# GenEM: Physics-Informed Generative Cryo-Electron Microscopy

Jiakai Zhang1,4⋆ Qihe Chen1,4⋆ Yan Zeng1,4 Wenyuan Gao1,4 Xuming He1 Zhijie
Liu2,3 Jingyi Yu1
1School of Computer Science and Technology, ShanghaiTech University
2iHuman Institute 3School of Life Science and Technology, ShanghaiTech
University
4Cellverse Co., Ltd

###### Abstract

In the past decade, deep conditional generative models have revolutionized the
generation of realistic images, extending their application from entertainment
to scientific domains. Single-particle cryo-electron microscopy (cryo-EM) is
crucial in resolving near-atomic resolution 3D structures of proteins, such as
the SARS-COV-2 spike protein. To achieve high-resolution reconstruction, AI
models for particle picking and pose estimation have been adopted. However,
their performance is still limited as they lack high-quality annotated
datasets. To address this, we introduce physics-informed generative cryo-
electron microscopy (GenEM), which for the first time integrates physical-
based cryo-EM simulation with a generative unpaired noise translation to
generate physically correct synthetic cryo-EM datasets with realistic noises.
Initially, GenEM simulates the cryo-EM imaging process based on a virtual
specimen. To generate realistic noises, we leverage an unpaired noise
translation via contrastive learning with a novel mask-guided sampling scheme.
Extensive experiments show that GenEM is capable of generating realistic cryo-
EM images. The generated dataset can further enhance particle picking and pose
estimation models, eventually improving the reconstruction resolution. We will
release our code and annotated synthetic datasets.

$\star$$\star$footnotetext: Authors contributed equally to this work.

##  1 Introduction

Over the past decade, deep generative models ranging from earlier variational
auto-encoders (VAEs) [24] and generative adversarial networks (GANs) [13] to
the more recent diffusion models [14] have shown successes in realistic image
generation. With inputs as simple as textual descriptions, techniques such as
Stable Diffusion [38] can generate photographic quality images far richer than
one can capture. Additional controls [49] can be imposed to produce tailored
visual effects, e.g., theatrical lighting [35], specific perspectives [30],
human poses [52], etc. In fact, the successes of image generation have gone
way beyond visual pleasantness, stimulating significant advances in scientific
explorations. Examples include brain magnetic resonance imaging (MRI) to
computational tomography (CT) translation [46], X-ray image generation [41],
cosmological N-body simulation [50], etc. All have long relied on expensive
imaging apparatus.

Figure 1: GenEM enhance AI models in cryo-EM via data augmentation. The
captured cryo-EM real dataset requires AI models to accurately pick particles
and estimate the poses of particles to perform a high-resolution 3D
reconstruction. However, the performance of AI models is limited due to the
lack of high-quality annotated datasets. To address this, GenEM takes an
initial result as input and generates a realistic synthetic dataset with
ground-truth annotations. Such a dataset can significantly enhance AI models’
performance on real datasets, leading to an improved result.

Different from “entertainment” applications that have largely focused on
perceptual soundness, generation techniques for scientific imaging should
faithfully follow the physical process: the generated results would eventually
be applied to downstream tasks such as medical image diagnosis and protein
structure reconstruction, where physical correctness overweighs visual
pleasantness. In this work, we extend image generation to a specific
biomolecular imaging technique, single-particle cryo-electron microscopy
(cryo-EM) to improve the performance of its downstream tasks. Cryo-EM aims to
recover the near-atomic resolution 3D structure of proteins, with its latest
application in recovering the SARS-COV-2 spike protein [47] structure to guide
drug development.

As shown in Figure 1, the cryo-EM pipeline starts with capturing 2D projection
images, termed as micrographs, of flash-frozen purified specimens. These
contain hundreds of thousands of target particles with unknown but varied
locations, poses, and shapes. High-resolution cryo-EM reconstruction, similar
to multi-view stereo [12] in computer vision, requires accurately picked
particles and precisely estimated poses. Recent AI models such as Topaz [4]
for particle picking and CryoFIRE [27] for pose estimation have been widely
used. Yet, their performances are limited due to the lack of high-quality
training datasets with expert annotations, which are very labor-intensive and
difficult to acquire. A potential solution is using ground-truth annotations
as controls to generate realistic synthetic images. However, it is quite
challenging to generate physically correct synthetic images with realistic
cryo-EM noises, which aims to enhance particle picking and pose estimation
models, eventually improving reconstruction resolution on real datasets.
Existing physical simulations [51] are computationally intensive to emulate
complex noises. State-of-the-art generative methods such as CycleGAN [55] and
Contrastive Unpaired Translation (CUT) [33] lack consideration of physical
constraints and cannot generate random realistic noises.

In this paper, we propose physics-informed generative cryo-EM (GenEM), the
first approach that integrates physical-based cryo-EM simulation with a
generative unpaired noise translation to produce realistic synthetic cryo-EM
datasets with annotations. Our synthetic dataset can significantly enhance
cryo-EM particle picking and pose estimation models. To introduce physical
constraints, GenEM starts from initial 3D biomolecule structures that can be
obtained from the initial reconstruction. Then GenEM simulates the real cryo-
EM imaging process on a virtual specimen that contains a variety of target
particles to obtain physical-based simulation results. Furthermore, real cryo-
EM images exhibit complex noises induced by limited electron beam dosage and
manufacturing errors. To generate realistic noises, we propose a novel
unpaired noise translation technique via contrastive learning to transform an
initial Gaussian noise to a more realistic noise learned from real cryo-EM
images. Finally, we find that randomly sampled negative patches are likely
semantically similar to the positive sample, conflicting with contrastive
learning assumptions. GenEM hence adopts a particle-background mask to guide
the sampling scheme, ensuring that negative and positive samples are
semantically different. Therefore, we improve the training efficiency and
performance via our novel mask-guided sampling scheme.

We evaluate the capabilities of our method in photo-realism, particle picking,
and pose estimation across three real cryo-EM datasets. The results show that
GenEM not only produces the most photo-realistic cryo-EM images but also
significantly improves the reconstruction resolution of the target molecule by
fine-tuning particle picking and pose estimation models. In terms of photo-
realism, we measure the similarity between generated and real micrographs. The
results show that our approach achieves the best realistic cryo-EM image
generation. In particle picking, the model augmented with GenEM data
demonstrates the highest increased accuracy and finally improves the
resolution of reconstruction in all three datasets. In pose estimation, we
develop an end-to-end training pipeline on synthetic datasets and evaluate
them on real datasets. The pose estimation module trained with our data
achieves the best reconstruction resolutions across all three datasets. GenEM
shows significant potential for generative methods in applications of
scientific fields like cryo-EM where high-quality annotated real datasets are
exhaustively required yet infeasible to obtain.

Figure 2: Overview of GenEM. We prepare a virtual specimen that contains
numerous initial reconstruction results and emulate the imaging process of
cryo-EM with physical priors including uneven ice thickness and optical
aberrations to produce a physical-based simulated image. We then translate
simple Gaussian noise to realistic noise via contrastive noise modeling on
physical-based simulation results.

##  2 Related Works

Our work aims to extend generation methods to the field of cryo-EM. We
therefore only discuss the most relevant works in respective fields.

Cryo-EM Simulations. Theoretical simulation techniques, based on physical
priors, combine atomic-level simulations with global projection to accurately
compute electron scattering during the imaging process [51, 42, 39, 29, 31].
Traditional simulation methods such as InsilicoTEM [51] model the interaction
between electrons by taking specimens as multi-slices to theoretically improve
the algorithm’s interpretability. However, they typically require expensive
computational resources and may require complex adjustments of parameters to
achieve ideal results. Our work also includes an efficient physical-based
simulation module to effectively present the structural information of
particles. Additionally, we generate realistic noises via a novel unpaired
noise translation technique.

Unpaired Image-to-image Translation. Deep generative models can be used to
fill the gap between simulated projections and real images without paired
data, known as unpaired image-to-image translation. The diffusion model has
been recently introduced to this task, such as UNIT-DDPM [40]. Although the
experiments demonstrate compelling results in low resolution, the current
forms of these methods do not perform sufficiently well at high resolutions
like $1024\times 1024$, preventing them from further applications in cryo-EM
image generation.

