# Cross-domain Speech Recognition with Unsupervised Character-level
Distribution Matching

###### Abstract

End-to-end automatic speech recognition (ASR) can achieve promising
performance with large-scale training data. However, it is known that domain
mismatch between training and testing data often leads to a degradation of
recognition accuracy. In this work, we focus on the unsupervised domain
adaptation for ASR and propose CMatch, a Character-level distribution matching
method to perform fine-grained adaptation between each character in two
domains. First, to obtain labels for the features belonging to each character,
we achieve frame-level label assignment using the Connectionist Temporal
Classification (CTC) pseudo labels. Then, we match the character-level
distributions using Maximum Mean Discrepancy. We train our algorithm using the
self-training technique. Experiments on the Libri-Adapt dataset show that our
proposed approach achieves 14.39% and 16.50% relative Word Error Rate (WER)
reduction on both cross-device and cross-environment ASR. We also
comprehensively analyze the different strategies for frame-level label
assignment and Transformer adaptations.

Index Terms: speech recognition, domain adaptation

##  1 Introduction

With the recent success of deep learning and high-quality, large-scale
transcribed corpora, the performance of end-to-end (E2E) automatic speech
recognition (ASR) has been significantly improved. Existing approaches are
built on the i.i.d. condition that training and testing data are from the same
distribution, e.g., from the same recording device or environment. However,
this assumption does not always hold in reality when the training and test
distributions are different. For instance, a well-trained ASR model based on
PlayStation Eye recordings is likely to have deteriorated performance when it
is used to recognize speech from the Matrix recordings. It is expensive and
time-consuming to collecting labeled speech data from massive domains
(distributions). In this paper, we tackle a more challenging scenario where
there are no labeled samples available in the test data (i.e., target domain).
Thus, our goal is to perform unsupervised domain adaptation (UDA) [1, 2] to
improve the cross-domain ASR performance on the unlabeled target domain using
the well-labeled source data.

UDA for ASR has been studied in the existing literature. Liang et al. [3]
proposed to combine data augmentation with representation matching to force
the model to learn noise-invariant representations between clean speech and
their augmented noisy counterparts. In [4], the authors were able to recover
60% to 80% of the word error rates (WER) on the target domain by introducing a
pseudo-label filtering approach based on the model’s uncertainty using dropout
for ASR UDA. Recent work [5, 6] used the domain-adversarial training [7] for
speech recognition where they adversarially trained domain discriminators to
distinguish the source and target samples.

(a) Before CMatch

(b) After CMatch

Figure 1: Feature centers of all characters in two domains are closer after
CMatch. Red and blue dots denote the source and target domains, respectively.
Some obvious bad cases in (a): {a, b, c, g, h, l, r, v, w, y} are aligned well
in (b).

However, existing methods typically treated a domain as one distribution when
matching distributions. They may ignore the fine-grained knowledge in cross-
domain ASR tasks that can be utilized for distribution matching. These fine-
grained knowledge are important to preserve the detailed distribution property
of different domains, such as the characters, phoneme, and word. Ignoring
these information is likely to result in unsatisfying results. This is also
validated in [8] where the images aligned in subdomains (i.e., domains split
by class labels) can generally achieve better adaptation performance than
traditional methods that align the whole domains.

In this paper, we propose CMatch, an algorithm to match the Character-level
distributions for cross-domain ASR. Compared to other types such as word-level
that are sparse and could lead to unstable performance, the character-level
distribution is more fine-grained and easier to compute in E2E ASR models.
Thus, it can preserve more fine-grained knowledge for each character than
existing methods that used one distribution for all characters. Figure 1 shows
that the distances of the same characters from two domains are smaller after
applying our algorithm.

To enable character-level distribution matching, CMatch consists of two steps.
First, we achieve frame-level label assignment to acquire the labels for each
encoded frame to compute the conditional distribution. This is done by using
the Connectionist Temporal Classification (CTC) [9] pseudo labels. Second,
CMatch can reduce the conditional distributions between each cross-domain
characters using the well-defined Maximum Mean Discrepancy (MMD) [10]. The
above two steps are jointly optimized in a self-training framework [11] in
Speech-Transformer [12]. Experiments on the Libri-Adapt dataset [13] show that
our proposed approach achieves 14.39% and 16.50% relative Word Error Rate
(WER) reduction on both cross-device and cross-environment ASR.

##  2 Preliminary

###  2.1 CTC-Attention Transformer ASR Model

In this work, we built our E2E ASR model based on the Speech-Transformer [12]
with joint CTC-attention structure [14] which has been successfully applied in
various ASR tasks [15, 16, 17, 18].

As model inputs, the acoustic features are 83-dimensional filter banks with
pitch features computed with 10 ms frame shift and 25 ms frame length. The
acoustic features are further subsampled by a factor of 4 by 2 convolution
layers with kernel size 3 and stride 2. The resulted features have a receptive
field of 100 milliseconds for each frame. Then the following encoder layers
process the subsampled features by self-attention and feed-forward as
illustrated in [19]. Apart from self-attention and feed-forward, the decoder
layers further accept the encoder outputs and perform cross-attention.

For the CTC-attention structure, an auxiliary CTC task [9] is introduced for
encoder outputs in order to encourage the monotonic alignment and accelerate
the convergence speed. In training, a multi-task learning scheme is adopted
consisting of two losses: the attention loss $\mathcal{L}_{\text{ATT}}$ and
the CTC loss $\mathcal{L}_{\text{CTC}}$:

$\mathcal{L}_{\text{ASR}}=(1-\lambda)\mathcal{L}_{\text{ATT}}+\lambda\mathcal{L}_{\text{CTC}},$
(1)

where $\lambda$ denotes the weight of the CTC module. Similarly, during
decoding, the beam search decoding results of the Transformer decoder are re-
scored by the CTC module:

$\hat{Y}=\arg\max_{Y\in\mathcal{Y}}(1-\lambda)\log
P_{\text{ATT}}(Y|X)+\lambda\log P_{\text{CTC}}(Y|X),$ (2)

where $X$ denotes the acoustic features, $\mathcal{Y}$ represents a set of the
decoding hypotheses.

##  3 Our Method: CMatch

The goal of cross-domain ASR is to predict the transcript for the target
speech given only paired source domain data $(X_{S},Y_{S})$ and the speech-
only target domain data $X_{T}$. Our key idea is to match the character-level
distributions, which can be computed via the conditional distributions
$P(Y|X)$ (i.e., features belonging to each character in ASR). To this end,
there are two technical challenges ahead. First, how to acquire the
conditional distributions for the speech features, i.e., obtain the features
for each character is difficult in this sequence-to-sequence framework since
we do not have one-to-one mapping for the encoded speech features and labels.
Second, how to reduce the distribution divergence between these conditional
distributions is another challenge.

###  3.1 Frame-level Label Assignment

Computing $P(Y|X)$ requires obtaining the labels for each input, which refers
to the frame-level labels in ASR as its inputs are frames. The feature of an
input $x$ extracted by Transformer encoder $f$ is $f(x)\in\mathbb{R}^{N\times
D}$, where $N$ denotes numbers of frames and $D$ denotes feature dimension.
The Transformer decoder will output $M$ labels using CTC, which is not mapped
to the $N$ frames, i.e., $N\neq M$. Thus, it is challenging to acquire frame-
level labels $y$ for computing conditional distributions $P(Y|X)$. Note that
this challenge exists for both the source and target domains. This is also
significantly different from image classification where we can easily get the
labels for each sample [20, 8].

Figure 2: Three strategies for frame-level label assignment. Symbol “-”
represents <blank>. Dotted labels are filtered out based on their predicted
confidence scores.

It is effective to use CTC forced alignment [21, 22] to take the labels from
the most probable path selected by CTC as the frame-level assignment (Figure
2(a)). However, this process is computationally expensive. It is also feasible
to use a dynamic window average (Figure 2(b)) strategy that assigns labels for
each frame by averaging, which can only work in a strict condition that the
character output is a uniform distribution. In our work, to obtain the frame-
level labels, we propose to use the CTC pseudo labels (Figure 2(c)) for both
efficiency and correctness guarantee. We assume that an ideal CTC model should
predict the label assignment with high accuracy and propose to directly
utilize the CTC predictions since the CTC module naturally predicts labels
frame by frame including <blank> symbol as the null label. More formally, the
pseudo label for the $n$-th frame $X_{n}$ can be obtained by:

$\hat{Y}_{n}=\arg\max_{Y_{n}}P_{\text{CTC}}(Y_{n}|X_{n}),\quad 1\leq n\leq N.$
(3)

We further filter out the CTC predictions with a threshold based on their
softmax scores to improve the accuracy. Only the labels with a softmax score
of over 0.9 are used.

###  3.2 Distribution Matching

For distribution matching, we adopt the popular Maximum Mean Discrepancy (MMD)
[23] for distribution measure. MMD is an efficient non-parametric criterion to
empirically evaluate the distribution divergence between two domains. MMD has
been widely applied in the fields of computer vision, natural language
processing, etc. [24, 25, 26], and its effectiveness has been theoretically
proved [10]. Given $X_{S}\sim P,X_{T}\sim Q$, the MMD distance between two
distributions $P$ and $Q$ can be formulated as follows:

$\text{MMD}(\mathcal{H}_{k},P,Q)=\sup_{||\phi||_{\mathcal{H}_{k}\leq
1}}\mathbb{E}_{X_{S}\sim P}\phi(X_{S})-\mathbb{E}_{X_{T}\sim Q}\phi(X_{T}),$
(4)

where $\mathcal{H}_{k}$ is the Reproduced kernel Hilbert space with Mercer
kernel $k(\cdot,\cdot)$, $||\phi||_{\mathcal{H}_{k}}\leq 1$ denotes its unit
norm ball and $\phi(\cdot)$ is the feature map.

