This manuscript presents a very elegant theory on DNP under slow-tumbling conditions in liquid solution via a mechanism that is historically known as the solid effect. For a paper discussing high-level spin dynamics theory, it is astonishingly clear. The reader is helped gently over the hurdles on the path to understanding (example: pointing out the outer product with identity operators or the identity matrix in Eq. (53)). The theory is sound and it explains observations recently made by Kuzhelev et al. Compared to an earlier approach that the author calls “more ambitious”, the new theory leads to much simpler results. Although still elaborate, it is one of these theories that appear straightforward after someone has found how to do it. I do not have many remarks and I did not find any serious problem. In principle, this manuscript could be published as is (just correct the typo). The author might want to consider whether any of the following remarks could still improve it.

At some points of my review I refer to the companion paper “Dynamic view of the solid-state DNP effect” (https://doi.org/10.5194/mr-2023-1).

1. Line 35: You refer to the companion paper and discuss the case where the mw nutation frequency approaches the nuclear Larmor frequency. This case is the basis of NOVEL DNP (https://doi.org/10.1016/0022-2364(88)90190-4). You may want to point this out.

2. Line 50: You describe thermal mixing as a DNP process that involves two electron spins and one nuclear spin. Common usage of this term is that it involves several electron spins, not necessarily only two.

3. I understand that the current manuscript needs to repeat some material of the companion paper in order to be self-contained. For my taste, you repeat too much. No derivations are required here. You could simply state the key results of the companion paper before Section 3, where the new derivations and results are described. I would substantially shorten Section 2. Basically, you only need to introduce notation and equations used in later Sections or for generating Figure 3.

4. Line 338: “partial derivative with respect to the time dependence, at fixed ζ.” The reader might be puzzled here, as for any molecule, ζ is generally time dependent, too. Of course, it is mathematically sound to do it this way and average over ζ in the end. However, maybe you want to remind the reader at this point that you are talking about evolution of an ensemble state that contains all orientations. Not all readers might be as familiar with the SLE that they find this obvious.

5. You rightly point out that, with the experimental parameters that you have, you cannot determine the contact distance with sufficient accuracy. However, in principle it may be feasible to determine all parameters that you need (with some effort for B1). Of course, the FFHS model is still a strong simplification. Nevertheless, determining the contact distance within this model may be of interest. Maybe you want to stress this.

Typos:

Line 436: “hallow” should read “hollow”

The presented manuscript “The solid-state DNP effect in liquids” by Deniz Sezer is devoted to the extremely valued and inappropriately underestimated subject of DNP solid effect (SE) in the presence of dynamics, which is assigned by the author to SE in the liquids. The interpretation of DNP studies in complex systems, especially liquids and soft matter, remains partially phenomenological despite a general understanding of the underlying principles dating back to Abragam’s times. However, only very few researchers have attempted to compute analytically, based on operator formalism, the shape and intensity of DNP spectra and their relation to the easily determined EPR parameters. This contribution is a most welcome addition to the field, and the author provides a systematic and thorough analysis of the effect conventionally labelled “Solid Effect” and its observation in liquids, relating it to the dispersive component of the EPR line which is mostly not considered explicitly in the interpretation of such DNP effects. The current contribution is a logical continuation of the companion paper “Dynamic view of the solid-state DNP effect”, which should be considered while reading through the actual paper. As the main results, the description of SE via the product of the dispersive component of the EPR spectrum and spectrum of electron-nuclear dipolar interaction, as well as the introduction of the force-free-hard-sphere model to all derivations, can be pointed out. In contrast to the static Abragam model, which is widely and perhaps the only used model to describe the solid effect, the newly derived theory allows describing of the solid effect in the system where dynamics plays a substantial role in modulating both an amplitude and DNP profile shape of solid effect. As the authors claimed, the theory presented in the paper is a more simple version of a theory developed by Korringa and colleagues (Papon et al., 1968; Leblond et al., 1971a), the main point of which was describing the transition between SE and Overhauser effect due to the changes of the dynamics in the system. Nevertheless, the way the author presents spin dynamics derivation and implementation of the new concepts is unique, and the paper absolutely deserves to be accepted after minor revision. On the other hand, the paper is remarkably long and dense, perhaps containing a lot of points for further discussion both from fundamental spin dynamics theory points of view as well as perspectives of application for real experimental cases. However, there are some comments and questions below, after answering which the manuscript is suggested to be accepted.

