# DNP Tutorial

### Cross-Effect DNP and Dynamic Hole Burning

by Mikhail Veshtort

IMPORTANT: Before looking up the answers or how-to instructions, always try to do it on your own first!

1. Go to the spinev/examples/dnp folder in your home directory and study the mas-ce-sing-cryst example. First, look at the output of the simulation (double-click on the html file). Then study the main input file. Understand everything that’s in it.

2. Make a copy of the mas-ce-sing-cryst file and name it ce, because we will be editing this file, and you want to keep the original, and because a shorter file name is more convenient to work with.

3. If the plot size in the html file is too large for your display size, add the line chart_height=500 to the Variables section of the ce file. The default value of this parameter is 750 (pixels). You can adjust it later if you want to.

4. We’ll be using the ce file from now on. Run this calculation yourself. To make it faster, use the steps of 20 MHz insteaad of 2 MHz.
How-to: Change 2 to 20 in the active scan_par line. If you have more than one CPU core on your notebook, add the -splitN option to the command line, where N is the number of cores you have. Run the simulation.

5. Look at the field profile you obtained. (It’s actually a MW frequency profile in our case, but it is very similar). Why does the profile look like this? Why is it so square? These are probably the first questions that should be popping into your mind. In this exercise, we’ll show you how this kind of questions can be answered by using SpinEvolution to methodically explore all relevant aspects of the system in question. And also how much you can discover on the way.

6. Keeping in mind our questions, let’s look at the details of how this polarization on the nucleus is built up for just one single point of the profile. Let’s choose a point where we see large enhancement: -500 MHz. Compute the full build-up curve for this point, sampling once every rotor period for 65536 rotor periods.
How-to: Remove the scan_par and the title/axes labeling lines. Remove “x” from the timing(usec) line, thus making the pulse sequence one-dimensional (it was zero dimensional before). Use freq_1_1_1=-500e3, or remove the freq_1_1_1 line altogether and put -500e3 at the freq_offs(kHz) line for channel 1.

7. Now look at the build-up curve using a finer, 1 usec, sampling, computing it up to only 10 ms.
How-to: change the pulse sequence length from 100 to 1 usec on both channels and use 10000 points instead of 65536.

8. Study the build-up curve you obtained. Zoom into it to look closer. What are these steps you see?
Answer: These are the rotor events. More specifically, these are the cross-effect (CE) rotor events.

9. Why the curve starts with a zero slope that quickly increases to some constant value?
Answer: This should become clear by the end of the exercise. If not, think about it after class.

10. When do these rotor events happen and why are they short “events”?
Answer: They happen at the CE resonance conditions: $\omega_{e_1}(t)-\omega_{e_2}(t) = \pm \omega_n$. They are short “events” because the resonance occurs only when $\omega_{e_1}(t)$ and $\omega_{e_2}(t)$, which are modulated by the rotation of the g-tensors, come to be exactly $\omega_n$ apart.

11. This equation for CE events makes it clear that existence or positions of these events do not depend in any way on the MW frequency offset. So CE events are exactly the same for all MW offsets in our field profile. How come then we have a MW frequency-dependent profile? What is the other factor that affects the profile?
Answer: CE events are basically occurrences of a Hamiltonian capable of driving polarization transfer. If we have the right Hamiltonian but don’t see polarization transfer, then we are obviously missing the right kind of polarization.

12. Now recall that polarization in DNP comes from the electrons. So we should study then what happens to the electrons. Let’s add I1z and I2z to your observables and perform the simulation for 10 rotor cycles, observing every 100 nanoseconds.
How-to: Change the rho0 line to I1z I2z I3z and change the pulse sequence length to 0.1 on both channels.

13. Let’s look at the results. What kind of rotor events do you see now? Zoom in on these events to see the details.
Answer: These are the MW events (occurring on electron 2 only!) and the flip-flop events.

14. Explain what happens during the MW events. Why the Iz magnetization of the resonant electron is always decreased (in magnitude) by the event?
Answer: Resonant MW field partially rotates the Iz magnetization to the transverse plane, where the transverse component gets dissipated by the very fast T2 relaxation. This leaves us with pure Iz polarization, except that it is of a smaller magnitude now.