A biased empirical estimate of MMD can be obtained by computing the empirical
expectations on the samples:

$\begin{split}&\text{MMD}(\mathcal{H}_{k},X_{S},X_{T})=\\\
&\sup_{||\phi||_{\mathcal{H}_{k}\leq 1}}\left(\frac{1}{|X_{S}|}\sum_{x_{s}\in
X_{S}}\phi(x_{s})-\frac{1}{|X_{T}|}\sum_{x_{t}\in
X_{T}}\phi(x_{t})\right).\end{split}$ (5)

The character-level distribution matching is computed as:

$\mathcal{L}_{\text{cmatch}}=\frac{1}{|\mathcal{C}|}\sum_{c\in\mathcal{C}}\text{MMD}(\mathcal{H}_{k},X_{S}^{c},X_{T}^{c}),$
(6)

where $X_{S}^{c}$, $X_{T}^{c}$ denotes the encoder features of the source and
target samples of the class $c$, $\mathcal{C}$ denotes the character set. Note
that we use CTC pseudo labels for both of the source and target domains,
instead of using ground-truth labels. For real implementation, we input the
features extracted by Transformer encoder to MMD instead of the raw inputs $X$
for computation.

###  3.3 Overall Loss

Finally, we apply self-training by utilizing the pseudo labeled target-domain
data $(X_{T},\hat{Y}_{T})$ with the pseudo ASR loss
$\mathcal{L}_{\text{ASR}}^{\text{tgt}}$ to assist the distribution matching.
Given the speech-only target domain data $X_{T}$ and the ASR model pre-trained
on source domain, we can obtain the beam search decoding results as pseudo
labels $\hat{Y}_{T}$ for the target speech data $X_{T}$. Denote $\gamma$ the
tradeoff hyperparameter for the distribution matching loss, the overall
formulation of CMatch is:

$\mathcal{L}=\frac{1}{2}\left(\mathcal{L}_{\text{ASR}}^{\text{src}}+\mathcal{L}_{\text{ASR}}^{\text{tgt}}\right)+\gamma\mathcal{L}_{\text{cmatch}}.$
(7)

To reduce the noise in pseudo labels, we filter out the last 30% utterances
based on their beam search confidence scores. The complete algorithm for
CMatch is illustrated in Algo. 1. Note that Step 2 in the algorithm (i.e.,
self-training) can be iterated while we found it is enough to do it once in
experiments to achieve satisfying results.

Algorithm 1 Learning algorithm of CMatch

Input: Source domain $(X_{S},Y_{S})$, target domain $X_{T}$.

1: Train network $M_{S}$ on source domain $(X_{S},Y_{S})$.

2: Obtain pseudo label $\hat{Y}_{T}$ with $M_{S}$.

3: while not done do

4: Obtain the frame-level labels using Eq. (3).

5: Joint optimization using Eq. (7).

6: end while

7: return Adapted model $M_{S\to T}$ and target transcripts.

##  4 Experimental Setup

###  4.1 Data Set

We employ the Libri-Adapt dataset [13] for the experiments. The Libri-Adapt
dataset is designed for unsupervised domain adaptation tasks and built on top
of the Librispeech-clean-100 corpus recorded using 6 microphones under 4
synthesized background noise conditions (Clean, Rain, Wind, Laughter) in 3
different accents (en-us, en-gb, and en-in). Since the cross-accent data has
not been not fully released by the author, in this work, we use the US accent
(en-us) as the main accent language and keep it unchanged to isolate the
influence brought by accents.

We constructed two types of cross-domain ASR tasks: (1) cross-device ASR
adaptation, i.e., the source and target devices are different; and (2) cross-
environment ASR adaptation, i.e., the source and target environment are
different. Specifically, we select 3 devices for cross-device experiments,
namely Matrix Voice (M), PlayStation Eye (P), and ReSpeaker (R), each two of
them can form a pair of tasks, leading to 6 tasks in total. For cross-
environment ASR, we select 3 types of noise (Rain, Wind, and Laughter) for
different target environments and the source environment remains clean,
leading to 3 tasks. In our experiments, we split the last 10% utterances in
the training set (source domain) for development. For each domain, the numbers
of training/validation/test utterances are 25685/2854/2600 corresponding to
hours of 93.77/10.71/5.60 hours, respectively. Note that we did not use any of
the labels on the target domain during training, and they are only used for
evaluation.

###  4.2 Baselines

We consider the following approaches for comparison:

  * •

Source-only: directly apply the pre-trained model from the source domain to
target domain without adaptation.

  * •

MMD-ASR: this is used in recent literature for cross-domain speech recognition
[27]. We minimize the MMD distance on the averaged encoder outputs to
encourage the domain-level feature alignment.

  * •

Domain Adversarial Training (ADV): this is adopted in recent work [5, 6]. An
adversarial domain classifier is trained on top of the averaged encoder
outputs to correctly classify the feature domains, while the encoder is
trained to maximize the domain classification loss. We also re-implemented
this idea.

###  4.3 Implementation Details

We implemented our experiments based on the ESPnet codebase [28]. The Speech-
Transformer has 12 encoder layers and 6 decoder layers with 4 attention heads
and an inner dimension of 2048. We use 31 characters including 26 alphabets
and 5 symbols (<’>, <space>, <eos>, <unk>, <blank>) as the modeling units. The
weight of the CTC module is set to 0.3 for both training and decoding. We set
$\gamma$ to 10.0 for our CMatch method tuned on the validation set. For the
baseline MMD and ADV methods, adaptation loss weight $\gamma$ is set to 10.0
and 0.3 based on the validation set, respectively. We set $\lambda=0.3$
following ESPnet [28]. For both pseudo labeling and decoding, beam size of 10
is used. We choose the primal kernel for the MMD kernel. We adapt the final
encoder layer of the Transformer, which is the same as MMD and ADV methods for
fair study. We train all the models for 100 epochs with batch size 64. Early
stopping is adopted with patience 10 and 5 epochs during pre-training and
adaptation, respectively. We optimize the models in the same way as [19] with
warmup steps 25000 and a maximum learning rate of 3.0. Code for CMatch is at
https://github.com/jindongwang/transferlearning/tree/master/code/ASR/CMatch.

##  5 Results

For comparison, we first show the word error rates (WER) of the in-domain ASR
results in Table 1 where the training and test data are from the same domain.

Table 1: Results (WER) on standard ASR in clean environment. Domain | WER
---|---
Matrix Voice (M) | 24.25
PlayStation Eye (P) | 20.07
ReSpeaker (R) | 23.78
Average | 22.70

###  5.1 Cross-domain Adaptation Results

Device adaptation. We conduct cross-device ASR experiments in Table 2.
Compared to Table 1, the baseline results show a severe performance
degradation caused by distribution shift. Compared to source-only method, both
adversarial training and MMD-ASR approaches obtain improvements on the target
domain with unlabeled target-domain speech only. MMD surpasses ADV on most
tasks, which demonstrates the effectiveness of the MMD. Our CMatch achieves
the lowest WERs on all tasks and obtains an average WER of 22.85% with a
relative reduction of $\mathbf{14.39\%}$ compared with the baseline.

Table 2: WER on cross-device ASR in clean environment. Task | Source-only | MMD | ADV | CMatch
---|---|---|---|---
M $\to$ P | 23.87 | 20.87 | 21.11 | 20.38
M $\to$ R | 25.21 | 22.21 | 22.27 | 21.77
P $\to$ M | 31.15 | 27.22 | 28.29 | 26.17
P $\to$ R | 23.99 | 21.90 | 21.74 | 20.43
R $\to$ M | 32.45 | 28.27 | 29.95 | 27.77
R $\to$ P | 23.48 | 21.09 | 21.23 | 20.58
Average | 26.69 | 23.59 | 24.10 | 22.85

Noise adaptation. We pick the ReSpeaker device for the noise adaptation
experiments, whose standard WER in the clean environment is 23.78%. The
results are presented in Table 3. We also observe similar performance of MMD
and ADV. Our CMatch achieves $\mathbf{16.50\%}$ relative WER reduction on
average, indicating its effectiveness for cross-environment adaptation. CMatch
still shows the best performance on all the tasks. It is worth noting that
when adapted to the wind environment, CMatch obtains very competitive results
without the real labels.

Table 3: WER on cross-environment adaptation (source: Clean) Target | Source-only | MMD | ADV | CMatch
---|---|---|---|---
Rain | 38.21 | 33.61 | 34.65 | 32.90
Wind | 29.70 | 26.06 | 26.73 | 23.12
Laughter | 33.36 | 29.85 | 30.41 | 28.55
Average | 33.76 | 29.84 | 30.60 | 28.19

###  5.2 Ablation Study

We conduct the ablation study in Table 4. Results indicate that compared to
baselines, our character-level matching loss and the self-training strategy
can independently produce competitive performance. Thus, they are critical in
our algorithm. We observe that self-training can generally perform well,
indicating its effectiveness in obtaining labels, which is important in our
unsupervised conditional distribution matching algorithm. Besides, we also
tested the combination of vanilla MMD and self-training and found its
performance not comparable to our results. By combining the advantage of fine-
grained distribution matching and self-training, our CMatch obtains the best
performance through fine-grained distribution matching.

Table 4: Ablation study on the CMatch method. Variant | Device | Noise
---|---|---
Source-only | 26.69 | 33.76
w/ self-training | 22.99 | 28.31
w/ distribution matching | 23.87 | 30.43
All | 22.85 | 28.19

###  5.3 Analyzing the Label Assignment in CMatch

In this section, we compare three label assignment approaches as mentioned in
Section 3.1. The results in Table 5 show that the CTC alignment method
achieves the best results. However, we found it computationally expensive
($2\times$ slower than others since the forced alignment requires computing
the forward-backward algorithm in each iteration). As its complement, frame
average also achieves good results, but it can generally works with uniform
speaking speed. Our pseudo CTC prediction has no assumption on the speaking
speed while obtaining similar results and more efficient than CTC alignment.

