# Zero-point entropies of spin-jam and spin-glass states in a frustrated
magnet

C. Piyakulworawat  Department of Physics, Faculty of Science, Mahidol
University, Bangkok, 10400, Thailand  A. Thennakoon  Department of Physics,
University of Virginia, Charlottesville, Virginia, 22904, USA  J. Yang
Department of Physics, University of Virginia, Charlottesville, Virginia,
22904, USA  Department of Physics, New Jersey Institute of Technology, Newark,
New Jersey, 07102, USA  H. Yoshizawa  Neutron Science Laboratory, Institute
for Solid-State Physics, The University of Tokyo, Kashiwa, Chiba, 277-8581,
Japan  D. Ueta  Neutron Science Laboratory, Institute for Solid-State Physics,
The University of Tokyo, Kashiwa, Chiba, 277-8581, Japan  T. J Sato  Institute
of Multidisciplinary Research of Advanced Materials, Tohoku University,
Sendai, Miyagi, 980-8577, Japan  K. Shen  Center for Condensed Matter
Sciences, National Taiwan University, Taipei, 10617, Taiwan  Taiwan Consortium
of Emergent Crystalline Materials, National Science and Technology Council,
Taipei, 10622, Taiwan  W.-T. Chen  Center for Condensed Matter Sciences,
National Taiwan University, Taipei, 10617, Taiwan  Taiwan Consortium of
Emergent Crystalline Materials, National Science and Technology Council,
Taipei, 10622, Taiwan  W.-W. Pai  Center for Condensed Matter Sciences,
National Taiwan University, Taipei, 10617, Taiwan  Taiwan Consortium of
Emergent Crystalline Materials, National Science and Technology Council,
Taipei, 10622, Taiwan  K. Matan<EMAIL_ADDRESS>Department of
Physics, Faculty of Science, Mahidol University, Bangkok, 10400, Thailand
ThEP, Commission of Higher Education, Bangkok, 10400, Thailand  S.-H. Lee
<EMAIL_ADDRESS>Department of Physics, University of Virginia,
Charlottesville, Virginia, 22904, USA

###### Abstract

Thermodynamics studies of a prototypical quasi-two-dimensional frustrated
magnet Ba2Sn2ZnCr7pGa10-7pO22 where the magnetic Cr3+ ions are arranged in a
triangular network of bipyramids show that the magnetic zero-point entropy for
$p=0.98$ is 55(1)% of the entropy expected when the Cr3+ moments are fully
disordered. Furthermore, when combined with a previous neutron scattering
study and the perimeter scaling entropy of a spin jam, the analysis reveals
that with decreasing $p$, i.e., doping of the nonmagnetic Ga3+ ions, the
variation in the magnetic zero-point entropy can be well explained by the
combined effects of the zero-point entropy of the spin jam state and that of
weakly coupled orphan spins, shedding light on the coexistence of the two
types of spin states in quantum magnetism.

##  I Introduction

Zero-point entropy, i.e., the entropy at absolute zero temperature, of a
macroscopic system has been a strenuously debated topic ever since the
introduction of the third law of thermodynamics. One example of magnetic
solids that could possess a finite zero-point entropy is the so-called spin
glasses. The spin-glass state can exist in dilute magnetic alloys in which
nonmagnetic metals are doped with magnetic ions at low concentrations. These
magnetic impurities can interact with one another through the Ruderman-Kittel-
Kasuya-Yosida (RKKY) interaction. Below the spin-glass transition temperature,
the magnetic moments of impurities freeze in random directions without long-
range ordering due to the randomness of the RKKY interactions, resulting in a
finite zero-point entropy. The zero-point entropy in spin glasses has been
estimated theoretically by Edwards and Tanaka, who predicted the values for
long-range-interacting Ising and $XY$ spin glasses to be 1.66 and 4.30
Jmol-1K-1, respectively [1, 2]. Experimentally, the zero-point entropy of a
dilute dipolar-coupled Ising spin glass LiHopY1-pF4 with $p=0.167$ was
measured and found to be close to 1.66 Jmol-1K-1, consistent with the
theoretical prediction [3].

