# SPEECH RECOGNITION BY SIMPLY FINE-TUNING BERT

###### Abstract

We propose a simple method for automatic speech recognition (ASR) by fine-
tuning BERT, which is a language model (LM) trained on large-scale unlabeled
text data and can generate rich contextual representations. Our assumption is
that given a history context sequence, a powerful LM can narrow the range of
possible choices and the speech signal can be used as a simple clue. Hence,
comparing to conventional ASR systems that train a powerful acoustic model
(AM) from scratch, we believe that speech recognition is possible by simply
fine-tuning a BERT model. As an initial study, we demonstrate the
effectiveness of the proposed idea on the AISHELL dataset and show that
stacking a very simple AM on top of BERT can yield reasonable performance.

Index Terms—  speech recognition, BERT, language model

##  1 Introduction

Conventional automatic speech recognition (ASR) systems consist of multiple
separately optimized modules, including an acoustic model (AM), a language
model (LM) and a lexicon. In recent years, end-to-end (E2E) ASR models have
attracted much attention, due to the believe that jointly optimizing one
single model is beneficial to avoiding not only task-specific engineering but
also error propagation. Current main-stream E2E approaches include
connectionist temporal classification (CTC) [1], neural transducers [2], and
attention-based sequence-to-sequence (seq2seq) learning [3].

LMs play an essential role in ASR. Even the E2E models that implicitly
integrate LM into optimization can benefit from LM fusion. It is therefore
worthwhile thinking: how can we make use of the full power of LMs? Let us
consider a situation that we are in the middle of transcribing a speech
utterance, where we have already correctly recognized a sequence of history
words, and we want to determine what the next word is being said. From a
probabilistic point of view, a strong LM, can then generate a list of
candidates where each of them is highly possible to be the next word. The list
may be extremely short that there is only one answer left. As a result, we can
use few to no clues in the speech signal to correctly recognize the next word.

There has been rapid development of LMs in the field of natural languages
processing (NLP), and one of the most epoch-making approach is BERT [4]. Its
success comes from a framework where a pretraining stage is followed by a
task-specific fine-tuning stage. Thanks to the un-/self-supervised objectives
adopted in pretraining, large-scale unlabeled datasets can be used for
training, thus capable of learning enriched language representations that are
powerful on various NLP tasks. BERT and its variants have created a dominant
paradigm in NLP in the past year [5, 6, 7].

In this work, we propose a novel approach to ASR, which is to simply fine-tune
a pretrained BERT model. Our method, which we call BERT-ASR, formulates ASR as
a classification problem, where the objective is to correctly classify the
next word given the acoustic speech signals and the history words. We show
that even an AM that simply averages the frame-based acoustic features
corresponding to a word can be applied to BERT-ASR to correctly transcribe
speech to a certain extent, and the performance can be further boosted by
using a more complex model.

Fig. 1: The input representation of the original BERT and the proposed BERT-
ASR. Fig. 2: Illustration of the decoding process of the proposed BERT-ASR.

##  2 BERT

BERT [4], which stands for Bidirectional Encoder Representations from
Transformers, is a pretraining method from a LM objective with a Transformer
encoder architecture [8]. The full power of BERT can be released only through
a pretraining–fine-tuning framework, where the model is first trained on a
large-scale unlabeled text dataset, and then all/some parameters are fine-
tuned on a labeled dataset for the downstream task.

The original usage of BERT mainly focused on NLP tasks, ranging from token-
level and sequence-level classification tasks, including question answering
[9, 10], document summarization [11, 12], information retrieval [13, 14],
machine translation [15, 16], just to name a few. There has also been attempts
to combine BERT in ASR, including rescoring [17, 18] or generating soft labels
for training [19]. In this section, we review the fundamentals of BERT.

###  2.1 Model architecture and input representations

BERT adopts a multi-layer Transformer [8] encoder, where each layer contains a
multi-head self-attention sublayer followed by a positionwise fully connected
feedforward network. Each layer is equipped with residual connections and
layer normalization.

The input representation of BERT was designed to handle a variety of down-
stream tasks, as visualized in Figure 1. First, a token embedding is assigned
to each token in the vocabulary. Some special tokens were added to the
original BERT, including a classification token ([CLS]) that is padded to the
beginning of every sequence, where the final hidden state of BERT
corresponding to this token is used as the aggregate sequence representation
for classification tasks, and a separation token ([SEP]) for separating two
sentences. Second, a segment embedding is added to every token to indicate
whether it belongs to sentence A or B. Finally, a learned positional embedding
is added such that the model can be aware of information about the relative or
absolute position of each token. The input representation for every token is
then constructed by summing the corresponding token, segment, and position
embeddings.

###  2.2 Training and fine-tuning

Two self-supervised objectives were used for pretraining BERT. The first one
is the masked language modeling (MLM), which is a denoising objective that
asks the model to reconstruct randomly masked input tokens based on context
information. Specifically, 15% of the input tokens are first chosen. Then,
each token is (1) replaced with [MASK] for 80% of the time, (2) replaced with
a random token for 10% of the time, (3) kept unchanged for 10% of the time.

During fine-tuning, depending on the downstream task, minimal task-specific
parameters are introduced so that fine-tuning can be cheap in terms of data
and training efforts.

##  3 Proposed Method

In this section, we explain how we fine-tune a pretrained BERT to formulate
LM, and then further extend it to consume acoustic speech signals to achieve
ASR.

Assume we have an ASR training dataset containing $N$ speech utterances:
$\bm{D}_{\text{ASR}}=\langle\bm{X}^{(i)},\bm{y}^{(i)}\rangle_{i=1}^{N}$, with
each $\bm{y}=(y_{1},...,y_{T})$ being the transcription consisting of $T$
tokens, and each $\bm{X}=(\bm{x}_{1},...,\bm{x}_{T^{\prime}})$ denoting a
sequence of $T^{\prime}$ input acoustic feature frames. The acoustic features
are of dimension $d$, i.e., $\bm{x}_{t}\in\mathbb{R}^{d}$, and the tokens are
from a vocabulary of size $V$.

###  3.1 Training a probabilistic LM with BERT

We first show how we formulate a probabilistic LM using BERT, which we will
refer to as BERT-LM. The probability of observing a symbol sequence $\bm{y}$
can be formulated as:

$\displaystyle
P(\bm{y})=P(y_{1},\dots,y_{T})=\prod_{t=1}^{T}{P(y_{t}|y_{1},\dots,y_{t-1})}.$
(1)

The decoding (or scoring) of a given symbol sequence then becomes an iterative
process that calculates all the terms in the product, which is illustrated in
Figure 2. At the $t$-th time step, the BERT model takes a sequence of
previously decoded symbols and the [CLS] token as input, i.e.,
$(\texttt{[CLS]},y_{1},\dots,y_{t-1})$. Then, the final hidden state
corresponding to [CLS] is sent into a linear classifier, which then outputs
the probability distribution $P(y_{t}|y_{1},\dots,y_{t-1})$.

To train the model, assume we have a training dataset with $N$ sentences:
$\bm{D}_{\text{text}}=\\{\bm{y}^{(i)}\\}_{i=1}^{N}$. An essential technique to
train the model is to exhaustively enumerate all possible training samples.
That is, each sentence with $T$ symbols is extended to $T$ different training
samples following the rule in Equation (2):

$\displaystyle(y_{1},...,y_{T})\to\begin{cases}(\texttt{[CLS]})\\\
(\texttt{[CLS]},y_{1})\\\ (\texttt{[CLS]},y_{1},y_{2})\\\ \dots\\\
(\texttt{[CLS]},y_{1},\dots,y_{t-1})\end{cases}.$ (2)

The training of the BERT-LM becomes simply minimizing the following cross-
entropy objective:

$\displaystyle\mathcal{L}_{\text{LM}}=-\sum_{i=1}^{N}\sum_{t=1}^{T}P(y^{(i)}_{t}|\texttt{[CLS]},y^{(i)}_{1},\dots,y^{(i)}_{t-1}).$
(3)

Fig. 3: Illustration of the alignment between the text and the acoustic
frames. Fig. 4: The average encoder.

Fig. 5: The conv1d resnet encoder.

###  3.2 BERT-ASR

We introduce our proposed BERT-ASR in this section. Since the time resolution
of text and speech is at completely different scales, for the model described
in Section 3.1 to be capable of taking acoustic features as input, we first
assume that we know the nonlinear alignment between an acoustic feature
sequence and the corresponding text, as depicted in Figure 3. Specifically,
for a pair of training transcription and acoustic feature sequence
$\langle(\bm{x}_{1},\dots,\bm{x}_{T^{\prime}}),(y_{1},\dots,y_{T})\rangle$, we
denote $\bm{F}_{i}$ to be the segment of features corresponding to $y_{i}$:
$\bm{F}_{i}=(\bm{x}_{t_{i-1}+1},\dots,\bm{x}_{t_{i}})\in\mathbb{R}^{t_{i}\times
d}$, which is from frame $t_{i-1}+1$ to $t_{i}$, and we set $t_{0}=0$. Thus,
the $T^{\prime}$ acoustic frames can be segmented into $T$ groups:
$\bm{X}=(\bm{F}_{1},\dots,\bm{F}_{T})$, and a new dataset containing segmented
acoustic feature and text pairs can be obtained:
$\bm{D}_{\text{seg}}=\langle\bm{F}^{(i)},\bm{y}^{(i)}\rangle_{i=1}^{N}$.

