M³ – Monologues, MRI, Miscellanea

Why is the slice rephaser just half the slice-selection gradient?

Or, to use slightly more precise language: “Why is the area of the slice-selection rephasing gradient just half the area of the slice-selection gradient?” The meaning of this question is visualized in the following figure, which shows the initial part of a typical, but simplified slice-selective MRI pulse sequence with a radio-frequency (rf) pulse and the slice-selection gradient amplitude, G_z (it’s simplified because the gradients do not have a trapezoidal shape, but appear to be switched on and off infinitely fast):

Some background explanations about slice selection in MRI can be found in two earlier posts from 2017-12-15 and 2020-09-10. What I didn’t mention there is that the slice-selection gradient, G_z, dephases (as a kind of unwanted side effect) the spins within the selected slice: Due to the slice-selection gradient, the magnetic field varies linearly along the z (slice-selection) direction. Therefore, the precession frequency ω(z) = γ(B₀ + z G_z) is different at different positions z within the selected slice, which means that the spins dephase and the transverse magnetization (i. e., the signal) is destroyed. To compensate for this (and recover the signal), the (light-blue) rephasing gradient is required.

Over the last months, I was asked the initially cited question about the area of the slice-selection rephaser a few times, and after some discussions I had to admit that my usual explanation is not fully satisfying (or, maybe, not satisfying at all …).

My usual explanation was: Well, the details of rf excitation are complicated, but as a first approximation let’s just assume that the excitation (e. g., the 90° flip of the magnetization) takes place instantaneously at the center of the rf pulse. So there is no dephasing before the center of the rf pulse and then there’s “normal” gradient-induced dephasing (along the slice direction) afterwards; this dephasing is exactly rephased by the standard 50 % rephasing gradient as in the diagram above.

And a completely justified reply to this is: Yeah, but the rf pulse has a certain duration and it should start turning the magnetization a bit out of the longitudinal direction as soon as it is switched on. So, dephasing should also start right away, shouldn’t it? Then, why don’t I need a full-size rephasing gradient (with the same area as the slice-selection gradient)?

At this point, I was running out of good and simple arguments …

But, of course, there should be some explanation (not necessarily a simple one), and to understand what’s going on, it is always possible to simulate the effect of an rf pulse on some magnetization vectors and look at the dephasing and rephasing for some simple cases. So, let’s do this now for three simple rf pulse shapes:

• A truncated sinc pulse (with 4 zeros on each side and Hann filter apodization) – this is a very typical rf pulse shape that has been used for years for slice-selective MRI, since it provides a nice, approximately rectangular slice profile (for more details see, e. g., the “Handbook of MRI pulse sequences” by Bernstein, King, and Zhou, chapter 2.2);
• a truncated Gaussian pulse (with a width of 3 standard deviations), which provides an approximately Gaussian slice profile (not ideal, but still usable for 2D imaging);
• a rectangular pulse (also called “hard” pulse), which is usually used only for non-selective excitation, since its sinc-like slice profile is not suited at all for slice-selective imaging. Nevertheless, this very simple pulse shape is really useful to learn something about the dephasing during excitation.

The following figure shows these three rf pulse shapes (in the top row, all with a total duration of 1 ms) and their frequency spectrum (blue in the bottom row, together with the FWHM bandwidth in gray). For small flip angles (e. g., 10°), the slice profile along the slice-selection direction will be approximately proportional to the shown frequency spectrum. As can be expected because of the different widths of the main lobes of the pulses in the time domain, their bandwidths are also quite different (note the different scales of the horizontal frequency axes).

Based on these pulse shapes, the actual rf pulse is obtained by multiplying an rf field with carrier frequency ω_rf by these shapes. The carrier frequency is given by the field strength at the center, z₀, of the desired slice, i. e., ω_rf = γ(B₀ + z₀ G_z).

