DOCSLIB.ORG

Lecture #3 Nuclear Spin Hamiltonian

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture #3 Nuclear Spin Hamiltonian Lecture #3! Nuclear Spin Hamiltonian • Topics – Liouville-von Neuman equation – Time-averaged versus instantaneous spin Hamiltonian – Chemical shift and J, dipolar, and quadrupolar coupling • Reading assignments – van de Ven: Chapters 2.1-2.2 – Levitt, Chapters 7 (optional) 1 Coherences • Nuclear spins can be thought of as weak magnetic dipoles, each with a well defined direction of polarization. • Summing the polarization across many spins can result in net phase coherences. ˆ S z ˆ ˆ I 2I zS z S € Net tendency for S spins to be €+z No net tendency for I or S spins to be ±z No net tendency for I spins in any direction If I or S is ±z, increased probability paired spin is ±z 2 The Spin Density Operator • Spin density operator, σ ˆ ( t ), is the mathematical quantity that describes a statistical mixture of spins and the associated phase coherences that can occur, as encountered in a typical NMR or MRI experiment. € • σ ˆ ( t ) is usually expressed as a linear combination of basis 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (the 16 two-spin product operators) operators, e.g. { 2 E, Ix, Sx, Iy, Sy,…, 2IzSz } • Coherences (signals) observable with an Rf coil: € M = γ Tr σˆIˆ = γ Iˆ ˆ ˆ ˆ x ! { x } ! x M y = γ!Tr{σ Iy } = γ! Iy • Some coherences not directly observable with an Rf coil: ˆ ˆ ˆ ˆ ˆ M z = γ! Iz Cxz = γ! 2IxSz Czz = γ! 2IzSz “antiphase x” “longitudinal two-spin order” • The goal is to describe how these quantities evolve in time. 3 Liouville-von Neumann Equation • Time evolution of σ ˆ ( t ) : Iˆ z σˆ (0) ˆ t ˆ σ ( ) I y ˆ ∂ ˆ ˆ ˆ H is the Hamiltonian operator σ = −iHσ and corresponds to the energy € € ∂t of the system (E / ! ). € ˆ I x • H ˆ can often be expressed as sum of a large static component plus ˆ ˆ ˆ € a small time-varying perturbation: H = H 0 + H 1 ( t ) , leading to… ˆ Relaxation superoperator (a function of H 1 ) ∂ ˆ ˆ ˆ ˆ ˆ ˆ σ = −iH0σ − Γ(σ −σ B ) (Compare with Bloch’s equations) ∂t Rotations Relaxation • The key, yet again, is finding the Hamiltonian! 4 Spin-Lattice Disconnect • Complete QM description of a molecule involves lots of terms in the Hamiltonian (nuclear spin, molecular motion, electron-nucleus interactions, etc). ˆ ˆ ˆ ˆ H = H l + H s + H i lattice spin interaction term • Previously, we assumed a negligible interaction between the nuclear spins and the lattice: ˆ ˆ ˆ ∂ ˆ H ≈ H l + H s and just solved σˆ = −iH σˆ ∂t s • Now we need to take a closer look at the interaction term, which includes effects such as spin operator coefficients that depend on molecular orientation, etc. 5 The Nuclear Spin Hamiltonian • H ˆ is the sum of different terms representing different physical interactions. ˆ ˆ ˆ ˆ H = H1 + H 2 + H 3 +! Examples: 1) interaction of spin with B0 2) interactions with dipole fields of other nuclei 3) nuclear-electron couplings • In general,€ we can think of an atomic€ nucleus as a lumpy magnet with a (possibly non-uniform) positive electric charge • The