We can then augment the BERT-LM into BERT-ASR by injecting the acoustic
information extracted from $\bm{D}_{\text{seg}}$ into BERT-LM. Specifically,
as depicted in Figure 1, an acoustic encoder, which will be described later,
consumes the raw acoustic feature segments to generate the acoustic
embeddings. They are summed with the three types of embeddings in the original
BERT, and further sent into BERT. Note that the acoustic embedding
corresponding to the current word to be transcribed is added to the [CLS]
token as shown in Figure 2.

The probability of observing a symbol sequence $\bm{y}$ in Equations (1) can
therefore be reformulated as:

$\displaystyle
P(\bm{y})=\prod_{t=1}^{T}{P(y_{t}|y_{1},\dots,y_{t-1},\bm{F}_{1},\dots,\bm{F}_{t})}.$
(4)

Note that the acoustic segment of the current time step is also taken into
consideration, which is essential for the model to correctly transcribe the
current word being said. The training objective can be derived by
reformulating Equation (3) as:

$\displaystyle\mathcal{L}_{\text{ASR}}=$
$\displaystyle-\sum_{i=1}^{N}\sum_{t=1}^{T}P(y^{(i)}_{t}|$
$\displaystyle\langle\texttt{[CLS]},\bm{F}_{t}\rangle,\langle
y^{(i)}_{1},\bm{F}_{1}\rangle,\dots,\langle
y^{(i)}_{t-1},\bm{F}_{t-1}\rangle).$ (5)

In a nutshell, the training of BERT-ASR involves three steps.

  1. 1.

Pretrain a BERT using a large-scale text dataset.

  2. 2.

fine-tune a BERT-LM on the transcriptions of the ASR training dataset, as
described in Section 3.1,

  3. 3.

fine-tune a BERT-ASR using both text and speech data of the ASR training
dataset.

###  3.3 Acoustic encoder

We now describe two kinds of architectures for the acoustic encoder mentioned
in Section 3.2. Formally, the acoustic encoder takes the whole acoustic frame
sequence $\bm{X}$ as input, and outputs the corresponding acoustic embeddings
$(AE_{1},\dots,AE_{T})$, where $AE_{t}\in\mathbb{R}^{d_{model}}$ with
$d_{model}$ being the BERT embedding dimension. The acoustic encoder must
contain the segment layer to obtain the acoustic segments.

####  3.3.1 Average encoder

We first consider a very simple average encoder, as depicted in Figure 4.
First, the segmentation is performed. Then, for each $\bm{F}_{t}$, we average
on the time axis, and then the resulting vector is passed through a linear
layer to distill the useful information, while scaling the dimension from $d$
to $d_{model}$. Simple as it seems, as we will show later, initial results can
already be obtained with this average encoder.

####  3.3.2 Conv1d resnet encoder

The drawback of the average encoder is that temporal dependencies between
different acoustic segments was not considered. Therefore, we investigate a
second encoder, which we will refer to as the conv1d resnet encoder, as
illustrated in Figure 5. While it has the identical segment and linear layers
as the average encoder, we add $L$ learnable residual blocks that operates on
$\bm{X}$. Each residual block contains two conv1d layers over the time axis
followed by ReLU activations. It is expected that taking the temporal
relationship between segments into account can boost performance.

Table 1: Results on the AISHELL-1 dataset. ”Orac.” and ”Prac.” denote the oracle decoding and practical decoding, respectively. ”Conv1d resnet X” denotes the conv1d resnet encoder with X resnet blocks. Best performance of the BERT-ASR are shown in bold. Model |  Acoustic encoder | Perplexity | CER (Orac.) | CER (Prac.) | SER (Orac.) | SER (Prac.)
---|---|---|---|---|---|---
Dev | Test | Dev | Test | Dev | Test | Dev | Test | Dev | Test
Trigram-LM | - | 133.32 | 127.88 | - | - | - | -
LSTM-LM | - | 79.97 | 78.80 | - | - | - | -
BERT-LM | - | 39.74 | 41.72 | - | - | - | -
BERT-ASR | Average | 5.88 | 9.02 | 65.8 | 68.9 | 96.4 | 105.8 | 60.3 | 63.5 | 91.5 | 100.3
Conv1d resnet 1 | 4.91 | 7.63 | 55.8 | 59.0 | 89.6 | 99.6 | 50.0 | 53.8 | 84.4 | 94.1
Conv1d resnet 2 | 4.77 | 6.94 | 54.6 | 58.8 | 89.7 | 99.1 | 49.5 | 53.6 | 84.6 | 93.5
Conv1d resnet 3 | 4.83 | 7.41 | 54.8 | 58.9 | 89.8 | 99.4 | 49.6 | 53.6 | 84.6 | 93.9
Conv1d-resnet 4 | 4.78 | 7.29 | 54.6 | 59.0 | 89.5 | 99.3 | 49.4 | 53.9 | 84.4 | 93.8
GMM-HMM | - | - | - | 10.4 | 12.2 | - | -
DNN-HMM | - | - | - | 7.2 | 8.4 | - | -

##  4 Experiments

###  4.1 Experimental settings

We evaluate the proposed method on the AISHELL-1 dataset [20], which contains
170 hr Mandarin speech. We used the Kaldi toolkit [21] to extract 80-dim log
Mel-filter bank plus 3-dim pitch features and normalized them. The training
data contained around 120k utterances, and the exhaustive enumeration process
described in Section 3.1 resulted in 1.7M training samples. For the first step
of the proposed BERT-ASR, i.e., pretraining a BERT model using a large-scale
text dataset (cf. Section 3.2), we adopt an updated version of BERT, which is
whole word masking (WWM), whose effectiveness was verified in [22]. The major
difference between the updated BERT and the classic BERT is in the masking
procedure of MLM training. If a masked token belongs to a word, then all the
tokens that complete the word will be masked altogether. This is a much more
challenging task, since the model is forced to recover the whole word rather
than just recovering tokens. we directly used the hfl/chinese-bert-wwm
pretrained model provided by [22]111https://github.com/ymcui/Chinese-BERT-wwm,
which was trained on Chinese Wikipedia. The modeling unit was Mandarin
character. We conducted the experiments using the HuggingFace Transformer
toolkit [23]. The alignment we used during training was obtained by forced
alignment with an HMM/DNN model trained on the same AISHELL-1 training set.

We considered two decoding scenarios w.r.t the alignment strategy. First, the
oracle decoding is where we assumed that alignment is accessible. Second, to
match a practical decoding setting, as a naive attempt, we assumed that the
alignment between each utterance and the underlying text is linear, and
partitioned the acoustic frames into segments of equal lengths. The length was
calculated as the average number of frames per word in the training set, which
was 25 frames. In both scenarios, we considered beam decoding with a beam size
of 10.

###  4.2 Main results

We reported the perplexity (PPL) and character error rate (CER) in Table 1,
with the former being a metric to compare the performance of LMs and the
latter to compare performance of ASR systems. As a reference, we first
compared the PPL between different LMs. It can be clearly observed that the
BERT-LM outperformed the conventional trigram-LM and LSTM-LM, again showing
the power of BERT as a LM.

We then compare BERT-ASR with BERT-LM. By using a simple average encoder, a
significantly lower PPL could be obtained, showing that using acoustic clues
can greatly help guide recognition. Moreover, models with a complex acoustic
encoder like the conv1d resnet encoder could further reduce PPL. Looking at
the CERs, we observed that even with the simple average encoder, a preliminary
success could still be obtained. Furthermore, the conv1d resnet encoders
reduced the CER by almost 10%, showing that it is essential to have access to
global temporal dependencies before segmentation.

We finally consider the practical decoding scenario. There is a significant
performance degradation with the equal segmentation, and it is evidence to the
nonlinear relationship of the alignment. Thus, finding an alignment-free
approach will be an urgent future work [24, 25]. The performance of two
conventional ASR systems directly from the original paper of AISHELL-1 [20]
are also listed, and a significant gap exists between our method and the
baselines, showing that there is still much room for improvement.
Nevertheless, to the best of our knowledge, this is the first study to obtain
an ASR system by fine-tuning a pretrained large-scale LM. Moreover, it is
worth noticing that the proposed method is readily prepared for n-best re-
scoring [17], though in this paper we mainly focus on building an ASR system.
We thus leave this as future work.

###  4.3 Error Analysis

In this section, we examine two possible reasons for the current unsatisfying
results.

####  4.3.1 Polyphone in Mandarin

Mandarin is a character-based language, where the same pronunciation can be
mapped to different characters. As our method is based on a character-based
BERT, it might be infeasible for the model to learn to map the same acoustic
signal to different characters. To examine if our model actually suffered from
this problem, the syllable error rates (SERs) were calculated and reported in
Table 1. It is obvious that the SERs are much lower than the CERs, showing the
existence of this problem. Thus, learning phonetically-aware representations
will be a future direction.