Now let’s simulate some magnetization vectors that are to be flipped by these three pulses by 90°. That’s rather boring if we consider only the magnetization exactly at the center, z₀, of the excited slice where the precession frequency, ω(z₀), of the magnetization is exactly on-resonance (i. e., exactly the carrier frequency, ω_rf, of the rf pulse). But if we look at spins that are still inside the excited slice, but positioned slightly off-center at z = z₀ + Δz, then their precession frequency will be different from the carrier frequency by Δω = γΔz G_z. This difference is the reason for the dephasing that needs to be undone after the excitation.

Perhaps surprisingly, the effect of the sinc pulse is that the resulting transverse magnetization (top row of the previous figure) is generated rather quickly around the center of the rf pulse (i. e., at t = 0 ms); the magnetization can be well approximated by a step function and is almost independent of the frequency offset (which corresponds to the almost rectangular frequency spectrum). With the Gaussian pulse, the spin flip appears less instantaneous (less steep slope), and shows less efficient flipping for the highest frequency offset. Finally, with the hard pulse, the transverse magnetization builds up continuously over the total duration of the pulse (except for the highest frequency offset, which behaves more irregularly).

Even more interesting is the central row of the previous figure that shows the phase evolution of the transverse magnetization (and, hence, contains the “simulation answer” to the original question). With the sinc pulse, there are some strange phase oscillations during the first half of the rf pulse (t ≤ 0 ms), but then – starting at a value of π / 2 – the magnetization dephases approximately linearly during the second half of the rf pulse (0 ≤ t ≤ 0.5 ms) and rephases after the rf pulse (0.5 ms ≤ t ≤ 1.0 ms), when the rephasing gradient is switched on. Apparently, in this case, the phase of the spins does actually behave quite similar to my initial explanation …

However, the evolution is quite different with the Gaussian pulse: The dephasing starts immediately as soon as the rf pulse begins (at t =  – 0.5 ms), slowly at the beginning, and then increasingly faster. At the end of the rf pulse, the resulting dephasing is approximately the same as for the sinc pulse and is almost completely rephased by the rephasing gradient.

Perhaps the most clarifying result is obtained with the rectangular pulse: The phase evolution is approximately linear during the rf pulse (starting simultaneously with the pulse), and the slope of the evolution is half of the slope (with opposite sign) seen during the rephasing gradient. So, the dephasing takes place over the whole duration of the rf pulse, but is exactly half as fast as the rephasing afterwards. Put differently: The effect of the slice-selection gradient is reduced by 50 % by the simultaneously acting rf pulse. And in my opinion, in this observation lies an alternative answer to the original question: During a constant rf pulse (independent of its amplitude), the dephasing takes place with half speed (compared to dephasing without rf pulse). During a non-constant, more complicate rf pulse (such as the sinc or the Gaussian pulses), the behavior is more complicated, but the result is about the same at the end of the pulse.

But why is the phase evolution slowed down to 50 % by the (constant) rf pulse? To see this, one needs to know a small factoid about the use of rotating reference frames in MR physics (that can be found, e. g., in the “Handbook of MRI pulse sequences” by Bernstein, King, and Zhou, chapter 1.2): The equation of motion for a magnetization vector, M(t), in a reference frame rotating with the carrier frequency, ω_rf = γ(B₀ + z₀ G_z), of the applied rf pulse is:

where B₁(t) is the envelope of the rf pulse and B_z = B₀ + z G_z is the static magnetic field strength at the position of the magnetization.

In the center of the excited slice at z = z₀, we have B_z = B₀ + z₀ G_z = ω_rf / γ, which means that the equation of motion simplifies to dM / dt = γM × B₁(t)e_x. This yields the precession of M around the x axis as illustrated here:

The B₁ vector is shown in red, pointing along the x axis, and the magnetization is turned from the longitudinal (z) axis by 90° around this vector to the y axis (orange trajectory). The projection of the trajectory onto the transverse plane is shown in blue. There is no phase evolution (i. e., the polar angle in the transverse plane stays constant along the blue trajectory).