spin Hamiltonian contains terms which describe the orientation dependence of the nuclear energy The nuclear magnetic moment interacts with magnetic fields Hˆ = Hˆ elec + Hˆ mag The nuclear electric charge interacts with electric fields 6 € Electromagnetic Interactions • Magnetic interactions magnetic moment ! ! ! ! Hˆ mag ˆ B Iˆ B = −µ ⋅ = −γ" ⋅ local magnetic field quadrapole • Electric interactions monopole dipole Nuclear€ electric charge distributions can be expressed as a sum of multipole components. ! (0) ! (1) ! (2) ! C (r ) = C (r ) + C (r ) + C (r ) +" Symmetry properties: C(n)=0 for n>2I and odd interaction terms disappear Hence, for spin-½ nuclei there are no electrical ˆ elec H = 0 (for spin I€= 1/2) energy terms that depend on orientation or internal nuclear structure, and they behaves exactly like ˆ elec H ≠ 0 (for spin I >1/2) point charges! Nuclei with spin > ½ have electrical quadrupolar moments. 7 € Motional Averaging • Molecular motion • Previously, we used averaging to simplify the Hamiltonian Molecular orientation depends on time and Hamiltonian terms can be written ˆ 0 as H int ( Θ ( t ) ). These terms were replaced by their time averages: τ ˆ 0 1 ˆ 0 ergodicity ˆ 0 ˆ 0 H int = τ ∫ H int (Θ(t))dt H int = H int p(Θ)dΘ 0 ∫ p(Θ)=probability € Secular ˆ isotropic 1 ˆ 0 Hamiltonian Isotropic materials: H H d density for molecule int = N ∫ int (Θ) Θ normalization having orientation Θ What does € • We no longer want to make this€ approximation. “secular” mean? Instead, the time variations will be analyzed as perturbations. € 8 Time-averaged Spin Hamiltonian Relative magnitudes Relative 9 Instantaneous Spin Hamiltonian Relative magnitudes Relative 10 Simplifications • In general, the nuclear spin Hamiltonian is quite complicated. • We’ll regularly make use of two simplifications. 1. For terms in the Hamiltonian that are periodic, we use a change of basis. ˆ ˆ ˆ Hˆ ! = e−iωtIz Hˆ = e−iωtIz HˆeiωtIz rotating frame laboratory frame Hˆ t Iˆ Iˆ cos t Iˆ sin t Hˆ Iˆ Iˆ ( ) = −ω0 z −ω1 ( x ω − y ω ) eff = −(ω0 −ω) z −ω1 x 2. The secular approximation 11 The Secular Approximation • The large B0 field dominates some of the internal spin interactions. • Consider the general case of Hˆ = Aˆ + Bˆ “large” “small” ˆ • Choosing the eigenkets, A i = a i i , as an orthonormal basis set, then A = diag(a1, a2..., an ). • Example: let the eigenvalues of A have the following pattern: In general, B is ! * * * * * * * $ Energy # & non-zero * * * * * * * everywhere. # & a5, a6, a7 # * * * * * * * & # & a nearly degenerate. B = * * * * * * * 1 # & # * * * * * * * & a a 2, 3 degenerate # * * * * * * * & # & a4 * * * * * * * " % 12 The Secular Approximation • The secular approximation of B is: Eigenvalues of A ! * 0 0 0 0 0 0 $ # & Energy # 0 * * 0 0 0 0 & # 0 * * 0 0 0 0 & a5, a6, a7 B0 = # 0 0 0 * 0 0 0 & # & a1 # 0 0 0 0 * * * & # & 0 0 0 0 * * * a2, a3 # 0 0 0 0 * * * & " % a4 ` 0 • Mathematically B = ∑bnn n n + ∑bmn m n n m≠n where the` indicates summation only over terms which connect degenerate or nearly degenerate states of A. • This is equivalent to omitting terms for which bmn << am − an . • The secular approximation for the Hamiltonian is: Hˆ ≈ Aˆ + Bˆ 0. 