Interpretation and visualization of magnetic resonance The following slides were presented at the MMCE conference, Slovakia 2011. They were adapted for web use, and made available for viewing and downloading at http://www.drcmr.dk/MMCE2011 You may reuse slides, graphs, ideas, etc, but please give reference to website or paper mentioned on slide 3. Online viewing: Click in window to proceed. Offline viewing: Use ”slide show”-mode for the slides to present well in powerpoint. MMCE: ”Magnetic Moments in Central Europe” is a remarkably nice conference aimed at providing ”aha-moments” for people working in the field of NMR.

Is quantum mechanics really needed to understand basic magnetic resonance? Lars G. Hanson Danish Research Centre for MR and Biomedical Engineering Group, DTU Denmark

Motivation for talk Wrote controversial paper. Claimed many introductions to magnetic resonance were overly complicated or even based on misinterpreted QM. Was well received. Many had similar thoughts. Others had been confused and were relieved. Yet others were provoked.

Example reader responses “…For many years now I have repeatedly and emphatically encouraged students in NMR to forget about the two-cone model entirely, but it felt like working against a cultural dogma that has infiltrated almost the whole basic NMR literature…” “…Your paper in conjunction with the book of Dr. Levitt switched my views from the incorrect depictions to the correct largely classical, views shown in your manuscript…” (author of well known book) ”…I am strongly opposed to your position and plan to write a critical reply…”

Purpose of talk Challenging a typical way of introducing MR. Typical quantum interpretations are problematic. Often based on misunderstood QM. Basic MR is perfectly understandable from classical mechanics, including coherence and spectral features. …and you may benefit from the classical description. Software may aid teaching significantly. Anticipated perception of talk: Simple and provocative. Likely provocatively simple... Provocation is not the aim, however. Main aim: Make you reconsider understanding and teaching of basics.

Definition Definition: In this talk… ”Magnetic Resonance” is Magnetic Resonance …in contrast to every effect worth knowing about when dealing with magnetic resonance e.g. spin, J-coupling, field properties, relaxation,… These effects all require elements of QM to be understood in detail, …but maybe less than you think. Additional important limitation in this talk: Only spin 1/2 and only introductory NMR.

Definition and apology Definition: In this talk… ”Radio waves” (or RF) are fluctuating magnetic fields. …in contrast to travelling waves, …though such are now used experimentally for MRI. David Hoult, in particular, have good papers about why I should not call it radio waves.

Important points QM is not obsolete for MR calculations... …but it is often misinterpreted. There is nothing new in this talk. see e.g. Feynman et al, 1956: QM of two-level dynamics reduces to classical mechanics. ”The fathers of MR” were perfectly aware. Classical intuition probably made them ”fathers”. Worth speaking about anyway: Numerous introductory books and lectures are flawed.

Nuclei have spin. Makes them magnetic. In absense of field, directional distribution is uniform. Common explanation of MR inspired by QM: Nuclei can only point near parallel or anti-parallel to field: This explanation is largely wrong (math is right; interpretation unsuported by QM). It opens more questions than it answers. Back to Basics This picture has severe flaws… …that are not important for NMR. I’ll go with it. In fact, I really, really like this picture… because of the predictive power it gives you,… despite the fact that the picture is wrong!

Why? Why would nuclei align anti-parallel to the field? Are nuclei forced into ”cone states” instantly? How can radio waves limit the angular spread? Can radio waves change the magnetization size? It seems so. Why don’t spin flips just equalize populations? Good questions that are not well-answered unless the student doesn’t need teaching. For most: Non-intuitive, no predictive power

A fix Can the picture be improved? Certainly – it does not show NMR. Rather photoluminescence

Predictive power? This is meaningful, if you know the Schrödinger equation well enough to realize that this picture describes vector dynamics. The picture is good and useful, but maybe not for providing the basic understanding. Little predictive power unless combined with the Schrödinger equation.