However, if we look at a magnetization vector at a different position z = z₀ + Δz ≠ z₀, then we have B_z = B₀ + z G_z, resulting in a so-called effective field vector appearing in the brackets of the displayed equation of motion above. This effective field vector has two components, lying in the xz plane, namely (B₁, B_z – ω_rf / γ) = (B₁, Δz G_z ). The direction of this effective field is shown in red again in the following figure, and the solution of the equation of motion is a precession of the magnetization around this new effective field:

The projection of the orange trajectory onto the transverse plan shows that the 360° precession around the red effective field vector corresponds to a phase evolution by 180° in polar coordinates (the thin blue vectors rotate basically once from the positive to the negative y axis, i. e., by 180°). This geometrical effect explains the 50 % reduction of the dephasing speed during the (constant) rf pulse. (To illustrate this effect, I’ve used a very large off-resonance frequency Δω = γ Δz G_z here, which means that the red B₁ vector is turned very far out of its original direction and that the magnetization is no longer tilted by 90° at the end of the rf pulse. The same effect with respect to the (blue) polar angle would be found for smaller off-resonance frequencies, but would be harder to visualize.)

So, in the end we have two complementary explanations (illustrated in the third figure above) for the area under the slice-selection rephasing gradient:

• For rf pulse shapes such as truncated sinc pulses, the highest pulse amplitudes are found at the center of the pulse in the central lobe of the sinc function. The actual tipping of the magnetization occurs in this central part of the pulse; the effects of the long preceding and trailing parts of the amplitude on the phase evolution are mostly canceled by the positive and negative components of the shape there. For these pulses, assuming an approximately instantaneous spin flipping at the center of the pulse is well justified by the simulation results.
• For simpler pulse shapes (without oscillations of the sign such as the Gaussian or rectangular pulses), the (monotonic) dephasing starts indeed immediately at the beginning of the pulse. However, the dephasing takes place (at least in average) with only half the precession frequency that one would expect for the given gradient, which means that at the end of the pulse the dephasing angle has only 50 % of the expected value.

Both explanations lead to the same area of the rephasing gradient, which is – in these cases – 50 % of the area of the slice-selection gradient.

View-angle tilting MRI (part 3): Fewer approximations

This is a third (and presumably final) text about view-angle tilting (or VAT) MRI concluding the explanations posted on 2020-09-10 and on 2020-09-15. I don’t really like the mathematical approximations that I used in my first text and, therefore, I would like to try again; the simple graphical visualization should correspond to an equally simple mathematical derivation …

Let’s start as before with the MRI signal equation for an excited slice without any distortions

with frequency-encoding direction x and frequency-encoding gradient G_x (phase encoding is generously omitted). The x integral is over the complete imaged object (but for a compact object that fits into our field of view, we can integrate from  – ∞ to ∞). The slice-selection direction is denoted by ẑ, where I use the hatted (“^”) variable to differentiate the undistorted (original) slice position ẑ from the distorted position z = z(x) analyzed below.

I will ignore the slice thickness, since the essence of the VAT technique can pretty well be explained without it. And everything gets really complicated if we consider varying slice thicknesses – or even slices split in several parts as illustrated in the following figure.

It’s useful to keep in mind the resonance relation between an excitation pulse with frequency ω_rf applied after switching on the slice-selection gradient G_z,slc and the slice position ẑ (without field inhomogeneities):

If we now consider inhomogeneity-induced distortions of a slice (due to a field inhomogeneity, ΔB(x,z)), then each original slice position ẑ (corresponding to the excitation frequency ω_rf) is “moved” to a new position, z(x), which is, in general, different for each x. The new position, z(x), is the solution of the equation

for each (fixed) value of x. The relation to the original slice position, ẑ, can be expressed (after division by γG_z,slc) as

which can be re-arranged to

(Of course, the previous equation is not an explicit solution describing z(x), since z(x) still appears also on the right-hand side of the equation.)

