lect2 continuous wave NMR

Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 1Continuous Wave Magnetic Resonance In this lecture we transform the Bloch equations to the rotating frame to simplify the description of the spin system in the presence of electromagnetic radiation. We then discuss how the mag-netization vector moves during resonant and slightly off-resonant irradiation. Up to this point,the physics is the same for NMR and EPR experiments. After this we consider the effect of con-tinuous irradiation of the spin system over times that are much longer than the transverse andlongitudinal relaxation time. This part is relevant mainly for EPR spectroscopy, where suchcontinuous-wave (CW) measurements are the standard experiment. We discuss shortly, how a CW EPR spectrometer works and why the derivative of the absorption spectrum rather than theabsorption spectrum itself is detected. 1 Motion of the Magnetization Vector during Irradiation 1.1 Bloch Equations in the Rotating Frame At thermal equilibrium, the magnetization vector is parallel to the magnetic field, i.e., it is di- rected along z. The components Mx and My are zero. According to equations (1.24)-(1.26), only the field components Bx and By can perturb this equilibrium. In other words, transitions between the two spin states are induced by a transverse field in the xy plane. Experimentally it is most simple to generate a linearly polarized field. If we chose the direction of this field along x, the sum of the static longitudinal and the oscillatory transverse magnetic fields is given by , (2.1) , (2.2) . (2.3) Solving the Bloch equations with such a time-dependent field is difficult. Therefore, we try to obtain a time-independent field by a coordinate transformation. First, we write the linearly po-larized field as a sum of right-handed and left-handed circularly polarized fields, , (2.4)B xt() 2B1ωosct ()cos = Byt() 0= Bzt() B0= B1t() 2B1 ωosct ()cos ex B1rt() B1lt() + == Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 2where , (2.5) . (2.6) One of these components rotates with the same sense as the magnetization vector precesses. Only this component is effective in inducing transitions. The other, counterrotating componentcan be neglected in a good approximation. Without loosing generality we keep and ne-glect . The time-dependent magnetic field is now given by the components , (2.7) , (2.8) . (2.9) Now we consider the situation in a coordinate frame that rotates counterclockwise with angular frequency ω osc about the z axis of the original laboratory frame. In this rotating frame , is time-independent, , .B1rt() ωosct ()cos exωosct ()sin ey+ = B1lt() ωosct ()cos ex ωosct ()sin ey – = B1rt() B1lt() xy spin precessionB1r B1l B0B1r+ Bxt() B1ωosct ()cos = Byt() B1ωosct ()sin = Bzt() B0= B1r Bx B1= By 0= Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 3As the rotating frame is not an inertial frame, we must be careful about converting . The mo- tion of the magnetization vector induced by the static field B0 in the laboratory frame (equations (1.20)-(1.23)) is a precession with frequency . As the rotating frame rotates with the samesense, the magnetization vector precesses in this frame only with the resonance offset . (2.10) Thus, the rotating frame Bloch equations are , (2.11) , (2.12) , (2.13) where (2.14) quantifies the strength of the oscillatory field. For (off-resonance irradiation) the spin system is not significantly perturbed. This is because the rotating frame precession with angular frequency averages the effect of the os-cillatory field along the x direction. In contrast, for the precession due to the static field vanishes in the rotating frame and only the oscillatory field drives the motion of the magnetiza-tion vector. Hence, the magnetization vector precesses with angular frequency ω 1 about the x axis of the rotating frame. This case corresponds to resonant irradiation , . (2.15)Bz ω0 Ω0ω0ωosc– = dMx dt---------- Ω0My–Mx T2-------– = dMy dt---------- Ω0Mxω1Mz–My T2-------– = dMz dt---------- ω1MyMzM0– T1------------------- - – = ω1 γB1 = Ω0ω1» Ω0 Ω0 0= ωoscω0 = Ω0⇒ 0= Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 4In general, the magnetization vector precesses about an effective field that can be pictured as the vector sum of and . The precession cone is tilted by an angle . 1.2 Continuous irradiation After an irradiation time that is much longer than the logitudinal and transverse relaxation timesT 1 and T2, the spin ensemble should attain a steady state . The magnetization vector in this steady state can be computed from the requirement that the derivatives of its components be zero. Fromequations (2.11)-(2.13) we find , (2.16) , (2.17) . (2.18) This system of three linear equations can be solved for the three variables .We ob- tain , (2.19)Ω 0 ω1 θ arctan ω1Ω0⁄() = ωeff ω1θ xyMz Ω0My –Mx T2-------–0 = Ω0Mxω1Mz–My T2-------–0 = ω1MyMzM0– T1------------------- - –0 = MxMyMz ,, Mx M0ω1Ω0T22 1Ω02T22ω12T1T2++------------------------------------------------ = Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 5 , (2.20) . (2.21) We first consider the case of resonant irradiation, . The magnetization component with the same phase as the irradiation, Mx, vanishes. The magnetization component that is pahse shifted by 90°, My, is proportional to the equilibrium magnetization M0, and, for (2.22) is also proportional to the irradiation field strength . Thus, My corresponds to the absorption signal. The condition expressed in equation (2.22) also ensures that , (2.23) i.e., the spin system is close to equilibrium. If condition (2.22) is violated, the signal My grows less than linearly with ω1, which is called a saturation effect . At sufficiently large irradiation field strengths the signal attains a maximum and if both and are fulfilled,the signal vanishes again! We now discuss the shapes of the resonance lines corresponding to M y and Mx, i.e., the depend- ence of these magnetization vector components on Ω0. These shapes do not depend on ω1 as long as the transition is not saturated, i.e., as long as condition (2.22) is fulfilled. For My we find . (2.24)MyM0ω1T2 1Ω02T22ω12T1T2++------------------------------------------------ –= MzM01Ω02T22+ 1Ω02T22ω12T1T2++------------------------------------------------ = Ω00= ω12T1T21« ω1 MzM0≈ ω1T11» ω1T21» MyΩ0() M0ω1T21 1Ω02T22 +---------------------- - = Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 6This lineshape is a Lorentzian absorption line with a full width at half height (FWHH) of . Note that this width corresponds to angular frequencies. On a frequency scale, the width is by afactor of 2 π smaller, i.e., the FWHH is . The signal corresponding to M x vanishes only exactly at resonance. We find , (2.25) which gives the shape of a Lorentzian dispersion line. Note that the dispersion line is much broader. Whenever possible, magnetic resonance experiments are therefore performed in a waythat provides pure absorption lines. 