EDUCATION Revista Mexicana de F´ısica E 63 (2017) 48–55 JANUARY–JULY 2017 Quantum-mechanical aspects of magnetic resonance imaging J.A. Soto, T. Cordova,´ and M. Sosa Division´ de Ciencias e Ingenier´ıas Campus Leon,´ Universidad de Guanajuato Loma del bosque no. 103, Col. Lomas del Campestre, Leon,´ Guanajuato, Mexico,´ phone:(+52)477-7885100 email: jasa@fisica.ugto.mx; theo@fisica.ugto.mx; modesto@fisica.ugto.mx S. Jerez Centro de Investigacion´ en Matematicas´ A.C. Jalisco S/N, Col. Valenciana, Guanajuato, Guanajuato, Mexico,´ phone: (+52)473-7327155 email: [email protected] Received 7 March 2016; accepted 23 September 2016 The Magnetic Resonance Imaging (MRI) is a non-invasive technique which uses the physical phenomenon of nuclear magnetic resonance to obtain structural and compositional information about human body regions. In this imaging study we use the radio-frequency and a powerful static magnetic field, which aligns the magnetization of hydrogen nuclei. Nowadays there are many types of clinical equipment that conduct MRI studies, which have intensities of magnetic fields from 0.2T to 7.0T. Moreover, liquid helium is required for the superconducting coil. This paper presents an analysis of the magnetic resonance phenomenon; by doing a review of the quantum-mechanical aspects as the spin and Zeeman effect. Keywords: Magnetic resonance; magnetic field; spin; magnetization; voxel. PACS: 01.40.Fk; 03.65.Ca; 41.20.Gz 1. Introduction electrons are fermions and are subject to the Pauli exclusion principle, being in a crystal lattice, the energy between them Nowadays, there are many techniques to observe and analyze becomes negative (attractive) so that pairs are created to min- the inside of the human body in order to obtain a better diag- imize the energy and behave as bosons. nosis. One non-invasive and high-resolution technique is the In this paper we discuss the basic physics involving mag- Magnetic Resonance Imaging (MRI), which takes advantage netic resonance, providing an analysis from the viewpoint of of hydrogen nuclei, a powerful static magnetic field, and a quantum mechanics. computer system to process and get images. The Nuclear Magnetic Resonance is a physical phe- 2. The Hydrogen atom nomenon in which certain atomic nuclei (with an odd num- ber of protons or neutrons) are placed under a high-intensity MRI uses the properties of hydrogen nuclei when they are magnetic field and can selectively absorb energy from elec- exposed to a high magnetic field and a radio-frequency field, tromagnetic waves in the radio-frequency range. Once the so it is important to analyze the physical characteristics of the nuclei have absorbed the energy, the excess energy returns to hydrogen atom [11]. the surroundings through a process called relaxation which is The hydrogen atom is the simplest atom since, there is a accompanied by a local magnetic variation, which induces a proton in its core and an electron orbiting it experimenting an signal to a receiving antenna for digital processing and thus attractive Coulomb potential. obtain an image or to perform a spectrometric analysis [6]. e2 V (r) = ¡ : (1) MRI equipment consists of a magnet (usually supercon- 4¼²0r ductor), radio frequency coils, magnetic field gradients, a To perform the analysis, the Schrodinger¨ equation indepen- bore or tunnel, and a computer for signal processing. MRI dent of time is used: requires the use of a high-intensity magnetic field. Clinical µ ¶ p2 equipment use field strengths ranging from 0.5-3.0 T, which Hª(r) = + V (r) ª(r) = Eª(r); (2) have been achieved by replacing the permanent magnets by 2¹ superconducting electromagnets resulting in a very wide line where ¹ is the reduced mass of the system and p, the momen- research. BCS theory [12–14] is the dominant physical the- tum operator ory of superconductivity and was proposed by John Bardeen, m ¢ m ¹ = p e ; (3) Leon Cooper, and Robert Schrieffer. The theory is based on mp + me the fact that the charge carriers are not free electrons but, rather, pairs of electrons known as Cooper pairs. Although p = ¡i~ 5 : (4) QUANTUM-MECHANICAL ASPECTS OF MAGNETIC RESONANCE IMAGING 49 Because there is a central potential V (r), natural coordinates multiplying the above equation by sin2θ: are spherical coordinates, and Eq. (2) becomes: sen2θ d2£ senθcosθ d£ · µ ¶ + ~2 @ @ £(θ) dθ2 £(θ) dθ ¡ sen θ r2 2¹r2sen θ @r @r 1 d2© µ ¶ ¸ + = ¡l(l + 1)sen2θ; (12) @ @ 1 @2 ©(Á) dÁ2 + sen θ + ª(r) @θ @θ sen θ @Á2 separating the