# Transformer Network for Semantically-Aware and Speech-Driven Upper-Face
Generation ††thanks: This work was performed within the Labex SMART
(ANR-11-LABX-65) supported by French state funds managed by the ANR within the
Investissements d’Avenir programme under reference ANR-11-IDEX-0004-02.

Mireille Fares  ISIR, STMS
Sorbonne Université
Catherine Pelachaud  CNRS
ISIR, Sorbonne Université Nicolas Obin  STMS
IRCAM, Sorbonne Université, CNRS

###### Abstract

We propose a semantically-aware speech driven model to generate expressive and
natural upper-facial and head motion for Embodied Conversational Agents (ECA).
In this work, we aim to produce natural and continuous head motion and upper-
facial gestures synchronized with speech. We propose a model that generates
these gestures based on multimodal input features: the first modality is text,
and the second one is speech prosody. Our model makes use of Transformers and
Convolutions to map the multimodal features that correspond to an utterance to
continuous eyebrows and head gestures. We conduct subjective and objective
evaluations to validate our approach and compare it with state of the art.

###### Index Terms:

Semantically and Speech Driven Gestures, Transformers, Visual Prosody,
Embodied Conversational Agents

##  I Introduction

###  I-A Context

The human face is a key channel of communication in human-human interaction.
During speech, humans spontaneously and continuously display various facial
and head gestures that convey a large panel of information to the
interlocutors. These gestures are known as “visual prosody” [1]: facial or
head movement are produced in conjunction with verbal communication. During
speech, the Fundamental Frequency (F0) variations and upper-facial movements
are highly correlated [2]; they are the results of linguistic and
conversational choices. Eyebrow motion can also occur during pauses. Another
kinematic–acoustic relation that happens during production of speech is
between F0 and head motion [2]. Natural and rhythmic head movements are one of
the key factors in producing natural animations [3, 4].

###  I-B Related works and our contributions

A large number of head motion generation systems has been proposed in previous
works [5, 6, 7, 8, 9, 10, 11]. A variety of generative statistical models
aimed to predict the multimodal behavior of a virtual agent. Hidden Markov
Models (HMM) [5], Recurrent Neural Networks (RNN) [11, 6], and Dynamic
Bayesian Networks (DBN) [9, 10] have been used to generate head motion from
speech; Generative Adversarial Networks (GAN) have been proposed to produce
facial gestures from speech [12, 8]. [13] generates continuous 3D hand
gestures based on acoustics and semantics. However, most of the aforementioned
approaches exploit as input one modality only, namely speech, and neglect to
render their approach using semantic information. Also, most of them focus
only on facial expressions while the correlation between facial expressions
and head movements are crucial to produce a natural behavior. For instance,
[5, 6, 7, 10] do not generate eyebrow motion along with head motion, which are
both correlated to F0 [2], and therefore are correlated to each other. On the
other hand, transformer networks and attention mechanisms have been recently
proved to be very efficient for sequence-to-sequence modelling, with
particular advances for modelling multimodal processes. For instance,
Transformers were previously used for translating speech to text (ASR)[14,
15], and multimodal learning of images based on text [16]. To overcome those
limitations, we propose a novel approach for upper-facial and head gestures
generation based on a multimodal transformer network. Our contributions can be
listed as follows: 1) a transformer network operating on multi-modal input
text and speech information in order to generate upper-facial and head
movements, and 2) a cross-attention module that can efficiently exploit
semantic and speech information. 111Video samples of our model’s gestures
predictions and other related material can be found in:
https://github.com/mireillefares/VAAnimation/blob/main/README.md

The paper is organised as follows. The next section describes the proposed
architecture of our model and its multi-modal features. Then our objective and
subjective evaluation experiments are presented including the LSTM-based
Baseline Model. We finally discuss our results.

##  II Proposed architecture

###  II-A Multimodal Input/Output Features

The upper-facial movements and head rotations are represented by mean of
Action Units (AUs) as defined in the Facial Action Coding Systems (FACS)[17]
and 3D head angles. The work presented in this paper only considers the AUs
that represent eyebrows movements which are: inner raise eyebrow _AU1_ , outer
raise eyebrow _AU2_ , frown _AU4_ , upper lid raiser _AU5_ , cheek raiser
_AU6_ , and lid tightener _AU7_. Head rotations have three degrees of freedom,
represented by the Euler angles _roll_ , _pitch_ and _yaw_. They are
represented by _RX_ , _RY_ and _RZ_ which are the rotation of the head with
respect to the _X_ , _Y_ and _Z_ axes. In this paper, F0 values were extracted
from the speech signal at a 5ms audio frame rate, linearly interpolated
between unvoiced segments, and clipped to the range of 50 to 550Hz which
represents the F0 range of human speech. Since F0, AU intensities (AU) and
head rotations (R) are continuous, they were quantized to produce a finite set
of discrete values to reduce the model size and energy consumption [18].
Finally, BERT word embeddings were extracted from the text transcription.

(a)

(b)

(c)

Figure 1: Face Gesture Generation Model Architecture

###  II-B Face Gesture Generation Model Architecture

The proposed architecture aims at mapping multimodal speech and text feature
sequence into continuous facial and head gestures. This problem is treated as
a multimodal sequence-to-sequence problem, for which a transformer network
operating at the word level is presented as illustrated in Fig.1 (a). The
inputs and outputs of the transformer network consist of one feature vector
for each word _W_ of the input text sequence. The input text corresponds to an
Inter-Pausal Unit (IPU) which is a sequence of words separated by silent
pauses longer than 0.2 seconds. F0 input sequence as well as AU intensities
_AU_ and head rotations _R_ output sequences have a variable length. We set
the maximum F0 input sequence length to 100, and the maximum _AU/R_ output
length to 124. Shorter sequences were padded to the maximum length, and longer
ones were truncated. In order to handle continuous flow of input and output
information with different timing, the transformer is wrapped with a F0
encoder module at its input and to a _AU/R_ decoder at its output. The
objective of the F0 encoder is to encode the continuous F0-values at the word
level and the objective of the _AU/R_ decoder is to reconstruct the continuous
values for _AU_ and _R_ from a word-level encoding. Each spoken word _W_ is
represented by a word-level F0 embedding vector _X ${}^{(1)}_{W}$_ which at
its turn corresponds to the sequence of F0-values
${\textbf{X}^{(1)}}=(F0_{1},\dots,F0_{N_{W}})$, where $N_{W}$ is the number of
F0-values corresponding to the spoken word _W_ ; and _X ${}^{(2)}_{W}$_ is the
BERT word embedding vector corresponding to the spoken word _W_ \- including
silences. Silences less than 0.2 secs may belong to IPUs. They do not have a
contextual BERT embedding. Hence, we replaced them by a comma - _”,”_. Each
spoken word _W_ is also represented by a word-level _AU_ vector
${\mathbf{Y}^{(j)}_{W,{AU}}}$ (resp. _R_ vector
${\mathbf{Y}^{(j)}_{{W},{R}}}$) which in turn corresponds to the sequence of
values
d${\textbf{Y}^{(j)}_{{AU}}=(AU_{1}^{(j)},\dots,AU_{N^{\prime}_{W}}^{(j)})}$
(resp. ${\textbf{Y}_{R}^{(j)}=(R_{1}^{(j)},\dots,R_{N^{\prime}_{W}}^{(j)})}$)
where $j$ denotes the $j^{th}$ _AU/R_ and $N^{\prime}_{W}$ the number _AU/R_
-values corresponding to the word _W_.

F0 Encoder: As depicted in Fig.1 (a), for each _W_ , three one-dimensional
convolutional layers are applied to project the input F0 sequence _X
${}^{(1)}_{W}$_ into a word-level representation of F0 contours covering local
context of F0 variations. These convolutional layers include _64_ filters,
with a _kernel size_ equal to _3_. The generated output vector of the latter
layers _X ${}^{(1)}_{W}$_ is then fed as an input to the _Transformer
Encoder_.

Transformer Encoder: The transformer encoder architecture is depicted in Fig.1
(b); it is similar to the one proposed in [19]. In our work, it is composed of
a stack of _N = 4_ identical layers. Each layer has two sub-layers: the first
one is a multi-head self attention mechanism with _4 attention heads_ , and
the second one is a position-wise fully connected feed-forward network. As the
original transformer encoder, we employ a residual connection around each of
the 2 sub-layers, followed by layer normalization.

Cross-Modality Attention Module (CMAM): The output of the transformer encoder
_Z_ , as well as _X ${}^{(2)}_{W}$_ are fed as inputs to the Cross-Modality
Attention Module (see Fig.1 (a)). This Module has the same structure as the
Transformer Decoder in [19]. It generates a representation that can take into
account both modalities, text and speech. The representation learning is done
in a master/slave manner, where one modality - the master - is used to
highlight the extracted features in the other modality - the slave. This
module takes _X ${}^{(2)}_{W}$_ \- Text Modality - as master, and _Z_ \-
Speech Modality - as slave. Thus, it performs cross-attention such that the
attention mask is derived from text modality, and is harnessed to leverage the
latent features from the speech modality.

Transformer Decoder: The decoder is composed of _N = 4_ identical decoding
layers, with _4 attention heads_. Similar to the one proposed in [19], it is
composed of residual connections applied around each of the sub-layers,
followed by layer normalization. As depicted in Fig.1 (c), the self-attention
sub-layer in the decoder stack is modified to prevent positions from attending
to subsequent positions. The output predictions which are offset by one
position, and this masking ensure that the predictions for position index _j_
depend only on the known outputs at positions less than _j_. We use 6
Transformer Decoders, one for each _AU/R_. For simplicity, Fig.1 (a) only
illustrates one decoder.

AU/R decoder: As depicted in Fig.1 (a), the Transformer Decoder outputs are
concatenated together, then fed to 3 one-dimensional convolutional layers that
include 64 filters, with a kernel size equal to 3, to learn the correlation
between the 6 output features, and therefore the correlation between facial
and head movements. Finally, a Dense layer with a Softmax activation function
is applied on each of the outputs, to convert the outputs to predicted next-
token probabilities. The final output sequences are ${\mathbf{Y}_{{AU}}}$ and
${\mathbf{Y}_{{R}}}$.

Transformer Sub-Layers and Hyperparameters: the transformer encoder and
decoders have attention sub-layers, and contain fully connected feed-forward
networks which are applied to each position separately and identically.
Similarly to other sequence to sequence models, we use learned embeddings to
convert the input tokens and output tokens to vectors of dimension dmodel =
64. All sub-layers and embedding layers therefore use this dimension. The
inner feed-forward layers are of dimension dff = 400. Positional encodings are
applied to the inputs of the transformer encoder and decoders. They have the
same dimension as the embeddings, so that they can be added together. We use
sine and cosine functions, similar to [19].

##  III Experiments

###  III-A Material and Experimental Setups

We trained it on a subset of the TED dataset collected in [20], containing
preprocessed AUs/R, F0s, and BERT embeddings of filtered shots where speakers’
face and head are visible and close to the camera. Our subset consists of the
features of 200 videos. Videos vary between 2 and 25 minutes, with a frame
rate of 24 FPS, the total numbers of IPUs is 919, and of words is 62307. We
shuffled all the IPUs, then split them into: training set (80%), validation
set (10%) and test set (10%). There are two test conditions: SD (Speaker
Dependent) and SI (Speaker Independent). The SD condition aims to assess to
what extent the model can generalize on new sentences pronounced by a speaker
seen during training - training set included multiple speakers. The SI
condition aims to assess the extent to which gestures predictions can be
extrapolated to unseen speakers. Each training batch contained 128 pairs of
word embeddings, F0 sequences, and their corresponding AUs, RX, RY and RZ. The
loss function used is the categorical cross-entropy between predicted and
actual values. We used Adam optimizer with $\beta_{1}=0.9,\beta_{2}=0.98$ and
$\epsilon=10^{9}$. We used a learning rate scheduler as in [19], with _warmup
steps = 4000_. We applied a _dropout = 0.1_ to the output of each sub-layer of
the transformer, and to the sums of the positional encodings in the
Transformer encoder and decoder stacks. All features values were normalized
between 0 and 1. The total number of the model’s parameters is 2051133.

