Simona Korenjak-Černe and Nataša Kejžar 11institutetext: University of
Ljubljana, School of Economics and Business,
and Institute of Mathematics, Physics and Mechanics, Slovenia
11email<EMAIL_ADDRESS>22institutetext: University of Ljubljana,
Faculty of Medicine,
Institute for Biostatistics and Medical Informatics, Slovenia
22email<EMAIL_ADDRESS>
# Identifying patterns of main causes of death
in the young EU population

Simona Korenjak-Černe 11 Nataša Kejžar 22

###### Abstract

The study of mortality patterns is a popular research topic in many areas. We
are particularly interested in mortality patterns among main causes of death
associated with age-gender combinations. We use symbolic data analysis (SDA)
and include three dimensions: age, gender, and patterns across main causes of
death. In this study, we present an alternative method to identify clusters of
EU countries with similar mortality patterns in the young population, while
considering comprehensive information on the distribution of deaths among the
main causes of death by different age-gender groups. We explore possible
relationships between mortality patterns in the identified clusters and some
other sociodemographic indicators. We use EU data of crude mortality rates
from 2016, as the most recent complete data available.

###### keywords:

mortality pattern, the main cause of death, young population, symbolic data
analysis, adapted clustering methods

##  1 Introduction

Mortality data on causes of death provide important information about the
public health in the observed geographic area. Consequently, the study of
mortality patterns is a very popular research topic in many areas (e.g.,
demography, health economics, public policy, and actuarial science), and
therefore several approaches to the analysis of these data have been developed
depending on specific needs. For an overview of these approaches with many
references, see for example the paper of Van Raalte [21]. As highlighted in
this paper, most life course studies are still conducted at the individual
level, while the main challenge is to link life course studies more closely to
mortality patterns at the aggregate level [21] (p. S107). In this paper, we
present a possible approach to the analysis of mortality data at the aggregate
level.

Our exploratory study focuses on the leading causes of death in the young
population. Our study was motivated by the Centres for Disease Control and
Prevention (CDC) publication summarising 10 leading causes of death by age
group in the United States. The three leading causes of death among the young
population in the United States are unintentional injuries, suicide, and
homicide. In the fourth place are malignant neoplasms. Since the most common
causes of death in the young population aged 20-39 are external (i.e.,
accidents, especially traffic accidents, suicides, and assaults), which are
preventable, we focus on mortality patterns in the young population in EU
countries.

It is well-known that the number of deaths is strongly related to age, gender,
and cause of death. All three factors can be captured by selecting all
possible combinations of these factors as variables and then applying
classical multivariate data analysis techniques. However, in this case
internal relationships among these variables are usually not taken into
account. These relationships may be described with an empirical distribution
of deaths among different causes of death. To include them simultaneously in
the analysis, more advanced analysis methods are needed.

The data that we deal with are aggregated deaths, and we analyze them using
symbolic data analysis (SDA) [6, 5, 7]. SDA methods consider internal
variability; in our case, variability among deaths by cause of death.
Following this path, we use symbolic data descriptions [18] and consider data
as symbolic objects, where the units of interest are EU countries and the
symbolic variables are five-year age-gender groups described with
distributions of the (expected) number of deaths across selected causes of
death. We name these distributions mortality patterns across causes of death.
To find groups of EU countries with similar mortality patterns, we adapt the
hierarchical and clustering method presented in [17] and implemented in the R
package clamix, which we modify to include the requirements of our specific
problem.

In addition to preserving the relationship structure, another question that we
want to tackle is how to sufficiently transform mortality rates to make them
directly comparable between countries.

The objectives of this study are twofold: (1) to present an alternative method
to identify clusters of EU countries with similar mortality patterns in the
young population that takes into account more comprehensive information on the
three dimensions of the data: age, gender, and distribution across causes of
death; and (2) to explore possible relationships between mortality patterns in
the identified clusters and some other sociodemographic indicators, which can
serve as a starting point for further detailed quantitative and qualitative
investigations of possible associations. We use EU data on crude mortality
rates from 2016, which are the most recent complete data available.

The rest of the paper is organized as follows. In the second section, we
present an additional rationale for our data selection and its preparation. In
the third section, we explain the adaptations of the clustering method to our
symbolic descriptions of the data. In the fourth section, we present and
comment on some of the results. The final section will give the concluding
remarks of this study.

##  2 Data

The young population is generally considered to be very healthy, and therefore
the leading causes of death in this population are often associated with risky
behaviors. The leading causes of death in the general population in EU
countries in 2016 and 2017 were circulatory diseases and various cancers;
followed by respiratory diseases; while external causes of death, including
accidents, suicides, homicides, and other violent causes of death, ranked
fourth [10, 19]. The leading causes of death in the young population are very
different [11], with external causes being by far the most common causes of
death (the percentage varies by age group but usually comprises more than half
of the deaths).

The boundaries of the age groups that comprise the young population are not
clearly defined. In this work, we focus on the ages from 20 to 39 years
because this is the time when most people face major changes in their lives
(usually starting a new job, becoming independent from parents, finding a new
place to live, planning own family, raising children, etc.). This selection
also coincides with the psychological developmental stage of young adulthood
that was introduced by Erikson [9]. An additional reason for this particular
data selection is that the generation of ages from 20 to 39 years is often
overlooked in specific health studies because their causes of death are
associated with risky behaviors that are often considered to be personal
choices. Meanwhile, risky behaviors are closely related to mental status, with
stress being one of the most important factors. Most of these causes of death
are preventable, and consequently we should not overlook these important
issues in our society. It is therefore important to develop appropriate
prevention programs for this target group because they face very specific
challenges in life and represent about a quarter of the total population of
the EU.

###  2.1 Symbolic representation of mortality data

For easier understanding and interpretation, mortality data are usually
presented with crude death rate (CDR), which is calculated as the ratio
between the (total) number of deaths and the (total) population size, and is
usually expressed per 100,000. Thus, it represents the number of deaths per
100,000 persons at risk in the observed region. Because mortality data are
affected by age and gender, age- and gender-adjusted death rates are more
appropriate for comparisons between regions with different population age and
gender structures. Another factor strongly associated with age and gender, in
addition to the number of deaths, is the cause of death.

Given the key recommendations in the work of Anderson and Rosental [2] to
obtain as much of the age- and gender-cause-specific information and to make
the data comparable across countries, we decided to convert the observed
number of deaths to the expected number of deaths. To do this, we consider
age- and gender-specific death rates for a common (standard) population, which
we also recalculated on a two-dimensional (age-gender) structure for this
purpose.

We took the latest complete mortality data that was available from Eurostat,
which is the statistical office of the EU [12], for the year 2016. Thus, we
originally dealt with 28 units (i.e., EU countries) and eight symbolic
variables (for each gender we had four five-year age groups: 20-24, 25-29,
30-34, and 35-39).

We considered the data as symbolic objects [6, 5, 7], where the units of
interest are the EU countries and the symbolic variables are five-year age-
gender groups related to young adults, which are described with the
distributions of deaths among the selected causes of death. They are called
”symbolic” because they represent aggregations of deaths and are described
with adjusted ”symbolic” variables that contain information on the number of
deaths and their distributions among causes of death. Some of this information
would be lost if the data were presented in a classical way.

####  2.1.1 Data description with distributions

We consider each country as an observation unit (class) $s_{i}$ and describe
it with a symbolic description $S_{i}$. Each country is in our case described
with eight symbolic variables $Y_{j},j=1,...,8$, representing mortality data
for the following age-gender groups: $Y_{1}$: 20-24 male, $Y_{2}$: 20-24
female, $Y_{3}$: 25-29 male, $Y_{4}$: 25-29 female, $Y_{5}$: 30-34 male,
$Y_{6}$: 30-34 female, $Y_{7}$: 35-39 male, and $Y_{8}$: 35-39 female.

More generally: a unit $S_{i}$ from a data set of $n$ units $\mathbf{S}$ is a
list of values of modal valued variables $Y_{j},\;j=1,\ldots,p$. A symbolic
variable $Y_{j}$ has $m_{j}$ categories (subsets). Each variable is described
by a list of values of corresponding categories:

$Y_{j}=\left[L_{j1}:y_{j1},\ldots,L_{jm_{j}}:y_{jm_{j}}\right].$

In the description, $L_{j\ell}$ is a category label and $y_{j\ell}$ is its
value.