Table 2: Development set results with a ratio of the leading characters correctly recognized. Model | Ratio | CER | SER
---|---|---|---
Conv1d resnet 3 | 0 | 54.8 | 49.6
1/3 | 61.9 | 55.3
1/2 | 57.3 | 51.4

####  4.3.2 Error propagation

BERT benefits from the self-attention mechanism and is therefore known for its
ability to capture global relationship and long-term dependencies. The full-
power of BERT may not be exerted with a relatively short context. That is to
say, our BERT-ASR can be poor at the earlier decoding steps. As a result, the
error in the beginning might propagate due to the recursive decoding process.
To examine this problem, we assume that the starting characters up to a
certain ratio are correctly recognized, and start the decoding process
depending on those characters. Although we expected the error rates to
decrease as the ratio increases, as shown in Table 2, the CERs and SERs were
not lower. Thus, we conclude that error propagation was not a major issue.

##  5 Conclusion

In this work, we proposed a novel approach to ASR by simply fine-tuning BERT,
and described the detailed formulation and several essential techniques. To
verify the proposed BERT-ASR, we demonstrated initial results on the Mandarin
AISHELL-1 dataset, and analyzed two possible sources of error. In the future,
we will investigate more complex model architectures and the possibility of
multi-task learning, in order to close the gap between our and conventional
ASR systems. We also plan to evaluate the BERT-ASR on other languages, and
apply the proposed method for n-best re-scoring [17].

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00footnotetext: Received 14 March 2009

# Topological index associated with transverse axial vector and vector
anomalies in QED††thanks: Supported by National Natural Science Foundation of
China (No.10775059)

SUN Yi-Qian1;1) LU Pin1;2) BAO Ai-Dong1,2,3;3<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1 Center for Theoretical Physics,
Department of Physics, Jilin University Chang chun 130023,china
2 Department of Physics, Northeast Normal University, Chang chun 130024, China
3 Department of Physics, Kashi Normal University, Xin jiang 844000, China

###### Abstract

It is shown that a novel anomaly associated with transverse Ward-Takahashi
identity exists for pseudo-tensor current in QED, and the anomaly gives rise
to a topological index of Dirac operator in terms of Atiyah-Singer index
theorem.

###### keywords:

anomaly, Ward-Takahashi, topological index, Atiyah-Singer index theorem,
pseudo-tensor current

###### pacs:

1

1.40.Ha, 11.30.Rd

00footnotetext: $\scriptstyle\copyright$2013 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd

##  1 Introduction

Some time ago Takahashi made the argument for the plausible existence of
transverse Ward Takahashi(WT) relation in canonical field theory, which has
the potential to restrict the transverse vertex function from gauge symmetry
alone[1]. Subsequently these transverse WT relations for the fermion-boson
vertex in coordinate space (or in momentum space) are cast in four-dimensional
Abelian gauge theory by computing the curl of the time ordered products of
three-point Green functions [2,3]. In addition, the proposed transverse WT
relation holds at one-loop order level in four dimensions gauge theory[4]. Up
to the effect of quantum anomaly, the possible anomaly for the transverse
Ward-Takahashi relations in four dimensional gauge theories is studies by He
using the point-splitting method[5]. Recently, the anomaly issue reexamined by
means of perturbative method. The conclusion is that there are no transverse
anomalies for both the axial vector and vector current [6]. Also the path-
integral derivation of the transverse WT relation for the vector vertex is
presented due to a set of infinitesimal transformation of field variable in
QED in Refs.[7,8], wherein topological property of the axial-vector current
has been illustrated in terms of Atiyah-Singer index theorem. Based on the
validity of Fujikawa’s analysis, we reevaluate in detail the transverse
anomaly of the transverse WT identity for the axial-vector and vector vertex
in the QED, we find that a careful application of Fujikawa’s approach leads to
a transverse quantum anomaly for the axial vector current.[9]. In this paper,
we point out that the mathematical explanation of such a anomaly is directly
related to Atiyah-Singer index theorem. The topological index for Dirac
operator corresponding to the anomaly will be expressed.

##  2 Calculation of anomaly factor in Ward-Takahashi identity

From the point of view of path-integral formulation, we proposed a
infinitesimal transverse transformation of field variables to derive the WT
identities[7][9]. Let us consider a set of infinitesimal local transformation
in the QED

$\displaystyle\psi^{{}^{\prime}}(x)$ $\displaystyle=$ $\displaystyle
e^{-\frac{1}{4}\theta(x)\omega_{\mu\nu}\sigma^{\mu\nu}}\psi(x)$
$\displaystyle\overline{\psi}^{{}^{\prime}}(x)$ $\displaystyle=$
$\displaystyle\overline{\psi}(x)\gamma^{0}e^{-\frac{1}{4}\theta(x)\omega_{\mu\nu}\sigma^{\mu\nu\dagger}}\gamma^{0}$
$\displaystyle B^{{}^{\prime}}_{\mu}(x)$ $\displaystyle=$ $\displaystyle
B_{\mu}(x)+\omega_{\mu\nu}\partial^{\nu}\theta(x).$ (1)

where $\omega_{\mu\nu}$ stands for the antisymmetry tensor,$\psi(x)$ and
$B_{\mu}(x)$ are the fermion and gauge fields,respectively.
In principle, the variation of the generating functional itself under the
transformation of field variables Eq.$(1)$ can lead to Ward-Takahashi type’s
identities .The change of the function integral due to the transformation
(choosing $\delta B_{\mu}(x)=0$ for simplicity) gives the relation in momentum
space in QED

$\displaystyle-
iq^{\mu}\Gamma^{\nu}_{V}(p_{1},p_{2})+iq^{\nu}\Gamma^{\mu}_{V}(p_{1},p_{2})+(p_{1\lambda}+p_{2\lambda})$
$\displaystyle\times\varepsilon^{\lambda\mu\nu\rho}q_{\lambda}\Gamma_{A\rho}(p_{1},p_{2})++S^{-1}(p_{2})\sigma^{\mu\nu}+S^{-1}(p_{1})\sigma^{\mu\nu}$
$\displaystyle-\int\frac{d^{4}k}{(2\pi)^{4}}2k_{\lambda}\epsilon^{\lambda\mu\nu\rho}\Gamma_{A\rho}(p_{1},p_{2};k)=0.$
(2)

This WT relation for the vector current has been listed in Ref.[9]. The
integral term in Eq.(2) may be called the integral-term involving the vertex
function $\Gamma_{A\rho}(p_{1},p_{2};k)$ with the internal momentum $k$ of the
gauge boson appearing in the Wilson line$[10]$. The Fourier transformation for
vertex function $\Gamma_{A\rho}(p_{1},p_{2};k)$ is defined as

$\displaystyle\int~{}d^{4}xd^{4}x_{1}d^{4}x_{2}e^{i(p_{1}x-p_{2}x_{2}-qx)}\langle
0|T(\overline{\psi}(x)\varepsilon^{\lambda\mu\nu\rho}\gamma_{\rho}\gamma^{5}\psi(x)\overline{\psi}(x_{1})\psi(x_{2})U_{P}(x^{{}^{\prime}},x))|0\rangle$
$\displaystyle=(2\pi)^{4}\delta^{4}(p_{1}-p_{2}-q)iS_{F}(p_{1})\Gamma_{A\rho}(p_{1},p_{2},k)iS_{F}(p_{2}).$
(3)

where $q=(p_{1}-k)-(p_{2}-k)$. Obviously, the full vector function and the
full axial -vector function are coupled with each other.As shown is Ref[3],
the apparent feature of this transverse identity Eq.(2) is that the vertex
function $\Gamma^{\mu}_{V}$ (fermion’s three point function) has the
transverse component of itself .

