DOCSLIB.ORG

Chapter 8 Magnetic Resonance

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 8 Magnetic Resonance Chapter 8 Magnetic Resonance 9.1 Electron paramagnetic resonance 9.2 Ferromagnetic resonance 9.3 Nuclear magnetic resonance 9.4 Other resonance methods TCD March 2007 1 A resonance experiment involves a specimen placed in a uniform magnetic field B0 B0 and applying an AC magnetic 2b1cos!t field in the perpendicular direction 2b1cos!t B0 2b1cos!t A magnetic resonance experiment TCD March 2007 2 Larmor frequency B m = "l m = m x B µ ! 0 ! = dl/dt m m x B d /dt = -" 0 = m x B ! != µ"B0 NB. The electron precesses counterclockwise because of the negative charge, " is Solution is m(t) = m ( sin# cos! t, sin# sin! t, cos# ) where ! = "B L L L 0 negative. eB Torque ! cause µ to precess about B with the Larmor frequency# = me B /2 Magnetic moment precesses at the Larmor precession frequency fL = " 0 " The Larmor precession is half the cyclotron frequency for orbital moment, but " = -e/2me equal to it for spin moment. " = -e/me TCD March 2007 3 An alternating field along the x-axis can be decomposed into two counter-rotating fields. b = 2b1cos !t y b = b1[exp!t + exp-!t] -!t x !t TCD March 2007 4 m = "hS H Z = - "!B0Sz Ei = - "!B0MS MS = S, S-1, … S = 1/2 MS 1 0 -1 Zeeman-split enegy levels for an electronic system with S = 1 Splitting is "!B0; ! = "B0 TCD March 2007 5 Why does the AC field have to be applied perpendicular to B0 ? H = -"!(B0Sz + 2b1Sx) If the field is applied in the z-direction, the Hamiltonian is diagonal so there is no mixing of different Ms states However, Sx has nonzero off-diagonal elements (n, n±1). The second term mixes states with $MS = ±1. Electronic energy levels; Electronic Paramagnetic Resonance (EPR) GHz range Nuclear energy levels; Nuclear Magnetic Levels (NMR) MHz range Ferromagnetic moment precession Ferromagnetic Resonance (FMR) GHz range TCD March 2007 6 TCD March 2007 7 9.1 Electron paramagnetic resonance (EPR) Larmor precession frequency for electron spin is 2% fL = !L = (ge/2m)B0 -1 fL = 28.02 GHz T . TCD March 2007 8 Microwave cavity delivers b1 in a TM100 mode. ! ! X-band radiation, 9 GHz, B0 300 mT. Energy splitting of ±1/2 levels is 0.2 K. Polarization of the spin system is P = (n& - n')/ (n& + n') = [1 - exp(-gµBB0/kT)]/ [1 + exp(-gµBB0/kT])] ! gµBB0/2kT At RT in 300 mT this is only 7 10-4. TCD March 2007 9 EPR lineshape. Fix frequency ! and amplitude b1, scan magnetic field at a constant rate. Absorption line is measured by modulating the field B0 with a small ac field and using lockin detection Integrated lorentzian lineshape Derivative lineshape TCD March 2007 10 MS E = h( 1/2 -1/2 Microwave power w Switch off power; relaxation time is T1 spin-lattice relaxation n t TCD March 2007 11 EPR works best for S-state ions with half-filled shells. 2 Free radicals S1/2 2+ 3+ 6 Mn Fe S5/2 3+ 8 Gd S7/2 Ions should be dilute in a crystal lattice to diminish dipole-dipole interactions. The outer electrons in these shells interact strongly with surroundings. Crystal-field interactions may mix different MS states. Second order $MJ ± 2 Fourth order $MJ ± 4 Sixth order $MJ ± 6 TCD March 2007 12 TCD March 2007 13 Spin hamiltonian TCD March 2007 14 2 Zero-field splitting DSz 2 H spin = DSz - "!B0Sz TCD March 2007 15 Hyperfine interactions in epr These interactions are ! 