If we go back to the MRI signal equation and include the distorted slice geometry, z(x), the integral changes to

where, ω(x,z) is used to describe the Larmor frequencies during readout. Without field inhomogeneities, this is simply (and independent of z) ω(x,z) = γG_x x. The inhomogeneity, ΔB(x,z) changes the Larmor frequencies to ω(x,z(x)) = γ (G_x x + ΔB(x, z(x))). If we also include the additional VAT readout gradient, G_z,VAT, we get

We can now insert the expression for z(x) derived above into the third addend of this last expression, which gives

So, by setting G_z,VAT = G_z,slc (as proposed for VAT MRI), the inner parenthesis vanishes and neither ΔB nor z(x) (which implicitly contains ΔB as well) appears in the expression for the Larmor frequency. This means that the artifacts in readout direction (due to changed readout frequencies) are removed:

Other artifacts caused by the changed slice geometry, z(x), obviously remain (as indicated by the term ρ(x, z(x)) in the integral) and are not corrected by the VAT approach.

Since I’ve omitted the integration (∫dz …) over the finite slice thickness, the meaning of the slice coordinate ẑ in the final results is not completely obvious. Actually, the appearance of ẑ during readout corresponds to a tilted readout (that’s why it’s called view-angle tilting), which becomes clearer if one includes the omitted integration over the slice direction in the resulting MRI signal equation.

View-angle tilting MRI (part 2): Explanations and visualizations

Here I would like to extend the introduction to view-angle tilting (or VAT) MRI posted on 2020-09-10. Using the same field inhomogeneity and slice geometry as in the previous text, I’m trying to provide some more (visual) explanations about what’s going on.

Let’s consider two sample slices as shown in the following figure. Both slices have the same thickness Δz and are really meant to be at the same z position (with respect to the field inhomogeneity that I’m going to add later on); they are only plotted at different z positions to combine them into one diagram.

The top slice is a homogeneous slice with the same signal (magnetization density, color-coded in gray) for all x positions (except for the periphery without signal). The bottom slice has alternating signals with two low-signal areas; there is a change from dark to bright signal exactly at the center of the slice. Below each slice, the signal intensity, I(x) – obtained by a readout in x direction – is plotted.

Using the field inhomogeneity defined in the previous text, both slices can be plotted with inhomogeneity-induced distortions in z direction, which is illustrated in the next figure. If we assume (which is generally not the case in real life) that the signal of the underlying object doesn’t vary in slice (z) direction, then the signal intensity I(x) remains exactly the same as before (i. e., the projection of the signals onto the x axis is unchanged). (This is true at least as long as we ignore that the thickness of the excited slice might also change as function of x.)

However, as previously explained the distorted slice geometry is not the only effect of the field inhomogeneity. Field inhomogeneities during readout influence the spatial encoding in readout (x) direction. This is illustrated in the next figure, in which the slice distortion effect in z direction is omitted for better visualization and only in-slice effects are visualized.

The voxels are scaled in x direction (some are wider, others narrower than in the reference plot above). Since the number of protons in each voxel remains the same, the signal, which is proportional to the proton density, is scaled inversely proportionally to the voxel width, so narrow voxels appear brighter. The strongly increased signal intensity, I(x), in the right half of the plot is called “signal pile-up”.

In addition, the in-slice geometry is distorted which is nicely illustrated in the blue intensity curve: the change from low to high signal is clearly shifted from the exact center to the right-hand side of the plot; this is, of course, a direct consequence of the wider voxels in the left half of the slice.

Both effects, the distorted slice geometry as well as the in-slice signal artifacts are shown together in the following figure. The spatial shift in x direction becomes very obvious as a shift of the minimum of the slice shape to the right. The signal intensity after readout, I(x) remains the same as in the previous figure.

This situation now is the starting point for artifact correction with the view-angle tilting approach. The artifacts shown in the signal intensity curves, I(x), can – perhaps surprisingly – be corrected by a simple change of the view angle as illustrated below.

Just by projecting the distorted spin density onto a tilted axis (shown at the bottom of the following plot), the signal artifacts (areas with low signal or signal pile-up) are corrected. In this example, the projection is numerically evaluated and yields the almost perfectly straight intensity curve in the bottom plot.