Compared with diffusion models, GAN-based methods are more efficient and
lightweight for high-resolution image generation. CycleGAN [55] and DualGAN
[48] introduce two generative adversarial networks to calculate cycle
consistency losses, allowing for training on unpaired data [19, 25, 34, 22].
DiscoGAN [23] extended this idea to maintain consistency in both directions.
UNIT [28] uses shared latent representations for unsupervised image-to-image
translation, while MUNIT[16] extends this concept to achieve multi-modal
translation. TraVeLGAN [2], DistanceGAN [3], and GcGAN [11] achieve this goal
through various methods, making one-way translation possible while avoiding
traditional cycle consistency. Recently, Contrastive Unpaired Translation
(CUT) [33] introduces a new generative framework via contrastive learning [21,
43, 53]. The core idea of contrastive loss is to encourage the generated
images to be consistent with the original images in a low-dimensional
embedding space, thereby improving image consistency and quality. However,
existing methods do not address the physical process of electron microscopy
projection. Our method introduces a physical-based simulation to add physical
constraints. Further, we propose a novel unpaired noise translation based on
contrastive learning with a mask-guided sampling scheme.

Neural Noise Generation. In response to the constraints posed by basic
synthetic noise models, extensive work has been dedicated to developing
various generative noise models. These models aim to generate complex noise
patterns to tackle the challenges associated with real-world image-denoising
tasks or other downstream tasks. Some recent methods focus on signal-dependent
noise modeling [26, 18, 15, 5], while others [1, 8, 9] use metadata such as
camera parameters to generate noise with specific distributions. These studies
aim to enhance the performance of image denoising and noise synthesis to meet
the demands of real-world scenarios. However, the current methods for
generating signal-dependent noise follow a deterministic process, lacking the
ability to vary from a single content to multiple real images. We instead
incorporate Gaussian noise with physically simulated images to introduce
randomness to the networks, achieving the translation from simple Gaussian
noise distribution to the complex randomness presented in real cryo-EM images.

##  3 Physics-informed Generative Cryo-EM

We propose the GenEM method, which for the first time integrates physics-
informed generative cryo-EM image generation with a novel contrast noise
generation technique, as shown in Figure 2. Our approach first prepares a
virtual specimen containing a large number of uniformly distributed target
protein structures (Section 3.1). Based on this, we emulate the cryo-EM
imaging process to introduce physical constraints (Section 3.2). To generate
realistic cryo-EM noises, we use an unpaired noise translation via contrastive
learning (Section 4.1) with a novel mask-guided sampling scheme (Section 4.2).
Finally, we detail our training scheme of GenEM (Section 4.3).

Figure 3: Mask-guided unpaired noise translation. We translate a simple
Gaussian noise to a realistic cryo-EM noise through generative networks. In
the training process, we introduce a mask-guided sampling scheme to improve
training efficiency and performance. The positive and negative sampled patches
are then encoded to multi-scale features to calculate mask-guided noise
contrastive estimation loss. We achieve higher photo realism by minimizing an
additional adversarial loss between synthetic cryo-EM images and real cryo-EM
images.

###  3.1 Virtual Specimen Preparation

To simulate the cryo-EM imaging process, we first prepare the virtual
specimen, a large volume containing numerous low-resolution target particles.
Particles inside the position, rotation, and conformation state of the
specimen are randomly presented. To obtain an initial model of these
particles, we employ cryoSPARC [36] for coarse particle picking and pose
estimation. The picked particles with estimated poses then input to cryoDRGN
[54] for continuous dynamic reconstruction of an initial 3D neural volume of
density
$V(\mathbf{p},\mathbf{w}):\mathbb{R}^{3}\times\mathbb{R}^{8}\mapsto\mathbb{R}$,
where $\mathbf{p}=(x,y,z)^{\text{T}}\in\mathbb{R}^{3}$ represents the spatial
coordinates, $\mathbf{w}\in\mathbb{R}^{8}$ denotes a high-dimensional vector
in cryoDRGN representing particle conformation states.

During the construction of the virtual specimen $S(x,y,z)$, we place $N$
particles $V(\mathbf{p};\mathbf{w}_{i})$ with random conformations
$\mathbf{w}_{i}$ into an initially empty virtual specimen. This placement
involves randomly sampled rotation matrices $R_{i}\in\mathbb{R}^{3\times 3}$
and translations $\mathbf{t}_{i}\in\mathbb{R}^{3}$, which can be further used
as ground-truth annotations. To prevent unrealistic overlapping due to random
placement, particles are iteratively inserted. The inserted particle area in
the virtual specimen is marked as occupied to avoid further placement. The
virtual specimen can be expressed as:

$S(x,y,z)=\sum_{i=1}^{N}V(R_{i}\mathbf{p}+\mathbf{t}_{i};\mathbf{w}_{i}),$ (1)

where the number of particle $N\sim\mathcal{U}(N_{\min},N_{\max})$, minimum
$N_{\min}$ and maximum $N_{\max}$ are derived from real data to ensure content
fidelity.

###  3.2 Emulating the Imaging Process

Based on a virtual specimen, we can mimic the cryo-EM imaging process. This
process includes electron-specimen interaction, uneven ice thickness, and
aberrations of cryo-EM “optical” devices. The electron-specimen interaction
can be modeled as a projection of the specimen density using the weak phase
object approximation [44] (WPOA). In the actual imaging process, we must
consider the local attenuation of imaging intensity caused by uneven ice layer
thickness. Thus, we utilize a re-implemented version of Icebreaker [32] to
generate a weight map $W(x,y)$ representing the local attenuation of electron
maximum intensity. Lastly, we model the aberrations of optical devices as a
point spread function $g$, which can be estimated by CTFFIND4 [37] from cryo-
EM images. Consequently, we can represent the 2D physical-based result
$I_{\text{phy}}(x,y)$ as

$I_{\text{phy}}(x,y)=g*\left[W(x,y)\cdot\int_{\mathbb{R}}S(x,y,z)\mathrm{d}z\right].$
(2)

##  4 Noise Generation via Contrastive Learning

Real cryo-electron microscopy (cryo-EM) images contain complex, random noises.
Accurately generating realistic noises on synthetic cryo-EM images is crucial
for downstream tasks. As shown in Figure 3, our method translates a simple
Gaussian noise to a realistic cryo-EM noise via an unpaired noise translation
(Section 4.1). In the training process, we introduce a mask-guided sampling
scheme (Section 4.2) to improve training efficiency and performance.

###  4.1 Unpaired Noise Translation

The generation of noise can be modeled as an unpaired image-to-image
translation task using a conditional generative model. The input,
$I_{\text{phy}}\sim X$, learns realistic noises from real cryo-EM images
$I_{\text{real}}\sim Y$, to produce synthetic cryo-EM images
$I_{\text{syn}}\sim\hat{Y}$. The generative model includes a discriminator
$D$, and a generative function $G=G_{\text{dec}}\circ G_{\text{enc}}$, where
$G_{\text{enc}}$ is a encoder and $G_{\text{dec}}$ is a decoder.

The original contrastive learning method lacks randomness, producing only one
electron microscopy noise image per physically simulated image, limiting the
generation of varied synthetic datasets for downstream tasks. To overcome
this, we introduced randomness in training process by adding zero-mean
Gaussian random noise $\mathcal{G}$ with variance $\sigma^{2}$ to physical
microscopy images, producing intermediate results
$I_{\text{inter}}=I_{\text{phy}}+\mathcal{G}$, which is passed to generative
function to produce $I_{\text{syn}}=G(I_{\text{inter}})$. In all our
experiments, we adjust the variance of Gaussian noise by signal-noise-ratio
(SNR$\,\,=0.1$). Following this, we randomly select paired patches in
$I_{\text{inter}}$ and $I_{\text{syn}}$ as the positive sample and query,
respectively. We then choose $K$ patches in $I_{\text{inter}}$ as negative
samples. The encoder $G_{\text{enc}}$ maps these to normalized vectors
$\boldsymbol{v}$, $\boldsymbol{v}^{+}$, and
$\boldsymbol{v}^{-}=\left\\{\boldsymbol{v}_{k}^{-}\right\\}_{k=1}^{K}$,
respectively. Our objective is to maximize the likelihood of selecting a
positive sample by minimizing the cross-entropy loss:

$\displaystyle\ell(\boldsymbol{v},\boldsymbol{v}^{+},\boldsymbol{v}^{-})=$ (3)
$\displaystyle-\log\left[\frac{\exp(\boldsymbol{v}\cdot\boldsymbol{v}^{+}/\tau)}{\exp(\boldsymbol{v}\cdot\boldsymbol{v}^{+}/\tau)+\sum_{k=1}^{K}\exp(\boldsymbol{v}\cdot\boldsymbol{v}_{k}^{-}/\tau)}\right],$

where $\tau$ is a temperature factor that scales the distance between the
query and samples.