In our case, values $L_{j\ell}$ are the causes of death and $y_{j\ell}$ are
the proportions of deaths by the corresponding cause. We have the same
categories for all age-gender combinations ($m_{j}=7$ for all eight symbolic
variables $j,j=1,...,8$). We selected leading causes of death for EU countries
in the age groups 20 to 39 years. These are

  * •

Neoplasms (_Neop_),

  * •

Nerves (_Nerv_) (Diseases of the nervous system and sense organs (G00-H95)),

  * •

Circulatory (_Circ_) (Diseases of the circulatory system (I00-I99)),

  * •

Respiratory (_Resp_) (Diseases of the respiratory system (J00-J99)),

  * •

Accidents (_Acc_) (V01-X59, Y85, Y86) and

  * •

Suicides (_Suic_) (Intentional self-harm).

The codes in parentheses refer to the ICD classification [15]. We did not
include homicide as a separate cause because it represents a very small part
of the causes of death among the young population in EU countries.

A description of symbolic object $s_{i}$ is a vector of modal values
$S_{i}=[\mathbf{y}_{i1},\ldots,\mathbf{y}_{ip}]$, where
$\mathbf{y}_{ij}=\left[L_{j1}:y_{ij1},\ldots,L_{jm_{j}}:y_{ijm_{j}}\right]$ is
the modal value corresponding to symbolic variable $Y_{j}$ for symbolic object
$s_{i}$.

For example, if $s_{i}$ = EU28, the whole European union, and $Y_{j}$ =
$Y\textsubscript{20-24 male}$, then

$\small\mathbf{y}\textsubscript{EU28, 20-24 male}=$
$\tiny=\left[\text{Neop}:0.093,\text{Nerv}:0.046,\text{Circ}:0.054,\text{Resp}:0.022,\text{Acc}:0.387,\text{Suic}:0.218,\text{Oth}:0.180\right].$

From this description we clearly see that the two leading causes of death in
the male population in the age group from 20 to 24 years are accidents
($38.7\%$) and suicides ($21.8\%$).

Given that for a symbolic variable $Y_{j}$ the categories are fixed, we fix
their order and omit them from the modal value list—the category is determined
by the value’s position
$\mathbf{y}_{ij}=\left[y_{ij1},\ldots,y_{ijm_{j}}\right]$. For the example
above, we have:

$\small\mathbf{y}\textsubscript{EU28, 20-24
male}=[0.093,0.046,0.054,0.022,0.387,0.218,0.180]$.

Note that in our case we considered the following constraint
$\sum_{\ell=1}^{m_{j}}y_{ij\ell}=1$ for all symbolic objects
$s_{i},i=1,...,n$, and all symbolic variables $Y_{j},j=1,...,p$. If we do not
include any additional information (e.g., weights representing levels, such as
the number of deaths), then our symbolic variables for each age-gender
combination represent compositions [20]. Each of these compositions represents
the mortality pattern across causes of death, which are related to the
specific age-gender combination.

####  2.1.2 Selection of weights

Age-gender-specific rates were obtained from the Causes of Death – Crude Death
Rate table by NUTS 2 regions of residence, 3-year average [12]. We intended to
compare mortality rates across EU countries. However, each EU country has its
own age-gender structure. Thus, we had to convert these rates. We used a
standard population for this purpose.

Unfortunately, the ESP (European Standard Population) that is used by Eurostat
is only one-dimensional; that is, only for age groups, a breakdown by gender
is unavailable [13] (p. 15). Therefore, we used the standard population from
the Federal Health Monitoring Information System website [16] as a basis and
recalculated the separate male and female age distribution of the new European
Standard Population into a two-dimensional standard population by age and
gender.

The calculation follows the basic idea of the age-adjusted death rate, which
is based on an age-gender-specific death rate (i.e., the number of deaths per
100,000 people in the age-gender group relative to the standard population).
We define weight $w_{xj}$ as the expected number of deaths in the age-gender
group $j$ in country $X$ by

$w_{xj}=\frac{D_{xj}}{P_{xj}}\cdot P_{std,j},$ (1)

where $P_{std,j}$ denotes the number of persons in the age-gender group $j$ in
the standard population, $D_{xj}$ is the number of deaths in the age-gender
group $j$ in country $X$, and $P_{xj}$ the number of persons in the age-gender
group $j$ in the country $X$.

A description of mortality data for the entire young population in the EU is
given in Appendix 1. We easily see the differences in weight by age and
gender, as well as different mortality patterns in terms of causes of death in
different age-gender combinations.

##  3 Adapted clustering methods

We used the clustering method from [17] to obtain clusters of EU countries
with similar patterns of the main causes of death in the young population
based on the presented data description, which we adjusted with our data
constraints. For this purpose, we adapted the R package clamix to our specific
problem.

In the description of a general model for the leader method [17], where the
set of feasible clusterings is _a set of partitions_ of the finite set of
units $\mathbf{S}$ in $k$ clusters, the criterion function is the sum of all
cluster ”errors”. The ”error” of a cluster $\text{err}(C)$ is the sum of the
dissimilarities of its units with respect to the optimal representative, which
is called the leader $R_{C}$ of the cluster:

$\text{Err}(P)=\sum_{C\in P}\text{err}(C),\textrm{where }P\textrm{ is a
partition and }\text{err}(C)=\sum_{X\in C}d(X,R_{C}).$

Let the description of the unit $X$ (representing the symbolic object $s$) for
the symbolic variable $Y_{j},j=1,...,p,$ be a pair $(w_{xj},\mathbf{x}_{j})$,
where
$\mathbf{x}_{j}=\left[L_{j1}:x_{j1},\ldots,L_{jm_{j}}:x_{jm_{j}}\right]$,
$\sum_{\ell=1}^{m_{j}}x_{j\ell}=1$. If the order of categorical values is
fixed, then it can be simplified to
$\mathbf{x}_{j}=\left[x_{j1},\ldots,x_{jm_{j}}\right]$. The weights $w_{xj}$
are related to the unit $X$ and the $j$-th variable.

Compared to the more general formulation in [17], here we have an additional
condition for each symbolic variable; that is, the sum of the values equals 1.

For each variable $Y_{j},j=1,...,p,$ we assume that the leader $R$ is
represented by the list of non-negative vectors of size $m_{j}$:
$\mathbf{r}_{j}=\left[r_{j1},\ldots,r_{jm_{j}}\right],\mbox{ where }0\leq
r_{j\ell}\leq 1\mbox{ for all }\ell,\ell=1,...,m_{j},\mbox{ and
(again)}\sum_{\ell=1}^{m_{j}}r_{j\ell}=1.$ We denote the set of all possible
representatives by ${\cal R}$. The representation space is due to the
constraints limited to the ${\cal
R}=[0,1]^{m_{1}}\times[0,1]^{m_{2}}\times\dots\times[0,1]^{m_{p}}$.

We choose the squared Euclidean distance for the basic dissimilarity because
it allows for easy computation of the leaders while emphasizing the largest
value and also because there are zeros in some categories in some EU countries
that we wanted to retain. Thus, in our case, the dissimilarity measure between
the unit $X$ and the cluster representative $R$ is defined as

$d(X,R)=\sum_{j=1}^{p}w_{xj}d_{E}^{2}(\mathbf{x}_{j},\mathbf{r}_{j})=\sum_{j=1}^{p}w_{xj}\sum_{\ell=1}^{m_{j}}||x_{j\ell}-r_{j\ell}||^{2},\quad
w_{xj}\geq 0,$

where $d_{E}^{2}(\mathbf{x}_{j},\mathbf{r}_{j})$ is the squared Euclidean
distance and $w_{xj}$ are weights with respect to the description of the unit
$X$ by the $j$-th symbolic variable.