13 Secular Approximation Example • Consider a spin in a large field B0 to which we add an x gradient Gx. ! • We actually can’t create the field G x x z alone (see Laplace’s Eqn) ! ! ! ! ! • We really have: B = B0z +Gx xz + ΔBx (x, y, z) x + ΔBy (x, y, z) y In MRI, these are often called “Maxwell” or “concomitant gradient” terms. ! ! ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Hence: H = −γB⋅ I = −γB0Iz −γGx xIz −γΔBx Ix −γΔByIy = A + B # & " % γG x γ ΔB +iΔB γB0 0 % x ( x y ) ( ˆ A 1 $ ' Bˆ B = 1 where A = 2 and 2 % ( $ 0 −γB0 ' γ ΔB −iΔB −γG x # & $% ( x y ) x '( • However, B 0 > > G x x , Δ B x , Δ B y è a1 − a2 = γB0 >> γ ΔBx ±iΔBy " γG x 0 % hence B0 = 1 $ x ' 2 0 G x #$ −γ x &' ! ! ˆ ˆ ˆ ˆ 0 ˆ • The secular approximation is: H = −γB⋅ I ≈ A + B = −γ (B0 +γGx x)Iz i.e. we can safely ignore the “Maxwell” terms as is routinely done in MRI (with just a few exceptions, particularly at low field) 14 B0-Electron Interactions When a material is placed in a magnetic field it is magnetized to some degree and this modifies the field… Electrons in an atom circulate about B0, Shielding: generating a magnetic moment opposing the applied magnetic field. • Global effects: magnetic susceptibility s B0 = (1− χ)B0 field inside sample bulk magnetic susceptibility applied field Hereafter we’ll use “B0” to refer to the internal field. € • Local effect: Chemical Shift shielding constant (Don’t confuse with the spin density operator!) Different atoms experience B B (1 ) different electron cloud densities. = 0 −σ 15 € The Zeeman Hamiltonian ! • The interaction energy! between! the magnetic field, B , and the magnetic moment, µ = γ I , is given by the Zeeman Hamiltonian. ! ! ! !ˆ Classical: E = −γB ⋅ I QM: Hˆ = −γB ⋅ I zeeman€ • The formal€ correction for chemical shielding is: !ˆ ! Hˆ I (1 )B σ = 3× 3 shielding tensor € zeeman = −γ −σ € where • In vivo, rapid molecular tumbling averages out the non-isotropic components. σ = σ iso = Tr(σ /3) € ! € • Hence for B = [0, 0, B0 ]: E ˆ ˆ €H Zeeman = −γ(1−σ )B0Iz 16 B = 0 B = B0 € € A little foreshadowing… Chemical Shielding Tensor • Electron shielding is in general anisotropic, i.e. the degree of shielding depends on the molecular orientation. • The shielding tensor can be written as the sum of three terms: ! $ σ xx σ xy σ xz ! 1 0 0 $ # & # & See Kowalewski, pp # & (1) (2) 105-6 for details. σ = σ yx σ yy σ yz = σ iso # 0 1 0 &+σ +σ # & # 0 0 1 & symmetric and # σ zx σ zy σ zz & " % traceless " % antisymmetric 1 = 3 (σ xx +σ yy +σ zz ) • Both σ(1) and σ(2) are time-varying due to molecular tumbling. • σ(2) gives rise to a relaxation mechanism called chemical shift anisotropy (CSA). (to be discussed later in the course) • σ(1) causes only 2nd order effects and is typically ignored. 17 J-Coupling: Mechanism • At very small distances (comparable to the nuclear radius), the dipolar interaction between an electron and proton is replaced by an isotropic interaction called “Fermi contract interaction”.