Completely analogous to the calculations above, let us consider the other
transverse transformation

$\displaystyle\psi^{{}^{\prime}}(x)$ $\displaystyle=$ $\displaystyle
e^{-\frac{1}{4}\theta(x)\omega_{\mu\nu}\sigma^{\mu\nu}\gamma^{5}}\psi(x)$
$\displaystyle\overline{\psi}^{{}^{\prime}}(x)$ $\displaystyle=$
$\displaystyle\overline{\psi}(x)\gamma^{0}e^{-\frac{1}{4}\theta(x)\omega_{\mu\nu}\gamma^{5\dagger}}\sigma^{\mu\nu\dagger}\gamma^{0}$
(4)

we obtain the identity for the axial -vector current in momentum space as

$\displaystyle-
iq^{\mu}\Gamma^{\nu}_{A}(p_{1},p_{2})+iq^{\nu}\Gamma^{\mu}_{A}(p_{1},p_{2})+(p_{1\lambda},p_{2\lambda})\varepsilon^{\lambda\mu\nu\rho}q_{\lambda}\Gamma_{V\rho}(p_{1}+p_{2})+$
$\displaystyle+S^{-1}(p_{2})\sigma^{\mu\nu}\gamma^{5}-S^{-1}(p_{1})\sigma^{\mu\nu}\gamma^{5}-\int\frac{d^{4}k}{(2\pi)^{4}}2k_{\lambda}\epsilon^{\lambda\mu\nu\rho}\Gamma_{V\rho}(p_{1},p_{2};k)=0.$
(5)

According to Fujikawa’s interpretation, it is argued that the appearance of
the quantum anomaly in WT identity is a symptom of the impossibility of
defining a suitably invariant functional integral measure due to the relevant
transformations on fermionic field variables.The regularization procedure for
the variations of the integral measure can provide access to a wider class of
such anomaly objects [11][12][13][14].To see how the change of the measure
corresponding to the transverse transformation Eq.(1)and Eq.(4) gives rise to
a possible anomaly factor, let us consider an Abelian gauge field to show our
argument. The Lagrangian density for massive QED, which is of the form

$\displaystyle
L_{eff}=\overline{\psi}(x)\gamma^{\mu}(\partial_{\mu}-ieB_{\mu}(x))\psi(x)-\overline{\psi}(x)m\psi(x)-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\xi}(\partial^{\mu}B_{\mu})^{2}.$
(6)

where $e$ and $m$ denote, respectively, the charge and mass of the electron.
In this case, the gauge field is just the photon field $B_{\mu}(x)$. Thus
jacobian $J^{[\alpha\beta]}(x)$ of integral measure due to the transformations
Eq.$(1)$ is evaluated below

$\displaystyle
J^{[\alpha\beta]}(x)=e^{\int~{}d^{4}xA^{[\alpha\beta]}(x)\omega_{[\alpha\beta]}\theta(x)}e^{\int~{}d^{4}(x)\bar{A}^{[\alpha\beta]}(x)\omega_{[\alpha\beta]}\theta(x)}.$
(7)

This is what we set out to calculate. Due to the transverse
transformation(2.1), the anomaly functions can be written as the limit of a
manifestly convergent integral

$\displaystyle A^{[\alpha\beta]}(x)$ $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}(\frac{-\sigma^{\alpha\beta}}{4})f\big{(}(\frac{i\gamma^{\mu}D_{\mu}}{M})^{2}\big{)}e^{ikx}$
$\displaystyle\overline{A}^{[\alpha\beta]}(x)$ $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\gamma^{0}(\frac{-\sigma^{\alpha\beta}}{4})^{\dagger}\gamma^{0}f\big{(}(\frac{i\gamma^{\mu}D_{\mu}}{M})^{2}\big{)}e^{ikx}.$
(8)

where $D_{\mu}=\partial_{\mu}-ieB_{\mu}(x)$ is the covariant derivative. In
addition, the transformation of the field $B_{\mu}(x)$ is a translation, so
that its Jacobian is trivial. The anomaly function $A^{[\alpha\beta]}(x)$
requires regulation,which is achieved by inserting a regulator

$\displaystyle f\big{(}\frac{-(\gamma^{\mu}D_{\mu})^{2}}{M^{2}}\big{)}$
$\displaystyle=$
$\displaystyle\big{(}e^{-\frac{(\gamma^{\mu}D_{\mu})^{2}}{M^{2}}}\big{)}^{2}$
(9)

The expression of the transverse vector anomaly function
$A^{[\alpha\beta]}(x)$ can be put in the regulating form

$\displaystyle A^{[\alpha\beta]}(x)$ $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}(\frac{-1}{4}\sigma^{\alpha\beta})f\big{(}(\frac{i\gamma^{\mu}D_{\mu}}{M})^{2}\big{)}e^{ikx}$
(10) $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\big{[}(\frac{-1}{4}\sigma^{\alpha\beta})(\sum_{n}\frac{1}{n!}f^{(n)}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma^{\mu},\gamma^{\nu}]F_{\mu\nu})^{n})$
$\displaystyle\times\big{[}\sum_{m}\frac{1}{m!}f^{(m)}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma^{\rho},\gamma^{\sigma}]F_{\rho\sigma})^{m}]e^{ikx}$
$\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\big{[}(\frac{-1}{4}\sigma^{\alpha\beta})(f^{(^{\prime})}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma_{\mu},\gamma_{\nu}]F_{\mu\nu})^{1})(f^{(^{\prime})}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma_{\rho},\gamma_{\sigma}]F_{\rho\sigma})^{1})$
$\displaystyle=$
$\displaystyle\frac{1}{128\pi^{2}}\big{(}g^{\alpha\mu}g^{\beta\rho}g^{\nu\sigma}-g^{\alpha\mu}g^{\beta\sigma}g^{\nu\rho}+g^{\alpha\nu}g^{\beta\sigma}g^{\mu\rho}-g^{\alpha\nu}g^{\beta\rho}g^{\mu\sigma}+$
$\displaystyle+g^{\beta\mu}g^{\alpha\sigma}g^{\nu\rho}-g^{\beta\mu}g^{\alpha\rho}g^{\nu\sigma}+g^{\beta\nu}g^{\alpha\rho}g^{\nu\sigma}-g^{\beta\nu}g^{\alpha\sigma}g^{\mu\rho}\big{)}Tr(F_{\mu\nu}F_{\rho\sigma}).$

In terms of the symmetry of metric and antisymmetry of 4-dimensional field
strength tensor, we expand the anomaly function and find that it equals zero.
Thus the Jacobian Eq.$(7)$ becomes

$\displaystyle J^{[\alpha\beta]}(x)$ $\displaystyle=$ $\displaystyle
e^{2\int~{}d^{4}xA^{[\alpha\beta]}(x)\omega_{[\alpha\beta]}\theta(x)}$ (11)
$\displaystyle=$ $\displaystyle 1.$

By the parallel procedure, for the case of the transformation Eq.(4), the
transverse axial vector anomaly function is given by

$\displaystyle A^{[\alpha\beta 5]}(x)$ $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}(\frac{-1}{4}\sigma^{\alpha\beta}\gamma^{5})f\big{(}(\frac{i\gamma^{\mu}D_{\mu}}{M})^{2}\big{)}e^{ikx}$
(12) $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\big{[}(\frac{-1}{4}\sigma^{\alpha\beta}\gamma^{5})(\sum_{n}\frac{1}{n!}f^{(n)}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma_{\mu},\gamma_{\nu}]F_{\mu\nu})^{n})$
$\displaystyle\times\big{[}\sum_{m}\frac{1}{m!}f^{(m)}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma^{\rho},\gamma^{\sigma}]F_{\rho\sigma})^{m}]e^{ikx}$
$\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int\frac{d^{4}k}{(2\pi)^{4}}Tr\big{[}(\frac{-1}{4}\sigma^{\alpha\beta}\gamma^{5})(f^{(^{\prime})}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma^{\mu},\gamma^{\nu}]F_{\mu\nu})^{1})(f^{(^{\prime})}(-\frac{D^{2}}{M^{2}})(\frac{i}{4M^{2}}[\gamma^{\rho},\gamma^{\sigma}]F_{\rho\sigma})^{1})$
$\displaystyle=$
$\displaystyle\frac{i}{128\pi^{2}}\big{(}-g^{\alpha\mu}\epsilon^{\beta\nu\rho\sigma}+g^{\alpha\nu}\epsilon^{\beta\mu\rho\sigma}+g^{\beta\mu}\epsilon^{\alpha\nu\rho\sigma}-g^{\beta\nu}\epsilon^{\alpha\mu\rho\sigma}+$
$\displaystyle+g^{\nu\rho}\epsilon^{\alpha\beta\mu\sigma}-g^{\mu\rho}\epsilon^{\alpha\beta\nu\sigma}+g^{\mu\sigma}\epsilon^{\alpha\beta\nu\rho}-g^{\nu\sigma}\epsilon^{\alpha\beta\mu\rho}\big{)}Tr(F_{\mu\nu}F_{\rho\sigma})$
$\displaystyle=$ $\displaystyle-\bar{A}^{[\alpha\beta 5]}(x).$

The corresponding Jacobian is non-trivial in the form

$\displaystyle J^{[\alpha\beta
5]}(x)=e^{2\int~{}d^{4}xA^{[\alpha\beta]}(x)\omega_{[\alpha\beta]}\theta(x)}.$
(13)

Obviously the Eq.(11) and Eq.(13) is perfectly consistent with result of
derivation of transverse vector and axial vector anomalies in four-dimensional
$U(1)$ gauge theory using perturbative methods[5].

