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 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
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
How is the longitudinal relaxation described?
described by a rate equation
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 is the transverse rate equation?
The change in the transverse manetisation per unit time is directly proportional to Mxy
What are the equations for the observed decay of transverse magnetisation?
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
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 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