Load more

Recommended publications
  • Relaxation 11/26/2020 | Page 2
    RUPRECHT-KARLS- UNIVERSITY HEIDELBERG Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Master‘s Program in Medical Physics 11/26/2020 | Page 1 Physics of Imaging Systems Basic Principles of Magnetic Resonance Imaging III Prof. Dr. Lothar Schad Chair in Computer Assisted Clinical Medicine Faculty of Medicine Mannheim University of Heidelberg Theodor-Kutzer-Ufer 1-3 D-68167 Mannheim, Germany [email protected] www.ma.uni-heidelberg.de/inst/cbtm/ckm/ RUPRECHT-KARLS- UNIVERSITY HEIDELBERG Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Relaxation 11/26/2020 | Page 2 Relaxation Seite 1 1 RUPRECHT-KARLS- UNIVERSITY HEIDELBERG Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Magnetization: M and M 11/26/2020 | Page 3 z xy longitudinal magnetization: Mz transversal magnetization: Mxy transversal magnetization: Mxy - phase synchronization after a 90°-pulse - the magnetic moments of the probe start to precede around B1 leading to a synchronization of spin packages → Mxy - after 90°-pulse Mxy = M0 RUPRECHT-KARLS- UNIVERSITY HEIDELBERG Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: M and M 11/26/2020 | Page 4 z xy source: Schlegel and Mahr. “3D Conformal Radiation Therapy: A Multimedia Introduction to Methods and Techniques" 2007 Seite 2 2 RUPRECHT-KARLS- UNIVERSITY HEIDELBERG Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Longitudinal Relaxation Time: T1 11/26/2020 | Page 5 thermal equilibrium excited state after 90°-pulse: -N-1/2 = N+1/2 and Mz = 0, Mxy = M0 after RF switched off: - magnetization turns back to thermal equilibrium - Mz = M0, Mxy = 0 → T1 relaxation longitudinal relaxation time T1 spin-lattice-relaxation time T1 RUPRECHT-KARLS- UNIVERSITY HEIDELBERG Computer Assisted Clinical Medicine Prof.
    [Show full text]
  • Proton Relaxation Times in Paramagnetic Solutions. Effects of Electron Spin Relaxation N
    Proton Relaxation Times in Paramagnetic Solutions. Effects of Electron Spin Relaxation N. Bloembergen and L. O. Morgan Citation: The Journal of Chemical Physics 34, 842 (1961); doi: 10.1063/1.1731684 View online: http://dx.doi.org/10.1063/1.1731684 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/34/3?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 75.183.112.71 On: Fri, 29 Nov 2013 20:41:28 842 L. C. SNYDER AND R. G. PARR cause the computed value of the property to depend of the test magnetic dipole. The presence of these on the origin taken for the vector potential. It also difficulties in the perturbation method should en­ is clear that caution must be exercised in applying sum courage the continued investigation of variation or 13 17 rules to estimate the excited state parts. With a single other methods - as the way to achieve quantitative average excited state energy the excited state part of computation of the magnetic properties of molecules. the magnetic susceptibility for a hydrogen atom can be ,estimated for any origin of the vector potential. How­ J3 M. J. Stephen, Proc. Roy. Soc. (London) A242, 264 (1957). ever, no single average excited state energy can give a J4 B. R. McGarvey, J. Chem. Phys. 27, 68 (1957). correct sum rule estimate of the excited state contribu­ J. J. A. Pople, Proc.
    [Show full text]
  • NMR Experiments for Structure Determination
    NMR Experiments for Structure Determination Dr Michael Thrippleton Introduction Nuclear Overhauser Effect (NOE) J-coupling •through space •through bond •J-spectroscopy •1D NOE •DQF COSY •2D NOESY •z-COSY •ROESY •HMBC The NOE Self-Relaxation pulse(s) relaxation Relaxation is the process by which magnetisation returns to equilibrium • longitudinal relaxation rate R1 • Transverse relaxation rate R2 The NOE Inversion recovery τ Magnetisation returns to +z axis during at rate R 1 •figures reproduced from Understanding NMR spectroscopy , by James Keeler The NOE Cross-relaxation The NOE Cross-relaxation Perturbation of spin I from equilibrium causes spin S to grow/shrink σ • cross-relaxation rate constant 12 • spins must be close in space The NOE Transient NOE •Target spin inverted and allowed to relax to equilibrium •Cross-relaxation generates NOE on neighbouring spin “mixing time” irradiated spectrum − Sirr Sref reference spectrum η = Sref difference spectrum NOE enhancement •figures reproduced from Understanding NMR spectroscopy , by James Keeler The NOE Transient NOE irradiated spin difference spectrum reference spectrum •figures reproduced from Stott et al., J. Magn. Reson. 125 , 302-324 The NOE NOESY •2D version of 1D transient NOE •takes longer, but contains more information •row from NOESY looks like 1D experiment •spectrum reproduced from Understanding NMR spectroscopy , by James Keeler The NOE Relaxation Mechanisms •Dipole-dipole coupling nearly averaged out by molecular tumbling •Remainder responsible for relaxation τ •Rate depends on timescale of motion c and distance between spins ( r-6 ) Small molecules, e.g. quinine τ •rapid motion / short c Large molecules, e.g. proteins τ tumbling •slow motion / long c The NOE How big, how fast? ω τ = 5 “zero crossing” 0 c 4 •slow / absent for v.