###  III-B Objective Evaluation

To assess the quality of the generated gestures, we used the following
measures: _Root Mean Squared Error_ (_RMSE_), _Pearson_ Correlation
Coefficient (_PCC_), _Activity Hit Ratio_ (_AHR_) and _Non-Activity Hit Ratio_
(_NAHR_). AHR and NAHR were proposed by Freeman et al. [21], to evaluate the
performance of Voice Activity Detector (VAD) systems. We also considered them
since evaluating AU activity looks similar to VAD evaluation. We considered an
AU as _“Activated”_ when its value is greater than _0.5_ , otherwise it is
_“Not-Activated”_. _AHR_ is the percentage of predicted AU activation with
respect to ground truth. If it is greater than 100%, it means that the model
is predicting more activation than the amount of activation that is in the
ground truth. _NAHR_ is the same but for non-activity. We assessed the full
model using the _SD_ test set. We additionally evaluated its capacity to
generalize on the _SI_ set. To evaluate the different parts of our
architecture, we conducted an ablation study as follows: _(1)_ speech
ablation, _(2)_ text ablation, _(3)_ CMAM ablation, and _(4)_ AU/R decoder
ablation. The ablation study was evaluated using RMSE metric.

| Proposed Transformer Mode (SD) | SOTA Baseline (SD)
---|---|---
| RMSE | PCC | AHR | NAHR | RMSE | PCC | AHR | NAHR
AU01 | 0.082 | 1.0 | 98.0 | 101.0 | 0.20 | -0.012 | 42.56 | 115.95
AU02 | 0.028 | 0.99 | 100.0 | 100.0 | 0.48 | -0.002 | 34.15 | 107.08
AU04 | 0.037 | 1.0 | 99.0 | 130.0 | 0.53 | -0.012 | 60.31 | 121.85
AU05 | 0.023 | 1.00 | 100.0 | 100.0 | 0.50 | -0.011 | 21.12 | 135.95
AU06 | 0.060 | 1.0 | 99.3 | 102.6 | 0.48 | -0.002 | 33.11 | 135.05
AU07 | 0.10 | 1.0 | 98.0 | 104.1 | 0.33 | -0.053 | 20.21 | 131.52
RX | 0.16 | 1.0 | NA | NA | 0.53 | 0.018 | NA | NA
RY | 0.24 | 0.99 | NA | NA | 0.97 | -0.024 | NA | NA
RZ | 0.30 | 0.94 | NA | NA | 0.22 | 0.003 | NA | NA
TABLE I: Objective Evaluation: comparison of proposed transformer model vs.
SOTA baseline model

###  III-C Comparing to LSTM-based Baseline Model

We compared our approach to a sequence to sequence LSTM-based model [20] to
predict upper-face movements based on speech and text. This LSTM model[20] was
developed only for predicting eyebrows movements _(AUs)_. We extended it to
include head movements _(R)_. This extended model consists of mapping
sequences of F0 and the Bert embedding that correspond to a Word _(W)_ , to
the sequences of _(AUs/R)_ of that corresponding _W_. It employs 2 layers of
Bidirectional LSTMs to encode the concatenation of word-level F0 contours and
Bert embeddings. The hidden internal states are then transmitted to 6 decoders
to produce the corresponding AUs and R for a given input. The decoders are
followed by _Dense Layers_ with _Softmax Activation_. The hyperparameters of
the extended LSTM model are as follows: in both encoder and decoders, the
first layer of bidirectional LSTM has 200 units, and the second one has 100
units. The activation function used in these layers is _LeakyReLU_ with _alpha
= 0.01_. We trained and tested this model using the same training, validation
and test sets described in Section III - A. We trained it on 300 epochs, using
a _batch size = 128_. We used _Root Mean Squared Propagation (RMSProp)_
optimizer, and _Categorical Cross Entropy Loss_. We used the same metrics
described in III.B to compare this SOTA Baseline model to our Transformer-
based model.

###  III-D Subjective Evaluation

To investigate human perception of the facial gestures produced by our model,
we conducted two different experimental studies using the virtual agent Greta
[22]. We followed the recommendations proposed in [23], by adapting them to
facial gesture generation and assessed the _naturalness_ , _coherence_ , and
_human-likeness_ of the virtual agent’s gestures. Since we are not evaluating
deictic and iconic gestures, we did not use the metrics _appropriateness_ ,
and _intelligibility_ as proposed in [23]. We added the metrics
_synchronization_ , and _alignment_ to evaluate the gestures’ temporal
property with speech. Participants in both studies were fluent in English,
with a University degree, and recruited on Prolific, a crowd sourcing website.
We added attention checks at the beginning of our perceptual evaluations, to
filter out inattentive participants. The first study was done by 35
participants, and consisted of presenting 16 videos: each video showed the
virtual agent saying a sequence of words that corresponds to a sequence of
IPUs. We considered 4 conditions: 4 videos (condition M) used our full model
of _SD_ gestures predictions; 4 videos (condition GT) were simulated using the
gestures extracted from TED videos, which serve as ground truth; 4 videos of
the virtual agent were simulated using the LSTM-based Baseline model of _SD_
predictions (condition SOTA). The remaining 4 videos were produced using
predicted gesture animation of IPUs with the sound of other IPUs (condition
E). The second study was conducted by 55 participants. The goal of the second
study was to evaluate our model when simulated with SI data, and therefore its
capability to generalise to new speakers. It included 8 videos: 4 were
simulated with our model’s SI predictions, and 4 using SI gestures extracted
from _SI_ set (described in _III.A_), which serve as ground truth. For each
video in both studies, participants were asked to rate the 5 factors, namely
_naturalness_ , _coherence_ , _human-likeness_ , _synchronization_ , and
_alignment_ of the virtual agent’s gestures on a 1 to 7 likert scale [23]. The
questions were listed in a random order. The agent’s mouth movements were
blurred to prevent participants from getting distracted by these gestures
which were not inferred by our model, and therefore focus on the model’s
generated gestures.

##  IV Results and Discussion

Table I reports the model’s as well as the SOTA’s objective evaluation results
using the same _SD_ test set. Results reveal that RMSE errors are much smaller
for M ($0.0229\leq$_error_ $\leq 0.3$) than SOTA ($0.2046\leq$_error_
$\leq$0.9786). On the other hand, PCC coefficients show that M’s predictions
(0.94$\leq$_PCC_ $\leq$1.0) are more correlated than SOTA’s predictions
(-0.0525$\leq$_PCC_ $\leq$0.0179) to GT. AHR and NAHR were calculated to
measure the activation of AUs only, since they are not applicable for R.
Results show that M predicts better the activation rate AHR (_AHR_ $\geq$98.0)
than SOTA (20.211$\leq$_AHR_ $\leq$60.314). The non-activation rate is higher
for SOTA (107.084$\leq$_NAHR_ $\leq$135.95) than for M(100$\leq$_NAHR_
$\leq$130). This constitutes objective validation that M gives better results
than SOTA in terms of _error_ , _correlation_ , and AU’s _activation rate_.
Ablation studies conducted on our model resulted in higher RMSE errors when
performing Speech and Text Ablation for some AUs/R. For instance, AU01 RMSE
error (0.0819 for full model) increased to 0.0836 with text ablation and to
0.09 with speech ablation. On the contrary, RZ RMSE score (0.3036 for full
model) increased to 0.3089 with text ablation, and 0.3111 with speech
ablation. AU/R Decoder ablation resulted in even higher RMSE errors especially
for head rotations. AUs and R RMSE scores increased after CMAM ablation (i.e.
AU01 RMSE increased to 0.0901). This constitutes objective validation that the
use of multi-modal inputs (speech and text modalities) in M improves
predictions. Thus we can also conclude that CMAM module is an efficient and a
key component of our model, as it improves the generation accuracy of face
gestures and head rotations. As mentioned previously, we also tested our model
on the _SI_ set. RMSE errors are between 0.301 and 0.89 for AUs, and between
0.25 and 0.93 for R. As we could expect, we got higher errors than the errors
we had for the _SD_ condition (Table I) since the speakers in _SI_ set were
not seen by our model during the training phase.

(a)

(b)

Figure 2: Subjective Evaluation Results

For our first perceptive study conducted on _SD_ data, Fig. 2 (a) shows the
mean scores obtained on the 5 factors for the 4 conditions: our model M, the
baseline SOTA, the ground truth GT, and the error E. For the 5 factors, M is
perceived as much closer to GT than SOTA and E, especially in terms of
_Alignment_ and _Synchronization_ between speech and gestures. The mean
difference between M and GT is 0.72 for _Alignment_ and 0.74 for
_Synchronization_ (Fig.2 (a)). SOTA is perceived as the closest to E
especially in terms of _Coherence_ , and _Alignment_ : the mean difference is
0.4 and 0.44 respectively (Fig.2 (a)). We also conducted _ANOVA Between-
Subjects_ to test whether there are significant differences between the 4
conditions along all the 5 factors. Results showed that for all factors, tests
were significant (_p_ $<$0.001). We additionally performed a post-hoc
_Fisher’s LSD_ Test to do pair-wise comparisons of the means between the
factors of all conditions. Significant results (_p_ $<$0.001) were found for
all the factors when comparing the condition GT with the 3 other conditions
and when comparing our model M with SOTA and E. In particular, there were
significant differences between M and SOTA in terms of the 5 factors (_p_
$<$0.007). This constitutes experimental validation that when used with _SD_
data, condition M is perceived significantly better than SOTA and E for all
the factors. The difference for the 5 factors between M and GT is not
significant. For our second perceptive study conducted on _SI_ data, Fig. 2
(b) shows the means scores obtained on all factors for conditions M and GT:
for the 5 factors M is perceived as close to GT. As our data were not normally
distributed (Shapiro test’s p$<$0.5), we conducted a post-hoc unpaired
Wilcoxon test on each factor for the two conditions. For all tests, we could
not find significant differences (p$>$0.05). Thus our model when used with
_SI_ data and GT received similar values for the 5 factors.

##  V Conclusion

We have presented a new approach for modelling upper facial and head gestures
using Transformer model. We conducted objective and subjective evaluations
that showed that our model produces animations that are close to the ground
truth in term of expressivity while ensuring that speech and computed gestures
are aligned and synchronized. In a next future, we plan to expand our model to
learn different speaker styles for gesture generation.

## References

  * [1] H. P. Graf, E. Cosatto, V. Strom, and Fu Jie Huang,  “Visual prosody: facial movements accompanying speech,”  in IEEE Int Conf on Automatic Face Gesture Recognition, 2002, pp. 396–401.
  * [2] H Yehia, T Kuratate, and E Vatikiotis-Bateson,  “Linking facial animation, head motion and speech acoustics,”  Journal of Phonetics, vol. 30, no. 3, pp. 555–568, 2002.
  * [3] Y Ding, C Pelachaud, and T Artieres,  “Modeling multimodal behaviors from speech prosody,”  in Intelligent Virtual Agents, 2013, pp. 217–228.
  * [4] L Chen, G Cui, C Liu, Z Li, Z Kou, Y Xu, and C Xu,  “Talking-head generation with rhythmic head motion,”  in Eu Conference on Computer Vision, 2020, pp. 35–51.
  * [5] G Hofer and H Shimodaira,  “Automatic head motion prediction from speech data,”  in Interspeech, 2007.
  * [6] K Haag and H Shimodaira,  “Bidirectional LSTM networks employing stacked bottleneck features for expressive speech-driven head motion synthesis,”  in Intelligent Virtual Agents, 2016, pp. 198–207.
  * [7] J Lu and H Shimodaira,  “Prediction of head motion from speech waveforms with a canonical-correlation-constrained autoencoder,”  arXiv preprint arXiv:2002.01869, 2020.
  * [8] K Vougioukas, S Petridis, and M Pantic,  “Realistic speech-driven facial animation with gans,”  Int Journal of Computer Vision, pp. 1–16, 2019.
  * [9] S Mariooryad and C Busso,  “Generating human-like behaviors using joint, speech-driven models for conversational agents,”  IEEE Trans on Audio, Speech, & Language Processing, vol. 20, no. 8, 2012.
  * [10] N Sadoughi and C Busso,  “Speech-driven animation with meaningful behaviors,”  Speech Communication, vol. 110, pp. 90–100, 2019.
  * [11] S Wang, L Li, Y Ding, C Fan, and X Yu,  “Audio2head: Audio-driven one-shot talking-head generation with natural head motion,”  preprint arXiv:2107.09293, 2021.
  * [12] T Karras, T Aila, S Laine, A Herva, and J Lehtinen,  “Audio-driven facial animation by joint end-to-end learning of pose and emotion,”  ACM Trans on Graphics (TOG), vol. 36, no. 4, pp. 1–12, 2017.
  * [13] T Kucherenko, P Jonell, S van Waveren, G E Henter, S Alexandersson, I Leite, and H Kjellström,  “Gesticulator: A framework for semantically-aware speech-driven gesture generation,”  in ACM Int Conf on Multimodal Interaction, 2020, pp. 242–250.
  * [14] O Hrinchuk, M Popova, and B Ginsburg,  “Correction of automatic speech recognition with transformer sequence-to-sequence model,”  in ICASSP. IEEE, 2020, pp. 7074–7078.
  * [15] A Mohamed, D Okhonko, and L Zettlemoyer,  “Transformers with convolutional context for asr,”  arXiv preprint arXiv:1904.11660, 2019.
  * [16] S Yao and X Wan,  “Multimodal transformer for multimodal machine translation,”  in ACL, 2020, pp. 4346–4350.
  * [17] P Ekman and EL Rosenberg,  What the face reveals: Basic and applied studies of spontaneous expression using the Facial Action Coding System (FACS),  Oxford University Press, USA, 1997.
  * [18] Y Guo,  “A survey on methods and theories of quantized neural networks,”  arXiv preprint:1808.04752, 2018.
  * [19] A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, L Kaiser, and I Polosukhin,  “Attention is all you need,”  arXiv preprint arXiv:1706.03762, 2017.
  * [20] M Fares,  “Towards multimodal human-like characteristics and expressive visual prosody in virtual agents,”  in ICMI, 2020, pp. 743–747.
  * [21] W Q Ong, A W C Tan, V V Vengadasalam, C H Tan, and T H Ooi,  “Real-time robust voice activity detection using the upper envelope weighted entropy measure and the dual-rate adaptive nonlinear filter,”  Entropy, vol. 19, no. 11, 2017.
  * [22] C Pelachaud,  “Greta: a conversing socio-emotional agent,”  in ACM SIGCHI Int WS on ISIAA, 2017, pp. 9–10.
  * [23] P Wolfert, N Robinson, and T Belpaeme,  “A review of evaluation practices of gesture generation in embodied conversational agents,”  arXiv preprint arXiv:2101.03769, 2021.


analysis, the latter is subject to $1/m_{c}$ corrections which might well be
sizeable.