Table 5: Comparison of different label assignment strategy. Task | PseudoCTCPred | FrameAverage | CTCAlign
---|---|---|---
M $\to$ P | 20.38 | 20.21 | 20.23
M $\to$ R | 21.77 | 21.80 | 21.75
P $\to$ M | 26.17 | 26.02 | 25.84
P $\to$ R | 20.43 | 20.36 | 20.44
R $\to$ M | 27.77 | 27.94 | 27.73
R $\to$ P | 20.58 | 20.55 | 20.52
Average | 22.85 | 22.81 | 22.75

###  5.4 Adapting Encoder and Decoder

A fundamental step in cross-domain ASR is to determine the adaptation
contribution of the encoder and decoder. In our experiments, we adapt the last
encoder layer. It is intuitive and necessary to ask: Will adapting the encoder
and decoder lead to better performance? In this section, we empirically answer
these questions via extensive experiments. Table 6 shows our results, where
“first”, “last”, and “all” denote we adapt the first / last / all decoder
layers, respectively. It implies that in ASR, it is more necessary to adapt
the encoder layers. Since the decoders are not task-specific in our
experiments (all standard English ASR), reducing their distribution gap may
not be effective and could generally hurt the performance. The conclusions are
similar on cross-device experiments. In addition, we did not get positive
results by adapting more encoder layers. Therefore, adapting the last encoder
layer is sufficient in our task.

Table 6: CMatch with / without decoder adaptation. “first”, “last” and “all” correspond to decoder layer(s) for adaptation. Target | w/o decoder | first | last | all
---|---|---|---|---
Rain | 32.90 | 32.92 | 32.85 | 33.12
Wind | 23.12 | 23.18 | 23.18 | 23.28
Laughter | 28.55 | 28.66 | 28.56 | 28.63
Average | 28.19 | 28.25 | 28.20 | 28.34

##  6 Conclusion

In this paper, we proposed CMatch for cross-domain speech recognition. Our key
motivation is to match the character-level distributions from the source and
target domain to leverage the fine-grained information for better adaptation.
Experiments on cross-device and cross-environment ASR have shown the
superiority of our CMatch. Moreover, we also conduct extensive experiments to
empirically analyze the contribution of encoders and decoders in Speech-
Transformer architectures and analyze different frame-level label assignment
strategies. In the future, we plan to extend our research into other speech
adaptation applications such as dataset adaptation and speaker adaptation.

## References

  * [1] S. J. Pan and Q. Yang, “A survey on transfer learning,” _IEEE Transactions on knowledge and data engineering_ , vol. 22, no. 10, pp. 1345–1359, 2009.
  * [2] V. M. Patel, R. Gopalan, R. Li, and R. Chellappa, “Visual domain adaptation: A survey of recent advances,” _IEEE signal processing magazine_ , vol. 32, no. 3, pp. 53–69, 2015.
  * [3] D. Liang, Z. Huang, and Z. C. Lipton, “Learning noise-invariant representations for robust speech recognition,” in _IEEE Spoken Language Technology Workshop (SLT)_. IEEE, 2018, pp. 56–63.
  * [4] S. Khurana, N. Moritz, T. Hori, and J. L. Roux, “Unsupervised domain adaptation for speech recognition via uncertainty driven self-training,” _arXiv preprint arXiv:2011.13439_ , 2020.
  * [5] S. Sun, C.-F. Yeh, M.-Y. Hwang, M. Ostendorf, and L. Xie, “Domain adversarial training for accented speech recognition,” in _IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 4854–4858.
  * [6] R. Duan, “Unsupervised feature adaptation using adversarial multi-task training for automatic evaluation of children’s speech,” in _ICASSP_ , 2020\.
  * [7] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky, “Domain-adversarial training of neural networks,” _The journal of machine learning research_ , vol. 17, no. 1, pp. 2096–2030, 2016.
  * [8] Y. Zhu, F. Zhuang, J. Wang, G. Ke, J. Chen, J. Bian, H. Xiong, and Q. He, “Deep subdomain adaptation network for image classification,” _IEEE transactions on neural networks and learning systems_ , 2020.
  * [9] A. Graves, S. Fernández, F. Gomez, and J. Schmidhuber, “Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks,” in _Proceedings of the 23rd international conference on Machine learning_ , 2006, pp. 369–376.
  * [10] A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. Smola, “A kernel two-sample test,” _The Journal of Machine Learning Research_ , vol. 13, no. 1, pp. 723–773, 2012.
  * [11] R. Masumura, M. Ihori, A. Takashima, T. Moriya, A. Ando, and Y. Shinohara, “Sequence-level consistency training for semi-supervised end-to-end automatic speech recognition,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 7054–7058.
  * [12] L. Dong, S. Xu, and B. Xu, “Speech-transformer: a no-recurrence sequence-to-sequence model for speech recognition,” in _IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 5884–5888.
  * [13] A. Mathur, F. Kawsar, N. Berthouze, and N. D. Lane, “Libri-adapt: a new speech dataset for unsupervised domain adaptation,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 7439–7443.
  * [14] S. Kim, T. Hori, and S. Watanabe, “Joint ctc-attention based end-to-end speech recognition using multi-task learning,” in _2017 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2017, pp. 4835–4839.
  * [15] S. Karita, N. Chen, T. Hayashi, T. Hori, H. Inaguma, Z. Jiang, M. Someki, N. E. Y. Soplin, R. Yamamoto, X. Wang _et al._ , “A comparative study on transformer vs rnn in speech applications,” in _2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU)_. IEEE, 2019, pp. 449–456.
  * [16] W. Hou, Y. Dong, B. Zhuang, L. Yang, J. Shi, and T. Shinozaki, “Large-Scale End-to-End Multilingual Speech Recognition and Language Identification with Multi-Task Learning,” in _Proc. Interspeech 2020_ , 2020, pp. 1037–1041. [Online]. Available: http://dx.doi.org/10.21437/Interspeech.2020-2164
  * [17] W. Hou, Y. Wang, S. Gao, and T. Shinozaki, “Meta-adapter: Efficient cross-lingual adaptation with meta-learning,” in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 7028–7032.
  * [18] W. Hou, H. Zhu, Y. Wang, J. Wang, T. Qin, R. Xu, and T. Shinozaki, “Exploiting adapters for cross-lingual low-resource speech recognition,” _arXiv preprint arXiv:2105.11905_ , 2021.
  * [19] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” _Advances in Neural Information Processing Systems_ , vol. 30, pp. 5998–6008, 2017.
  * [20] Y. Pan, T. Yao, Y. Li, Y. Wang, C.-W. Ngo, and T. Mei, “Transferrable prototypical networks for unsupervised domain adaptation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 2239–2247.
  * [21] H. Sak, A. Senior, K. Rao, O. Irsoy, A. Graves, F. Beaufays, and J. Schalkwyk, “Learning acoustic frame labeling for speech recognition with recurrent neural networks,” in _2015 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2015, pp. 4280–4284.
  * [22] L. Kürzinger, D. Winkelbauer, L. Li, T. Watzel, and G. Rigoll, “Ctc-segmentation of large corpora for german end-to-end speech recognition,” in _International Conference on Speech and Computer_. Springer, 2020, pp. 267–278.
  * [23] K. M. Borgwardt, A. Gretton, M. J. Rasch, H.-P. Kriegel, B. Schölkopf, and A. J. Smola, “Integrating structured biological data by kernel maximum mean discrepancy,” _Bioinformatics_ , vol. 22, no. 14, pp. e49–e57, 2006.
  * [24] S. J. Pan, I. W. Tsang, J. T. Kwok, and Q. Yang, “Domain adaptation via transfer component analysis,” _IEEE Transactions on Neural Networks_ , vol. 22, no. 2, pp. 199–210, 2010.
  * [25] E. Tzeng, J. Hoffman, N. Zhang, K. Saenko, and T. Darrell, “Deep domain confusion: Maximizing for domain invariance,” _arXiv preprint arXiv:1412.3474_ , 2014.
  * [26] J. Zhang, W. Li, and P. Ogunbona, “Joint geometrical and statistical alignment for visual domain adaptation,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 1859–1867.
  * [27] P. G. Shivakumar and P. Georgiou, “Transfer learning from adult to children for speech recognition: Evaluation, analysis and recommendations,” _Computer speech & language_, vol. 63, p. 101077, 2020.
  * [28] S. Watanabe, T. Hori, S. Karita, T. Hayashi, J. Nishitoba, Y. Unno, N. Enrique Yalta Soplin, J. Heymann, M. Wiesner, N. Chen, A. Renduchintala, and T. Ochiai, “Espnet: End-to-end speech processing toolkit,” in _Proc. Interspeech_ , 2018.


# Determination of the single-ion anisotropy energy in a $S$ = 5/2 kagome
antiferromagnet using x-ray absorption spectroscopy.

M. A. de Vries<EMAIL_ADDRESS>Laboratory for Quantum Magnetism,
École Polytechnique Fédérale de Lausanne, Station 3, CH-1015 Lausanne,
Switzerland  CSEC and School of Chemistry, The University of Edinburgh,
Edinburgh EH9 3JZ, UK  T. K. Johal  Daresbury Laboratory, Warrington WA4 4AD,
UK  A. Mirone  ESRF, 6 rue Jules Horowitz, F-38000 Grenoble, France  J. S.
Claydon  Department of Engineering Materials, University of Sheffield,
Sheffield S1 3JD, UK  G. J. Nilsen  Laboratory for Quantum Magnetism, École
Polytechnique Fédérale de Lausanne, Station 3, CH-1015 Lausanne, Switzerland
CSEC and School of Chemistry, The University of Edinburgh, Edinburgh EH9 3JZ,
UK  H. M. Rønnow  Laboratory for Quantum Magnetism, École Polytechnique
Fédérale de Lausanne, Station 3, CH-1015 Lausanne, Switzerland  G. van der
Laan  Daresbury Laboratory, Warrington WA4 4AD, UK  Diamond Light Source,
Chilton, Didcot OX11 0DE, UK  A. Harrison  CSEC and School of Chemistry, The
University of Edinburgh, Edinburgh EH9 3JZ, UK  Institut Laue-Langevin, 6 rue
Jules Horowitz, F-38000 Grenoble, France

###### Abstract

We report x-ray absorption and x-ray linear dichroism measurements at the Fe
$L_{2,3}$ edges of the geometrically frustrated systems of potassium and
hydronium iron jarosite. Comparison with simulated spectra, involving ligand-
field multiplet calculations modelling the $3d$-$2p$ hybridization between the
iron ion and the oxygen ligands, has yielded accurate estimates for the ligand
metal-ion hybridization and the resulting single-ion crystal field anisotropy
energy. Using this method we provide an experimentally verified scenario for
the appearance of a single-ion anisotropy in this nominally high-spin $3d^{5}$
orbital singlet ${}^{6}S$ system, which accounts for features of the spin-wave
dispersion in the long-range ordered ground state of potassium iron jarosite.