    [Show full text]
  • Optical Detection of NMR J-Spectra at Zero Magnetic Field
    1 Optical detection of NMR J-spectra at zero magnetic field M. P. Ledbetter1, C. W. Crawford2, A. Pines2,3, D. E. Wemmer2, S. Knappe4, J. Kitching4, and D. Budker1,5 1Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300 USA, 2Department of Chemistry, University of California, Berkeley, California 94720 USA, 3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720 USA, 4Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado, 80305 USA, 5Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,USA Scalar couplings of the form JI1·I2 between nuclei impart valuable information about molecular structure to nuclear magnetic-resonance spectra. Here we demonstrate direct detection of J-spectra due to both heteronuclear and homonuclear J-coupling in a zero-field environment where the Zeeman interaction is completely absent. We show that characteristic functional groups exhibit distinct spectra with straightforward interpretation for chemical identification. Detection is performed with a microfabricated optical atomic magnetometer, providing high sensitivity to samples of microliter volumes. We obtain 0.1 Hz linewidths and measure scalar-coupling parameters with 4-mHz statistical uncertainty. We anticipate that the technique described here will provide a new modality for high-precision “J spectroscopy” using small samples on microchip devices for multiplexed screening, assaying, and sample identification in chemistry and biomedicine. 2 Nuclear magnetic resonance (NMR) endures as one of the most powerful analytical tools for detecting chemical species and elucidating molecular structure. The fingerprints for identification and structure analysis are chemical shifts and scalar couplings (1,2) of the form JI1 ·I2.
    [Show full text]
  • Spin-Lattice Relaxation Times Reading Assignment
    Spin-Lattice Relaxation Times Reading Assignment : T. D. W. Claridge, High Resolution NMR Techniques in Organic Chemistry , Chapter 2; E. Breitmaier, W. Voelter, Carbon 13 NMR Spectroscopy,3rd Ed., 3.3.2. Purpose Use the inversion recovery method to measure the T 1 relaxation times for the different chemical environments in a molecule. The T 1 values will be used to assess motion in different parts of the molecule. Introduction Spin-Lattice Relaxation When molecules absorb light they are transferred from lower energy states to higher energy states. This leaves the system out of equilibrium, and the system must undergo transitions from the upper states to lower states to get back to equilibrium. This process is illustrated in Figure 1. For nuclear magnetic resonance of protons, the energy level diagram has two energy levels, the lower state with protons "spin up", or aligned with the external field, and the higher state with protons "spin down", or aligned against the external field. Light in the radiofrequency region of the spectrum causes protons to "flip", or jump to higher energy levels. The process whereby the system returns to equilibrium involves spin flips to return to the lower energy state, and the excess energy is lost to the surroundings in the form of heat. The surroundings in magnetic resonance experiments is called the "lattice", therefore the name spin- lattice relaxation. The characteristic life-time of a spin in the upper state is called the spin-lattice relaxation time T 1. T 1 is the average length of time that a proton remains in the same energy level.