The new leader $R_{C}$ of the cluster $C$ is determined with the optimization
with constraints

$R_{C}=\mathop{\rm arg\ min}\nolimits_{R}\sum_{X\in
C}d(X,R)=\big{[}\mathop{\rm arg\ min}\nolimits_{\mathbf{r}_{j}}\sum_{X\in
C}w_{xj}d_{E}^{2}(\mathbf{x}_{j},\mathbf{r}_{j})\big{]}_{j=1}^{p}.$

The optimal representative, which is the leader of the cluster $C$, is
described by the symbolic variable $Y_{j}$, which is described by the vector
$\mathbf{r}_{j}=[r_{j1},\ldots,r_{jm_{j}}]$,
where

$r_{j\ell}=\frac{1}{\sum_{X\in C}w_{xj}}\sum_{X\in
C}w_{xj}x_{j\ell}\mbox{\;\;\;for all \;\;}\ell,\ell=1,...,m_{j}.$

Thus, the components are the weighted average of $\ell$-th components
$x_{j\ell}$ over all units $X$ in cluster $C$. Its derivation can be found in
Appendix 2.

The dissimilarity in the agglomerative hierarchical clustering compatible with
the criterion function $\text{Err}(P)=\sum_{C\in P}\text{err}(C)$ is [17]:

$D(C_{u},C_{v})=\text{err}(C_{u}\cup
C_{v})-\text{err}(C_{u})-\text{err}(C_{v}),$

and the dissimilarity between the merged clusters for our chosen data
description and the previously defined optimization problem with the optimal
cluster representative is

$D(C_{u},C_{v})=\frac{1}{p}\sum_{j=1}^{p}\frac{w_{uj}\cdot
w_{vj}}{w_{uj}+w_{vj}}d_{E}^{2}(\mathbf{u},\mathbf{v}),$

where $\displaystyle\mathbf{u}\mbox{ and }\mathbf{v}$ are the leaders of the
clusters $C_{u}$ and $C_{v}$, respectively,
and $\displaystyle w_{uj}=\sum_{X\in C_{u}}w_{xj},w_{vj}=\sum_{X\in
C_{v}}w_{xj}$. If we are only interested in mortality patterns, then we select
$w_{xj}=1$ for all countries $X$ and all symbolic variables
$V_{j},j=1,j=1,...,p.$

##  4 Results

In this section, we present some of the results obtained with the clustering
methods adapted to our symbolic data descriptions. In particular, we present
the results obtained with weights (see Eq. (1)). The data consist of $n=28$ EU
countries (classes) in 2016. To reduce random variation, we considered
mortality rates as 3-year averages for the number of deaths. We have $p=8$
symbolic variables corresponding to eight age-gender combinations, based on
four five-year age groups for young adults: 20-24, 25-29, 30-34, and 35-39;
$m_{j}=m=7$. The selected categories of the six leading causes of death for EU
countries for young adults (four diseases and two external causes, listed with
ICD-10 codes) are:

  * DISEASES:
Neop – Neoplasms (C00-D48),
Nerv – Diseases of the nervous system and sense organs (G00-H95),
Circ – Diseases of the circulatory system (I00-I99),
Resp – Diseases of the respiratory system (J00-J99),

  * EXTERNAL CAUSES OF DEATH:
Acc – Accidents (V01-X59, Y85, Y86), and
Suic – Suicides (Intentional self-harm (X60-X84, Y87)),

  * OTHER CAUSES
Oth – all other causes of death fall into a final category called Oth.

From the dendrogram shown in Fig. 1, we can easily identify three large
(although not very homogeneous) clusters. From GapMinder [8], we obtained and
plot under the dendrogram the following external sociodemographic factors that
could be related to mortality:

  * MD1000 – Medical doctors per 1000 people (last year in the data 2006),

  * ALC – Alcohol consumption per adult (last year in the data 2008), and

  * GOV – Government health spending of total gov spending percent (last year in the data 2010).

  * We added the number of expected deaths in the entire age group 20-39 years per 100,000 of the standard population (CDR).

The values are replaced by colours (empirical deciles), with a darker colour
representing a higher value. The clustering procedure did not include
sociodemographic factors.

Figure 1: Dendrogram of 28 EU countries based on the differences in mortality
patterns across causes of death with some sociodemographic factors displayed
at the bottom

The mortality patterns across causes of death of cluster leaders for three
main clusters are shown in Fig. 2, where we see the largest absolute
differences in neoplasms, circulatory diseases, suicides, and the other
category.

Figure 2: Mortality patterns across causes of death for eight age-gender
combinations for three main clusters. Note: The order of the categories is
fixed. The graphs should not be regarded as classic line charts – broken lines
are only used to facilitate comparison of mortality patterns.

Given that the cluster leader is described with a composition that is
considered in our study to be a mortality pattern (see explanation on page
2.1.1), it is of primary interest to compare the clusters’ patterns with the
overall mortality pattern. A comparison of the descriptions of the obtained
clusters and the description of the whole data set may be measured with an
index comparing each symbolic variable $Y_{j}$ in each cluster $C$. Batagelj
et al. [4] defined a version of such an index, called specificity. In this
work, we change it slightly (by using squared differences instead of absolute
values) to agree with the distance used in the clustering process. We define
it as

$\mathrm{spec}(Y_{j},C)={1\over
2}\sum_{\ell=1}^{m_{j}}(r_{\mathbf{S}j\ell}-r_{Cj\ell})^{2},$ (2)

where $r_{\mathbf{S}j\ell}$ is the $\ell$-th component of the symbolic
variable $Y_{j}$ for the leader of the whole set of units $\mathbf{S}$, and
$r_{Cj\ell}$ is the $\ell$-th component of the symbolic variable $Y_{j}$ for
the leader of cluster $C$. Note that this index shares the same
characteristics as the one from [4], namely $0\leq\mathrm{spec}(Y_{j},C)\leq
1$.

To additionally identify the most characteristic components $\ell$ of the
symbolic variable $Y_{j}$ on the cluster $C$, we compute the _contrast_
indices (also proposed in [4])

$\mbox{contrast}_{j\ell}=\begin{cases}\frac{r_{Cj\ell}}{r_{\mathbf{S}j\ell}}&\mbox{if
}r_{Cj\ell}\geq r_{\mathbf{S}j\ell}\\\
-\frac{r_{\mathbf{S}j\ell}}{r_{Cj\ell}}&\mbox{otherwise}\end{cases},\quad\ell=1,\ldots,m_{j}$
(3)

and select those with the highest absolute values. By definition,
$|\mbox{contrast}_{j\ell}|\geq 1$. A value of $|\mbox{contrast}_{j\ell}|$
close to 1 suggests that this component has almost the same value/probability
in cluster $C$ as in the whole set $\mathbf{S}$ (overall value). The higher
the value, the greater the contrast between the values of the cluster leader
and the representative of all countries.

The calculated contrasts and specificities for the three main clusters are
given in Table 1, with the contrasts with values at least 1.25 highlighted. We
see that the highest contrasts occur for suicides, with the second cluster
(from left-hand to right-hand in Fig. 1) having values below the overall value
in all age-gender combinations, while in the first cluster all contrasts are
above the overall value. We also observe that deaths due to respiratory
diseases are more pronounced in the second cluster, especially in the youngest
age group, while contrasts for respiratory disease in the other two clusters
are below the overall value in all age-gender combinations. Additionally,
contrasts for deaths due to circulatory disease are all above the overall
value with a value of at least 1.25 in the second cluster; while there seem to
be less neoplasms in cluster three.

Additionally, we explored possible relationships between mortality patterns in
the identified clusters and external sociodemographic indicators (which are
displayed at the bottom of the dendrogram from Fig. 1). Comparisons of values
between groups with classical one-way ANOVA and post-hoc multiple comparison
Bonferroni tests show the most significant differences between the first and
the third group (from left-hand to right-hand in Fig. 1) in the number of
expected deaths in the entire age group 20-39 years per 100,000 of the
standard population (CDR $\mbox{mean difference }=-14.04,p<0.01$). Note that
this variable is not completely independent from the clustering variables. A
significant difference is also found between the first two clusters in the
proportion of government spending in health (GOV $\mbox{mean difference
}=3.34,p<0.1$).

