Magnetic Moment of a Spin, Its Equation of Motion, and Precession B1.1.6
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Basic Magnetic Measurement Methods
Basic magnetic measurement methods Magnetic measurements in nanoelectronics 1. Vibrating sample magnetometry and related methods 2. Magnetooptical methods 3. Other methods Introduction Magnetization is a quantity of interest in many measurements involving spintronic materials ● Biot-Savart law (1820) (Jean-Baptiste Biot (1774-1862), Félix Savart (1791-1841)) Magnetic field (the proper name is magnetic flux density [1]*) of a current carrying piece of conductor is given by: μ 0 I dl̂ ×⃗r − − ⃗ 7 1 - vacuum permeability d B= μ 0=4 π10 Hm 4 π ∣⃗r∣3 ● The unit of the magnetic flux density, Tesla (1 T=1 Wb/m2), as a derive unit of Si must be based on some measurement (force, magnetic resonance) *the alternative name is magnetic induction Introduction Magnetization is a quantity of interest in many measurements involving spintronic materials ● Biot-Savart law (1820) (Jean-Baptiste Biot (1774-1862), Félix Savart (1791-1841)) Magnetic field (the proper name is magnetic flux density [1]*) of a current carrying piece of conductor is given by: μ 0 I dl̂ ×⃗r − − ⃗ 7 1 - vacuum permeability d B= μ 0=4 π10 Hm 4 π ∣⃗r∣3 ● The Physikalisch-Technische Bundesanstalt (German national metrology institute) maintains a unit Tesla in form of coils with coil constant k (ratio of the magnetic flux density to the coil current) determined based on NMR measurements graphics from: http://www.ptb.de/cms/fileadmin/internet/fachabteilungen/abteilung_2/2.5_halbleiterphysik_und_magnetismus/2.51/realization.pdf *the alternative name is magnetic induction Introduction It -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging Permalink https://escholarship.org/uc/item/2fw343zm Author Halpern-Manners, Nicholas Wm Publication Date 2011 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging by Nicholas Wm Halpern-Manners A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Alexander Pines, Chair Professor David Wemmer Professor Steven Conolly Spring 2011 Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging Copyright 2011 by Nicholas Wm Halpern-Manners 1 Abstract Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging by Nicholas Wm Halpern-Manners Doctor of Philosophy in Chemistry University of California, Berkeley Professor Alexander Pines, Chair Magnetic resonance has evolved into a remarkably versatile technique, with major appli- cations in chemical analysis, molecular biology, and medical imaging. Despite these successes, there are a large number of areas where magnetic resonance has the potential to provide great insight but has run into significant obstacles in its application. The projects described in this thesis focus on two of these areas. First, I describe the development and implementa- tion of a robust imaging method which can directly detect the effects of oscillating electrical currents. This work is particularly relevant in the context of neuronal current detection, and bypasses many of the limitations of previously developed techniques. -
Thomas Precession and Thomas-Wigner Rotation: Correct Solutions and Their Implications
epl draft Header will be provided by the publisher This is a pre-print of an article published in Europhysics Letters 129 (2020) 3006 The final authenticated version is available online at: https://iopscience.iop.org/article/10.1209/0295-5075/129/30006 Thomas precession and Thomas-Wigner rotation: correct solutions and their implications 1(a) 2 3 4 ALEXANDER KHOLMETSKII , OLEG MISSEVITCH , TOLGA YARMAN , METIN ARIK 1 Department of Physics, Belarusian State University – Nezavisimosti Avenue 4, 220030, Minsk, Belarus 2 Research Institute for Nuclear Problems, Belarusian State University –Bobrujskaya str., 11, 220030, Minsk, Belarus 3 Okan University, Akfirat, Istanbul, Turkey 4 Bogazici University, Istanbul, Turkey received and accepted dates provided by the publisher other relevant dates provided by the publisher PACS 03.30.+p – Special relativity Abstract – We address to the Thomas precession for the hydrogenlike atom and point out that in the derivation of this effect in the semi-classical approach, two different successions of rotation-free Lorentz transformations between the laboratory frame K and the proper electron’s frames, Ke(t) and Ke(t+dt), separated by the time interval dt, were used by different authors. We further show that the succession of Lorentz transformations KKe(t)Ke(t+dt) leads to relativistically non-adequate results in the frame Ke(t) with respect to the rotational frequency of the electron spin, and thus an alternative succession of transformations KKe(t), KKe(t+dt) must be applied. From the physical viewpoint this means the validity of the introduced “tracking rule”, when the rotation-free Lorentz transformation, being realized between the frame of observation K and the frame K(t) co-moving with a tracking object at the time moment t, remains in force at any future time moments, too. -