##  3 Topological index for the anomaly

From the topological viewpoint, the topological property of the quantum
anomaly Eq.(13) is addressed by Atiyah-Singer index theorem for Dirac operator
in Eq.(6) in a gauge background [15]. Since $D$ is an elliptic operator, the
Atiyah-Singer local index formula for the topological index $Ind_{t}D$ of $D$
in terms of the chern character of $ChE$ and the $\widehat{A}$-genus of
$E^{4}$ gives

$\displaystyle Ind_{a}D$ $\displaystyle=$ $\displaystyle Ind_{t}D$ (14)
$\displaystyle=$ $\displaystyle\int\widehat{A}\left(M\right)\cdot
Ch\left(E\right).$ (15)

Here the more natural definition of$\widehat{A}$ -genus form with the
Riemannian curvature $R$ is

$\displaystyle\widehat{A}\left(M\right)=det^{\frac{1}{2}}\frac{\frac{R}{2}}{\sinh\left(\frac{R}{2}\right)}.$
(16)

And the chern character form is

$\displaystyle Ch\left(E\right)=Str\left(e^{-\triangledown^{2}}\right).$ (17)

Substituting this into Eq.(14) we have

$\displaystyle Ind_{t}D=-\frac{4}{32\pi^{2}}\int
d^{4}xTr\partial_{\mu}\left(\varepsilon^{\mu\nu\rho\sigma}B_{\nu}\partial_{\rho}B_{\rho}\right).$
(18)

Obviously the quantity on the right of the Eq.(18) is known as chen-pontrjagin
term.The Eq.(18) can be expressed by a complete of eigenstates $\phi_{n}$ of
Dirac operater $D$

$\displaystyle Ind_{t}(D)$ $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int d^{4}x\langle
x|\Gamma^{[5]}f(-\frac{D^{2}}{M^{2}})|x\rangle$ (19) $\displaystyle=$
$\displaystyle\int d^{4}xA^{[5]}(x).$

Here,the result in Eq.(19) is no other than the topological index for the
corresponding anomaly function $A^{[5]}(x)$ of axial-vector current[8].
Furthermore we generalize the above approach to the pseudo-tensor current.
According to the local Atiyah-Singer index theorem, the existence of an
asymptotic expansion for $\langle x|\Gamma^{[\alpha\beta
5]}f(-D^{2}/M^{2})|x\rangle$ at $M\to\infty$ is presented by the index
$Ind(\Gamma^{[\alpha\beta 5]},D)$ of Dirac operator $D$ on the manifold $M$ ,
Which is calculated below

$\displaystyle Ind(\Gamma^{[\alpha\beta 5]},D)$ $\displaystyle=$
$\displaystyle\underset{M\to\infty}{lim}\int d^{4}x\langle
x|\Gamma^{[\alpha\beta 5]}f(-\frac{D^{2}}{M^{2}})|x\rangle$ (20)
$\displaystyle=$ $\displaystyle\int d^{4}xA^{[\alpha\beta 5]}(x).$

Here the kernel function $A^{[\alpha\beta 5]}(x)$ in Eq.(20) is no other than
the anomaly function Eq.(12) for Jacobian of functional measure Eq.(8). This
is what we do. It is shown that the index theorem for the operator
$\Gamma^{[\alpha\beta 5]}e^{(-D^{2}/M^{2})}$ associated with the square of
Dirac operator D is a statement about the relationship between the kernel
$A^{[\alpha\beta 5]}(x)$ and the topological character in gauge theory.
Fortunately, the asymptotic expansion of the kernel $A^{[\alpha\beta]}(x)$ of
the operator $\Gamma^{[\alpha\beta 5]}e^{(-D^{2}/M^{2})}$ on the right of
Eq.(5) has evaluated in the Eq.(12) by Fujikawa’s method for these fermion
pseudo-currents.

##  4 concluding Remarks

As already described, the topological index of Dirac operator reflecting the
property of fermion current coupling gauge field is indeed understand on
Atiyah-Singer index theorem in quantum field theory as a consequence of the
fact that the Jacobian of integration measure possesses anomaly terms, which
is associated with Ward-Takahashi identity for pseudo- tensor current in QED.

##  5 References

## References

  * [1]
  * [2] Takahashi Y, Mancini F. In Quantum Field Theory, Amsterdam: Elsevier Scince Publisher, 1986, 19
  * [3] KondoK-I,Maris. Phys.Rev.D,1995, 52: 1212—1233
  * [4] HE H X., Khanna F C and Takahashi Y. Phys.Lett.B, 2000,480: 222—228
  * [5] Pennnington M R , Williams R. J.Phys.G:Nucl.Part.Phys., 200632:2219—2233
  * [6] He H X. Phys.Lett.B, 2001, 507:,351—355
  * [7] SUN W S , ZONG H S , CHEN X S et al. Phys. Lett.B,2003, 569(2):,211—238
  * [8] BAO A D , Wu S S. Int.J.Theor.Phys., 2007,46:12—27
  * [9] BAO A D ,YAO H B and WU S S. Chinese Phys.C,2009, 33(3):,177—182
  * [10] BAO A D ,SUN Y Q, and WAN D. Chin. Phys.Lett, 2012, 29:,121101—121104
  * [11] HE H X. An Introduction to Nuclear Chromodynamics ,China, China University of Sci and Tech Press, 2009. 12
  * [12] HE H X. Int.J.Mod.Phys.A, 2007,22:,2119—2132
  * [13] Fujikawa K. Phys.ReV.Lett. 1997,42:,1195—1198
  * [14] EinhornM B ,T Jones D R. Phys.Rev.D, 1984 29:,331—333
  * [15] Umezswa M , Phys.Rev.D ,1989,39:,3672—3683
  * [16] Nicole Verline, Ezra Getzle and Michele, Vergne. Heat Kernels and Dirac operators, Veilag Berlin Heidelberg: Springer 2004.18–169


# Evolution of low-lying M1 modes in germanium isotopes

S. Frauendorf<EMAIL_ADDRESS>Department of Physics, University of Notre
Dame, Notre Dame, Indiana 46556, USA  R. Schwengner  Helmholtz-Zentrum
Dresden-Rossendorf, 01328 Dresden, Germany

###### Abstract

Magnetic dipole strength functions are determined for the series of germanium
isotopes from $N=Z=32$ to $N$ = 48 on the basis of a large number of
transition strengths calculated within the shell model. The evolution of the
strength with increasing neutron number in the $1g_{9/2}$ orbital is analyzed.
A bimodal structure comprising an enhancement toward low transition energy and
a resonance in the region of the scissors mode is identified. The low-energy
enhancement is strongest near closed shells, in particular at the almost
completely filled $1g_{9/2}$ orbital, while the scissorslike resonance is most
pronounced in the middle of the open shell, which correlates with the
magnitude of the also deduced electric quadrupole transition strengths. The
results are consistent with previous findings for the shorter series of iron
isotopes and prove the occurrence and correlation of the two low-lying
magnetic dipole modes as a global structural feature.

###### pacs:

21.10.Pc, 21.60.Cs, 23.20.Lv, 27.50.+e

##  I Introduction

The excitation and deexcitation of the nucleus by electromagnetic radiation at
high excitation energy and high level density are described by means of
$\gamma$-ray strength functions ($\gamma$-SF) which represent average
transition strengths in a certain energy range. The experimental determination
and the theoretical understanding of the properties of $\gamma$-SF has
attracted increasing interest because of their importance for the accurate
description of photonuclear reactions and the inverse radiative-capture
reactions, which play a central role in in the synthesis of the elements in
various stellar environments arn07 ; kap11 . The region of low transition
energies is important for these processes. Traditionally, the $\gamma$-SF has
been associated with electric dipole ($E1$) transitions. Recently a strong
component of $M1$ character has been observed as an upbend of the $\gamma$-SF
toward transition energy $E_{\gamma}=0$.

First observed in 56,57Fe voi04 , the upbend was found also in various
nuclides in other mass regions, such as in Mo isotopes gut05 , in 105,106Cd
lar13 , and in Sm isotopes sim16 ; sim19 . The experiments used light-ion
induced reactions such as (3He,3He’), and the data were analyzed with the so-
called Oslo method to extract level densities and $\gamma$-SFs. This method
was also applied in connection with $\beta$ decay to 76Ga spy14 . A dominant
dipole character of the low-energy strength was demonstrated in Ref. lar13L ,
and an indication for a magnetic dipole ($M1$) character was discussed for the
case of 60Ni voi10 .

Shell-model calculations revealed that a large number of $M1$ transitions
between excited states produces an exponential increase of the $\gamma$-ray
strength function that peaks at $E_{\gamma}\approx 0$ and describes the low-
energy enhancement of dipole strength observed in Mo isotopes around the
neutron shell closure at $N$ = 50 sch13L . In these calculations, large
reduced transition strengths $B(M1)$ appear for transitions linking states
with configurations dominated by both protons and neutrons in high-$j$
orbitals, the spins of which recouple. The low-energy enhancement was
confirmed in shell-model calculations for 56,57Fe bro14Fe , 46,50,54Ti sie17a
and 44Sc sie17 . In the latter work, also the electric dipole ($E1$) strength
function was calculated, which does not show an upbend comparable to that of
the $M1$ strength. A correlation between the low-energy $M1$ strength (LEMAR -
Low Energy Magnetic Radiation) and the scissors resonance (SR), a fundamental
$M1$ excitation occurring in deformed nuclei around 3 MeV hey10 , was found in
shell-model calculations for the series of isotopes from 60Fe to 68Fe sch17 .
It was found that the low-energy $M1$ strength decreases and the scissors
strength develops when going into the open shell. The simultaneous appearance
of the two $M1$ modes is in accordance with experimental findings in Sm
isotopes sim16 ; sim19 . Later on, $M1$ strength functions were calculated for
isotopic series in various mass regions kar17 ; sie18 ; mid18 . The study in
Ref. mid18  confirmed that the low-energy $M1$ strength is strongest in
nuclides near shell closures.

