Can T2 Be Longer Than T1?

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetic resonance imaging. A key process in MRI is when the magnetization vector M is rotated away from the static magnetic field and is then allowed to relax back to equilibrium. To be specific, let’s assume that the static field is in the z direction, and the magnetization is rotated into the x-y plane. The magnetization M_z along the static field returns to its equilibrium value M₀ exponentially with time constant T₁. The M_x and M_y components relax to zero with time constant T₂. Russ and I write

The transverse relaxation time [T₂] is always shorter than T₁. Here is why. A change of M_z requires an exchange of energy with the [thermal] reservoir. This is not necessary for changes confined to the xy plane... M_x and M_y can change as M_z changes, but they can also change by other mechanisms, such as when individual spins precess at slightly different frequencies, a process known as dephasing.

Is T₂ always less than T₁? Let me start by giving you the bottom line: T₂ is usually less than T₁, and for most purposes we can assume T₂ < T₁. But Russ and I wrote “always,” meaning no exceptions. It’s not always true that T₂ < T₁.

“Relaxation: Can T₂ Be Longer Than T₁?”
by Daniel Traficante.

To see why, look at the 1991 article by Daniel Traficante in the journal Concepts in Magnetic Resonance (Volume 3, Pages 171–177), “Relaxation: Can T₂ Be Longer Than T₁?” Traficante begins by analyzing the relaxation equations introduced in Section 18.4 of IPMB,

      dM_x/dt = − M_x/T₂     dM_y/dt = − M_y/T₂    dM_z/dt = (M₀ − M_z)/T₁ .

If we start at t = 0 with M_x = M₀ and M_y = M_z = 0 (the situation after a 90° radiofrequency pulse), the magnetization is

       M_x = M₀ e^−t/T₂          M_y = 0                    M_z = M₀ (1 − e^−t/T₁) . 

(For the experts, this is correct in the frame of reference rotating with the Larmor frequency.) We are particularly interested in how the magnitude of the magnetization vector |M| changes (or, to avoid taking a square root, how the square of the magnetization changes, M² = M_x² + M_y² + M_z²). In our example, we find

                M²/M₀² = e^−2t/T₂ + (1 − e^−t/T₁)².

Traficante claims that many researchers mistakenly believe that |M| is equal to M₀ at all times; the vector simply rotates in the x-z plane, with its tip following the blue dashed arc in each figure below. Figure 18.5 in IPMB proves that Russ and I did not make that mistake. For the usual case when T₂ << T₁, the x-component decays quickly, while the z-component grows slowly, so |M| starts at M₀, quickly shrinks to a small value, and then slowly rises back to M₀. In the x-z plane, the tip of M follows the red path shown below. Clearly |M| is always less than M₀ (the red curve is well under the blue arc).

The path of the tip of M, for T₂ << T₁. 

 

If T₂ equals T₁, Traficante shows that in the x-z plane the tip of M follows a straight line, and again |M| is less than M₀.

The path of the tip of M, for T₂ = T₁.

What if T₂ >> T₁? Then M_z would rapidly rise to its equilibrium value M₀ while M_x would slowly fall to zero. 

The path of the tip of M, for T₂ >> T₁.

In this case, |M| would become larger than M₀ (the red curve passes outside of the blue arc). Traficante argues that an increase in |M| above M₀ would be unphysical (I suspect it would violate one of the laws of thermodynamics), so T₂ cannot be much larger than T₁.

Can T₂ be just a little larger than T₁? The straight-line plot for T₂ = T₁ suggests that |M| stays less than M₀ with room to spare. I tried to make a new homework problem asking you to find the relation between T₁ and T₂ that would prevent |M| from ever rising above M₀. The analysis was more complicated than I expected, so I skipped the homework problem. Below is my hand-waving argument to find the largest allowed value of T₂.

You can use a Taylor series analysis to show that |M| is less than M₀ for small times (corresponding to the lower right corner of the plots above), regardless of the values of T₁ and T₂. For longer times, I’ll suppose that |M| might become larger than M₀, but it can’t oscillate back-and-forth, going from smaller to larger to smaller and so on (I haven’t proven this, hence the hand waving). So, what we need to focus on is how |M| (or, equivalently, M²) behaves as t goes to infinity (corresponding to the upper left corner of the plots). If M² is less than M₀² at large times, then it should be less than M₀² at all times and we have not violated any laws of physics. If M² is greater than M₀² at large times, then we have a problem.

A little algebra applied to our previous equation gives

                       M²/M₀² = 1 + e^–2t/T₂  + e^–2t/T₁ – 2e^–t/T₁ .

At long times, the term with –2t/T₁ in the exponent must be smaller than the term with –t/T₁, so we can ignore it. That leaves two terms to compete, a positive term with –2t/T₂ in the exponent and a negative one with –t/T₁. The term with the smaller decay constant will ultimately win, so M² will never become greater than M₀² if T₂ < 2T₁.

I admit, my argument is complicated. If you see an easier way to prove this, let me know.

Traficante concludes

It is a common misconception that after a pulse, the net magnetization vector simply tips backwards toward the z axis, while maintaining a constant length. Instead, under the normal conditions when T₂* [for now, let’s ignore the difference between T₂ and T₂*] is less than T₁, the resultant first shrinks, and then grows back toward its initial value as it tips back toward the z axis. This behavior is clearly shown by examining the basic equations that describe both the decay of the magnetization in the xy plane and its growth up along the z axis. From these equations, the magnitudes of the xy and z components, as well as their [vector] sums, can be calculated as a function of time. This same behavior is demonstrated even when T₂* is equal to T₁—the resultant still does not maintain a constant value of 1.0 as it tips back. 

The resultant does not exceed 1.0 at any time during the relaxation if the T₂/T₁ ratio does not exceed 2. However, experimental evidence has been obtained that shows that the ratio can be greater than 1.

Spin Dynamics,
by Malcom Levitt

Malcolm Levitt, in his book Spin Dynamics: Basics of Nuclear Magnetic Resonance, comes to the same conclusion.

The following relationship holds absolutely

        T₂ < 2 T₁ (theoretical limit).

In most cases, however, it is usually found that T₂ is less than, or equal to, T₁:

        T₂ < T₁ (usual practical limit).

The case where 2T₁ > T₂ > T₁ is possible but rarely encountered.

  In a footnote, Levitt expands on this idea.

The case where T₂ > T₁ is encountered when the spin relaxation is caused by fluctuating microscopic fields which are predominately transverse rather than longitudinal.

I would like to thank Steven Morgan for calling this issue to my attention. Russ and I now address it in the errata. In general, we appreciate readers finding mistakes in Intermediate Physics for Medicine and Biology. If you find something in our book that looks wrong, please let us know.