###  4.2 Mask-Guided Contrastive Learning

Randomly sampled negative patches are likely semantically similar to the
positive sample, conflicting with contrastive learning assumptions [10]. To
address this, we propose a mask-guided sampling scheme. We binarize a
physical-based image to a particle-background mask $M(x,y)$, with $M(x,y)=1$
denoting particle and $M(x,y)=0$ denoting background. Guided by this mask, we
randomly select the positive and the negative samples at locations where the
mask labels are opposite, improving training efficiency and quality of
results.

Similar to CUT [33], our method leverages multiple intermediate layers of
$G_{\text{enc}}$ to extract multi-scale features from input patches. Notably,
we align the receptive field with particle size by selecting the
$G_{\text{enc}}$’s first $L$ layers. Feature map from each layer is then
passed through compact two-layer MLP networks $H_{l}$ ($l\in\\{1,2,...,L\\}$)
to generate a feature
$\boldsymbol{z}_{l}=H_{l}(G_{\text{enc}}^{l}(I_{\text{inter}}))$, where
$G_{\text{enc}}^{l}$ is the $l$-th layer. In all our experiments, we set $L$
to 5.

For particle-background mask locations, we define those on particles as $p\in
P$, and those on the background as $b\in B$, where $P$ represents all
locations with $M=1$ and $B$ represents all locations with $M=0$. The feature
of $l$-th layer at a particle position $p$ is denoted as
$\boldsymbol{z}^{p}_{l}$, and the set represented as
$\boldsymbol{z}^{P}_{l}=\left\\{\boldsymbol{z}^{p}_{l}\right\\}_{p\in P}$.
Similarly, for background positions, we denote the $l$-th layer feature as
$\boldsymbol{z}^{b}_{l}$, with the set represented as
$\boldsymbol{z}^{B}_{l}=\left\\{\boldsymbol{z}^{b}_{l}\right\\}_{b\in B}$.
Features encoded from the output $I_{\text{syn}}$ are represented as
$\hat{\boldsymbol{z}}$, where
$\hat{\boldsymbol{z}}_{l}=H_{l}(G_{\text{enc}}^{l}(I_{\text{syn}}))$. We
propose a novel mask-guided Noise Contrastive Estimation (NCE) loss for
contrastive learning:

$\displaystyle\mathcal{L}_{\text{MaskNCE}}(G,H,X|\mathcal{G})=$ (4)
$\displaystyle\mathbb{E}_{I_{\text{inter}}\sim
X|\mathcal{G}}\sum_{l=1}^{L}\Biggl{[}\sum_{p\in
P}\ell\left(\boldsymbol{z}_{l}^{p},\hat{\boldsymbol{z}}^{p}_{l},\hat{\boldsymbol{z}}^{B}_{l}\right)+\sum_{b\in
B}\ell\left(\boldsymbol{z}_{l}^{b},\hat{\boldsymbol{z}}^{b}_{l},\hat{\boldsymbol{z}}^{P}_{l}\right)\Biggr{]}.$

By minimizing MaskNCE loss, we effectively differentiate particle features and
background features.

Figure 4: Qualitative comparison results. Our approach achieves the most
authentic noise generation across all datasets. The Gaussian noise model
succeeds in preserving the structural information while lacking photo-realism.
CycleGAN and CUT introduce severe artifacts on the Ribosome dataset as they
cannot generate random noises.

###  4.3 Training Scheme

We introduce randomness to the physical-based intermediate image by adding a
random sampled Gaussian noise to a physical-based clean image. The
intermediate image then inputs into the generative function to get a synthetic
output image. Following the mask-guided sampling scheme, we extract multi-
scale features to calculate MaskNCE loss, with query in the input domain,
positive and negative in the output domain. Additionally, we introduce an
adversarial loss function, $\mathcal{L}_{\text{GAN}}$, to encourage the
network to produce more realistic simulated cryo-EM images, $I_{\text{syn}}$,
as follows:

$\displaystyle\mathcal{L}_{\text{GAN}}(G,D,$ $\displaystyle
X|\mathcal{G},Y)=\;\mathbb{E}_{I_{\text{real}}\sim Y}[\log
D(I_{\text{real}})]$ $\displaystyle+\mathbb{E}_{I_{\text{inter}}\sim
X|\mathcal{G}}[\log(1-D(G(I_{\text{inter}})))].$ (5)

The overall training loss function is:

$\mathcal{L}_{\text{GAN}}(G,D,X|\mathcal{G},Y)+\lambda\mathcal{L}_{\text{MaskNCE}}(G,H,X|\mathcal{G}),$
(6)

where $\lambda=10.0$ is a experimental hyperparameter for adjusting loss
weights.

##  5 Experiments

In this section, we evaluate our GenEM on a variety of challenging datasets.
We run all experiments on a single NVIDIA GeForce RTX3090 GPU and it takes
about 2 hours to train a GenEM with 200 epochs. We choose PatchGAN [17] as our
discriminator and UNet implemented by [33] as our generator. During each
training iteration, we resize the image resolution to $1024\times 1024$.
Guided by the particle-background mask, we evenly sample 256 queries on
particles and 256 on the background, ensuring that their corresponding
negative samples are located where the labels are opposite. The training
dataset consists of 100 real images and 100 physical-based simulated results.
We also do a data augmentation by rotating images by 90, 180, and 270 degrees.

Datasets. We evaluate GenEM on three challenging cryo-EM datasets. Each
includes expert-curated particle picking results for high-resolution
reconstruction, regarded as annotated particles. 1) T20S Proteasome
(EMPIAR-10025) [6] contains 196 images and 49,954 annotated particles. The
high density of these particles leads to occlusions, posing significant
challenges in particle picking. 2) 80S Ribosome (EMPIAR-10028) [45] contains
1,081 images with 105,417 annotated particles. The complexity of these
particles corresponding to a complex structure makes it difficult to estimate
their poses. 3) Asymmetric $\alpha\text{V}\beta 8$ Integrin (EMPIAR-10345) [7]
includes 1,644 micrographs and 84,266 annotated particles with varied
conformations, requiring more advanced processing for heterogeneous
reconstruction.

Figure 5: Qualitative comparison results of particle picking. We compare our
method against Pre-trained Topaz, Gaussian Noise, CycleGAN, and CUT. The blue
circles indicate matches with manual picking results, while the red circles
represent misses or excess picks by the model. Our approach achieves the
visually best performance compared to baselines. Further, we compare the
reconstructed results by cryoSPARC. Our reconstruction result reveals the most
detailed double helix structures, achieving the highest resolution.

###  5.1 Comparison

Similar to traditional image generation tasks, we begin with evaluating the
similarity between our generated data and actual cryo-EM data. To validate our
approach in the context of a cryo-EM data processing pipeline, we apply the
generated data to two critical downstream tasks: particle picking and pose
estimation. In the particle picking task, a state-of-the-art particle picking
model augmented by GenEM data achieves the highest performance improvement,
leading to better reconstruction resolution. In the pose estimation task, we
train a cryo-EM pose estimation model with an augmented dataset. The results
show that our model achieves the best reconstruction resolution on real
datasets. Across all experiments, GenEM outperforms existing baselines both
qualitatively and quantitatively.

Baselines. To evaluate the performance of GenEM compared to other existing
methods, we select a traditional baseline approach by adding Gaussian noise to
physically simulated results (denoted as Gaussian Noise). We also choose
existing state-of-the-art unpaired image-to-image translation works, CycleGAN
and CUT, as baselines to represent the performance of current deep generative
models.

best second-best
---
Metric | FID$\downarrow$
Dataset | Proteasome | Ribosome | Integrin
Gaussian Noise | 89.87 | 177.01 | 174.94
CycleGAN | 89.33 | 114.20 | 54.83
CUT | 46.61 | 101.54 | 72.30
Ours | 42.96 | 11.17 | 42.46

Table 1: Quantitative comparison of photo-realism. Compared with Gaussian
Noise, CycleGAN and CUT, our approach achieves the best performance in FID
metric.

