Magnetic Resonance Flashcards
How does magnetic field strength change gap between spin states?
As B increases the energy gap increases
@ high B the frequency required is higher
What does magnetic momentum interact with?
Interacts with applied field to give resonance
orbit:
μs~l = γel
e- spin: μs~g0s = 2γes
nuclear spin (with spin angular momentum, uppercase i):
μN= γI
How does value of I change magnetic moment of a particle?
Non-zero spin quantum number I required to have a non-zero spin angular momentum I and therefore a magnetic moment μ
What are the Zeeman Eigenstates?
A single-spin (1/2) has 2 eigenstates of angular momentum on z-axis
ψ = |I,mI>
ψ+/- 1/2 = |1/2, +/- 1/2>
where + is α and - is β
where I (uppercase i) = spin angular momemntum qn
mI = spin proj q.n
What are the eigenstates of the Zeeman eigenstates?
Izψ = mI hbar ψ
Izψ1/2 = +1/2 hbar ψ1/2
Izψ-1/2 = -1/2 hbar ψ-1/2
What is the interaction energy between a field and a spin?
Classic:
E = -γ(hbar)mIB
QM:
H^onespin = -γB0I^z
H^onespinψ results in
Eα = -1/2 (hbar)γB0
Eβ = 1/2 (hbar)γB0
What is the resonance frequency from state α -> β?
ν = γB0/2π
What are allowed transitions between spin states?
ΔmI = +/- 1
What is the difference in population between α and β states?
Nβ/Nα = exp(-ΔE/kT) ~ 1 - ΔE/kT
ΔE small so can make approx, and population difference is minimal
Therefore insensitive technique
How does population difference of spin states effect sensitivity and how is it effected by field?
Net absorption depends on difference in population - therefore NMR insenstitive
Population diff inreases with B or γ, and decreasing temperature
What is the origin of shielding?
Orbiting e- creates magnetic fields which can oppose or reinforce B0
What is difference in energy and transition freq including shielding?
ΔE = hbar γB0 (1-σ)
v = B0γ(1-σ)/2π
What is lenz’ law?
ε = -N(ΔφB/Δt)
Induced emf gives rise to a current whose mag field opposes original change in flux (is why -ve)
where ε is emf, and the next part is change in flux
What is diamagnetic and paramagnetic shielding?
Diamagnetic - opposes B and so causes shielding, due to spherical charge distribution
Paramagnetic - can add to B so deshield, due to non-spherical charge distribution
What is the Lamb formula for diamagnetic shielding?
σd = (μ0e2/3me) ∫ rρ(r) dr
where ρ(r) is e- density
Assumes spherical elec distribution
What are some features of diamagnetic shielding?
Caused by spherically distributed ground state atom/molecules (from s orbitals)
Opposes B0
Proportional to e- density
1/r dependence
What are some features of paramagnetic shielding?
Caused by non-spherical distribution, from p and d e-
Augments B0 (is -ve)
Proportional to 1/ΔE and 1/r3
What is chemical shift?
Measure to relative inner standard (usually TMS)
δ = 106(σref - σ) = 106 (v-vref)/vref
Independent of B0
How does paramagnetic cause different shifts in TM complexes?
Paramagnetic shielding dominates, proportional to 1/ΔE
where ΔE is usually between t2g and eg
As stronger field ligands then smaller chemical shifts
What are some contributions to shielding other than dia/para?
Neighbouring group anisotropy
Ring current effects
Elec effects
Intermolecular interactions - H bonds, solvent interactions, etc.
H is e- poor and ΔE large so must consider these effects
What is anisotropy?
Property of being directionally dependent
Includes p,d, and f orbitals
Opposite is isotropy
What is magnetisation dependent on?
M = χ H
where M is magnetisation, H is magnetic field strength, χ is susceptibility
χ is directionally dependent so magnetisation is too
chemical shielding of nucleus hence depends on it too
What are neighbouring group contirbutions?
Occur due to currents induced in nearby groups of atoms
Effect is shield/deshield nucleus depending on geometry of group/nucleus
How is a point dipole different in each axis?
Bμy = 0
Bμz is where the spin is quantised
What is μpar and μper?
μpar is parallel to symm axis (z), and is larger
μper is perpendicular to symm axis (z)
How do ring currents cause shielding/deshielding?
What is the neighbouring group anisotropy in alkynes?
What is the neighbouring group anisotropy in alkenes?
Protons are deshielded
Above/below is shielded
What is the shielding that occurs due to ring currents from an aromatic ring?
