VI Relaxation. 1. Measuring T 1 and T 2. The amount of decay depends on the time TI we d wait. We can solve the Bloch equations:

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1 VI Relaxation Lecture notes by Assaf al 1. Measuring 1 and Inversion Recovery o measure 1 of water, consider the following exeriment: I Acquire A B C D Let s go through what haens to the magnetiation at each of the oints outlined above. A.) he magnetiation is at thermal equilibrium, B.) A hard -ulse is used to fli the magnetiation onto the axis. C.) We wait a time I. Longitudinal ( 1) relaxation kicks into effect. D.) We excite the sin onto the xy-lane and measure. For the sake of simlicity, we can take the magnitude of the initial signal. he amount of decay deends on the time I we d wait. We can solve the Bloch equations: dm M M dt 1 initial condition: M ( t ) M o solve, substitute: so: Y M M dy dm dt dt Y( ) M ( ) M 2M 1 dy Y Y t dt t / 1 2M e Substituting back Y in terms of M, we recover the solution: t1 1 2 / M t M e his will determine the amlitude of the signal after waiting a time I. We can imagine a set of exeriments done with different Is. In each exeriment, the maximal value of the signal is taken: Excite & measure I= (A.) hermal eq. (B.) After -ulse s() t M sin t Excite & measure I = (C.) After time I (D.) Precession (measure) / s() t M e sint

2 g Excite & measure of sins. For examle, if we had sins, the u state would have slightly more than the down state: I = / s() t M e sint /2- /2+ Excite & measure I = / s() t M e sint hese diagrams reresent the oulations, or diagonal terms of the density matrix which we ve seen. Uon any disturbance of the system out of equilibrium say, by excitation the system will re-align itself within a time ~ 1. ext, you can imagine taking the initial amlitude of each decay and grahing it. You will then be able to directly observe the decay of M and deduce 1: initial signal By fitting this decay curve to t / M t M e I Sin Echo Exeriment Imagine having a samle with sins having different offsets due to a combination of chemical shifts and inhomogeneity of B. Once you excite the sins from thermal equilibrium, they begin recessing at different rates, and eventually sread out in the xy-lane, due to both B inhomogeneity and a sread in chemical shifts. his means that, if you were to acquire their signal, it would slowly die out because the sins would end u ointing in all sorts of directions and add u destructively (remember, the signal is a vector sum of the sins in the xy-lane): you can find 1. his is called an inversion recovery (IR) exeriment. If our samle has multile chemical shifts, a Fourier transform will yield a set of eaks, each recovering with its own unique 1 rate constant. 1.2 An Energy Level Look At 1 Relaxation Following excitation, all sins oint in the same direction However, since each sin recesses at a different rate (= different colors), they end u dehasing he Bloch shere icture can be eschewed in favor of a more energy-level-diagram look at relaxation. he sin-1/2 system we ll be looking at has two ossible states, u and down, reflecting its alignment or anti-alignment with resect to the main B field. Each level has a different energy which leads to a different Boltmann distribution What would haen if we were to aly a 18 ulse along, say, the x-axis, after a time? he ulse would invert our sins:

3 Before 18 x ulse After 18 x ulse However, note the interesting art: if we were to wait an additional time, the sins would end u re-aligning along the x-axis: his might come about because, for examle, your main field is not erfectly homogeneous, B Br, leading to a recession frequency r B r (er chemical shift). his is sometimes called inhomogeneous broadening. Each of these rocesses is characteried by its own decay constant. he microscoic decay is described by 2 which we ve already met. Inhomogeneous broadening leads to exonential-like decay in many cases and is denoted by 2. he combined rate, denoted 1/ 2*, is under most circumstances given by the sum of rates: overall decay rate inhomogeneous broadening microscoic * ' After 18 x ulse After additional time he reason for this can be understood by thinking of a articular sin: suose a articular sin acquired some hase just rior to the 18 ulse. After the ulse, its hase would be -. After a time its hase would increase by once again, so its hase at the end would be (-)+ =, i.e., it s back at the x-axis. If we d continue acquiring throughout this exeriment, we d end u seeing the signal revive back again. his is called a sin echo. In terms of ulse sequences: Signal RF 9 -y 18 x ow, the above drawing is a bit of a lie: in reality, the echo would be somewhat smaller than the original signal intensity. o see why, we need to divide the decay mechanisms into two: 1. Decay due to microscoic 2 effects, which cannot be reversed with a sin echo. 2. Decay due to a satial sread of recession frequencies in the samle, as described above. Only the inhomogeneous broadening is refocused by the 18 ulse. he microscoic fluctuations are unaffected, meaning 2 decay will be refocused but 2 will not, leading to: What would haen if we were to give successive 18 x ulses, saced 2 aart? One might initially think this attern would reeat itself indefinitely, since the sins would dehase, get flied (by the 18), rehase, dehase again, get flied (by the 18), rehase, dehase,... ad infinitum; in effect, there is relaxation that needs to be taken into account. But what relaxation? Because the 18 ulse refocuses sins with different recession frequencies, there are no B -inhomogeneity effects in the overall decay. Only the true microscoic decay, 2, lays a role here: e -t/2* Enveloe e -t/

