Magnetic Resonance Spectroscopy ( )

Transcription

1 Magnetic Resonance Spectroscopy In our discussion of spectroscopy, we have shown that absorption of E.M. radiation occurs on resonance: When the frequency of applied E.M. field matches the energy splitting between two quantum states. Magnetic resonance differs from these other methods in the sense that we need to immerse the same in a magnetic field in order to see the levels that we probe with an external (rf or µwave) field. (Two fields: Static magnetic and E.M.) We will be probing the energy levels associated with the spin angular momentum of nuclei and electrons: NMR--nuclear magnetic resonance and ESR/EPR--electron spin resonance. Angular momentum: In our treatment of rotational energy levels, we said that the energy levels depended on the rotational angular momentum, L, which was quantized: L = J J+ J = 0,, rot. quant. number Degeneracy of J was ( m J = 0,, ±J) ( J +) We related L to the energy levels L Erot = BJ J+ I Actually, all angular momentum is quantized. If a particle can spin, it has A.M. and quantized E levels. In particular, we also have to be concerned with the spin of individual nuclei and electrons Lecture Notes: Magnetic Resonance Spectroscopy Page

2 You already know that electrons have ORBITAL angular momentum M = + = 0,, orbital angular momentum quantum number degeneracy of orbitals: + from m =, + magnetic quantum number m represents the quantization of the components of M : Projection of M onto ˆ z axis: M Z = m + z M = M = (How we choose ˆ z doesn t matter until we apply a magnetic field.) 0 Now, the angular momentum that we are concerned with is: Electron Spin Angular Momentum = + s: electron spin quantum number = ½ S s s for each unpaired e S z = m s m: s ± ( S, S +,, + S) one unpaired e Two paired electrons: s = 0. Two unpaired electrons (triplet): s = Lecture Notes: Magnetic Resonance Spectroscopy Page

3 Nuclear spin angular momentum I = II+ I: nuclear spin quantum number I z = m I m I : I, I +,, I What is I? Each proton/neutron has a spin quantum number of /. Spin of many nucleons add up vectorily to give I. Spins pair up so that even number of spins are paired I = 0.. For even number of protons plus even number of neutrons: I = 0. 6 C, O. For one unpaired nucleon I = m I = ± degeneracy I + = H, 3 C, 5 N So the proton and electron are similar both spin particles. We ll talk about these two particles more specifically 3. Two unpaired nucleons I = m I = 0,± H, 4 N 5.33 Lecture Notes: Magnetic Resonance Spectroscopy Page 3

4 For spin ½ particles (I or S = ½) there are two degenerate energy states. When you put these in a magnetic field you get a splitting. A low energy state aligned with the field, and A high energy state aligned against the field Classical picture: Think of our electron or nucleus as a charged particle with angular momentum, M. A circulating charge produces a magnetic field. This charge possesses a magnetic dipole moment µ that can be affected by an applied magnetic field. m,q M µ M = m( v r) µ= Q( v r ) /c The dipole lies along M and the strength of µ is proportional to M : v r Q µ = M γ M mc γ is the gyromagnetic ratio Quantum: For electrons: e µ e = g e S m c = γ e e S g factor (.003) For nuclei: + e µ N = g m c =γ N N I N I g N = 5.6 for H g is a relativistic quantum mechanical correction Lecture Notes: Magnetic Resonance Spectroscopy Page 4

5 If we immerse this system in a magnetic field, B, which is oriented along the z axis The interaction potential is Eint = µ B The spins align along the magnetic field: Eint = µ zb Since µ z = γeszand µ =γ z N I z, using Sz = msand Iz = mi we have For electrons: Eint = γe msb ms = ± For nuclei: Eint = γn mib mi = ± So, as we increase the magnetic field strength, the two energy levels originally degenerate split, one increasing in energy and one decreasing in energy. This is known as the Zeeman effect. Nuclei Electrons Eint m I = / m s = +/ 0 E E = γn B E e =γ B Selection Rules: m s = ± m I = ± 0 B m I = +/ m s = / Now we have a system that can absorb E.M. radiation on resonance: E=hν ν is the applied frequency (in the radio frequency range) Frequency domain spectrometer: Typically sweep B and hold ν constant. γb ν= (Larmor frequency) π 5.33 Lecture Notes: Magnetic Resonance Spectroscopy Page 5