The Magnetic Moment of a Bar Magnet and the Horizontal Component of the Earth’S Magnetic Field
260 16-1 EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH’S MAGNETIC FIELD I. THEORY The purpose of this experiment is to measure the magnetic moment μ of a bar magnet and the horizontal component BE of the earth's magnetic field. Since there are two unknown quantities, μ and BE, we need two independent equations containing the two unknowns. We will carry out two separate procedures: The first procedure will yield the ratio of the two unknowns; the second will yield the product. We will then solve the two equations simultaneously. The pole strength of a bar magnet may be determined by measuring the force F exerted on one pole of the magnet by an external magnetic field B0. The pole strength is then defined by p = F/B0 Note the similarity between this equation and q = F/E for electric charges. In Experiment 3 we learned that the magnitude of the magnetic field, B, due to a single magnetic pole varies as the inverse square of the distance from the pole. k′ p B = r 2 in which k' is defined to be 10-7 N/A2. Consider a bar magnet with poles a distance 2x apart. Consider also a point P, located a distance r from the center of the magnet, along a straight line which passes from the center of the magnet through the North pole. Assume that r is much larger than x. The resultant magnetic field Bm at P due to the magnet is the vector sum of a field BN directed away from the North pole, and a field BS directed toward the South pole. -
PH585: Magnetic Dipoles and So Forth
PH585: Magnetic dipoles and so forth 1 Magnetic Moments Magnetic moments ~µ are analogous to dipole moments ~p in electrostatics. There are two sorts of magnetic dipoles we will consider: a dipole consisting of two magnetic charges p separated by a distance d (a true dipole), and a current loop of area A (an approximate dipole). N p d -p I S Figure 1: (left) A magnetic dipole, consisting of two magnetic charges p separated by a distance d. The dipole moment is |~µ| = pd. (right) An approximate magnetic dipole, consisting of a loop with current I and area A. The dipole moment is |~µ| = IA. In the case of two separated magnetic charges, the dipole moment is trivially calculated by comparison with an electric dipole – substitute ~µ for ~p and p for q.i In the case of the current loop, a bit more work is required to find the moment. We will come to this shortly, for now we just quote the result ~µ=IA nˆ, where nˆ is a unit vector normal to the surface of the loop. 2 An Electrostatics Refresher In order to appreciate the magnetic dipole, we should remind ourselves first how one arrives at the field for an electric dipole. Recall Maxwell’s first equation (in the absence of polarization density): ρ ∇~ · E~ = (1) r0 If we assume that the fields are static, we can also write: ∂B~ ∇~ × E~ = − (2) ∂t This means, as you discovered in your last homework, that E~ can be written as the gradient of a scalar function ϕ, the electric potential: E~ = −∇~ ϕ (3) This lets us rewrite Eq. -
Ph501 Electrodynamics Problem Set 8 Kirk T
Princeton University Ph501 Electrodynamics Problem Set 8 Kirk T. McDonald (2001) [email protected] http://physics.princeton.edu/~mcdonald/examples/ Princeton University 2001 Ph501 Set 8, Problem 1 1 1. Wire with a Linearly Rising Current A neutral wire along the z-axis carries current I that varies with time t according to ⎧ ⎪ ⎨ 0(t ≤ 0), I t ( )=⎪ (1) ⎩ αt (t>0),αis a constant. Deduce the time-dependence of the electric and magnetic fields, E and B,observedat apoint(r, θ =0,z = 0) in a cylindrical coordinate system about the wire. Use your expressions to discuss the fields in the two limiting cases that ct r and ct = r + , where c is the speed of light and r. The related, but more intricate case of