Fundamentals of Magnetic Resonance Flashcards
What is nuclear spin?
Nuclei have an intrinsic total angular momentum - the operator for nuclear spin is a bold captial I with a hat to show it is a vector and an operator
The nuclei spin quantum number (total angular momentum quantum no.) is I with units bof angular momentum per nucleus
Total angular momentum is measured in units of ħ (h/2π)
What is Planck’s constant?
a fundamental quantity that is equal to the energy per unit frequency of a quantum of electromagnetic radiation
E = hv = ħω
h = 6.626 x 10-34 J/Hz
ħ = h/2π
What values can nuclear spin take?
Can be an integar or half integar
When protons = even and neutrons = even and mass number is even the spin is 0
When protons = odd and neutrons = even and mass no = odd the spin is half integar
When protons = even and neutrons = odd and mass no. = odd the spin is half intear
When protons = odd, neutrons = odd and mass no. = even the spin is an integar value
What is total angular momentum?
A vector
I = {Ix, Iy, Iz}
What are the eigenvalues and eigenfunctions of Iz?
Izψm = mħψm
ψm - eienfunction
mħ - eigenvalue
m - quantum number for Iz
The number of values of m are 2I + 1
the values of m are {-I, -I+1 …… +I}
What is the nuclear magnetic momentum?
Nuclei with nuclear spin have a nuclear magnetic moment μ (with a hat)
μ = {μx, μy, μz) - vector operator
μ is proportional to anuglar momentum: μ = γI
γ is the constant of proportionality - gyromanetic ratio which is a property of the nucleus measured in rad s-1 T-1 (or MHzT-1 if it is γ/2π)
What is the Zeeman interaction?
The interaction between the nuclear magnetic moment and anyapplied magnetic field B0
The energy of this interaction is described by the Zeeman Hamiltonian Hz
- H*z = -μ x B = -γI x B = -γB0Iz
- H*z is the Zeeman Hamiltonian, μ is the magnetic moment, B is the field vector, I is the angular momentum, γ is the gyromanetic ratio, B0 is the applied magnetic field defined in the z direction, Iz is the z component of the angular momentum
How do we find the eigenvalues and eigenfunctions of the Zeeman Hamiltonian?
Solve the time-independent Schrodinger equation: Hzψ = Eψ
- γB0mħψ = Eψ
- γB0mħψm = Emψm
Eigenfunctions: ψm
Eigenvalues: Em = -γB0mħ
What is the difference in energy between the two spin states?
|ΔE| = | (-1/2γB0ħ) - (1/2γB0ħ) | = γB0ħ
What is the frequency of energy required to make a transition between ther spin states?
ΔE = γB0ħ
ΔE = ħω
.’. γB0ħ = ħω - cancel the ħ
γB0 = ω (rad s-1)
This is the Larmor frequency
ω is different for different nuclei in the same field
ω is the same for higher spin nuclei
What is the selection rule for NMR and why?
Δm = ±1
so that the Larmor frequency is the same for all values of I
The Larmor frequency of 1H is 42.577 (γ) x 9.4 (B0) = 400 MHz
Why do nuclei of the same type in different chemical environments have different Larmor frequencies (why does chemical shift change)?
The electron density around the nucleus generates a weak induced magnetic field at the nucleus
Bind = -σiB0
Bind is the induced field, σi is the isotropic shielding constant and B0 is proportional to the main field
The -ve sin means the induced field is in the opposite direction to the main field
What is the effective field felt by each nucleus?
The sum of the applied field, B0, and the induced field, Bind.
Beff = B0 + Bind = (1 - σi)B0
The larmor frequency is proportional to the local field at the nucleus - Beff
ω = γBeff = γ (1 - σ) B0 = (1 - σi) ω0 where ω0 is the shift in the larmor frequency
Why do we use ppm?
The isotropic chemical shift parameter σi is a chemical property but the Larmor frequency that is measured is an experimental property because it depends on the exact B0
ppm is a standard way to describe chemical shifts that is independant of the applied field
ppm is a relative scale - a fraction of the Larmor frequency
How do we define chemical shift in ppm?
For 1H and 13C chemical shift is defined in terms of TMS as δ = 0
All chemical shift is measured relative to a reference compound
How do you convert ppm to frequency in Hz?
Multiply the chemical shift by the reference frequency in MHz
relative frequency = chemical shift (ppm) x Larmor frequency (MHz)
(v - vref) = δ x vref (MHz)
How do you calculate the difference between two peaks in Hz?