Photo-realism. We compare GenEM with several baselines in terms of the realism
of synthesized cryo-EM images. As illustrated in Figure 4, our approach
consistently achieves the best quality across all datasets in terms of overall
noise distribution and the preservation of particle structural information.
While Gaussian noise effectively preserves structural information, it fails to
mimic authentic noise distributions. CycleGAN struggles with convergence on
the Proteasome dataset and has severe artifacts on background regions on the
Ribosome dataset due to the breakdown of its bijection assumption. CUT can
achieve better photo-realism compared to CycleGAN but its performance is still
limited due to its random sampling scheme. As shown in Table 1, we also
employed Frechet Inception Distance (FID) which is widely used in generative
tasks to measure the similarity between generated and real data. Our results
significantly outperform existing baseline methods in all three datasets.

best second-best
---
Metric | AUPRC$\uparrow$ | Res(Å) $\downarrow$
Dataset | Proteasome | Ribosome | Integrin | Proteasome | Ribosome | Integrin
Gaussian Noise | 0.471 | 0.485 | 0.259 | 2.50 | 3.76 | 7.63
CycleGAN | 0.200 | 0.308 | 0.414 | 6.83 | 4.14 | 6.42
CUT | 0.415 | 0.001 | 0.335 | 2.57 | 5.94 | 5.99
Pre-trained Topaz | 0.302 | 0.679 | 0.526 | 2.59 | 3.28 | 5.08
Topaz* | 0.599 | 0.749 | 0.650 | 2.53 | 3.27 | 5.36
Ours | 0.490 | 0.797 | 0.606 | 2.48 | 3.25 | 4.85

Table 2: Quantitative comparison of particle picking. Compared with Gaussian
Noise, CycleGAN, CUT, Pre-trained Topaz, and Topaz*, our approach consistently
achieves the best or the second best in AUPRC and Res(Å) metrics.

best second-best
---
Metric | Res(px)$\downarrow$
Dataset | Proteasome | Ribosome | Integrin
Gaussian Noise | 2.97 | 6.00 | 6.86
CycleGAN | 3.02 | 4.74 | 5.94
CUT | 2.66 | 5.40 | 6.34
Ours | 2.60 | 4.27 | 5.25

Table 3: Quantitative comparison of pose estimation. Compared with Gaussian
Noise, CycleGAN, and CUT, our approach achieves the best performance in
Res(px) metric.

Particle Picking. We fine-tune a popular particle picking model, Pre-trained
Topaz [4], with generated datasets with ground-truth annotations from
different methods. As shown in Figure 5, we use the fine-tuned Topaz to pick
particles on the Ribosome dataset. Our method outperforms the Pre-trained
Topaz and other baselines in terms of visual picking results and
reconstruction accuracy - our reconstruction result reveals the most detailed
double helix structures, achieving the best resolution. While other baseline
methods show a noticeable domain gap between their generated data and real
data, reducing Topaz’s performance. Notably, although the baseline methods
significantly impacted Topaz’s performance, the reconstruction results may be
not markedly decreased due to the sufficient quantity of data.

Table 2 presents the quantitative particle picking results, where we utilized
the Area Under the Precision-Recall Curve (AUPRC) metric. This metric is
particularly suitable for datasets with incomplete positive labeling, a
scenario evident in our manual annotations. Additionally, we include a version
of Topaz that has been fine-tuned on roughly 1000-2000 particle annotations
from 10 micrographs, referred to as Topaz*. The quantitative comparison
results on AUPRC demonstrate that our method enhances Topaz’s performance and
outperforms Topaz* with manual annotations on the Ribosome dataset. Taking
each particle picking result as an input, cryoSPARC [36] reconstructs a 3D
volume with its spatial resolution Res(Å). The quantitative comparison results
on Res(Å) show that our approach consistently achieves the best resolution.

Pose Estimation. In the cryo-EM reconstruction pipeline, the accuracy of pose
estimation is crucial for the resolution of the final reconstruction. Due to
the unknown ground-truth particle poses in real cryo-EM images, existing cryo-
EM reconstruction AI models, such as cryoFIRE [27], resort to self-supervised
learning to predict particle poses directly from images during reconstruction.
However, by leveraging the ground-truth particle pose information, we can
supervise particle poses for the pre-trained pose estimation module of
cryoFIRE. This approach improves the prediction of real particle poses, as
shown in Table 3.

best second-best
---
Dataset | Ribosome
Task | Photo-realism | Particle Picking | Pose Estimation
Metric | FID$\downarrow$ | AUPRC$\uparrow$ | Res(Å)$\downarrow$ | Res(px)$\downarrow$
w/o Physical Priors | 16.63 | 0.789 | 3.29 | 4.88
w/o Gaussian Noise | 50.74 | 0.735 | 3.31 | 4.24
Third Layer Noise | 15.83 | 0.764 | 3.31 | 4.45
Fifth Layer Noise | 21.30 | 0.777 | 3.26 | 4.46
w/o Mask | 12.61 | 0.721 | 3.29 | 4.44
Ours | 11.17 | 0.797 | 3.25 | 4.27

Table 4: Quantitative result of ablation study. Our complete model
consistently achieves the best or the second-best performance in FID, AUPRC,
Res(Å), and Res(px) metrics.

###  5.2 Ablation Study

To generate realistic cryo-electron microscopy images, our method incorporates
several key designs, including the integration of physical priors, the
introduction of randomness in the generative model, and a mask-guided
contrastive learning sampling strategy. We validated the effectiveness of
these designs on the Ribosome dataset. First, we remove physical priors
including ice thickness weighting and optical aberration simulation, denoted
as w/o Physical Priors. We also remove Gaussian noise addition (w/o Gaussian
Noise) to validate the effectiveness of introducing randomness. Further, we
explore introducing randomness at the feature level by adding Gaussian noise
at the third layer of the encoder (Third Layer Noise) and the fifth layer of
the encoder (Fifth Layer Noise). Lastly, we remove the mask-guided sampling
scheme (w/o Mask) to show how significantly it helps. As illustrated in Figure
6, the qualitative results show that removing modules or changing the position
of Gaussian noise struggles insignificant CTF effects, severe artifacts on
background noise, or content inconsistency. Quantitative results in Table 4
also show that our complete model consistently achieves the best or the
second-best performance in all metrics.

Figure 6: Qualitative evaluation of ablation study. The w/o Physical Prior
struggles insignificant CTF effects. The w/o Gaussian Noise, Third Layer
Noise, and Fifth Layer Noise create artifacts on the background. The w/o Mask
suffers content inconsistency.

##  6 Conclusion

Limitations. As the first trial to achieve authentic cryo-EM image generation
with a novel combination of physical-based simulation and unpaired noise
translation via contrastive learning, GenEM has some limitations. First, our
physical-based simulation relies on an initial 3D model, which may be hard to
obtain in some challenging cases, such as when the target molecule is too
small to detect or is too flexible to estimate poses. Existing protein
sequence prediction models like AlphaFold 2 [20] are helpful in giving an
initial 3D structure. Also, GenEM lacks the generalization capability like
changing input particles while still preserving realistic noises as we train
GenEM per scene. In the future, we are willing to train a generalized version
of GenEM to unlock more real applications in cryo-EM.

Conclusion. We have presented the first approach, GenEM, that marries
physical-based simulation with a contrastive noise generation, to improve
cryo-EM reconstruction results. To simulate a physical-based image, we first
prepare a virtual specimen that contains enormous target particles, then we
simulate the cryo-EM imaging process including projection, uneven ice
thickness, and optical aberrations. Then we adopt an unpaired noise
translation to transform simple Gaussian noises into complex realistic noises.
Finally, we significantly improve the training efficiency and performance by
proposing a novel mask-guided sampling scheme. Extensive experiments have
shown that GenEM’s generated dataset with ground-truth annotations can
effectively improve particle picking and pose estimation models, eventually
improving reconstruction results. We believe that GenEM serves as a critical
step for high-fidelity cryo-EM image generation, with essential cryo-EM
applications such as particle picking and pose estimation.

