NUCLEAR MAGNETIC RESONANCE


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1 19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic moments of nuclei. Theoy: In addition to its wellknown popeties of mass, chage, and intinsic angula momentum (spin), the atomic nucleus possesses in geneal a magnetic moment. The classical analogy explaining the existence of a nuclea magnetic moment is that of a spinning, chaged sphee. The classical magnetic moment of a spinning sphee with chage q and mass M unifomly distibuted thoughout its volume is: µ = q 2 M L whee µ is the magnetic moment and L is the angula momentum If this chaged, spinning sphee (a magnetic dipole with magnetic moment µ is placed in a unifom magnetic field B, the (classical) enegy is E= µ B To obtain the coect (quantum mechanical) expession fo the enegy due to the nucleus being placed in a magnetic field, the coect elation between µ and L must be used, and the quantization of angula momentum must be consideed. The coect elation between µ and L is µ = g e = γ 2M L L whee g (the Lande splitting facto) and γ (the gyomagnetic atio) ae numbes chaacteistic of a paticula nuclea isotope in a paticula nuclea enegy state. The symbol M denotes the poton mass and e the poton chage. It is usual to expess the angula momentum in tems of I, the nuclea spin vecto, using the elation I = L / h whee h = h/2π and h = Planck s constant. Thus: µ = γ h I The lagest obsevable value of I in a given diection, denoted I, and called the spin (o spin quantum numbe), is integal o halfintegal. That is, it can only have values such as 0, ½, 1, 3/2,.... Each nuclea gound state is chaacteized by just one value of I. Substituting into the enegy expession,
2 19 Jul 04 NMR.2 E = µ B= γh I B = γ hb I B$ whee $B is a unit vecto in the diection of B. Howeve, the quantization of spin (angula momentum) leads to discete enegy levels athe than a continuous ange of enegies. The expession I B $ can be intepeted as the component of I in the diection of B. Accoding to quantum mechanics, the only pemitted values of I B $ ae I, I + 1,..., I 1, I. The expession I B $ (i.e. the component of I in a pefeed diection) is given the symbol m and called the spin magnetic quantum numbe. Thus, substituting into the enegy equation, E = γ h m B m = I, I + 1,..., I 1, I Conside a sample of potons (hydogen nuclei) placed in a magnetic field. Since the spin of the poton is ½, thee ae two enegy states, coesponding to m = ±½, which ae occupied when potons ae placed in a magnetic field. The enegy diffeence between these two states is E = γ hb( ½) ( γ hb( ½)) = γ hb E = ( s 1 T 1 )( J s) B E = ( J/T) B If adiation of the fequency E/h (= o ) is supplied to the potons in the magnetic field B, tansitions between the two possible enegy states will occu. This esonance condition can be witten as J/T o = B h o = ( s 1 T 1 ) B o = (4.257 MHz/kG) B Fom a quantum mechanical calculation, it is found that fo potons in a magnetic field, being supplied with electomagnetic enegy of the esonance fequency, the pobability of a tansition fom the lowe enegy state to the highe (absoption) is exactly equal to the pobability of a tansition fom the highe enegy state to the lowe (stimulated o induced emission). If the numbe of nuclei (potons) in each state is the same, then no net absoption of powe will occu and no esonance can be obseved. Howeve, if the potons ae in equilibium with thei suoundings, then each enegy state will be populated accoding to the Boltzmann distibution. Thus thee will be a (small) excess of potons in the lowe enegy state. It is this excess that allows obsevation of esonance. Resonance can be obseved by measuing the absoption of the supplied electomagnetic enegy as a function of its fequency. At the esonance fequency, this absoption should abuptly incease.