At last let us mention that we performed a non-trivial test of the
identification in Eq. (A.81). Approximating our FF-expression to the pole
part, inserting it into our rate in Eq. (2.15), and then comparing to the rate
in Ref. [41] (cf. their Eq. (25)), we can confirm that Eq. (A.81) is
consistent with both rates. This is a strong hint of the correctness of the
treatment in our work and theirs.

####  A.5.2 The dispersion representation of the $B\to\gamma^{*}$ off-shell
form factors

The assumed $q^{2}=0$ is well below the various $m_{B}^{2}$-type poles and
does not affect the computation. However, in the variable $k^{2}$ there are
the previously mentioned $\rho/\omega$ ($\phi$)- and $\Upsilon$-resonances
(cf. Table 7) which are far away from our region of interest $k^{2}\approx
m_{B}^{2}$ and therefore have little impact. If one wanted to fit the FFs at
lower $k^{2}$ then a dispersion ansatz, e.g. (A.2), could be combined with the
$z$-expansion.

$\begin{array}[]{ l rrr
}&m_{B_{q}}=m_{0^{-}}&m_{B^{*}}=m_{1^{-}}=m_{\perp}&m_{B_{q1}}=m_{1^{+}}=m_{\parallel}\\\
\hline\cr\hline\cr{b\to
s}&5.367\,\textrm{GeV}&5.415\,\textrm{GeV}&5.829\,\textrm{GeV}\\\ {b\to
d,u}&5.280\,\textrm{GeV}&5.325\,\textrm{GeV}&5.726\,\textrm{GeV}\\\
\hline\cr\end{array}$
Table 7: Lowest resonance masses for FFs [66]. The mass $m_{0^{-}}$ is the
$B$-meson mass in $B_{q}\to\gamma$. The Form Factors $V,T_{\perp}$ and
$V,T_{\parallel}$ are associated with the resonances $J^{P}=1^{-}$ and $1^{+}$
respectively.

####  A.5.3 Fit ansatz and $z$-expansion

The procedure to fit the FFs and how to include the correlation of
uncertainties largely follows Ref. [29]. Based on the previous part of this
section let us first motivate the fit-ansatz before summarising the essence of
the $z$-expansion. There are four on-shell FFs and at $q^{2}=0$ there are five
off-shell FFs,

on-shell:
$\displaystyle\\{V^{B\to\gamma}_{\perp,\parallel}(q^{2}),T^{B\to\gamma}_{\perp,\parallel}(q^{2})\\}\;,$
off-shell:
$\displaystyle\\{P^{B\to\gamma^{*}}(0,k^{2})\;,V^{B\to\gamma^{*}}_{\perp,\parallel}(0,k^{2}),\;T^{B\to\gamma^{*}}_{\perp,{\mathbb{L}}}(0,k^{2})\\}\;.$
(A.82)

  * •

The on-shell FFs are parameterised

$F_{n}^{B\to\gamma}(q^{2})=\frac{1}{1-q^{2}/m_{R}^{2}}\left(\alpha_{n0}+\sum_{k=1}^{N}\alpha_{nk}(z(q^{2})-z(0))^{k}\right)\;,$
(A.83)

using the knowledge of the presence of the first pole $m_{R}$ (A.5.1), cf.,
Table 7. The remaining part in brackets are supposed to take into account
higher states in the spectrum. Specifically the $\alpha_{nk}$-coefficients are
to be determined from a fit and $z(q^{2})$ is defined further below. The
constraint of the residue, cf. (A.5.1) and Table 6, is implemented by

$r_{V_{\perp}}=\alpha_{V_{\perp}0}+\sum_{k=1}^{N}\alpha_{V_{\perp}k}(z(m_{B^{*}_{q}}^{2})-z(0))^{k}\;,$
(A.84)

and similarly for other FFs. Further to that the constraint (A.11) is imposed
by

$T^{B\to\gamma}_{\perp}(0)=T^{B\to\gamma}_{\parallel}(0)\;\;\Leftrightarrow\;\;\alpha_{T_{\perp}0}=\alpha_{T_{\parallel}0}\;.$
(A.85)

  * •

The off-shell FFs are simply parameterised by

$F_{n}^{B\to\gamma^{*}}(0,k^{2})=\alpha_{n0}+\sum_{k=1}^{N}\alpha_{nk}(z(k^{2})-z(0))^{k}\;,$
(A.86)

The constraint $V_{\parallel}^{*}(0,m_{B_{q}}^{2})=2P^{*}(0,m_{B_{q}}^{2})$
(A.26) is imposed

$\alpha_{V_{\parallel}^{*}0}+\sum_{k=1}^{N}\alpha_{V_{\parallel}^{*}k}(z(m_{B^{*}_{q}}^{2})-z(0))^{k}=2\big{(}\alpha_{P^{*}0}+\sum_{k=1}^{N}\alpha_{p^{*}k}(z(m_{B^{*}_{q}}^{2})-z(0))^{k}\big{)}\;.$
(A.87)

The fit-ansatz (A.86) could easily be improved including the information on
the $\rho/\omega$ ($\phi$)-like resonances from the dispersion representation
(A.2).202020The extension to fit the two-variable FF
$F_{n}^{B\to\gamma^{*}}(q^{2},k^{2})$ is not straightforward but one would
best proceed by building an ansatz from a double dispersion relation in
$q^{2}$ and $k^{2}$ and in addition force the constraints (A.9,A.11,A.1.2).

Let us now describe the $z$-expansion in order to remain self-consistent. The
function $z(t)$ is defined by

$z(t)=\frac{\sqrt{t_{+}-t}-\sqrt{t_{+}-t_{0}}}{\sqrt{t_{+}-t}+\sqrt{t_{+}-t_{0}}}\;,$
(A.88)

where $t_{0}\equiv t_{+}(1-\sqrt{1-t_{-}/t_{+}})$ and
$t_{\pm}\equiv(m_{B_{q}}\pm m_{\rho})^{2}$. The $\rho$-mass,
$m_{\rho}=770\mbox{MeV}$, is just an arbitrary reference scale and the values
of $m_{B_{q}}$ are given in Table 7.

The coefficients $\alpha_{nk}$ are determined by fitting $N=200$ random points
at each integer value of $q_{i}^{2}$ (in $\textrm{GeV}^{2}$-units) in a
specific interval. Uncertainties in input parameters, $p\pm\delta p$, as for
example $m_{b}$, are accounted for by sampling them with a normal distribution
$N(p,\delta p)$, which accounts for the same correlations as in Ref. [29]. The
$N=200$ random samples of $F_{I}=F_{i}(q_{j}^{2})$, where $I=(i,j)$ denotes
the collective index for the FF-type and the momentum, determine the
$(ij)\times(ij)$ covariance matrix

$C_{IJ}=\langle F_{I}F_{J}\rangle-\langle F_{I}\rangle\langle F_{J}\rangle\,.$
(A.89)

Angle brackets denote the average over random samples. The coefficients
$\alpha_{nk}$ are then found by minimising the function

$\chi^{2}(\\{\alpha\\})=\sum_{IJ}(F_{I}^{\text{sample}}-F_{I}^{\text{fit}}(\\{\alpha\\}))C_{IJ}^{-1}(F_{J}^{\text{sample}}-F_{J}^{\text{fit}}(\\{\alpha\\}))\;,$
(A.90)

for each random sample, where the correlation matrix remains constant for all
samples. The fitted values of $\alpha$ are then averaged over all the samples
and errors are calculated from the standard deviation, which is justified
because each of the samples are statistically independent.

  * •

The computation of the four on-shell FFs [35] are limited to roughly
$q^{2}<14\mbox{GeV}^{\,2}$. The $200$ sample points are generated for each
integer interval in $q^{2}\in[-5,14]\mbox{GeV}^{\,2}$ to which the
$\alpha_{n}$’s are then fitted to the ansatz (A.86).

  * •

Since we only need the off-shell FFs in the region
$k^{2}\in[(4.9\mbox{GeV})^{2},m_{B_{d,s}}^{2}]\mbox{GeV}^{\,2}$ we restrict
our fitting procedure to this region.

##  Appendix B The $B_{q}\to\ell\ell\mathbb{V}$ for searches beyond the
Standard Model

Figure 7:  Illustration of the three-body decay of a $B_{q}$ to a pair of
leptons and a new, light, beyond-the-standard-model vector $\mathbb{V}$
induced by a flavour-violating vector coupling ($\bar{b}\not{\mathbb{V}}q$,
see Lagrangian in Eq. (B.1)).

The main focus of this work has been to propose a search for the QCD axion via
the three-body decay $B_{q}\to\ell\ell a$ and the computation of the FFs
contributing to this beyond-the-SM decay channel as well as to the radiative
$B_{q}\to\ell\ell\gamma$ decay. However, the computation of the off-shell FFs
allows us to extend this analysis to searches for light, beyond-the-SM vector
bosons ($\mathbb{V}$) with flavour violating couplings. In this appendix, we
present the $B_{q}\to\ell\ell\mathbb{V}$ rate, which is relevant for a search
via $B_{q}\to\mu\mu$, and briefly discuss the search’s sensitivity.

We parametrise the relevant flavour violating couplings via the effective
Lagrangian

$\displaystyle\mathscr{L}_{\mathbb{V}}$
$\displaystyle=\left(\mathbb{V}_{\mu}\bar{q}\gamma^{\mu}\left(g^{V}_{bq}\frac{m_{\mathbb{V}}^{2}}{\Lambda^{2}}+g^{A}_{bq}\frac{m_{\mathbb{V}}^{2}}{\Lambda^{2}}\gamma_{5}\right)b+\partial_{\mu}\mathbb{V}_{\nu}\bar{q}\sigma^{\mu\nu}\left(\frac{g_{bq}^{T}}{\Lambda}+\frac{g_{bq}^{T_{5}}}{\Lambda}\gamma_{5}\right)b+\text{h.c.}\right)-\frac{1}{2}m_{\mathbb{V}}^{2}\mathbb{V}^{\mu}\mathbb{V}_{\mu}\;,$
(B.1)

where $\Lambda$ is the heavy NP scale. Above, we opted to factorise out the
term $m_{\mathbb{V}}^{2}/\Lambda^{2}$ in the dimension-four vector- and axial-
vector couplings to demonstrate the “restoration” of gauge invariance in the
limit $m_{\mathbb{V}}\to 0$. We further introduce the quantities

$\displaystyle\hat{g}_{bq}^{V}\equiv\frac{m_{\mathbb{V}}^{2}}{\Lambda^{2}}g_{bq}^{V}\,,$
$\displaystyle\hat{g}_{bq}^{A}\equiv\frac{m_{\mathbb{V}}^{2}}{\Lambda^{2}}g_{bq}^{A}\,,$
$\displaystyle\hat{g}_{bq}^{T}\equiv\frac{m_{B_{q}}}{\Lambda}g_{bq}^{T}\,,$
$\displaystyle\hat{g}_{bq}^{T_{5}}\equiv\frac{m_{B_{q}}}{\Lambda}g_{bq}^{T_{5}}\,.$
(B.2)

which simplify some of the intermediate formulae.