a solenoid with a linearly rising current is considered in http://physics.princeton.edu/~mcdonald/examples/solenoid.pdf Princeton University 2001 Ph501 Set 8, Problem 2 2 2. Harmonic Multipole Expansion A common alternative to the multipole expansion of electromagnetic radiation given in the Notes assumes from the beginning that the motion of the charges is oscillatory with angular frequency ω. However, we still use the essence of the Hertz method wherein the current density is related to the time derivative of a polarization:1 J = p˙ . (2) The radiation fields will be deduced from the retarded vector potential, 1 [J] d 1 [p˙ ] d , A = c r Vol = c r Vol (3) which is a solution of the (Lorenz gauge) wave equation 1 ∂2A 4π ∇2A − = − J. (4) c2 ∂t2 c Suppose that the Hertz polarization vector p has oscillatory time dependence, −iωt p(x,t)=pω(x)e . -
Units in Electromagnetism (PDF)
Units in electromagnetism Almost all textbooks on electricity and magnetism (including Griffiths’s book) use the same set of units | the so-called rationalized or Giorgi units. These have the advantage of common use. On the other hand there are all sorts of \0"s and \µ0"s to memorize. Could anyone think of a system that doesn't have all this junk to memorize? Yes, Carl Friedrich Gauss could. This problem describes the Gaussian system of units. [In working this problem, keep in mind the distinction between \dimensions" (like length, time, and charge) and \units" (like meters, seconds, and coulombs).] a. In the Gaussian system, the measure of charge is q q~ = p : 4π0 Write down Coulomb's law in the Gaussian system. Show that in this system, the dimensions ofq ~ are [length]3=2[mass]1=2[time]−1: There is no need, in this system, for a unit of charge like the coulomb, which is independent of the units of mass, length, and time. b. The electric field in the Gaussian system is given by F~ E~~ = : q~ How is this measure of electric field (E~~) related to the standard (Giorgi) field (E~ )? What are the dimensions of E~~? c. The magnetic field in the Gaussian system is given by r4π B~~ = B~ : µ0 What are the dimensions of B~~ and how do they compare to the dimensions of E~~? d. In the Giorgi system, the Lorentz force law is F~ = q(E~ + ~v × B~ ): p What is the Lorentz force law expressed in the Gaussian system? Recall that c = 1= 0µ0. -
Electron Spins in Nonmagnetic Semiconductors
Electron spins in nonmagnetic semiconductors Yuichiro K. Kato Institute of Engineering Innovation, The University of Tokyo Physics of non-interacting spins Optical spin injection and detection Spin manipulation in nonmagnetic semiconductors Physics of non-interacting spins 2 In non-magnetic semiconductors such as GaAs and Si, spin interactions are weak; To first order approximation, they behave as non-interacting, independent spins. • Zeeman Hamiltonian • Bloch sphere • Larmor precession ∗ • , , and • Bloch equation The Zeeman Hamiltonian 3 Hamiltonian for an electron spin in a magnetic field magnetic moment of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free electrons) : Bohr magneton 58 eV/T , ) : electronic charge : Planck constant : free electron mass : spin operator : Pauli operator The Zeeman Hamiltonian 4 Hamiltonian for an electron spin in a magnetic field magnetic moment of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free electrons) : Bohr magneton 2 58 eV/T : electronic charge : Planck constant : free electron mass : spin operator 2 : Pauli operator The energy eigenstates 5 Without loss of generality, we can set the -axis to be the direction of , i.e., 0,0, 1 Spin “up” |↑ |↑ 2 1 2 2 1 Spin “down” |↓ |↓ 2 1 2 2 Spinor notation and the Pauli operators 6 Quantum states are represented by a normalized vector in a Hilbert space Spin states are represented by 2D vectors Spin operators are represented by 2x2 matrices. -
4 Nuclear Magnetic Resonance
Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I. -
Optical Detection of Electron Spin Dynamics Driven by Fast Variations of a Magnetic Field