In the present work we study the low-energy $\gamma$-SFs for the chain of the
Ge isotopes. The relatively small configuration space allows us to carry out
shell model calculations covering completely the open neutron shell $32\leq
N\leq 48$. We demonstrate for the first time that the low-energy $M1$ strength
is concentrated in the LEMAR spike at the bottom of the shell, it is partially
moved into the SR in the middle of the shell, and is again concentrated in the
LEMAR spike at the top of the shell.

##  II Shell-model calculations

The shell-model calculations for the germanium isotopes were carried out in
the jj44pn model space with the jj44bpn Hamiltonian hon09 ; bro ; lis04  using
the code NuShellX@MSU bro14Nu . The model space included the proton and
neutron orbitals $(1f_{5/2},2p_{3/2},2p_{1/2},1g_{9/2})$. At first, we
calculated the energies of the yrast states and the reduced transition
strengths of the linking electric quadrupole ($E2$) transitions for varied
limitations of occupation numbers to test at which numbers the $B(E2)$ values
do not change further and a convergence is achieved. This is in particular
important for the mid-shell isotopes. For example, an increase of the allowed
maximum occupation number (upper limit) in the neutron $1g_{9/2}$ orbital from
four to six in 70Ge does not change the $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$
value and, thus, the application of an upper limit of four is appropriate. In
74Ge, a change of this number from six to eight changes the
$B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ value from 357 to 369 e2fm4, while the
further increase to ten neutrons does not cause any further change. At the
same time, the allowed minimum occupation numbers (lower limits) in the
neutron $2p_{3/2}$ and $1f_{5/2}$ orbitals were set to two. A decrease of
these lower limits to zero results in $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ =
371 e2fm4. In the full calculations including all transition strengths, the
following limits of occupation numbers were applied to truncate the
configuration space and, hence, make the calculations feasible and efficient.
Up to four protons were allowed to occupy each of the $1f_{5/2}$ and
$2p_{3/2}$ orbitals while up to two could be lifted to each of the $2p_{1/2}$
and $1g_{9/2}$ orbitals. The same holds for the neutrons in 64Ge, while there
can be up to six neutrons in the $1f_{5/2}$ orbital in 66Ge. In 70Ge, at least
two neutrons are in each of the $1f_{5/2}$ and $2p_{3/2}$ orbitals and up to
four can be excited to the $1g_{9/2}$ orbital. The possible occupation numbers
of neutrons in the $1g_{9/2}$ orbital are further increased in the heavier
isotopes, ranging from two to eight in 74Ge, from six to ten in 78Ge, and from
eight to ten in 80Ge. For the calculation of the reduced electric quadrupole
transition strengths $B(E2)$, standard effective charges of $e_{\pi}=1.5e$ and
$e_{\nu}=0.5e$ were used and for the $B(M1)$ strengths, effective $g$ factors
of $g^{\rm eff}_{s}=0.7g^{\rm free}_{s}$ were applied.

The full calculations were performed for the lowest 40 states of each spin
from $J_{i},J_{f}$ = 0 to 10 and each parity. The reduced transition strengths
$B(M1)$ were calculated for all transitions from initial to final states with
energies $E_{f}<E_{i}$ and spins $J_{f}=J_{i},J_{i}\pm 1$. This resulted in
more than 24000 $M1$ transitions for each parity, which were sorted into 0.1
MeV bins of transition energy $E_{\gamma}=E_{i}-E_{f}$. The average $B(M1)$
value for one energy bin was obtained as the sum of all $B(M1)$ values divided
by the number of transitions within this bin. Average $B(E2)$ values were
deduced in an analogous way, but include also the $\Delta J$ = 2 transitions.
$M1$ strength functions were deduced according to

$f_{M1}(E_{\gamma},E_{i},J_{i},\pi)=\\\ 16\pi/9~{}(\hbar
c)^{-3}~{}\overline{B}(M1,E_{i}\rightarrow
E_{f},J_{i},\pi)~{}\rho(E_{i},J_{i},\pi),$ (1)

where the $\overline{B}(M1,E_{i}\rightarrow E_{f},J_{i},\pi)$ are averages in
considered $(E_{i},E_{f})$ bins for given $J_{i}$, $\pi$, and
$\rho(E_{i},J_{i},\pi)$ are the level densities derived from the present
calculations. The strength functions $f_{M1}(E_{\gamma})$ were obtained by
averaging step-by-step over $E_{i}$, $J_{i}$, and $\pi$.

##  III Results for the yrast region

To check the reliability of the shell-model calculations, we studied the yrast
regions of the Ge isotopes. The calculated energies of the ground-state and
first excited bands in 74Ge are compared with the experimental ones sun14  in
Figs. 1 and 2. They represent the results obtained with the just discussed
upper limits of six and eight $1g_{9/2}$ neutrons, respectively. In both
cases, the experimental bands are well described by the calculations. A
similarly good description of the experimental yrast and yrare bands by the
present calculations is achieved for all other isotopes, which are presented
in Figs. 11 and 12 of the appendix.

Figure 1:  Experimental (red circles) and calculated (black squares)
excitation energies of the yrast states and the states of the first excited
band in 74Ge. The lines represent the linking $E2$ transitions with large
strengths. The numbers at the lines are calculated $B(E2)$ values in e2fm4.
The configuration space allowed up to six neutrons in the $1g_{9/2}$ orbital
(see text). Figure 2:  As Fig. 1, but with a configuration space allowing up
to eight neutrons in the $1g_{9/2}$ orbital (see text).

In accordance with the experiment, the energies $E(J)$ increase in a regular
way, forming quasi-rotatonal bands. The crossing of the ground-state band with
an excited band at $J=6$ is reproduced. The energy ratios $E(J)/E(2^{+})$
deviate substantially from the rotor rule $J(J+1)$, The calculated ratios
$E(4^{+}_{1})/E(2^{+}_{1})$ of 2.44, 2.59, 2.59, 2.38, 2.32, 2.22 for
64,66,70,74,78,80Ge compare well with experimental ratios of 2.28, 2.27, 2.07,
2.46, 2.54, 2.64, respectively. They characterize the Ge isotopes as soft
nuclei in the transitional region between spherical and deformed shapes,
because they are well below the rotor ratio of 3.33. The calculated $B(E2)$
values increase toward high spin up to the crossing region. For 74Ge, the
experimental ratio $B(E2,4^{+}_{1}\rightarrow
2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})=1.33(8)$ is reproduced by the
calculated one of 1.35.

The results for the $2^{+}_{1}$ states are compared with the experimental
values in Table 1. The energies of the $2^{+}_{1}$ states are reproduced
within 100 keV, except the high experimental value for 70Ge. The experimental
$B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ values indicate a maximum of the
quadrupole collectivity in the middle of the shell, which is reproduced by the
calculations. However, the calculated peak is much shallower than in the
experiment. A similar shallow peak is obtained for $J=4$ and 6 (see Figs.11
and 12 of the appendix.).

One should be aware that the determination of the quadrupole collectivity from
the $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ values only is based on the
assumption of a rotational behavior of the yrast states, which is not realized
for the soft nuclei under consideration. Instead, the sums of the
$B(E2,0^{+}_{1}\rightarrow 2^{+}_{n})$ values of all transitions from the
ground state are more appropriate Kum72 ; pov20  and also given in Table 1.
These sums of all $E2$ transitions into the ground state are however only
little larger and follow the trends of the $B(E2,2^{+}_{1}\rightarrow
0^{+}_{1})$ values. An even more comprehensive indicator of the collectivity
may be the consideration of average $B(E2)$ values between all the states
considered here. The further discussion of $B(E2)$ values in Sec. IV therefore
takes into account these values.

Table 1: Experimental and calculated reduced $E2$ transition probabilities for the $2^{+}_{1}\rightarrow 0^{+}_{1}$ transitions in 64,66,70,74,78,80Ge. |  $E(2^{+}_{1}$) |  $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1}$) | $\Sigma B(E2,2^{+}_{n}\rightarrow 0^{+}_{1})$
---|---|---|---
| (keV) | (e2fm4) | (e2fm4)
| EXP111The values for 64,66,70,74,78,80Ge were taken from Refs. nds64 ; nds66 ; nds70 ; nds74 ; nds78 ; nds80 , respectively. | CALC | EXP111The values for 64,66,70,74,78,80Ge were taken from Refs. nds64 ; nds66 ; nds70 ; nds74 ; nds78 ; nds80 , respectively. | CALC | CALC
64Ge32 | 902 | 883 |  | 296 | 308
66Ge34 | 957 | 828 | 190(36) | 300 | 321
70Ge38 | 1040 | 649 | 356(7) | 336 | 382
74Ge42 | 596 | 704 | 609(7) | 369 | 384
78Ge46 | 619 | 782 | 455(79) | 320 | 343
80Ge48 | 659 | 871 | 279(55) | 233 | 280

Figure 3:  Results for 64Ge in energy bins of 0.1 MeV. (a): Average reduced
$M1$ transition strengths (blue circles) and their orbital contributions
($g_{s}$ = 0) only (red dashed line) for positive-parity states. (b): The $M1$
strength function including both parities. (c): Average reduced $E2$
transition strengths (green squares) and average $B(E2)$ values for stretched
transitions with $J_{i}=J_{f}+2$ only (red circles) for positive-parity
states.