1.3 Physics of detection On resonance or near resonance the sample absorbs part of the irradiation and converts it to heatby spin-lattice relaxation. Due to the small population difference between the two spin levels2T 2⁄ 1πT2()⁄ -10 -5 0 5 100MT01 2ω Ω02T Ω02T∆ω= 2/T2 ∆ν π= 1/ T2half height -10 -5 0 5 10My Mxabsorption dispersion MxΩ0() M0ω1T2Ω0T2 1Ω02T22 +---------------------- - = R L CVt0 cosωsample Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 7this absorption is weak, i.e., only a small part of the total power is absorbed. Such a weak ab- sorption is best measured by detecting the influence of the sample on a resonant circuit. The frequency of such a circuit is given by . (2.26) It is usually tuned to the radiofrequency (or microwave frequency) of the irradiation. The am- plitude of the driven oscillation then depends on the driving amplitude V0 and the quality factor . (2.27) If the spin ensemble is in resonance with the irradiation, there is an additional resistance Rsample due to the absorption of the irradiation by the sample. This causes a decrease in the Q value which in turn leads to changes in several characteristics of the RLC resonant circuit. This de-crease of the Q value is detected. In combination with a circulator (see below), it is possible to separate the small change from the large total power. Sensitivity of the detection thus increasestremendously. 2 The Continuous Wave EPR Spectrometer In NMR spectroscopy, continuous wave (CW) techniques have been almost completely super- seeded by the more versatile and often more sensitive Fourier transform (FT) techniques dis-cussed in lecture 3. The advantage of FT techniques is to some extent due to the narrowdispersion of resonance frequencies of a given isotope in NMR. In such a situation the wholespectrum can be excited by a short irradiation pulse. In EPR spectroscopy, resonance frequen-cies vary over a much broader range, so that usually only a small part of the spins can be excitedby a pulse. CW EPR measurements are thus often more sensitive than pulse measurements. Fur-thermore, CW EPR is applicable over a broader range of relaxation times and is technically eas-ier to realize, in particular at frequencies larger than 200 GHz. For all these reasons, CW EPRwill not be completely superseeded by FT EPR or other pulse techniques in the near future.ω LC1 LC-----------= Q1 R---L C---- = Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 82.1 The CW EPR Spectrometer The basic setup for CW EPR consists of a microwave source, a magnet with variable field, a resonant cavity, magnetic field modulation coils, a microwave diode as detector, a phase-sensi-tive detector, and a computer for data acquisition, storage, and processing. The cavity substi-tutes for the RLC resonant circuit at microwave frequencies, where conventional coils andcapacitors would have impractical dimensions. A cavity is basically an empty metal box withdimensions on the order of the wavelength of the microwave irradiation (~cm to ~mm range). The experiment is done at fixed microwave frequency to avoid retuning of the cavity during the sweep. Instead of the frequency, the magnetic field is swept, which is equivalent as the reso-nance condition is . (2.28) At a field where the spin system is off resonance, the cavity is tuned to the microwave frequency and the impedance (complex resistance) of the cavity is matched to the impedance of thewaveguide (usually 50 Ω). Under these conditions, no microwave is reflected by the cavity. In other words, all the power that is transmitted from the source to the cavity is absorbed by theresistance R of the resonance circuit. Therefore, no microwave leaves the cavity towards the cir- culator and no power is transmitted from the circulator towards the microwave detector. Whenthe spin system is brought into resonance by changing the magnetic field, it starts to absorb mi-crowave. Thus, the impedance and Q value of the cavity change. This causes reflection of aN SMagnetCirculator microwave sourcemicrowave detectorphase- sensitivedetector sample modulationcoilswaveguide & cavity field modulationsource ν0gµB h----------B0 = Structure determination II (NMR, EPR) , G. Jeschke, WS 2003/04 2. Continuous Wave Magnetic Resonance L2 - 9small part of the microwave power. The circulator guides this reflected microwave power to- wards the detector and a signal arises. The setup described so far would not be very sensitive as affordable microwave detectors (di- odes) are very broadband and therefore gather noise over a broad frequency range. To overcomethis problem the magnetic field is modulated at a low frequency (typically 100 kHz). This caus-es a modulation of the reflected microwave and thus of the detector output current with the samelow frequency. The phase sensitive detector basically measures the amplitude of this signalmodulation. Since most of the microwave noise does not oscillate at this low frequency and withthe same phase as the field modulation, it is suppressed. As a consequence of this kind of detec-tion, changes in the signal are measured that are caused by the field modulation. The signal ac-quired during the field sweep therefore corresponds to the derivative of the absorption line. Although the derivative absorption line looks similar as a dispersion line, it is much narrower. In fact, resolution in derivative absorption spectra is significantly better than in pure absorptionspectra, while resolution in dispersion spectra is significantly worse. field modulationsignal modulation Ω02Tabsorption line first derivative 2 3T2------------ -