Protons above/below ring shielded
Protons in plane of ring are deshielded
What are electric effects in chemcial shielding?
Charged/polar groups modify diamagnetic and/or paramagnetic currents by polarising local e- distributions and perturbation of ground and excisted state wavefn
What occurs to give splitting due to scalar spin-spin proton coupling?
What is the E difference between two spin-spin coupled atoms?
If two nuclei A and X
E = hJAXmAmX
where J is spin-spin coupling const (can be +ve or -ve)
What is the spin-spin coupling const proportional to?
J α γA γX
γ can be +ve/-ve, so J can be too
What is the splitting for A and X nulcei?
Weak coupling regime
How does J relate to Δv for two nuclei in different regimes?
Weak coupling: JAX «_space; |vx - vA|
Mag equivalent:
JAX > |vx - vA|
Strong coupling:
JAX >= |vx - vA|
What is chemical equivalence?
2x nuclei which are:
1. spins of same isotopic species
2. molec symm which exchanges two spins
What is magnetic equivalence?
Nuclei which are:
1. Chemically equivalent
AND one of
all coupling const of nucleus i and j with all other nuclei of molecule l (this is an L)
Jil (that is an L) = Jjl
OR
No other spins in the molecule
What are the eigenvalues and splitting in weak coupling limit?
Result is two doublets
What are the eigenstates when there are equivalent nuclei?
ΔS = 0 means only two energy differences with the same frequency, gives one peak
What is the hamiltonian for a two spin system (I and S)?
H^ = -ωII^z - -ωSS^z + (2πJ/hbar) I^ . S^
Where first two terms are interaction with field and 3rd is the interaction with eachother
Remember I is uppercase i
What is the dot product of two spins I^.S^?
I^.S^ = I^xS^x + I^yS^y + I^zS^z
= 1/2(I^+S^-) + 1/2(I^-S^+) + I^zS^z
How do you show the ket of a two spin system?
First term in ket is I and second is S
States include:
|αIαS>
|αIβS>
|βIαS>
|βIβS>
How do operators I^z and S^z act on kets?
Where the ket doesnt matter
I^z|> = +1/2 | >
S^z|> = -1/2 | >
How do the operators I^+/- and S^+/- work on different kets?
I^+|αI> = 0
I^-|αI> = |βI>
S^+|βS> = |αS>
S^-|βS> = 0
What are the eigenstates in the strong coupling case?
|4> = |βIβS>
|3> = -sinθ|αIβS> + cosθ|βIαS>
|2> = cosθ|αIβS> + sinθ|βIαS>
|1> = αIαS>
What is the coupling parameter?
tan2θ = J/δ
when θ=0 then is no coupling and when θ=3π/4
What are the transitions seen in strong (or general) coupling?
Gives roofing
What is the permutation operator and how is used for mag equivalent?
P^12 the hamiltonian for two mag equivalent nuclei (1&2) will remain unchanged
Exchange labels of two nuclei and then see what occurs
What is the fermi contact interaction?
Is the origin of spin couplings
Mag interaction between an e- and an atomic nucleus
Crucially dependent upon s-electron char of ground and 1st excited state
How is 3J dependent on dihedral angle?
3J = A + Bcosθ + Ccos2θ
This is karplus relationship
How is the fermi contact interaction dependent on mag field and tumbling?
Mag field - not affected by strength or direction, so independent of spec frequency
Isotropic - no dependence on molec tumbling
What is the range of 1J C-H coupling?
100-250 Hz
What does through space coupling occur in solution and solid?
Solution: anisotropic quantity averages out due to tumbling
Bμx =! Bμy =! Bμz
Solids: doesn’t average out, leads to
∫ (3cos2θ-1)sinθdθ
for (3cos2θ-1) = 0, then θ=54.7 which gives a singlet
What does the notation KAX denote?
Splitting in spectrum of X caused by dipolar coupling to A
What does the fourier transform do?
Converts spectra from time to frequency domain
Results in product function which in peaks rather than a progression
How does the life-time of a signal determine the signal broadness?
Signal broadens as lifetime decreases
What are the advantages of fourier-transform NMR?
- Increased sensitivity
- Time saved means more spectra can be done and better signal:noise
- Allows for using pulse-sequences
What occurs during a pulse to spin states?
Phases of spins are random initially random
Mag field aligns the spins
What is a rotating frame?
View spins from a frame that rotates with the field
Means the view is an apparent static field
Why is a rotating frame required?