Figure 4:  As Fig. 3, but for 66Ge.

Figure 5:  As Fig. 3, but for 70Ge.

Figure 6:  As Fig. 3, but for 74Ge.

Figure 7:  As Fig. 3, but for 78Ge.

Figure 8:  As Fig. 3, but for 80Ge. Note the different vertical scale in (a)
compared with the corresponding graphs for the other isotopes.

##  IV Results for the strength functions

The average $B(M1)$ and $B(E2)$ values for positive parity states and the $M1$
strength functions including both parities are shown for all considered Ge
isotopes in Figs. 3,4,5,6,7,8. The $N=Z=32$ nucleus 64Ge shows a fluctuating,
but on average flat distribution of the $B(M1)$ strength as a function of
$E_{\gamma}$ with an even slight decrease toward $E_{\gamma}=0$, which is
similar to predictions for the $N=Z$ nuclei 48Ca sie17a  and 108Xe sie18 .
This seems to point to a more general feature of the low-energy $M1$ strength
in $N=Z$ nuclei. It is suggested below that isospin conservation quenches the
LEMAR spike. In the $N=Z+2$ nuclide ${}^{66}_{32}$Ge34, a gradual enhancement
of the $M1$ strength toward $E_{\gamma}=0$ is seen. The behavior resembles the
one in ${}^{60}_{26}$Fe34 sch17 , but is less pronounced. Both these nuclei
are localized near the bottom of the neutron $(fpg)$ shell. For
${}^{70}_{32}$Ge38, the bimodal structure of a LEMAR peak at $E_{\gamma}=0$
and a broad SR peak around $E_{\gamma}=3$ MeV appears. A similar bimodal
distribution is seen in ${}^{74}_{32}$Ge42. This bimodal strength distribution
is characteristic for nuclei located well in the open shell, as for example
also in ${}^{64,68}_{~{}~{}~{}26}$Fe38,42 sch17  and in nuclides with $A>$ 100
sie18 . The SR peak becomes weak in ${}^{78}_{32}$Ge46 and disappears in
${}^{80}_{32}$Ge48, when approaching the top of the neutron shell. A similar
suppression of the SR peak toward the next higher neutron shell was also found
in $N\approx 80$ nuclei sie18 . The present calculations find a maximum of the
SR strength in the middle of the neutron shell that correlates with the clear
maximum of the experimental $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ values.
Unlike the experiment, the calculated $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$
values have a very shallow mid-shell maximum, as was also obtained in the
calculations for the Fe isotopes sch17 .

However, a different behavior is seen for the average $B(E2,J\rightarrow J-2)$
values shown in the panels (c) of Figs. 3,4,5,6,7,8. Here, a peak around 1 MeV
develops toward the middle of the shell (70,74Ge). This peak indicates
enhanced collectivity in the $\Delta J=2$ sequences and clearly correlates
with the maximum of the SR strength in the middle of the shell. It can be
interpreted as the appearance of damped rotational transitions (see e.g. Ref.
mat97 ) as a consequence of building-up quadrupole collectivity. The
calculations include states up angular momentum 10 $\hbar$ with equal weight.
The average transition energy of 1 MeV and the average angular momentum of
$5~{}\hbar$ correspond to a moment of inertia of about 10 $\hbar^{2}$/MeV,
which is somewhat smaller than the rigid-body value of 14 $\hbar^{2}$/MeV for
$A=70$. In the same way, from Fig. 4 of Ref. sch17  one derives a moment of
inertia of 8 $\hbar^{2}$/MeV for ${}^{68}_{26}$Fe42. In Ref. sim16 , the
increased SR strength of the $\gamma$-SF of 151,153Sm could be reproduced by
replacing the ground state moment of inertia by the rigid-body value in the
phenomenological expression of Ref. end05 , which was developed for the
excitation of the SR from the ground state.

We calulated the total $B(M1)$ strengths in certain energy ranges by a
numerical integration of the $M1$ strength functions:

$B(M1)_{\rm tot}=9/(16\pi)~{}(\hbar c)^{3}~{}\sum f_{M1}(E_{\gamma})\Delta
E_{\gamma}.$ (2)

The results for the LEMAR region ($E_{\gamma}<2$ MeV), the SR region (2 MeV
$\leq E_{\gamma}<5$ MeV) and their sums are compiled in Table 2. As visualized
in the panels (b) of Figs. 3,4,5,6,7,8, one also quantitatively observes a
shift of strength to the SR region when going into the open shell
(${}^{70,74}_{~{}~{}~{}32}$Ge38,42) and a shift back to the LEMAR region when
approaching the $N$ = 50 shell closure (${}^{78,80}_{~{}~{}~{}32}$Ge46,48),
while the sum of the two remains roughly constant. Only the $N=Z$ nuclide 64Ge
does not fit the systematics for reasons discussed below. In the calculations
of Ref. sch17 , a similar redistribution of the $M1$ strength was found for
the isotopes ${}^{60,64.68}_{~{}~{}~{}~{}~{}~{}26}$Fe34,38,42 when moving into
the open shell by adding neutrons. However, the integrated strength up to 5
MeV, which is the sum of the LEMAR and SR strength, is about 1 $\mu_{N}^{2}$
for the Ge isotopes and about 10 $\mu_{N}^{2}$ for the Fe isotopes. The
authors of Ref. sch13L  suggested that the low-energy $M1$ radiation is
generated by the reorientation of the valence nucleons on high-$j$ orbitals.
This mechanism is particular efficient if protons are hole-like and neutrons
are particle-like (or vice versa). Then the transverse magnetic moments add
up, which generates strong $M1$ radiation. An analogous mechanism generates
the ”shears bands” manifesting ”magnetic rotation” MagRot . In the case of the
Fe isotopes one has active $1f_{7/2}$ proton holes, which favorably combine
with the active $1g_{9/2}$ neutrons. In the case of the Ge isotopes the active
$1g_{9/2}$ neutrons combine with the $1f_{5/2}$ protons, which have a small
magnetic moment, and $1g_{9/2}$ protons, which have a magnetic moment with the
opposite sign. The factor of 10 in the integrated low-energy $M1$ strength
reflects the different valence proton configurations of the Fe and Ge
isotopes. In case of the $Z>50$, $N\geq 80$ nuclides the integrated strength
up to 4 MeV is 0.5 - 1 $\mu_{N}^{2}$ (see Fig. 3 of Ref. sie18 ). The small
number is expected because active protons and neutrons are particle-like and
do not occupy the high-$j$ orbitals.

Also shown in Table 2 are the integrated $B(E2)$ strengths up to $E_{\gamma}$
= 5 MeV, which were determined analogously to the integrated $B(M1)$
strengths. The values are maximal at the mid-shell nuclei 70,74Ge in
accordance with the SR strengths, which proves the correlation of SR strength
and collectivity.

The excitation of the SR from the ground state, which appears as a bunch of
$1^{+}$ states around 3 MeV, has been extensively studied and reviewed in Ref.
hey10 . Therein, the SR is considered as being dominated by exciting the
orbital angular momentum of the protons. In contrast, our calculations for the
Ge isotopes show a reduction of the low-energy strength by a factor of about
two when the spin part of the magnetic dipole operator is set equal to zero,
which is illustrated in the panels (a) of Figs. 3,4,5,6, 7,8. An equal
reduction appeared in our earlier calculations for the Mo and Fe isotopes as
displayed in Figs. 9 and 10. The strong dependence of the $M1$ strength on the
spin part indicates that the reorientation of high-$j$ orbitals such as
$1g_{9/2}$ and $1h_{11/2}$ must play a central role in generating the strong
$M1$ radiation (see Ref. sch17 ). In the calculations for 130Te in Ref. sie18
, the SR peak is similarly quenched when the spin part of the $M1$ operator is
set equal to zero whereas the LEMAR spike remains unchanged (see Figs. 2c and
2d therein).

Figure 9:  As Fig. 3(a), but for 68Fe. The corresponding shell-model
calculations are described in Ref. sch17 .  Figure 10:  As Fig. 3(a), but for
94Mo. The corresponding shell-model calculations are described in Ref. sch13L
.

The SR strength in the medium and heavy nuclei has been found proportional to
the square of the quadrupole deformation, which is maximal in the middle of
the open shell hey10 . As discussed in Ref. sim16 , the strength of the SR in
the $\gamma$-SF seems to be related to the nuclear ground-state deformation in
a similar way, though being a factor of three larger. The calculations in Ref.
sie18  show the built-up of a SR around 4 MeV when moving into the open shell
by adding neutrons in the case of the Te isotopes and when adding protons in
the case of the $N=80$ isotones, where the SR strength increases with the
$B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ values (compare Figs. 1 and 3 in Ref.
sie18  and the experimental values for the studied nuclides given in Ref.
ram01 ). The same correlation between the increase of experimental
$B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ and of the increase of the SR strength
was found for the Fe isotopes in the calculations in Ref. sch17 .

