III Spins and their hermodnamics On the menu:. Relaation & thermal equilibrium. Relaation: phenomenolog (, ) 3. Relaation: microscopic description 4. * dephasing 5. Measuring, and * We ve remarked: So: or. Be between and.. Sum to. ( ) ( ) ( ) Pr + Pr +... + Pr N = E / k EN / k e +... + e = his allows us to solve for : E / k E / k n = e +... + e. hermal Equilibrium of Spin-/ Ecite Wait his is the onl point in our lectures where we ll use the quantum-mechanical nature of spin. Quantum mechanics tells us a spin-/ in a field B has two energ levels: he three questions of relaation:. What is the equilibrium state?. How do the spins return to it? (macro) 3. Wh do the spins return to it? (micro) E E = = γ B γ B (aligned along B (aligned against B his does NO mean a spin must point along B in thermal equilibrium. It onl means that statisticall. ) ). hermal Equilibrium. Boltmann s Principle B γ γ he whole of statistical phsics rests on the following Boltmann hpothesis: at thermal equilibrium, the probabilit of the sstem being in a state with energ E is: / Pr( E) = e E k where is a constant number independent of the energ or k. If ou have N states with energies E,,E N, then the probabilit of being in state i is: Pr i / () E k i = e We want this to be an actual probabilit, i.e., Pr( ) Pr( ) We now ask: what is the average magnetic moment of (one) spin ½ at equilibrium? < m >=< m >= Pr ( ) Pr( ) < m >= γ γ If we knew that, then for N non-interacting spins (and spins are non interacting in our world, let me assure ou) at equilibrium, M= N < m > ˆ
which would be our equilibrium magnetic moment. So we reall just need to compute <m >. Now, γb γb k k = e + e his unpleasant epressions can be simplified considerabl if we remember that, for small a (<<): a e + a. e ( e.g.. 99) In our case, at room temperature (homework!), so we can simplif: Hence, γ B k << ( γb ) ( ) γb k k + + = Pr Pr γ B k e γ B ( ) = ( + k ) γ B k e γ B ( ) = ( k ) So, what is the average magnetic moment of a spin ½ at equilibrium? You will show in the homework that, for a cm 3 voel with just water (tissue has ~ 7-8% water, so that s reasonable), M 37. 8 Joule esla Note this has units of energ per unit field. You can think of it as the amount of energ ou create when ou put the water in a field of such and such esla. It s quite small! Some notes:. M is small.. he signal we measure will be proportional to M. 3. Hdrogen nuclei (water!) have the largest g and hence the largest signals. How luck we are that nature somehow turned out that wa! 4. Proportional to B : want a larger signal? Go to higher fields. 5. Careful when using this formula. N is basicall determined b the natural abundance, etc. For eample, if ou have water molecules then N should be, because ou have Hdrogen atoms per molecule. Actuall it should be more like *.99985 if ou d like to take into account Hdrogen s natural abundance (99.985%).. Phenomenolog ( γ ) = ( ) γ Pr( ) < m >= γ and, for N spins, Pr γ B γ B ( + ) γ 4 ( k ) γ 4 k = B 4k. and Ecite Wait ( γ ) B N M = 4k Equilibrium magnetic moment What happens here? It turns out that the component of magnetiation perpendicular to B (-ais) and the component parallel to it rela differentl.
Ever magnetiation vector M can be decomposed to a parallel component (to B, here parallel to ) and a perpendicular one: B. Bloch Equations Bloch found that the equations we ve found last chapter, = γ M B, in components: It turns out:. he transverse magnetiation gets eaten up and eventuall disappears with a time constant. Usuall ~ tens of ms in tissue.. he longitudinal magnetiation gets built up back to its equilibrium value, with a time constant which is alwas larger (but not alwas b much!) than. 3. Remember, alwas. his means the transverse magnetiation gets eaten up faster than the longitudinal magnetiation gets built up. Here are some tpical values (taken from Haacke 999): (ms) (ms) GM 95 WM 6 8 Muscle 9 5 CSF 45 Fat 5 6 Blood (venous) Here s a movie of what ll happen in a tissue with >> such as GM or WM (so transverse relaation occurs before longitudinal): = γm B γm B = γm B γm B = γm B γm B can be modified to describe the relaation phenomenologicall: M = γmb γmb M = γmb γmb M M = γmb γmb Notes:. M is the thermal equilibrium magnetiation, computed last section.. hese equations hold in the rotating frame as well. 3. Note how M, M has new terms with and M has a new term with, epressing the fact the longitudinal and transverse relaations occur with different time rates. What can we make of these? Let s look at a simple case:. We ve just ecited our spins.. We re on resonance (no offset, B =). 3. No irradiation (B =B =). Our equations become: Initiall relaation relaation
M = M = with the initial condition M( t = ) = M ˆ M M = We can solve these equations. First, M = initiall and it will sta so, because M = solves the nd equation. How about M? his is a tpical differential equation =a with a solution (t)= e a. Hence, the solution is: M () t = M e t / M is slightl trickier to solve for, and I ll leave that as a homework problem. he solution is: Plotted (for M =): () ( t/ = ) M t M e. Meanwhile, and act in the background, causing the magnetiation to slowl return to its equilibrium. 3. A Look Under the Hood 3. Microscopic Mechanism What causes relaation? We have three mechanisms to account for: M = γmb γmb M = γmb γmb M M = γmb γmb. Red: wh do M, M deca?. Green: wh does M changes with a different time constant,? 3. Blue: wh does buildup occur? he buildup part (#3) is difficult to eplain and we ll have to leave it at that. he deca of M, with different transverse & longitudinal times, will be eplained net. he nuclear magnetic moments each create their own fields: Relaation in Gre Matter H =95 ms, = msl a. u...8.6.4 M HtL M HtL. ime Hms L 5 5 5 3 In general:. Your spins rotate with the applied eternal fields. Now think about the following eperiment: take two spins, fi one and move the other one about. Just b virtue of moving, the field felt b the spin changes:
isochromat would all rotate with the same Larmor frequenc, ω =gb. Now imagine each spin feels a fluctuating field along the -direction, so its precession frequenc also becomes time dependent: ω(t)=g(b +B D, (t)) (Red: magnetic field) Since all water molecules keep tumbling and moving around (a.k.a. brownian motion), each sees the main field + a fluctuation field created b the other spins. hese fluctuating fields are what cause relaation. You can think of the spin of a single molecule as tumbling on the surface a sphere: otal field = Main B field (large, ~ esla) Smaller dipolar fluctuating longitudinal field (~ Without fluctuating With fluctuating fields: precession fields: precession + erratic jumps (not drawn to scale, etc.) with the end result being the total magnetic moment decas back to equilibrium. he fluctuating fields B D felt b a spin can also be composed into transverse & longitudinal components: B () t = B () t + B () t D D, D, he longitudinal fluctuating field causes transverse relaation and the transverse fluctuating field causes the longitudinal relaation. 3. ransverse Relaation Imagine eciting a spin onto the plane. Without the fluctuating field, it would just eecute precession and make a phase f=gb. With the fluctuating field along the precessing frequenc fluctuates as well, with the end result being a slightl different precessing frequenc at the end, f+δf, where Δf depends on the eact nature of the fluctuations (imagine turning a wheel with a shaking hand): f No fluctuations ( Firm hand ) f+δf With fluctuations ( Shak hand ) Note here Δf< Now imagine a number of spins. In the absence of fluctuations the would all make the same angle. In the presence of fluctuations, the would fan out (remember, each spin feels a different fluctuation): Wh does the transverse magnetiation get eaten up? Let s work in the lab frame. Imagine first no fluctuating fields. A bunch of spins in an
f Man spins, no fluctuations. (microscopic view) Man spins, fluctuations. (microscopic view) his is what happens microscopicall. Now, the macroscopic magnetiation is the (vector) sum of the microscopic magnetiation. What happens when ou sum vectors that don t point in the same direction? he (partiall) cancel out. Eample: fluctuating field slower fluctuations mean fewer collisions and hence a less dense environment, leading to greater diffusion (dephasing, in our case). his directl relates to molecule sies, because: Large molecules umble slowl Slow fluctuations Small (fast deca) Small molecules umble fast Fast fluctuations Small (slow deca) Hence we can draw this graph: Slow deca Fast Deca Slow / Fast / Large mol. Small mol. umbling op: summing 4 vectors not pointing in the same direction. Bottom: all 4 vectors point in the same direction. In both cases, the mini-vectors (blue) all have the same sie. You can now see wh the magnetiation in the plane decas: In tissue, water can be free (A) or in the vicinit of large macromolecules (B), which slow it down and lengthens its : he fluctuating -field causes the spins to spread out (dephase), and hence add up destructivel, leading to a deca of the macroscopic magnetiation vector, M. How fast does M deca what determines? Quite simpl: the rate of fluctuations. Fast fluctuations will result in lesser dephasing and hence slower deca. An analog might help see this: think of diffusion. Molecules randoml change their direction upon colliding with each other. It should be intuitivel apparent that, the lower the concentration of our sample, the larger the diffusion. Here the stor is the same: ou can think of the spin as diffusing under the action of the Fast, long Slower, shorter In a solid, where there is almost no motion, is etremel short:
making the macromolecule concentration lower, making the water molecules tumble faster, increasing and. his is wh, e.g., bone cannot be imaged (it s a solid with ~. ms - it returns to equilibrium too fast for us to measure it!). 3.3 Longitudinal Relaation he & components of the fluctuating fields cause longitudinal relaation. his can be easil understood if ou can think of these fields as tin RF pulses that tilt the magnetiation. Remember the idea: for an RF pulse 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 frequenc as the spin, Cancer Swelling Lower macromol. concentration Faster tumbling of water molecules Larger, (slower deca) 4. * Dephasing An additional source that causes the macroscopic magnetiation M to diminish in sie is known as dephasing. It is caused b spatial field variations along the -direction: ( ) Bˆ B +ΔB ˆ his leads to dephasing, this time between spins at different locations (adjacent or not): ω fluctuations = ω = gb the tin RF pulses tilt the magnetiation back to equilibrium much more efficientl, hence making shorter. oo fast or too slow and ou won t be on