###### pacs:

75.25.+z, 71.70.Ch, 71.70.Ej, 75.10.Dg

##  I Introduction

The jarosite group of minerals has received much attention for being close to
ideal realizations of the kagome antiferromagnet. The kagome topology, which
is named after a Japanese basket weaving pattern, frustrates the
antiferromagnetic Néel ordering Diep:04 . Due to the resulting
“underconstraint” neither in the quantum case of $S$ = 1/2 nor in the
classical limit is the kagome antiferromagnet expected to show a symmetry-
breaking transition, even at $T=0$ K Reimers:91 ; Chalker:92 ; Reimers:93 .
Physical realizations of this system are highly valued because they allow for
the investigation of the Mott insulating phase in the absence of Néel-like
magnetic order. The jarosite group of general stoichiometry
$AM_{3}$(SO4)2(OH)6, where $A$ can be—amongst others—K, Na, Rb, Ag, NH3, or
H3O, provides physical models for the kagome antiferromagnet with spin $S$ =
3/2 and 5/2, where the $M$ site is occupied by trivalent Cr KerenJar:96 ;
Inami:01 ; Morimoto:03  and Fe Takano:68 ; Townsend:86 ; Wills:96 ;
WillsThesis ; KerenJar:96 ; WillsKjar:00 ; WillsKjar:01 ; Grohol:05 ,
respectively. $M=$ V is a $S$ = 1 system with ferromagnetic near-neighbor
interactions Grohol:02 ; Grohol:03van . Fig. 1 illustrates the jarosite
structure of well separated kagome layers consisting of $M$O6 octahedra.
Potassium iron jarosite is representative for most of the iron analogues. In
this system a transition to a non-collinear long-range ordered state is
observed at 64 K, despite that the magnitude of the Curie-Weiss temperature,
$\Theta_{\rm{CW}}$, is as high as $-$800 K Grohol:03 ; WillsKjar:00 . The
ground state has shown to be a so-called $q=0$ state of positive chirality, in
which the spins lie within the kagome plane and all point either in or out of
each shared kagome triangle Townsend:86 ; Inami:00 ; WillsKjar:01 ; Inami:03 ;
Grohol:05 . The large figure $\Theta_{\text{CW}}/T_{\rm{N}}=12.5$ is an
indication of relatively strong geometric frustration Ramirez:94 , but clearly
the observed long-range ordered ground state points to additional terms in the
Hamiltonian of this kagome antiferromagnet, beyond near-neighbor exchange. It
has been shown that the $q=0$ ground state can arise due to magnetic
anisotropies such as easy-plane single-ion crystal field (CF) anisotropy
$D_{z}\hat{S}_{z}^{2}-E_{xy}(\hat{S}_{x}^{2}-\hat{S}_{y}^{2}),$ Nishiyama:03 ;
Yildirim:06  where $D_{z}$ is the zero-field splitting parameter, or a
Dzyaloshinsky-Moriya interaction (DMI) Elhajal:02 ; Yildirim:06
$D_{ij}\hat{S}_{i}\times\hat{S}_{j}$. The DMI term has shown to be symmetry
allowed on the kagome lattice and has been argued to prevail over the CF
anisotropy in the spherical ($L=0$) 3$d^{5}$ Fe3+ ion Elhajal:02 . Fits to the
spin-wave dispersion curves measured with neutron spectroscopy on single
crystal Matan:06  and powder samples Coomer:06  of potassium jarosite have
found to be slightly better in a model with DMI compared to the model with CF
anisotropy. The evidence so far has, however, been inconclusive.

Given the similarity in crystal structure of potassium (iron) jarosite and
hydronium (iron) jarosite the above effects could expected to be equally
important in the latter. However, hydronium jarosite has a spin-glass ground
state with a freezing temperature of $T_{g}=17(2)$ K Bisson:unpub07 . The
Weiss temperature for hydronium jarosite is less accurately known ($-$700 to
$-$1400 K) because the inverse susceptibility does not enter a linear regime
sufficiently far below the decomposition temperature of the compound Wills:96
. It has been suggested that the spin-glass state is the result of proton
transfer from the hydronium groups to the Fe-(O${}_{\text{eq}}$H)-Fe super-
exchange mediating hydroxy groups Grohol:03 . One indication that this could
happen might be that the OH groups at the crystallographically analogous
location in the related kagome compound zinc paratacamite can form a muonium
state by chemically binding a positive muon Mendels:07 . The aim of the work
described here is to investigate the origins of the magnetic anisotropy (of
either single-ion or DMI character) in the iron jarosites, and to understand
the essential differences between the salts such as potassium jarosite with a
long range ordered ground state and hydronium jarosite.

Figure 1: (color online) The jarosite structure viewed perpendicular to the
$c$ axis (left picture) and along the $c$ axis (right picture), revealing the
kagome network of magnetic ions. The $M$O6 octahedra are drawn in transparent
brown with the $M^{3+}$ ions at the center in dark brown. The SO4 tetrahedra
are transparent yellow. The large spheres at the origin of the unit cells in
blue are the $A$ site ions, and the hydrogen of the OH groups are drawn as
small white spheres.

Using Mössbauer spectroscopy Afanasev:74 ; Bonville:06  the spin state of the
Fe3+ in the iron jarosites was found to be $S$ = 5/2, thus nominally high-spin
$3d^{5}$. The free-ion 3$d^{5}$ state is ${}^{6}S$, i.e. $L=0$. Hence, there
is no orbital angular momentum due to the spin-orbit coupling in the presence
of a crystal field as long as the system is high-spin 3$d^{5}$. This holds at
least up to third order perturbation theory of the trigonal crystal-field
distortions, spin-spin and spin-orbit couplings111It has been shown
Watanabe:57  that in 4th, 5th and 6th order perturbation theory of the crystal
field and spin-spin interactions a small orbital angular momentum and single-
ion anisotropy arises.. Even in a $D_{4h}$ crystal field the 3$d^{5}$ $S$ =
5/2 spin should therefore be practically isotropic apart from the possible
effects of dipole-dipole interactions. Both DMI and single-ion anisotropy
arise only when the spin-orbit operator $L\cdot S$ can induce a finite orbital
angular momentum Moriya:60 ; Watanabe:57 . That $\langle L\rangle>0$ in the
iron jarosites is clear from Curie-Weiss fits to the high temperature magnetic
susceptibility which indicate $\mu_{\text{eff}}$ between 6.2 and 6.7
$\mu_{\text{B}}$ in potassium iron jarosite Grohol:05  and 6.6(2)
$\mu_{\text{B}}$ in the hydronium analogue Wills:96 ; WillsH3Ojar:00
(compared to a spin only value of the effective moment of 5.92
$\mu_{\text{B}}$). Clearly, this opens up the possibility for both DMI and
single ion anisotropies. However, the origin of the orbital angular momentum
itself has so far received little attention.

A very strong crystal field, such as in iron phthalocyanine Thole:88Spinmix
can give rise to a low-spin ground state in which the spin-orbit coupling will
in general reinstate part of the orbital angular momentum of the free-ion
configuration which is, in zero order, quenched by the crystal field. It could
be that in weaker crystal fields, such as those present in iron oxides, some
orbital angular momentum appears due to a mixing in of the low-spin state into
the ground state. The crystal field alone does, however, not provide matrix
elements between high-spin and low-spin states. The most likely explanation is
therefore that an Fe2+ configuration mixes in due to charge transfer Zaanen:85
from the surrounding ligands. For hematite, e.g., the effective valence of the
Fe ion has been estimated at 2.43 due to charge transfer from the oxygen
ligands Coey:71 . These effects have earlier been shown to explain zero field
splitting in Mn2+ and Fe3+ substituted crystals Wan-Lun:89 ; Wan-Lun:94  and
in the Mn2+ compounds MnF3 Fransisco:88 , MnPS3 and MnPSe3 Jeevanandam:99 ,
despite the lower covalence of the fluoride and phosphorus-containing ligands
compared to oxygen.

Transition metal $2p\to 3d$ x-ray spectroscopy studies have played a key role
in the understanding of the effect of charge-transfer and crystal-field in
determining the electronic and magnetic properties in transition metals and
their compounds Zaanen:85 ; vanderLaan:86 ; FdG:94Rev ; FdG:05Rev . We have
measured the x-ray absorption and x-ray linear dichroism (XLD) at the Fe
$L_{2,3}$ ($2p\to 3d$) edges on single-crystalline and powder samples of
potassium and hydronium iron jarosite. By comparison with atomic multiplet
calculations, taking into account ligand-field effects and multiple ionic
configurations, we obtain accurate values for the charge transfer and single-
ion anisotropy in the $S$ = 5/2 kagome antiferromagnet. The results are
compared to magnetic susceptibility and neutron spectroscopy measurements on
in particular potassium iron jarosite.

##  II Experimental

Single crystals of potassium iron jarosite, KFe3(OH)6(SO4)2, were grown using
a hydrothermal reduction-oxidation method as described in Ref. Grohol:03, ;
4.88 g (28.0 mmol) K2SO4 and 2.2 mL (40 mmol) H2SO4 were dissolved in 50 mL
distilled water, and transferred into a 125 mL PTFE liner of a stainless steel
bomb. 0.56 g (10 mmol) iron wire with a diameter of 2 mm was added to the
solution. The bomb was placed in an oven at 202∘C for 4 days, then cooled down
to room temperature at a rate of 0.3∘C/min. The precipitate was washed,
filtered and dried, yielding 0.37 g, which is 22% based on Fe. In order to
obtain sufficiently large crystals it was important that the surface area of
the solution exposed to air inside the vessel was minimized. This was achieved
using a glass container which fitted inside the PTFE liner, with only a small
hole at the top. Using this method single crystals of up to 0.7 mm in diameter
were obtained, which was sufficiently large for the application in polarized
x-ray spectroscopy. A number of crystals were characterized using single-
crystal x-ray diffraction. These crystals were identified as potassium iron
jarosite and face-indexed.

