Martin-Luther-Universität Halle-Wittenberg FB Physik Advanced Lab Course V27: RF Spectroscopy ) Electron spin resonance (ESR) Investigate the resonance behaviour of two coupled LC circuits (an active rf oscillator closely coupled to a passive circuit). Calculate the magnetic flux density of a pair of Helmholt coils along the coil axis. Use I =. A as intensity of current, n = 32 turns per coil, and R = 6.8 cm as (average) radius of the coils. Measure the magnetic flux density along the coil axis and compare the experimental results with the numerical ones obtained in the preceding task. Estimate experimentally the Landé factor g of the unpaired electron in the diphenyl picrylhydrail (DPPH) molecule and compare it with the value which is expected theoretically. Give the maximum energy splitting of this free radical which can be obtained with the given experimental setup. 2) Nuclear Magnetic Resonance (NMR; here cw mode only) Calculate the magnetic flux densities required for proton ( H) and 9F resonance frequencies of 9 MH, respectively. Find the proton magnetic resonance of a glycerol sample at 9 MH by variation of the stationary magnetic field (fast sweep mode) and explain its shape. (Why there are oscillations which lead to another shape than that of a simple minimum observed in the ESR case?) Display the resonance curve in the slow sweep mode. Explain why the shape changes (no more oscillations are observed). Find the proton resonances for water and polystyrene. Explain possible changes in the shape of the resonance curves compared to that of the glycerol signal. Find the 9 F resonance of Teflon (polytetrafluorethylene, PTFE). Literature: P.W. Atkins: Physikalische Chemie H. Friebolin, Nuclear magnetic resonance Bergmann-Schaefer: Experimentalphysik VI H. Kumany: Festkörperspektroskopie
Questions for testing your knowledge: Explain the phenomenon called resonance. Give the quantiation rules for angular momentum. Explain the principle of Magnetic Resonance with a classical vector model as well as at quantum-mechanical basis. Why ESR and still more NMR give a rather weak signal compared to that of other spectroscopic methods? What is the meaning of cw spectroscopy and pulse spectroscopy? What is the advantage of the latter? Explain longitudinal and transverse relaxation. Which phenomenon is characteried by the g factor? Why we are seeking for best homogeneity of the magnetic field in NMR spectroscopy? What happens if the field would be inhomogeneous? Are there applications for the latter case?
. Magnetic resonance Basics Electron spin resonance (ESR) and nuclear magnetic resonance (NMR) spectroscopy are research methods which belong to the field of rf spectroscopy. They use of the behaviour of the magnetic dipoles of electrons and nuclei in magnetic fields for investigations of structure and dynamics of molecules or solids. The phenomenon of magnetic resonance can be explained in two ways: Classically or quantum-mechanically. The classical approach can be used only if the total magnetiation is considered which can be represented by a macroscopic vector (Examples: Effect of rf pulses on the magnetiation, description of certain kind of spin echoes). Once individual features of the spins are important, as for example the coupling between neighboured spins, the quantum-mechanical description is required. First of all, in chapter the common features of are introduced. Following this specific aspects of ESR and NMR are mentioned in chapters 2 and 3, respectively.. Quantum-mechanical description: spins in the magnetic field Magnetic resonance is based on the magnetic moments belonging to electrons or nuclei which posess a spin. (Therefore, paired electrons in a common orbital compensating their spin or nuclei without spin cannot be detected by this method.) The angular momentum p is quantied with respect to magnitude as well as orientation: p = II ( + ) ; I= ; 3 2 ; ; 2 ;... p = m ; m = -I;...; I -; I () I is the spin quantum number, and m is the magnetic quantum number. - - - I = /2 I = I = 3/2 Fig. : Possible angular-momentum vectors for various spin quantum numbers The angular momentum is connected to a magnetic moment µ. The quotient between them is called gyromagnetic ratio γ: µ γp µ γp γm If an magnetic field (flux density B ) is applied, each m belongs to a discrete energy value (this is also the reason for the Zeeman effect observed in optical spectroscopy): (2) E µ B γm m B (3)
m=-/2 +/2 B = B > m=- B = B > m=-3/2 -/2 +/2 +3/2 B = B > Fig.2: Energy levels of spins I = /2,, and 3/2 with/without magnetic field B. Therefore we obtain (2I+) energy levels. The energy differences between the neighbouring levels m and m Δ E γb mm, belong to the transition frequency ω ΔE mm, γb bw. f 2π This is the central equation of magnetic resonance. γ B (Larmor equation) (4) It means: If a system of spins is exposed to a magnetic field, and if an electromagnetic field is irradiated the frequency of which coincides with the Larmor frequency, transitions between the levels can be induced. Transitions from lower to higher levels lead to absorption of energy, in the reverse direction energy will be emitted. If the number of transitions are equal for both directions (because both levels are equally populated; this situation is called saturation ), emission and absorption compensate each another. In this case a macroscopic instrument cannot receive any signal. The larger the population difference is the more intensive appears the absorption or emission to a macroscopic viewer..2 Classical description: Behaviour of the magnetiation The magnetiation of the sample (sum of all magnetic moments per unit volume) can be treated by classical electrodynamics because it is a macroscopic quantity: M = å µ V n n If the magnetiation would be non parallel with respect to the magnetic field, a torque occurs which tries to rotate the magnetiation into the field direction. However, the magnetic moments are non-detachably connected to the angular momentum (see above). Conservation of angular momentum then requires a precession motion, i.e. the magnetiation vector rotates at the surface of a cone the axis of which is the magnetic field.