1. Certainly, a detailed derivation of the formalism is beyond the scope, or the abilities, of quite some readers; the author, however, succeeds in making the essential results available in a graphical way by plotting different contributions, and parameter dependences, in figures 3 and 4. Figure 5 is a bit less intuitive. Maybe a short (one-sentence) description of the meaning of “perturbative” within the caption would help.
2. Figures 7 and 8 are meaningful and convincing descriptions of a particular experimental case, with an “unusual” DNP enhancement spectrum that is not straightforward to interpret by not using the approach in this work. Are there any arguments that the interpretation given in this paper (despite the good fit) is more realistic than the one in the original JACS paper of Kuzhelev (thermal mixing)? Are these just two alternative interpretations of the spectrum, or are there good reasons why thermal mixing can be ruled out? Can the frequency separation of small offset DNP peaks on J-band DNP spectra of BDPA in lipid bilayers, which seems to be smaller than v_n~400 MHz, be the reason to exclude thermal mixing or cross-effect from consideration?
3. Although the author provides some key quantities (like eq. 87) and follows with a general description, it would be helpful for the reader if it were clearly outlined, possibly in the Conclusion section, which are the conditions when a significant deviation from the classical SE effect is expected; when this might be safely neglected; and possibly when an additional contribution by CE or TM or even OE exists, in other words, how to easily identify or predict the contributions to the DNP spectrum. This could be a separate Section, apart from the existing Discussion (5) that puts this work into context.
4. Perhaps a short overview of DNP effects, in addition to solid effect and thermal mixing mentioned in the introduction, could help non-expert readers understand the differences between them. Also, the explanation presented in paragraph around line 55 for thermal mixing fits more to the cross effect.
5. Another aspect is related to Figure 2. The design of this diagrammatic representation may be a major simplification of the derivation (in terms of equations), or it may confuse the reader; only time will tell. Since the diagrams are derived and explained in the other paper mr-2023-1, it is suggested to leave fig. 2 out in this paper and rather remove any suggestion for the superficial reader to work through 70 pages instead of only 35. The two papers should remain independent of each other if that is possible.
6. Line 485: Perhaps, authors could describe for which types of dynamics monoexponential correlation function fits better than for translational diffusion, e.g. rotational dynamics, complex formation etc., with corresponding references.
7. Lines 489, eq. 66: D is a relative diffusion coefficient, which is the sum of diffusion constants of molecules carrying either electron or nuclear spins. Perhaps, it should be mentioned in the text.
8. Table 1 and corresponding fitting in Figures 7-9: T1s depends on the B1. Is there any explanation for this phenomenon? Or is it just a fitting procedure drawback? Otherwise, could the author add some more detailed explanation around line 665.
9. Line 670 and perhaps 635: How does the radical concentration affect b (nm), which is sometimes called the minimal distance of approach, which should not depend on the concentration, but the structure of diffusion molecules?
10. Figure 7. Was the fitting of s_x and T_ix done separately, or only their product was scaled with amplitude after adding the broadening?
11. Figure 7, green line: The residual of the fitting exhibits some “peak” around zero offset, which seems to be a result of either an additional DNP effect or a slight misplacing of the offset axis. How was the offset axises correspondence between experimental data and calculated DNP spectra done? Is the Overhauser effect expected at J-band with dynamics corresponding to tau~7 ns?
12. Line 658: Having an average value of tau~7 ns, is it possible to calculate the diffusion coefficient using eq. (66)? How does it agree with the literature?
13. While the approach presented in the contribution is developed for the ideal case of one homogeneously broadened EPR line, which can be the case for BDPA and trityl-like radicals, how is the situation changed when anisotropy of g-factor and hyperfine interaction are introduced, e.g. for nitroxides or DPPH radicals?

Additionally, no critical grammar mistakes or typos were found in the text of the manuscript, except the one on line 293, where “states” instead of “sates” was, probably, meant. In some places, missed punctuation can be noticed, but without cutting or changing of any sense, e.g. missed comma before and/or after the adverb “however” or “which”, etc.