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 B 0 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 L z = I z   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  B with a related energy of E =  µ·B. The gyromagnetic (actually magnetogyric) ratio  is defined by µ =  L. The z component of the nuclear magnetic moment is µ z =  L z =  I z    m . The energy for I = 1/2 is split into 2 Zeeman levels E m =  µ z B 0 =   m  B 0 =    B 0 /2 =   L   /2. Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet.

4. Larmor frequency Joseph Larmor described in 1897 the precession of electron orbital magnetization in an external magnetic field. Classical model: the torque T 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, B 0 ) and choose M(t  0)  |M| (sin , 0, cos  ). Then we obtain M x  |M| sin  cos  L t, M y  |M| sin  sin  L t, M z  |M| cos  with  L  =   B 0. The rotation vector is thus opposed to B 0 for positive values of . The Larmor frequency is most commonly given as an equation of magnitudes:  L =  B 0 or

5. Macroscopic magnetization h L « 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 w L = B  ½,+½ w L, where B refers to the Einstein coefficients for induced transitions and w L 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 L  500 MHz and T  300 K, h L /kT  8  10  is very small, and the exponential function can be expanded to the linear term:

6. Longitudinal relaxation time T 1 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 B 0 at time t  0 (difficult to do in practice) with T 1 is the longitudinal or spin-lattice relaxation time an n 0 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. T 1 depends upon the transition probability P as 1/T 1  = 2P  2B  ½,+½ w L.

7. To measure T 1 by IR The inversion recovery (IR) by  -  /2 By setting the parentheses equal to zero, we get  0  T 1 ln2 as the passage of zero. 00

8. Rotating coordinate system and the offset For the case of a static external magnetic field B 0 pointing in z-direction and the application of a rf field B x (t) = 2B rf cos(  t) in x-direction we have for the Hamilitonian operator of the external interactions in the laboratory sytem (LAB) H 0 + H rf =   L I z + 2   rf cos(  t)I x, where  L = 2  L =  B 0 denotes the Larmor frequency, and the nutation frequency  rf is defined as  rf =  B rf. The transformation from the laboratory frame to the frame rotating with  gives, by neglecting the part that oscillates with the twice radio frequency, H 0 i + H rf i =   I z +   rf I x, where  =  L   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  =  rf  or  =  t. Quadratur detection yields value and sign of .

9. Bloch equation and stationary solutions We define B eff  (B rf, 0, B 0  /  ) and introduce the Bloch equation: Stationary solutions to the Bloch equations are attained for dM/dt  0:

10. Correlation time  c, relaxation times T 1 and T 2 The relaxation times T 1 and T 2 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  c   0 exp(E a /kT). It thus holds that T 1  T 2  1/  c when  L  c « 1 and T 1   L 2  c when  L  c » 1. T 1 has a minimum of at  L  c  0,612 or L  c  0,1.

11. Hahn echo  /2 pulse FID,  pulse around the dephasingaround the rephasing echo y-axis x-magnetization x-axis x-magnetization  (r,t) =  (r)·t  (r,t) =   (r,  ) +  (r)·(t   )

12. Line width and T 2 A pure exponential decay of the free induction (or of the envelope of the echo, see next page) corresponds to G(t) = exp(  t/T 2 ). The Fourier-transform gives f Lorentz = const.  1 / (1 + x 2 ) with x = (    0 )T 2, 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(  t 2 M 2 / 2) and a line form f Gaussian = exp (  2 /2M 2 ), blue dotted line above, where M 2 denotes the second moment. A relaxation time can be defined by T 2 2 = 2 / M 2. Then we get

13. T 2 and T 2 *

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 3cos 2   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.