    [Show full text]
  • Normal Values of Magnetic Relaxation Parameters of Spine Components with the Synthetic MRI Sequence
    ORIGINAL RESEARCH SPINE Normal Values of Magnetic Relaxation Parameters of Spine Components with the Synthetic MRI Sequence X M. Drake-Pe´rez, X B.M.A. Delattre, X J. Boto, X A. Fitsiori, X K.-O. Lovblad, X S. Boudabbous, and X M.I. Vargas ABSTRACT BACKGROUND AND PURPOSE: SyMRI is a technique developed to perform quantitative MR imaging. Our aim was to analyze its potential use for measuring relaxation times of normal components of the spine and to compare them with values found in the literature using relaxometry and other techniques. MATERIALS AND METHODS: Thirty-two spine MR imaging studies (10 cervical, 5 dorsal, 17 lumbosacral) were included. A modified multiple-dynamic multiple-echo sequence was added and processed to obtain quantitative T1 (millisecond), T2 (millisecond), and proton density (percentage units [pu]) maps for each patient. An ROI was placed on representative areas for CSF, spinal cord, intervertebral discs, and vertebral bodies, to measure their relaxation. RESULTS: Relaxation time means are reported for CSF (T1 ϭ 4273.4 ms; T2 ϭ 1577.6 ms; proton density ϭ 107.5 pu), spinal cord (T1 ϭ 780.2 ms; T2 ϭ 101.6 ms; proton density ϭ 58.7 pu), normal disc (T1 ϭ 1164.9 ms; T2 ϭ 101.9 ms; proton density ϭ 78.9 pu), intermediately hydrated disc (T1 ϭ 723 ms; T2 ϭ 66.8 ms; proton density ϭ 60.8 pu), desiccated disc (T1 ϭ 554.4 ms; T2 ϭ 55.6 ms; proton density ϭ 47.6 ms), and vertebral body (T1 ϭ 515.3 ms; T2 ϭ 100.8 ms; proton density ϭ 91.1 pu).
    [Show full text]
  • Design, Characterization and Molecular Modeling of New Fluorinated Paramagnetic Contrast Agents for Dual 1H/19F MRI
    magnetochemistry Article Design, Characterization and Molecular Modeling of New Fluorinated Paramagnetic Contrast Agents for Dual 1H/19F MRI Emilie Hequet 1,Céline Henoumont 1, Vera Djouana Kenfack 1, Vincent Lemaur 2, Roberto Lazzaroni 2,Sébastien Boutry 1,3, Luce Vander Elst 1, Robert N. Muller 1,3 and Sophie Laurent 1,3,* 1 General, Organic and Biomedical Chemistry Unit, Nuclear Magnetic Resonance (NMR) and Molecular Imaging Laboratory, University of Mons, 7000 Mons, Belgium; [email protected] (E.H.); [email protected] (C.H.); [email protected] (V.D.K.); [email protected] (S.B.); [email protected] (L.V.E.); [email protected] (R.N.M.) 2 Laboratory for Chemistry of Novel Materials, University of Mons, 7000 Mons, Belgium; [email protected] (V.L.); [email protected] (R.L.) 3 CMMI–Center for Microscopy and Molecular Imaging, 6041 Gosselies, Belgium * Correspondence: [email protected] Received: 18 December 2019; Accepted: 7 February 2020; Published: 11 February 2020 Abstract: One major goal in medical imaging is the elaboration of more efficient contrast agents (CAs). Those agents need to be optimized for the detection of affected tissues such as cancers or tumors while decreasing the injected quantity of agents. The paramagnetic contrast agents containing fluorine atoms can be used for both proton and fluorine magnetic resonance imaging (MRI), and they open the possibility of simultaneously mapping the anatomy using 1H MRI and accurately locating the agents using 19F MRI. One of the challenges in this domain is to synthesize molecules containing several chemically equivalent fluorine atoms with relatively short relaxation times to allow the recording of 19F MR images in good conditions.