Take the differencebetween peaks in ppm and multiply by the reference frequency
Δv = (δ1 - δ2) x vref (MHz)
What is the difference between an NMR experiment and an MRI test?
In a normal NMr experiment a very homogenous magnetic field is used to make sure all nuclei experience the same field and small differences in the Larmor frequency due to chemical shift can be detected
In MRIs we deliberately apply a magnetic field that varies as a function of position - magnetic field radient Gx and has units of T m-1
Btotal(x) = B0 + Gx
What do we obtain if a magnetic field gradient is used on an NMR experiment?
An image of the sample is obtained because the Larmor frequency is directly proportional to the position of each nucleus
ω(x) = γBtotal(x) = ω0 + γGxx
where ω is the Larmor frequency which depends on position and x is the position
What is the probability of a nucleus being in a spin state ψm?
How do we determine how many nuclei are in each spin state?
Using Boltzmann statistics
Nuclei populate both energy states almost equally
What is nuclear polarisation, P?
The difference in population between the spins states
What is the energy level picture for NMR?
useful way to think about frequencies in an NMR spectrum
Cannot think about NMR in terms of nuclei in the ground state being excited to a higher state then relaxing to ground state
Transition between the states are continuous even in equilibrium
There is a slightly higher probability for transition from -1/2 to 1/2 so a equilibrium there is a population imbalance with an excess of spins in the low energy state
Is NMR relaxation provide the NMR signal?
Relaxation estabilishes the population difference but does not gve rise to the NMR signal
Not enough absorption for detection- requires resonance amplification
What is a vector and how are they defined?
Quantities that have a magnitude and a direction - we define a vector in terms of a coordinate system
Which coordinate systems can be used for defining vectors?
Cartesian and cylindrical
In cartesian coordinates the mantgetic moment has components along the x, y and z
In cylindrical the magnetic moment is divided into two components μz, μx,y and a phase φ
μz = μz - longitudinal coordinates
μx = μx,ycosφ
μy = μx,ysinφ
How does the magnetic moment interact with the magnetic field?
The magnetic moment interacts with the applied magnetic field B0.
This interaction generates a torque (τ) - tisting force acting perpendicular to the gravitational field
τ = μ x B = γI x B
Where does the torque come from?
The torque comes from the cross product between the magnetic moment and the field
It acts in the direction that is perpendicular to both the magnetic moment and the field
This causes the nuclear magnetic moment to rotate or process around the field
What is the frequency of the procession of the magnetic moment?
The frequency of the procession is the Larmor frequency
ω = γB0
What happens to the nuclear spins in the absence of a magnetic field?
There are millions of nuclei in a sample
THe associated magnetic moments are randomly oriented
The Zeeman states are degenerate with no preference for any particular direction
There is no net magnetisation - the magnetic moments cancel each other out
What happens to the nuclei when a magnetic field is applied?
Interaction between the magnetic moments and magnetic field causes procession
Slight preference to point with the field
In cylindrical coordinates: Component along the z axis is Mz - sum of all magnetic moments just along B0
Component in the xy plane: Mxy with phase φ - sum of magnetic moments in the xy plane
How do we find the net magnetisation along the z axis?
There is a slight preference for magnetic moments to be aligned with the magnetic field rather than against it due to Zeeman interactions
The excess of spins in low energy state is the polarisation
What is the equation for total magnetisation along the z axis in general?
What is transverse magnetisation?
The magnetisation in the xy plane
Has boith magnitude, Mxy, and direction (phase: φ)
Each magnetic moment precesses around the magnetic field at the Larmor frequency
What is the phase of each manetic moment in transverse magnetisation?
Each magnetic moment is a vector changing with time
φ(t) = φ(0) + ωt
where φ(0) is the initial phase at t=0 and ωt is the procession phase that changes with time
How is the magnetisation vector written in terms of exponentials and complex numbers?
M0exp(iφ) = M0 cosφ + i M0 sin φ
where M0 is how big the circle is, i is the imaginary component and φ is the ohase
Mxy = M0exp (i φ(t)) = M0exp(i (φ0 + ωt) = M0exp(i φ0) exp(i ωt)
where M0exp(i φ0) is the initial phase and exp(i ωt) is the time dependent part (procession)
If all the nuclei process at the same frequency how they add up depends on their ohase - the direction they point in at a point in time t=0
What is phase coherence?
At equilibrium entropy tells us here is no reason for the nuclei to process in phase with one another
If we add up all the nuclei with random phases on average they cancel out
There can only be net magnetisation in the xy plane if there is phase coherenec
Phase coherence is when the nuclear magnetic moments precess in sync with one another with both the same frequency and the same initial phase
Which direction does the total magnetisation lie in at equilibrium?