An interesting question that arises is what will happen to the zero-point
entropy if, unlike in the dilute magnetic alloys, the magnetic ions are
densely populated and strongly interact with each other. The so-called
geometrically frustrated magnets are the case in point. For example,
pyrochlore rare-earth oxides $A_{2}B_{2}$O7 which exhibit the so-called spin-
ice state at low temperatures have similar degenerate ground-state
configurations to water ice in which two spins must point inwards while the
other two point out of the tetrahedron [4]. Surprisingly, the zero-point
entropies of the Ho2Ti2O7 and Dy2Ti2O7 spin ices have been reported to exhibit
a value close to that of water ice [5, 6, 7, 8]. CuAl2O4 and CuGa2O4 spinels
with magnetic ions residing in the diamond sublattice have also been found to
manifest a finite zero-point entropy [9, 10]. Other frustrated lattices can
have local zero-energy modes in the mean-field level, i.e., the weathervane
modes in the two-dimensional kagome antiferromagnets [11, 12, 13, 14] and the
antiferromagnetic hexagon modes in the three-dimensional spinel ZnCr2O4 [15],
which can induce macroscopic ground-state degeneracy and thus a finite zero-
point entropy.

These densely populated geometrically frustrated magnets can exhibit a
magnetic glassy state at low temperatures that is called a spin jam [16].
While the canonical spin-glass state arises due to the random RKKY
interactions, the spin-jam state can arise from quantum fluctuations [16]. The
essential distinction between the two glassy states is in their energy
landscape topology. Quantum fluctuations render the energy landscape of the
spin jam to be non-hierarchical and have a flat but rugged shape. On the
contrary, for the spin glass, the energy landscape is hierarchical and has a
rugged funnel shape [17]. To date, the crossover between these glassy states
has been observed in the dynamic susceptibility measurements and the memory
effects when the spin density is varied in the systems [18, 19]. In this
letter, we report experimental evidence of the zero-point entropy in the
glassy state of a QS ferrite-derived compound Ba2Sn2ZnCr7pGa10-7pO22 (BSZCGO)
[20, 21, 22, 23], a realization of the frustrated triangular network of
bipyramids. The material can be viewed as a stacking of two types of blocks,
the nonmagnetic ‘Q’ block and the magnetic ‘S’ block, alternating with each
other. Furthermore, through our analysis, we show how the spin-jam state
crosses over to the spin-glass state as the spin density $p$ varies in terms
of the low-lying excitations and the zero-point entropy using DC magnetic
susceptibility and heat capacity measurements down to 0.5 K.

Since its discovery, SrCr9pGa12-9pO19 (SCGO), a cousin compound to BSZCGO, has
been a good model system for the triangular network of bipyramids or
pyrochlore slab [see Fig. 1(a)] [24, 25, 26, 27, 28, 29, 30, 18]. The system,
however, has triangular layers of spin dimers formed by Cr3+ spins [orange
spheres in Fig. 1(a)], residing between the pyrochlore slabs [27]. The
existence of the extra magnetic layers of dimers complicates the physics of
the pure pyrochlore slab. BSZCGO, on the other hand, does not comprise the
spin dimer layers. The crystal structure of BSZCGO is characterized by the
hexagonal system with the space group $P\bar{3}m1$ and lattice parameters $a$
= $b$ = 5.8568(1) Å and $c$ = 14.2537(3) Å for the sample with $p$ = 0.97
[31]. The magnetic $s=\frac{3}{2}$ Cr3+ ions form the pyrochlore slabs, and
the successive slabs are separated by about 10 Å, making the pyrochlore slabs
well isolated and quasi-two-dimensional [see Fig. 1(b)]. There are, however,
two types of intrinsic disorder in BSZCGO. Firstly, nonmagnetic Ga3+ ions
inevitably share $6i$ and $1a$ sites with Cr3+ ions leading to the highest
possible value of the spin density $p$ to be about 0.97 [32]. Furthermore,
Ga3+ ions also share the $2d$ site with Zn2+ ions in a 1:1 ratio which causes
structural strains and in turn renders bond disorders between Cr3+ ions [21,
33]. Despite these disorders, BSZCGO is the best model system to explore the
physics of frustration in the triangular network of bipyramids due to its
robustness against small disorders [16, 34]. BSZCGO exhibits a freezing
transition with $T_{\mathrm{f}}$ around 1.5 K for $p$ = 0.97 [20, 35]. In this
clean limit, $p\to 1$, the magnetic heat capacity $C_{\mathrm{mag}}$ has been
observed to show a $T^{2}$ dependence below $T_{\mathrm{f}}$ [20], indicative
of the unconventional glassy state.

