SCPY152 Lecture 21 Magnetic Resonances Udom Robkob, Physics-MUSC March 18, 2015 Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Today Topics Electron orbit an its magnetic moment Magnetic interaction Electron spin angular momentum Electron spin magnetic moment Electron spin resonance Nuclear spin magnetic moment Nuclear magnetic resonance Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Electron Orbit and Its Magnetic Moment Let us determine an electron under circular motion with ~ = m~r × ~v . corresponding angular momentum L With electric charge −e of an electron, an orbiting electron produces electric current i around the orbit but in opposite direction as i= ev eme vr −e =− =− T 2πr 2πme r2 Udom Robkob, Physics-MUSC (1) SCPY152Lecture 21 Magnetic Resonances Electron Orbit and Its Magnetic Moment The magnetic moment of current loop i is defined to be ~ A = πr2 − loop area µ ~ = iA, (2) Then the magnetic moment of electron orbit is µ ~ =− eme vrˆ n e ~ =− L 2m2 2me (3) ~ = me vrˆ where n ˆ is direction of the orbitng plane and L n= me~r × ~v . Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Magnetic Interaction The magnetic interaction of magnetic moment µ ~ with ~ = B zˆ is defined to be uniform magnetic field B ~ = U = −~ µ·B eB Lz . 2me (4) In quantum physics, this energy will quantized according quantization of Lz to the values; U =m e~ B = mµb B, m = 0, ±1, ±2, ..., ±l. 2me (5) where µB = 5.79 × 10−5 eV/T is known as Bohr magneton. Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Magnetic Interaction Diagram of magnetic interaction energy at l = 1 appear in the following figure; Exercise: Draw the energy digrams of magnetic interaction at l = 0, 2, 3 for arbitrary energy E and magentic field B. Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Electron Spin Angular Momentum Electron spin angular momentum is observed in Stern-Gerlach expeiment. Suppose µ ~ S is a magnetic moment of electron, interaction energy with non-uniform magnetic field B(z)ˆ z is ~ = −mz B(z) U = −~ µS · B Udom Robkob, Physics-MUSC (6) SCPY152Lecture 21 Magnetic Resonances Electron Spin Angular Momentum There will be force from magnetic non-unifomity in z-direction acts on the magnetic moment; dU dB(z) Fz = − = µSz (7) dz dz The result of Stern Gerlach experiment made us to define ~ with electron spin angular momentum, or spin, S corrsesponding magnetic moment gµB ~ µ ~S = S. (8) ~ ~ where g ' 2 is known as g-factor. The quantization of S ~ as will follow the same rule of L S 2 = s(s + 1)~2 , Sz = ms ~. (9) where s is spin quantum number and ms is spin magnetic moment quantum nimber and ms = −s, −(s − 1), ..., (s − 1), s Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Electron Spin Angular Momentum From Stern Gerlach experiment, we have to assign the value of electron spin to be 1 1 s = , ms = ± . 2 2 (10) Then we have 1 S = 2 2 √ 1 3 2 2 ~ = 3 ~, + 1 ~ = ~ → |S| 2 4 2 1 Sz = ± ~ 2 (11) (12) Let us assign an abstract spin state χs,ms , then χ 1 , 1 is 2 2 known as ”spin-up” state and χ 1 ,− 1 is known as 2 2 ”spin-down” state. Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Electron Spin Angular Momentum The spin angular momentum orientation will appear as in the following figure; Magnetic interaction of electron spin magnetic moment ~ = B zˆ will be with magnetic field B ~ = − gµB B Sz = ∓µB B. U = −~ µS · B ~ Udom Robkob, Physics-MUSC (13) SCPY152Lecture 21 Magnetic Resonances Electron Spin Resonance Energy diagram of this interaction energy is; The ”resonance” of photon energy to make transitions between these two states will be ~ω = hc = ∆E = 2µB B λ Udom Robkob, Physics-MUSC (14) SCPY152Lecture 21 Magnetic Resonances Nuclear Spin Magnetic Moment Inside the atomic nucleus we have nucleons, proton p and neutron n. Nuclear spin magnetic moment comes from spin of the nucleon. Spin quantum number of proton and neutron is s = 1/2. Their corresponding magnetic moments will be µ ~ =g µN ~ S ~ (15) where g = 2.7928 for proton and g = −1.9130 for neutron. e~ µN = 2m = 3.153 × 10−8 eV/T is known as nuclear p magneton. Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Nuclear Spin Magnetic Moment The alignments of proton and neutron magnetic moments with respect their spin appear in the following figures. Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Nuclear Magnetic Resonance Let us determine proton nuclear magnetic moment µ ~ in ~ uniform magnetic field B = B zˆ. The magnetic interaction energy will be ~ =− U = −~ µ·B 1 gµN B Sz = ∓ gµN B ~ 2 (16) The corresponding emission or absorption photon will have energy hc ~ω = = gµN B (17) λ Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances Nuclear Magnetic Resonance Example - calculate photon wavelength resonances to proton spin magnetic moment energy in uniform magnetic field of magnitude 1.5 tesla. λ= hc 1.25 × 10−6 = = 9.42m. gµN B 2.793 × 3.15 × 10−8 × 1.5 It is in the range of radiowaves Exercise - calculate photon wavelength resonances to electron spin magnetic moment energy in magnetic field of magnitude 0.5 tesla. Udom Robkob, Physics-MUSC SCPY152Lecture 21 Magnetic Resonances
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