A second visual “proof” of the VAT approach are the gray projection lines (one for each voxel) that appear equidistant in the tilted projection, but very non-equidistant in the conventional projection shown at the top of the figure.

In addition, the sinc-smoothing effect of the VAT pulse sequence is visualized at the periphery of the projected slice, where the signal increases (and decreases) in a relatively long ramp instead of a sharp step.

Finally, it’s interesting to see how the strength, G_z, of the slice-selection gradient influences the VAT approach (assuming that the readout gradient, G_x, is not changed). A lower slice-selection gradient results in more severe through-slice distortions (since the field inhomogeneities are now stronger relative to the gradient). But the increased slice distortion has a surprising advantage in terms of in-slice artifacts as illustrated in the following figure: increased through-slice (z) distortions shift the slice away from the maximum of field inhomogeneities and, thus, reduce the remaining field inhomogeneity in the slice (along the x direction) after excitation.

The reduced in-slice artifacts are shown in the next figure together with the uncorrected signal intensity (top) and VAT-corrected intensity (bottom).

Obviously, the voxels are less distorted along the readout (x) direction, and the uncorrected signal demonstrates reduced signal pile-up. The angle of the tilted axis at the bottom is smaller than above; it’s always arctan(G_z / G_x) which gives 45° in the first case and about 26.565° now. The numerical projection on this axis shows strongly reduced artifacts (compared to the uncorrected readout) as well as reduced sinc-smoothing compared to the VAT readout above.

View-angle tilting MRI: Theory and approximations

This text is first of all an explanation and note to myself – but some parts of it might be of more general interest (if this is not the case for you, please just stop reading …).

Metallic implants can cause severe image artifacts in MRI. The problem of these implants in MRI is their magnetic susceptibility, χ, that influences the static magnetic field B₀ in their neighborhood. Consequently, the (ideally) perfectly homogeneous static field B₀(x,y,z) = B₀ e_z becomes slightly (or not so slightly) inhomogeneous: B₀(x,y,z) = B₀ + ΔB(x,y,z) (considering only the z component from here on). A simplified two-dimensional example of such an inhomogeneity ΔB(x,z) is shown in the following figure (together with another kind of desired inhomogeneity, namely a linear gradient field).

MR imaging usually relies on the assumption that B₀ is homogeneous; the presence of ΔB(x,y,z) results in image artifacts such as voxels moved to wrong positions, signal pile-up, and signal voids.

To simplify things, I ignore the phase-encoding direction (y) and restrict this discussion to the frequency-encoding direction (x, with frequency-encoding gradient G_x; displayed horizontally) and the slice direction (z, with slice-selection gradient G_z; displayed vertically). Slice selection in MRI works by applying a slice-selection gradient G_z during excitation. This means that each position z corresponds to a unique Larmor frequency ω(z) = γz G_z, which is used to excite the spins at the desired z positions. A slice is selected by applying a radio-frequency (rf) pulse with a certain frequency (ω_rf) and bandwidth (Δω_rf). Thus, all spins with Larmor frequencies ω_rf ±Δω_rf / 2 are excited. For a perfectly linear gradient G_z (i. e., a gradient over a homogeneous background field), these spins form perfect planes (illustrated at the left hand side of the following figure for three different excitation frequencies). In the presence of inhomogeneities, spins are no longer excited in linear planes, but in often very nonlinear structures (illustrated by the central slices on the right hand side).

Now look at the central, most severely deformed slice: During slice selection, all Larmor frequencies in this slice (independent of x) were exactly centered around ω_rf (only these Larmor frequencies were in resonance with the excitation pulse). However, after switching off the slice-selection gradient (as shown in the next figure), the Larmor frequencies in this slice are no longer the same at all positions: Obviously, the frequencies close to the center are higher due to the field inhomogeneity.

