M³ – Monologues, MRI, Miscellanea

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.