azimuth part, e2 ¡ ª(r) = Eª(r); (5) 1 d2© 4¼²0r = ¡m: (13) ©(Á) dÁ2 re-arranging the equation above: The solution to Eq. (13) is given as: 2 µ 2 ¶ 2 @ ª @ª 2¹ re 2 r 2 + 2r + 2 + Er ª ¡imÁ imÁ @r @r ~ 4¼²0 ©(Á) = Ae + Be ; (14) @2ª cosθ @ª 1 @2ª where A and B are constants. Equation (12) is as follows: + 2 + + 2 2 = 0: (6) @θ senθ @θ sin θ @Á sen2θ d2£ senθcosθ d£ + ¡ m2 = ¡l(l + 1)sen2θ; (15) To solve Eq. (6) the separation of variables is: £(θ) dθ2 £(θ) dθ ª(r; θ; Á) = R(r)Y (θ; Á); (7) multiplying the equation by £(θ) we get: d2£ d£ replacing the solution proposed in Eq. (6) and dividing by sen2θ + senθcosθ R(r)Y (θ; Á) we obtain: dθ2 dθ £ 2 2¤ 2 2 µ 2 ¶ + l(l + 1)sen θ ¡ m £(θ) = 0: (16) r @ R(r) 2r @R(r) 2¹ re 2 2 + + 2 + Er R(r) @r R(r) @r ~ 4¼²0 Making the following change in the Eq. (16) cos θ ! x and 1 @2Y (θ; Á) cosθ @Y (θ; Á) £ ! y, to get: = ¡ ¡ Y (θ; Á) @θ2 Y (θ; Á)senθ @θ d£ d£ = ¡senθ ; 1 @2Y (θ; Á) dθ dx ¡ : (8) Y (θ; Á)sen2θ @Á2 d2£ d2£ dθ = sen2θ ¡ cosθ ; dθ2 dx2 dx We can separate the Eq. (8) into two with a separation con- 2 2 stant. For reasons that will eventually become clear, we will sen θ = 1 ¡ x : write this separation constant as l(l + 1), i.e. Performing the above changes, Eq. (16) is written as: 1 @2Y (θ; Á) cosθ @Y (θ; Á) · ¸ + d2y dx m2 Y (θ; Á) @θ2 Y (θ; θ) θ @θ (1 ¡ x2) ¡ 2x + l(l + 1) ¡ y = 0: (17) sen dx2 dy 1 ¡ x2 1 @2Y (θ; Á) + = ¡l(l + 1); (9) Equation (17) is known as Associated Legendre Differential Y (θ; Á)sen2θ @Á2 Equation, and also reveals why we chose the constant above r2 @2R(r) 2r @R(r) to be l(l+1). That is, its solutions are given by the Associated + R(r) @r2 R(r) @r Legendre Polynomials: µ ¶ 2¹ re2 m l+m + + Er2 = l(l + 1): (10) m (¡1) 2 m=2 d 2 l 2 Pl (x) = (1 ¡ x ) (x ¡ 1) : (18) ~ 4¼²0 2ll! dxl+m For the solution of Eq. (9) the separation of variables method The solution Y (θ; Á) = £(θ) ¢ ©(Á), comprising the angular is used again, this time making Y (θ; Á) = £(θ) ¢ ©(Á), to part of the Eq. (8) is of the form: get: m imÁ Yl;m(θ; Á) = NPl (cosθ)e ; (19) 1 d2£ cosθ d£ + £(θ) dθ2 £(θ)senθ dθ where N is a normalization constant defined as: s 2 1 d © (2l + 1)!(l ¡ 1)! + = ¡l(l + 1); (11) N = ; (20) ©(Á)sen2θ dÁ2 4¼(l + 1)! Rev. Mex. Fis. E 63 (2017) 48–55 50 J.A. SOTO, T. CORDOVA,´ M. SOSA, AND S. JEREZ and thus it ensures that the functions Yl;m(θ; Á) are orthonor- 3. Nuclear magnetic resonance mal. Equation (19) is known as Spherical Harmonics. Re- turning to the radial part of Schrodinger¨ Eq. (10), rearranging Nuclear magnetic resonance is a physical phenomenon asso- it to be: ciated with the intrinsic angular momentum of the spin and the magnetic properties of atomic nuclei. When a nucleus d2R(r) 2 dR is placed in a magnetic field an interaction occurs between + dr2 r dr the magnetic moment of the nucleus and the field, resulting · µ ¶ ¸ 2¹ e2 l(l + 1) in an energy splitting. By the absorption and emission of + + E ¡ R(r) = 0; (21) photons with the right frequency, transitions between these ~2 4¼² r r2 0 energy states may occur. whole solution is given by the Associated Laguerre Polyno- Local magnetic changes produced by the absorption and mials, which meet the following normalization condition: emission of photons are detected by an antenna that sends the signal to a computer for decoding and image generating. Z1 2n [(n + l)!]3 e¡½½2l[L2l+1(½)]2½2d½ = ; (22) 3.1. Nuclear Spin n+l (n ¡ l ¡ 1)! 0 The particles that make up the atomic nuclei (protons and neutrons) have the intrinsic quantum mechanical property of so the solution of Eq. (8) for the radial part is expressed as: spin. In nuclear physics the total angular momentum that s the nucleus has is called nuclear spin [18], though the term (n ¡ l ¡ 1)! R(r) = should not be confused with spin of each nucleon or total 2n[(n + l)!]3 spin as the sum of all nucleons, because this is just one of the µ ¶3=2 two contributions to the nuclear spin, the other is the angular 2 ¡½=2 l 2l+1 momentum of the nucleons. So each nucleon will have a net £ e ½ Ln¡l¡1(½); (23) na0 angular momentum such that: 2 where ½ = 2r=na0 y a0 = ~=me . ji = li + si; (26) Since we have the solutions of each of the equations de- then the nuclear spin will be: pending only on one variable, we proceed to build the exact X X solution of the Schrodinger¨ equation independent of time for J = ji = (li + si); (27) the hydrogen atom, i i s or just: µ ¶3=2 (n ¡ l ¡ 1)! 2 = + : ª(r; θ; Á) = J L S (28) 2n[(n + l)!]3 na 0 Each of these vectors have a similar quantum number.