The original solvothermal synthesis method Dutrizac:76 ; WillsThesis  for the
jarosites was used for the preparation of H3OFe3(OH)6(SO4)2. In this
preparation 6.6 g (22 mmol) of Fe2(SO4)${}_{3}\cdot$5H2O was dissolved in 50
mL water. This solution was transferred to a 125 mL PTFE liner of a stainless
steel bomb. The solution was heated in the bomb to 140∘C for 12 hours. The
hydronium iron jarosite which precipitated during the reaction was washed,
filtered and dried, yielding $\sim$0.27 g of product.

Polarized soft x-ray absorption measurements were carried out using the liquid
helium cryostat, housed in the high-field superconducting magnet on ID08 at
the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Total-
electron-yield spectra were measured by recording the drain current from the
sample as a function of photon energy in the region of the Fe $L_{2,3}$
absorption edges. Potassium iron jarosite is strongly insulating, so the
resonance enhanced total-electron-yield signal at the iron edge is much
smaller than the signal from the surrounding metal of the sample holder. To
avoid this source of background intensity a potassium iron jarosite single
crystal of 0.7 mm diameter was attached using silver glue to the end of a
graphite tip, which in turn was glued to the end of a standard ESRF sample
holder. The sample was aligned in the x-ray beam with the crystallographic $c$
axis pointing in the horizontal plane and perpendicular to the incident beam.
This allowed us to measure absorption spectra with the incident x-ray
polarization both perpendicular and parallel to the $c$ axis. Spectra were
recorded with alternating horizontal and vertical polarization, and for each
polarization averages were taken over at least eight scans. In this way the
polarization dependent spectra were taken at a number of temperatures between
20 and 290 K, in zero external field.

Isotropic absorption spectra at the Fe $L_{2,3}$ edges were obtained on
beamline 5U.1 of the Synchrotron Radiation Source (SRS) at Daresbury
Laboratory, UK, using potassium and hydronium iron jarosite powder dispersed
over double-sided UHV-compatible carbon tape.

##  III Results

Figure 2 shows the spectra obtained from single-crystal and powder samples of
potassium iron jarosite at 290 K. The upper curves in Fig. 2 display the x-ray
absorption spectra from a single crystal measured on beamline ID08 at the
ESRF, with the incident x-ray polarization parallel to the $c$ axis, $I_{c}$
(black circles), and perpendicular to the $c$ axis, $I_{\text{ab}}$ (red
crosses). The curves in the middle of the figure give the corresponding
isotropic absorption spectrum
$I_{\text{iso}}=\frac{2}{3}I_{\text{ab}}+\frac{1}{3}I_{c}$ (black line) and
the isotropic spectrum directly measured from a powder sample on station 5U.1
at the SRS (broken red line). The curve at the bottom gives the x-ray linear
dichroism $I_{\text{xld}}$ defined as $I_{\text{ab}}-I_{\text{c}}$ (thin black
line). The $2p\to 3d$ transition consists of two edges, the first one at
$\sim$$709$ eV ($L_{3}$ edge) corresponds to the creation of a $2p_{3/2}$ core
hole. The transition giving rise to a $2p_{1/2}$ core hole ($L_{2}$ edge) is
at $\sim$$12$ eV higher photon energy due to the $2p$ spin-orbit coupling. The
inset of Fig. 2 shows the $I_{\text{c}}$ and $I_{\text{ab}}$ spectra in close
up at the $L_{3}$ edge, which is the most significant region of the spectrum.

Figure 2: (color online) x-ray absorption spectra of potassium iron jarosite
at 290 K measured with x-ray polarization parallel and perpendicular to the
$c$ axis, $I_{\text{c}}$ (black circles) and $I_{\text{ab}}$ (red crosses),
respectively. The lower trace shows the corresponding values of the isotropic
absorption, $I_{\text{iso}}=\frac{2}{3}I_{\text{ab}}+\frac{1}{3}I_{\text{c}}$
(black line) and the x-ray linear dichroism,
$I_{\text{xld}}=I_{\text{ab}}-I_{\text{c}}$ (thin black line). Also shown
$I_{\text{iso}}$ from a powder sample measured on 5U.1 at SRS Daresbury with a
lower energy resolution (broken, red line). The inset shows a close up of
$I_{\text{c}}$ and $I_{\text{ab}}$ at the $L_{3}$ edge.

As seen in Fig. 3, the structure at the $L_{3}$ edge becomes more prominent in
the isotropic spectrum as the temperature is lowered. At each stabilized
temperature, the reproducibility of up to eight spectra demonstrated the
absence of any temporal variation in the spectral line shape or intensity,
therefore charging effects and drift in sample alignment can be eliminated as
the cause of the observed temperature dependence. These isotropic spectra
obtained by weighted averaging of $I_{\text{ab}}$ and $I_{\text{c}}$ are in
good agreement with the spectra from powder samples measured for the SRS at
all temperatures. That this change in line shape occurs in the powder averaged
spectrum might be an indication of a slight change in the electronic structure
as the temperature is lowered. Only very small changes were observed in the
XLD spectral line shape on cooling the sample, as shown in Fig. 4. This change
in the spectral line shape is accompanied by a slight reduction in the
integrated XLD spectrum Scho:98

$\int I_{\text{xld}}(\omega)d\omega/\int I_{\text{iso}}(\omega)d\omega$ (1)

corresponding to a small change of the total quadrupole moment of 0.05(2).
This might be interpreted as that the Fe $3d$ shell gains between 3 and 7% net
$z^{2}-r^{2}$ character at base temperature.

Figure 3: (color online) Temperature dependence of the isotropic x-ray
absorption measured at the Fe $L_{2,3}$ edges from a synthesized potassium
iron jarosite single crystal (black lines), compared to calculated spectra (as
discussed in Secs. IV and V) with simulated temperatures as listed in the
legend.

The room temperature spectra of hydronium iron jarosite were identical to the
spectra obtained on potassium iron jarosite within the experimental resolution
as measured at the SRS (the red, broken line in the main panel of Fig. 2 for
comparison).

Figure 4: (color online) Temperature dependence of the x-ray linear dichroism
measured at the Fe $L_{2,3}$ edges from a synthesized potassium iron jarosite
single crystal (black lines), compared to calculated linear dichroism
corresponding to the isotropic simulated spectra of Fig. 3, scaled in
intensity to fit the experimental spectra by factors ranging from $1/2$ at 290
K down to $1/3$ for the 60 and 40 K data. Possible reasons for this mismatch
are discussed in the text.

##  IV Ligand-field multiplet calculations

Due to the strong electrostatic interactions between the $2p$ core hole and
the $3d$ levels in the final state, the $2p$ $\to$ $3d$ absorption spectrum is
not simply proportional to the density of unoccupied $3d$ levels vanderlaan:91
as a function of energy. For ferromagnets and magnetically soft materials,
where all the magnetic moments can be aligned by applying an external field,
the x-ray magnetic circular dichroism (XMCD) and x-ray magnetic linear
dichroism (XMLD) sum rules can be applied to obtain the expectation values of
the orbital and spin magnetic moments Thole:92  and the magneto-crystalline
anisotropy vanderlaan:99 , respectively. In the case of the jarosites it is
not possible to align the magnetic moments parallel with an external field due
to the large energy scale of the antiferromagnetic interactions
($\Theta_{\rm{CW}}\approx-800$ K). The only way to obtain accurate information
about the relevant expectation values is by explicitly calculating the
transition probabilities

$f(E_{\hbar\omega},\mathbf{q})=\frac{|\bra{\psi_{g}}\mathbf{\hat{r}_{q}}\ket{\psi_{e}}|^{2}}{E_{\hbar\omega}-\Delta
E-i\Gamma/2},$ (2)

between the ground state $\psi_{g}$ and excited state $\psi_{e}$, where
$\mathbf{\hat{r}_{q}}$ is the electric-dipole operator for x-ray absorption
with polarization $\mathbf{q}$, and $\Gamma$ is the lifetime broadening, which
is treated as a fitting parameter. The experimental x-ray spectra were used to
obtain good approximations for $\psi_{g}$ and $\psi_{e}$ with the aid of the
Hilbert++ code Mirone:00 ; Mirone:code . The Hilbert space for
$\ket{\psi_{g}}$ is spanned by determinants of the lowest atomic $LSJ$
configurations of $\ket{3d^{n}}$ and $\ket{3d^{n+1}\underline{L}}$, where
$\underline{L}$ denotes a ligand hole. Likewise, the Hilbert space for
$\ket{\psi_{e}}$ is spanned by determinants of the atomic configurations of
$\ket{2p^{5}3d^{n+1}}$ and $\ket{2p^{5}3d^{n+2}\underline{L}}$. The Slater
integrals for the $3d$-$3d$ and $2p$-$3d$ interactions and the $2p$ and $3d$
spin-orbit parameters for the ground state and final state configurations
calculated using Cowan’s atomic multiplet program Cowan:81  are tabulated in
Ref. vanderlaan:92, . Approximate solutions for the eigenfunctions of the iron
ion in the compound are then obtained by diagonalization of the Hamiltonian
including charge-transfer and crystal-field terms arising from the surrounding
oxygen ligands Mirone:code