Figure 1: Magnetic lattices of SCGO and BSZCGO. A triangular network of
bipyramids consists of two kagome layers (blue spheres) sandwiching an
intermediate triangular layer (red spheres). Bonds shown in different colors
have different lengths. (a) In SCGO, a triangular network of dimers (orange
spheres) separates the successive pyrochlore slabs. (b) In BSZCGO, successive
pyrochlore slabs are well separated, and there are no Cr3+ ions in between.
Axes represent the crystallographic axes of the lattices.

##  II Experimental details

Ten powder samples of BSZCGO with $0.44\leq p\leq 0.98$ and a nonmagnetic
sample with $p=0$ (Ba2Sn2ZnGa10O22) were prepared with standard solid-state
reactions. A stoichiometric mixture of BaCO3, SnO2, ZnO, Ga2O3, and Cr2O3 were
intimately ground and pelleted. The pellet was put in an alumina crucible and
sintered in air at 1400 °C for 48 hours with intermediate grinding. X-ray
diffraction was performed at room temperature for each sample to verify the
crystal structure and to determine Cr3+ concentration within the sample (see
Section III in [36] for details). The temperature dependence of the DC
magnetic susceptibility was measured using a commercial SQUID magnetometer
from 0.5 K up to 20 K with an applied magnetic field of 0.01 T. The
measurements were done with both field-cooled (FC) and zero-field-cooled (ZFC)
methods. Susceptibility data of samples with $p<0.67$ were not taken as these
samples have transition temperatures lower than 0.5 K. The temperature
dependence of molar heat capacity was measured with a commercial physical
property measuring system utilizing a thermal relaxation technique. Pelleted
powder samples ranging in mass from 1 to 7 mg were affixed using Apiezon
grease to a platform equipped with a heater and thermometer. The molar heat
capacity from 0.5 K to 10 K was measured with the 3He option and from 3 K to
50 K (up to room temperature for the $p=0.98$ and $p=0$ samples) with the 4He
option in a zero magnetic field (see Figs. S1 and S3 in [36] for all raw heat
capacity data). As shown in Fig. S1, above $\sim$50 K, the molar heat capacity
of the $p=0.98$ and $p=0$ samples coincides with one another indicative of
vanishing magnetic contribution to the magnetic sample above such temperature.
The magnetic heat capacity was obtained by subtracting the interpolated molar
heat capacity of the nonmagnetic ($p=0$) sample from that of the magnetic
($p\neq 0$) samples without artificially rescaling the high-temperature data.