The sum of all the magnetic moments is along the z axis (same direction as field
Mz = M0
Mxy = 0
What happens if the magnetisation vector is rotating?
It will generate an alternating current in an approximately oriented coil of wire through a proces called induction
The current will oscillate at the same frequency as the rotaring magnetisation vector
The oscillating voltae provides a signal
How does rotating the magnetisation vector affect the nuclei?
Magnetic moments are precessing in sync so that they add up in the transverse plane
The longitudinal magnetisation is destroyed and so this must mean the energy levels are populated equally
What is the rotating frame?
A reference frame used to underrstand how the magnetisation vector is moved into the xy plane
The z axis rotates at the larmor frquency so the magnetisation vector is static - does not precess
x’, y’ and z’ describe the axes of this frame
The magnetisation vector doe snot rotate so it does not experience the B0 so B0 disappears- This only happens when frame is completely in sync with Larmor precession
What does the rotating frame look like at thermal equilibrium
How do we induce transitions between spin states?
Need to irradiate at the larmor frequency which is in the radio region
We apply a magnetic field, B1, that oscillates at the Larmor frequency and is perpendicular to B0 -B1 is much smaller than B0
The magnetisation vector precesses around B1 - rotating frequency = gyromagnetic ratio multiplues by the the field B1
If the pulse is applied for the a given time we can rotate the M0 into the xy plane
How do we generate an RF pulse?
An oscillating voltage passed through a coil of waire will generate an oscillating magnetic field
What happens to the magentisation vector after the pulse?
After the pulse the magnetisation vector is in the transverse plane
In the rotating plane it is now static
In the lab frame is precesses around B0, main field
What is Free induction decay?
The coil of wire used to generate the RF pulse is used to detect the precession and aquire the NMR signal - this can only be done in the transverse plane as it is in the plane of the wire
This is free (detect signal after the RF pulse) Induction (detection method) Decay (detection method)
What affects the amplitude of the NMR signal, S0?
S0 is proportional to:
- amplitude of precessing magnetisation Mxy
Mxy = Mzsinα where Mz is the initial magnetisation along the z axis and α is the rotation angle
- precession frequency: Larmor frequency: ω0 = γB0
S0 ∝ ω0 x Mxy = (γB0) x Mzsinα
Induction process is more efficient at higher frequencies
What are the factors affect the size of the NMR signal?
N = number of nuclei
γ3 - gyromagnetic ratio cubed - polarisation, size magnetic moment, larmor frequency
B02 - magnetic field squared - polarisation, larmor frequency
sinα - sine iof the RF pulse angke - experimental
T-1 - inverse of temp - experimental parameter
What is NMR receptivity?
a measure of how easy it is to detect the NMR signal of a given nucleus
quoted relative to another nucleus - usually 1H or 13C for the same field (B0) and temeprature
We compare the same field, temperature and flip angle:
S0 ∝ Nγ3 I(I +1) where N is the no. spins (natural isotopic abundance) and I is the spin quantum number
What affects receptivity?
Ratio of natural abundance
Ratio of gyromagnetic ratio cubed
Size of magnetic moment
Polarisation
Amplitude of NJR signal per unit magnetisation (Larmor frequency)
What kind of relaxation are there?
Longitudinal T1 relaxation: along z, process that generates equilibrium population difference across the energy levels and hence estabilishes equilibrium along z axis
Transverse T2 relaxation: the process that cause the transverse magnetisation to decay to zero. Transverse magnetisation comes from phase coherence. Therefore T2 relaxation causes Mxy to decay through dephasing - it destroys the phase coherence by randomising the phases of the spins - T2 ≤ T1
How is the longitudinal relaxation described?
described by a rate equation
How does rate of change of longitudinal magnetisation change?
is proportional to the difference between the current polarisation and the equilibrium polarisation
The further from equilibrium the faster the rate of change
How does magnetisation along z affect Mz
If magnetisation along z is smaller than M0, Mz increases with time due to T1 relaxation - this is the case immediately follwing an RF pulse - want short T1 - generates magnetisation quickly
If Mz is larger than M0 (hyperpolarisation), then Mz will decay with time due to T1 relaxation - want long T1 to keep hyperpolarisation
How long does it take to reach recover equilibrium using T1 relaxation?
5T1
The key role of T1 is the determination of the time it takes to establish equilibrium magnetisation along z
What is the time between each RF pulse called?