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathcal{H}_{\text{atom}}$ (3) $\displaystyle+$
$\displaystyle\sum_{b}[t_{\sigma,b}d^{+}_{3\tilde{z}^{2}-r^{2}}p^{-}_{\tilde{z}}+t_{\pi,b}(d^{+}_{\tilde{x}\tilde{z}}p^{-}_{\tilde{x}}+d^{+}_{\tilde{y}\tilde{z}}p^{-}_{\tilde{y}})+\text{cc}]$
$\displaystyle+$
$\displaystyle\sum_{b}[V_{\sigma,b}d^{+}_{3\tilde{z}^{2}-r^{2}}d^{-}_{3\tilde{z}^{2}-r^{2}}+V_{\pi,b}(d^{+}_{\tilde{x}\tilde{z}}d^{-}_{\tilde{x}\tilde{z}}+d^{+}_{\tilde{y}\tilde{z}}d^{-}_{\tilde{y}\tilde{z}})]$
$\displaystyle+$
$\displaystyle\epsilon_{p}\sum_{b}(p^{+}_{\tilde{x}}p^{-}_{\tilde{x}}+p^{+}_{\tilde{y}}p^{-}_{\tilde{y}}+p^{+}_{\tilde{z}}p^{-}_{\tilde{z}}).$

In this Hamiltonian, the first term represents the atomic Hamiltonian. The
second term gives the hybridization between metal $3d$ and oxygen $2p$
orbitals, where $t_{\sigma,b}$ and $t_{\pi,b}$ are the Slater-Koster hopping
parameters. For each bond, $t_{\sigma,\pi}$ are rescaled with the bond length
using $t_{\sigma,\pi,b}=t_{\sigma,\pi}(R_{\text{b}}/R_{\text{ref}})^{\alpha}$
where $R_{b}/R_{\text{ref}}$ is the normalized length of Fe-O bond $b$ and the
rescaling exponent $\alpha$ is treated as a free parameter. The third term
gives the contribution of the electrostatic crystal field which was not used
in our case. The last term gives the energy of the $2p$ electrons in the
oxygen valence band, where $\epsilon_{p}$ is a simulation parameter related to
the charge-transfer gap $\Delta_{pd}$ via
$\Delta_{pd}=U_{3d^{5}}-\epsilon_{p}$. The total $dd$ Coulomb energy for the
addition of a sixth electron in the Fe $3d$ shell
$U_{3d^{5}}=\sum_{n=1}^{5}U_{n}$ was found to be 44.3 eV by considering the
case $t_{\sigma,\pi}=0$, where no charge transfer occurs unless the Fe2+
$d^{6}\underline{L}$ configuration has a lower energy than the Fe3+ $d^{5}$
configuration. The sum is made over all bonds $b$, where the local coordinate
frame $\tilde{x},\tilde{y},\tilde{z}$ is oriented with the $\tilde{z}$ axis
along the bond direction. Numerous comparisons between experiment and theory
Thole:PRB85 ; FdG:05Rev  have shown that this approach works well, provided
that the Slater integrals for the Coulomb interactions and spin-orbit
couplings are reduced to 70-80% of the atomic Hartree-Fock values
Thole:88Spinmix . Here, all $pd$ and $dd$ Slater integrals were reduced to 80%
of the atomic values calculated using Cowan’s program. The lifetime broadening
$\Gamma$ was set to $0.24$ eV for the $L_{3}$ edge and $0.3$ eV for the
$L_{2}$ edge. These are values as commonly used in the calculation of x-ray
absorption spectra, with $L_{2}$ always slightly larger than $L_{3}$. This
spectral broadening is due to the coupling with the (infinite dimensional)
electromagnetic field.

Figure 5: (color online) Experimentally obtained isotropic spectrum and x-ray
linear dichroism (black lines) at 290 K and simulated XAS and XLD (red lines)
Boltzmann averaged at 110 K. The simulated XLD is roughly twice as large as
the experimentally obtained XLD which is ascribed to mis-alignments and sample
imperfections.

The crystal field at the iron atom in potassium iron jarosite is approximately
of $D_{4h}$ symmetry, with an elongation of the bond length $R_{\text{ap}}$
along the fourfold symmetry axis ($c^{\prime}$) of $\sim$4% with respect to
the bond length $R_{\text{eq}}$ in the equatorial plane Grohol:03 . The local
symmetry axes of the FeO6 octahedra are canted with respect to the
crystallographic $c$ axis by an angle of 19.2∘, resulting in buckled FeO4
planes forming the kagome layers.

Our simulation shows that the spectra are mainly determined by the strongly
anisotropic $\sigma$-type hybridization between the iron $3d$ and oxygen $2p$
orbitals. The electrostatic part of the crystal field could be left out. An
excellent agreement between experimental and simulated spectra for $I_{iso}$
was obtained using only four simulation parameters, namely the charge-transfer
gap $\Delta_{pd}$, $t_{\sigma}$, $t_{\pi}$ and the rescaling constant $\alpha$
as defined in
$t_{\pi,\sigma}^{\text{ap}}=t_{\pi,\sigma}^{\text{eq}}(R_{\text{ap}}/R_{\text{eq}})^{\alpha}$
(with $R_{\text{ap}}/R_{\text{eq}}=1.035$ in the present case). When the
simulated isotropic spectra is in good agreement with the experimental
spectrum, at the same time optimal agreement is obtained for the linear
dichroism spectrum. At high temperature there is good agreement between the
experimental and simulated linear dichroism spectrum, apart from an intensity
scaling factor of 0.5 as shown in Figs. 4 and 5. With the fitting parameters
for the simulation there is no way to reduce the overall intensity of the
linear dichroism without radically modifying the isotropic spectrum. For up to
6% the intensity difference might be attributable to the local canting of the
FeO6 octahedra with respect to the $ab$-plane. In addition to this there is an
inevitable loss of intensity in the XLD due to misalignments and sample
imperfection. Reduced energy resolution can also lead to a reduction of the
measured XLD. We conclude that the difference in overall intensity of the
experimental and calculated $I_{\mathrm{xld}}$ is not due to fundamental
shortcomings of the simulation. For reasons we do not quite understand, the
agreement between theory and experiment as evident in the temperature
dependence of the isotropic spectra is not reflected in the XLD spectra at
lower temperatures. We expect that these spectra, which are averages over
increasingly fewer states, are also increasingly sensitive to shortcomings in
our simulations. Since the XLD is an orientational energy dependence it will
be particularly sensitive for any shortcomings in our theoretical model. It
should be emphasised that, despite the differences between the experimental
and simulated spectra at low temperatures, the overall agreement between
experiment and simulation as obtained here is remarkably good. The parameters
as found in our simulation are given in Table 1 (left-hand side). The large
error bar on the value for the charge-transfer gap of 3.5(2.0) eV reflects the
adjustment of the total $dd$ Coulomb energy on the iron site
$U_{dd}^{\text{tot}}=U_{3d^{5}}(\langle n\rangle-5)$ to any increase in
$\epsilon_{p}$ by transfer of additional electron weight into the Fe ion. This
is reflected in the total error in $\langle n\rangle$ of $\sim$0.1 electron.

The values found here for $\Delta_{pd}$ and $t_{\sigma}$ are similar to those
found previously for other Fe3+ compounds FdG:03JACS ; FdG:05Rev . Fe2O3,
which is in many respects probably the compound most closely related to iron
jarosite, was previously found to have a band gap of predominantly charge-
transfer character of 2 to 2.5 eV Ciccacci:91 . With an unusually small value
for $t_{\pi}$ of 0.3(2) eV the lowest energy peak, at 707.8 eV is reproduced
in the simulated spectrum. The main peak at 709 eV splits into two peaks with
the introduction of a trigonal distortion of the FeO6 octahedron. A common
value for the rescaling exponent $\alpha$ is $-$3 and the small value found
here ($-$7.5) is an indication that the charge anisotropy is larger than the
crystal field anisotropy as defined by the oxygen positions around the iron.
The lowest energy low-spin configuration, corresponding to the
${}^{4}F_{11/2}$ free-ion configuration, lies 1.2 eV higher in energy and is
not found to mix into the ground state.

Table 1: Results from simulations of the Fe $L_{2,3}$ spectra of potassium iron jarosite, with ligand-field multiplet calculations using the Hilbert++ codeMirone:code . The left-hand side lists all the parameters varied in the simulation. The right-hand side gives the resulting expectation values for the calculated ground-state properties of the Fe ion. The errors indicate the range over which the parameters could be varied whilst maintaining a good agreement with the data. The right-hand side of the lowest panel gives the effective moment in the case of full orbital polarization, while the calculation indicates only a very weak orbital polarization of the added electron weight due to charge transfer. We use the convention that upper case $J,L,S$ represent operators and lower case $j,l,s$ represent the quantum numbers given by $\langle L^{2}\rangle=l(l+1)$ etc.  Simulation pars. |  | Expectation values |
---|---|---|---
$\Delta_{pd}$ (eV) | 3.5(2.0) |  $D_{z}$ (meV) | 0.5(1)
$t_{\pi}$ (eV) | 0.3(2) | $\sqrt{\langle S^{2}\rangle}=\langle S\rangle$ | 2.77(6)
$t_{\sigma}$ (eV) | 3.0(2) | $\leadsto s=2.32$ |
$\alpha$ |  $-$7.5 | $\sqrt{\langle L^{2}\rangle}(\neq\langle L\rangle)$ | 1.5(2)
|  | $\leadsto l=1.11$ |
|  | but $\langle L\rangle$ | 0.008(3)
|  | $\langle n\rangle$ | 5.39(7)
From $t_{\sigma,\pi}$ and $\alpha$: | (in eV) | Using $\langle L\rangle$ we obtain |
$t_{\sigma}^{\text{eq}}$ | 3.0(2) | $j=s(+l\approx 0)$ | 2.3
$t_{\sigma}^{\text{ap}}$ | 2.3(2) | $g_{J}$ | $\sim 2$
$t_{\pi}^{\text{eq}}$ | $<0.5$ | $\mu_{\text{eff}}$ | $5.5(1)\mu_{\text{B}}$
$t_{\pi}^{\text{ap}}$ | $<0.4$ | and using $\sqrt{\langle L^{2}\rangle}$ |
|  | $j=s+l$ | 3.4
|  | $g_{J}$ | 1.67(5)
|  | $\mu_{\text{eff}}$ | $6.51(7)\mu_{\text{B}}$

The single-ion anisotropy $D_{z}$ is obtained as the difference between the
energy of the $S_{z}$ = $\pm 1/2$ and $\pm 5/2$ levels and was found to be of
the easy-plane type. This easy-plane anisotropy was also confirmed by rotating
the spin quantisation axis to various directions with respect to the crystal
field. In this way we have also attempted to obtain estimates for $E_{xy}$.
The $D_{4h}$ symmetry of the ligand field was lowered to $C_{2}$, to model the
actual positions of the oxygen anions around the cation as obtained from
crystallographical data Grohol:03 ; WillsKjar:00 . However, this had no effect
on the agreement of the calculated spectra with experiment, nor on the values
as listed in Table 1 and hence we consider an estimate for $E_{xy}$ from our
data presently not attainable. It should in this respect be noted that the
value for $D_{z}$ was found to change with the splitting of the main peak at
709 meV, introduced by the trigonal distortion of the ligand field. Its value
is therefore tightly constrained by the experimentally obtained spectra and
the 3d spin-orbit coupling strength. As is detailed in Table 1 our calculation
indicates that for the isolated FeO6 cluster most of the orbital angular
momentum of the Fe 3$d^{6}$ configuration is quenched by the ligand field,
with $\langle L\rangle=0.008(3)$ while the maximum orbital angular momentum
which could in this case be realised as $\sqrt{\langle L^{2}\rangle}=1.5(2)$.