So, a first effect of field inhomogeneities are deformed slice geometries. A second effect results from the remaining field inhomogeneity in the excited slice shown above. This second effect becomes relevant during frequency encoding, when the spin localizations, x, are encoded by Larmor frequencies that are assumed to depend linearly on x: ω(x) = γx G_x. However, in the selected slice, the position x is no longer proportional to the Larmor frequency, ω(x), as shown in the following figure.

The deviation from the linear dependency shifts voxel locations in readout direction resulting in distortion artifacts. This effect (but not the deformed slice geometry) is (at least approximately) corrected by the view-angle tilting (VAT) MRI pulse sequence suggested by Z. H. Cho et al. (1988).

Mathematically, these effects can be analyzed using the basic signal equation of MRI. Without field inhomogeneities, the signal S(t) of the magnetization density ρ(x,z) during the readout is:

Frequently ignored, but here made explicit, is the integral over the slice thickness Δz from z₁ to z₂ = z₁ + Δz. As written above, the position and thickness of the slice are defined (in the case without field inhomogeneities) by the slice-selection gradient G_z and by the frequency (ω_rf) as well as bandwidth (Δω_rf) of the excitation rf pulse:

The signal equation also shows that each position x is uniquely encoded by a corresponding Larmor frequency ω(x) = γx G_x.

With additional field inhomogeneities, the Larmor frequencies change proportional to ΔB(x,z): during excitation, the frequencies are ω(x,z) = γ(z G_z + ΔB(x,z)) and, during readout, they are ω(x,z) = γ(x G_x + ΔB(x,z)). In both cases, the field inhomogeneity can be directly transformed into spatial shifts:

and

In both cases, the distortions (Δz(x,z) and Δx(x,z)) decrease when stronger gradients (G_z, G_x) are used.

So, in the presence of field inhomogeneities, the signal equation must be changed accordingly by adding an x-dependency to the slice positions z₁ and z₂ (describing, e. g., the curvy slice shape shown above) as well as using the modified readout frequency:

The slice positions z_1,2(x) – and, more generally, the position(s) z(x) corresponding to an excitation frequency ω_rf – are given by the following implicit equation (for each value of x):

which can give very nonlinear solutions for z(x), since ΔB(x,z) can be a complicated function of x and z. The slice can even split into several components centered at different z positions, which means that the z integral is not over a single interval, but over several intervals.

The view-angle tilting sequence (shown in the following figure) adds an additional readout gradient in slice-selection direction with the same magnitude G_z as the original slice-selection gradient.

The additional gradient during readout must be included into the signal equation as additional component γG_z z of the Larmor frequency:

For the following, a significant approximation is made: The complicated implicit equation for the slice position, z(x), given above is substantially simplified by replacing the x-dependent slice position z(x) as argument inside ΔB(x, z(x)) with the original slice position, ẑ:

The last part of this equation can be easily solved, giving:

where the relation between excitation frequency, ω_rf and original slice position, ẑ, was used in the final step: γẑG_z = ω_rf, i. e., ẑ = ω_rf / (γG_z).

So, in this approximation, the slice is simply shifted by  – ΔB(x, ẑ) / G_z and the slice thickness remains unchanged, i. e.

(A slightly less approximated calculation is presented by K. M. Koch et al. (2010), who introduce a first-order Taylor series:

where D(x, z) = ∂ΔB(x,z) / ∂z is the first derivative of the inhomogeneity function. Then

yielding

This describes shifting and scaling (by 1 + D(x,ẑ) / G_z) of the slice variable, z, and, thus, includes now scaling of the slice thickness.)

Using the simpler (zeroth-order) version z_1,2(x) = ẑ_1,2 – ΔB(x, ẑ) / G_z, the signal equation reads

and this can be transformed (by substituting z′ = z – ΔB / G_z) to

The innermost parentheses now contain the desired correction term that compensates the shifts in readout (x) direction:

This last equation describes a readout in a tilted readout direction (not longer along G_x, but along the vector (G_x, G_z)), without any distortions (which were related to the canceled term ΔB / G_x) along the readout direction.

