Therefore, the total angular momentum of the Ag atom is due entirely to the spin of the outer electron (\(s = 1/2\)). But how does the electron obtain a magnetic moment if it has zero angular momentum and therefore produces no "current loop" to produce a magnetic moment? Suppose that the angular momentum for the orbital motion is L. Then the orbital magnetic dipole moment is. would, if non zero, be the anapole moment. where Find (i) Speed of electron, (ii) radius of circular path, (iii) time period. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. F The frequency of light emitted is proportional to the energy (\(\Delta E\)) difference between these two states. are the gamma matrices (known as Dirac matrices) and i is the imaginary unit. 0 One consequence is that an external magnetic field exerts a torque on the electron magnetic moment depending on its orientation with respect to the field. See Landé g-factor for details. In classical terms, a ball of charge could have a magnetic moment if it were spinning such that the charge at the edges produced an effective current loop. 0000009502 00000 n The potential difference that has accelerated the electron could be used to calculate the velocity acquired by the electron. 0000008934 00000 n In a hydrogen atom, the electron magnetic moment can interact with the magnetic field produced by the orbital angular momentum of the electron, a phenomenon called spin-orbit coupling. ] {\displaystyle -F_{3}(0)/[\,2\,m_{\rm {e}}\,]} . Legal. An electron moving at constant velocity generates a steady magnetic field, but (like a stationary magnet in a coil of wire) a constant magnetic field won't result in another electric field. 4 The electron magnetic moment couples with the external magnetic field. According to classical predictions, the angular momentum (and, therefore, the magnetic moment) of the Ag atom can point in any direction, so one expects, instead, a continuous smudge on the screen. 0000002323 00000 n 0000003240 00000 n ( The spin magnetic dipole moment is approximately one μB because [2] It is. When an electron accelerates through a potential difference, the potential energy of an electron-field system in converted into kinetic energy of the electron. Since this electron has zero orbital angular momentum (orbital quantum number l=0), one would expect there to be no interaction with an external magnetic field. If we double the strength of the magnetic field, holding all other things constant, the frequency of the radiation doubles and its wavelength is cut in half. In a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. p With this evidence, we say that the electron has spin 1/2. The electron is a charged particle with charge −1e, where e in this context is the unit of elementary charge. Two types of experimental evidence which arose in the 1920s suggested an additional property of the electron. One consequence is that an external magnetic field exerts a torque on the electron magnetic moment depending on its orientation with respect to the field. Quantization of angular momentum had already arisen for orbital angular momentum, and if this electron spin behaved the same way, an angular momentum quantum number s = 1/2 was required to give just two states. Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. Substituting this expression into the first equation gives after some rearrangement. a) What kind of path does the electron follow? {\displaystyle q^{\mu }=p_{f}^{\mu }-p_{i}^{\mu }} The deflection can be shown to be proportional to the spin and to the magnitude of the magnetic field gradient. F The electromagnetic field propagates at the speed of light (in fact, this field can be identified as light) and interacts with charges and currents. 0000000936 00000 n [ is the momentum transfer from the current to the electron. If a charge particle is moving in a close orbit, quantization condition is given by the Bohr-Sommerfeld relation: where p is the total momentum of the free electron… Thus, in vector form, the spin magnetic moment is, \[\vec{\mu} = \left(\frac{e}{m_e}\right)\vec{S}.\], The z-component of the magnetic moment is (from Equation \ref{eq2}), \[\begin{align} \mu_z &= - \left(\frac{e}{m_e}\right) S_z \\[5pt] &= - \left(\frac{e}{m_e}\right) m_s \hbar. 2. p provides the formal definion of the electron's electric dipole moment. 0000006538 00000 n {\displaystyle g_{\rm {s}}\approx 2} The potential energy of the electron spin magnetic moment in a magnetic field applied in the z direction is given by. \end{align*}\]. The necessity of introducing half-integral spin goes back experimentally to the results of the Stern–Gerlach experiment. The total magnetic dipole moment resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum J by a similar equation: The g-factor gJ is known as the Landé g-factor, which can be related to gL and gS by quantum mechanics. 0 q This allows the determination of hyperfine splitting of electron shell energy levels in atoms of protium and deuterium using the measured resonance frequency for several transitions.[6][7]. p m 2 μ These states are shown in Figure \(\PageIndex{1}\). Transitions between these states (spin-flip transitions) result in the emission of a photon with a wavelength of \(\lambda = 21\) cm (in the radio range). Transition from each of these two states to a lower-energy level results in the emission of a photon of slightly different frequency. On leave of absence from Department of Physics, University of Bucharest, Bucharest, Romania. 0000001350 00000 n . [ "article:topic", "authorname:openstax", "Fine structure", "hyperfine structure", "spin-flip transitions", "spin-orbit coupling", "spin projection quantum number ( msms )", "spin projection quantum number", "spin quantum number (s)", "license:ccby", "showtoc:no", "program:openstax" ], 8.3: Orbital Magnetic Dipole Moment of the Electron, 8.5: The Exclusion Principle and the Periodic Table, Electron Spin-Nuclear Spin Coupling (Hyperfine Structure), Creative Commons Attribution License (by 4.0), Express the state of an electron in a hydrogen atom in terms of five quantum numbers, Use quantum numbers to calculate the magnitude and direction of the spin and magnetic moment of an electron, Explain the fine and hyperfine structure of the hydrogen spectrum in terms of magnetic interactions inside the hydrogen atom. An electron moving at constant velocity generates a steady magnetic field, but (like a stationary magnet in a coil of wire) a constant magnetic field won't result in another electric field. and As we will see later, quantum numbers help to classify subatomic particles and enter into scientific models that attempt to explain how the universe works. Its quantum counterpart is one of the four fundamental forces of nature (the others are gravitation, weak … The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. Classical notions such as the center of charge and mass are, however, hard to make precise for a quantum elementary particle. 2 ) The magnetic field diverts the spin up atoms in one direction and the spin-down atoms in another direction. f A hydrogen atom in the ground state is placed in an external uniform magnetic field (\(B = 1.5 \, T\)). 0000004121 00000 n q − The most accurate value for the electron magnetic moment is, The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. = It is the field described by classical electrodynamics and is the classical counterpart to the quantized electromagnetic field tensor in quantum electrodynamics. m A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Here gL is the electron orbital g-factor and μB is the Bohr magneton. If the electron is visualized as a classical charged particle rotating about an axis with angular momentum L, its magnetic dipole moment μ is given by: where me is the electron rest mass. accelerating), however, generates a CHANGING magnetic field, which WILL produce a changing electric field, which produces a changing magnetic field, etc. Trapped particles in magnetic fields are found in the Van Allen radiation belts around Earth, which are part of Earth’s magnetic field. (According to nuclear theory, this moment is due to the orbital motion of quarks within the proton.) Its angular momentum comes from two types of rotation: spin and orbital motion. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. form factor The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. The names, symbols, and allowed values of these quantum numbers are summarized in Table \(\PageIndex{4}\). The magnetic moment of an electron is approximately twice what it should be in classical mechanics. the frequency of the classical motion. An angular momentum and a magnetic moment could indeed arise from a spinning sphere of charge, but this classical picture cannot fit the size or quantized nature of the electron spin. 0000008458 00000 n An electron with a kinetic energy of 300 eV enters a region with a uniform magnetic field of 0.3T. Here S is the electron spin angular momentum.

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