resonance anmore, diminishing the relaation. As before, we can draw: Slow deca In bio. tissue we are in the fast regime he spins in the bod are ecited and left to evolve. in variations in the main field mean spins at different positions precess with slightl different angular velocities and eventuall get out of snc (i.e. dephase) Fast deca Slow Large mol. Fluc. rate Fast Small mol. Contrast this with, which is caused b temporal field variations: Field variations cause relaation: Spatial, constant emporal fluctuations In solids, for eample, we saw is ver short, but will be ver long. As an interesting application, let s appl our microscopic insight to understand wh and values in tumors are larger than in regular tissue. Cancer is usuall edematous: cells swell with water, What causes these field non-homogeneities?. Hardware imperfections of B.. Susceptibilit artifacts: the eternal field B = B affects the electron spins in different parts of the bod slightl differentl, depending on the properties
of the tissue. his creates small variations in the -field B. In the Bloch equations, both and pla a smilar role. For eample, for the component, instead of we d have: M = γmb γmb Deca due to field fluctuations M M = γmb γmb ' π I Acquire A B C D Let s go through what happens to the magnetiation at each of the points outlined above. A.) he magnetiation is at thermal equilibrium, B.) A hard π-pulse is used to flip the magnetiation onto the ais. C.) We wait a time I. Longitudinal relaation kicks into effect. D.) We ecite the spin onto the -plane and measure. For the sake of simplicit, we can take the magnitude of the initial signal. Deca due to nonhomogeneit In general, one introduces a new deca quantit, *, that combines the effects of both and : = + * ' (A.) hermal eq. (B.) After π-pulse his wa we can rewrite the bloch equations e.g., for the -component: M = γmb γmb * (C.) After time I (D.) Precession (measure) akes into account both field fluctuations (temporal) and inhomogeneities (spatial) 5. Measuring,, and * 5. - Inversion Recover o measure of water, consider the following eperiment: he amount of deca depends on the time I we d wait. We can solve the Bloch equations: M M = initial condition: M ( t = ) =M o solve, substitute: Y = M M dy = Y( ) = M ( ) M =M
so: dy Y = Y t = t/ () M e Net, ou can imagine taking the initial amplitude of each deca and graphing it. You will then be able to directl observe the deca of M and deduce : Substituting back Y in terms of M, we recover the solution: initial signal t () = / M t M e I his will determine the amplitude of the signal after waiting a time I. We can imagine a set of eperiments done with different Is. In each eperiment, the maimal value of the signal is taken: B fitting this deca curve to t () = / M t M e I= I = Δ I = Δ Ecite & measure Ecite & measure Ecite & measure ( ω ) s() t =M sinδ t ( ω ) Δ/ s() t = M e sinδ t ( ω ) Δ/ s() t = M e sinδ t ou can find. his is called an inversion recover (IR) eperiment. 5. Spin Echo Eperiment Imagine having a sample with spins having different offsets. his can come about in several was, and here are two:. Non-homogeneit ( ) of B.. A gradient is turned on. Once ou ecite the spins from thermal equilibrium, the begin precessing at different rates, and eventuall spread out in the -plane. his means that, if ou were to acquire their signal, it would slowl die out because the spins would end up pointing in all sorts of directions and add up destructivel (remember, the signal is a vector sum of the spins in the -plane): Ecite & measure I = NΔ Following ecitation, all spins point in the same direction However, since each spin precesses at a different rate (= different colors), the end up dephasing ( ω ) NΔ/ s() t = M e sinδ t
What would happen if we were to appl a 8 pulse along, sa, the -ais, after a time? he pulse would invert our spins: there is relaation that needs to be taken into account. But what relaation? his spins are in the plane, but it s not *; rather, it is. Because the 8 pulse refocuses spins with different precession frequencies, there are no effects in the overall deca. Onl the true microscopic deca,, plas a role here: Envelope e -t/ Before 8 pulse After 8 pulse However, note the interesting part: if we were to wait an additional time, the spins would end up re-aligning along the -ais: 9 8 8 8 8 After 8 pulse After additional time he deca after the ecitation is determined b * (b both microscopic field fluctuations and field non-homogeneities), but the overall deca of the echoes is determined b alone. his furnishes us with a method of measuring the true microscopic deca of a sample. he reason for this can be understood b thinking of a particular spin: suppose a particular spin acquired some phase φ just prior to the 8 pulse. After the pulse, its phase would be -φ. After a time its phase would increase b φ once again, so its phase at the end would be (-φ)+φ =, i.e., it s back at the -ais. If we d continue acquiring throughout this eperiment, we d end up seeing the signal revive back again. his is called a spin echo. In terms of pulse sequences: Signal RF 9-8 What would happen if we were to give successive 8 pulses, spaced apart? One might initiall think this pattern would repeat itself indefinitel, since the spins would dephase, get flipped (b the 8), rephase, dephase again, get flipped (b the 8), rephase, dephase,... ad infinitum; in effect,