4 he decay after the excitation is determined by 2 * (by both microscoic field fluctuations and field non-homogeneities), but the overall decay of the echoes is determined by 2 alone. his furnishes us with a method of measuring the true 2 microscoic decay of a samle. 1.4 Homonuclear Sin Echoes and J-Couling It is very imortant to realie that J-couling evolution continues to evolve during a train of ulses given on a homonuclear system, and is not refocused by them. his makes quantifying the 2 decay of J-couled secies tricky. We will not send any time on this toic, but you should kee in mind it s a non-trivial toic. ow think about the following exeriment: take two sins, fix one and move the other one about. Just by virtue of moving, the field felt by the sin changes: 2. A Microscoic Look at 1 and Relaxation is Caused By Fluctuating Fields What causes relaxation? We have three mechanisms to account for: dm dt dm dt dm dt x y M MyB MBy 2 M MBx MxB M M MxBy MyBx 2 x y 1 (Red: magnetic field) Since all water molecules kee tumbling and moving around (a.k.a. brownian motion), each sees the main field + a fluctuation field created by the other sins. hese fluctuating fields are what cause relaxation. You can think of the sin of a single molecule as tumbling on the surface a shere: 1. Red: why do M x, M y decay? 2. Green: why does M changes with a different time constant, 1? 3. Blue: why does buildu occur? he buildu art (#3) is difficult to exlain and we ll have to leave it at that. he decay of M, with different transverse & longitudinal times, will be exlained next. he nuclear magnetic moments each create their own fields: Without fluctuating With fluctuating fields: recession fields: recession + erratic jums (not drawn to scale, etc.) with the end result being the total magnetic moment decays back to equilibrium.

5 he fluctuating fields B D felt by a sin can also be comosed into transverse & longitudinal comonents: f f+f BD BD, BD, () t () t () t he longitudinal fluctuating field causes transverse relaxation and the transverse fluctuating field causes the longitudinal relaxation. o fluctuations ( Firm hand ) With fluctuations ( Shaky hand ) ote here f< 2.2 ransverse Relaxation ( 2 ) Why does the transverse magnetiation get eaten u? Let s work in the lab frame. Imagine first no fluctuating fields. A bunch of sins in an isochromat would all rotate with the same Larmor frequency, =gb. ow imagine each sin feels a fluctuating field along the -direction, so its recession frequency also becomes time deendent: ow imagine a number of sins. In the absence of fluctuations they would all make the same angle. In the resence of fluctuations, they would fan out (remember, each sin feels a different fluctuation): f (t)=g(b +B D, (t)) Many sins, no fluctuations. (microscoic view) Many sins, fluctuations. (microscoic view) otal field = Main B field (large, ~ esla) Smaller diolar fluctuating longitudinal field (<<main) his is what haens microscoically. ow, the macroscoic magnetiation is the (vector) sum of the microscoic magnetiation. What haens when you sum vectors that don t oint in the same direction? hey (artially) cancel out. Examle: Imagine exciting a sin onto the xy lane. Without the fluctuating field, it would just execute recession and make a hase f=gb t after recessing for a time t. With the fluctuating field along the recessing frequency fluctuates as well, with the end result being a slightly different recessing frequency at the end, f+f, where f deends on the exact nature of the fluctuations (imagine turning a wheel with a shaking hand): o: summing 4 vectors not ointing in the same direction. Bottom: all 4 vectors oint in the same direction. In both cases, the mini-vectors (blue) all have the same sie. You can now see why the magnetiation in the lane decays:

6 he fluctuating -field causes the sins to sread out (dehase), and hence add u destructively, leading to a decay of the macroscoic magnetiation vector, M. How fast does M decay what determines 2? Quite simly: the rate of fluctuations. Fast fluctuations will result in lesser dehasing and hence slower decay. An analogy might hel see this: think of diffusion. Molecules randomly change their direction uon colliding with each other. It should be intuitively aarent that, the lower the concentration of your samle, the larger the diffusion. Here the story is the same: you can think of the sin as diffusing under the action of the fluctuating field slower fluctuations mean fewer collisions and hence a less dense environment, leading to greater diffusion (dehasing, in our case). his directly relates to molecule sies, because: Large molecules umble slowly Slow fluctuations Short 2 (fast decay) Small molecules umble fast Fast fluctuations Long 2 (slow decay) Hence, large molecules such as roteins have short 2 s, and as a result suffer from both broad linewidths (leading to a lack of sectral resolution) and smaller signal intensities (leading to lesser SR). his is one of the reasons why the study of large roteins can be very challenging. We can draw this grah: Slow decay 2 Large mol. Small mol. In tissue, water can be free (A) or in the vicinity of large macromolecules (B), which slow it down and lengthens its 2: Fast, long 2 Slower, shorter 2 In solids, where motion is greatly reduced, 2 can be extremely short. 2.3 Longitudinal Relaxation ( 1 ) he x & y comonents of the fluctuating fields cause longitudinal relaxation. his can be easily understood if you can think of these fields as tiny RF ulses that tilt the magnetiation. Remember the idea: for an RF ulse to be successful, it needs to be on resonance. he rate of fluctuations determines whether the RF is on resonance or not: when the field fluctuates at the same frequency as the sin, fluctuations = = gb the tiny RF ulses tilt the magnetiation back to equilibrium much more efficiently, hence making 1 shorter. oo fast or too slow and you won t be on resonance anymore, diminishing the relaxation. As before, we can draw: In bio. tissue we are 1 in the fast regime Slow decay Fast Decay Slow / Fast / umbling Fast decay Slow Large mol. Fluc. rate Fast Small mol.

7 In solids, for examle, we saw 2 is very short, but 1 will be very long. 2.4 Alication: Relaxation in Cancer As an interesting alication, let s aly our microscoic insight to understand why 2 and 1 values in tumors are larger than in regular tissue. Cancer is usually edematous: cells swell with water, making the macromolecule concentration lower, making the water molecules tumble faster, increasing 2 and 1. Cancer Swelling Lower macromol. concentration Faster tumbling of water molecules Larger 1, 2 (slower decay) 3. Cross Relaxation 3.1 Phenomenology We have so far assumed a sin relaxes because of its "environment". In reality, this environment could very well be another nuclear sin in the same molecule. wo nuclear sins which are close enough can induce cross relaxation, in which the random tumbling of one creates fluctuating fields at the osition of the other and vice-versa. Let's look at longitudinal cross-relaxation. Assume two sins in the same molecule. In the absence of RF fields, assuming we are onresonance, and neglecting cross-relaxation, each sin would relax with its own 1 time constant: where R 1=1/ 1. he resence of cross-relaxation imlies the off-diagonal terms in the matrix are non-ero: R1 1 M M R 1 d M. dt M M M Just like with many other things in nature, cross relaxation is recirocal, meaning the rate at which sin (1) relaxes because of sin (2) is the same as the rate at which (2) relaxes because of (1). Hence Cross relaxation can also occur between transverse magnetiation states. Cross relaxation has two very imortant roles in MR: first, it is distance deendent. his makes it a highly useful tool in investigating molecular structure, and it is for this reason Kurt Wutrich was awarded the 22 obel rie in chemistry. When a articular rotein folds, for examle, its different arts come close together and we can observe cross relaxation between eaks; while when it is unfolded, the cross-relaxation effects disaear. For examle, in the folding rotein diagram below, oints A and B are initially far aart and no cross-relaxation will occur. In the final folded state they draw close together and cross-relaxation can be observed: B dm dt dm dt M M M M 2 1 his can be recast in matrix form: R1 M M R 1 d M, dt M M M A he roximity-deendent cross-relaxation effect is usually interreted qualitatively and not quantitatively. It is observed when the distance is ~ 5 angstroms or less. he second imortant use is for enhancing the olariation of a system. Unlike IEP, this enhancement is not coherent and relies on modifying the thermal equilibrium state of the sins. We study this effect next.