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# Magnetic structure in square cupola compound Ba(TiO)Cu4(PO4)4: a 31P NMR
Study

Riho Rästa  Ivo Heinmaa  National Institute of Chemical Physics and
Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia  Kenta Kimura  Tsuyoshi
Kimura  Department of Advanced Materials Science, University of Tokyo,
Kashiwa, Chiba 277-8561, Japan  Raivo Stern  National Institute of Chemical
Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia

(August 27, 2024)

###### Abstract

The magnetic structure of the antiferromagnetic square cupola compound
Ba(TiO)Cu4(PO4)4 with the tetragonal structure is studied with 31P nuclear
magnetic resonance techniques. The magnetic hyperfine shift $K$ shows a clear
splitting at the Néel temperature TN = 9.5 K, where the resonance splits into
two lines when an external magnetic field is oriented along the $c$ axis and
into four lines when the field is along the $a$ axis. In the paramagnetic
region $K(T)$ follows temperature dependence of the magnetic susceptibility
$\chi(T)$. From $K$ vs $\chi$ plot we determined nearly isotropic hyperfine
field values H${}_{\text{hf}}^{a}$ = 765 mT/$\mu_{B}$ and
H${}_{\text{hf}}^{c}$ = 740 mT/$\mu_{B}$ for the magnetic field oriented along
$a$ and $c$, respectively. From the rotation of the single crystal in the
external magnetic field we determined eight different orientations of
$K$-tensor in the paramagnetic region. In the antiferromagnetic state at $T$ =
6 K we found that the local field at phosphorus is mainly due to dipolar field
of coppers. Here the rotation of the single crystal shows eight different
orientations of the local field $B_{\text{int}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 35.6$ mT. The orientations correspond to the calculation
of dipolar fields at phosphorus assuming magnetic quadrupolar configuration of
magnetic moments $\varGamma_{3}(1)$ described previously [Nat. Commun. 7,
13039 (2016); Phys. Rev. B 96, 214436 (2017)].

###### pacs:

76.60.-k, 75.47.Lx, 75.50.Ee

##  I Introduction

Figure 1:  Description of BTCPO unit cell with up and down Cu4O12 square
cupola (blue), PO4 tetrahedra (green), and TiO5 pyramids (brown), and Ba ions
as large grey spheres along the $c$ (left panel) and $a$ (middle panel) axis.
Two possible arrangements of the spins $\varGamma_{3}(1)$ and
$\varGamma_{3}(2)$ (right panel).

Recently, an interesting class of compounds,$AE$(TiO)Cu4(PO4)4 ($AE$TCPO; $AE$
= Ba, Sr, or Pb), was newly synthesized Kimura2016magneto ; Kimura2018a  . The
structure of $AE$TCPO is tetragonal with the space group $P$4212 and consists
of the layers of up and down square cupola Cu4O12 (FIG. 1). Each square cupola
is made of four corner sharing CuO4 plaquets. The layers are separated by $AE$
ions and TiO5 pyramids. These compounds undergo an antiferromagnetic (AF)
transition at low temperatures. For example, Ba(TiO)Cu4(PO4)4 (BaTCPO)
exhibits an AF ordered state at temperatures below the Néel temperature TN =
9.5 K. Below TN the arrangement of magnetic moments of coppers shows an
antiferroic quadrupolar order. Because of the antiferroic order, BaTCPO does
not exhibit a magnetoelectric effect where an electric (magnetic) field causes
magnetization (electric polarization). Instead, this compound exhibits a
remarkable magnetodielectric effect. Sr(TiO)Cu4(PO4)4 (SrTCPO) also shows
similar magnetic and magnetodielectric properties below TN = 6.2 K SrPRRB2019
, while Pb(TiO)Cu4(PO4)4 (PbTCPO) exhibits a ferroic quadrupolar order
resulting in a linear magnetoelectric effect below TN = 6.5 K PbPRMat2018 ;
PbJPSJ2019 .

Magnetic structure of BaTCPO was carefully studied by neutron scattering
experiments Kimura2016magneto . These studies yielded two possible
arrangements of the spins: one, denoted $\varGamma_{3}(1)$, where the moments
are confined approximately in the CuO4 planes; and the other,
$\varGamma_{3}(2)$, where the moments are approximately perpendicular to the
CuO4 planes forming two-in-two-out-type structure (FIG. 1). The latter
arrangement had smaller R-factor, thus being the most probable arrangement.
Later Babkevich et al. Babkevich2017  confirmed the two possible structures
using spherical neutron polarimetry, inclining by a discernible, albeit small
advantage towards the $\varGamma_{3}(2)$ spin structure.

In the present report we will use 31P NMR techniques to study the local
magnetic fields in a single crystal of BaTCPO. Phosphorus ions in PO4
tetrahedra see four coppers in two cupolas as next-nearest-neighbours.
Therefore, the local field at phosphorus is definitely influenced by the
magnetic arrangement of coppers. Previous NMR studies of SrTPCO Islam2018  and
BaTPCO Kumar2019  have used powder samples where detailed information about
the magnetic structure is difficult to get.

##  II Experimental details

Single crystals of Ba(TiO)Cu4(PO4)4 were grown with the flux method by Kimura
et al. Kimura2018a ; Kimura2016inorganic . The sample crystal used in the
experiments sized 1.9 x 2.0 x 3.9 mm and weighed 61.84 mg. NMR measurements
were conducted using the spectrometer MAGRes2000 attached to a B = 4.7 T
superconducting magnet, 31P resonance frequency 80.97 MHz. He-flow cryostat
(JANIS Research Inc.) allowed measurements in the temperature range of 4.5 to
300 K. The NMR probe was our own made and had a single-axis goniometer.
LakeShore CERNOX calibrated sensors and Model 332 temperature controller were
used for temperature regulation. Spin-lattice relaxation T1 was measured using
inversion-recovery method. The magic angle spinning (MAS) room temperature
spectrum was recorded using a home-built probe with the $1.8$ mm O.D. rotor
spinning $25$ kHz. The magnetic shifts are given relative to the resonance of
H3PO4.

##  III Results and analysis

###  III.1 Magnetization

The temperature dependence of magnetic susceptibility was measured with the
applied magnetic field B = 4.7 T along the [001] and [100] directions as shown
in FIG. 2. At temperature region T${}^{\text{max}}\approx$ 17 K there is a
broad maximum of $\textstyle\chi$ vs T curve, which indicates onset of short
range order.

Figure 2: Temperature dependence of magnetic susceptibility
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(T)$
in an applied magnetic field B = 4.7 T, measured in two different directions:
B$\parallel$[100] (black, filled) and B$\parallel$[001] (red, empty). The cyan
line represents the Curie-Weiss fit. The inset shows a detailed behavior of
susceptibilities below T${}_{N}\leavevmode\nobreak\ =\leavevmode\nobreak\ 9.5$
K.

At high temperature
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(T)$
followed the Curie-Weiss law:

$\displaystyle{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(T)={\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{0}+\frac{C}{T-\theta_{\text{CW}}}.$
(1)

Here
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{0}$
is temperature-independent susceptibility. At T $>$ 100 K the fitting gives
the following parameters: for the magnetic field along the [001] direction:
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{0}\leavevmode\nobreak\
=\leavevmode\nobreak\ -7.3(4)\times 10^{-5}$ cm3/mole per Cu,
$C\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.460(1)$ cm3 K per mole Cu,
$\theta_{\text{CW}}\leavevmode\nobreak\ =\leavevmode\nobreak\ -29.0(4)$ K; and
for the field along the [100] direction:
${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{0}\leavevmode\nobreak\
=\leavevmode\nobreak\ -3.5(2)\times 10^{-5}$ cm${}^{3}/$mole-Cu,
$C\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.454(7)\leavevmode\nobreak\
\times 10^{-4}$ cm3 K $/$mole-Cu, and $\theta_{\text{CW}}\leavevmode\nobreak\
=\leavevmode\nobreak\ -29.3(2)$ K. From the Curie constant one gets the
effective copper magnetic moment as $\mu_{\text{eff}}=\sqrt{3k_{B}C/N_{A}}$,
where $k_{B}$ is the Boltzmann factor, and $N_{A}$ is Avogadro’s number. We
get $\mu_{\text{eff}}\leavevmode\nobreak\ =\leavevmode\nobreak\
1.920\leavevmode\nobreak\ \mu_{B}$ and 1.911 $\mu_{B}$ for [001] and [100]
directions, respectively. Here, $\mu_{B}\leavevmode\nobreak\ $ is the Bohr
magneton. Using $g\leavevmode\nobreak\ =\leavevmode\nobreak\
\sqrt{S(S+1)}\mu_{\text{eff}}$ we get almost equal Lande $g$-factor values as
$g\leavevmode\nobreak\ =\leavevmode\nobreak\ 2.22$ and $g\leavevmode\nobreak\
=\leavevmode\nobreak\ 2.20$ for [001] and [100], respectively, which are
common for Cu2+ ions.