0.1 K. They represent coupling of the spin of the nucleus to the magnetic field produced by the atomic electrons. Nuclear spin I. MI = I, I-1 ……… -1. mn = gnµN MI Hyperfine Hamiltonian Hhf = A I.S TCD March 2007 16 Hyperfine interactions in epr TCD March 2007 17 9.2 Ferromagnetic resonance (FMR) Resonance frequencies are similar to those for EPR. The coupled moments are due to electrons. # = -(e/m) TCD March 2007 18 Kittel equation TCD March 2007 19 Ferromagnetic resonance can give values of Ms and K as well as ", without the need to know the dimensions or mass of the sample. TCD March 2007 20 TCD March 2007 21 9.2.1 Spin-wave resonance t Spin-wave dispersion. !! = Dk2 K = n%/t TCD March 2007 22 9.2.2 Antiferromagnetic resonance TCD March 2007 23 9.2.2 Damping Two forms of the damping; Landau-Lifschitz and Gilbert TCD March 2007 24 TCD March 2007 25 TCD March 2007 26 TCD March 2007 27 9.2.3 Domain wall resonance z 1/2 )w = %(A/K 1) d#/dx = sin #/ )w Apply a field B along Oz. Pressure on the wall is 2BMs The TCD March 2007 28 TCD March 2007 29 9.3 Nuclear magnetic resonance (NMR) TCD March 2007 30 NMR experiment MI E = h( -1/2 1/2 TCD March 2007 31 Chemical shift Proton resonance spectrum of an organic compound Knight shift Shift in resonance due to shielding of the applied field by the conduction electrons. ! 1 % TCD March 2007 32 9.3.1 Hyperfine interactions Hyperfine field has contact, orbital and dipolar contributions eQ nuclear quadrupole moment eq = Vzz electric field gradient at the nucleus Vxx 0 0 V + V + V = 0 efg xx yy zz 0 Vyy 0 * = (Vxx - Vyy)/Vzz 0 0 Vzz TCD March 2007 33 TCD March 2007 34 9.3.2 Relaxation T1 Spin lattice relaxation TCD March 2007 35 T2 Spin-spin relaxation TCD March 2007 36 Bloch’s Equations TCD March 2007 37 9.3.2 Rotating frame TCD March 2007 38 Bloch’s equations in the rotating frame TCD March 2007 39 TCD March 2007 40 9.3.3 Pulsed nmr TCD March 2007 41 TCD March 2007 42 TCD March 2007 43 Spin echo TCD March 2007 44 TCD March 2007 45 TCD March 2007 46 TCD March 2007 47 A typical free induction decay, and its spectrum TCD March 2007 48 9.4 Other resonance methods 9.4.1 Mossbauer effect 2 2 Recoilless fraction f = exp -k" <x > F is the probability of a zero-phonon emission or absorption event in a solid 2 source. E "= hk" <x2> is rms displacement of the nucleus TCD March 2007 49 TCD March 2007 50 Conversion electron Mossbauer spectroscopy Electron detector 57Co (t 250d) "-ray surface 1/2 Emitted electron t interface substrate 7.3 keV conversion electron 5/2 14.4 keV "-ray 57Fe 3/2 14.4 keV "-ray 3/2 1/2 1/2 Source Absorber TCD March 2007 51 9.4.2 Muon spin rotation A muon is an unstable particle with spin 1/2 Charge ± e Mass 250 me Half-life +µ = 2.2 microseconds. Pions are produced in collisions of high-energy protons with a target. They decay in 26 ns to give muons + + % , µ + (µ Neutrino, muon have their spin antiparallel to their momentum, S%= 0 The MeV muons are rapidly thermalized in a solid specimen. After time t, probability of muon decay is 1 - exp(-t/ +µ) + + µ , e + (e + (’e The direction of emission of the positron is related to the spin direction of the muon. The muon precesses around the local field at 135 GHz T-1 TCD March 2007 52 TCD March 2007 53 TCD March 2007 54 TCD March 2007 55 TCD March 2007 56 TCD March 2007 57 TCD March 2007 58 TCD March 2007 59 TCD March 2007 60 TCD March 2007 61 TCD March 2007 62 TCD March 2007 63 TCD March 2007 64 TCD March 2007 65 TCD March 2007 66 TCD March 2007 67 TCD March 2007 68 TCD March 2007 69 TCD March 2007 70 TCD March 2007 71 8.5 Superparamagnetism TCD March 2007 72 TCD March 2007 73 8.6 Bulk nanostructures Recrystallization of amorphous Fe-Cu-Nb- Si-B to obtain a two-phase crystalline/ amorphous soft nanocomposite TCD March 2007 74 The hysteresis loop spontaneous magnetization remanence coercivity virgin curve initial susceptibility major loop The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to a magnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains. The magnet cannot be in thermodynamic equilibrium anywhere around the open part of the curve! M and H have the same units (A m-1). TCD March 2007 75 TCD March 2007 76 Magnetostatics Poisson’s equarion Volume charge Boundary condition en 2. air + 1. solid + M + M( r) , H( r) BUT H( r) , M( r) Experimental information about the domain structure comes from observations at the surface. The interior is inscruatble. TCD March 2007 77.
