Class Note for CHEM 777 at UMass(4)
Class Note for CHEM 777 at UMass(4)
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Date Created: 02/06/15
Ch 777 Spring 03 page 1 These lecture notes are incomplete to encourage you to remain engaged by actively annotating them I Will sometimes hand these out and other times expect you to check the course website and download them yourselves before class 1 Nuclear spins and NMR 2 2 Vector diagrams 5 3 Detection of NMR signals 8 4 Bloch equations amp relaxation 10 5 Product operator formalism 11 6 Relaxation gt NOE Nuclear Overhauser Effect 15 7 Physical interpretations of product operators amp manipulations 16 8 J coupling product operator calculations Spineecho effects on homo amp heteronuclear J 17 9 INEPT 19 10 Chemical shift 19 11 J coupling W Indirect Spinespin coupling 20 12 NMR spectrometers 21 13 Signal acquisition amp processing parameters 22 14 Solidestate NMR W Introduction 26 Ch 777 Spring 03 page 2 1 Nuclear spins and NMR Nuclear spin and magnetic moments Many nuclei have spin angular momentum J or P which leads to a magnetic moment and NMR transitions see below Spin angular momentum quantum number l empirical rules mass no atomic no 1 examples odd even or odd n2 1H 13C 15N 31F 19F 112 170 152 35C 132 even even 0 12C 150 even odd n 2H14N 11 10 NMR silent l 12 simplest NMRquot only 2 energy levels I 2 12 more complex electric quadrupole moment broad lines 7 discuss as special topic Nature was kind to us only 1H visible and simple among common organic nuclei 1H 12C 160 Can introduce as nonperturbing probes 13C 15N 2H Spin gt magnetic moment Classical analogy but nucleus not really spinning A spinning charge is an electric current ring 7 generates a magnetic moment L parallel to the angular momentum J L yJ y gyromagnetic ratio Quantum mechanics Angular momentum magnitude IJ hl111 where l spin angular momentum quantum number The magnitude of the angular momentum along any chosen axis by convention z JZ hmI where m1 I 171 fl magnetic quantum number There are 211 values of m1 and thus 21 I allowed orientations or quantum states of the nucleus All have the same energy in the absence of a magnetic field Ch 777 Spring 03 page 3 Effect of static magnetic field B0 in z direction gt energies and precession Energy levels Jz the magnitude of the component of angular momentum along the field has certain allowed values 1 12 has 2 states m1 ilZ 11 has 3 states m1 7101 80 BO A A 1 m 12 m PZ4 2l a PZIF1 PZ 2v P20 m0 m l2i Inf 4 1 39 m l Means certain orientations of magnetic moment Hz YJZ mam BO Energy of magnetic moment in a magnetic field E l o 30 ll 0cose pZBO yhmIBO cose H Back to the classical picture H 2 lowest energy aligns p parallel to B0 cose1 minimizes E But Quantum picture says only certain M1 rgt certain Jz Note that JZ hmI sh1ltfh 111 so Jz 3 J Zeeman energy levels I l 2 I l m 1 E1 yhBO m 123 Ep 1 2yhBO m 0 E0 0 m 1 2a Ea 12yhBo m 1 E1 39yhBO Only Am 1 transitions are quotallowedquot Energy difference between two neighboring states AE AmlyhBo 77130 Ch 777 Spring 03 page 4 E splitting of nuclear spin states in 1 2 magnetic field is known as Zeeman splitting 5 m12 lf supply right energy can induce transitions on quotResonancequot E QC BO hVapplied AE E Precession of magnetic moments at m l 2 Field applies torque on nuclear dipoles A 1 X B0 so nuclei precess around BO field B0 M not to scale gt X p 5 Precession frequency is called the Larmor frequency characteristic of nucleus vL Zl ia0 in Hz 1 730 in rad s NMR spectrometers are referred to by the proton Larmor frequency ie 500 MHZ spectrometer l 17 Tesla field proton precesses at 500 MHZ u 2i 0 267519 gtlt107 rad T39ls X 117 T 500MHz 291 21 rad cycle Other nuclei at same field sensitivity N 13 yof13C 14yof 1H so 14500MHZ 125 MHz 0016 7 of 15N m 110 7 of 1H so 110500MHz 50 MHz 0001 Observing an NMR transition To get transitions between energy levels in a spectroscopy experiment you need a difference in population of the energy levels radiation on resonance induces transitions in both directions with equal probability So if NOC NB then absorption emission and there is no signal observed Populations given by Botzmann distribution Ch 777 Spring 03 page 5 N AE 713 i 6 m 1 1 b m1 10 6 very small population difference because Na E kT AE ltlt kT thermal energy so energy levels are nearly equally populated thermal