Here we note, that in regular, collinear antiferromagnets the $\textstyle\chi$
vanishes in the direction where the local magnetic moments are aligned
parallel or antiparallel to an external magnetic field and stays constant in
the directions where the external field is perpendicular to the local magnetic
moments Singer1956 . Thus the local moments here may prefer the [001]
direction, but the non-vanishing $\textstyle\chi$ could also just be a
signature of the non-collinear order.

###  III.2 31P Knight shift

Figure 3: Temperature dependence of 31P Knight shift as measured when the
single crystal is oriented with $a$ axis parallel to the external field $B$
(a, filled blue dots) and with $c$ axis parallel to $B$ (b, empty circles).
The insets show the Clogston-Jaccarino plots giving quite close values to the
hyperfine field in both the directions.

Temperature dependence of 31P Knight shift is presented in FIG. 3. Above the
Néel temperature TN = 9.5 K, the Knight shift $K$(T) follows perfectly the
magnetic susceptibility curve $\textstyle\chi$(T), as given Clogston1961  by
Eq.(2):

$K(T)=K_{0}+\frac{\textit{H}_{\text{hf}}}{N_{A}\mu_{B}}{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}.$
(2)

Here, $K_{0}$ is the temperature-independent shift, the chemical shift, and
H${}_{\text{hf}}$ is the hyperfine coupling constant. From the $K$vs
$\textstyle\chi$ plots (insets of FIG. 3) we found the hyperfine field values
as H${}_{\text{hf}}$ = 7.65(5) kOe/$\mu_{B}$ for $a\parallel B$ direction and
slightly smaller value, H${}_{\text{hf}}$ = 7.40(5) kOe/$\mu_{B}$ for
direction $c\parallel B$.

Figure 4: Temperature development of the splitting of the resonance lines
below the ordering temperature in the orientation $a\parallel B$ (upper panel)
and $a\parallel B$ (lower panel). The solid lines are fittings by Eq. 3 .

Below TN, due to the onset of local magnetic fields, the resonance line splits
into four lines in the orientation $a\parallel B$ and into two lines when the
crystal is oriented with $c\parallel B$. Following the approach given in Ref.
Islam2018  we will estimate the critical order parameter $\beta$ of the phase
transition from the growth of the internal field $B_{int}$ in the vicinity of
the ordering temperature. As a measure of the internal field we use the
frequency difference between the most shifted resonance lines instead of the
internal ones, which is proportional to the internal field by gyromagnetic
ratio ${}^{31}\gamma/2\pi$ = 17.237 MHz/T. The temperature dependences of the
frequency differences in the two orientations of the single crystal are given
in FIG. 4.

The temperature dependence of the frequency difference is fitted by the
formula

$\Delta F(T)=\Delta F_{0}\left(1-\frac{T}{T_{n}}\right)^{\beta}.$ (3)

The best fit was obtained with TN = 9.48 K, $\beta$ = 0.26 for the orientation
$a\parallel B$, and TN = 8.82 K, $\beta$ = 0.20 for $c\parallel B$. The
critical exponent $\beta$ values, if compared to some theoretical values (as
given e. g. in Ref. Nath2009 ), indicate that the ordering scheme is the
closest to the 2D XY case.

###  III.3 31P NMR of powder sample

Figure 5: (a): 31P MAS NMR of BaTCPO powder sample at T = 305 K with a
spinning frequency $\nu$ = 25 kHz. The NMR spectrum of rotating sample
consists of a center band at isotropic Knight shift K${}_{\text{iso}}$ =
0.1643 %, and of a number of spinning sidebands denoted by asterisks at
multiples of the spinning frequency from the center band. (b): 31P NMR of a
static powder sample at slightly lower temperature T = 295 K. The red line is
a fit to the powder line with the eigenvalues of nearly axially symmetric
Knight shift tensor: K${}_{11}\leavevmode\nobreak\ =\leavevmode\nobreak\
0.1952$ %, K${}_{22}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.1832$ %,
K${}_{33}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.1298$ %.

Before starting 31P analysis of single crystal we recorded the spectrum of a
powder sample. FIG. 5(a) shows 31P NMR spectrum recorded with magic angle
spinning MAS-NMR_reference  of the sample. The spectrum shows a single sharp
line at isotropic magnetic shift. The spectrum of a static sample (FIG. 5(b))
shows a typical powder line shape with singularities at the principal values
of the Knight shift tensor:

$K_{PAS}=\begin{pmatrix}1+K_{11}&0&0\\\ 0&1+K_{22}&0\\\
0&0&1+K_{33}\end{pmatrix},$ (4)

with K${}_{11}=0.1952$ %, K${}_{22}=0.1832$ %, K${}_{33}=0.1298$ %. These
values give us the reference for the interpretation of the rotation patterns
of the single crystal.

###  III.4 Orientation of the 31P Knight shift tensor

Determination of tensor orientation in a single crystal is not very easy task
Mehring1976 . For that, in general case, one needs to record resonance
frequencies rotating the sample around three different axes. Depending on
symmetry or having some principal values of the shift tensor pre-determined,
the number of necessary rotation patterns may be smaller. After obtaining
those rotation patterns, one needs to find unitary transformation that will
transform the principal axis system (PAS) into the crystal frame. As usual,
the Hamiltonian of spin-$\frac{1}{2}$ nucleus consists of the Zeeman and the
Knight shift interaction

$\mathcal{H}=\mathcal{H}_{Z}+\mathcal{H}_{K}=\gamma\textbf{H}_{0}\cdot\textbf{K}\cdot\textbf{I},$
(5)

where $\textbf{I}=(I_{x},I_{y},I_{z})^{T}$, $\textbf{H}_{0}=(0,0,H_{0})$, and
$I_{x},I_{y},I_{z}$ are Pauli matrices:

$I_{x}=\frac{\hbar}{2}\begin{pmatrix}0&1\\\
1&0\end{pmatrix};I_{y}=\frac{\hbar}{2}\begin{pmatrix}0&i\\\
-i&0\end{pmatrix};I_{z}=\frac{\hbar}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.\\\ $ (6)

Three successive rotations, each characterized by three Euler angles,
transform the Hamiltonian from the principal axes frame (PAS) into the
laboratory frame:

$\displaystyle\begin{array}[]{c}K_{\scriptstyle PAS}\\\
\scriptstyle(1,2,3)\end{array}\underrightarrow{\scriptstyle
R_{p}(a_{1},a_{2},a_{3})}\begin{array}[]{c}K^{*}\\\
\scriptstyle(a,b,c)\end{array}\scriptstyle\underrightarrow{R_{i}(b_{1},b_{2},b_{3})}\begin{array}[]{c}K_{g}\\\
\scriptstyle(x_{g},y_{g},z_{g})\end{array}\scriptstyle\underrightarrow{R_{g}(c_{1},c_{2},c_{3})}\begin{array}[]{c}K\\\
\scriptstyle(x,y,z)\end{array}$

There are four frames of reference: PAS with diagonal tensor ($K_{PAS}$),
crystal frame ($K^{*}$), goniometer frame ($K_{g}$), and laboratory frame (K).
The transformations between them are $R_{p}$, $R_{i}$, $R_{g}$ with the
corresponding Euler angles. We used the conventional Euler ZYZ rotation of the
Hamiltonian easyspin .

In BaTCPO there are eight different positions of the phosphorus ions in the
unit cell, each giving the resonance corresponding to different transformation
$R_{p}$. The rotation patterns around the $c$ and $a$ axis at temperatures T =
295 K and T = 18 K are shown in FIG. 6. The angles $b_{1}$, $b_{2}$, $b_{3}$
and $c_{1}$, $c_{2}$, $c_{3}$ are unique for each experiment; $a_{1}$,
$a_{2}$, $a_{3}$ are desired eight sets of the Euler angles transforming the
tensor in PAS to the crystal frame for each phosporus site. It turns out that
the principal axis $K_{33}$ of the Knight shift tensor is tilted by 45 degrees
from the crystal $c$ axis. Schematics of the Knight tensor’s eight
orientations are presented in Fig. 7.