Load more

Recommended publications
  • Few Electron Paramagnetic Resonances Detection On
    FEW ELECTRON PARAMAGNETIC RESONANCES DETECTION TECHNIQUES ON THE RUBY SURFACE By Xiying Li Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Adviser: Dr. Massood Tabib-Azar Co-Adviser: Dr. J. Adin Mann, Jr. Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY August, 2005 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents TABLE OF CONTENTS ................................................................................................................................. II LIST OF FIGURES ...................................................................................................................................... IV ABSTRACT............................................................................................................................................... VII CHAPTER 1 INTRODUCTION .................................................................................................................1
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Magnetic Moment of a Spin, Its Equation of Motion, and Precession B1.1.6
    Magnetic Moment of a Spin, Its Equation of UNIT B1.1 Motion, and Precession OVERVIEW The ability to “see” protons using magnetic resonance imaging is predicated on the proton having a mass, a charge, and a nonzero spin. The spin of a particle is analogous to its intrinsic angular momentum. A simple way to explain angular momentum is that when an object rotates (e.g., an ice skater), that action generates an intrinsic angular momentum. If there were no friction in air or of the skates on the ice, the skater would spin forever. This intrinsic angular momentum is, in fact, a vector, not a scalar, and thus spin is also a vector. This intrinsic spin is always present. The direction of a spin vector is usually chosen by the right-hand rule. For example, if the ice skater is spinning from her right to left, then the spin vector is pointing up; the skater is rotating counterclockwise when viewed from the top. A key property determining the motion of a spin in a magnetic field is its magnetic moment. Once this is known, the motion of the magnetic moment and energy of the moment can be calculated. Actually, the spin of a particle with a charge and a mass leads to a magnetic moment. An intuitive way to understand the magnetic moment is to imagine a current loop lying in a plane (see Figure B1.1.1). If the loop has current I and an enclosed area A, then the magnetic moment is simply the product of the current and area (see Equation B1.1.8 in the Technical Discussion), with the direction n^ parallel to the normal direction of the plane.
    [Show full text]
  • 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.
    [Show full text]
  • Localized Ferromagnetic Resonance Using Magnetic Resonance Force Microscopy
    LOCALIZED FERROMAGNETIC RESONANCE USING MAGNETIC RESONANCE FORCE MICROSCOPY DISSERTATION Presented in Partial Ful¯llment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jongjoo Kim, B.S.,M.S. ***** The Ohio State University 2008 Dissertation Committee: Approved by P.C. Hammel, Adviser D. Stroud Adviser F. Yang Graduate Program in K. Honscheid Physics ABSTRACT Magnetic Resonance Force Microscopy (MRFM) is a novel approach to scanned probe imaging, combining the advantages of Magnetic Resonance Imaging (MRI) with Scanning Probe Microscopy (SPM) [1]. It has extremely high sensitivity that has demonstrated detection of individual electron spins [2] and small numbers of nuclear spins [3]. Here we describe our MRFM experiments on Ferromagnetic thin ¯lm structures. Unlike ESR and NMR, Ferromagnetic Resonance (FMR) is de¯ned not only by local probe ¯eld and the sample structures, but also by strong spin-spin dipole and exchange interactions in the sample. Thus, imaging and spatially localized study using FMR requires an entirely new approach. In MRFM, a probe magnet is used to detect the force response from the sample magnetization and it provides local magnetic ¯eld gradient that enables mapping of spatial location into resonance ¯eld. The probe ¯eld influences on the FMR modes in a sample, thus enabling local measurements of properties of ferromagnets. When su±ciently intense, the inhomogeneous probe ¯eld de¯nes the region in which FMR modes are stable, thus producing localized modes. This feature enables FMRFM to be important tool for the local study of continuous ferromagnetic samples and structures.
    [Show full text]
  • 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.