Based on Eq. (B.1) we compute the $B_{q}\to\ell\ell\mathbb{V}$ rate using the
helicity formalism. The $B_{q}\to\ell\ell\mathbb{V}$ decay differs from the
semileptonic and flavour-changing-neutral-current decays $B\to\rho\ell\nu$ and
$B\to K^{*}\ell\ell$ in that the vector meson is emitted from the weak vertex
and the lepton pair originates from an off-shell photon cf. Figure 7. The
helicity amplitude for the decay reads

${\cal
A}_{\lambda}\propto\omega^{*}_{\mu}(\mathbb{V},k,\lambda)\epsilon_{\rho}^{*}(\gamma^{*},q,\lambda)~{}\left(\hat{g}^{V}_{bq}M^{\rho\mu}_{V}+\hat{g}^{A}_{bq}M^{\rho\mu}_{A}+\hat{g}^{T}_{bq}M^{\rho\mu}_{T}+\hat{g}^{T_{5}}_{bq}M^{\rho\mu}_{T_{5}}\right)$
(B.3)

where $g^{T}_{bq}\equiv m_{B_{q}}/\Lambda^{T}_{bq}$ and analogous for $T_{5}$
as this leads to transparent formulae. Explicit polarisation vectors are given
in Eq. (A.1.3). The helicity amplitudes are then given by

$\begin{split}{\cal
A}_{\perp}&\;=\;(\lambda^{(\mathbb{V})}_{B_{q}})^{1/2}F_{\perp}\;,\\\\[2.84544pt]
{\cal
A}_{\parallel}&\;=\;(m_{\mathbb{V}}^{2}-m_{B_{q}}^{2})F_{\parallel}\;,\\\\[2.84544pt]
{\cal
A}_{0}&\;=\;\frac{m_{\mathbb{V}}}{2\sqrt{2q^{2}}(m_{B_{q}}^{2}-q^{2})}\left(\lambda^{(\mathbb{V})}_{B_{q}}F_{\mathbb{L}}-(m_{B_{q}}^{2}-m_{\mathbb{V}}^{2})(m_{B_{q}}^{2}-m_{\mathbb{V}}^{2}+3q^{2})F_{\parallel}\right)\;,\end{split}$
(B.4)

with
$\lambda^{(\mathbb{V})}_{B_{q}}=\lambda(m_{B}^{2},q^{2},m_{\mathbb{V}}^{2})$
and the FFs

$F_{\perp}=\hat{g}^{V}_{bq}V^{*}_{\perp}(m_{\mathbb{V}}^{2},q^{2})+\hat{g}^{T}_{bq}T^{*}_{\perp}(m_{\mathbb{V}}^{2},q^{2})\;,\quad
F_{\parallel,\mathbb{L}}=\hat{g}^{A}_{bq}V^{*}_{\parallel,\mathbb{L}}(m_{\mathbb{V}}^{2},q^{2})-\hat{g}^{T_{5}}_{bq}T^{*}_{\parallel,\mathbb{L}}(m_{\mathbb{V}}^{2},q^{2})\;.$
(B.5)

The minus sign in the last expression originates from the minus sign in Eq.
(2.1).

Our FFs provide a good description in the case where $m_{\mathbb{V}}$ is much
smaller than all other scales since we have evaluated them for
$m_{\mathbb{V}}=0$. Three observations on the helicity amplitudes are in
order. Firstly, in the $m_{\mathbb{V}}\to 0$ limit the pseudoscalar part in
the longitudinal component survives since
$V_{\mathbb{L}}^{*}(m_{\mathbb{V}}^{2},q^{2})=-2m_{B_{q}}^{2}/m_{\mathbb{V}}^{2}\hat{V}^{*}(m_{\mathbb{V}}^{2},q^{2})$
in general and $\hat{V}^{*}(0,q^{2})=P^{*}(0,q^{2})$ in particular. Secondly,
in the $m_{\mathbb{V}}\to 0$ limit the part of the amplitude proportional to
$\hat{g}_{bq}^{T}$ happens to be identical to the part proportional to
$\hat{g}_{bq}^{T_{5}}$, i.e., both are proportional to the same FF, namely
$T^{*}_{\perp}(0,q^{2})$ (see Eq. (A.34)). Finally, it is observed that at the
kinematic endpoint $q^{2}\to(m_{B}-m_{\mathbb{V}})^{2}$ one has ${\cal
A}_{0}={\cal A}_{+}={\cal A}_{-}$ (where $\sqrt{2}{\cal
A}_{\perp(\parallel)}={\cal A}_{+}\mp{\cal A}_{-}$) as a result of the
restoration spherical symmetry [68].

For the differential rate we find

$\frac{d\Gamma}{dq^{2}}(\bar{B}_{q}\to\ell\ell\mathbb{V})=\frac{\alpha^{2}Q_{\ell}^{2}(\lambda^{(\mathbb{V})}_{B_{q}})^{1/2}\lambda_{\gamma}^{1/2}}{24\pi
m_{B}^{5}q^{6}}(q^{2}+2m_{\ell}^{2})\left(|{\cal A}_{\perp}|^{2}+|{\cal
A}_{\parallel}|^{2}+|{\cal A}_{0}|^{2}\right)\;,$ (B.6)

with $\lambda^{(\mathbb{V})}_{B_{q}}$ defined above and
$\lambda_{\gamma}\equiv\lambda(q^{2},m_{\ell}^{2},m_{\ell}^{2})=q^{2}(q^{2}-4m_{\ell}^{2})$.

The sensitivity study for searching for such light, flavour-violating vectors
at the tail of $B_{q}\to\mu\mu$ is analogous to the axion study presented in
Section 3. As an illustration we show here the expected $90\%$ CL exclusion
limits with the full LHCb data set of $300$ fb-1 for the different cases

$\displaystyle\frac{m_{\mathbb{V}}^{2}}{\Lambda^{2}}{g}^{V}_{bs}$
$\displaystyle<9\times 10^{-7}\,,$
$\displaystyle\frac{m_{\mathbb{V}}^{2}}{\Lambda^{2}}{g}^{V}_{bd}$
$\displaystyle<4\times 10^{-6}\,,$
$\displaystyle\frac{m_{\mathbb{V}}\,m_{B_{s}}}{\Lambda^{2}}{g}^{A}_{bs}$
$\displaystyle<2\times 10^{-7}\,,$
$\displaystyle\frac{m_{\mathbb{V}}\,m_{B_{d}}}{\Lambda^{2}}{g}^{A}_{bd}$
$\displaystyle<6\times 10^{-7}\,,$
$\displaystyle\frac{g^{T/T_{5}}_{bs}}{\Lambda}$
$\displaystyle<\frac{10^{-6}}{\text{GeV}}\,,$
$\displaystyle\frac{g^{T/T_{5}}_{bd}}{\Lambda}$ $\displaystyle<\frac{3\times
10^{-6}}{\text{GeV}}\,.$

The bounds are computed for the case of $m_{\mathbb{V}}\to 0$ so contain only
the leading in $m_{\mathbb{V}}/m_{B_{q}}$ term.