Figure 6:  31P NMR resonance frequencies by rotating BaTCPO single crystal around the $a$ (a) and $c$ axis (b) at T=295 K (upper panels) and T=18 K (lower panels). The full lines correspond to the angle dependencies according to the parameters given in Table 1. Due to symmetry, eight different sites in the unit cell contribute only to two different rotation patterns. The red lines in panel (a) correspond to the sites 1, 3, 6 and 7 (see Table 1), and to the sites 2, 3, 6 and 8 in panel (b), the rest of the sites are given by blue lines. Figure 7:  Orientations of the Knight shift tensor at eight phosphorus locations of the unit cell at temperature T = 295 K. The tensors have three principal vectors - green, orange, dark-violet, which correspond to the $K_{PAS}$ values $K_{11}$, $K_{22}$, $K_{33}$, respectively. Table 1: (a) and (b): the Euler angles for transforming BaTCPO single crystal shift tensors from crystal frame to the laboratory frame. Tables (c) and (d) show the Euler angles for transforming the tensors from PAS to the crystal frame. Tensor components of PAS ($K_{11}$,$K_{22}$,$K_{33}$) are in $\%$ units of $\omega_{L}$. (a) T=295K | T |  | (c) 295K |  |  | (d) 18K |  | T=295K
---|---|---|---|---|---|---|---|---
| $c_{1}$ | $c_{2}$ | $c_{3}$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | nr | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $K_{11}$ | $K_{22}$ | $K_{33}$
c$\parallel$zg | (-12:372) | 90 | 0 | 85 | 0 | 0 | 1 | 30 | 45 | -45 | 25 | 45 | -40 | $0.198$ | $0.185$ | $0.128$
a$\parallel$zg | (-12:372) | 90 | 0 | 0 | 90 | 0 | 2 | -30 | 45 | 45 | -25 | 45 | 50 |  |  |
|  |  |  |  |  |  | 3 | 30 | 45 | 135 | 25 | 45 | 140 |  |  |
(b) T=18K | 4 | -30 | 45 | -135 | -25 | 45 | -130 | T=18K
| $c_{1}$ | $c_{2}$ | $c_{3}$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | 5 | 30 | 135 | 45 | 25 | 135 | 50 | $K_{11}$ | $K_{22}$ | $K_{33}$
c$\parallel$zg | (-12:372) | 90 | 0 | 80 | 0 | 0 | 6 | -30 | 135 | -45 | -25 | 135 | -40 | $1.14$ | $1.05$ | $0.67$
a$\parallel$zg | (-12:372) | 90 | 0 | 0 | 90 | 0 | 7 | 30 | 135 | -135 | -25 | 135 | 140 |  |  |
|  |  |  |  |  |  | 8 | -30 | 135 | 135 | 25 | 135 | -130 |  |  |

###  III.5 Spin-lattice relaxation results

Spin-lattice relaxation $T_{1}$was measured with inversion-recovery pulse
sequence at magnetic fields along [001] and [100] directions (Fig. 8). The
magnetization recovery was exponential throughout all the measurements:

$M(\tau)=M_{0}(1-A\exp(-\tau/T_{1})),$ (8)

where $M(\tau)$ is the magnetization at delay $\tau$ after inversion, $M_{0}$
is the equilibrium magnetization, and $A\leavevmode\nobreak\ \leq$ 2 is a
constant depending on the accuracy of the inversion.

Figure 8:  Temperature dependence of 31P spin-lattice relaxation rate in
directions [100] (red dots) and [001] (blue dots). The inset shows that
relaxation rate 1/T1 is proportional to T7 below TN.

The relaxation rate at T $>$ 60 K is almost constant which is typical for a
paramagnetic material, where the relaxation is caused by the fluctuation of
the magnetic moments. Before the phase transition at TN, a sharp spike occurs
in the relaxation speed which is connected to the rapid slowing of the
fluctuations. Below TN, relaxation speed decreases sharply proportional to T7.
In case of $c\leavevmode\nobreak\ \parallel\leavevmode\nobreak\ B$, the
relaxation rate seems to have a discontinuity in close vicinity of TN. Here we
note, that 1/T1 $\propto$ T7 was also observed for SrTCPO Islam2018 .

In paramagnetic temperature region we can use the Moriya’s theory of
relaxation Moriya , where:

$\frac{1}{T_{1}}=\frac{\gamma^{2}_{N}\sqrt{2\pi}S(S+1)}{3\omega_{E}z^{\prime}}H^{2}_{hf},$
(9)

Here $\gamma_{N}$ is the nuclear gyromagnetic ratio, $S$ is the nuclear spin,
$H_{hf}$ is the hyperfine field as in Eq. 2, $z^{\prime}=2$ is the number of
nearest Cu2+ neighbors for the nucleus and
$\omega_{E}=(|J|k_{B}/\hbar)\sqrt{2zS(S+1)/3}$ is the Heisenberg exchange
frequency (in units rad-1), where $z$ = 2 is the number of Cu2+ ions as
nearest neighbors to 31P, $S$ = 1/2 is the electronic spin, and $J$ is the
exchange interaction.

With the hyperfine field value of $H_{hf}$ = 7.650 kOe/$\mu_{B}$ and the
relaxation rate for the paramagnetic region 1/T1 = 1410 s-1 we get an estimate
for the exchange interaction inside the square cupola $J/k_{B}$ = 35 K with an
exchange frequency of $\omega_{E}$ = 4.5$\cdot 10^{12}$ rad/s. This value is
in coherence with previous results of $J$ = 3.0 meV = 34.8 K Kimura2016magneto
.

###  III.6 Local magnetic structure in the ordered state

The resonance frequencies by rotation of the single crystal around the [001]
and [100] at T = 6 K are given in Fig. 9. The results are in coherence with
the Knight shift temperature dependence (FIG. 3). Once the magnetic field is
turned around the $c$ axis, one can see that when $a\parallel$ B, there are
four different magnetic field projections in the AFM region as given in FIG.
3(a). When the crystal is oriented $c\parallel$ B, there are two different
local field projections to the external field direction as given in FIG. 3(b).
Rotating the sample around $c$ and $a$ gives eight different rotation patterns
for eight phosphorus ions in the unit cell. Each rotation pattern can be
described by the equation:

$F=K+L\cos(\alpha-\alpha_{1})+M\cos(2(\alpha-\alpha_{2})),$ (10)

where the constant term $K$ is the Larmor frequency plus average chemical
shift; the second term describes the angle dependence of the local field
projection to the external field direction. The third term describes the angle
dependence of the resonance frequency due to turning of the chemical shift
tensor.

The rotation patterns around the [001] axis show two sets of lines (blue and
red). The phase shift within the set is 90 degrees. The red lines are shifted
from $b\parallel$ B by +16 degrees, while the blue lines are shifted -16
degrees. We can assign the blue lines to the phosphorus ions in “up” cupola
and the red lines to the ions in “down” cupola. The rotation patterns of the
crystal around the [100] (FIG. 9(b), Table 2b) are not so well resolved. Here,
the approximation of the frequencies by Eq. 10 is not particularly good. A
possible reason might be that the magnetic structure in 4.7 T magnetic field
at $c\parallel$ B direction is not yet well settled at temperature 6 K. We
found above (see FIG. 4) that the ordering temperature in the direction
$c\parallel$ B was TN = 8.8 K, while in case of $b\parallel$ B we had TN = 9.5
K.

Despite of that, one can clearly see the rotation patterns with two different
amplitudes as expected for two different local field projections along the $b$
axis. The assignment of the resonances to “up” and “down” cupola is not
unique. For example, we cannot distinguish the cases, where all the local
field directions of one cupola have positive projection to the $c$ axis and
the moments of the other cupola have negative projection, from the case, where
the ions of one cupola have two positive and two negative projections and the
field on ions of the other cupola have, respectively, two negative and two
positive projections to the $c$ axis. The assignment in FIG. 9(b) corresponds
to the local field configuration as given below in FIG. 11.