    [Show full text]
  • Ultrafast Acoustics in Hybrid and Magnetic Structures Viktor Shalagatskyi
    Ultrafast acoustics in hybrid and magnetic structures Viktor Shalagatskyi To cite this version: Viktor Shalagatskyi. Ultrafast acoustics in hybrid and magnetic structures. Physics [physics]. Uni- versité du Maine, 2015. English. NNT : 2015LEMA1012. tel-01261609 HAL Id: tel-01261609 https://tel.archives-ouvertes.fr/tel-01261609 Submitted on 25 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Viktor SHALAGATSKYI Mémoire présenté en vue de l’obtention du grade de Docteur de l’Université du Maine sous le label de L’Université Nantes Angers Le Mans École doctorale : 3MPL Discipline : Milieux denses et matériaux Spécialité : Physique Unité de recherche : IMMM Soutenue le 30.10.2015 Ultrafast Acoustics in Hybrid and Magnetic Structures JURY Rapporteurs : Andreas HUETTEN, Professeur, Bielefeld University Ra’anan TOBEY, Professeur associé, University of Groningen Examinateurs : Florent CALVAYRAC, Professeur, Université du Maine Alexey MELNIKOV, Professeur associé, Fritz-Haber-Institut der MPG Directeur de Thèse : Vasily TEMNOV, Chargé de recherches, CNRS, HDR, Université du Maine Co-directeur de Thèse : Thomas PEZERIL, Chargé de recherches, CNRS, HDR, Université du Maine Co-Encadrante de Thèse : Gwenaëlle VAUDEL, Ingénieur de recherche, CNRS, Université du Maine Contents 0 Introduction 9 1 Ultrafast carrier transport at the nanoscale 13 1.1 Two Temperature Model for bimetallic structure .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • II Spins and Their Dynamics
    For the spins to precess, and for us to detect them, II we need to somehow force them away from Spins and their equilibrium – for example, make them perpendicular to the main field: Dynamics Lecture notes by Assaf Tal B0 1. Excitation 1.1 WhyU Excite? If we do this and let them be, two things will Let us recant the facts from the previous lecture: happen: 1 1. We’re interested in the bulk 1. They will precess about B0 at a rate gB0. (macroscopic) magnetization. 2. Eventually they will return to equilibrium 2. When put in a constant magnetic field, because of thermal relaxation. This this bulk magnetization tends to align usually takes ~ 1sec, but can vary. itself along the field: Until they return to thermal equilibrium their signal is “up for grabs”, which is precisely the idea of a basic NMR/MRI experiment: B0 1. Excite the spins. 2. Measure until they return to sum equilibrium. What to exactly do with the measured signal and how to recover an image from it is something we’ll discuss in subsequent lectures. Meanwhile, let’s Microscopic Macroscopic just ask ourselves how can one excite a spin? That is, how can one tilt it from its equilibrium position along the main field and create an angle between 3. The macroscopic magnetization precesses them (usually 90 degrees, but not always)? about the external magnetic field. 1.2 TheU RF Coils In MRI, we can only detect a signal from the spins if they precess and therefore induce a current in Fortunately (or perhaps unfortunately), our our MRI receiver coils by Faraday’s law.
    [Show full text]
  • Hybrid Perfect Metamaterial Absorber for Microwave Spin Rectification
    www.nature.com/scientificreports OPEN Hybrid perfect metamaterial absorber for microwave spin rectifcation applications Jie Qian1,2, Peng Gou1, Hong Pan1, Liping Zhu1, Y. S. Gui2, C.‑M. Hu2 & Zhenghua An1,3,4* Metamaterials provide compelling capabilities to manipulate electromagnetic waves beyond the natural materials and can dramatically enhance both their electric and magnetic felds. The enhanced magnetic felds, however, are far less utilized than the electric counterparts, despite their great potential in spintronics. In this work, we propose and experimentally demonstrate a hybrid perfect metamaterial absorbers which combine the artifcial metal/insulator/metal (MIM) metamaterial with the natural ferromagnetic material permalloy (Py) and realize remarkably larger spin rectifcation efect. Magnetic hot spot of the MIM metamaterial improves considerably electromagnetic coupling with spins in the embedded Py stripes. With the whole hybridized structure being optimized based on coupled‑mode theory, perfect absorption condition is approached and an approximately 190‑fold enhancement of spin‑rectifying photovoltage is experimentally demonstrated at the ferromagnetic resonance at 7.1 GHz. Our work provides an innovative solution to harvest microwave energy for spintronic applications, and opens the door to hybridized magnetism from artifcial and natural magnetic materials for emergent applications such as efcient optospintronics, magnonic metamaterials and wireless energy transfer. Metamaterials ofer a great avenue to control the absorption,
    [Show full text]