3 19 Jul 04 NMR.3 As powe is supplied to the system at the esonance fequency, the excess numbe of potons in the lowe enegy state will decease and soon no moe absoption will be obseved. This is called satuation. Howeve, the spinlattice inteaction between the potons and thei suoundings, causing an enegy tansfe fom the spin system to the suoundings due to Bownian motion and lattice vibation, tends to estoe the Boltzmann distibution, and allow futhe powe absoption. The spinlattice inteaction (also called elaxation) is chaacteized by a time constant T 1. The shote T 1, the moe powe can be supplied without eaching satuation, and hence the stonge the esonance signal. Thee is also an inteaction between neighbouing nuclea spins (spinspin inteaction) which causes a slight shift in the enegy states fo each poton in the applied magnetic field. This causes a boadening of the esonance signal, since thee is no longe a welldefined fequency at which powe absoption will occu. In ode to facilitate obsevation of the esonance signal, paamagnetic ions may be added to the sample to boaden the esonance signal. Signal boadening also esults fom inhomogeneity of the extenal magnetic field. Caeful study of the signal obtained in a nuclea magnetic esonance expeiment leads to infomation concening cystal stuctue and chemical binding in the sample. The technique of obsevation of nuclea magnetic esonance is also used to study magnetic moments of vaious substances. Finally, a moe pactical application of NMR is the accuate measuement of magnetic fields. An accuacy of 1 pat in 10 5 can be obtained fo magnetic fields fom 10 2 to 10 5 gauss (0.01 to 10 T). A elatively ecent application of NMR is in diagnostic medical imaging (MRI magnetic esonance imaging). MRI is based on two pinciples: 1. the magnetic esonance fequency depends on the value of the magnetic field; 2. the elaxation time depends on the natue of the suoundings of the esonating nuclei. In MRI the hydogen nuclei (poton) concentation is measued. By poviding a magnetic field that vaies unifomly acoss the subject, potons at diffeent positions will have diffeent esonance fequencies. Measuement of the elaxation times of these hydogen nuclei gives an indication of the types of tissues suounding the hydogen nuclei. Thus by applying the appopiate magnetic field gadients and detecting the poton esonant signal in sufficient detail, MRI povides a pictue of a twodimensional slice though the subject at any desied location. Advantages of MRI ove xay imaging ae its moedetailed images of soft tissues and its nonionizing natue. Appaatus: The sample pobe consists of a 7 mm glass tube filled with glyceine (a good souce of hydogen nuclei (potons)). The tube is suounded by a coil to povide the high fequency electomagnetic enegy.
4 19 Jul 04 NMR.4 To obseve the esonance signal, the extenal magnetic field is modulated. A pai of Helmholtz coils poweed at ac line fequency is used fo this, one coil placed on eithe side of the sample. The entie aangement is placed between the polefaces of an electomagnet (see Figue 1). Figue 1 A maginal oscillato (Alpha Scientific Laboatoies, Inc. Model AL 675 NMREPR) is used to poduce the adio fequency (f) electomagnetic powe and to detect esonance. If the maginal oscillato is appoximated as a constant cuent souce, then at esonance the powe loss inceases suddenly due to absoption in the sample. This is detected as a voltage dop which is amplified and detected on an oscilloscope. The advantages of the maginal oscillato ae compactness and simplicity but it cannot be used to study the esonance signal in detail. To bette undestand how the appaatus is used, efe to Figue 2, a plot of enegy vs. time.
5 19 Jul 04 NMR.5 Figue 2 Since the Helmholtz coils ae connected to a powe unit (A.C. Sweep AL 679) oscillating at 60 Hz, the net magnetic field expeienced by the sample (the sum of the Helmholtz field and the electomagnet field) is B net = B + B sin[(377 ad/s) t] Thus the enegy diffeence between spin states of the sample, DE, also oscillates sinusoidally, between the values γh(b + B ) and γh(b B ) The magnitude of B with espect to B is geatly exaggeated in Figue 2. Suppose f powe of fequency 1 is fed to the sample. Since, as shown in Figue 2, the f enegy (h 1 ) is always lowe than the enegy between states, no absoption can occu and no esonance signal is obtained. If the f fequency is inceased to 2 then, as shown in Figue 2, absoption can occu and a esonance signal is poduced, since the enegy supplied (h 2 ) equals the enegy between states twice fo each cycle of the modulated magnetic field. By adjusting the fequency of the f enegy, the esonant fequency fo the electomagnet field alone can be detemined (i.e. h 0 = γhb).
