Theoretical calculation of Electron Paramagnetic Resonance (EPR) spectra

Question

I have read that rather than holding the static magnetic field constant and varying the frequency of the oscillating field in Electron Paramagnetic Resonance/Electron Spin Resonance (EPR/ESR) spectroscopy, the oscillating field is held constant while the static field varies in strength. I understand why this might be convenient for the purposes of physical implementation, but it seems to be that it would make it much harder to perform theoretically calculations of EPR structure (which is necessary to derive physical meaning from the results of the experiment). Specifically, if the spin Hamiltonian with external magnetic field strength $B$ is $H(B)$, it seems to me that the following things are true:

• If the static field $B_0$ is held constant while the oscillating field $B_1$ is varied, it seems like we can just calculate the eigenvalues of $H(B_0)$. Then, the resonant frequencies at which $B_1$ is observed will correspond to transitions between these energy eigenvalues.
• If the oscillating field $B_1$ is held at constant frequency while the strength of the static field is varied, it seems that it would be necessary to find the eigenvalues of $H(B_0)$ for every possible field strength $B_0$, so that one can check when the frequency of the oscillating field happens to align with one of the transition energies. This seems like it would require a massive amount of additional computational overhead.

So are my instincts correct - does calculating the frequency of EPR spectra require much more computation as a result of the experimental conventions? Or can anyone explain to me where I am confused?

Answer 1

Yes in theory, while varying $B_0$ might initially seem more computationally intensive than varying B₁, modern EPR equipment and software have made both approaches relatively straightforward. The actual computational effort will often depend more on the sample's complexity and the desired precision than on the experimental convention chosen. EPR spectrometers use FPGAs which are specifically designed to do the mathematics quickly.

So in any EPR experiment we have two magnetic fields to consider:

1. $B_0$ the static magnetic field
2. $B_1$ the oscillating magnetic field.

$B_0$ is used to split the energy levels of the unpaired electrons due to the Zeeman effect. And since $B_1$ is applied perpendicular to $B_0$, it is used to induce the transitions between these energy levels of the electrons.

Considering $B_0$ is held constant while $B_1$ is varied:

You had this one correct. if $B_0$ is held constant, then we can calculate the eigenvalues of the spin Hamiltonian with that particular $B_0$. Once these energy levels are determined, $B_1$ is applied, and if its frequency matches resonance occurs and is detected as absorption in the EPR spectrum.

Considering $B_0$ is varied while $B_1$ is held constant:

In this case, as you vary $B_0$, the energy levels of the unpaired electrons change. If at any particular $B_0$ value, the energy difference between the two levels matches the frequency of the constant $B_1$, resonance will occur. The method of varying $B_0$ while keeping $B_1$ constant can be seen as a 'sweep' across different possible energy level differences, looking for a match with B₁.

This approach requires calculating the eigenvalues of the spin Hamiltonian for every value of $B_0$ to determine the corresponding resonant frequencies. Since you're "sweeping" through a range of B₀ values, this method might appear computationally intensive. But, in practice, modern EPR spectrometers and software tools are optimized for this, making the process efficient.

The complexity of the sample being studied usually matters more at influencing the computational effort. For instance, samples with multiple interacting spins or those where additional interactions (like spin-orbit coupling) are significant and can require more sophisticated computational models.

Answer 2

Your instincts are largely correct, but I'll clarify and provide some additional context for electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy. In EPR/ESR spectroscopy, it's indeed common to hold the oscillating field (B1) constant and vary the strength of the static magnetic field (B0), and this experimental convention does introduce some computational challenges for theoretical calculations.

When B1 is held constant, and B0 is varied, as you mentioned, you can calculate the eigenvalues of the spin Hamiltonian H(B0) for the given static field strength. This allows you to determine the energy levels and transitions between them. The frequency of the oscillating field (resonance frequency) will correspond to the energy difference between the eigenstates. Calculating these eigenvalues for different B0 values can be computationally demanding, especially if you need to explore a wide range of magnetic field strengths. This is because you'd essentially need to perform these calculations for every B0 value of interest, which can be a substantial computational task.

When B0 is held constant, and B1 is varied, the situation is more straightforward in terms of the computational effort. You can indeed calculate the eigenvalues of H(B0) once for the given static field strength, and then vary the frequency of the oscillating field to find resonant frequencies corresponding to transitions between the energy eigenstates. The computational effort is primarily in calculating the energy eigenvalues, and once that is done, you can analyze EPR spectra at different B1 frequencies.

In practice, researchers often use a combination of both approaches. For example, they might first determine the static field strength that provides the most informative EPR spectrum, and then vary B1 to fine-tune the measurement. Additionally, computational tools and software have been developed to streamline the calculations of EPR spectra. These tools can automate the process and efficiently calculate the energy levels for a range of B0 values.

In summary, the experimental convention of holding B1 constant and varying B0 can indeed require more computational effort, especially if you want to explore a wide range of magnetic field strengths. However, the field benefits from computational tools and strategies that make these calculations more manageable and less time-consuming.