilon]] [less than and similar to] 1, the quadrupole shift is on the order of 1 Hz. 6. Appendix A. Angular Averaging of the Quadrupole Shift For the purpose of describing the quadropole shift, the orientation of the laboratory (quantization) axis with respect to the principal-axis frame is defined by the angles [Beta] and [Alpha]. In the principal-axis coordinate system, a unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1 (the unit length). A unit vector is often written with a superscribed caret or “hat”, like this along the laboratory z axis is Axis I Psychiatry A classification dimension used with DSM-IV, which includes clinical disorders and syndromes and/or other areas of concern. See DSM-IV, Multiaxial system. defined in terms of [Beta] and {Alpha] by z[acute sign over it] = (sin [Beta] cos [Alpha], sin [Alpha], cos [Beta] (60) We wish to show that thtr angular dependence of the quadropole shift is such that the diagonal matrix elements given by Eq. (46) average to zero, for Z[acute sign over it] along any three mutually perpendicular directions An arbitrary set of three mutually perpendicular unit vectors e[sub.1], e[sub.2],and e[sub.3] can be parameterized by the set of angles [phi],[empty set], and [varphi] in the follwing way: e[sub.1] = sin [pi]cos [empty set], sin [pi], sin [empty set], cos [pi] (61) e[sub.2]=(cos[pi]cos[emptyset]cost[varphi]-sin[empty set]sin[varphi], sin[empty set]cos[pi]cos[var[hi]+cos[emptyset]sin[varphi]-sin[pi] cos[varphi](62) e[sub.3]=9-cos[emptyset]cos[pi]sin[varphi]-sin[emptyset]cos[varphi], -sin[emptyset]cos[pi]sin[varphi] +cos[emptyset]cos[varphi], sin[pi]sin[varphi](63) It can be verifed by direct computaiton that e[sub.i][dot]e[sub.j] = [[delta][sub.ij]. The quadropole shift can be evaluated for each of these three unit vectors subsituted for z[acute sign over it][Eq.(60)] and the average taken. First consider the average of the quantity (3 cos[sup.2] [Beta-1] that appears in Eq.(46): We use the fact that cos [Beta] is the third component of z[acute sign over it], so the average is: <3 cos[sup.2][Beta]-1>= cos[sup.2][pi] + sin[sup.2][pi]cos[sup.2][varphi] + sin[sup.2][pi]sin[sup.2][varphi]-1,(64) =cos[sup.2][[pi] + sin [sup.2][pi]-1(65) =0,(66) for arbitrary [pi][emptyset], and [varphi]. Similarly, the average of the other angle-dependent term in Eq. (46), sin[sup.2 [Beta](cos[sup.2[alpha] -sin[sup.2[alpha]), is calculated by making use of the fact that sin [Beta] cos[alpha]is the first component of z[acute sign over it], and sin [Beta] sin[alpha]is the second: <sin[sup.2] [Beta](cos [sup.2][alpha]-sin[sup.2[alpha])> = 1/3[sin[sup.2][pi]cos[sup.2][emptyset]-sin[sup.2][pi]sin[sup.2][emptyset] +[(cos [emptyset]cos[pi]co[varphi]-sin[emptyset]sin[varphi])[sup.2]] -[(sin[emptyset]cos[pi]cos[varphi]+cos[emptyset]sin[varphi])[sup.2]] +[(cois[emptyset]cos[pi]sin[varphi]+sin[emptyset]cos[varphi])[sup.2]] -[(sin[emptyset]cos[pi]sin[varphi]-cos[emptyset]cos[varphi])[sup.2]]](67) =0 (68) for arbitrary [pi],[emptyset] amd [varphi]. Hencem the matrix elements of H[sub.q] given by Eq.(46) average to zero for any three mutuallyu perpendicular orientations of the laboratory quantization axis. Acknowledgments We thank Dr. C was a fictional scientist from the TV series Cro. She and her companion, Mike, went to the Arctic and thawed out a mammoth, who could talk. That mammoth now tells stories of life in the stone age with his friend, Cro, and his fellow mammoths. . J. Sansonetti for making available the results of Ref. [9] prior to publication. We acknowledge financial support from the U.S. Office of Naval Research The U.S. Office of Naval Research (ONR), headquartered in Arlington, Virginia (Ballston), is the office within the U.S. Department of the Navy that coordinates, executes, and promotes the science and technology programs of the U.S. . About the author: Wayne M. Itano is a physicist in the Time and Frequency Division of the NIST Physics Laboratory. The National Institute of Standards and Technology National Institute of Standards and Technology, governmental agency within the U.S. Dept. of Commerce with the mission of "working with industry to develop and apply technology, measurements, and standards" in the national interest. is an agency of the Technology Administration. U.S. Department of Commerce. 