## References

  * [1] S. Weinberg, _A New Light Boson?_ , _Phys. Rev. Lett._ 40 (1978) 223–226.
  * [2] F. Wilczek, _Problem of Strong p and t Invariance in the Presence of Instantons_ , _Phys. Rev. Lett._ 40 (1978) 279–282.
  * [3] R. D. Peccei and H. R. Quinn, _CP Conservation in the Presence of Instantons_ , _Phys. Rev. Lett._ 38 (1977) 1440–1443.
  * [4] R. D. Peccei and H. R. Quinn, _Constraints Imposed by CP Conservation in the Presence of Instantons_ , _Phys. Rev._ D16 (1977) 1791–1797.
  * [5] J. Preskill, M. B. Wise and F. Wilczek, _Cosmology of the Invisible Axion_ , _Phys. Lett._ B120 (1983) 127–132.
  * [6] L. F. Abbott and P. Sikivie, _A Cosmological Bound on the Invisible Axion_ , _Phys. Lett._ B120 (1983) 133–136.
  * [7] M. Dine and W. Fischler, _The Not So Harmless Axion_ , _Phys. Lett._ B120 (1983) 137–141.
  * [8] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner and K. A. van Bibber, _Experimental Searches for the Axion and Axion-Like Particles_ , _Ann. Rev. Nucl. Part. Sci._ 65 (2015) 485–514, [1602.00039].
  * [9] F. Wilczek, _Axions and Family Symmetry Breaking_ , _Phys. Rev. Lett._ 49 (1982) 1549–1552.
  * [10] L. Calibbi, F. Goertz, D. Redigolo, R. Ziegler and J. Zupan, _Minimal axion model from flavor_ , _Phys. Rev._ D95 (2017) 095009, [1612.08040].
  * [11] Y. Ema, K. Hamaguchi, T. Moroi and K. Nakayama, _Flaxion: a minimal extension to solve puzzles in the standard model_ , _JHEP_ 01 (2017) 096, [1612.05492].
  * [12] F. Björkeroth, L. Di Luzio, F. Mescia and E. Nardi, _$U(1)$ flavour symmetries as Peccei-Quinn symmetries_, _JHEP_ 02 (2019) 133, [1811.09637].
  * [13] L. Di Luzio, F. Mescia, E. Nardi, P. Panci and R. Ziegler, _Astrophobic Axions_ , _Phys. Rev. Lett._ 120 (2018) 261803, [1712.04940].
  * [14] F. Björkeroth, L. Di Luzio, F. Mescia, E. Nardi, P. Panci and R. Ziegler, _Axion-electron decoupling in nucleophobic axion models_ , _Phys. Rev. D_ 101 (2020) 035027, [1907.06575].
  * [15] K. Saikawa and T. T. Yanagida, _Stellar cooling anomalies and variant axion models_ , _JCAP_ 03 (2020) 007, [1907.07662].
  * [16] J. Martin Camalich, M. Pospelov, P. N. H. Vuong, R. Ziegler and J. Zupan, _Quark Flavor Phenomenology of the QCD Axion_ , _Phys. Rev. D_ 102 (2020) 015023, [2002.04623].
  * [17] R. D. Peccei, _The Strong CP Problem_ , _Adv. Ser. Direct. High Energy Phys._ 3 (1989) 503–551.
  * [18] M. S. Turner, _Windows on the Axion_ , _Phys. Rept._ 197 (1990) 67–97.
  * [19] J. L. Feng, T. Moroi, H. Murayama and E. Schnapka, _Third generation familons, b factories, and neutrino cosmology_ , _Phys. Rev._ D57 (1998) 5875–5892, [hep-ph/9709411].
  * [20] F. Björkeroth, E. J. Chun and S. F. King, _Flavourful Axion Phenomenology_ , _JHEP_ 08 (2018) 117, [1806.00660].
  * [21] ATLAS collaboration, M. Aaboud et al., _Study of the rare decays of $B^{0}_{s}$ and $B^{0}$ mesons into muon pairs using data collected during 2015 and 2016 with the ATLAS detector_, _JHEP_ 04 (2019) 098, [1812.03017].
  * [22] CMS collaboration, S. Chatrchyan et al., _Measurement of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ Branching Fraction and Search for $B^{0}\to\mu^{+}\mu^{-}$ with the CMS Experiment_, _Phys. Rev. Lett._ 111 (2013) 101804, [1307.5025].
  * [23] LHCb collaboration, R. Aaij et al., _Measurement of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching fraction and effective lifetime and search for $B^{0}\to\mu^{+}\mu^{-}$ decays_, _Phys. Rev. Lett._ 118 (2017) 191801, [1703.05747].
  * [24] F. Dettori, D. Guadagnoli and M. Reboud, _$B^{0}_{s}\to\mu^{+}\mu^{-}\gamma$ from $B^{0}_{s}\to\mu^{+}\mu^{-}$_, _Phys. Lett._ B768 (2017) 163–167, [1610.00629].
  * [25] CMS collaboration, _Combination of the ATLAS, CMS and LHCb results on the $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ decays_, .
  * [26] Belle-II collaboration, W. Altmannshofer et al., _The Belle II Physics Book_ , _PTEP_ 2019 (2019) 123C01, [1808.10567].
  * [27] A. A. Alves Junior et al., _Prospects for Measurements with Strange Hadrons at LHCb_ , _JHEP_ 05 (2019) 048, [1808.03477].
  * [28] LHCb collaboration, R. Aaij et al., _Search for the rare decay $D^{0}\to\mu^{+}\mu^{-}$_, _Phys. Lett._ B725 (2013) 15–24, [1305.5059].
  * [29] A. Bharucha, D. M. Straub and R. Zwicky, _$B\to V\ell^{+}\ell^{-}$ in the Standard Model from light-cone sum rules_, _JHEP_ 08 (2016) 098, [1503.05534].
  * [30] M. Beneke, T. Feldmann and D. Seidel, _Systematic approach to exclusive $B\to Vl^{+}l^{-}$, $V\gamma$ decays_, _Nucl. Phys._ B612 (2001) 25–58, [hep-ph/0106067].
  * [31] J. Lyon and R. Zwicky, _Isospin asymmetries in $B\to(K^{*},\rho)\gamma/\ell^{+}\ell^{-}$ and $B\to K\ell^{+}\ell^{-}$ in and beyond the standard model_, _Phys. Rev._ D88 (2013) 094004, [1305.4797].
  * [32] A. Kozachuk, D. Melikhov and N. Nikitin, _Rare FCNC radiative leptonic $B_{s,d}\to\gamma l^{+}l^{-}$ decays in the standard model_, _Phys. Rev._ D97 (2018) 053007, [1712.07926].
  * [33] D. Melikhov and N. Nikitin, _Rare radiative leptonic decays $B(d,s)\to l^{+}l^{-}\gamma$_, _Phys. Rev._ D70 (2004) 114028, [hep-ph/0410146].
  * [34] M. Beneke, C. Bobeth and Y.-M. Wang, _$B_{d,s}\to\gamma\ell\bar{\ell}$ decay with an energetic photon_, _JHEP_ 12 (2020) 148, [2008.12494].
  * [35] T. Janowski, B. Pullin and R. Zwicky, _Charged and Neutral $\bar{B}_{u,d,s}\to\gamma$ Form Factors from Light Cone Sum Rules at NLO_, 2106.13616.
  * [36] B. Pullin and R. Zwicky, _Radiative Decays of Heavy-light Mesons and the $f_{H,H^{*},H_{1}}^{(T)}$ Decay Constants_, 2106.13617.
  * [37] C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou and M. Steinhauser, _$B_{s,d}\to\ell^{+}\ell^{-}$ in the Standard Model with Reduced Theoretical Uncertainty_, _Phys. Rev. Lett._ 112 (2014) 101801, [1311.0903].
  * [38] D. Guadagnoli, M. Reboud and R. Zwicky, _B ${}_{s}^{0}$ $\to\ell^{+}$ $\ell^{-}\gamma$ as a test of lepton flavor universality_, _JHEP_ 11 (2017) 184, [1708.02649].
  * [39] J. Lyon and R. Zwicky, _Resonances gone topsy turvy - the charm of QCD or new physics in $b\to s\ell^{+}\ell^{-}$?_, 1406.0566.
  * [40] D. M. Straub, _flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond_ , 1810.08132.
  * [41] Y. G. Aditya, K. J. Healey and A. A. Petrov, _Faking $B_{s}\to\mu^{+}\mu^{-}$_, _Phys. Rev._ D87 (2013) 074028, [1212.4166].
  * [42] C. Kane, C. Lehner, S. Meinel and A. Soni, _Radiative leptonic decays on the lattice_ , in _37th International Symposium on Lattice Field Theory (Lattice 2019) Wuhan, Hubei, China, June 16-22, 2019_ , 2019, 1907.00279.
  * [43] C. T. Sachrajda, M. Di Carlo, G. Martinelli, D. Giusti, V. Lubicz, F. Sanfilippo et al., _Radiative corrections to semileptonic decay rates_ , in _37th International Symposium on Lattice Field Theory (Lattice 2019) Wuhan, Hubei, China, June 16-22, 2019_ , 2019, 1910.07342.
  * [44] LHCb collaboration, R. Aaij et al., _Measurement of the $b$-quark production cross-section in 7 and 13 TeV $pp$ collisions_, _Phys. Rev. Lett._ 118 (2017) 052002, [1612.05140].
  * [45] LHCb collaboration, LHCb-CONF-2013-011, _Updated average $f_{s}/f_{d}$ $b$-hadron production fraction ratio for 7 TeV $pp$ collisions_, .
  * [46] A. J. Buras, J. Girrbach, D. Guadagnoli and G. Isidori, _On the Standard Model prediction for $\text{BR}(B_{s,d}\to\mu^{+}\mu^{-})$_, _Eur. Phys. J._ C72 (2012) 2172, [1208.0934].
  * [47] A. J. Buras, R. Fleischer, J. Girrbach and R. Knegjens, _Probing New Physics with the $B_{s}\to\mu^{+}\mu^{-}$ Time-Dependent Rate_, _JHEP_ 07 (2013) 77, [1303.3820].
  * [48] R. D. Cousins and V. L. Highland, _Incorporating systematic uncertainties into an upper limit_ , _Nucl. Instrum. Meth._ A320 (1992) 331–335.
  * [49] A. Danilina, N. Nikitin and K. Toms, _Decays of charged $B$-mesons into three charged leptons and a neutrino_, _Phys. Rev. D_ 101 (2020) 096007, [1911.03670].
  * [50] A. Bharucha, B. Kindra and N. Mahajan, _Probing the structure of the $B$ meson with $B\to\ell\ell\ell^{\prime}\nu$_, 2102.03193.
  * [51] M. Beneke, P. Böer, P. Rigatos and K. K. Vos, _QCD factorization of the four-lepton decay $B^{-}\rightarrow\ell\bar{\nu}_{\ell}\ell^{(\prime)}\bar{\ell}^{(\prime)}$_, 2102.10060.
  * [52] M. A. Ivanov and D. Melikhov, _Theoretical analysis of the leptonic decays $B\to\ell\ell\ell^{\prime}\bar{\nu}_{\ell^{\prime}}$_, 2107.07247.
  * [53] F. Kruger and D. Melikhov, _Gauge invariance and form-factors for the decay $B\to\gamma\ell^{+}\ell^{-}$_, _Phys. Rev._ D67 (2003) 034002, [hep-ph/0208256].
  * [54] M. Wirbel, B. Stech and M. Bauer, _Exclusive Semileptonic Decays of Heavy Mesons_ , _Z. Phys._ C29 (1985) 637.
  * [55] T. Janowski, B. Pullin and R. Zwicky, _$B_{d,u,s}\to\gamma$ Form Factors from LCSR at NLO – to appear_.
  * [56] J. Gratrex, M. Hopfer and R. Zwicky, _Generalised helicity formalism, higher moments and the $B\to K_{J_{K}}(\to K\pi)\bar{\ell}_{1}\ell_{2}$ angular distributions_, _Phys. Rev._ D93 (2016) 054008, [1506.03970].
  * [57] M. Dimou, J. Lyon and R. Zwicky, _Exclusive Chromomagnetism in heavy-to-light FCNCs_ , _Phys. Rev._ D87 (2013) 074008, [1212.2242].
  * [58] S. Weinberg, _The Quantum theory of fields. Vol. 1: Foundations_.  Cambridge University Press, 2005.
  * [59] R. Zwicky, _A brief Introduction to Dispersion Relations and Analyticity_ , in _Proceedings, Quantum Field Theory at the Limits: from Strong Fields to Heavy Quarks (HQ 2016): Dubna, Russia, July 18-30, 2016_ , pp. 93–120, 2017, 1610.06090, DOI.
  * [60] V. Prochazka and R. Zwicky, _Finiteness of two- and three-point functions and the renormalization group_ , _Phys. Rev._ D95 (2017) 065027, [1611.01367].
  * [61] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, _QCD and Resonance Physics. Theoretical Foundations_ , _Nucl. Phys._ B147 (1979) 385–447.
  * [62] R. Mertig, M. Bohm and A. Denner, _FEYN CALC: Computer algebraic calculation of Feynman amplitudes_ , _Comput. Phys. Commun._ 64 (1991) 345–359.
  * [63] V. Shtabovenko, R. Mertig and F. Orellana, _New Developments in FeynCalc 9.0_ , _Comput. Phys. Commun._ 207 (2016) 432–444, [1601.01167].
  * [64] M. Jamin and B. O. Lange, _f(B) and f(B(s)) from QCD sum rules_ , _Phys. Rev._ D65 (2002) 056005, [hep-ph/0108135].
  * [65] B. Pullin and R. Zwicky, _Effective Couplings of Heavy-light Mesons to Photons: $g_{HH^{*}\gamma}$ and Decay Constants $f_{H,H^{*},H_{1}}^{(T)}$_.
  * [66] Particle Data Group collaboration, M. e. a. Tanabashi, _Review of particle physics_ , _Phys. Rev. D_ 98 (Aug, 2018) 030001.
  * [67] C. McNeile, A. Bazavov, C. Davies, R. Dowdall, K. Hornbostel, G. Lepage et al., _Direct determination of the strange and light quark condensates from full lattice QCD_ , _Phys. Rev. D_ 87 (2013) 034503, [1211.6577].
  * [68] G. Hiller and R. Zwicky, _(A)symmetries of weak decays at and near the kinematic endpoint_ , _JHEP_ 03 (2014) 042, [1312.1923].


# Stroboscopic Time-of-Flight Neutron Diffraction in Long Pulsed Magnetic
Fields

Taro Nakajima<EMAIL_ADDRESS>The Institute for Solid State
Physics, the University of Tokyo, Kashiwa, Chiba, 277-8581, Japan.  RIKEN
Center for Emergent Matter Science (CEMS), Saitama 351-0198, Japan.  Masao
Watanabe  J-PARC Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195,
Japan.  Yasuhiro Inamura  The Institute for Solid State Physics, the
University of Tokyo, Kashiwa, Chiba, 277-8581, Japan.  J-PARC Center, Japan
Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan.  Kazuki Matsui  Tomoki
Kanda  Tetsuya Nomoto  The Institute for Solid State Physics, the University
of Tokyo, Kashiwa, Chiba, 277-8581, Japan.  Kazuki Ohishi  Yukihiko Kawamura
Neutron Science and Technology Center, Comprehensive Research Organization for
Science and Society (CROSS), Tokai, Ibaraki 319-1106, Japan.  Hiraku Saito
The Institute for Solid State Physics, the University of Tokyo, Kashiwa,
Chiba, 277-8581, Japan.  Hiromu Tamatsukuri  J-PARC Center, Japan Atomic
Energy Agency, Tokai, Ibaraki 319-1195, Japan.  Noriki Terada  National
Institute for Materials Science, Sengen 1-2-1, Tsukuba, Ibaraki 305-0047,
Japan.  Yoshimitsu Kohama  The Institute for Solid State Physics, the
University of Tokyo, Kashiwa, Chiba, 277-8581, Japan.

###### Abstract

We present proof-of-principle experiments of stroboscopic time-of-flight (TOF)
neutron diffraction in long pulsed magnetic fields. By utilizing electric
double-layer capacitors, we developed a long pulsed magnet for neutron
diffraction measurements, which generates pulsed magnetic fields with the full
widths at the half maximum of more than $10^{2}$ ms. The field variation is
slow enough to be approximated as a steady field within the time scale of a
polychromatic neutron pulse passing through a sample placed in a distance of
the order of $10^{1}$ m from the neutron source. This enables us to
efficiently explore the reciprocal space using a wide range of neutron
wavelength in high magnetic fields. We applied this technique to investigate
field-induced magnetic phases in the triangular lattice antiferromagnets
CuFe1-xGaxO2 ($x=0,0.035$).

††preprint: APS/123-QED

##  I INTRODUCTION

Exploring new quantum states of matter in extreme conditions, such as low
temperatures, high magnetic fields, and high pressures, is one of the central
topics in condensed matter physics. Among them, novel field-induced phases
have been extensively investigated; for example, spin-lattice-coupled
magnetization plateaus in frustrated spin systems [1, 2, 3, 4], field-induced
flop of the spin-driven electric polarization in multiferroics [5, 6], spin-
nematic states in quantum spin systems [7, 8] etc. Neutron scattering is one
of the most powerful techniques to study these exotic phenomena because it can
probe Fourier-transformed time-space correlation functions of atomic positions
and magnetic moments; specifically crystal and magnetic structures are
determined by the elastic scattering, and phonons and magnons are measured by
the inelastic scatterings. However, the highest magnetic field for neutron
scattering instruments is limited to approximately 15 T even for state-of-the-
art superconducting magnets. Although there was a superconducting and non-
superconducting hybrid magnet for neutron scattering with the highest magnetic
field of 26 T in the Helmholtz Zentrum Berlin (HZB)[9], it is no longer
available since the research reactor in HZB was shut down in 2019.

An alternative is a pulsed magnetic field[10, 11, 12, 13]. For instance,
Nojiri and co-workers successfully observed a magnetic Bragg peak in a field-
induced phase of multiferroic MnWO4 by time-of-flight (TOF) pulsed neutron
diffraction measurements with pulsed magnetic fields up to 30 T[11]. However,
the widths of the pulsed magnetic fields are often limited to several
milliseconds, which are shorter than a time spread of a polychromatic neutron
pulse after flying a typical source-to-sample distance in existing TOF neutron
diffractometers, as we explain later. In this case, the high magnetic field is
achieved only for a limited wavelength range of the neutrons, and this
situation makes explorations of new magnetic peaks in the reciprocal space
difficult.

To overcome this problem, in the present study, we have developed a long
pulsed magnet for neutron scattering experiments. By utilizing electric
double-layer capacitors (EDLC)[14], we successfully generated pulsed magnetic
fields whose full widths at the half maximum exceed 100 ms. We established the
low-temperature neutron scattering environment with this long pulsed magnet,
and performed stroboscopic TOF neutron diffraction measurements on the
triangular lattice antiferromagnets CuFe1-xGaxO2 ($x=0,0.035$). By virtue of
the extended time scale of the magnetic field, we can use a wide wavelength
range of the incident neutron beam, which enables us to map out the neutron
diffraction intensities in the reciprocal space in high magnetic fields.