What is the problem? Whats wrong with starting from a QM derivation of the ”classical” Bloch equations and then explaining MR from there? Nothing, but do it right! Level diagrams are fine if presented correctly. Don’t introduce myths.

Alternatives How can we give students an intuitive basic understanding? Magnetic Resonance day 1: http://www.drcmr.dk/JavaCompass

A useful analogy? Obviously too simple, but does it catch the essentials? Yes and no. If nuclei behaved like compass needles… …we would still be able to do imaging, for example, …even using the exact same equipment! But the angular momentum of nuclei matters.

The role of angular momentum

The polarisation process Field causes precession and alignment,.. …but nuclear interactions randomise orientations

Origin of common misconception Quantum Mechanics: When a measurement is performed on a system, it’s state ”collapses” into an eigenstate. Apparent consequence: If we measure the polarization, each nucleus is forced into an eigenstate. In fact not: The ensemble as a whole is forced into an eigenstate. This is very different.

Relevance of the cone picture? So what would force the magnetization into a state like this? Single spin measurements of each and every nucleus with no subsequent interaction. This is not at all what we are doing. Instead: Measurement on ensemble. Projection onto an enormous subspace. Result: Insignificant change of near-isotropic distribution. So how does the quantum picture look?

Magnetic Resonance Spins precess around B0 and around… …a weak B1-field that rotates around B0. Understanding of arrows: Illustrating a distribution.

Use of movie The movie showed the common classical and quantum mechanical evolution of the spin ensemble during excitation. It represents a more correct visualization than the corresponding cone pictures. However, since it is enough to keep track of the net magnetization, the picture can be simplified immensely. The resonance phenomenon can conveniently be visualized using software.

Visualizing NMR Available at http://www.drcmr.dk/bloch. Runs in browser.

But wait a minute… Why is the spherical distribution more correct than the cone picture ??? Two distributions with the same density matrix: Observations depend only on the density matrix, so aren’t the descriptions equally good?

Magnetic resonance made complicated Elements of truth: ”Cones” represent states with well-defined energy. If nuclei started out in the eigenstates, observations would be unchanged. But if you accept the cones, you also have to accept rotated versions. Wrong! QM: Homogenous fields (B0 and B1) cannot change relative orientations!

QM illustration of NMR Spins precess around B0 and around… …a weak B1-field that rotates around B0. Understanding of arrows: Illustrating a distribution.

Myths and operational truths Truth: Beyond reach. Physics only provides descriptions. Operational truth: As good as it gets: Descriptions that facilitates correct predictions. Preferably simple.

Typical myths Nuclei can only be in the spin-up or the spin-down state (cone picture). The Bo field and QM are somehow responsible. RF brings the precessing spins in phase, thus creating coherence. Quantum jumps play a significant role in MR. In particular, the spectra reflects sudden jumps between energy eigenstates. Magnetic Resonance is a quantum phenomenon, i.e. necessitates a QM explanation.

What is a quantum phenomenon? A phenomenon where understanding requires QM. Example: ”Atom formation” Only QM correctly predicts that atoms are stable. Hierarchical definition: Though atoms require QM to be understood,… …not all phenomena involving atoms are quantum. Similarly: MR is a classical phenomenon… …relying on spin, which is quantum-relativistic phenomenon.

Facts contradicted by the myths Important truths that can be derived from QM: MR is a classical phenomenon. In contrast to spin, exchange coupling,… Homogeneous magnetic fields (including RF) can only rotate the spin-distribution as a whole. Quantum jumps play little, if any, role in NMR. The near spherical spin distribution is only skewed weakly by the Bo field. T1-relaxation is the true source of coherence.

Validity of the classical description? Does this contradict QM? Not at all. Classical mechanics is, after all, a special case of QM, describing most of what we see. A QM-calculation shows that… Fields do not force spins into eigenstates. Relative orientations of spins are unchanged by homogeneous fields. Field-assisted nuclear interactions skew the spin distribution towards north. The above contradicts the interpretation of QM provided earlier.