Assuming that ρ(x, z′ – ΔB / G_z) does not vary significantly over the the z′ integration interval (say, a thin slice), it can be replaced by ρ(x, ẑ – ΔB / G_z). The integral over z′ can now be performed independently of ρ, resulting in a sinc-shaped modulation of the signal over the time t or, after reconstruction, a certain amount of blurring.

If additional corrections of the slice distortions along the z axis are required, techniques such as “Slice Encoding for Metal Artifact Correction” (SEMAC) as proposed by Lu et al. (2009) can be used in combination with the VAT approach (however, at the cost of substantially prolonged scan times).

Why are T₁ and T₂ of water what they are?

Relaxation is one of the most basic and obvious effects in nuclear magnetic resonance (NMR) experiments. Relaxation describes how the nuclear magnetization returns to its equilibrium state after an excitation by a pulsed radio-frequency field. The transverse magnetization (precessing in the plane perpendicular to the external magnetic field B₀ and yielding the measurable NMR signal) decays exponentially with a time constant called T₂; the longitudinal magnetization (parallel to B₀) recovers exponentially with a time constant called T₁.

For pure (distilled) water, these relaxation times are in the order of 1 to 5 seconds as established by numerous experiments. However, the calculation of these relaxation time constants is known to be notoriously complicated and difficult (requiring detailed quantum mechanical analysis of the spin ensembles and the spin interactions); a typical result of such an analysis is shown in the figure below.

So I wondered if the NMR relaxation time constants T₁ or T₂ of pure water can be roughly estimated using only some simple assumptions and considerations such that the resulting quantities are at least in the correct order of magnitude.

The main reason for transverse and longitudinal relaxation is the magnetic dipole-dipole interaction between spins (other effects such as interaction with the thermal radiation field or electric interactions with the electrons can be neglected for the water protons). The dipole field of a magnetic moment m is

or, which is enough for our order-or-magnitude estimations,

where μ₀ = 4π × 10^–7 T² m³ / J is the vacuum permeability.

The magnetic moment m of the proton can be easily expressed by the gyromagnetic ratio γ ≈ 267.5 × 10⁶ rad / (sT), which is defined as the ratio between magnetic moment and angular momentum I, m = γI. Approximating the angular momentum I of the proton by I ~ ℏ ≈ 1.055 × 10^–34 Js leads to |m| ~ γℏ ≈ 3 × 10^–26 J / T and

A first idea about T₂ or transverse relaxation (introduced already 1946 in the seminal NMR paper by F. Bloch) is to consider spins (protons) in a rigid lattice, where the magnetic field of the nearest neighboring proton induces a phase shift at the spin of interest leading to signal dephasing. Using the shortest distance that is relevant for the nuclei in water molecules, namely the intramolecular proton-proton distance d_pp ≈ 0.15 × 10^–9 m, the additional, dephasing dipole field of a single neighboring proton is δB = B_dipole(d_pp). This yields a dephasing angular frequency δω = γδB = γB_dipole(d_pp) (in addition to the Larmor frequency due to the external B₀ field). Assuming that transverse relaxation requires an additional phase angle of the order of 1, the time constant T₂ can be estimated to be of the order

Well, this value for T₂ ~ 5 μs is obviously way too small for water – by about 6 orders of magnitude! So we are not even close … but there is still something to learn from this: The very short T₂ is in fact quite exactly what we find in solids (as e.g. in water ice with T₂ ≈ 4 μs as measured by T. G. Nunes et al.), which shouldn’t be too surprising since we have assumed a rigid spin configuration above. Consequently, we’ve learned now that the constant random motion of water molecules in liquid water must play a very important role for the quantitative understanding of relaxation.

To obtain at least an order-of-magnitude estimation of the actual T₂ relaxation time (and in fact also of T₁) of liquid water, we have to include some additional information about the random motion of water molecules. A handy physical parameter to describe the effects of this random motion is the correlation time τ_c, which can be used to describe, e. g., a fluctuating magnetic field component ⟨B(t)B(t + τ)⟩ = ⟨B(t)²⟩ exp(–|τ| / τ_c), where ⟨ · ⟩ denotes the ensemble average over all spins and the result is assumed to be independent of the time t. So basically, after the correlation time τ_c the fluctuating field strength becomes statistically independent of its former values.