8 3.2 uclear Overhauser Effect (OE) he OE effect was first suggested by Albert Overhauser in 1953, who suggested enhancing nuclear olariation by irradiating the electron sin transition in metals. His initial suggestion was met with great sketicism in the community, but was verified later that year by Carver and Slichter. Although initially suggested for electron-nuclear diolar sin couling, the OE mechanism is equally valid for nuclear-nuclear diolar sin couling. In the basic OE exeriment there are two transitions, A and X. When X is saturated by a continuous RF irradiation ulse, the magnitude of A changes. When it increases we say the OE effect is ositive; when it decreases, we say the OE effect is negative. One often runs two exeriments, one in which no irradiation is alied and one in which X is irradiated, and takes the difference sectrum: A Initially, both sins are not radiated rior to acquisition. X RF he OE is best thought of in terms of energy level diagrams. For a single sin-1/2, relaxation amount to transitions between the "u" and "down" states: When we saturate a transition with an RF ulse we equilibrate the oulations leading to no signal: /2- /2+ /2 /2 he MR signal we observe (after excitation) will be roortional to the difference between the two levels. In the first case we ll observe the regular MR line, and in the second case we won t observe a line at all. Of course, after we sto irradiating the oulations of the second case will relax back to their Boltmann distribution described by the first diagram and we ll see an MR line again. wo uncouled sin-1/2s - say, a heteronuclear hydrogen air - would be described by two such uncouled diagrams. Here, sins would be divided in roortion to their Boltmann constants at the high temerature aroximation, i.e. (neglecting normaliation): he X sin is irradiated, and the magnitude of A changes. Here it increases, meaning a ositive OE effect E1 E2 k k E1 E2 k k E1 E2 k k E1 E2 k k aking the difference sectrum allows you to examine the OE effect in greater detail. ow, since E 1 and E 2 only differ by the chemical shift (which, for homonuclear sins is extremely tiny), we can aroximate E1 E2 E, and obtain:

9 E k 2E k his means that, if we had sins, they would have the following distributions: and the corresonding energy level diagram would be: /4 (2) (1) /4- ZQ DQ /4+ (1) (2) /4 he red arrow indicates transitions of system (2), while the green arrow indicates transitions of system (1). hese are called single quantum transitions since in each transition only one sin changes direction while the other remains unchanged. he sectrum, which will consist of two lines, reresents the two ossible single quantum transitions, (1) and (2). he resence of cross-relaxation can be thought of as rocesses in which both sins change simultaneously. he blue line indicates a ero quantum transition, also called a "fli flo" transition, in which both sins change direction simultaneously. he total sin doesn't change, hence the name, "ero quantum transition." he orange line is a double quantum transition in which the sins fli from down/down to u/u and vice-versa. Once again, the name is self evident: the total angular momentum changes by two quanta. When we irradiate the transition corresonding to system (2), we equalie the corresonding oulations and the corresonding line in the sectrum disaears: he oulation differences across (2) become, but the differences across (1) remain. ote the total number of nuclei is conserved and equals. However, as we try to take system (2) out of thermal equilibrium it tries to return to it, and it does so through all three ossible mechanisms: the single quantum transition of system (2), the ZQ and the DQ transitions; the latter two also affect the oulation of system (1). o be more recise, system (2) has too many down sins and too few u sins. o re-establish the oulation difference between the and states, and the and states, the system can do one of three things: Relax via the single quantum transition (2). Relax via the ZQ transition. In this case, sins for system (2) would move from down to u, causing sins of system (1) to move from u to down, causing artial loss of signal for eak (1), leading to a negative OE. /4-/2 /4-/2 (2) /4-/2 /4+/2 /4-/2 ZQ /4+/2 (2) /4+/2 /4+/2

10 Relax via the DQ transition. In this case, sins for system (2) would move from down to u, causing sins of system (1) to move from down to u with them. his would increase the sie of eak (1), leading to a ositive OE. /4-/2 /4-/2 DQ /4+/2 /4+/2 Whether the OE is ositive or negative will thereform deend on the dominant relaxation mechanism. It turns out that in small, fast tumbling molecules, the DQ transition dominates and the OE is ositive. In large molecules, like roteins, the tumbling is slow and the ZQ transition dominates, and the OE is negative. How big is the OE enhancement? his will deend at the nuclei observed and the molecular sies. For examle, for fast tumbling molecules, one can show that: 1 OE deending on the nuclei investigated, so: OE OE OE HH HC H15 ~1.5 ~3 ~ 4 (the negative sign at the end is because the gyromagnetic ratio of 15 is negative)