Table 1: Specificities of eight selected age-gender combinations and cause-of-death contrasts for selected three clusters of EU countries variable | specificity | Neop | Nerv | Circ | Resp | Acc | Suic | Oth
---|---|---|---|---|---|---|---|---
CLUSTER 1: |  |  |  |  |  |  |  |
F.Y20-24 | 0.14 | -1.10 | -1.01 | -1.13 | -1.46 | -1.04 | 1.32 | -1.00
F.Y25-29 | 0.13 | 1.07 | 1.09 | -1.17 | -1.03 | -1.02 | 1.22 | -1.14
F.Y30-34 | 0.21 | 1.11 | 1.04 | -1.26 | -1.19 | -1.10 | 1.34 | -1.14
F.Y35-39 | 0.16 | 1.08 | -1.02 | -1.17 | -1.21 | -1.02 | 1.33 | -1.16
M.Y20-24 | 0.09 | -1.01 | 1.10 | 1.02 | -1.29 | -1.03 | 1.14 | -1.17
M.Y25-29 | 0.17 | 1.12 | 1.16 | -1.26 | -1.27 | -1.02 | 1.19 | -1.19
M.Y30-34 | 0.26 | 1.25 | 1.22 | -1.21 | -1.34 | -1.00 | 1.23 | -1.24
M.Y35-39 | 0.30 | 1.24 | 1.21 | -1.18 | -1.35 | 1.01 | 1.30 | -1.23
CLUSTER 2: |  |  |  |  |  |  |  |
F.Y20-24 | 0.43 | 1.23 | -1.14 | 1.26 | 1.76 | 1.00 | -2.02 | -1.01
F.Y25-29 | 0.23 | 1.13 | 1.08 | 1.27 | 1.38 | -1.13 | -1.59 | -1.00
F.Y30-34 | 0.29 | 1.11 | 1.15 | 1.31 | 1.47 | -1.15 | -1.79 | -1.11
F.Y35-39 | 0.26 | 1.05 | -1.01 | 1.33 | 1.54 | -1.04 | -2.13 | -1.15
M.Y20-24 | 0.69 | 1.38 | 1.13 | 1.25 | 2.02 | 1.10 | -1.75 | -1.06
M.Y25-29 | 0.65 | 1.31 | -1.11 | 1.36 | 1.93 | 1.09 | -1.77 | -1.03
M.Y30-34 | 0.37 | 1.19 | -1.10 | 1.37 | 1.68 | -1.02 | -1.57 | -1.06
M.Y35-39 | 0.45 | 1.21 | -1.14 | 1.33 | 1.57 | -1.06 | -1.72 | -1.11
CLUSTER 3: |  |  |  |  |  |  |  |
F.Y20-24 | 0.04 | -1.03 | 1.11 | -1.01 | -1.07 | 1.06 | -1.15 | 1.01
F.Y25-29 | 0.26 | -1.24 | -1.21 | 1.02 | -1.27 | 1.11 | -1.08 | 1.18
F.Y30-34 | 0.47 | -1.29 | -1.18 | 1.09 | -1.10 | 1.20 | -1.21 | 1.23
F.Y35-39 | 0.40 | -1.18 | 1.04 | -1.03 | -1.15 | 1.06 | -1.10 | 1.28
M.Y20-24 | 0.13 | -1.31 | -1.27 | -1.23 | -1.63 | -1.03 | 1.10 | 1.22
M.Y25-29 | 0.12 | -1.44 | -1.15 | 1.04 | -1.37 | -1.03 | 1.02 | 1.19
M.Y30-34 | 0.22 | -1.52 | -1.21 | -1.01 | -1.10 | 1.01 | -1.05 | 1.22
M.Y35-39 | 0.28 | -1.51 | -1.15 | -1.04 | -1.06 | 1.02 | -1.06 | 1.23

##  5 Conclusion

In this paper, we presented alternative clustering methods to identify
clusters of EU countries with similar mortality patterns in the young
population based on the more informative symbolic data description as one of
the possible approaches to analyze mortality data at the aggregate level. In
doing so, we considered more comprehensive information on the distribution of
deaths among the main causes of death by different age-gender groups. A
compositional data approach might also be suitable with this kind of data
representation and a comparison between the methods would be reasonable, but
both are out of the scope of this paper.

The main advantages of the symbolic description are that this representation
retains information at two levels for each age and gender group: the
distribution among the main causes of death and also the number of deaths
(which can be included as weights of the distributions). The symbolic
description of the data requires the analysis tools to be adapted. To this
end, we presented the adaptation of compatible leader and agglomerative
hierarchical clustering methods.

We applied clustering methods to mortality data of the young EU population and
identified the main patterns across causes of death. We presented some results
that were obtained with hierarchical clustering with weights. The main
differences between the three major clusters in different age-gender
combinations were found in different causes of death for suicides, respiratory
and circulatory diseases, and neoplasms. Some connections with the external
sociodemographic indicators were found. To confirm these associations (i.e.,
mortality patterns and government spending for health), a more detailed
quantitative and qualitative analysis should be performed.

## Acknowledgment

This work was partially supported by the Ministry of Higher Education, Science
and Technology of Slovenia, Grants P1-0294 and P3-0154.

##  Appendix A Appendix 1

Table 2: Symbolic description with eight symbolic variables for the young adults in the whole European Union (causes of deaths that do not fall in any of the first six categories are grouped under the ”Oth” category) |  Neop | Nerv | Circ | Resp | Acc | Suic | Oth | Weight
---|---|---|---|---|---|---|---|---
$\mathbf{y}\textsubscript{EU28, 20-24 male}$ |  $=[$ 0.093, | 0.046, | 0.054, | 0.022, | 0.387, | 0.218, |  0.180 $]$ | 2.305
$\mathbf{y}\textsubscript{EU28, 25-29 male}$ |  $=[$ 0.104, | 0.037, | 0.079, | 0.023, | 0.322, | 0.214, |  0.221 $]$ | 2.734
$\mathbf{y}\textsubscript{EU28, 30-34 male}$ |  $=[$ 0.127, | 0.033, | 0.118, | 0.026, | 0.255, | 0.182, |  0.259 $]$ | 3.324
$\mathbf{y}\textsubscript{EU28, 35-39 male}$ |  $=[$ 0.154, | 0.030, | 0.160, | 0.033, | 0.200, | 0.145, |  0.278 $]$ | 4.358
$\mathbf{y}\textsubscript{EU28, 20-24 female}$ |  $=[$ 0.168, | 0.066, | 0.082, | 0.040, | 0.245, | 0.146, |  0.252 $]$ | 0.757
$\mathbf{y}\textsubscript{EU28, 25-29 female}$ |  $=[$ 0.241, | 0.055, | 0.090, | 0.038, | 0.163, | 0.128, |  0.284 $]$ | 0.961
$\mathbf{y}\textsubscript{EU28, 30-34 female}$ |  $=[$ 0.328, | 0.043, | 0.110, | 0.036, | 0.123, | 0.102, |  0.259 $]$ | 1.310
$\mathbf{y}\textsubscript{EU28, 35-39 female}$ |  $=[$ 0.411, | 0.038, | 0.122, | 0.033, | 0.087, | 0.074, |  0.235 $]$ | 1.956

##  Appendix B Appendix 2

For the derivation of the leader (i.e., the optimal cluster’s representative
to the constrained problem), we assume that the symbolic variables are
independent so that we can optimize separately for each symbolic variable
$Y_{j},j=1,...,p,$ and therefore omit the index $j$ from the derivations. We
want to find such a vector with non-negative components
$\mathbf{r}=[r_{1},\ldots,r_{m}]$ with the constraint
$\sum_{\ell=1}^{m}r_{\ell}=1$, which minimizes the following function
$F(\mathbf{r})=\sum_{X\in C}w_{x}\sum_{\ell=1}^{m}(x_{\ell}-r_{\ell})^{2}$,
where $\sum_{\ell=1}^{m}x_{\ell}=1\mbox{ for all units }X$. Using the Lagrange
multiplier method, we have to solve the following optimization problem:
$\min_{\mathbf{r}}\left[F(\mathbf{r})-\lambda\sum_{\ell=1}^{m}r_{\ell}\right].$
With the partial derivation to $r_{\ell}$, we get