Figure 9: Rotation patterns of 31P NMR frequencies of BaTCPO single crystal at
temperature T = 6 K rotating around [001] (a) and [100] (b). The lines are
approximations with Eq. 10 using data in Table 2. The main directions of a
crystal $a\parallel$B, $b\parallel$B, and $c\parallel$B, $b\parallel$B are
noted in the respective $x$ axes. The blue symbols are assigned to the sites
in “up” cupola, the red symbols to the sites in “down” cupola. Table 2: The
fitting parameters $K$, $L$, $M$, $\alpha_{1}$, and $\alpha_{2}$ of the
rotation patterns according to Eq. 10 The upper table corresponds to the
rotation of the crystal around the $c$ axis (as shown on FIG. 9(a)), lower
table corresponds to rotation around the $a$ axis (FIG. 9(b)) at temperature T
= 6 K.

nr | $K$ | $L$ | $\alpha_{1}$ | $M$ | $\alpha_{2}$
---|---|---|---|---|---
1 | 81.59 | 0.56 | 347 | 0.075 | 155
2 | 81.59 | 0.56 | 20 | 0.075 | 25
3 | 81.58 | 0.56 | 80 | 0.075 | 50
4 | 81.59 | 0.56 | 110 | 0.075 | 125
5 | 81.59 | 0.55 | 170 | 0.075 | 155
6 | 81.595 | 0.56 | 198 | 0.065 | 30
7 | 81.59 | 0.575 | 258 | 0.078 | 50
8 | 81.60 | 0.56 | 288 | 0.065 | 130

nr | $K$ | $L$ | $\alpha_{1}$ | $M$ | $\alpha_{2}$
---|---|---|---|---|---
1 | 81.47 | 0.62 | 294 | 0.17 | 87
2 | 81.47 | 0.62 | 246 | 0.17 | 97
3 | 81.45 | 0.29 | 212 | 0.17 | 72
4 | 81.44 | 0.29 | 148 | 0.17 | 115
5 | 81.47 | 0.62 | 114 | 0.17 | 84
6 | 81.47 | 0.62 | 66 | 0.17 | 95
7 | 81.45 | 0.29 | 32 | 0.17 | 69
8 | 81.45 | 0.29 | 328 | 0.17 | 111

Figure 10:  Scheme of local field direction on a phosphorus ion in BaTCPO
crystal. B${}_{\text{int}}$ is the local field, Ba, Bb, and Bc are the
projections of B${}_{\text{int}}$ to the crystal $a$, $b$, and $c$ axis,
respectively. B1, B2, and B3 are the projections of B${}_{\text{int}}$ to the
$bc$, $ac$, and $bc$ planes, respectively. The latter amplitudes can be found
from the rotation pattern parameters given in Table 2.

The analysis of the data given in Table 2 can be carried out using the scheme
of the local field direction as given in FIG. 10. Three cosine amplitudes $L$
in Table 2 correspond to the local field projections B1, B2, and B3 in FIG.
10. Using the gyromagnetic ratio of 31P $\gamma/2\pi$ = 17.237 MHz/T, we find
B1 = 36 mT, B2 = 32.5 mT, and B3 = 16.8 mT. The angles $\alpha$ = $\pm$ 16,
$\beta$ = $\pm$ 66, and $\gamma$ = $\pm$ 32 degrees. It is not difficult to
calculate the projections Ba = 8.9 mT, Bb = 31.2 mT, and Bc = 14.6 mT, as well
as the module of the internal field B${}_{\text{int}}$ = 35.6 mT.

As noted above, unique assignment of the resonances to certain phosphorus in
the unit cell is not possible. One possible local field configuration is given
in FIG. 11.

Figure 11: Possible configuration of induced static magnetic fields at
phosphorus ions in BaTCPO unit cell. Red arrows represent the directions and
size of the static magnetic fields, plus and minus sign mean the field
direction to the front or to the back of the figure plane. Blue squares are
Cu2+ ions and the purple tetrahedrons show P ions.

The internal field at phosphorus ions consists of two components:

$B_{\text{int}}=B_{\text{hf}}+B_{\text{dip}},$ (11)

where $B_{\text{hf}}$ is transferred hyperfine field, and $B_{\text{dip}}$ is
the dipolar field from the magnetic moments of Cu2+ ions. In the following we
assume that $B_{\text{hf}}$ is very well cancelled in AF ordered state, and
the local magnetic field at phosphorus is due to dipolar field of Cu2+ ions.

We did calculate the dipolar field at each phosphorus ion of the unit cell
assuming the two magnetic structures proposed in Refs. Kimura2016magneto ;
Babkevich2017 . In the calculation we did sum the dipolar field from every
Cu2+ inside a sphere of 50 Å around given phosphorus. At that we took into
account that the unit cell of magnetic structure is doubled along $c$
direction, i. e. the magnetic moments of every other layer along the $c$ axis
were reversed. As a result, we obtain the following dipolar field projections
along the $a$, $b$, and $c$ axis

a) $\displaystyle B_{a}=37.5\leavevmode\nobreak\ \text{mT;
}B_{b}=90.1\leavevmode\nobreak\ \text{mT; }B_{c}=11.2\leavevmode\nobreak\
\text{mT, }$ $\displaystyle B_{\text{int}}=97.7\leavevmode\nobreak\
\text{mT;}$

for the structure noted $\varGamma_{3}(1)$, and

b) $\displaystyle B_{a}=42.2\leavevmode\nobreak\ \text{mT;
}B_{b}=36.0\leavevmode\nobreak\ \text{ mT; }B_{c}=0.75\leavevmode\nobreak\
\text{ mT, }$ $\displaystyle B_{\text{int}}=55.5\leavevmode\nobreak\ \text{
mT;}$

for the structure noted $\varGamma_{3}(2)$.

In comparison, our experiments above give the values:

$\displaystyle B_{a}=8.9\leavevmode\nobreak\ \text{mT;
}B_{b}=31.2\leavevmode\nobreak\ \text{mT; }B_{c}=14.6\leavevmode\nobreak\
\text{mT; }$ $\displaystyle B_{\text{int}}=35.6\leavevmode\nobreak\
\text{mT.}$

Figure 12:  Calculated dipolar field direction at phosphorus ions of the
BaTCPO unit cell (a): assuming structure of Cu2+ magnetic moments
$\varGamma_{3}(1)$ (given in Ref. Kimura2016magneto  ), and (b): assuming the
structure $\varGamma_{3}(2)$. The red arrows represent the dipole fields. The
length of the arrows corresponds to the relative value of dipolar field in
both cases and that in FIG. 11. Blue arrows represent directions of the Cu2+
magnetic moments and their sizes in the two cases.

Calculated dipolar field directions at the phosphorus ions are given in FIG.
12.

Comparison of the experimental field pattern to the calculated dipolar fields
gives remarkable similarity to the case calculated for $\varGamma_{3}(1)$
structure - the calculated dipolar field directions are close to the
experimental values in FIG. 11, although the calculated Bc value is relatively
small and the calculated local field is 2.7 times larger than the experimental
value. Three times larger value of the calculated dipolar field compared to
the experimentally determined local field was reported earlier for dipolar
field at Ba site in antiferromagnetic YBa2Cu3O6.05 Ref. Lombardi1996 . The
authors ascribed this controversy to the possible effect of delocalization of
the copper $d$ \- electron. Dipolar field calculated for $\varGamma_{3}(2)$
structure is quite different. It is almost confined to the $ab$ plane, with
nearly equal Ba and Bb components. Therefore, we can conclude that the NMR
data are better consistent with the magnetic structure $\varGamma_{3}(1)$.

##  IV Conclusion

We performed 31P NMR of the antiferromagnetic square cupola compound
Ba(TiO)Cu4(PO4)4 in the applied magnetic field B = 4.7 T and provided an in-
depth overview of the local magnetic environment around the Cu2+ cupolas. From
the 31P NMR frequency dependence of the single crystal orientation we
successfully determined the principal values of the 31P magnetic shift tensor
and the orientation of eight magnetic tensors in the unit-cell at room
temperature and at temperature T = 6 K. The Knight shift temperature
dependence in comparison of that of the bulk magnetic susceptibility enabled
to determine the hyperfine field on 31P nuclei H${}_{\text{hf}}$ = 7.65(5)
kOe/$\mu_{B}$ for $a\parallel B$ and H${}_{\text{hf}}$ = 7.40(5) kOe/$\mu_{B}$
for $c\parallel B$. The temperature dependence of 31P spin-lattice relaxation
resulted in an approximation for the exchange interaction constant between
Cu2+ ions $J\leavevmode\nobreak\ =\leavevmode\nobreak\ 35K$. 31P NMR frequency
dependence on the single crystal orientation in the antiferromagnetic state
gave a clear picture of local magnetic fields at 31P ions. The static magnetic
field at every phosphorus was determined as
$B_{\text{int}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 35.6$ mT.
Experimental configuration of the local field was compared to the calculated
dipolar field for several magnetic arrangement of the copper magnetic moments.
We found that the magnetic structure $\varGamma_{3}(1)$ determined by the
previous neutron diffraction studies Kimura2016magneto ; Babkevich2017  is
most consistent with the NMR data.

## Acknowledgemnts

This research was supported by the Estonian Science Council Grants IUT23-7 and
PRG4, the European Regional Development Fund Grant TK134, JSPS KAKENHI Grant
Numbers JP17H01143, JP19H05823 and JP19H01847, and by the MEXT Leading
Initiative for Excellent Young Researchers (LEADER).

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