The TR time (repetition time) - 2T1
This means full equilibrium is not achieved along z before a futher RF pulse makes it 0 again - The magnetisation observed is called Meff
How is the observed magnetisation, Meff, related to the equilibrium magnetisation, M0 and T1 relaxation?
The observed magnetisation will be proportional to T1?
What happens if species have different T1 within a sample and how lon does TR need to be to remove weighting by T1 in the intensity the peaks?
Integrals depend on T1
TR > 5T1 - longest T1 in the sample
What is the transverse rate equation?
The change in the transverse manetisation per unit time is directly proportional to Mxy
What is T2 relaxation characterised by?
the T2 time constant
What causes T2 relaxation
Mxy comes from phase coherence - decay of Mxy is due to the loss of phase coherence (dephasing)
In order for two spins to precess in sync at all points in time they must have bioth the same initial phase and be precessing at the same frequency
What does the T2 time constant describe?
the effects of irreversible dephasing that is caused by changes in the initial phase associated with spin transitions
Mxy can also decay by reversible dephasing which comes from nuclei experiencing different Larmor frequencies
Nuclei could experience different local magnetic fields due to: magnetic field inhomogeneity of the spectrometer itself ir differences in the magnetic suseptibility which induces inhomogeneity
What are the equations for the observed decay of transverse magnetisation?
What is the effect of T2 relaxation on NMR peaks?
Short T2 relaxation times leads to broad peaks - reduced resolution and sinal noise
What is shimming?
The process of improving the homogeneity of B0 by applyin small additional fields to camcel out small variations across the sample
What is the nuclear Hamiltonian?
The energy operator associated with nuclear inetractions
Often given in frequency units (rad s-1 or Hz)
The constant of proportionality between two units is Planck’s constant
Whatb is chemical shift?
Indirect magnetic interaction with B0 mediated by electrons
100 - 1000s Hz field dependant (ppm)
What are dipole dipole coupling?
Direct manetic interaction between two nuclei (magnetic moment).
Through space
Depends on orientation
Typically 10s - 1000s of Hz
What is J coupling?
Indirect interaction between nuclei meadiated by the electrons
Works through bonds
Field independant
Usually 200 Hz
What is a quadropolar interaction?
Only applies when I > 1/2
Interaction of electric quadropole moment of spins reater than 1/2
electric field gradients around the nucleus (can be MHz)
Which interactions have the greatest effects?
Zeeman > Quadropolar > dipole dipole > chemical shift > J coupling
What affects the J coupling constant?
the angular momentum operators for both spin states: IA and IB
The product of the angular momentum for spin A and B:
HJ = JAB IA x IB = JAB(IxAIxB + IyAIyB + IzAIzB)
What are the characteristics of the J coupling?
Guven in Hz and are independent of field
Proportional to the size of each magnetic moment and depend on the product of the gyromagnetic ratios of the spins
Coupling constants are sensitive to bonding because it depends on shared electron density - dependent on the number of bonds and bond angles - info on structure
J coupling acts throiugh bonds and can be heteronuclear or homonuclear
What is ther selectiopn rule for which transitions are observed in NMR?
Only observe single quantum transitions
For a two spin system which spin transitions are possible?
Which transitions in a two spin system are observed?
Active spin - spin that flips
PAssive spins - do not change
Peaks occur at the chemical shioft of the active spin
What happens if the frequencies of the two atoms are the same?
This is called chemical equivalence - two spins with the same chemical shift
You get a singlet
Where is J coupling observed?
Often only pbserved between chemically inequivalent nuclei but may be observed between two nuclei with the same chemical shift if they are magnetically inequivalent
How can J coupling change with chemical equivalence?
When two atoms are chemically eqivalent J coupling is not observed unless they are magnetically inequivalent
When the two atoms are completely chemically inequivalent J coupling is observed and patterns can be determined through 2n + 1 rule - explained using active and passive spins
When chemical equivalence is close coupling cannot be determined by the 2n+1 rule because J coupling interaction needs to be included in the hamiltonian
What is the equation for the chemical shift Hamiltonian?
HCS = -I x σ x B0
The chemical shift tensor can be can be divided into two parts: isotropic and anisotropic:
σ = σi + σCSA
The chemical shift Hamiltonian is a sum of the isotropic and anisotropic part:
HCS = Hiso+ HCSA = -σiIzB0 - I x σCSA x B0
Describe dipole dipole coupling?
Direct magnetic interaction between nuclear magnetic moments - nucleus A experiences the field of nucleus B and vice versa
The interaction is through space and is used to obtain distance information but does not include structural information
Can be calculated very precisely
Position / orientation dependent
No isotropic component and so is averaged to 0 in the solution state
When is electric quadropole moment present?