7. References (1.) P. L. Bender. J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger, and J. B. West, Bull. Am. Phys. Soc. 21, 599 (1976). (2.) W. M. Itano, J. C. Bergquist, R. G. Hulet, and D. J. Wineland, Phys. Rev. Lett. 59, 2732 (1987). (3.) J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, and G. Leuchs, Phys. Rev. Lett. 55, 1567 (1985). (4.) J. C. Bergquist, W. M. Itano, and D. Wineland, Phys. Rev. A 36, 428 (1987). (5.) R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett. 85, 2462 (2000). (6.) J. Rcichert, M. Niering, R. Holzwarth, M. Weitz, Th. Udem, and T. W. Hansch, Phys. Rev. Lett. 84, 3232 (2000). (7.) S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. T. Udem, and T. W. Hansch, Phys. Rev. Lett. 84, 5102 (2000). (8.) R. D. Cowan, The Theory of Atomic Structure and Spectra, Univ. California Press, Berkeley, CA (1981). (9.) C. J. Sansonetti and J. Reader, Phys. Scripta 63, 219 (2001). (10.) M. Wilson, Atomic Spectra and Oscillator Strengths for Astrophysics astrophysics, application of the theories and methods of physics to the study of stellar structure, stellar evolution, the origin of the solar system, and related problems of cosmology. and Fusion Research, J. E. Hansen, ed., North-Holland, Amsterdam (1990). (11.) D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 80, 2089 (1998). (12.) J. C. Bergquist, unpublished experimental results. (13.) O. Loebich and A. Steudel, Z. Phys. 166, 56 (1961). (14.) W. M. Itano, J. C. Bergquist. and D. J. Wineland, J. Opt. Soc. Am. B 2, 1392 (1985). (15.) Th. A. M. Van Kleef and M. Fred, Physica 29, 389 (1963). (16.) B. Cagnac, Ann. Phys. (Paris) 6, 467 (1961). (17.) A. Abragam and J. H. Van Vleck, Phys. Rev. 92, 1448 (1953). (18.) W. J. Childs and L. S. Goodman, Phys. Rev. 141, 176 (1966). (19.) J. R. P. Angel and P. G. H. Sandars, Proc. Roy. Soc. A 305, 125 (1968). (20.) W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233 (1982). (21.) T. Brage, C. Proffitt, and D. S. Leckrone, Ap. J. 513,524 (1999). (Updated tables of oscillator strengths were taken from the website http://aniara.gsfe.nasa.gov/sam/sam.html.) (22.) M. Henderson, L. J. Curtis, R. Matulioniene, D. G. Ellis, and C. E. Theodosiu, Phys. Rev. A 56, 1872 (1997). (23.) D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland. J. Appl. Phys. 83, 5025 (1998). (24.) J. R. P. Angel. P. G H. Sandars, and G. K. Woodgate, J. Chem. Phys. 47, 552 (1967). (25.) P. G. H. Sanders San´ders n. 1. An old name of sandalwood, now applied only to the red sandalwood. See under Sandalwood. and A. J. Stewart, Atomic Physics atomic physics Scientific study of the structure of the atom, its energy states, and its interaction with other particles and fields. The modern understanding of the atom is that it consists of a heavy nucleus of positive charge surrounded by a cloud of light, negatively 3, S. J. Smith and G. K. Walters, eds., Plenum In a building, the space between the real ceiling and the dropped ceiling, which is often used as an air duct for heating and air conditioning. It is also filled with electrical, telephone and network wires. See plenum cable. Press, New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of (1973) P. 429. (26.) N. F. Ramsey, Molecular Beams, Oxford Univ. Press, London (1956) Chap. III. (27.) L. S. Brown and G. Gabrielse, Phys. Rev. A 25, 2423 (1982). (28.) A. R. Edmonds, Angular Momentum in Quantum Mechanics quantum mechanics: see quantum theory. quantum mechanics Branch of mathematical physics that deals with atomic and subatomic systems. It is concerned with phenomena that are so small-scale that they cannot be described in classical terms, and it is , Princeton Univ. Press (1974). (Earlier printings include some incorrect equations involving the rotation operators The introduction to this article provides insufficient context for those unfamiliar with the subject matter. Please help [ improve the introduction] to meet Wikipedia's layout standards. You can discuss the issue on the talk page. . See Ref. [29].) (29.) A. A. Wolf, Am. J. Phys. 37, 531 (1969). 6. Appendix A. Angular Averaging of the Quadrupole Shift For the purpose of describing the quadrupole shift, the orientation of the laboratory (quantization) axis with respect to the principal-axis frame is defined by the angles [beta] and [alpha]. In the principal-axis coordinate system, a unit vector along the laboratory z axis is defined in terms of [beta] and [alpha] by z = (sin [beta] cos [alpha], sin [beta] sin [alpha], cos [beta]). (60) We wish to show that the angular dependence of the quadrupole shift is such that the diagonal matrix elements given by Eq. (46) average to zero, for z along any three mutually perpendicular directions. An arbitrary set of three mutually perpendicular unit vectors [e.sub.1], [e.sub.2], and [e.sub.3] can be parameterized by the set of angles [theta], [phi], and [psi PSI - Portable Scheme Interpreter ] in the following way: [e.sub.1] = (sin 0 cos [phi], sin [theta] sin [phi], cos [theta]), (61) [e.sub.2] = (cos [phi] cos 0 cos [psi] - sin [phi] sin [psi], sin [phi] cos [theta] cos [psi] + cos [phi] sin [psi], -sin [theta] cos [psi]), (62) [e.sub.3] = (-cos [phi] cos [theta] sin [psi] - sin [phi] cos [psi], -sin [phi] cos [theta] sin [psi] + cos [phi] cos [psi], sin [theta] sin [psi]). (63) It can be verified by direct computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. that [e.sub.i], [e.sub.j] = [[delta].sub.ij]. The quadrupole shift can be evaluated for each of these three unit vectors substituted for z [Eq. (60)] and the average taken. First consider the average of the quantity (3 [cos.sup.2] [beta] - 1) that appears in Eq. (46): We use the fact that cos [beta] is the third component of z, so the average is: (3 [cos.sup.2] [beta] - 1) = [cos.sup.2][theta] + [sin.sup.2][theta] [cos.sup.2][psi] + [sin.sup.2][theta] [sin.sup.2][psi] - 1, (64) = [cos.sup.2][theta] + [sin.sup.2][theta] - 1, (65) = 0. (66) for arbitrary 0, [phi], and [psi], Similarly, the average of the other angle-dependent term in Eq. (46), [sin.sup.2] [beta]([cos.sup.2] [alpha] - [sin.sup.2] [alpha]), is calculated by making use of the fact that sin [beta] cos [alpha] is the first component of z, and sin [beta] sin [alpha] is the second: ([sin.sup.2] [beta]([cos.sup.2] [alpha] - [sin.sup.2] [alpha])) = 1/3[[sin.sup.2] 0 [cos.sup.2] [phi] - [sin.sup.2] [theta] [sin.sup.2][phi] + [(cos [phi] cos [theta] cos [psi] - sin [phi] sin [psi]).sup.2] - [(sin [phi] cos [theta] cos [psi] + cos [phi] sin [psi]).sup.2] + [(cos [phi] cos [theta] sin [psi] + sin [phi] cos [psi]).sup.2] - [(sin [phi] cos [theta] sin [psi] - cos [phi] cos [psi]).sup.2]], (67) = 0, (68) for arbitrary [theta], [phi], and [psi]. Hence, the matrix elements of [H.sub.Q] given by Eq. (46) average to zero for any three mutually perpendicular orientations of the laboratory quantization axis. |
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