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This Motion Is Called the Larmor Precession. N Magnetic Materials 2. Diamagnetism Numan Akdoğan [email protected] Gebze Institute of Technology Department of Physics Nanomagnetism and Spintronic Research Center (NASAM) Magnetic moments of electrons There are two kinds of electron motion, orbital and spin, and each has a magnetic moment associated with it. In the Bohr model of the atom (1913), electron moves in a circular orbit of radius r with a velocity v. The orbital motion of an electron around the nucleus may be likened to a current in a loop of wire having no resistance; both are equivalent to a circulation of charge. Remember Eq. 7 in the first lecture. µ = iA In cgs the charge of an electron is given as e/c, where e is the elemantary electric charge and c is the velocity of light). The current i, or charge passing a given point per unit time, is then (e/c)(v/2πr) (cgs) or ev/2πr (SI) Therefore the magnetic moment produced by the circular motion of the electron in its orbit is given by N. Akdoğan 2. Diamagnetism Magnetic moments of electrons e v 2 evr µorbit = πr = (cgs) (1) c 2πr 2c ev evr µ = πr 2 = (SI) orbit 2πr 2 Another postulate of the Bohr’s atomic model was that the orbital motion of the electron is quantized, so that only discrete orbits can exist. In other words, the angular momentum of the electron must be an integral multiple of h/2π, where h is Planck’s constant. h L = n = mvr (2) 2π n is the principal quantum number and m is the mass of electron. N. Akdoğan 2. Diamagnetism Magnetic moments of electrons eh µorbit = (cgs) (3) 4πmc for the magnetic moment of the electron in the first (n=1) Bohr orbit. The spin of the electron was postulated in 1925 in order to explain certain features of the optical spectra of hot gases, particularly gases subjected to a magnetic field (Zeeman effect), and it later found theoretical confirmation in wave mechanics. Spin is an universal property of electrons in all states of matter at all temperatures. The electron behaves as if it were in some sense spinning about its own axis, and associated with this spin are definite amounts of magnetic moment and angular momentum. It is found experimentally and theoretically that the magnetic moment due to electron spin is equal to eh µspin = (cgs) (4) 4πmc N. Akdoğan 2. Diamagnetism Magnetic moments of electrons eh µ = spin 4πmc (4.80×10−10 esu)(6.62×10−27 erg ⋅sec) µ = spin 4π (9.11×10−28 g)(3×1010 cm / sec) −21 erg/Oe or emu in cgs µB = µspin = 9.27×10 (5) which is called the Bohr magneton (µB). −24 J/T or Am2 in SI µB = 9.27×10 N. Akdoğan 2. Diamagnetism Magnetic moments of atoms Atoms contain many electrons, each spinning about its own axis and moving in its own orbit. The magnetic moment associated with each kind of motion is a vector quantity, parallel to the axis of spin and normal to the plane of the orbit, respectively. The magnetic moment of the atom is the vector sum of all its electronic moments, and two possibilities arise: μorbit μspin electron nucleus The magnetic moments of all the electrons are so oriented that they cancel one another out, and the atom as a whole has no net magnetic moment. This condition leads to diamagnetism. The cancellation of electronic moments is only partial and the atom is left with a net magnetic moment. Such an atom is often referred to, for brevity, as a magnetic atom. Substances composed of atoms of this kind are para-, ferro-, antiferro-, or ferrimagnetic. N. Akdoğan 2. Diamagnetism Diamagnetism We have studied two contributions to the magnetic moment of atoms - the electron spin and orbital angular momenta. Next we are going to investigate the third (and final) contribution to the magnetic moment of a free atom. This is the change in orbital motion of the electrons when an external magnetic field is applied. The classical theory of this effect was first worked out by Paul Langevin in a noted paper published in 1905 [P. Langevin, Ann. Chemie et Physique, 5 (1905) p. 70–127]. The theory considers that the effect of an applied field on a single electron orbit is to reduce the effective current of the orbit, and so to produce a magnetic moment opposing the applied field. N. Akdoğan 2. Diamagnetism Diamagnetism When an external field is applied, the electron is accelerated by this field, and the velocity changes. Therefore, the change in the centrifugal force acting on the electron is given by 2 mv 2mv (6) ∆F = ∆ = ∆v r r which is just balanced by an increase in the Lorentz force e ∆F = v∆H (7) c In other words, the orbit precesses about the applied field without changing its shape, with angular velocity v e wL = = H (8) r 2mc This motion is called the Larmor precession. N. Akdoğan 2. Diamagnetism Diamagnetism Mechanism of atomic diamagnetism. Larmor precession of a tilted orbit. The magnetic moment produced by the motion shown on the left is given by e ∆v e∆vr µ = iA = − πr 2 = − (9) c 2πr 2c Using Eqs. 6 and 7 for ∆ν er ∆v = ∆H and we have (10) 2mc e2r 2 µ = − H (11) 4mc2 N. Akdoğan 2. Diamagnetism Diamagnetism In the case of a closed shell, electrons are distributed on a spherical surface with radius a, so that r2 in Eq. 11 is replaced by x2 +y2, where the z-axis is parallel to the magnetic field. Considering spherical symmetry, we have a2 x2 = y 2 = z 2 = 3 2 r 2 = x2 + y 2 = a2 3 Therefore Eq. 11 becomes N. Akdoğan 2. Diamagnetism Diamagnetism e2a2 µ = − H cgs (12) 6mc2 When a unit volume of the material contains N atoms, each of which has Z orbital electrons, the magnetic susceptibility is given by M µ e2a2 NZe2 χ = = = − = − a2 H VH 6Vmc2 6mc2 NZe2 χ = − 2 emu cgs (13) 2 a 6mc Oe⋅cm3 where a2 is the average a2 for all the orbital electrons. This relationship holds fairly well for materials containing atoms or ions with closed shells. N. Akdoğan 2. Diamagnetism Diamagnetism In fact diamagnetism is such a weak phenomenon that only those atoms which have no net magnetic moment as a result of their shells being filled are classified as diamagnetic. In other materials the diamagnetism is overshadowed by much stronger interactions such as ferromagnetism or paramagnetism. http://www.ru.nl/hfml/research/levitation/diamagnetic/ N. Akdoğan 2. Diamagnetism Diamagnetism M χ 0 H T 0 N. Akdoğan 2. Diamagnetism.
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