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With this evidence, we say that the electron has spin 1/2. 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. = b) What is the magnitude of the magnetic force (in N) on the 0000010120 00000 n Classically one would expect all possible orientations of the dipoles so that a continuous smear would be produced on the photographic plate, but they found that the field separated the beam into two distinct parts, indicating just two possible orientations of the magnetic moment of the electron. μ For example, quarks within an atomic nucleus are also spin-half particles. 52 0 obj<> endobj Note that the intrinsic quantum numbers introduced in this section ($$s$$ and $$m_s$$) are valid for many particles, not just electrons. q The magnetic field diverts the spin up atoms in one direction and the spin-down atoms in another direction. c) What is the radius (in m) of the electron’s path? A hydrogen atom in the ground state is placed in an external uniform magnetic field ($$B = 1.5 \, T$$). A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. Electrons have intrinsic angular momentum characterized by quantum number 1/2. [2] It is. non-quantum) field produced by moving electric charges. This intrinsic electron property gives: Experimental evidence like the hydrogen fine structure and the Stern-Gerlach experiment suggest that an electron has an intrinsic angular momentum, independent of its orbital angular momentum. An electron of 1 0 eV energy is revolving around circular path in magnetic field of 1 × 1 0 − 4 weber / metre 2. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. For the electron spin, the most accurate value for the spin g-factor has been experimentally determined to have the value, Note that it is only two thousandths larger than the value from the Dirac equation. On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units ħ = c = 1). e 0000028086 00000 n {\displaystyle F_{1}(0)=-e} u 0000025988 00000 n This is similar to the quantization of $$L$$, except that the only value allowed for $$s$$ for an electron is $$s = 1/2$$. The z-component of magnetic moment associated with the electron spin would then be expected to be. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 0000005687 00000 n \end{align}\]. The angle between the initial velocity of the electron and the magnetic field is 60°. . The intrinsic magnetic dipole moment of an electron $$\mu_e$$ can also be expressed in terms of the spin quantum number. 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]. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with Lz = −1, 0, and +1. startxref The spin projection quantum number is $$m_s = \pm 1/2$$, so the z-component of the magnetic moment is, $\mu_z = \pm \left(\frac{e}{2m_e} \right) = \pm \mu_B \hbar. 0 e The time period is given as T = 2πm / Bq where mass of the electron, m = 9.1 X 10-31 kg magnetic field strength, B = 0.2 mT = 0.2 X10-3 T charge on the electron, q = 1.6 X10-19 C If the magnetic moment of the electron and orbital magnetic moment of the electron are antiparallel, the potential energy from the magnetic interaction is relatively high, but when these moments are parallel, the potential energy is relatively small. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. Using the relationship of force to potential energy gives. 0 �WPP�������QH�Prq���e@B��LJ�`�f�� Using the relationship of force to potential energy gives . ( Stern and Gerlach directed the beam of silver atoms into a region of nonuniform magnetic field (see experiment sketch). where g is the electron spin g-factor and mB is the Bohr magneton. ) By the end of this section, you will be able to: In this section, we consider the effects of electron spin. The electromagnetic field propagates at the speed of light (in fact, this field can be identified as light) and interacts with charges and currents. Figure 5 (a) PC spectra of X 0 in a magnetic field from 0 to 9 T in the Faraday geometry. The potential energy of the electron spin magnetic moment in a magnetic field applied in the z direction is given by. 0000002084 00000 n a uniform magnetic field of 0.3T. 0 of the electromagnetic current operator between two on-shell states. Determine the frequency of radiation produced in a transition between the spin-up and spin-down states of the electron. Here This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0). (According to nuclear theory, this moment is due to the orbital motion of quarks within the proton.) 0000002689 00000 n Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. Example $$\PageIndex{1}$$: Electron Spin and Radiation in An external Magnetic field. In analogy to the orbital angular momentum, the magnitude of the electron magnetic moment is, \[\mu_s = \left(\frac{e}{2m_e}\right)S.$, According to the special theory of relativity, this value is low by a factor of 2. 1 The frequency of radiation produced by a transition between these states is proportional to the energy difference. An electron is magnetic, so we expect the electron to interact with other magnetic fields. An electron with a kinetic energy of 300 eV enters a region with a uniform magnetic field of 0.3T. A magnetic dipole moment will experience a force proportional to the field gradient since the two "poles" will be subject to different fields. 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. Classical notions such as the center of charge and mass are, however, hard to make precise for a quantum elementary particle. The quantum numbers associated with electron spin follow the characteristic pattern: Since the electron displays an intrinsic angular momentum, one might expect a magnetic moment which follows the form of that for an electron orbit. 0000001609 00000 n 0000001350 00000 n 0000027847 00000 n The electron is said to be a “spin-half particle.” The spin projection quantum number $$m_s$$ is associated with the z-components of spin, expressed by, In general, the allowed quantum numbers are, $m_s = -s, -s + 1, . Copyright © 2020 Elsevier B.V. or its licensors or contributors. the frequency of the classical motion. This made a major contribution to the development of the quantum theory of the atom. μ Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. It is the field described by classical electrodynamics and is the classical counterpart to the quantized electromagnetic field tensor in quantum electrodynamics. ] 0 q The spin projection quantum number has just two values ($$m_s = \pm1/2$$), so the z-component of the magnetic moment also has just two values: \[\mu_z = \pm \left(\frac{e}{2m_e}\right) = \pm \mu_B\hbar,$. = 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. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. The angle between the initial The small correction is known as the anomalous magnetic dipole moment of the electron; it arises from the electron's interaction with virtual photons in quantum electrodynamics. Starting from here the charge of the electron is e < 0 . 0000025620 00000 n As a result, the classical result is corrected by multiplying it with a dimensionless correction factor g, known as the g-factor: It is usual to express the magnetic moment in terms of the reduced Planck constant ħ and the Bohr magneton μB: Since the magnetic moment is quantized in units of μB, correspondingly the angular momentum is quantized in units of ħ. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure. 0000008458 00000 n ¯ i Both were discovered by looking at the fine structure of atomic spectra. The experiment passes a stream of silver (Ag) atoms through an external, nonuniform magnetic field. ¯ provides the formal definion of the electron's electric dipole moment. 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. This kind of reasoning led to the use of "electron spin" to describe the intrinsic angular momentum. In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. By using the Furry picture, the external field will be taken into account exactly in the expressions of the wave functions and propagators, and only the electron-photon interaction will be treated as a perturbation. Terms accelerating), however, generates a CHANGING magnetic field, which WILL produce a changing electric field, which produces a changing magnetic field, etc. The 21-cm line in atomic spectroscopy is a “fingerprint” of hydrogen gas. 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. This made a major contribution to the development of the quantum theory of the atom.