6 19 Jul 04 NMR.6 By measuing 0 and knowing γ a value fo B can be calculated. Altenatively, measuing 0 and B at esonance yields infomation concening the magnetic moment of the sample. Figue 3 is a block diagam of the electonics used in this expeiment. Figue 3
7 19 Jul 04 NMR.7 Pocedue and Expeiment: 1. Ensue that the dials on the Powe Supply and Regulato ae set to zeo. 2. Tun on the wate supply to cool the magnet windings. 3. Tun on the digital multimete and set it to the 10 A DC scale. 4. Tun on the Powe Supply and Regulato, and adjust the Regulato so that the pointe is within the black (contolled) egion of the mete. The Regulato contols the cuent though the magnet, and the Powe Supply is used to adjust the Regulato fo pope egulation. Vay the cuent though the magnet using the contols of the two devices, always keeping the Regulato pointe within the black egion, until you ae familia with the opeation of the equipment. 5. Befoe continuing with the expeiment, the magnet must be demagnetized. This is done in a simila manne as was used fo the βay Spectomete magnet. The cuent diection can be changed by placing the Regulato output cable in the othe output socket. Using the Powe Supply dial alone (i.e. don t woy about keeping the Regulato pointe in the egulation zone) incease the magnet cuent to 2.5 A, decease to 0, change output cable connection, continue deceasing to 2.5 A, then incease to 0. Repeat fo cuent limits of 2.3 A and 2.3 A and poceed in 0.2 A steps until the cycle A A 0 has been completed. The magnet has now been demagnetized. When demagnetizing, do not change the Regulato output cable connection until the Powe Supply is set to Tun on the oscilloscope. Set the scales at 0.1 V/cm (unless othewise advised) and 2 ms/cm. Use line o ac tiggeing. 7. Tun on the AC Sweep unit and set the dial just slightly less than fully clockwise. 8. Tun on the fequency counte. 9. Connect the kg pobe, containing the glyceine sample, to the NMREPR maginal oscillato and the AC Sweep unit. Set the oscillato selecto switch to C. (Fo the othe two pobes (labelled and ) use position B.) Adjust the oscillato cuent contol until noise is obseved on the oscilloscope tace.
8 19 Jul 04 NMR.8 MEASUREMENT OF MAGNETIC FIELDS USING NMR Ensue that the pobe is placed in the cente of the polefaces of the electomagnet. Set the cuent in the magnet to 0.5 A and slowly adjust the oscillato fequency contol until a esonance signal is obtained. When the esonance signals ae equally spaced on the oscilloscope tace, the oscillato fequency is equal to the esonance fequency 0 coesponding to the magnetic field of the electomagnet. Why? Detemine the hysteesis cuve of the electomagnet using the following pocedue: a) Incease the magnet cuent, measuing the esonance fequency evey 0.1 A. (When esonance can no longe be obtained with a paticula pobe, eplace it with the next one.) b) Continue until the esonance fequency coesponding to a magnet cuent of 2.0 A has been measued. c) Now decease the magnet cuent, again measuing the esonance fequency evey 0.1 A. d) As pat of you analysis of the data, plot magnetic field vesus cuent and comment on the shape of the gaph. Show clealy which eadings wee obtained with inceasing cuent and which with deceasing cuent. Detemine the homogeneity of the electomagnet s field as follows: a) Fo a cuent value of 1.0 A, measue the magnetic field as a function of position, x, of the pobe with espect to the polefaces. Use small incements of x when the pobe is appoaching the poleface edge. Choose x = 0 to coespond to the cente of the polefaces. b) Measue the diamete of the polefaces. c) As pat of you analysis, plot the magnetic field (B) vs. pobe position (x), showing the poleface edge locations on the gaph. MEASUREMENT OF MAGNETIC MOMENTS USING NMR Recall fom the Theoy that fo a nucleus with spin ½, the spacing between enegy levels is E = γhb Also, the magnetic moment of a nucleus is given by whee I is the spin µ = γhi Fo a constant magnetic field: E1 E2 = γ1h γ 2 h whee the subscipts 1, 2 efe to two diffeent nuclea species with spin ½.
9 19 Jul 04 NMR.9 h1 h2 = γ h γ h 1 2 h2 γ 2h= γ 1h h Since the two nuclea species have the same spin (½), 1 2 γ 2hI = γ 1hI 1 2 µ 2 = γ 1hI F µ F = γ phi whee p denotes poton, and F denotes a Fluoine nucleus. Using the value of γ p given in the Theoy, 1 p F µ F = ( J/T) p Fo a given magnet cuent, measue the poton esonance fequency. Replace the glyceine sample with the solid Teflon od and measue the Fluoine esonance fequency. Compae the calculated value of the Fluoine magnetic moment with the accepted value of µ F = J/T. Be sue to tun off all equipment at the end of the expeiment, especially the ammete and the oscillato since they ae batteypoweed. Refeences: Fette, Intoduction to Expeimental Physics, QC 41 Halliday, Intoductoy Nuclea Physics, QC 173 Melissinos, Expeiments in Moden Physics, QC 33 Pake, Ameican Jounal of Physics, 18, (1950), p. 438, QC 1
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