Figure 1: Schematic illustration of the side view of the long pulsed magnet
and CCR in the Al vacuum chamber.

CuFeO2 has been extensively investigated as a geometrically frustrated system
[15, 16]. Crystal structure of this compound belongs to the space group of
$R\bar{3}m$, and consists of triangular lattice layers of magnetic Fe3+ ions,
which are rhombohedrally stacked along the $c$ axis. In zero magnetic field,
this compound exhibits a collinear four-sublattice (4SL) antiferromagnetic
order, in which the spins are arranged to form an up-up-down-down sequence
running along the [110] direction, below 11 K. The spins in the adjacent
layers are antiferromagnetically aligned, and thus the magnetic modulation
wave vector ($q$-vector) is described as
$q=(\frac{1}{4},\frac{1}{4},\frac{3}{2})$. The 4SL antiferromagnetic order
breaks the threefold rotational symmetry about the $c$ axis of the crystal,
and thus results in three magnetic domains corresponding to the $q$-vectors of
$q=(\frac{1}{4},\frac{1}{4},\frac{3}{2})$,
$(-\frac{1}{2},\frac{1}{4},\frac{3}{2})$ and
$(\frac{1}{4},-\frac{1}{2},\frac{3}{2})$, which are interconverted to each
other by the threefold rotation operation. Previous x-ray diffraction studies
revealed that each magnetic domain exhibits a monoclinic lattice distortion
associated with the $q$-vector[17, 18]. For convenience, we employ the
hexagonal basis when referring $q$-vectors and positions in the reciprocal
space in the rest of this paper, although they could be properly indexed using
the monoclinic basis.

By applying a magnetic field along the $c$ axis at low temperatures, CuFeO2
exhibits field-induced successive phase transitions[19, 20, 21]. The first-
field-induced phase is known to have an incommensurate $q$-vector and spin-
driven ferroelectricity[22]. The second field-induced phase exhibits a
$\frac{1}{5}$-magnetization plateau. In this paper, we refer to these phases
as ferroelectric incommensurate-magnetic (FE-ICM) and $\frac{1}{5}$-plateau
phases, respectively. Both phases were studied by previous neutron scattering
experiments with vertical-field superconducting magnets[20, 21]. However, the
accessible scattering plane was limited to the $(h,k,0)$ plane, where the
fundamental magnetic reflections were not observed due to the finite $l$
component of the $q$-vectors. It is also known that the critical field between
the 4SL and FE-ICM phases can be reduced to zero by substituting a small
amount of nonmagnetic ions for Fe3+ [23, 24]. Thus, the magnetic structure of
the FE-ICM phase was investigated by neutron diffraction measurements on
CuFe${}_{1-x}A_{x}$O2 ($A=$Al, Ga), and determined to be the incommensurate
screw-type magnetic structure [25, 26]. Although the nonmagnetic-substitution-
induced phase was considered to be the same as the field-induced FE-ICM phase
in pure CuFeO2, the direct evidence is still lacking because the fundamental
magnetic Bragg reflections in the field-induced phase of pure CuFeO2 were not
directly observed.

Field-induced magnetic phase transitions in CuFe1-xGaxO2 were also studied by
means of magnetization, electric polarization, ESR and heat capacity
measurements[27, 28, 29]. At $x=0.035$, where the ground state is already
replaced with the FE-ICM phase, the first field-induced phase is considered to
be the $\frac{1}{5}$-plateau phase. However, the plateau in magnetization is
smeared with increasing $x$, and therefore the direct observation of the
$q$-vectors by neutron diffraction is necessary to unambiguously determine the
spin arrangements in the field-induced phase of the $x=0.035$ sample.

Figure 2: The TOF diagram of TAIKAN in MLF. The color map shows the
wavelengths of the incident neutrons.

##  II EXPERIMENTAL DETAILS

###  II.1 Sample, Cryostat and Magnet

A single crystal of CuFeO2 was grown by the floating zone method [30], and cut
into a cylinder shape with the diameter and length of 5 and 6 mm,
respectively. The sample was glued on a sapphire sample holder, which was
connected to the cold head of a 4He closed-cycle refrigerator (CCR).

We prepared a pulsed magnet consists of inner solenoid of a 18-layer and 11
turn and outer solenoid of 10-layer and 16 turn; both were wound with a copper
wire having a rectangular cross-section of $2.4\times 1.0$ mm2. The magnet
produces a horizontal magnetic field, and has two conical windows with an
opening angle of 60∘, as shown in Fig. 1. The inner bore of the magnet is 20
mm in diameter. The magnet was cooled by liquid N2, and was thermally isolated
from the sample by vacuum. The coil inductance, coil resistance, and the field
factor at 77 K are 3.9 mH, 57.5 m$\Omega$, and 7.85 T/kA, respectively. The
sample was set at the field center. The direction of the magnetic field was
parallel to the $c$ axis, namely perpendicular to the triangular lattice
plane. The [110] direction was selected to be the horizontal direction
perpendicular to the field direction. The magnet and sample were put in an Al-
chamber, which was also pumped to high vacuum. Note that the Al-chamber and
CCR were originally designed for the short pulsed magnet system in the
Materials and Life-science experimental Facility (MLF) of J-PARC [31], and are
also compatible with the long pulsed magnet.

A single crystal of CuFe1-xGaxO2 ($x=0.035$) was also grown by the floating
zone method [30], and cut into a rectangular shape with the dimensions of
4.9$\times$2.9$\times$3.5 mm3. The sample environments and the alignment of
the crystal were the same as those for the $x=0$ sample.

###  II.2 Time Structure of A Neutron Pulse

The neutron diffraction experiments were carried out at the small-angle and
wide-angle neutron TOF diffractometer TAIKAN in MLF of J-PARC [32]. Figure 2
shows a typical TOF diagram for TAIKAN. Since neutron velocity is inversely
proportional to its wavelength, a sharp polychromatic neutron pulse created at
the neutron source has a wavelength spread after flying the source-to-sample
distance ($L_{1}$), which is 14.35 m at TAIKAN. Taking into account that the
interval between two adjacent neutron pulses is 40 ms in MLF, TAIKAN normally
uses the wavelength range from 0.7 to 7.7 Å. In the present study, we focus on
the wavelength range from 1 to 3 Å to measure magnetic Bragg reflections of
CuFe1-xGaxO2, as we show in the following sections. This wavelength range
corresponds to the time spread of 7.25 ms at the sample position, as shown in
Fig. 2. If the width of the pulsed magnetic field is much larger than this
time spread, we can fully utilize the wavelength range to map out the neutron
diffraction intensities in the reciprocal space.

Figure 3: (a) Circuit diagram of the EDLC-based pulse power supply for the
generation of long pulsed magnetic fields. (b) Comparisons among the TOF
diagram, the temporal profile of the long pulsed magnetic field generated with
a charging voltage of 270 V, and the kicker signals for neutron pulse
generation. The neutron pulses are triggered by the rising edges of the kicker
signals.  Figure 4: (a) A schematic illustration of the top view of the
instrument layout in the present experiment. (b) A color map showing the
neutron wavelengths for measuring intensities on the $(h,k,\frac{1}{2})$ plane
in the scattering geometry shown in (a).

###  II.3 Generation of Long Pulsed Magnetic Fields

We have constructed an electrical circuit for generating long pulsed magnetic
fields which consists of an EDLC bank, a DC power supply, a trigger circuit,
two thyristors (Th1 and Th2), a discharge resistance ($R_{\rm dis}$), and a
pulsed magnet, as shown in Fig. 3(a). The EDLC bank consists of a series
connection of 120 EDLC cells (Nippon Chemi-Con) and has a total capacitance of
30 F and an internal resistance of $37$ m$\Omega$ with a maximum charged
voltage of 300 V. The DC power supply (TDK-Lambda Americas) was used to charge
the EDLC bank. The trigger circuit controls the timing of closing Th1 and Th2.
The current flow through the pulsed magnet, which is proportional to the field
strength, starts at the timing of closing Th1.

In Fig. 3(b), we show a temporal profile of the long pulsed magnetic field
generated by a charging voltage of $V_{c}=270$ V. The width of the field pulse
is more than one order of magnitude larger than the time spread corresponding
to the wavelength range from 1 to 3 Å which is indicated by the light-blue
rectangles in Fig. 3(b). This means that the long pulsed magnetic field can be
approximated as a steady field within the time scale of the neutron pulses. We
note here that there is a sudden drop of the magnetic field upon the field
decreasing process. At this point, Th2 is intentionally closed so that the
electric current flows not only to the main-circuit but also to the sub-
circuit for discharge. While the intentional reduction of the field strength
slightly reduce the width of the pulsed field, this suppresses unnecessary
heating of the coil, and thus reduces a waiting time for applying next
magnetic field pulse, which was approximately 9 minutes at maximum in the
present experiment.

Figure 5: Stroboscopic neutron diffraction results on CuFeO2. [(a)-(c)]
Intensity maps on the $(h,k,\frac{1}{2})$ plane in the (a) 4SL, (b) FE-ICM,
and (c) $\frac{1}{5}$-plateau phases. These data are obtained from the #5, #4,
and #3 frames, respectively, measured with the long pulsed magnetic field
generated with the charging voltage of 270 V, as shown in Fig. 3. The
application of the magnetic field was repeated for 122 times. Solid arrows
indicate the positions of the magnetic Bragg peaks in each phase. [(d)-(f)]
Line profiles along the $(\delta,1-2\delta,\frac{1}{2})$ direction in the (d)
4SL, (e) FE-ICM, and (f) $\frac{1}{5}$-plateau phases. Inset of (f) shows the
spin arrangement of the $\frac{1}{5}$-plateau phase cited from Ref. 20.
Figure 6: (a) Comparisons among the TOF diagram, the temporal profile of the
long pulsed magnetic field generated with a charging voltage of 142 V, and the
kicker signals for neutron pulse generation. [(b)-(d)] Line profiles along the
$(\delta,1-2\delta,\frac{1}{2})$ direction in CuFeO2 measured by the frame (b)
#3, (c) #4, and (d) #5, shown in (a). (e) Field dependence of the magnetic
modulation wave number $q$ in CuFeO2, which was determined from the peak
position of the line profile along $(\delta,1-2\delta,\frac{1}{2})$.

###  II.4 Stroboscopic TOF Neutron Diffraction

Despite the extended time scale of the pulsed magnetic fields, the intensity
of one neutron pulse is still not strong enough to precisely measure the
reflections from the sample. We thus repeated the application of a magnetic
field pulse to accumulate the data. The application of a field pulse was
triggered by one of the kicker signals for the neutron pulse generation. The
data measured by each neutron pulse were numbered starting from this trigger,
as shown in Fig. 3. In the following, we refer to these numbers as frame
numbers. We also introduced a delay time $t_{d}$ to adjust the peak position
of the magnetic field with respect to the neutron pulses. After repeating the
field applications, we extracted and accumulated the data having the same
frame number by UTSUSEMI software [33]. Finally, we obtained stroboscopic
neutron diffraction patterns in the long pulsed magnetic fields.

Figure 4(a) shows the experimental configuration. The angle between the
incident neutron beam and the magnetic field was set to be 23 deg. We mainly
used the high-angle detector bank of TAIKAN, which has a horizontal coverage
of the scattering angle from 24∘ to 48∘. Note that we can also measure
reflections out of the horizontal plane owing to the vertical coverage of the
high-angle detector bank. Figure 4(b) is the color map of the neutron
wavelengths for measuring intensities on the $(h,k,\frac{1}{2})$ plane in the
configuration mentioned above. Specifically, the scattering vector
$\mbox{\boldmath$Q$}(=\mbox{\boldmath$k$}_{i}-\mbox{\boldmath$k$}_{f})$, where
$\mbox{\boldmath$k$}_{i}$ and $\mbox{\boldmath$k$}_{f}$ are wavevectors of the
incident and scattered neutrons, respectively, for an elastic scattering event
is determined by the wavelength and the direction of the scattered neutron.
When the scattering vector is in the colored region in Fig. 4(b), the elastic
scattering condition is satisfied with the wavelength indicated by the color,
and the direction of the scattered neutron is covered by the high-angle
detector bank. In this region, we can observe satellite reflections with
$q$-vectors indexed as $(q,-2q,\frac{3}{2})$, which is equivalent to
$(q,q,\frac{3}{2})$ under the threefold rotation about the $c$ axis, from the
reciprocal lattice point of $(0,1,-1)$. The wavelength range to cover this
region is from 1 to 3 Å.