6 Typical numbers: Nuclear Magnetic Resonance Electron Spin Resonance µ H = (± ½) erg/gauss µ e = (± ½) erg/gauss for B = 0 kg: ν = 4.6 MHz ν = 8 GHz Typical: 300 MHz/70 kg 9.5 GHz/3.4 kg Typical population difference N = (Ν N + )/(Ν + N + ) 0.008% The Chemical Shift Thus far you would think that all H absorb at same frequency. Yet, in practice different nuclei absorb at different frequencies. The resonance frequency depends on the effective magnetic field that a proton feels. This can differ for different types of H due to local electron currents that counteract the applied field. Shielding eff app B = B σ σ: screening constant typically 0-6 Eint 0 E No shielding With shielding (shifted to lower ν) E = µ B = γ mb int N eff N I app ( σ) B app γ ν= σ π ( ) 0 Bapp Shift of frequency due to screening: Chemical Shift Typically you measure the chemical shift due to screening relative to a standard (TMS) Lecture Notes: Magnetic Resonance Spectroscopy Page 6

7 γb ν ν = σ σ π app i ref ref i 6 i ref Measure positions in δ: ppm. δ = ( σ σ ) 0 ν ν i ref i νref The magnitude of the shielding depends on how the motion of electrons modifies the local field. (Interaction between field and electron angular momentum) Qualitatively, proton NMR spectra can be interpreted by considering electronegativity of bound functional groups. Greater E.N. draws electrons away from H, lowering the resonance field and giving a larger δ. Other nuclei have larger chemical shifts because there are more electrons. Couplings between Nuclear and Electron Spins Based on the effects of screening (chemical shift), we would expect one line for each nucleus in an NMR spectrum. Actually there is usually additional structure, with each line split into several others. (Both in NMR and ESR) Splittings arise when different magnetic spins on nuclei and/or electrons interact with each other. The interactions change the spin energies to give new lines. Understanding the interaction allows you to reveal structural information such as connectivity. The couplings between two magnetic dipoles can be written as Ecoupling µ µ Couplings between unpaired electrons (a,b,c ) and nuclear spins (i,j,k ) are: nuclear-nuclear ( J ) coupling (NMR) N N electron-electron ( fine ) coupling µ e( a) µ e( b) = γeγesa Sb electron-nuclear ( hyperfine ) coupling (ESR) µ µ = γ γ µ i µ j = γ i γ j I I N N i j a i i S I e N e N a i 5.33 Lecture Notes: Magnetic Resonance Spectroscopy Page 7

8 NMR Interaction between protons: () () E = h J m i m j m = ± Jcoupling ij I I J: nuclear spin coupling constant; typically -0 Hz. I Total nuclear interaction energy with spins: E = E + E int nuclear Zeeman J coupling () () () () ν = B γ i m i σ + J m i m j i app N I i ij I I i j ESR Dominant effect: hyperfine coupling coupling bj s E = a m b m j I a: hyperfine coupling constant; typically -00 MHz Total interaction energy of electron and nuclear spins: E = E + E + E + E int electron Zeeman hyperfine coupling nuclear Zeeman J coupling ν = B γ m b σ + a m b m i + b app e s b bj s I i,j,... Sum of electron Zeeman effect and hyperfine interaction. Nuclear Zeeman energy and J coupling is relatively small in comparison. ESR Example: H atom proton and electron Eint = Eelectron Zeeman + Ehyperfine coupling + Enuclear Zeeman = Bγ m + a m m e s s I 5.33 Lecture Notes: Magnetic Resonance Spectroscopy Page 8

9 With no coupling: two states, with splitting as before. With coupling: ms =± ; mi =± and mm s I = ± 4 Eint ms +/ +/ mi +/ / /4 /4 E ms= mi=0 / / / +/ 0 B B B At zero field: two states with energies E=+ a and 4 E= a 4 (These correspond to F= 0 and F=, where F= S+ I is the total spin angular momentum). With increasing field, degenerate states split. By sweeping field with constant resonance frequency E=hν, we see resonances at two fields, B and B. E a B = γ e E+ a and B = γe so that the field splitting gives the hyperfine coupling: a = γ ( B B ) e Alternatively, you could stay in a fixed field B and sweep the rf frequency E, and you would observe two resonances: E/ =γ B± a e 5.33 Lecture Notes: Magnetic Resonance Spectroscopy Page 9

10 Again the frequency splitting gives the hyperfine coupling. Splitting Patterns For multiple nuclei coupled to an unpaired electron, we can expect each hyperfine interaction to split the remaining transition into a pair of peaks split by a. The overall spectrum can be predicted diagrammatically by a pattern of splittings in which one electron resonance is sequentially split in frequency by each hyperfine coupling interaction: Two inequivalent couplings a and a : Two equivalent couplings (a = a ): 5.33 Lecture Notes: Magnetic Resonance Spectroscopy Page 0