# 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 along the static field returns to its equilibrium value exponentially with time constant . The and components relax to zero with time constant . Russ and I write

The transverse relaxation time [T2] is always shorter thanT1. Here is why. A change of requires an exchange of energy with the [thermal] reservoir. This is not necessary for changes confined to thexyplane… and can change as changes, but they can also change by other mechanisms, such as when individual spins precess at slightly different frequencies, a process known asdephasing.

Is *T*2 always less than *T*1? Let me start by giving you the bottom line: *T*2 is *usually* less than *T*1, and for most purposes we can assume *T*2 < *T*1. But Russ and I wrote “always,” meaning no exceptions. It’s not *always* true that *T*2 < *T*1.

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 *T*2 Be Longer Than *T*1?” Traficante begins by analyzing the relaxation equations introduced in Section 18.4 of *IPMB*,

*dMx*/*dt* = −* Mx*/*T*2, *dMy/dt* = −* My*/*T*2, *dMz/dt* = (*M0* − *Mz*)/*T*1 .

If we start at *t* = 0 with *Mx *= *M0* and *My *= *Mz *= 0 (the situation after a 90° radiofrequency pulse), the magnetization is

*Mx* = *M0* e−*t*/*T*2, *My* = 0, *Mz* = *M0* (1 − e−*t*/*T*1) .

(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, *M*2= *Mx*2+ *My*2+*Mz*2). In our example, we find

*M*2/*M0*2= e−2*t*/*T*2 + (1 − e−*t*/*T*1)2.

Traficante claims that many researchers mistakenly believe that |**M**| is equal to 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 *T*2 << *T*1, the *x*-component decays quickly, while the *z*-component grows slowly, so **|M|** starts at , quickly shrinks to a small value, and then slowly rises back to . In the *x*- *z* plane, the tip of **M** follows the red path shown below. Clearly **|M|** is always less than (the red curve is well under the blue arc).

If *T*2 equals *T*1, Traficante shows that in the *x*-*z* plane the tip of **M** follows a straight line, and again |**M**| is less than *M0*.

What if *T*2 >> *T*1? Then *Mz* would rapidly rise to its equilibrium value *M0* while *Mx* would slowly fall to zero.

In this case, |**M**| would become larger than (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 *T*2 cannot be much larger than *T*1.

Can *T*2 be just a little larger than *T*1? The straight-line plot for *T*2 = *T*1 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 *T*1 and *T*2 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 *T*2.

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 *T*1 and *T*2. 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, *M*2) behaves as *t* goes to infinity (corresponding to the upper left corner of the plots). If *M*2 is less than *M0*2 at large times, then it should be less than *M0*2 at all times and we have not violated any laws of physics. If *M*2 is greater than *M0*2 at large times, then we have a problem.

A little algebra applied to our previous equation gives

*M*2/*M0*2= 1 + e-2 *t*/*T*2 + e-2 *t*/*T*1 - 2e- *t*/*T*1 .

At long times, the term with -2 *t*/*T*1 in the exponent must be smaller than the term with -*t*/*T*1, so we can ignore it. That leaves two terms to compete, a positive term with -2*t*/*T*2 in the exponent and a negative one with -*t*/*T*1. The term with the smaller decay constant will ultimately win, so *M*2 will never become greater than *M0*2 if *T*2 < 2 *T*1.

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 thezaxis, while maintaining a constant length. Instead, under the normal conditions whenT2* [for now, let’s ignore the difference betweenT2 andT2*] is less thanT1, the resultant first shrinks, and then grows back toward its initial value as it tips back toward thezaxis. This behavior is clearly shown by examining the basic equations that describe both the decay of the magnetization in thexyplane and its growth up along thezaxis. From these equations, the magnitudes of thexyandzcomponents, as well as their [vector] sums, can be calculated as a function of time. This same behavior is demonstrated even whenT2* is equal toT1-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 theT2/T1 ratio does not exceed 2. However, experimental evidence has been obtained that shows that the ratio can be greater than 1.

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

The following relationship holds absolutely

T2 < 2T1 (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 2

T1 >T2 >T1 is possible but rarely encountered.

In a footnote, Levitt expands on this idea.

The case whereT2 >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.

*Originally published at **http://hobbieroth.blogspot.com**.*