##  III RESULTS AND DISCUSSIONS

###  III.1 CuFeO2

We performed the stroboscopic neutron diffraction in long pulsed magnetic
fields on CuFeO2 at 4 K, at which the lower critical fields of the FE-ICM and
1/5-plateau phases, namely $H_{c1}$ and $H_{c2}$, are 7 and 13 T,
respectively. We repeated the application of the long pulsed magnetic field
with the charging voltage of 270 V for 122 times, and then extracted the
diffraction patterns of #3, #4 and #5 frames, which correspond to the
1/5-plateau, FE-ICM, and 4SL phases, respectively, as shown in Fig. 3. The
intensities are normalized to the incident neutron flux taking into account
its wavelength dependence, and mapped onto the $(h,k,\frac{1}{2})$ plane after
integrating them with respect to $l$. The integration range for $l$ is from
$0.45$ to $0.55$.

Figure 5(a) shows the intensity map of the 4SL phase. We observed a magnetic
Bragg reflection at $(\frac{1}{4},\frac{1}{2},\frac{1}{2})$, which is assigned
as a satellite reflection having the $q$-vector of
$(\frac{1}{4},-\frac{1}{2},\frac{3}{2})$ from the reciprocal lattice point of
$(0,1,-1)$. We also observed several magnetic reflections having the same
distance from the $c^{*}$ axis. They are magnetic Bragg peaks arising from
multiply twinned crystal domains in which the directions of the $c^{*}$ axes
are common to each other, indicating that the sample is not an ideal single
crystal. In Fig. 5(d), we show a line profile along the
$(\delta,1-2\delta,\frac{1}{2})$ direction in the 4SL phase [34]. A sharp peak
at the commensurate position of $\delta=\frac{1}{4}$ is consistent with the
four-sublattice antiferromagnetic order.

In the FE-ICM phase, we observed two peaks as indicated by the two blue arrows
in Fig. 5(b). From the line profile shown in Fig. 5(e), the peak positions are
determined to be $\delta=0.203$ and $0.297$. The former is assigned as a
satellite reflection from the reciprocal lattice point of $(0,1,-1)$ using the
$q$-vector of $(q_{\rm ICM},-2q_{\rm ICM},\frac{3}{2})$ where $q_{\rm
ICM}=0.203$. The latter peak can also be assigned as a satellite reflection
from $(\frac{1}{2},0,2)$, which becomes a crystallographic zone center due to
the lattice distortion associated with the magnetic order[35, 36], using the
same $q$-vector. The two peaks at $\delta=q_{\rm ICM}$ and $\frac{1}{2}-q_{\rm
ICM}$ and their intensity distributions are quite similar to those in
CuFe1-xGaxO2 with $x=0.035$ in zero magnetic field[37], confirming that the
nonmagentic-substitution-induced FE-ICM phase is equivalent to the field-
induced FE-ICM phase in pure CuFeO2.

In the $\frac{1}{5}$-plateau phase, a sharp magnetic Bragg reflection was
observed at $(\frac{1}{5},\frac{3}{5},\frac{1}{2})$, which corresponds to the
commensurate $q$-vector of $(\frac{1}{5},-\frac{2}{5},\frac{3}{2})$, as shown
in Figs. 5(c) and 5(f). This means that there are commensurate three-up-two-
down modulations running along the [110] and its equivalent directions on the
triangular lattice plane. The strong magnetic Bragg peak on the
$(h,k,\frac{1}{2})$ plane indicates that the majority of the spins maintain
the antiferromagnetic coupling along the $c$ axis. This is consistent with the
magnetic structure model proposed in the previous study by Mitsuda et al.[20],
which is shown in the inset of Fig. 5(f).

The slopes of the long pulsed magnetic fields can be tuned by changing the
charging voltage. Figure 6(a) shows a profile of a long pulsed magnetic field
generated by a charging voltage of 142 V. Though the maximum of the field is
reduced to 9 T, the field variation becomes slower, so that we can measure the
diffraction patterns upon the phase transition between the FE-ICM and 4SL
phases in detail. We repeated the field application with $V_{c}=142$ V for 40
times, and extracted the diffraction profiles of #3, #4, and #5 frames as
shown in Figs. 6(b), 6(c), and 6(d), respectively. Among them, the frame #4
clearly shows the coexistence of the magnetic Bragg peaks of the FE-ICM and
4SL phases as shown in Fig. 6(c). The field variation within this frame is
approximately 0.2 T, which is small enough to capture the phase coexistence
during the first-order phase transition between the two phases.

We also performed the measurements with different charging voltages, and
obtained the positions of the magnetic peaks on the
$(\delta,1-2\delta,\frac{1}{2})$ line as functions of magnetic field, as shown
in Fig. 6(e). Here, we refer to the in-plane magnetic modulation wavenumber as
$q$. The $q$-values of the 4SL and $\frac{1}{5}$-plateau phases coincide with
the commensurate values, while that of the FE-ICM phase continuously changes
with magnetic field. This is consistent with the results obtained by the
previous neutron diffraction measurements with steady magnetic field, in which
the $q$ values were deduced from the higher-harmonic magnetic reflections on
the $(h,k,0)$ plane [20, 21]. The present results also demonstrate that the
stroboscopic neutron diffraction with long pulsed fields is suitable for
investigating field-dependent incommensurate magnetic modulations, such as
spin-density-wave states of frustrated quantum spin systems in high magnetic
fields [38].

###  III.2 CuFe1-xGaxO2 ($x=0.035$)

The field-induced phase transition in the $x=0.035$ sample was also
investigated in the same manner. We applied pulsed magnetic fields generated
by the charging voltage of 254 V at 4 K, and repeated the field application
for 80 times. In addition, we also carried out the same set of the
measurements with a different delay time to increase the number of data points
with different magnetic fields.

Similarly to the results of the $x=0$ sample, we observed a pair of magnetic
Bragg reflections on the $(h,k,\frac{1}{2})$ plane in the FE-ICM phase, as
shown in Fig. 7(a). We note here that the $x=0.035$ sample did not contain any
crystal twinning in contrast to the $x=0$ sample. Therefore, the signal-to-
noise ratio was much improved in both the intensity map and the line profile
along the $(\delta,1-2\delta,\frac{1}{2})$ direction shown in Fig. 7(c).

Figure 7: Stroboscopic neutron diffraction results on CuFe1-xGaxO2 with
$x=0.035$. [(a)-(b)] Intensity maps on the $(h,k,\frac{1}{2})$ plane (a) above
and (b) below the critical field $H_{c2}^{\prime}$. These data are obtained by
applying the long pulsed magnetic field generated with the charging voltage of
270 V for 80 times at 4 K. Solid arrows indicate the positions of the magnetic
Bragg peaks in each phase. [(c)-(d)] Line profiles along the
$(\delta,1-2\delta,\frac{1}{2})$ direction (c) above and (d) below the
critical field $H_{c2}^{\prime}$. [(e)-(h)] Magnetic field dependence of the
integrated intensities at (e) $\delta=q$ and (f) $\frac{1}{2}-q$, (g) the
magnetic modulation wavenumber $q$, and (h) the half width at the half maximum
of the peak at $\delta=q$. Open and close symbols denote the data measured
field-increasing and decreasing process, respectively.

The critical field of the FE-ICM phase in the $x=0.035$ sample
($H_{c2}^{\prime}$) is approximately 10 T, above which we found that the peak
at $\delta=\frac{1}{2}-q$ disappeared as shown in Figs. 7(b) and 7(d). We
obtained integrated intensities of the magnetic Bragg peaks at $\delta=q$ and
$\frac{1}{2}-q$ as functions of magnetic field as shown in Figs. 7(e) and
7(f), respectively, confirming that the intensity at $\delta=\frac{1}{2}-q$
drops to zero at $H_{c2}^{\prime}$. The diffraction pattern above
$H_{c2}^{\prime}$ is quite similar to that in the $\frac{1}{5}$-plateau phase
in pure CuFeO2; in fact, the phase diagram established by previous bulk
measurements suggested that the $\frac{1}{5}$-plateau phase appears as the
first-field-induced phase in the $x=0.035$ sample[27, 28, 29]. However, the
present results show that the wavenumber $q$ was not exactly on the
commensurate position of $\frac{1}{5}$ as shown in Fig. 7(g).

One possible explanation for the incommensurate nature would be the effects of
magnetic domain walls, which may shift the phase of the magnetic modulations.
A similar peak-shift effect was reported in a commensurate Ising
antiferromagnet [39], and became more significant when the system was divided
into small magnetic domains. Actually, the half width at the half maximum
(HWHM) of the peak at $\delta=q$ slightly increases in the field induced phase
as shown in Fig. 7(h), which implies the magnetic correlation length becomes
shorter upon the field-induced phase transition.

##  IV CONCLUSIONS AND OUTLOOK

We have established the stroboscopic TOF neutron diffraction measurements with
long pulsed magnetic fields at low temperatures. The widths of the magnetic
fields are more than one order of magnitude larger than the time scale of a
polychromatic neutron pulse passing through the sample. As a result, we can
utilize a wide range of neutron wavelength to measure diffraction patterns in
high magnetic fields. We applied this technique to study the field-induced
magnetic phase transitions of the triangular lattice antiferromagnets
CuFe1-xGaxO2 ($x=0,0.035$), and successfully observed the field evolutions of
the magnetic Bragg reflections up to 14 T. As for the $x=0$ sample, we
directly observed the magnetic Bragg reflections in the field-induced FE-ICM
phase, confirming that the peak positions and intensity distribution are the
same as those in the nonmagnetic-substitution-induced FE-ICM phase [25, 26].
We also observed the diffraction patterns of the first field-induced phase of
the $x=0.035$ sample, which were found to be quite similar to that in the
$\frac{1}{5}$-plateau phase of the $x=0$ sample. The tiny peak shift might be
explained by the phase shift of the magnetic modulations at the magnetic
domain walls, which is inferred from the broadening of the peak profile.

One of the biggest advantages of this technique is that one can analyze the
neutron diffraction data in the same manner as the analysis for the steady-
field measurements. Each neutron data frame contains a wide range of
wavelength, which enables us to explore the three-dimensional reciprocal
space. This feature is essential to explore unknown magnetic phases in high
magnetic fields. It would also be possible to quantitatively investigate field
variations of diffuse scattering patterns up to high magnetic fields.

Another advantage is the potability of the experimental setup. As shown in
Fig. 1, the cryostat and magnet are relatively small, and can be moved from
one beamline to another. The capacitor bank and voltage charger are also
relatively small; they are approximately 80 cm in width and less than 2 m in
height. In addition, the stray field form the magnet is also small;
specifically, it is less than 3 Oe in a distance of 20 cm from the center of
the magnet when the field reaches 14 T. Taking into account that the pulsed
magnetic field is shielded by the Al-radiation shield and Al-vacuum chamber,
it does not affect external instruments. These features provide opportunities
of high-field experiments for a variety of instruments and their subject
matter. In fact, high-field environments are necessary not only for physics
but also material science and its applications; such as developments of
ferromagnetic materials for industrial and medial uses [40, 41].

The highest magnetic field of the present setup is 14 T. However, it was
already demonstrated that long pulsed magnetic fields with the height of 24.3
T and the width at the half maximum of more than 100 ms can be generated by
using EDLC banks [14]. Therefore, it is possible to develop a long pulsed
magnet for neutron scattering whose highest magnetic field exceeds that of
existing superconducting magnets for scattering experiments. Although a major
drawback of the pulsed magnetic field is the relatively long time-interval for
repeating the field applications, a recent study shows that the interval can
be reduced by using a liquid-helium-cooled coil made of high-purity copper
wires, which remarkably reduces Joule heating effect [42]. These technological
developments will extend the high-field capability of neutron scattering.

## Acknowledgements

The authors thank Y. Ihara for enlightening discussions. This work was partly
supported by Grants-in-Aid for Scientific Research (Grant No. 22H00104 ) from
JSPS. The neutron scattering experiment at J-PARC MLF was carried out along
the proposals (No. 2021B0070, 2022L0301).