energy is enough to populate upper E level Means NMR is an inherently insensitive spectroscopic technique Evisible light gtgt Eradio frequency so Apopulationelectronic states gtgt Apopulationnuclear spin states lnsensitivity need 10 mg quantities of protein MINIMUM 05 ml lmM for sol39n r thats 15mg of SOkDa protein 20 mg for solids AE yhBO gt enhanced by larger AE largery and B0 2 Vector diagrams Bulk magnetization vectors Sum the magnetic moments of all the spins 7 random orientations in xy directions cancel small excess in lower E state aligned with the field Macroscopic magnetization M or M bulk magnetic moment is aligned with the field 2 direction We39ll manipulate it with NMR experiments MZw What happens to M in the NMR experiment CW NMR Analogy to simple spectroscopies ie Bo UVrVis absorption Could use constant frequency sweep field CW EPR p constant field sweep frequency CWNMR l Magnetic moments p precessing B B around B0 To induce flips between x 1 I 1 two states use a magnetic field B1 perpendicular to B0 B1 x direction It exerts a torque causing precession around x so p can flip between up and down states 281 How get B1 x coil along x current radio frequency vL rgt B1 I Bl x oscillating at vL What about precession of p 0 Linear alternating magnetic field sum of 2 counterrotating magnetic l 2 B1 Z fields of equal intensity and frequency and opposite direction just as linearly polarized light is Ch 777 Spring 03 page 6 equivalent to the sum of left and right circularly polarized light B1 is the one rotating in same direction as p precesses lf B1 rotation frequency is only near precession frequency of p B1 causes at most some wobbling or nutation of p The counterrotating component is too far off resonance to have an effect on p lf B1 rotation frequency is equal to Larmor precession frequency of p r on resonance 7 it exerts torque in phase and causes precession around B1 7 get flip of p from low to high energy state B 1 field oscillating at vL perpendicular to static B0induces transitions of p Rotating Frame simplifies picturing things Laboratory frame of reference xyz r B1 actually rotating at vL 7 too complex to draw quotRotating framequot x39y39z axes rotating at v1 7 B1 stays along x39 axis I39ll drop the prime designations but always be talking about the rotating frame B0 BO 2 B1 X A X 1 Detection l flipping p moving a magnet in a coil induces signal in coil Before precession of p were randomly distributed doesn39t induce signal B1 gives precession coherence so that it39s detectable 7 this was method used by Bloch initially OR 2 detect absorption of E simple spectrometer components magnet sample coil transmit and receive signal to amplifiers computer etc Pulsed NMR enhance sensitivity by signal averaging can39tjust scan super slow 7 will saturate resonance equalize populations no net absorption or flip repeated fast scans n scans 3 2 signal oc n 2 noise oc nl2 some cancellation 3 Z SN 2 I ll2 but time spent scanning through baseline regions is wasted faster method 7 pulse excitation of entire spectrum simultaneously Ch 777 Spring 03 page 7 Pulse burst of rf of Larmor frequency but actually contains a band of frequencies square wave m Fourier series of sines and cosines W t P bandwidth of frequencies m vL r ltp Where tp is the length of the pulse short pulses have large bandwidths r puselengths N ps excite entire spectrum tp 10113 10 5 s ltp 105 Hz 100 kHz 1H 10 ppm X 500 Hzppm 5 kHz 13C 200 ppmx 125 Hzppm 25 kHz extra bandwidth needed for uniform excitation long pulses have narrow bandwidths r can use for selective irradiation Pulse tip angle analogous to Larmor precession r vL i130 angular frequency wL 21w L 730 for eldstrength B1 pulselengtht rotation angle wtp yBltP angle of rotation proportional to magnitude of B at right frequency 90 and 180 pulses Choose pulselength to rotate M 90 around x39 90xv rotates Mz into My 7 max signal maximum value of Mxy precessing coherently around 2 B0 which gives the maximum signal induced in the coil amp detected Twice the 90 pulselength will rotate 180 180xv rotates Mz into MZ 7 inversion Magnetization vector picture Signal intensity vs pulselength 90 3 max positive 180 3 zero 270 max negative 90 Signal intensity 180 360 Pulselength 270 tp0 e30ps Experimentally use 180 or 360 to null signal and determine 90
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