The integrated strengths up to a transition energy of 5 MeV in Table 2 are
about 1 $\mu_{N}^{2}$. They are much larger than the sums of the strengths of
transitions from $1^{+}$ states down to the ground state, which are also given
in Table 2. As seen in Table II of Ref. sch17 , the integrated strengths up to
a transition energy of 5 MeV in 60,64,68Fe are about 10 $\mu_{N}^{2}$ to be
compared with 0.33, 0.55, 0.58 $\mu_{N}^{2}$ for the respective sums of all
transitions from $1^{+}$ states down to the ground state. In Ref. sim19 , the
experimental integrated $M1$ strengths up to 5 MeV for transitions in the
quasicontinuum of 147,149,151,153Sm were found to be about 8 $\mu_{N}^{2}$,
which has to be compared with summed strengths of 0.16, 0.32, 0.80, 0.81
$\mu_{N}^{2}$ from $1^{+}$ states to the ground states of 148,150,152,154Sm,
respectively end05 . In Ref. sch17 , the enhancement was attributed to the
quenching of the pair correlations with increasing excitation energy, i.e. the
thermal quenching of pairing.

According to the collective model of Ref. end05 , the $M1$ strength of the SR
on the ground state scales $\propto A\delta^{2}$, where $\delta$ is the
deformation parameter and $A$ the mass number. The sums $\sum
B(M1,1^{+}\rightarrow 0^{+}_{1})$ of transition down to the ground states of
the Ge, Fe,Te and Sm isotopes roughly follow the scaling (cf. Table 2 and
Refs. sch17 ; sie18 ; sim19 ). The simple collective behavior seems to be
caused by the pair correlations. The integrated $M1$ strengths for the
transitions in the quasicontinuum do not obey it. Once the pair correlations
are quenched, the bimodal LEAMAR-SR structure appears in the strength
function, the total strength of which depends strongly on the individual
magnetic properties of the valence nucleons. This is in analogy to the moments
of inertia. At low spin, when the the pair correlations are strong, the
moments of inertia behave in a systematic manner, being $\propto
A^{5/3}\delta^{2}$. At high spin, when the Coriolis force overcomes the pair
correlation, the individuality of the valence nucleons comes to light.

The experimental summed strengths for the transitions from $1^{+}$ states to
the ground state given in Table 2 are smaller than the calculated ones.
However, the experimental values represent only a lower limit. In the
calculations, a large number of weak transitions contributes to the summed
strengths, whereas experiments as the ones in Ref. jun95  detect only the
strongest transitions. It has been demonstrated that a large number of weak
transitions, which are hidden in a quasicontinuum, may substantially enlarge
the $M1$ strength function, as seen for example in Ref. mas14 .

The $M1$ operator has approximately isospin $T=1$ character. In $N=Z$ nuclei
the low-lying states have $T=0$. The $T=1$ states lie substantially higher. In
case of 64Ge, the experimental energy of the lowest $T=1$ state is 6.2 MeV.
Thus, the sum $\sum B(M1,1^{+}\rightarrow 0^{+}_{1})$ = 0.001 $\mu_{N}^{2}$ in
Table 2 includes only transitions between $T=0$ states, which are isospin
forbidden. The very small value for 64Ge reflects that isospin conservation
nearly quenches $M1$ transitions between the $T=0$ states. For $N=Z+2$ nuclei
the low-lying states have $T=1$. Transitions between $T=1$ states are allowed,
which results in $\sum B(M1,1^{+}\rightarrow 0^{+}_{1})=0.25\mu_{N}^{2}$ for
66Ge. One expects that the same mechanism works for higher excitation
energies. Transitions between the $T=0$ states are nearly forbidden. The $T=1$
states lie on the average substantially above the $T=0$ states that are
connected by the $M1$ operator, which prevents transition energies close to
zero. For $N>Z$ nuclides, the transitions between the states with the same
isospin $T>0$ are allowed and the LEMAR spike appears. The author of Ref.
sie18  suggested an alternative explanation: $N=Z$ nuclei have a particular
large deformation that moves $M1$ strength from the LEMAR spike to the SR,
which results in a flat distribution. At variance, the $N$ dependences of the
$E2$ strength in Figs. 3 to 8 and Table 2 indicate little $E2$ collectivity
for 64Ge.

Table 2: Summed $B(M1)$ strengths in ranges of transition energy (LEMAR: $E_{\gamma}<2$ MeV, Scissors: 2 MeV $\leq E_{\gamma}<5$ MeV, $\sum$: the sum of the two), summed $B(E2)$ strengths for $E_{\gamma}<$ 5 MeV, and summed strengths of all discrete transitions from $1^{+}$ states to the ground states in 64,66,70,74,78,80Ge. |  $B(M1)_{\rm tot}$111Integrated $M1$ strength calculated according to Eq. (2). |  $B(E2)_{\rm tot}$222Integrated $E2$ strength calculated for positive-parity states in analogy to Eqs. (1) and (2). |  $\sum B(M1,1^{+}\rightarrow 0^{+}_{1})$333Summed $M1$ strength of transitions from the $1^{+}$ states below 5 MeV to the ground state.
---|---|---|---
| $(\mu_{N}^{2})$ | ($e^{2}$fm4) | $(\mu_{N}^{2})$
| LEMAR | SR | $\sum$ |  | EXP444Value taken from Ref. jun95 . | CALC
64Ge32 | 0.30 | 0.54 | 0.84 | 155 |  | 0.001
66Ge34 | 0.35 | 0.35 | 0.70 | 185 |  | 0.25
70Ge38 | 0.54 | 0.62 | 1.16 | 219 | 0.04(1) | 0.49
74Ge42 | 0.44 | 0.50 | 0.94 | 241 | 0.30(3) | 0.57
78Ge46 | 0.63 | 0.49 | 1.12 | 181 |  | 0.48
80Ge48 | 0.84 | 0.28 | 1.12 | 123 |  | 0.31

##  V Summary

Shell-model calculations were performed for the series of germanium isotopes
with neutron numbers from $N$ = 32 to $N$ = 48. Average $B(M1)$ and $B(E2)$
strengths were determined from a large number of transitions linking states of
spins from 0 to 10. The average $B(M1)$ strengths and the associated $M1$
strength functions are strongly enhanced near zero transition energy, which is
the LEMAR spike observed before. The LEMAR spike develops with increasing
neutron number and is strongest at $N=80$. It is suppressed at $N=Z$, which is
attributed to isospin conservation. In the mid-shell nuclei, a bump around 3.5
MeV appears, which is interpreted as the scissors resonance. The strength of
the SR correlates with the quadrupole collectivity, as reflected by the
experimental $B(E2,2^{+}_{1}\rightarrow 0^{+})_{1}$ values and the integrated
average $E2$ strength of quasicontinuum transitions. The sum of the LEMAR and
SR strengths depends only weakly on the neutron number. These characteristics
are consistent with those found for the series of iron isotopes and with the
experimental observation of LEMAR and SR strengths in samarium isotopes as
well. They exhibit the important role of high-$j$ orbitals, such as $1g_{9/2}$
and $1h_{11/2}$, for the evolution of the low-lying modes. Spin and orbital
contributions to the $M1$ strength appear nearly equal at low energy in most
isotopes, while there are stronger orbital contributions above 4 MeV of
transition energy in the mid-shell isotopes 70,74Ge. The present systematic
analysis of low-lying $M1$ strength in a relatively long isotopic series
demonstrates that the correlated appearance of the two $M1$ modes is a
phenomenon that occurs across various mass regions.

##  VI Acknowlegdments

We thank B. A. Brown for his support in using the code NuShellX@MSU. The
allocation of computing time through the Centers for High-Performance
Computing of Technische Universität Dresden and of Helmholtz-Zentrum Dresden-
Rossendorf are gratefully acknowledged. S. F. acknowledges support by the DOE
Grant DEFG02-95ER4093.

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##  Appendix A Yrast properties of the remaining isotopes

Figure 11:  Excitation energies versus spin of experimental (red circles) and
calculated (black squares) yrast states in 64,66,70Ge, and of experimental
(red triangles up) and calculated (blue triangles down) states built on the
second $2^{+}$ states in 66,70Ge. The lines represent the linking $E2$
transitions. The numbers at the lines are calculated $B(E2)$ values in e2fm4.
The experimental data were taken from Refs. far03 ; ste03 ; muk01 ,
respectively.

Figure 12:  Excitation energies versus spin of experimental (red circles) and
calculated (black squares) yrast states in 78,80Ge, and of experimental (red
triangles up) and calculated (blue triangles down) states built on the second
$2^{+}$ state in 78Ge. The lines represent the linking $E2$ transitions. The
numbers at the lines are calculated $B(E2)$ values in e2fm4. The experimental
data were taken from Refs. for18 ; pod06 , respectively.