Class Note for CHEM 777 at UMass(2)
Class Note for CHEM 777 at UMass(2)
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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 18 views.
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Date Created: 02/06/15
Ch 777 Spring 03 page 8 3 Detection of NMR signals What happens after 90 pulse On resonance signals precess at vL 7 same as rotating frame so appear fixed in rotating frame 7 M stays along y Offeresonance signals precessing at vL Av in rotating frame M rotates off of y at rate Av Receiver coil along y detects My39 signal proportional to My39 is induced in coil Actually induced signal vL r Av nuclear frequency is mixed with transmitted signal vLpulse carrier frequency to get a difference signal r Av kHz frequencies instead of MHZ On resonance frequency detected as 0 Hz others at vL r Av are detected as r Av Time course of My with precession max positive zero max negative zero etc plus fanning out exponentially decreases amplitude DRAW FID free induction decay time domain spectrum decay of transverse magnetization free 7 free of B1 field after pulse induction 7 observing current induced in coil decay r of transverse magnetization to equilibrium 7 T2 relaxation draw FID W single frequency on resonance off resonance multiple frequencies FT Fourier transform 7 mathematical conversion of time domain to frequency domain spectrum fw j fte quotdt ft is the time domain spectrum the FID f m is the frequency domain spectrum contains real and imaginary components which differ in phase by 90 real signal 7 absorption 7 convention in 1D to display real signals imaginary signal 7 dispersion after FT phase correction of spectrum manipulate linear combination of real amp imaginary to get pure absorption real magnitude Nreal2 imaginary2 absorptionrtype signals all positive but broader base Quadrature detection Ch 777 Spring 03 page 9 Problem Detecting Av from carrier frequency 3 r v will look the same Could put carrier frequency at one end of spectrum 7 drawbacks uniform excitation not centered over spectrum half of spectrum is only noise which is folded back into spectrum decreasing SN by V2 Solution quadrature detection 2 phasersensitive detectors 90 out of phase from each other mix signal with carrier frequency vs same phase shifted 90 gt Carrier M39Xer Real gt Signal gt Mixer gtmaginary A p 90 gt Detect My and Mx r v will have different behavior Mx goes or 7 first One signal is real FID starts max other imaginaryFlD starts zero Phasing choosing right linear combination of the two to get pure absorption signal Ch 777 Spring 03 page 10 4 Bloch equations amp relaxation Sum of effects of static field pulse and relaxation Mxi Myf Mz Mok T 1 W a a a a E vMxBovMxBlr 2 Simplify by 1 transforming to rotating frame 2 Look at behavior after B1 pulse turned off ie relaxation plus B0 only dM M2 M0 2 3M2 MZ0 M0el M0 7 T1 F6111 rwm MMl Yl T1 T1 er39 Mr39 M M0e39Tz dt 2 dM M r y y 3M M0e 7T2 dt T2 y y Limits t 02M MZ0 M M0 My My0 t 003 M2 M0M My 0 Exponential decay to equilibrium values with time constants T1 and T2 2A My 39V39o 90X gt y t X 2A M2 M0 180X gt t y VO x Y Above assumes on resonance offeresonance My is FID envelope Tzquot decay MZ Ch 777 Spring 03 page 11 5 Product operator formalism Goal What happens to the signal after a pulse sequence predict the effect of each pulse amp delay Bloch equations and vector diagrams 7 look at the evolution of bulk M only 7 for instance they don39t incorporate population transfer Density matrix formalism o density operator representing state of spin system 7 rigorous treatment of the evolution of the spin system through an exp BUT little qualitative insight cumbersome for gt2 spins Example 2 spin 12 nuclei each is or or 3 represent system with matrix IISIGG 30 M3 quot35 Ian Ilia IaB quot35 Diagonal elements populations of these states Offrdiagonal elements associated with transitions coherences zero quantum coherence ZQIS single quantum coherences 1Q1 IQS 2 of each double quantum coherence ZQIS Start with initial state at equilibrium 60 only diagonal elements Apply pulses multiply by rotation matrices evolution calculate evolution shift amp J of each nonzero element To get density matrix of final state 6f Product operator formalism r insight because simpler valid in the weak coupling limit J ltlt A6 Treat as independent weakly coupled spin systems Ignores relaxation for simplicity density matrix treatments do too Density operator 6t describes the state of the system represented by density matrix linear combination of a basis set of operators 60 cllz czSz c1 and c2 are constants We will use Cartesian basis set quotproduct operatorsquot Express starting state of spin system 60 product of the spin angular momentum component of each nucleus Ix 1y 11 Product operators product of angular momentum components each nucleus Ch 777 Spring 03 page 12 How to use product operators to predict effects of rf pulses and evolution with effects of chemical shift amp Jrcoupling Steps 1 Express starting state of spin system 60 product of the spin angular momentum component of each nucleus Ix ly ll 2 Predict evolution by looking at effects of each pulse and each evolution using rules below 60 gt 61 gt 62 gt 63 Simple rules I Pulses CL flip angle Ht oclx or ocly IZL zcosu Iy sinu QL ymsa iIZsina x LIX 1 IZ Lgt IZ cosu i Ix smu 1y gt 5 1 7 I L x cosoc IZ sma ll Precession 7 chemical shif Ht 9111 t I 9122 I 1x LIX cosQIt 1y sinQIt 1y iny cosQIt IxsinQIt lll Precession J coupling Ht 2njlslez t I ZMISIzgt I 1x cos7rJISt ZIySZ sinnJI I 1y 1y cosIIJISt ZIXSZ sinnJISt ZIZSZM IZSZ 25g Mn cosJ39I39JISt 1y sinJ39rJISt 21131 s ZIySZ gt21ySZ cosa39JISt Ix sina1JISt Swapping l and S in last 2 equations gives 2 more rules Simplifying points 1 You can break up simultaneous processes and compute each separately in series the order doesn39t matter Example chemical shift on I on S and JIS 9 Z 9 2 awn4 is equivalent to the cascade 9122 93322 27rd 1312322 gt gt gt 01 0392 03 0 4 2 Rotations affect only operators of same spin selectivity of pulses
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