www.nature.com/scientificreports OPEN Optical detection of electron spin dynamics driven by fast variations of a magnetic feld: a simple ∗ method to measure T1 , T2 , and T2 in semiconductors V. V. Belykh1*, D. R. Yakovlev2,3 & M. Bayer2,3 ∗ We develop a simple method for measuring the electron spin relaxation times T1 , T2 and T2 in semiconductors and demonstrate its exemplary application to n-type GaAs. Using an abrupt variation of the magnetic feld acting on electron spins, we detect the spin evolution by measuring the Faraday rotation of a short laser pulse. Depending on the magnetic feld orientation, this allows us to measure either the longitudinal spin relaxation time T1 or the inhomogeneous transverse spin dephasing time ∗ T2 . In order to determine the homogeneous spin coherence time T2 , we apply a pulse of an oscillating radiofrequency (rf) feld resonant with the Larmor frequency and detect the subsequent decay of the spin precession. The amplitude of the rf-driven spin precession is signifcantly enhanced upon additional optical pumping along the magnetic feld. Te electron spin dynamics in semiconductors can be addressed by a number of versatile methods involving electromagnetic radiation either in the optical range, resonant with interband transitions, or in the microwave as well as radiofrequency (rf) range, resonant with the Zeeman splitting1,2. For a long time, the optical methods were mostly represented by Hanle efect measurements giving access to the spin relaxation time at zero magnetic feld3. New techniques giving access both to the spin g factor and relaxation times at arbitrary magnetic felds have been very actively developed. -
Magnetic & Electric Dipole Moments
Axion Academic Training CERN, 1 December 2005 Magnetic & Electric Dipole Moments. Yannis K. Semertzidis Brookhaven National Lab •Muon g-2 experiment d c •EDMs: What do they probe? q •Physics of Hadronic EDMs θ •Probing QCD directly (RHIC), & indirectly (Hadronic EDM) •Experimental Techniques dsr v r r = μr × B + d × E dt Building blocks of matter Force carriers Muons decay to an electron and two neutrinos with a lifetime of 2.2μs (at rest). dsr v r r = μr × B + d × E Axion Training, 1 December, 2005 Yannis Semertzidis, BNL dt Quantum Mechanical Fluctuations •The electron particle is surrounded by a cloud of virtual particles, a …soup of particles… •The muon, which is ~200 times heavier than the electron, is surrounded by a heavier soup of particles… dsr v r r = μr × B + d × E Axion Training, 1 December, 2005 Yannis Semertzidis, BNL dt A circulating particle with r charge e and mass m: μr, L r • Angular momentum e, m L = mvr e r • Magnetic dipole μr = L moment 2m μ = IA dsr v r r = μr × B + d × E Axion Training, 1 December, 2005 Yannis Semertzidis, BNL dt For particles with intrinsic angular momentum (spin S): e r μr = g S 2m The anomalous magnetic moment a: g − 2 a = 2 dsr v r r = μr × B + d × E Axion Training, 1 December, 2005 Yannis Semertzidis, BNL dt In a magnetic field (B), there is a torque: r τr = μr × B Which causes the spin to precess in the horizontal plane: dsr r =×μr B dt r ds v r r = μr × B + d × E Axion Training, 1 December, 2005 Yannis Semertzidis, BNL dt Definition of g-Factor magnetic moment g ≡ eh / 2mc angular momentum h From Dirac equation g-2=0 for point-like, spin ½ particles. -
Griffith's Example 4.3: Larmor Precession We Will Be Considering the Spin of an Electron
Griffith's Example 4.3: Larmor Precession We will be considering the spin of an electron. The operator for the component of spin in the rˆ = (sin cosijˆˆ sin sin cosˆ ) k cos sin ei cos(½ ) leading to Sˆ which has an eigen-spinor rˆ rˆ i i sin e cos sin(½ )e i rˆ sin(½ )e with eigenvalue ½ and with eigenvalue -½ . We can imagine a cos(½ ) state in which the spin is initially in the x-z plane and makes an angle relative to the cos(½ ) z axis so (0) . sin(½ ) First we consider an applied magnetic field in the z direction providing interaction ˆ ˆ energy: H Bkozo SB Bmos. Using the standard eigenstates of Sz, ½0 B ˆ o H 0½ Bo Note that the difference between the energies of the spin up and spin down states is Bo. We expect things to oscillate at frequencies corresponding to the energy differences so the frequency for this problem is Bo, the Larmor frequency. The frequencies associated with the two states have a magnitude of one-half of the Larmor frequency; the difference between the frequencies is the relevant frequency for oscillation of probability density or, in this case, precessing. The hamiltonian is diagonal so the spin z up and down states are the eigenfunctions. a ½it ½0 Bo a cos(½ )e Hiˆ it () t t b ½it 0½ Bo b t sin(½ )e where = Bo . The expectation values can be computed with the results: Sz(t) = cos() ( /2); Sx(t) = sin() ( /2) sin(t);Sy(t) = -sin() ( /2) cos(t) Exercise: Compute Sz(t) and Sx(t) for(t) given above.