$\sum_{X\in C}w_{x}2(x_{\ell}-r_{\ell})(-1)-\lambda=0\mbox{ i.e.,
}r_{\ell}=\frac{\sum_{X\in C}w_{x}x_{\ell}}{\sum_{X\in
C}w_{x}}+\frac{\lambda}{2\sum_{X\in C}w_{x}}$

for each $\ell,\ell=1,...,m$. Considering the constraint
$\sum_{\ell=1}^{m}r_{\ell}=1$, we get $\lambda=0$ and thus we obtain for the
$\ell$-th component of the optimal representation of the cluster $C$ with
$\displaystyle r_{\ell}=\frac{1}{\sum_{X\in C}w_{x}}\sum_{X\in
C}w_{x}x_{\ell}.$

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# Adiabatic demagnetization refrigeration to mK temperatures with the
distorted square lattice magnet NaYbGeO4

U. Arjun1,2<EMAIL_ADDRESS>K. M. Ranjith3 A. Jesche2 F. Hirschberger2 D. D.
Sarma1 P. Gegenwart2<EMAIL_ADDRESS>1Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore-560012,
India  2Experimental Physics VI, Center for Electronic Correlations and
Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg,
Germany  3Laboratoire National des Champs Magnétiques Intenses-EMFL, CNRS,
Université Grenoble Alpes, 38042 Grenoble, France

###### Abstract

We report the synthesis, characterization, low-temperature magnetic, and
thermodynamic measurements of the novel milli-Kelvin adiabatic demagnetization
refrigeration (mK-ADR) candidate material NaYbGeO4 which exhibits a distorted
square lattice arrangement of YbO6 magnetic units. Magnetization and specific
heat indicate weakly interacting effective spin-1/2 moments below 10 K, with a
Curie-Weiss temperature of only 15 mK, that can be polarized by magnetic
fields of order 1 T. For the ADR performance test, we start the
demagnetization from 5 T at a temperature of $\sim 2$ K and reach a minimum
temperature of 150 mK at zero field. The warming curve indicates a sharp
magnetic transition in the heat capacity at 210 mK, implying only weak
magnetic frustration. The entropy density of $S_{\rm GS}\simeq 101$ mJ K-1cm-3
and hold time below 2 K of 220 min are competitive while the minimal
temperature is higher compared to frustrated Ytterbium-oxide ADR materials
studied under similar conditions.

††preprint: APS/123-QED

##  I Introduction

Ultra-low temperatures below 1 K are often prerequisites for the occurrence of
quantum phenomena in condensed matter and, respectively, key for modern
quantum technologies. The global helium shortage [1, 2, 3, 4, 5] and the
rising need for milli-Kelvin (mK) refrigeration lead to a resurrection of
adiabatic demagnetization refrigeration (ADR), a long-established helium-free
cooling technique [6, 7, 8]. ADR cooling has the advantage of having less
environmental impact and saving almost one-third of energy [9], and it can be
used in various applications such as quantum computation [10], space
technologies [11, 12, 13, 14, 15], etc. For mK-ADR applications, materials
with a significant entropy change (in the temperature range of 20 mK to 5 K)
and a strong magneto-caloric effect (MCE) [16] (in a minimal applied magnetic
field) are preferred. In ADR, the coolant material is pre-cooled in a magnetic
field of a few teslas and then adiabatically demagnetized. Due to the low
entropy of the material after pre-cooling in the field, adiabatic
demagnetization requires a further temperature reduction to mK. Recent
advances in continuous mK-ADR address the ineffectiveness of ADR compared to
3He-$4$He dilution refrigeration for continuous refrigeration [12, 17].
Commercial continuous refrigerators based on ADR are already available on the
market, suggesting their potential as a significant refrigeration advancement
[18].

For cooling down to mK, hydrated paramagnetic salts such as
CrK(SO4)2$\cdot$12H2O (CPA), which orders at $30$ mK, are typically used in
commercial ADR setups [19]. However, the necessary dilution of magnetic
moments in hydrated paramagnetic salts drastically reduces the volumetric
cooling power. Conversely, materials with higher concentrations of magnetic
ions may exhibit enhanced exchange couplings, leading to magnetic order. The
presence of magnetic ordering hinders further adiabatic refrigeration, as it
reduces entropy towards zero and sets a temperature limit for ADR [20]. While
an ideal refrigerant would be a perfect paramagnet with maximum entropy at
zero-field around 0 K, actual paramagnetic substances inevitably have weak
interactions that affect their refrigeration performance by limiting the
minimal ADR temperatures. In addition, hydrated organic paramagnetic salts
need vacuum-tight encapsulation due to high water vapor pressure and there is
a limitation in properly baking them above $100~{}^{\circ}$C for UHV uses.
Additionally, they have very poor intrinsic thermal conductivity, which
requires the elaborate incorporation of metal wires into the encapsulated ADR
pills [21].

To overcome the disadvantages of hydrated paramagnetic salts for mK-ADR,
different other material classes were explored. These include frustrated [22,
23, 24, 20, 25, 26, 27, 28], quantum critical [29, 30] and molecular magnets
[31, 32], as well as metallic rare-earth compounds [33, 34]. Among them,
geometrically frustrated Ytterbium-oxides like KBaYb(BO3)2 [20] and
(Na/K)YbP2O7 [28], combine several advantages: minimal ADR temperatures well
below 50 mK, in combination with larger volumetric entropy density compared to
hydrated paramagnetic salts, and, most importantly, chemical stability. ADR
pellets based on these materials can easily be prepared by pressing
polycrystals with fine silver powder (to ensure optimal thermal contact at low
temperatures) and upscaled to the kilogram range.

As a result of spin orbit coupling and crystal electric field (CEF) splitting
of the Yb3+ states in an octahedral environment, a Kramers doublet with
effective spin 1/2 moments arises in these materials at low temperatures. The
effect of geometrical frustration in triangular rare-earth magnets has
recently been investigated in YbMgGaO4 and AYb$X_{2}$ ($A$=alkali metal, $X=$
Se, S, O) [35, 36, 37, 38, 39, 40, 41, 42]. Absence of long-range order
together with broad excitation continua was related to the effect of
geometrical frustration. In addition, the structural randomness of Mg2+ and
Ga3+ in the former material gives rise to broad CEF splittings and broad
distributions of the g-factors and exchange interactions. The structural
randomness is therefore detrimental to long-range order and could mimic spin
liquid behavior [43]. KBaYb(BO3)2, with 40% larger Yb-Yb distance compared to
YbMgGaO4 and no direct Yb-O-Yb links, features similar structural randomness
of K+ and Ba2+ atoms, occupying the same position in the crystal lattice. This
randomness could, together with frustration, further enhance the ADR
performance [20]. For isostructural KBaGd(BO3)2, which orders magnetically at
0.26 K, a minimal temperature of 0.12 K was obtained [27]. Indeed the phase
transition in this material is strongly broadened and of triangular instead of
$\lambda$-type shape, implying that the entropy release has been shifted by
structural randomness towards much lower temperatures. More generally, the
previous results motivate the investigation of other than triangular lattices
without structural randomness with respect to their mK-ADR performance.

In the spin-1/2 2D frustrated square lattice (FSL) model or $J_{1}$-$J_{2}$
model, frustration may arise due to competing nearest-neighbor $J_{1}$ (along
the edge) and next-nearest-neighbor $J_{2}$ (along the diagonal) interactions
[44, 45, 46, 47, 48, 49]. If both $J_{1}$ and $J_{2}$ are antiferromagnetic
(AF), frustration arises, leading to interesting phases. The spin-1/2 FSL
exhibits a rich theoretical phase diagram with various ground states depending
on $J_{1}$ and $J_{2}$ or the frustration parameter $\alpha=J_{1}/J_{2}$ [49,
50]. The model exhibits classical ground states characterized by ferromagnetic
(FM) order, Néel AF order, and columnar AF order. At the boundary between the
two latter a QSL state has been predicted for $J_{2}/J_{1}\simeq 0.4$ to
$0.6$. Furthermore, a spin nematic phase is predicted between the FM and
columnar AF phase [49, 50]. Although there are several spin-1/2 FSL candidate
systems based on transition metals that fit well within the theoretical phase
diagram [49, 50], there are yet no good examples of rare-earth-based spin-1/2
FSL candidate systems.