Due to the lower resolution at which the hydronium jarosite powder spectrum
has been measured we cannot provide detailed information about this system,
apart from a confirmation that the general crystal field is similar, with
comparable charge-transfer and ($\sigma$-type) hybridization. This is a good
indication that the magnetic exchange in hydronium jarosite is likely to be
comparable with that of the other jarosites, i.e. with a Weiss temperature
around $-$800 K. There is also a slight evidence that the $D_{4h}$ ligand-
field anisotropy and hence the zero-field splitting, is slightly smaller in
hydronium jarosite.

##  V Discussion

The expectation values of the ground-state obtained in our calculation are
given in Table 1 (right-hand side). The corresponding free-ion configuration
is $d^{5}$ ${}^{6}S_{5/2}$ with an additional electron weight of $\sim$0.4
electron that is mainly in the $d(x^{2}-y^{2})$ orbital. The spin-orbit
coupling is much smaller than the $t_{\sigma}$ hopping bandwidth and hence the
fraction of the electron transferred to the ligands to the Fe cation is only
very weakly orbitally polarized, with $\langle L\rangle=0.008(3)$. This value
is much smaller than the orbital angular momentum as inferred from high-
temperature magnetic susceptibility data, which indicate that
$\mu_{\text{eff}}\approx 6.5$ $\mu_{\text{B}}$ Grohol:05 . As shown in the
lower panel of Table 1 (right-hand side) the experimental value is remarkably
close to the case where the added electron weight due to charge transfer is
fully orbitally polarized. It is not expected that in this $3d$ transition
metal compound $l$ and $j(=s+l)$ are good quantum numbers, but there is
sufficient evidence that $\langle L\rangle$ is significantly larger than zero
222Iron jarosite is in this sense not a unique case. For $3d$ transition metal
compounds with more than half-filled shells (such as e.g. Herbertsmithite
deVries:07 ) the effective moments are in general significantly larger than
the spin-only moment.. It might be that comparison of experimental results
with the Mott-Hubbard-like calculations as described here, but with larger
clusters containing two or three iron centres, can provide an explanation for
the discrepancy between the calculated and experimentally obtained orbital
angular momentum.

The energy of $D_{z}$ = 0.5(1) meV for the easy-plane single-ion anisotropy is
in good agreement with the values found by Matan  _et al._ Matan:06  (0.428(5)
meV) and Coomer  _et al._ Coomer:06  (0.47(2) meV) in their spin-wave analysis
of inelastic neutron data from single crystals and powders, respectively. This
implies that an easy-plane anisotropy, along with further neighbor
interactions, can explain the magnetic ground state in potassium iron
jarosite. Further evidence might be found in the temperature dependence of the
experimental and calculated isotropic spectra. The spectra in Fig. 5 were
taken at room temperature and were compared with the calculated spectra for
the Boltzmann averaged configurations of the FeO6 cluster. As is visible in
Fig. 3 the relative intensity of the two maxima in the split peak at 709 eV
changes in a similar way as was observed in the experimentally obtained
spectra. At the same time an (unexplained) increase in intensity of the peak
at 707.8 eV peak relative to the intensity of the main peak is observed in the
experimental spectra. We have not been able to account for the latter (which
was also observed in powder samples) either by changing the temperature of the
calculated spectra or with simulations with new values for $\delta_{pd}$,
$t_{\sigma,\pi}$ and $\alpha$. The change in the main peak at 709 eV where we
obtain good agreement, is a direct result of the spin-orbit coupling which
causes a change in the shape of the electronic $3d$ orbital as the spins align
in the easy plane. The change in shape of the main peak in the experimental
spectra was found to occur between 200 K and 40 K. This is in agreement with
previous experiments which have shown that in potassium jarosite the alignment
of spin with the kagome planes sets in above 120 K, i.e. well above the
transition temperature of 65 K Grohol:05 . The temperature dependence in the
experimentally obtained spectra was simulated with the temperature (given in
units of eV) used for the Boltzmann averaging over excited states as the only
extra parameter, as shown in Fig. 3. All other simulation parameters were
fixed to the values given in Table 1. In this way we found the expectation
values for $\langle|S_{z}|\rangle$ at each corresponding experimental
temperature, arising from the thermal population of the $m_{S}$ levels in the
$S=5/2$ multiplet. The temperature dependence of $\langle|S_{z}|\rangle$ (Fig.
6) shows a gradual alignment of the spins into the kagome planes, at
temperatures well above the transition temperature to a long-range ordered
state at 64 K. This is in rough agreement with the degree of co-planarity of
the spins as measured using neutron diffraction on large single crystals
Grohol:05 .

Figure 6: The expectation values $\langle|S_{z}|\rangle$ of the simulated
spectra, against the temperature of the corresponding experimental spectrum.
For a single $S=5/2$ spin the minimum for $\langle|S_{z}|\rangle$ is 0.5 and
the maximum is 1.5 (equal population of all $m_{S}$. The gradual increase of
$\langle|S_{z}|\rangle$ with temperature is in agreement with earlier neutron
measurements which show a gradual alignment of the spins in the kagome plane
at temperatures well above the transition temperature of 64 K Grohol:05 .

The simulated linear dichroism spectra at the lowest temperatures,
corresponding to states with the spins aligned in the equatorial plane (the
states $\ket{s,m_{z^{\prime}}}=\ket{2.5,\pm 0.5}$) do not agree well the
experimentally obtained spectra as shown in Fig. 4, although some features in
the temperature dependence of the XLD are reproduced in the simulated spectra.
Without success we have attempted to improve on this situation by calculating
the spectra for the Fe spin aligned along the $x^{\prime}$ axis in the
equatorial plane (i.e. $\ket{s,m_{x^{\prime}}}=\ket{2.5,\pm 2.5}$) which
resembles more closely the actual ground state in potassium jarosite. The
remaining disagreement could be in part due to experimental error due to
sample imperfections and in part due to intrinsic limitations of the
simulation.

It might be naive to expect that the ligand-field can be inferred from
crystallographic data alone. A first indication for this is the large value of
$\alpha$ obtained here, which amplifies the effect of the crystallographic
trigonal distortion of the FeO6 octahedra. Even when the symmetry of the
ligand field is further lowered from $D_{4h}$ and accurately modelled on the
locations of the oxygen anions around the central iron 333Though all
Fe-O${}_{\text{eq}}$ bond lengths are the same, the equatorial plane is
rectangular rather than square, with the short side pointing into the kagome
triangles. Furthermore, the Fe-O${}_{\text{ap}}$ axis is canted by $4^{\circ}$
away from the local $c^{\prime}$ axis. neither the calculated spectra nor the
simulation parameters change significantly. Therefore, it is possible that the
actual charge anisotropy follows more closely the $xy$ plane rather than the
equatorial oxygens. The easy-plane magnetic anisotropy of $D_{z}=0.5(1)$ meV
found here (see Table 1), is then not directly related to the trigonal
distortion of the crystal field but more to the 2D character of the electronic
hopping and magnetic exchange, the virtual absence of $\pi$ symmetry
hybridization and the Fe $3d$ spin-orbit coupling strength. In that case the
easy plane could coincide more closely with the kagome plane. The strong 2D
character can in part be attributed to the sulphate groups separating the
kagome layers. The formal valence of the sulphur cation is 6+ with the [Ne]
electronic configuration, but this large electrostatic charge is likely to
attract electron weight from the neighboring oxygen anions, including the
oxygens apical in the FeO6 octahedra. The hopping parameters $t_{\sigma,\pi}$
from these oxygens to the iron is then further reduced with respect to the in-
plane hopping. This translates into the large value for $\alpha$ found here.

The large difference between $t_{\pi}$ and $t_{\sigma}$ obtained here might be
explained by the triangular arrangement of the neighboring FeO6 octahedra.
This dramatically lowers the symmetry of the local crystal field around the
oxygen ligands and reduces the $\pi$ symmetry overlap of the oxygen $2p$
orbitals with the iron $3d$ orbitals. Due to the low symmetry of the oxygen
sites the symmetry of the crystal field can be expected to be even lower than
indicated by the crystallographic positions of the anions. It is in this sense
fortunate that the situation can be modelled so well by the anisotropic $\pi$
and $\sigma$ hopping whose ratio is equal for all six anions around the
central iron. It could well be that the last differences between the
calculated and experimental spectra, such as the low temperature x-ray linear
dichroism, some of the difference in the overall intensity of the XLD and the
missing bump just above 707 eV, could be brought into agreement by allowing a
different balance between $t_{\pi}$ and $t_{\sigma}$ for the apical oxygen
anions.

