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A few fundamentals of NMR

A few fundamentals of NMR. Dieter Freude. Harry Pfeifer's NMR-Experiment 1951 in Leipzig. H. Pfeifer: Über den Pendelrückkoppelempfänger (engl.: pendulum feedback receiver) und die Beobachtungen von magnetischen Kernresonanzen, Diplomarbeit, Universität Leipzig, 1952.

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A few fundamentals of NMR

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  1. A few fundamentals of NMR Dieter Freude

  2. Harry Pfeifer's NMR-Experiment 1951 in Leipzig H. Pfeifer: Über den Pendelrückkoppelempfänger (engl.: pendulum feedback receiver) und die Beobachtungen von magnetischen Kernresonanzen, Diplomarbeit, Universität Leipzig, 1952

  3. Nuclear spin I = 1/2 in an magnetic field B0 Many atomic nuclei have a spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is The component in the direction of an applied field is Lz = Iz m  =  ½  for I = 1/2. Atomic nuclei carry an electric charge. In nuclei with a spin, the rotation creates a circular current which produces a magnetic moment µ. An external homogenous magnetic field B results in a torque T = µ Bwith a related energy of  E = -µ·B. The gyromagnetic (actually magnetogyric) ratio g is defined by µ = gL. The z component of the nuclear magnetic moment is µz =  gLz =  g Izg m . The energy for I = 1/2 is split into 2 Zeeman levels Em = -µz B0 = -g mB0 = gB0/2 = wL/2. Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet.

  4. Larmor frequency Classical model: the torqueT acting on a magnetic dipole is defined as the time derivative of the angular momentum L. We get By setting this equal to T = µ  B , we see that The summation of all nuclear dipoles in the unit volume gives us the magnetization. For a magnetization that has not aligned itself parallel to the external magnetic field, it is necessary to solve the following equation of motion: We define B =(0, 0, B0) and chooseM(t= 0) = |M| (sina, 0, cosa). Then we obtain Mx= |M| sina coswLt, My= |M| sina sinwLt, Mz= |M| cosa with wL =-gB0. The rotation vector is thus opposed to B0for positive values of g. The Larmor frequency is most commonly given as an equation of magnitudes: wL =gB0 or Joseph Larmor described in 1897 the precession of electron orbital magnetization in an external magnetic field.

  5. Macroscopic magnetization hnL« kT applies at least for temperatures above 1 K and Larmor frequencies below 1 GHz. Thus, spontaneous transitions can be neglected, and the probabilities P for absorption and induced emission are equal. It follows P = B+½,-½ wL= B-½,+½ wL, where B refers to the Einstein coefficients for induced transitions and wL is the spectral radiation density at the Larmor frequency. A measurable absorption (or emission) only occurs if there is a difference in the two occupation numbers N. In thermal equilibrium, the Boltzmann distribution applies to N and we have If nL= 500 MHz and T =300 K, hnL/kT 8  10-5 is very small, and the exponential function can be expanded to the linear term:

  6. Longitudinal relaxation time T1 All degrees of freedom of the system except for the spin (e.g. nuclear oscillations, rotations, translations, external fields) are called the lattice. Setting thermal equilibrium with this lattice can be done only through induced emission. The fluctuating fields in the material always have a finite frequency component at the Larmor frequency (though possibly extremely small), so that energy from the spin system can be passed to the lattice. The time development of the setting of equilibrium can be described after either switching on the external field B0 at time t =0 (difficult to do in practice) with T1 is the longitudinal or spin-lattice relaxation time an n0 denotes the difference in the occupation numbers in the thermal equilibrium. Longitudinal relaxation time because the magnetization orients itself parallel to the external magnetic field. T1 depends upon the transition probability P as  1/T1 =2P =2B-½,+½wL.

  7. t0 By setting the parentheses equal to zero, we get t0=T1 ln2 as the passage of zero. To measure T1 by IR The inversion recovery (IR) by p-p/2

  8. Rotating coordinate system and the offset For the case of a static external magnetic field B0 pointing in z-direction and the application of a rf field Bx(t) = 2Brf cos(wt) in x-direction we have for the Hamilitonian operator of the external interactions in the laboratory sytem (LAB) H0 + Hrf = hwLIz+ 2hwrf cos(wt)Ix, where wL = 2pnL = -g B0 denotes the Larmor frequency, and the nutation frequency wrf is defined as wrf = -g Brf. The transformation from the laboratory frame to the frame rotating with w gives, by neglecting the part that oscillates with the twice radio frequency, H0 i + Hrf i = hDwIz+ hwrfIx, where Dw= wL -w denotes the resonance offset and the subscript i stays for the interaction representation. Magnetization phases develop in this interaction representation in the rotating coordinate system like b = wrft or a = Dwt. Quadratur detection yields value and sign of a.

  9. Bloch equation and stationary solutions We define Beff= (Brf, 0, B0-w /g) and introduce the Bloch equation: Stationary solutions to the Bloch equations are attained for dM/dt= 0:

  10. Correlation time tc, relaxation times T1 and T2 The relaxation times T1 and T2 as a function of the reciprocal absolute temperature 1/T for a two spin system with one correlation time. Their temperature dependency can be described by tc=t0 exp(Ea/kT). It thus holds that T1=T2 1/tc when wLtc « 1 and T1wL2 tc when wLtc » 1. T1 has a minimum of at wLtc 0,612 or nLtc 0,1.

  11. Hahn echo p/2 pulseFID, p pulse around the dephasing around the rephasing echo y-axisx-magnetizationx-axisx-magnetization a(r,t) = Dw(r)·t a(r,t) = -a(r,t) + Dw(r)·(t -t)

  12. Line width and T2 A pure exponential decay of the free induction (or of the envelope of the echo, see next page) corresponds to G(t) = exp(-t/T2). The Fourier-transform gives fLorentz = const.  1 / (1 + x2) with x = (w-w0)T2, see red line. The "full width at half maximum" (fwhm) in frequency units is Note that no second moment exists for a Lorentian line shape. Thus, an exact Lorentian line shape should not be observed in physics. Gaussian line shape has the relaxation function G(t) = exp(-t2 M2 / 2) and a line form fGaussian = exp (-w2/2M2), blue dotted line above, where M2 denotes the second moment. A relaxation time can be defined by T22 = 2 / M2. Then we get

  13. T2 and T2*

  14. EXSY, NOESY, stimulated spin echo

  15. Magic-angle spinning • Rotation frequency should be much greater than: • heterogeneous line broadening • homogeneous line broadening • rate of mobility of the species • Magnetic resonance shifting with the geometry factor 3cos2-1 is caused by: • homo-nuclear dipole-dipole interaction, • heteronuclear interaction, • anisotropy of the chemical shift, • first-order quadrupole interaction, • sample microinhomogenieties. • The shift gives rise to a signal broadening in powder materials.

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