Freq of applied RF-field B1(t) close to spin resonance freq
The effect of the pulse tilts magnetisation vector away from z axis
B1(t) oscillates itself so rotating frame to view B1(t) required
How does the rotating frame work?
Detail linear field as sum of counter-rotating circularly polarised components
Only component that rotates in same sense as Larmor preccesion of the spins is retained
What occurs to magnetisation during a pulse in the rotating frame?
Rotates about z-axis with ang freq ωrf, a field B1
What is nutation freq?
Angular freq of precession
ω1 = -γB1
What is the angle through which M rotates?
β = -ω1tp
where tp is the length of pulse
What is a 90-degree pulse?
What is a 180-degree pulse?
How do spins cause coherence?
Coherent EM radiation induces coherence amongst spins and causes orientations of individual mag moments in the x-y plane to not be random
What is relaxation in NMR?
The population after a pulse returning to thermal eqm
How does magnetisation increase with time?
Mz(t) = Meq (1 - exp[-(t-ton)/T1]
where T1 is the spin-lattice or longitudinal relaxation time in same direction as field (z)
How does magnetisation decreases to eqm?
Mz(t) = Meq (1 - exp[-(t-toff)/T1]
where T1 is the spin-lattice or longitudinal relaxation time in same direction as field (z)
What is the dependence of emission (relaxation) in NMR?
From Einstein coefficient:
A α v3
Therefore very slow for NMR
How does magnetic interactions effect spin-lattice relaxation?
Instant local mag field might induce a radiationless transition if @ correct freq
Therefore can cause spin-lattice relaxation
What is rotational correlation time, τc?
Time taken for root-mean-square deflection of the molecules to be about 1 radian
t «_space;τc - close to original position
t»_space; τc - lost all memory of position
What is τc-1?
Root-mean square rotational frequency to reach 1 radian
What is spectral density function, J(ω)?
J(ω) = 2τc/(1 + ω2τc2)
Chance that molecular tumbling leads to right freq to make a transition between energy levels
What is the plot of spectral density function and tumbling rate?
Intermediate tumbling causes fastest relaxation
What does spin-lattice relaxation time, T1, depend on?
Prob that local mag fields are oscillating at resonant NMR freq J(ω)
What is spin-spin relaxation time, T2?
Time quantifies rate of decay of manetization within xy plane
Doesnt affect total amount of z-magnetization
How do T1 and T2 depend on τc?
Same when fast tumbling wrt frequency
T1 >= T2
As full magnetization along z cannot be recovered unless all phase coherence has gone with T2
What is observed orbital-wise when I > 1/2?
Nucleus possesses an electric quadrupolar moment additional to magnetic dipole moment
Leads to distribution of nuclear charge being ellipsoidal, so elec energy varies with orientation
What do electric quadrupoles interact with?
Interacts with field gradients
NOT with uniform elec field
What occurs to quadrupolar interaction in high symm?
High symm environ causes electric field gradients to cancel
Means no net quad interaction, so no effect seen
How does quadrupolar effect spins?
Induces spin relaxation so causes line broadening
Loses multiplet splitting
How can you measure spin-lattice relaxation?
Inversion-recovery experiment
What is the pulse sequence in an inversion-recovery experiment?
Can measure τ1,2,3,4
Plot NMR signal vs τ
How can you get T1 from an inversion-recovery measurement?
Plot ln[I - I(τ)] against τ
where I is the fully relaxed signal intensity
Then use following to get T1:
Mz(τ) = M0(1-2exp[-τ/T1])
How can you measure the spin-spin relaxation time (T2)?
Spin-echo experiment
What is the pulse sequence in a spin-echo experiment?
What is the intensity of signal from a spin-echo experiment?
I(2τ) = I(0) exp(-2τ/T2)
How does relaxation change the echo amplitude?
Random mag fields destroy phase coherence (which due to random molec motion)
Not refocused by p pulse
What is the equation for line broadening according to spin-lattice relaxation?
Δv = 1/πT1
What is the line broadening due to spin-spin relaxation?
Δv = 1/πT2
What occurs to broadening at slow exchange rate?
Δv = kex = 1/πτ
2 separate sharp peaks observed
What occurs to line broadening in intermediate exchange?
kex = πδv/Sqrt[2] ~ 2.2δv
A broad peak is observed (two lines merged)
What occurs to line broadening in fast exchange?
Δv = π(δv)2/2kex = (π/2)(δv)2τ
Gives one sharp line (for two environments)