In neutron spin-wave studies Matan:06 ; Yildirim:06 ; Coomer:06  the crystal
field anisotropy scenario has been compared in close detail with that of DMI.
In the present context such a discussion would require calculations on
clusters containing two to three iron centres. Such a calculation would
certainly be of great interest, but has not been attempted here, since the
present approach already accounts so well for the experimental observations on
potassium iron jarosite. The present result closes the last missing causal
link, from electronic structure to collective magnetic ground state.

##  VI Conclusion

The isotropic and x-ray linear dichroism spectra taken at the Fe $L_{2,3}$
edges in potassium iron jarosite indicate that an orbital angular momentum and
zero-field splitting arise due to an anisotropic, mainly $\sigma$ type,
charge-transfer from the oxygen ligands. This closely resembles the scenario
described in Ref. Wan-Lun:94,  to explain the zero-field splitting as observed
in Mn2+ salts.

The zero-field splitting parameter of the crystal field anisotropy of 0.5 meV
as found here is in excellent agreement with the value found from fits of a
crystal field anisotropy model to the spin-wave spectrum in the ground state
of potassium iron jarosite Matan:06 ; Yildirim:06 ; Coomer:06 . However, the
most important result obtained here is that we have verified in detail the
mechanism by which this magnetic anisotropy arises, closing the remaining
missing link between the electronic structure in potassium iron jarosite and
its magnetic ground state. We have even measured a change in the electronic
structure as the magnetic ground state becomes co-planar, with the change of
the isotropic line shape over the relevant temperature range. Of course, an
additional in-plane anisotropy $E_{xy}$ and further- neighbor interactions are
needed to fully explain the magnetic ground state. The present study does not
provide more hints to the origin of the glassy state in hydronium jarosite. We
found that the energy scale of the magnetic interaction in hydronium jarosite
should be comparable to that in potassium jarosite. It might be that
temperature dependent spectra taken at a higher resolution could provide new
insights in the case of hydronium jarosite. A final interesting observation is
that the maximum orbital angular momentum polarization possible for the amount
of charge-transfer observed here, but not the actually calculated orbital
polarization, is in rough agreement with the experimentally observed effective
moment.

##  VII Acknowledgement

Simon Parsons and Clivia Hejny (Edinburgh University) are acknowledged for
their help with the characterization and alignment of single crystals of
potassium jarosite. We would like to thank Nicholas Harrison (Imperial
College, London and Daresbury Laboratory) and Barry Searle (Daresbury
Laboratory) for fruitful discussions about LSDFT calculations on potassium
jarosite. Peter Bencok (ESRF and Diamond) and Nicola Farley (Daresbury
Laboratory) are acknowledged for their help during x-ray spectroscopy
measurements. We further thank Claudine Lacroix (CNRS, Grenoble), Andrew Wills
(UCL), Frank de Groot (Utrecht University) and Paul Attfield (Edinburgh
University) for fruitful discussions. MdV gratefully acknowledges financial
support from the Centre for Materials Physics and Chemistry (Science and
Technology Facility Council, UK) and the Highly Frustrated Magnetism network
of the European Science Foundation.

## References

  * (1) Frustrated spin systems, Ed. H. T. Diep (World Scientific, Singapore, 2004).
  * (2) J. N. Reimers, A. J. Berlinsky, and A.-C. Shi, Phys. Rev. B 43, 865 (1991).
  * (3) J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender, Phys. Rev. Lett. 68, 855 (1992).
  * (4) J. N. Reimers and A. J. Berlinsky, Phys. Rev. B 48, 9539 (1993).
  * (5) A. Keren, K. Kojima, L. P. Le, G. M. Luke, W. D. Wu, Y. J. Uemura, M. Takano, H. Dabkowska, and M. J. P. Gingras, Phys. Rev. B 53, 6451 (1996).
  * (6) T. Inami, T. Morimoto, M. Nishiyama, S. Maegawa, Y. Oka, and H. Okumura, Phys. Rev. B 64, 054421 (2001).
  * (7) T. Morimoto, M. Nishiyama, S. Maegawa, and Y. Oka, J. Phys. Soc. Jpn 72, 2085 (2003).
  * (8) M. Takano, T. Shinjo, M. Kiyama, and T. Takada, J. Phys. Soc. Jpn 25, 902 (1968).
  * (9) M. G. Townsend, G. Longworth, and E. Roudaut, Phys. Rev. B 33, 4919 (1986).
  * (10) A. S. Wills and A. Harrison, J Chem. Soc. Faraday Frans. 92, 2161 (1996).
  * (11) A. S. Wills, A. Harrison, C. Ritter, and R. I. Smith, Phys. Rev. B 61, 6156 (2000).
  * (12) A. S. Wills, Phys. Rev. B 63, 064430 (2001).
  * (13) D. Grohol, K. Matan, J. H. Cho, S. H. Lee, J. W. Lynn, D. G. Nocera, and Y. S. Lee, Nature Materials 4, 323 (2005).
  * (14) D. Grohol and D. Nocera, J. Am. Chem. Soc. 124, 2640 (2002).
  * (15) D. Grohol, Q. Huang, B. H. Toby, J. W. Lynn, Y. S. Lee, and D. G. Nocera, Phys. Rev. B 68, 094404 (2003).
  * (16) D. Grohol, D. G. Nocera, and D. Papoutsakis, Phys. Rev. B 67, 064401 (2003).
  * (17) T. Inami, M. Nishiyama, S. Maegawa, and Y. Oka, Phys. Rev. B 61, 12181 (2000).
  * (18) M. Nishiyama, S. Maegawa, T. Inami, and Y. Oka, Phys. Rev. B 67, 224435 (2003).
  * (19) A. P. Ramirez, S.-W. Cheong, and M. L. Kaplan, Phys. Rev. Lett. 72, 3108 (1994).
  * (20) N. Nishiyama and S. Maegawa, Physica B 329-333, 1065 (2003).
  * (21) T. Yildirim and A. B. Harris, Phys. Rev. B 73, 214446 (2006).
  * (22) M. Elhajal, B. Canals, and C. Lacroix, Phys. Rev. B 66, 014422 (2002).
  * (23) K. Matan, D. Grohol, D. G. Nocera, T. Yildirim, A. B. Harris, S. H. Lee, S. E. Nagler, and Y. S. Lee, Phys. Rev. Lett. 96, 247201 (2006).
  * (24) F. C. Coomer, A. Harrison, G. S. Oakley, J. Kulda, J. R. Stewart, J. A. Stride, B. Fåk, J. W. Taylor, and D. Visser, J. Phys.: Condens. Mat. 18, 8847 (2006).
  * (25) W. G. Bisson and A.S. Wills, to be published (2008).
  * (26) P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Harrison, F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and C. Baines, Phys. Rev. Lett. 98, 077204 (2007).
  * (27) A. M. Afanasev, V. D. Gorobchenko, D. S. Kulgawczuk, and I. I. Lukashevich, Phys. Status Solidi (A) 26, 697 (1974).
  * (28) P. Bonville et al., Hyperfine Interact. 168, 1085 (2006).
  * (29) T. Moriya, Phys. Rev. 120, 91 (1960).
  * (30) H. Watanabe, Progr. Theor. Phys. 18, 405 (1957).
  * (31) A. S. Wills, V. Dupuis, E. Vincent, J. Hammann, and R. Calemczuk, Phys. Rev. B 62, R9264 (2000).
  * (32) B. T. Thole, G. van der Laan, and P. H. Butler, Chem. Phys. Lett. 149, 295 (1988).
  * (33) J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 (1985).
  * (34) J. M. D. Coey and G. A. Sawatzky, J. Phys. C: Solid St. Phys. 4, 2386 (1971).
  * (35) YuWan-Lun, Phys. Rev. B 39, 622 (1989).
  * (36) YuWan-Lun and TanTao, Phys. Rev. B 49, 3243 (1994).
  * (37) E. Francisco and L. Pueyo, Phys. Rev. B 37, 5278 (1988).
  * (38) P. Jeevanandam and S. Vasudevan, J. Phys.: Cond. Mat. 11, 3563 (1999).
  * (39) G. van der Laan, J. Zaanen, G. A. Sawatzky, R. Karnatak,, and J. M. Esteva, Phys. Rev. B 33, 4253 (1986).
  * (40) F. de Groot, J. El. Spec. Rel. Phen. 67, 529 (1994).
  * (41) F. de Groot, Coordination Chemistry Reviews 249, 31 (2005).
  * (42) J. E. Dutrizac and S. Kaiman, Canadian Mineralogist 14, 151 (1976).
  * (43) A. S. Wills, Ph.D. thesis, The University of Edinburgh, 1996.
  * (44) P. F. Schofield, G. van der Laan, C. M. B. Henderson, and G. Cressey, Mineralogical Magazine 62, 65 (1998).
  * (45) G. van der Laan and B. T. Thole, Phys. Rev. B 43, 13401 (1991).
  * (46) B. T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992).
  * (47) G. van der Laan, Phys. Rev. Lett. 82, 640 (1999).
  * (48) A. Mirone, M. Sacchi, and S. Gota, Phys. Rev. B 61, 13540 (2000).
  * (49) A. Mirone, Hilbert++ manual, arXiv:0706.4170v1, 2007.
  * (50) R. D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, 1981).
  * (51) G. van der Laan and I. W. Kirkman, J. Phys.: Condens. Mat. 4, 4189 (1992).
  * (52) B. T. Thole, G. van der Laan, J. C. Fuggle, G. A. Sawatzky, R. C. Karnatak and J.-M. Esteva, Phys. Rev. B 32, 5107 (1985).
  * (53) E. C. Wasinger,F. M. F. de Groot, B. Hedman, K. O. Hodgson, and E. I. Solomon, J. Am. Chem. Soc. 125, 12894 (2003).
  * (54) F. Ciccacci, L. Braicovich, E. Puppin, and E. Vescovo, Phys. Rev. B 44, 10444 (1991).
  * (55) M. A. de Vries, K. V.Kamenev, W. A. Kockelmann, J. Sanchez-Benitez, and A. Harrison, Phys. Rev. Lett. 100, 157205 (2008).