## References

  * Ueda  _et al._ [2005] H. Ueda, H. A. Katori, H. Mitamura, T. Goto, and H. Takagi, Magnetic-Field Induced Transition to the $1/2$ Magnetization Plateau State in the Geometrically Frustrated Magnet $\mathrm{C}\mathrm{d}{\mathrm{C}\mathrm{r}}_{2}{\mathrm{O}}_{4}$, Phys. Rev. Lett. 94, 047202 (2005).
  * Penc  _et al._ [2004] K. Penc, N. Shannon, and H. Shiba, Half-magnetization plateau stabilized by structural distortion in the antiferromagnetic heisenberg model on a pyrochlore lattice, Phys. Rev. Lett. 93, 197203 (2004).
  * Matsuda  _et al._ [2007] M. Matsuda, H. Ueda, A. Kikkawa, Y. Tanaka, K. Katsumata, Y. Narumi, T. Inami, Y. Ueda, and S.-H. Lee., Spin-lattice instability to a fractional magnetization state in the spinel HgCr2O4, Nat. Phys. 3, 397 (2007).
  * Jaime  _et al._ [2012] M. Jaime, R. Daou, S. A. Crooker, F. Weickert, A. Uchida, A. E. Feiguin, C. D. Batista, H. A. Dabkowska, and B. D. Gaulin, Magnetostriction and magnetic texture to 100.75 Tesla in frustrated SrCu2(BO3)2, Proc. Natl. Acad. Sci. U.S.A. 109, 12404 (2012).
  * Kimura  _et al._ [2003] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Magnetic control of ferroelectric polarization, Nature 426, 55 (2003).
  * Lawes  _et al._ [2005] G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, T. Yildrim, M. Kenzelmann, C. Broholm, and A. P. Ramirez, Magnetically Driven Ferroelectric Order in Ni3V2O8, Phys. Rev. Lett. 95, 087205 (2005).
  * Kohama  _et al._ [2019] Y. Kohama, H. Ishikawa, A. Matsuo, K. Kindo, and Z. Hiroi, Possible observation of quantum spin-nematic phase in a frustrated magnet, Proc. Natl. Acad. Sci. U.S.A. 116, 10686 (2019).
  * Orlova  _et al._ [2017] A. Orlova, E. L. Green, J. M. Law, D. I. Gorbunov, G. Chanda, S. Krämer, M. Horvatić, R. K. Kremer, J. Wosnitza, and G. L. J. A. Rikken, Nuclear Magnetic Resonance Signature of the Spin-Nematic Phase in ${\mathrm{LiCuVO}}_{4}$ at High Magnetic Fields, Phys. Rev. Lett. 118, 247201 (2017).
  * Prokhnenko  _et al._ [2017] O. Prokhnenko, P. Smeibidl, W. D. Stein, M. Bartkowiak, and N. Stuesser, HFM/EXED: The high magnetic field facility for neutron scattering at BER II, JLSRF 3, A115 (2017).
  * Nojiri  _et al._ [1992] H. Nojiri, M. Motokawa, N. Nishida, and Y. Endoh, Neutron diffraction experiments in a pulsed high magnetic field., Physica B 180-181, 31 (1992).
  * Nojiri  _et al._ [2011] H. Nojiri, S. Yoshii, M. Yasui, K. Okada, M. Matsuda, J. S. Jung, T. Kimura, L. Santodonato, G. E. Granroth, K. A. Ross, J. P. Carlo, and B. D. Gaulin, Neutron Laue Diffraction Study on the Magnetic Phase Diagram of Multiferroic ${\mathrm{MnWO}}_{4}$ under Pulsed High Magnetic Fields, Phys. Rev. Lett. 106, 237202 (2011).
  * Fogh  _et al._ [2020] E. Fogh, T. Kihara, R. Toft-Petersen, M. Bartkowiak, Y. Narumi, O. Prokhnenko, A. Miyake, M. Tokunaga, K. Oikawa, M. K. Sørensen, J. C. Dyrnum, H. Grimmer, H. Nojiri, and N. B. Christensen, Magnetic structures and quadratic magnetoelectric effect in ${\mathrm{LiNiPO}}_{4}$ beyond 30 T, Phys. Rev. B 101, 024403 (2020).
  * Duc  _et al._ [2018] F. Duc, X. Tonon, J. Billette, B. Rollet, W. Knafo, F. Bourdarot, J. Béard, F. Mantegazza, B. Longuet, J. E. Lorenzo, E. Lelièvre-Berna, P. Frings, and L.-P. Regnault, 40-Tesla pulsed-field cryomagnet for single crystal neutron diffraction, Rev. Sci. Instrum. 89, 053905 (2018).
  * Matsui  _et al._ [2021] K. Matsui, T. Kanda, Y. Ihara, K. Kindo, and Y. Kohama, Compact megajoule-class pulsed power supply for generating long-pulsed magnetic fields, Rev. Sci. Instrum. 92, 024711 (2021).
  * Mitsuda  _et al._ [1991] S. Mitsuda, H. Yoshizawa, N. Yaguchi, and M. Mekata, Neutron Diffraction Study of CuFeO2, J. Phys. Soc. Jpn. 60, 1885 (1991).
  * Mekata  _et al._ [1993] M. Mekata, N. Yaguchi, T. Takagi, T. Sugino, S. Mitsuda, H. Yoshizawa, N. Hosoito, and T. Shinjo, Successive Magnetic Ordering in CuFeO2 -A New Type of Partially Disordered Phase in a Triangular Lattice Antiferromagnet-, J. Phys. Soc. Jpn. 62, 4474 (1993).
  * Terada  _et al._ [2006a] N. Terada, S. Mitsuda, H. Ohsumi, and K. Tajima, “Spin-Driven” Crystal Lattice Distortion in Frustrated Magnet CuFeO2: Synchrotron X-ray Diffraction Study, J. Phys. Soc. Jpn. 75, 023602 (2006a).
  * Ye  _et al._ [2006] F. Ye, Y. Ren, Q. Huang, J. A. Fernandez-Baca, P. Dai, J. W. Lynn, and T. Kimura, Spontaneous spin-lattice coupling in the geometrically frustrated triangular lattice antiferromagnet CuFeO2, Phs. Rev. B 73, 220404(R) (2006).
  * Mitsuda  _et al._ [2000a] S. Mitsuda, M. Mase, T. Uno, H. Kitazawa, and H. A. Katori, $H$-$T$ Magnetic Phase Diagram of a Frustrated Triangular Lattice Antiferromagnet CuFeO2, J. Phys. Soc. Jpn. 69, 33 (2000a).
  * Mitsuda  _et al._ [2000b] S. Mitsuda, M. Mase, K. Prokes, H. Kitazawa, and H. A. Katori, Field-Induced Magnetic Phase Transitions in a Triangular Lattice Antiferromagnet CuFeO2 up to 14.5 T, J. Phys. Soc. Jpn. 69, 3513 (2000b).
  * Petrenko  _et al._ [2000] O. A. Petrenko, G. Balakrishnan, M. R. Lees, D. M. Paul, and A. Hoser, High-magnetic-field behavior of the triangular-lattice antiferromagnet CuFeO2, Phys. Rev. B 62, 8983 (2000).
  * Kimura  _et al._ [2006] T. Kimura, J. C. Lashley, and A. P. Ramirez, Inversion-symmetry breaking in the noncollinear magnetic phase of the triangular-lattice antiferromagnet CuFeO2, Phys. Rev. B 73, 220401(R) (2006).
  * Seki  _et al._ [2007] S. Seki, Y. Yamasaki, Y. Shiomi, S. Iguchi, Y. Onose, and Y. Tokura, Impurity-doping-induced ferroelectricity in the frustrated antiferromagnet CuFeO2, Phys. Rev. B 75, 100403(R) (2007).
  * Kanetsuki  _et al._ [2007] S. Kanetsuki, S. Mitsuda, T. Nakajima, D. Anazawa, H. A. Katori, and K. Prokes, Field-induced ferroelectric state in frustrated magnet CuFe1-xAlxO2, J. Phys.: Condens. Matter 19, 145244 (2007).
  * Nakajima  _et al._ [2007] T. Nakajima, S. Mitsuda, S. Kanetsuki, K. Prokes, A. Podlesnyak, H. Kimura, and Y. Noda, Spin Noncollinearlity in Multiferroic Phase of Triangular Lattice Antiferromagnet CuFe1-xAlxO2, J. Phys. Soc. Jpn. 76, 043709 (2007).
  * Nakajima  _et al._ [2009] T. Nakajima, S. Mitsuda, K. Takahashi, M. Yamano, K. Masuda, H. Yamazaki, K. Prokes, K. Kiefer, S. Gerischer, N. Terada, H. Kitazawa, M. Matsuda, K. Kakurai, H. Kimura, Y. Noda, M. Soda, M. Matsuura, and K. Hirota, Comprehensive study on ferroelectricity induced by a proper-screw-type magnetic ordering in multiferroic CuFeO2: Nonmagnetic impurity effect on magnetic and ferroelectric order, Phys. Rev. B 79, 214423 (2009).
  * Seki  _et al._ [2009] S. Seki, H. Murakawa, Y. Onose, and Y. Tokura, Magnetic Digital Flop of Ferroelectric Domain with Fixed Spin Chirality in a Triangular Lattice Helimagnet, Phys. Rev. Lett. 103, 237601 (2009).
  * Fujita  _et al._ [2013] T. Fujita, S. Kimura, T. Kida, T. Kotetsu, and M. Hagiwara, High-Field Magnetism of the $S=5/2$ Triangular-Lattice Antiferromagnet CuFe1-xGaxO2 ($x=0-0.028$), J. Phys. Soc. Jpn. 82, 064712 (2013).
  * Song  _et al._ [2017] J. D. Song, X. M. Wang, Z. Y. Zhao, J. C. Wu, J. Y. Zhao, X. G. Liu, X. Zhao, and X. F. Sun, Low-temperature heat transport of ${\mathrm{CuFe}}_{1-x}{\mathrm{Ga}}_{x}{\mathrm{O}}_{2}$ $(x=0--0.12)$ single crystals, Phys. Rev. B 95, 224419 (2017).
  * Zhao  _et al._ [1996] T. R. Zhao, M. Hasegawa, and H. Takei, Crystal growth and characterization of cuprous ferrite (CuFeO2), J. Cryst. Growth 166, 408 (1996).
  * Watanabe  _et al._ [2023] M. Watanabe, T. Kihara, and H. Nojiri, Automated Pulsed Magnet System for Neutron Diffraction Experiments at the Materials and Life Science Experimental Facility in J-PARC, Quantum Beam Sci. 7, 1 (2023).
  * Takata  _et al._ [2015] S. Takata, J. Suzuki, T. Shinohara, T. Oku, T. Tominaga, K. Ohishi, H. Iwase, T. Nakatani, Y. Inamura, T. Ito, K. Suzuya, K. Aizawa, M. Arai, T. Otomo, and M. Sugiyama, The Design and $q$ Resolution of the Small and Wide Angle Neutron Scattering Instrument (TAIKAN) in J-PARC, JPS Conf. Proc. 8, 036020 (2015).
  * Inamura  _et al._ [2013] Y. Inamura, T. Nakatani, J. Suzuki, and T. Otomo, Development Status of Software “Utsusemi” for Chopper Spectrometers at MLF, J-PARC, J. Phys. Soc. Jpn. 82, SA031 (2013).
  * [34] To obtain the line profiles, the magnetic intensities from the misoriented crystal domains are projected on the $(\delta,1-2\delta,\frac{1}{2})$ direction taking into account the rotation about the $c$ axis.
  * Terada  _et al._ [2006b] N. Terada, Y. Tanaka, Y. Tabata, K. Katsumata, A. Kikkawa, and S. Mitsuda, Restoring Higher Symmetric Crystal Structure with Magnetic Field in Triangular Lattice Antiferromagnet CuFeO2, J. Phys. Soc. Jpn. 75, 113702 (2006b).
  * Terada  _et al._ [2008a] N. Terada, S. Mitsuda, Y. Tanaka, Y. Tabata, K. Katsumata, and A. Kikkawa, Field-induced Incommensurate Lattice Modulations in the Delafossite CuFeO2, J. Phys. Soc. Jpn. 77, 054701 (2008a).
  * Terada  _et al._ [2008b] N. Terada, T. Nakajima, S. Mitsuda, H. Kitazawa, K. Kaneko, and N. Metoki, Ga-substitution-induced single ferroelectric phase in multiferroic CuFeO2, Phys. Rev. B 78, 014101 (2008b).
  * Masuda  _et al._ [2011] T. Masuda, M. Hagihala, Y. Kondoh, K. Kaneko, and N. Metoki, Spin Density Wave in Insulating Ferromagnetic Frustrated Chain LiCuVO4, J. Phys. Soc. Jpn. 80, 113705 (2011).
  * Kobayashi  _et al._ [2004] S. Kobayashi, H. Okano, T. Jogetsu, J. Miyamoto, and S. Mitsuda, Domain growth kinetics in the isosceles triangular Ising antiferromagnet ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$, Phys. Rev. B 69, 144430 (2004).
  * Oba  _et al._ [2020] Y. Oba, N. Adachi, Y. Todaka, E. P. Gilbert, and H. Mamiya, Anomalous magnetic anisotropy and magnetic nanostructure in pure fe induced by high-pressure torsion straining, Phys. Rev. Res. 2, 033473 (2020).
  * Kobayashi  _et al._ [2023] S. Kobayashi, E. Nomura, M. Chiba, Y. Kawamura, K. Ohishi, K. Hiroi, and J. ichi Suzuki, Polarized neutron scattering study of hollow fe3o4 submicron spherical particles, J. Magn. Magn. Mater. 569, 170410 (2023).
  * Kohama  _et al._ [2023] Y. Kohama, Y. Ihara, Z. Yang, K. Matsui, and K. Kindo, Liquid helium-cooled high-purity copper coil for generation of long pulsed magnetic fields, Rev. Sci. Instrum. 94, 074701 (2023).