    [Show full text]
  • Hyperechoes Weigel.Pdf
    Magnetic Resonance in Medicine 55:826–835 (2006) Contrast Behavior and Relaxation Effects of Conventional and Hyperecho-Turbo Spin Echo Sequences at 1.5 and 3 T1 Matthias Weigel* and Juergen Hennig To overcome specific absorption rate (SAR) limitations of spin- advantages such as limited volume coverage, longer acqui- echo-based MR imaging techniques, especially at (ultra) high sition times, and a less well defined slice profile emerge. fields, rapid acquisition relaxation enhancement/TSE (turbo These limitations are not acceptable for most clinical ap- spin echo)/fast spin echo sequences in combination with con- plications. More promising solutions are a shorter ETL via stant or variable low flip angles such as hyperechoes and partially parallel acquisition (PPA) techniques such as TRAPS (hyperTSE) have been introduced. Due to the multiple GRAPPA, SMASH, and SENSE (3,4) or partial Fourier spin echo and stimulated echo pathways involved in the signal formation, the contrast behavior of such sequences depends reconstruction methods (5,6). Since SARϳRFPϳ␣ 2 the use of refocusing flip angles on both T2 and T1 relaxation times. In this work, constant and ref various variable flip angle sequences were analyzed in a volun- lower than 180° reduces RFP very efficiently (7). However, teer study. It is demonstrated that a single effective echo time a reduction of the (constant) refocusing flip angle leads to parameter TEeff can be calculated that accurately describes the a diminished signal-to-noise ratio (SNR) and also to subtle overall T2 weighted image contrast. TEeff can be determined by changes of contrast due to stimulated echo contributions means of the extended phase graph concept and is practically that are usually ignored.
    [Show full text]
  • Magnetic Resonance Imaging Biomarker Calibration Service: Proton Spin Relaxation Times
    NIST Special Publication 250-97 Magnetic Resonance Imaging Biomarker Calibration Service: Proton Spin Relaxation Times Michael A. Boss Andrew M. Dienstfrey Zydrunas Gimbutas Kathryn E. Keenan Anthony B. Kos Jolene D. Splett Karl F. Stupic Stephen E. Russek This publication is available free of charge from: https://doi.org/10.6028/NIST.SP.250-97 NIST Special Publication 250-97 Magnetic Resonance Imaging Biomarker Calibration Service: Proton Spin Relaxation Times Michael A. Boss Kathryn E. Keenan Karl F. Stupic Stephen E. Russek Applied Physics Division Physical Measurement Laboratory Andrew M. Dienstfrey Zydrunas Gimbutas Applied and Computational Mathematics Division Information Technology Laboratory Anthony B. Kos Quantum Electromagnetics Division Physical Measurement Laboratory Jolene D. Splett Statistical Engineering Division Information Technology Laboratory This publication is available free of charge from: https://doi.org/10.6028/NIST.SP.250-97 May 2018 U.S. Department of Commerce Wilbur L. Ross, Jr., Secretary National Institute of Standards and Technology Walter Copan, NIST Director and Undersecretary of Commerce for Standards and Technology Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedural or concept adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose. National Institute of Standards and Technology Special Publication 250-97 Natl. Inst. Stand. Technol. Spec. Publ. 250-97, 35 pages (May 2018) CODEN: NSPUE2 This publication is available free of charge from: https://doi.org/10.6028/NIST.SP.250-97 Abstract This document describes a calibration service to measure proton spin relaxation times, T1 and T2, of materials used in phantoms (calibration artifacts) to verify the accuracy of magnetic resonance imaging (MRI)-based quantitative measurements.