In this paper, we report a thorough investigation down to mK temperatures on
NaYbGeO4, with effective 1/2 spins on a structurally ordered distorted square
lattice. We do not find evidence for magnetic frustration. Instead the
material displays a sharp $\lambda$-type phase transition in the heat capacity
at 0.21 K. Indeed the minimal ADR temperature after demagnetization to zero
field under similar conditions to previous studies on KBaGd(BO3)2 [27] and
(Na/K)YbP2O7 [28] is only 0.15 K, while the warm-up time is enhanced compared
to that of the two latter systems, as could also be expected from the $\sim
60\%$ higher entropy density. Altogether, the comparison supports the previous
conjecture, that magnetic frustration and randomness are beneficial for
optimized ADR performance.

Figure 1: Crystal structure of NaYbGeO4. The YbO6 octahedra are linked via
GeO4 tetrahedra and span a distorted square lattice grid (left panel). Right
panel shows the possible connections between the square lattice grids.
Different Yb-Yb distances are color-coded.

##  II Methods

Synthesis: Polycrystalline samples of NaYbGeO4 were synthesized by solid-state
reaction, annealing the stoichiometric mixture of Na2CO3 (99.99%), Yb2O3
(99.99%), and GeO2 (99.99%) in an alumina boat at $1000$ ∘C for NaYbGeO4 for a
duration of $48$ h with one intermediate grinding and pelletization.

Powder X-Ray diffraction: The phase purity of the samples was confirmed by
powder x-ray diffraction (XRD, PANalytical powder diffractometer with Cu
$K_{\alpha}$ radiation, $\lambda_{\rm ave}=1.54182$ Å) at room temperature.
Rietveld refinement of the observed XRD patterns was performed using the
`FullProf` package [51] (see Figure 2), taking the initial parameters from
Ref. 52.

DC magnetization: DC magnetization ($M$) was measured as a function of
temperature $T$ and applied magnetic field $H$ using the SQUID magnetometer
with He-3 option [MPMS-3, Quantum Design].

Specific heat: Specific heat ($C_{\rm p}(T)$), was measured using the heat
capacity option of the PPMS (Quantum Design). For the low temperature ($0.4$ K
$\leq T\leq 2.2$ K) $C_{\rm p}$ measurements, the 3He probe (Quantum Design)
was utilized in the PPMS. Furthermore, we calculated the heat capacity below
0.4 K from the warming curve in the ADR experiment (see below).

To achieve strong thermal coupling, pellets made with NaYbGeO4 (grain diameter
$10-50~{}\mu$m) and fine silver powders (grain diameter $1~{}\mu$m) in a mass
ratio of $1:1$ were used for specific heat measurements. To obtain the
specific heat of the sample, the contribution of silver was subtracted from
the raw data. The specific heat $C_{\rm p}$ in a magnetic insulator has
significant contributions from phonons ($C_{\rm ph}$) and the magnetic moments
($C_{\rm m}$). At high temperatures, $C_{\rm p}(T)$ is entirely dominated by
$C_{\rm ph}$, while at low temperatures, it is dominated by $C_{\rm m}$. For
estimating the $C_{\rm ph}$, the zero-field data were fitted by a polynomial
function $C_{\rm ph}(T)=aT^{3}+bT^{5}+cT^{7}$ above $10$ K, where $a$, $b$,
and $c$ are appropriate constants [53]. Similar procedures have been adopted
earlier and proven to be efficient in cases where heat capacity data of non-
magnetic analogue compounds are unavailable [54, 55, 56, 57]. The fit was
extrapolated down to low temperatures, and $C_{\rm m}$ was obtained by
subtracting the fitted data from the experimental $C_{\rm p}$ data. The
magnetic entropy was estimated as $S_{\rm m}(T)=\int_{0.4~{}K}^{T}\frac{C_{\rm
m}(T^{\prime})}{T^{\prime}}dT^{\prime}$.

Adiabatic Demagnetization Refrigeration (ADR): A home-made ADR setup for the
PPMS was utilized [20, 27, 28]. A cylindrical pellet made from equal weights
of NaYbGeO4 (grain diameter $10–50~{}\mu$m) and silver powder (grain diameter
$1~{}\mu$m) was pressed weighing approximately $3.5$ g, with $15$ mm diameter
and $6$ mm thickness. The pellet was mounted on a plastic straw and thermally
isolated from the heat bath. A RuO2 thermometer was attached to the pellet and
connected through thin resistive manganin wires to minimize the heat flow. The
resistor was measured with a current of $1$ nA utilizing a Lake-Shore $372$ AC
Bridge. To minimize the effects of surrounding thermal radiation, a metallic
cap was utilized as a shield. The sample was cooled to $T\approx 2$ K at $5$ T
and then the high-vacuum mode (pressure $<10^{-4}$ mbar) was applied to
achieve thermal decoupling. Subsequently, the magnetic field was swept to zero
at a rate of $0.15$ T min-1. The pellet reached its lowest temperature and
then slowly warmed back to $2$ K via slow heat flow from the bath.

##  III Results

###  III.1 Structure

The compound NaYbGeO4 crystallizes in the orthorhombic crystal structure with
space group $Pnma$ (No. 62). The crystal structure of NaYbGeO4 contains a
distorted diagonally connected square lattice arrangement of YbO6 octahedra
carrying the magnetic moments connected through GeO4 tetrahedra along the
$bc$-plane (see left panel of Fig. 1). The possible interaction pathways
between these square lattice layers are shown in right panel of Fig. 1. From
the structure, the nearest neighbor exchange is through corner-shared YbO6
octahedra, while the second neighbor exchange is through GeO4 tetrahedra. Both
distances are smaller than $5.5$ Å, beyond which typically dipolar coupling
dominates in other Ytterbium-oxides [27].

###  III.2 Powder X-Ray diffraction

Figure 2: Powder x-ray diffraction pattern (open red circles) for NaYbGeO4 at
room temperatures. The solid line represents the Rietveld refinement, with the
vertical bars showing the expected Bragg peak positions and the lower solid
blue line representing the difference between observed and calculated
intensities.

Figure 2 shows the powder XRD pattern of NaYbGeO4 at room temperature, along
with the Rietveld refinement. The refinement is done using the orthorhombic
space group $Pnma$ (No. 62), taking initial parameters from Ref. 52. The
goodness of the fit is $\chi^{2}=4.19$. The obtained lattice parameters are
$a=11.2633(5)$ Å, $b=6.3756(3)$ Å, $c=5.2344(2)$ Å, and $V_{\rm cell}\simeq
375.89$ Å3. These values are in close agreement with the reported values [52].

###  III.3 Magnetization

Figure 3: Inverse magnetic susceptibility $1/\chi$ vs. temperature of NaYbGeO4
at a field of $1$ T. Dashed and solid lines represent the fits by Eq. 1 and
Eq. 2, respectively. Figure 4: Isothermal magnetization $M(H)$ of NaYbGeO4 at
a temperature of $0.4$ K. Dashed line represents linear contribution (see
text). The inset shows the low-temperature inverse magnetic susceptibility
after subtracting $\chi_{\rm 0}$ along with the CW fit.

Fig. 3 displays the inverse magnetic susceptibility $\chi(T)^{-1}=H/M$ of
NaYbGeO4 at 1 T field together with an extended Curie-Weiss (CW) fit according
to

$\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm CW}},$ (1)

which describes very well the high-$T$ region between 200 and 300 K. Here,
$\chi_{0}$ is the $T$-independent contribution consisting of the diamagnetic
susceptibility ($\chi_{\rm core}$) of core electron shells and the Van-Vleck
paramagnetic susceptibility ($\chi_{\rm VV}$) of the open shells of the Yb3+
ions. The second term in Eq. 1 is the CW law with the CW temperature
$\theta_{\rm CW}$ and Curie constant $C=N_{\rm A}\mu_{\rm eff}^{2}/3k_{\rm
B}$. Here, $N_{\rm A}$ is Avogadro’s number, $\mu_{\rm
eff}=g\sqrt{J(J+1)}$$\mu_{\rm B}$ is the effective magnetic moment, $g$ is the
Land$\acute{\rm e}$ $g$-factor, $\mu_{\rm B}$ is the Bohr magneton, and $S$ is
the spin quantum number. The fitting parameters are $\chi_{0}^{\rm HT}\simeq
4.34\times 10^{-4}$ cm3/mol, $C_{\rm HT}\simeq 2.19$ cm3K/mol, and
$\theta_{\rm CW}^{\rm HT}\simeq-47.1$ K. From the value of $C_{\rm HT}$, the
effective moment is obtained to $\mu_{\rm eff}^{\rm HT}\simeq 4.19\mu_{\rm
B}$, in good agreement with the expected spin-only value of $4.54$ $\mu_{\rm
B}$ for Yb3+ ($J=7/2$, $g=8/7$) ion in the $4f^{13}$ configuration.The large
value of the CW temperature arises from the CEF splitting. This is evident
from the curvature in $\chi^{-1}(T)$ below $200$ K leading to a change in
slope below 50 K, as excited CEF levels become thermally depopulated. Such
behavior is typical for Yb3+-based spin systems where CEF splits the $J=7/2$
multiplet into four Kramers doublets. Our entropy analysis below confirms the
presence of a Kramers doublet ground state at temperatures below $10$ K. Fig.
3 also includes a two-level fit [58] (cf. the red solid line) according to