Choosing a basis QM is conveniently expressed in a basis, e.g. |Ψ› = C↑ |↑› + C↓ |↓› Well-known concept: v = vx x + vy y For each direction in space, there is one set (C↑, C↓) Coordinate systems are connected by unitary transformations (rotations). What is along x in one system may be along y in another. Choosing a basis is a matter of convenience.

Role of the eigenstates The eigenstates form a convenient basis for math. The single spin eigenstates do not correspond to physical reality in NMR. Exception: Single spin NMR after measurement. Spectra are often interpreted as reflecting jumps between eigenstates, which is not quite true. The eigenstates are a mathematical convenience. One basis out of many. They are not even eigenstates of our measurement operator. Both classical physics and QM predict emission at eigenfrequencies – resonances.

Coupled pendulums: http://www.youtube.com/watch?v=RoSYKPTdlxs Eigenmodes of coupled pendulums (oscillators): In-phase and opposite-phase excited simultaneously in movie. These modes oscillate at different frequencies (2 peaks not corresponding to the individual pendulums). The superposition gives ”beats” in the ”FID”. No transitions between modes occur. Relevance: NMR itself: Energy is transferred back and forth between the RF field and the dipole, as between the resonant pendulums. J-coupling: Nuclei are coupled via electronic cloud. If pendulum strings had different lengths (uncoupled frequencies w1 and w2), and a weak coupling was introduced (strength J), the motional frequency content would have two doublets (splitting J) at frequencies w1 and w2 (4 eigenstates). Sounds familiar? Math is at http://fweb.wallawalla.edu/class-wiki/index.php/Coupled_Oscillator:_horizontal_Mass-Spring A classical example

Classical vs. QM calculations Notice: No jumps between states despite occurence of peaks in spectrum. Resonances, peaks, relaxation and splittings are expected classically, but there is no benefit of not using QM for doing NMR calculations. Quantitatively classical mechanics will often fail. But don’t let QM keep you from using classical intuition. Most effects are common sense. On the other hand, don’t expect all NMR effects to be explicable classically.

Coherence Relevance of basis choice for NMR: The basis choice affects interpretation. A unitary transformation converts density operator off-diagonal elements into population differences, i.e. they are formally the same. Real source of coherence is T1-relaxation. Coherence is generally correlations, non-random phase-relations, e.g. polarization.

Understanding relaxation

Reorientation by nuclear interactions Only changes matching Larmor frequency changes longitudinal mag.

Relaxation in classical terms T1-relaxation is caused by inelastic magnetic interactions only. Energy and therefore the longitudinal magnetization are preserved in elastic collisions. T2-relaxation is caused by both elastic and inelastic interactions, so T2 is always shorter than T1.

Finishing remarks

Why bother? Against: Problem is long known by many. Descriptions get back on track after a few pages. The math speaks for itself. MR is incomprehensible anyway. For: MR basics are sufficiently challenging without non-intuitive claims not rooted in QM. The myths contradict important facts.

Educational material Article and MR tutorial explaining basic MR and MRI accurately in non-technical terms. 42 English pages and supplementary animations at http://www.drcmr.dk/MR

Conclusions Basic NMR can be explained by QM but the interpretation is often problematic. Only QM gives the full picture. It is both needed and convenient for interpreting spectra. Even your parents can understand NMR, off-resonance effects, coherence and couplings, for example. Most aspects are as expected classically. A classical introduction to MR can provide students with intuition and predictive power. There are excellent reasons to teach QM formalism to those who need it. By far, QM takes you furthest.

Help beyond the very basics Available at http://www.drcmr.dk/bloch. Runs in browser.

Animations made with the Bloch simulator Example animations Spin-echo: One dimensional k-space imaging: Recommended: Interactive use for teaching Currently: No simulation of coupled nuclei, but ensembles, independent nuclei and gradients.

Teaching is often based on hand waving. Number of hands is a limiting factor. Computers offers enhanced visualisation. Numerous web resources. See e.g. Peter Lundbergs compilation http://eduNMRsoft.blogsome.com/ Summary, software