For the protons of water, this correlation time is extremely short, in the order of picoseconds: τ_c ≈ 5 × 10^–12 s. This value is in fact about the diffusion time of a water molecule over a region with diameter d_pp, which is d_pp² / (6D) ≈ 2 × 10^–12 s for D ≈ 2 × 10^–9m² / s. It is important to note that there is almost no dephasing over the correlation time, i. e., τ_c δω ≈ 10^–6 ≪ 1. This means that the static dephasing as assumed above cannot take place since the protons move too quickly and the static effects are averaged out. After about τ_c, the motion has completely changed the orientation of the vector distance r between the protons and, hence, also the magnetic dipole field direction and the magnitude of each field component.

To obtain an estimation of T₂ using this correlation time, we now consider the phase angles ϕ(t) within our ensemble of spins after a time t – more exactly, the additional phase angles (as seen in a rotating frame of reference) due to the fluctuating random magnetic fields δB(t). These phase angles exhibit a random distribution with mean value ⟨ϕ(t)⟩ = 0, but the width of this distribution ⟨ϕ(t)²⟩ increases proportional to the time ⟨ϕ(t)²⟩ = αt. Similarly as above, we estimate our relaxation time T₂ to be the time required for this standard deviation to become of order 1; i. e., T₂ ~ (1)/(α) = (t)/(⟨ϕ(t)²⟩).

What can we say about the value of α? The factor α must have the dimension of 1/time; it should increase with the local magnetic field strength δB = δω / γ and also with the correlation time τ_c (the longer the correlation time, the less effective becomes the averaging). So, the simplest approach to express α by the given quantities with the correct physical dimension is to set α = δω²τ_c. With this guess, we find

in surprisingly good agreement with the experimental data.

To obtain a very similar result by an actual calculation, we can quantitatively determine the width of the distribution ⟨ϕ(t)²⟩. The relation between the fluctuating magnetic field component δB(t) seen by an individual spin and the phase evolution of this spin is

yielding

or, by substituting τ = t″ – t′,

We can now use the correlation time relation from above for the fluctuating random magnetic field component δB(t), namely ⟨δB(t) δB(t + τ)⟩ = ⟨δB(t)²⟩ exp(–|τ| / τ_c) with the average squared field strength ⟨δB(t)²⟩ ~ B_dipole(d_pp)² ~ δω² / γ² to obtain

To solve this double integral without long calculations (which are nevertheless straightforward), we need the fact that τ_c ≪ t and therefore almost always τ_c ≪ τ (the second integration variable), so the exponential function in the integral is almost always 0.

Hence, the second (inner) integral can be approximated for almost all values of t′ as ∞∫–∞exp(–|τ| / τ_c) dτ = 2∞∫0exp(–τ / τ_c) dτ = 2τ_c. With this approximation, we find

and, consequently,

which is again an estimation that agrees well with the observed values.

A final short comment on T₁: In non-viscous liquids such as water, T₁ and T₂ are approximately the same. While loss of phase coherence is required for T₂ relaxation, energy transfer from individual spins to the liquid is causing T₁ relaxation. This energy transfer occurs due to transverse fluctuating magnetic fields δB_x,y(t) with spectral components agreeing with the Larmor frequency ω = γB₀. Since these frequencies are much lower than the inverse correlation time, i. e., γB₀ ≪ 1 / τ_c, they basically occur with the same probability as the dephasing T₂ effects considered above. A more detailed analysis of T₁ and T₂ requires quantitative estimations of the spectral density functions J(ω) of the dynamic processes involved.

Update (2019-02-29)

A concise introductory summary about some approaches and results of quantitative relaxation theory can be found in the text book “Principles of Nuclear Magnetic Resonance Microscopy” by Paul T. Callaghan, section 2.5.

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