    [Show full text]
  • 1 Nuclear Magnetic Resonance
    J. Rothberg October 12, 2001 Introduction to Quantum Mechanics – part 5 1 Nuclear Magnetic Resonance 1.1 Relaxation Times The Nuclear Magnetic Resonance setup includes a large static magnetic field, B0 in the z direction. The magnitude of this field is typically between 1 and 2 Tesla (10,000 to 20,000 gauss). For medical applications when large volumes and high fields are needed the field is produced with a superconducting solenoid. There are additional coils which modify this field to achieve spatial maps. The sample of nuclei with spin usually consists of protons in water or organic materials but also may include 19Fluorine, 13Carbon or other non-zero spin nuclei. 1.1.1 Spin-Lattice Relaxation - T1 When the B0 magnetic field is established a fraction of the protons go into the lower energy state with the magnetic moment aligned with the magnetic field. This is a thermal relaxation process depending on interactions and energy exchange with the medium. The fraction of spins that is aligned is only about one in a million at room temperature at 1 Tesla but this still provides about 1017 aligned spins in a typical sample. An important question is how long it takes to establish this spin alignment. The characteristic time for spins to align with the magnetic field is called T1 or the “Spin- Lattice” or the “Longitudinal” relaxation time or “energy” relaxation time. The growth of spin alignment or longitudinal magnetization follows an exponential of the form (1−exp−t/T1 ). Typical values for T1 in biological tissue range from 200 ms to two seconds.
    [Show full text]
  • Pulsed Nuclear Magnetic Resonance to Explore Relaxation Times
    Pulsed nuclear magnetic resonance to explore relaxation times Hannah Saddler, Adam Egbert, and Will Weigand (Dated: 26 October 2015) The method of pulsed nuclear magnetic resonance was employed in this experiment to allow for measurements of the Spin Lattice relaxation time (T1), the Spin Spin relaxation time (T2), and the Free Induction Decay ∗ time (T2 ) for glycerin and mineral oil. The Spin Lattice relaxation time was measured to be 33.9 ms ± 0.2 ms for glycerin and 45.5 ms ± 0.2 ms for mineral oil. The Spin Spin relaxation time was measured to be 24.3 ms ± 0.3 ms for glycerin and 34.2 ms ± 0.3 ms. The Free Induction Decay time were measured to be 3.3 ∗ ms ± 0.1 ms for glycerin and 3.2 ms ± 0.1 ms for mineral oil. The measured values for T1, T2, and T2 were then compared to two different equations for Free Induction Decay time; it was found that the Spin Lattice Relaxation time does not have a significant influence on the Free Induction Decay time. I. INTRODUCTION used to compare two different equations for FID time to see which quantities affected it. Nuclear magnetic resonance is response of magnetic The second section of this paper will cover the theory nuclei in a uniform magnetic field to a continuous radio necessary to understand the method of PNMR. Section frequency magnetic field as the field in tuned through res- III is an overview of the experimental design and appa- onance. The magnetic nuclei in a magnetic field absorb ratus used.
    [Show full text]
  • The Nuclear Overhauser Effect
    1/22/2017 <b>8.2 The Nuclear Overhauser Effect</b> 8.2 The Nuclear Overhauser Effect © Copyright Hans J. Reich 2016 All Rights Reserved University of Wisconsin An important consequence of DD relaxation is the Nuclear Overhauser Effect, which can be used to determine intra­ (and even inter­) molecular distances. The NOE effect is the change in population of one proton (or other nucleus) when another magnetic nucleus close in space is saturated by decoupling or by a selective 180 degree pulse. To understand this effect, we have to first consider the consequences of applying a second radio­frequency during an NMR experiment (decoupling). Double Resonance Experiments There are several types of NMR experiments that depend on the introduction of a second irradiation frequency (B1), i.e. irradiation of a nucleus other than the one being observed There are two direct consequences of irradiating an NMR signal using the decoupler: decoupling and saturation: 1. Decoupling. Irradiation of a signal at the resonance frequency interferes with any coupling of the nucleus to others in the molecule. The effects of decoupling are almost instantaneous ­ once the decoupler is turned on coupling disappears on the order of fractions of a millisecond (assuming the decoupler power is high enough), when the decoupler is turned off, the coupling reappears on a similar time scale. · If the B1 frequency is on resonance and the power is high enough, then coupling can be completely suppressed. At weaker powers complicated effects arise. The most common experiment of this type is homonuclear decoupling in proton NMR spectra (HOMODEC), which is a simple and effective technique for establishing coupling relationships among protons.
    [Show full text]