$1/\chi(T)=8(T-\theta_{\rm CW})\left(\frac{\mu_{\rm eff,1}^{2}+\mu_{\rm
eff,2}^{2}\cdot e^{(-\Delta/T)}}{1+e^{(-\Delta/T)}}\right)^{-1},$ (2)

where $\mu_{\rm eff,1}$ and $\mu_{\rm eff,2}$ correspond to energy levels
separated by an energy gap $\Delta$. The fit down to $25$ K yields
$\Delta\simeq 282$ K, $\mu_{\rm eff,1}\simeq 3.57~{}\mu_{\rm B}$, $\mu_{\rm
eff,2}\simeq 5.11~{}\mu_{\rm B}$, and $\theta_{\rm CW}\simeq-6.1$ K. The
deviation of the fit from the data below $30$ K indicates that the obtained
$\mu_{\rm eff,1}$ and $\theta_{\rm CW}$ do not represent the lowest Kramers
doublet. Using a fitting approach that incorporates four CEF levels and their
associated effective moments is challenging due to the excessive number of
adjustable parameters. Hence, we analyzed the magnetic susceptibility below
$2$ K separately using Eq. 1 to determine the properties associated with the
Kramers doublet ground state.

The magnetization isotherm $M(H)$ of NaYbGeO4 measured up to 7 T at $T=0.4$ K
shows a saturation at around $1.5$ T (Fig. 4). The saturation value $M_{\rm
sat}$ is estimated by fitting the high-field region above $5$ T with a
straight line reflecting $\chi_{0}H$ and extrapolating the line back to zero
field. This yields $\chi_{0}\simeq 5.58\times 10^{-3}$ cm3/mol and $M_{\rm
sat}\simeq 1.92~{}\mu_{\rm B}$, giving $g_{\rm eff}\simeq 3.84$ from $\mu_{\rm
sat}=g_{\rm eff}S_{\rm eff}~{}\mu_{\rm B}$.

After subtracting $\chi_{0}$, the $\chi(T)$ data below $2$ K are fitted by the
CW law $\chi(T)=C/(T-\theta_{\rm CW})$. The fitting yields $\theta^{\rm
LT}_{\rm CW}\simeq(15\pm 2)$ mK and $\mu_{\rm eff}^{\rm LT}\simeq
3.32~{}\mu_{\rm B}$. The obtained effective moment is in good agreement with
$\mu_{\rm eff}=g_{\rm eff}\sqrt{S_{\rm eff}(S_{\rm eff}+1)}~{}\mu_{\rm B}$ for
a pseudo spin-$1/2$ ground state with $g_{\rm eff}$ value of $\sim 3.84$,
consistent with the value estimated from $M_{\rm sat}$. The CW temperature
depends on the sum of all exchange interactions and thus the balance of AF and
FM contributions could result in a very small value of $\theta^{\rm LT}_{\rm
CW}$. Analysis of ADR curves, detailed later, evidences AF order at 210 mK,
which proves that the extremely small value of $\theta_{\rm{CW}}$ must arise
from the sum of AF and FM contributions. Note that a combination of nearest
neighbor AF and next-nearest neighbor FM interactions on the square lattice
excludes significant frustration.

###  III.4 Specific heat

Figure 5: Low-temperature magnetic specific heat $(C_{\rm m})$ of NaYbGeO4 at
several external magnetic fields (upper panel). Phonon contribution was
subtracted from the raw data using a polynomial fit [53]. The lower panel
displays the magnetic entropy $S_{\rm m}(T)$ obtained by integrating $C_{\rm
m}/T$ over temperature. For the $0$ and $1$ T data the integration constant
was adjusted such that a value of $R\ln 2$ (thin black line) is reached above
10 K. Two arrows show the ADR process.

Figure 5, upper panel, illustrates the magnetic specific heat for NaYbGeO4
under various applied fields, suggesting its potential for mK-ADR. The zero-
field specific heat data show no indication of magnetic long-range ordering
above $0.4$ K. A small anomaly at 2.23 K, cf. the difference between the data
and the solid blue line, arises from $0.6\%$ Yb2O3 impurities [53]. The
finite-field data show Schottky anomalies, resulting from the splitting of the
lowest Kramers doublet in field. This is confirmed by the fact that the
entropy approaches $R\ln 2$, as displayed in the lower panel. As indicated by
the two arrows, starting the ADR at $\sim 2$ K in a field of 5 T should reveal
a minimal temperature well below 0.4 K in zero field.

###  III.5 Adiabatic Demagnetization Refrigeration (ADR)

Figure 6: Cooling of NaYbGeO4 by ADR. The red and blue lines display the
temporal evolution of temperature (left y-axis) and magnetic field (right
y-axis). After precooling in a field of 5 T the sample space was evacuated and
subsequently the magnetic field was swept from $5$ to $0$ T at a rate of
$0.15$ T min-1. The inset enlarges the regime close to zero-field, where the
temperature reaches a minimum of 135 mK at 0.1 T and then saturates at 150 mK
in zero field. The warm-up time amounts to 220 min. Figure 7: Temperature
dependence of the total (magnetic) heat capacity of the NaYbGeO4-based ADR
pellet determined from the warming curve shown in Fig 6 by calculating $C_{\rm
ADR}=\dot{Q}/\dot{T}$ (red points). The constant heat input $\dot{Q}=0.58\mu$W
was used to obtain agreement with the measured heat capacity (black points)
above 0.4 K.

The cooling performance of NaYbGeO4 in our PPMS setup described above is shown
in Fig 6. Near 0.1 T the temperature trace passes a minimum at 135 mK. This
sign change of the adiabatic magnetocaloric effect indicates the crossing of a
magnetic phase boundary. Upon further reducing the field, the temperature
increases towards 150 mK in zero field. The subsequent warming results from
the finite heat input due to non-perfect adiabatic conditions. The time
dependence of the temperature $T(t)$ displays a clear anomaly near 100
minutes. To analyze this trace, we utilize the equation

$\dot{Q}=C_{\rm ADR}\cdot\dot{T},$ (3)

where $\dot{Q}$ represents a (constant) heat input per unit time, $C_{\rm
ADR}$ denotes the (magnetic) heat capacity of the ADR pellet and $\dot{T}$ is
the time derivative of temperature during warming. Taking $\dot{Q}=0.58\mu$W,
a perfect agreement between the directly measured heat capacity and $C_{\rm
ADR}$ is found down to $0.4$ K. This heat input mainly arises from the
residual He gas in our setup and here it was slightly lower than the
$0.71\mu$W determined previously for KBaGd(BO3)2 [27]. For KBaGd(BO3)2 we also
confirmed that the calculated heat capacity exactly equals the measured heat
capacity signature of the AF transition. This provides us strong confidence
that the sharp transition in $C_{\rm ADR}$ for NaYbGeO4 distinctly signifies
sharp AF order at $T_{\rm N}\simeq 210$ mK. Note, that the sign change of the
magnetocaloric effect in the field of 0.1 T at 130 mK is compatible with the
suppression of the AF order in field.