##  III Results and discussion

###  III.1 Low-lying excitations

Figure 2(a) shows the DC magnetic susceptibility data of five samples that
exhibit the freezing transitions at $T_{\mathrm{f}}$ indicated by the
bifurcation of FC and ZFC data. As shown in Fig. 2(a) (see also Fig. 4(a) for
$T_{\mathrm{f}}$ obtained from magnetic heat capacity data), $T_{\mathrm{f}}$
is found to decrease with increasing vacancy density (decreasing $p$), which
is consistent with the spin-jam theory [16, 18]. Note that for canonical spin
glasses, the impurity dependence of $T_{\mathrm{f}}$ behaves differently;
$T_{\mathrm{f}}$ increases with increasing magnetic impurity density [37, 38,
39]. The nature of the glassy states can be studied more carefully via the
behavior of the $T$-dependent magnetic heat capacity $C_{\mathrm{mag}}$.
Figure 2(b) shows $C_{\mathrm{mag}}/T$ as a function of $T$ in the low-
temperature region of six samples with $p\geq 0.67$ (see Fig. S4 in [36] for
the low-temperature data of all samples). For $p\geq 0.93$, $C_{\mathrm{mag}}$
exhibits a clear quadratic $T^{2}$-dependence. On the other hand, for $p\leq
0.86$, $C_{\mathrm{mag}}$ begins to deviate from the quadratic behavior. To
quantitatively analyze the data, we assume that the thermodynamics of the spin
fluctuations can be characterized by two modes; one is the hydrodynamic
Halperin-Saslow (HS) mode [40] that is a characteristic of the spin-jam state
and yields $C_{\mathrm{HS}}=AT^{2}+B$ for a two-dimensional system [16], where
$B$ is a temperature-independent term [25, 41], and the other is the localized
two-level (TL) system due to spin-glass clusters generated by the non-magnetic
doping and yields $C_{\mathrm{TL}}\propto T$ [42, 18]. The coefficient $A$ of
the $T^{2}$ term in $C_{\mathrm{HS}}$ is inversely proportional to the spin
wave velocity squared $v^{2}$ for a two-dimensional system,

$\displaystyle A=\frac{9\zeta(3)k_{\mathrm{B}}^{2}V_{\mathrm{c}}R}{\pi\hbar
v^{2}d},$ (1)

where $\zeta$ is the Riemann zeta function, $k_{\mathrm{B}}$ is Boltzmann’s
constant, $V_{\mathrm{c}}$ is the unit cell volume, and $d$ is the spacing of
successive bilayers [40, 41].

Figure 2: The T dependence of DC magnetic susceptibility and magnetic heat
capacity. (a) The DC magnetic susceptibility in the temperature range covering
the freezing transition of samples with p $\geq$ 0.67. Open symbols represent
ZFC data. Arrows mark Tf for each sample. The inset shows f as a function of
p, where f is the fractional population of the spin-jam state. The dashed line
in the inset is a guiding line. (b) Cmag/T data at low temperatures. Solid
lines are best fits to the two-state model with fitting parameters summarized
in Table 1.

Since the population ratio of the spin jam to the spin glass clusters can vary
with the spin density $p$, we have fitted the magnetic heat capacity of each
sample to the following phenomenological formula,
$C_{\mathrm{mag}}=fC_{\mathrm{HS}}+(1-f)C_{\mathrm{TL}}$, where $f$ and $1-f$
are the fraction of the spin-jam state and that of the spin-glass state,
respectively. The fitting range of $T$ is from the base temperature of 0.5 K
to about $T_{\textrm{f}}$. As shown by the solid lines in Fig. 2(b), the
phenomenological formula fits the data well for $p\geq 0.67$ while the data
for $p<0.67$ could not be fitted due to the lack of enough data points below
$T_{\mathrm{f}}$. The fitted parameters are summarized in Table 1. As the spin
density $p$ decreases below 0.93, $f$ gradually decreases roughly linearly as
shown in the inset of Fig. 2(a). In other words, the glassy state of BSZCGO
continually crosses over from a dominantly spin-jam state to a mixed state
with a considerable spin-glass state as $p$ decreases.

As another quantitative verification of the HS modes being dominant for large
values of $p$, an energy scale associated with this mode can be estimated from
the coefficient $A$ of the quadratic term of $C_{\mathrm{HS}}$. The spin
stiffness $\rho_{\mathrm{s}}$ and the spin wave velocity $v$ are related by
$v=\gamma\sqrt{\rho_{\mathrm{s}}/\chi}$, where $\chi$ is the magnetic
susceptibility and $\gamma$ is the gyromagnetic ratio, and $A$ is related to
$v$ via Eq. 1. From the spin stiffness $\rho_{\mathrm{s}}$, the HS energy
$E_{\mathrm{HS}}$ is expressed as