If I > 1/2 nuclear spin will posses an electric quadropole moment eQ
This interacts strongly with the electric field gradient (EFG) generated by the electron clouds
What affects quadropole coupling?
Electric quadropole moment, eQ - propoerty of the nucleus
Electric field gradient - molecular property that depends on the molecular structure and local chemical environment
If the environment is symmetric the gradient is small
If the environment is asymmetric the gradient is large
CQ ∝ eQ x EFG
How do anisotropic interactions show up on solid state NMR?
dipole dipole, anisotropic chemical shifts and quadropolar interactions give rise to broad peaks in solid state NMR
Each molecule gives rise to a peak at a frequency that depends on the orientation of the molecule relative to B0 - In a powdered solid there will be a range of orientations - instead of a single peak for each resonance there is a there are broad peaks with patterns corresponding to different molecular orientations
Peaks are analysed to extract parameters related to structure
How do anisotropic interactions show up in liquid state NMR?
In isotropic liquids molecules are constantly rotating and experience all orientations relatuive to B0
The timescale of tumbling is faster than anisotropic interactions
These orientation depenedent interactions average to 0 giveing narrow lines where position of the peaks rely on J constants and the isotropic part of chemical
anisotropic interactions still have an effect as they drive T1 and T2 relaxation
What drives longitudinal, T1 relaxation?
T1 establishes equilobrium population differnce between energy levels
Spin flips need to occur
This requires an exchange of energy with surroundings - energy comes from thermal energy in the form of molecular rotations
What drives transverse, T2 relaxation?
T2 causes the decay of transverse magnetisation due to irreversible dephasing of nuclei associated with spin transitions
When a nucleus undergoes a spin transition it loses phase memory and no longer has the same phase as the others resulting oin dephasing
What relaxation methods do single quantum transitions drive?
Transitions between spin up and spin down for single spin
Drives both T1 and T2 - changes population of energy levels and changes phase of the spin
Requires exchange of energy between spin and environement
Transitions caused by interaction between nuclei and local magnetic that fluctuate at Larmor frequency
Local fields driven by spin interactions like dipole dipole and field fluctuations driven by molecular motion
What relaxation methods do zero quantum transitions drive?
Simultaneous transition of two spins ‘flip flop’ - no overall change in relative population of energy levels and energy is conserved - no driving T1
Changes phase of both spins = drives T2
Transitions caused by interactions between nuclei - size of interaction affects how much relxation results T2 ≤ T1
How does molecular motion affect relaxation?
Any molecular motion has a correlation time τC - fast motion = short correlation time
Molecular motions are most efficient at causing single quantum transitions if the correlation time matches the Larmor frquency:
ωτC = 1
Fast molecular motions average anisotropic interactions that cause zero quantum transitions
The slower the motion the higher the probability of zero quantum transitions and the faster T2
What factors affect the time of T1 and T2?
Solvent viscocity
Temeprature
Size of anisotropic interactions
MAgnetic field strength (Larmor frequency)
What is NMR relaxometry?
measurement of relaxation parameters
Can be done in combination to give relaxation value for each resonance - protein NMR to give insight into local structure and dynamics
Useful in detecting the strength of ligand binding to a target protein
Can be done using cheap low field NMR devices
How is relaxation used in MRIs?
The contast in image is due to the concentration of water in an area and also due to differences in T1 and T2 relaxation
What is dipole dipole relaxation?
Direct interaction between neighbouring dipole moments - dominant in isotropic liquids
Dependent on: distance between spins (1/r3)
gyromagnetic ratio (generally low γ nuclei relax more slowly),
Orientation of spins relative to B0
What is quadropolar relxation?
Relaxation depends on size of quadropolar interaction and local electric field gradient
Coupling to quadropolar nuclei is rarely observed in liquid state due to rapid relaxation
Very small electric quadropolarcoupling constants and eQ can show coupling - 2H
What is paramagnetic relaxation?
Paramagnetic species mlostly unsuitable for NMR due to interaction between nuclei and unpaired electrons are very efficienmt at driving relaxation
Depends on symmetry, magnetic field strength and electron spin relaxation times and distance between nucleus and paramagnetic centre (∝ r-6)
What is magic angle spinning?
Technique used in solid state NMR to average anisotropic interactions to simplify data and recover isotropic chemical info
Similar to motion averaging
Rotation is applied at a fixed frequcy and a specific angle - 54.7°
Due to symmetry dipole dipole and CSA are zero and quadropolar interactions are reduced