B M This verbal conclusion will be supported in the following by a more mathematical treatment. The torque D of the field onto the total dipole is simultaneously the timederivative of the angular momentum: Because dp D µ n B (5) dt P pn μi it follows after division by V and multiplication by γ γ dm γ MB (6) dt This is the equation of motion of M in a non-moving frame (so-called lab frame). The kind of motion can be identified by means of a transformation into another frame which ro tates aroung the magnetic field; here for time derivatives is generally valid: dm dt LKS dm dt ω M (7) RKS ω is the angular velocity of the rotating frame. Inserting eq. (7) into eq. (6) we obtain the equation of motion in the rotating frame: dm dt RKS ω γ MB γ Now we can characterie the motion of M : M is constant (i.e. its time derivative is ero) in a frame which rotates with an angular velocity equal to the Larmor frequency: ω ω γ B. Therefore the magnetiation performs in the lab frame a motion at the surface of a cone with the angular velocity ω γ B. Within this classical framework, the resonance phenomenon is explained by the coincidence of the irradiated radio frequency with the precession frequency of the magnetiation..3 Basic experiments Cw (continous wave) method: Frequency of field strength are varied ( sweep ). The intensity of the absorption is recorded. Approaching the resonance increasing absorption of rf energy is observed. In case of too fast changes of f or B the decaying resonance signal interferes with the
irradiated radio frequency. The resonance curve is then superimposed with a decaying oscillation (so-called wiggles). Fourier spectroscopy: The spin system is excited by short pulse at a broad frequency band. The response of the spin system is recorded and will be converted to a spectrum by Fourier transformation. This method is particularly advantageous because of the short time requirement (seconds instead of minutes per spectrum). This is important in case of weak signals; here a large number of single signals have to be akkumulated. Moreover the most of modern experiments would not be possible in cw mode. Connection between spectrum and pulse response: Let S(ω) the frequency spectrum and F(t) the response of the spin system on pulse excitation (free induction decay; FID). S(ω) is the Fourier transform of F(t): exp( )d S ω F t iωt t Inverse Fourier transformation: F t S exp( i t)d 2 ω ω ω π In our experiment V27 we regard the cw method only. (Pulsed NMR is subject of the experiment V26.) Even though the most NMR experiments nowadays use pulse methods, cw method is nevertheless a good tool for demonstrating this kind of spectroscopy..4 Relaxation If the magnetiation was deflected out of its equilibrium state, it seeks to return again to equilibrium. This phenomenon is called magnetic relaxation. Most important forms are: Longitudinal relaxation or spin-lattice relaxation: Recovery of the longitudinal component of the magnetiation; time constant T Transversal relaxation or spin-spin relaxation: Decay of the transverse components of the non-equilibrium magnetiation; time constant T 2. Often the relaxation can be described by exponential functions: t t M M exp ; M M exp T T M fulfills Bloch s equations: In these cases 2 d M M M x / y Mx / y d M ; dt T dt T 2
2. ESR The gyromagnetic ratio can be represented for electrons by γ g μ /, where µ B B = 9,2742-24 J/T is Bohr s magneton and g is the LANDÉ-Faktor. The latter arises from the fact that spin S and orbital angular momentum L yield different contributions to the total magnetic moment. It can be calculated as J J S S LL g 2 J J (J total spin.) Special cases: S =, then J = L and it follows g =. L =, then J = S and it follows g = 2 (for example free electrons or s elektrons). S, L : g will be a rational number different from. Together with eq. (4) we obtain µ B f g B. (8) h In diphenyl-picrylhydrayl radical molecules, some of the nitrogen atoms are not able to use all outer electrons for bonds. The spatial arrangement of the atoms is responsible for that. Hence one 2s electron remains unpaired; this gives the possibility to receive a ESR signal from that substance. This compound serves for standard and reference experiments in ESR inverstigations and will be used here also. 3. NMR Atomic nuclei which belong to different isotopes possess different gyromagnetic ratios by what the resonance frequencies at same field strength vary over a wide range, see table below. A particular problem is the small sensitivity: The signal intensity is proportional to the difference in populations of the levels (see above). This can be described quantitatively by the Boltmann distribution. Because the energy differences between NMR levels are very small, the levels are nearly equally populated leading to weak signals. Furthermore, the smaller γ the smaller is E and the smaller is the intensity. Other reductons of the signal arise from low natural abundance of some isotopes (for example 3 C or 5 N).
gyromagnetic ratio Resonance frequency [MH] at Isotope Spin in 6 s - T - B =.47 T B =2.35 T B =9.4 T H /2 267,37 2,, 4, 2 H 4,95 3,63 5,35 6,26 2 C - - - - 3 C /2 67,23 5,29 25,44,57 4 N 9,3,445 7,224 28,896 5 N /2 27,9 2,27,33 4,532 6 O - - - - 7 O 5/2 36,25 2,72 3,557 54,228 9 F /2 25,53 8,85 94,77 376,38 23 Na 3/2 62,7 4,69 23,45 93,84 27 Al 5/2 69,67 52, 26,57 4,228 29 Si /2 53, 3,96 9,865 79,46 3 P /2 8,23 8,96 4,48 6,924 Tab.: Spin, gyromagnetic ratio and resonance frequencies of some nuclei The great variety of applications of NMR relies on spectral parameters which cannot be subject of this experiment. Chemical shift and spin-spin coupling constants belong to these parameters. They contain information about the inner structure of molecules and give the possibility of detailed investigations in chemical and physical research.