$\displaystyle\frac{E_{\mathrm{HS}}}{k_{\mathrm{B}}}=\frac{9\zeta(3)}{\pi}\frac{k_{\mathrm{B}}^{2}}{g^{2}\mu_{\mathrm{B}}^{2}}\frac{\chi}{A},$
(2)

where $g$ is the Landé factor and $\mu_{\mathrm{B}}$ the Bohr magneton [41].
The magnetic susceptibility $\chi$ is obtained from the measured
susceptibility below $T_{\mathrm{f}}$. As shown in Table 1, this formula
yields $E_{\mathrm{HS}}$ that is comparable to the freezing temperatures for
the two samples with the highest spin densities $p$, which supports our
interpretation of the dominant glassy state for $p\geq 0.93$ being the spin
jam. For $p<0.93$, the spin-glass population starts to grow, and its
susceptibility contributes significantly to the measured value, resulting in
the overestimation of $\chi$ used in Eq. 2 and hence the overestimation of
$E_{\mathrm{HS}}$ for $p<0.93$.

Table 1: Fitting parameters of the Halperin-Saslow modes in BSZCGO where p is the spin density obtained from the X-ray diffraction measurements (see Section III in [36]), f is the spin-jam population fraction, Tf, $\chi$ and Tf, $C_{\textrm{mag}}$ are the freezing temperatures extracted from the magnetic susceptibility and heat capacity, respectively, A is the coefficient of the quadratic term of CHS, and EHS/kB is the energy scale of the HS modes. Numbers in parentheses represent standard errors. The values of A for the last two samples have errors larger than itself. p |  f |  Tf, $\chi$ (K) |  Tf, $C_{\textrm{mag}}$ (K) |  A (JmolCr-1K-3) |  EHS/kB (K)
---|---|---|---|---|---
0.98(1) |  0.92(1) | 1.22(5) | 1.18(9) | 0.130(2) | 0.9(1)
0.93(2) | 0.92(1) | 1.24(5) | 1.18(9) | 0.120(5) | 1.0(1)
0.86(2) | 0.90(1) | 0.93(5) | 1.09(8) | 0.10(1) | 1.3(1)
0.83(2) | 0.79(2) | 0.83(5) | 1.01(8) | 0.08(1) | 1.8(2)
0.71(1) | 0.75(2) | - | 0.94(7) | - | -
0.67(1) | 0.65(4) | - | 0.87(7) | - | -

###  III.2 Zero-point entropies

The evolution of the glassy states as a function of spin density $p$ may also
be investigated in terms of entropy. In general, upon cooling, a magnetic
system gradually releases its magnetic entropy, and an ordinary magnet
releases all of its magnetic entropy when the system exhibits long-range order
below the ordering temperature. On the other hand, disordered magnets would
not release all their magnetic entropy due to strong frustrations, giving rise
to finite zero-point entropy. Also, it should be emphasized that the spin-jam
and spin-glass states may have different characteristic entropies.

Entropy can be estimated from the heat capacity data as

$\displaystyle S(T_{\mathrm{base}},T)=S_{0}+\Delta
S(T_{\mathrm{base}},T)=S_{0}+\int^{T}_{T_{\mathrm{base}}}\frac{C_{\mathrm{mag}}}{T}dT,$
(3)