15. Note that during the flip-flop event, the electrons are exchanging magnetization 100%. What is this phenomenon (of the 100% exchange) called and what are the conditions for things to happen this way?
Answer: This is called fully adiabatic transfer. It is possible only if the rate of sweep of the Hamiltonian through the resonance condition is slow compared to the square of the coupling (spin-spin coupling between the electrons in our case).

16. What happens to the adiabaticity of the flip-flops if we change the spinning frequency to 50kHz? Perform the computation to find out. Look at the results. Why is it so different now?
Answer: Because the Hamiltonian is now changing too quickly for the transfer to be adiabatic.

17. What other factors can destroy the adiabaticity of the flip-flip events in our system?
Answer: weaker spin-spin coupling and larger g-tensors.

18. Now lets’s change the spinning frequency back to 10kHz and re-run the simulation to see our original picture again. Can you notice something very particular and kind of unexpected about this picture?
Answer: One of the electrons always stays near its Boltzmann equilibrium baseline (at -656), wile the other one keeps getting its polarization destroyed towards zero, and they take turns at these roles. This phenomenon is analogous to the “hole burning” in a static experiment. Only here we have a “dynamic hole burning” because the two electrons are constantly changing the roles they play in forming a “hole”.

19. How does this affect the difference of the Iz polarizations of the electrons?
Answer: It grows very large in magnitude and alternates in its sign.

20. Now set the $\alpha$ angle of the CSA/g-tensor of the 1st electron to 50 degrees and run the simulation.
How-to: Change the first zero at the first csa_parameters line to 50.

21. Look at the result. What do you see?
Answer: It’s totally different now. Polarization on both electrons gets destroyed very quickly, never coming back to the Boltzmann baseline. There is no hole burning here, it’s a “total burning” now.

22. Now set the same $\alpha$ angle to 100 degrees and run the calculation again. What do you see now?
Answer: It’s similar to the $\alpha$ = 0 case, but now 1st electron is also participating in the MW events.

23. Now is the most important question: (Do not look at the answer until you thought about this very carefully!) Try to understand what makes such a huge difference for these three cases. Formulate the sufficient and necessary conditions leading to the dynamic hole burning, i.e. to the electron polarizations behaving as they do in the $$\alpha$$ = 100 case.
Answer: electrons will keep returning to the Boltzmann equilibrium baseline if and only if the following two conditions are met: 1) the flip-flops events are adiabatic, and 2) while the electron is at the baseline, no MW events happen to it.

24. Look at the picture where $\alpha$ = 100 to illustrate this statement and make sure you understand it. This is important. And this is actually our key finding for today.

25. Now go back to computing the long-term evolution and compute it for 10000 rotor cycles for the $\alpha$ angle equal to 0, 50, and 100 degrees. Look at the results. What can you conclude (or hypothesize)?
Answer: We get high enhancement only when large polarization difference builds up for the electrons.

26. This is a well-known fact actually, and it can be understood theoretically by looking at the CE Hamiltonian. But what we discovered in our quick exploration is that this difference in polarizations can easily become really huge, approaching the value of 656, and that there exist very simple sufficient (and probably necessary) conditions for this to happen, and we formulated them. We discovered that whether we’ll get large CE DNP enhancement or not, depends on how MW events are spaced with respect to the flip-flop events, and whether the flip-flops are adiabatic. People have been researching DNP for many years, but this crucial part of the CE DNP mechanism has gone unnoticed or misunderstood. But now SpinEvolution makes this kind insights easily discoverable.

27. What we have just discovered is actually much more interesting than the original question and probably has important consequence for the experimental DNP design and applications (helping us to answer questions such as how to make DNP more efficient at high spinning frequencies). And since we probably ran out of time already, try to answer our original question by continuing this play with SpinEvolution after class.
Hint: Using the -dumpeigs option, compute the eigenvalues of the Hamiltonian for some very large MW offset, say, -2500MHz, and then see what happens when you make the offset smaller. Remember that the eigenvalues are computed in the rotating frame for the electrons and the lab frame for the nuclei.