The warm-up time of $\sim 220$ min (3 hr 40 min) is approximately 4-6 times
longer than that of KYbP2O7 ($\sim 35$ min), NaYbP2O7 ($\sim 55$ min),
KBaYb(BO3)2 ($\sim 40$ min) [28] though much less than $\sim$8 hr for
KBaGd(BO3)2 [27]. This reflects that the entropy density 101 mJ/(K cm3) of
NaYbGeO4 is roughly 60% larger than that of the former materials but only half
of that in KBaGd(BO3)2. The latter material reaches a similar (even slightly
lower) ADR end temperature of 122 mK [27] compared to 135 mK in NaYbGeO4 and
is therefore even better suitable for applications in this target range.

##  IV Discussion

Table 1: Comparison of important parameters of different mK ADR materials: $T_{\rm N}$ is the magnetic ordering temperature, $T_{\rm min}$ is the minimum temperature attained, $S_{\rm GS}$ is the entropy of the ground-state multiplet, and $R$ is the universal gas constant. The abbreviations are MAS = Mn(NH4)2(SO4)${}_{2}\cdot 6$H2O (manganese ammonium sulfate), FAA = NH4Fe(SO4)$\cdot$$12$H2O (ferric ammonium alum), CPA = KCr(SO4)$\cdot 12$H2O (chromium potassium alum), CMN = Mg3Ce2(NO3)${}_{12}\cdot 24$H2O (cerium magnesium nitrate). |  | ADR materials |  |
---|---|---|---|---
Material | $T_{\rm N}$ | $T_{\rm min}$ | $S_{\rm GS}$ |  $S_{\rm GS}$/vol.
| (mK) | (mK) |  | [mJ/(K cm3)]
MAS [59] | $170$ | $<300$ | $R\ln 6$ | $70$
FAA [59] | $30$ | $<100$ | $R\ln 6$ | $53$
CPA [19] | $10$ | $<100$ | $R\ln 4$ | $42$
CMN [60] | $2$ | $<40$ | $R\ln 2$ | $16$
KBaGd(BO3)2 [27] | $263$ | $122$ | $R\ln 8$ | $192$
YbPt2Sn12 [34] | $250$ | $<200$ | $R\ln 2$ | $124$
Yb3Ga5O12 [24] | $54$ | $<200$ | $R\ln 2$ | $124$
NaYbGeO4 | 210 | 135 | R ln 2 | 101
KBaYb(BO3)2 [20] | $8$ | $<20$ | $R\ln 2$ | $64$
NaYbP2O7 [28] | $<45$ | $<45$ | $R\ln 2$ | $64$
KYbP2O7 [28] | $<37$ | $<37$ | $R\ln 2$ | $57$

The sharp AF phase transition at 210 mK indicates that NaYbGeO4 with distorted
square-lattice arrangement of Yb3+ moments is much weaker magnetically
frustrated compared to the triangular-lattices KBaYb(BO3)2 [20] and NaYbP2O7
[28], as well as the distorted hyperhoneycomb-lattice KYbP2O7 [28], which all
show no magnetic order above 35 mK ($T_{\rm N}=8$ mK in KBaYb(BO3)2 has been
deduced by comparison with KBaGd(BO3)2 [27]). The fact that the CW temperature
in NaYbGeO4 is much smaller compared to the ordering temperature could be
attributed to AF nearest neighbor and FM second neighbor exchange couplings.
In this configuration, no frustration arises, as both first and second
neighbor interactions are consistently satisfied simultaneously. If any
frustration exists, it would likely arise from interactions perpendicular to
the plane.

A comparison of the ADR performance of NaYbGeO4 with that of other ADR
materials, in particular geometrically frustrated Ytterbium-borates and
-diphosphates is therefore interesting, as this may allow to draw conclusions
on the importance of magnetic frustration in this respect. The crucial
parameters that determine the efficiency of the various known ADR materials
are compared in Table 1. The full entropy of the ground state is described as
$S_{\rm GS}=R\ln(2J+1)$, and the entropy density $S_{\rm GS}$/vol. is
calculated by dividing $S_{\rm GS}$ by the unit cell volume. A large value of
$S_{\rm GS}$ is always beneficial because the magnetic entropy changes
($\Delta S_{\rm m}$) of magnetic refrigerants act as the driving force of ADR.
However, it is worth noting that for practical purposes it is not the molar
entropy but the volumetric entropy density $S_{\rm GS}$ of the material that
is relevant.

As can be seen from the Table. 1, materials with high entropy density such as
YbPt2Sn or Yb3Ga5O12 exhibit magnetic orderings at $250$ mK and $54$ mK,
respectively, which limit their lowest attainable temperatures as the entropy
drops below the ordering temperatures. The high spin hydrated paramagnetic
salts such as MAS and FAA, which have a larger magnetic entropy $R\ln 6$ are
also affected by their much higher magnetic ordering temperatures. On the
other hand, low transition temperature materials such as CPA and CMN have low
magnetic moment density and hence low entropy density. A high entropy density
usually contradicts a low magnetic ordering temperature.

In this context, it has been reported that the frustrated Yb-based quantum
magnets $A$YbP2O7, KBaYb(BO3)2, exhibit these two mutually exclusive criteria,
such as a high volumetric entropy density combined with a very low ordering
[28, 20], allowing ADR to well below 50 mK. In KBaYb(BO3)2 both magnetic
frustration and structural randomness contribute to suppress magnetic
ordering. These oxides have been shown to be excellent anhydrous ADR
refrigerants and under the same experimental conditions in PPMS, out of these
three compounds, KYbP2O7 is attaining the lowest temperature of $37$ mK [28].
Utilizing a better adiabatic setup in the 3He-4He dilution refrigerator with
feedback control of the bath temperature following the sample temperature,
these materials can reach even lower temperatures well below $20$ mK, which
has been experimentally demonstrated for KBaYb(BO3)2 [20].

NaYbGeO4 has a large magnetic ion density ($\sim 10.6$ nm-3) and volumetric
entropy density ($\sim 101$ mJ/(K cm3)). These values are higher than that of
$A$YbP2O7 and KBaYb(BO3)2 and this is reflected in the significantly longer
warm up time under comparable conditions. However, the end temperature of 135
mK is much higher compared to the case of the frustrated materials, indicating
the importance of frustration for obtaining low end temperatures.

Compared to conventional mK ADR coolants based on hydrated paramagnetic salts,
NaYbGeO4 is stable at high-vacuum and can be heated up to $1000~{}^{\circ}$C
making it easy to handle and suitable for UHV applications. For our studies we
prepared ADR pellets by mixing powder samples with silver powder in $1:1$
ratio for getting excellent thermal contact. The NaYbGeO4 pellets, sintered at
$1000~{}^{\circ}$C, are mechanically stronger compared to the diphosphate
pellets which we sintered only at $600~{}^{\circ}$C [28]. Thus, mechanically
stable NaYbGeO4 ADR pills could also be made with much less or even without
silver powder.

##  V Conclusion

We synthesized and studied NaYbGeO4 with effective spin 1/2 moments on a
distorted square lattice with respect to its performance in milli-Kelvin (mK)
adiabatic demagnetization refrigeration (ADR). While the Curie-Weiss
temperature of 15 mK would suggest the absence of significant magnetic
interactions, the heat capacity, derived from the warming curve after ADR
shows a sharp AF phase transition at $T_{\rm N}\simeq 210$ mK. This suggests
AF nearest neighbor and FM second neighbor exchange couplings and a non-
frustrated magnetic state. Compared to geometrically frustrated Ytterbium-
borates and -diphosphates the minimal ADR temperature of NaYbGeO4 is
significantly larger. This supports the conjecture that magnetic frustration
is beneficial for mK-ADR.

Even though NaYbGeO4 has a limitation in the lowest attainable temperature
($\sim 135$ mK) due to the presence of sharp AF ordering, it has a comparably
high entropy density and respectively shows a much longer warm-up time than
Ytterbium-borates and -diphosphates. Chemical stability upon heating up to
$1000~{}^{\circ}$C, UHV compatibility, and the easy production of mechanically
stable sintered pellets make this material an interesting alternative to
hydrated paramagnetic salts for ADR applications above 200 mK.

###### Acknowledgements.

We thank Marvin Klinger and Yoshi Tokiwa for useful discussions. UA would like
to acknowledge DST, India, for financial support bearing sanction
(DST/INSPIRE/04/2019/001664). Work supported by the German Research Foundation
through project 514162746 (GE 1640/11-1). DDS thanks SERB, DST, and CSIR,
Government of India, for financial support.

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