where $S_{0}$ is the zero-point entropy and $T_{\mathrm{base}}$ is the base
temperature of 0.5 K. Thus, by investigating how $\Delta S$ evolves with
increasing $p$ we can study how the zero-point entropy $S_{0}$, i.e., the
entropy of the glassy state, evolves. From the $C_{\mathrm{mag}}/T$ data shown
in Fig. 3(a), we numerically calculated and plotted $\Delta S(T)$ in Fig.
3(b). To confirm that the magnetic heat capacity tends to zero above 50 K, the
heat capacity of the $p\sim 0.98$ and $p=0$ samples were measured up to room
temperature (see Fig. S1 in [36] for the raw data). At high temperatures above
50 K, the heat capacity data of the magnetic ($p\sim 0.98$) and nonmagnetic
($p=0$) samples coincide with each other, indicating that the magnetic
contribution to the heat capacity is negligible as shown in the inset of Fig.
3(a). The result leads to the conclusion that there is no further increase in
the magnetic entropy at high temperatures. The magnetic entropy change present
below 0.5 K, $\Delta S(T<0.5\;\mathrm{K})$, can be approximated by linear
extrapolation down to absolute zero temperature. Considering the $p=0.60$
sample, we obtained $\Delta S(T<0.5\;\mathrm{K})$ equal to about 0.07
Jmol${}_{\mathrm{Cr}}^{-1}$K-1 which is only $\sim$1% of $\Delta
S(50\;\mathrm{K})$. This value decreases for higher $p$, and is smaller than
the error bar of $\Delta S(50\;\mathrm{K})$, which is $\sim$0.4
Jmol${}_{\mathrm{Cr}}^{-1}$K-1. Hence, we conclude that the presence of
$\Delta S$ below 0.5 K is insignificant and can be safely ignored. The
shortfall of entropy is thus attributed to the zero-point entropy. We note
that the magnetic Cr3+ ion has spin $s=\frac{3}{2}$ with the expected maximum
magnetic entropy of $R\ln(2s+1)$ = 11.53 Jmol${}_{\mathrm{Cr}}^{-1}$K-1, which
is represented by the horizontal red dashed line in Fig. 3(b). Here, molCr
represents the unit of moles of Cr3+ ions.

Figure 3: The T dependence of Cmag/T and $\Delta$S(T). (a) Cmag/T for all
samples up to 50 K. The inset shows Cmag/T for the $p=0.98$ sample up to room
temperature. Open symbols denote the data from the low-temperature
measurements (up to 50 K) as shown in the main panel, while closed symbols
represent the data from the high-temperature measurements (up to room
temperature). (b) $\Delta$S(T) for all samples obtained by integrating Cmag/T
data. The red dashed line indicates Smax = R$\ln 4$.

As shown in Fig. 3(b), for $p>p_{\mathrm{c}}=0.5$, where $p_{\mathrm{c}}$ is
the percolation threshold for the magnetic lattice [43], the entropy released,
$\Delta S$(0.5 K, 50 K), between 0.5 K ($<T_{\mathrm{f}}$) and 50 K ($\gg
T_{\mathrm{f}}$) is only about half of the maximum magnetic entropy
$S_{\mathrm{max}}$. For instance, $\Delta S$(0.5 K, 50 K)
$=0.45(1)S_{\mathrm{max}}$ for BSZCGO(0.98). This result indicates that the
entropy that is not released down to 0.5 K is extensive;
$S_{0}(p=0.98)=0.55(1)S_{\mathrm{max}}=$ 6.34(14)
Jmol${}_{\mathrm{Cr}}^{-1}$K-1. On the other hand, for
$p=0.44<p_{\mathrm{c}}$, $S(T_{\mathrm{base}},T)$ at 50 K is close to
$S_{\mathrm{max}}$, $\Delta S$(0.5 K, 50 K) $\approx$ 0.8$S_{\mathrm{max}}$.
This result implies that the extensive zero-point entropy for
$p>p_{\mathrm{c}}$ is due to the collective frustrated interactions in the
quasi-two-dimensional triangular network of bipyramids. A similar observation
was reported for SCGO(0.89) in which at 100 K the magnetic entropy is
recovered by only 52% [44].

A close examination of $\Delta S$(0.5 K, 50 K) as a function of $p$ reveals an
interesting dependence on $p$. As shown in Fig. 4(b), as $p$ decreases from
0.98 to 0.79, $\Delta S$(0.5 K, 50 K) decreases by $\sim$25%. As a result, the
zero-point entropy $S_{0}$ increases as $p$ decreases from 0.98 to 0.79. Upon
further decreasing $p$ below 0.71, $\Delta S$(0.5 K, 50 K) increases again,
i.e., $S_{0}$ decreases. To understand the dip in $\Delta S$(0.5 K, 50 K) as a
function of $p$, we first note that our analysis of the $T$-dependent
$C_{\mathrm{mag}}$ data indicates that both spin-glass and spin-jam clusters
coexist and their fraction changes with the spin density $p$. This implies
that, since the spin-jam and the spin-glass states are expected to have
different zero-point entropies, the measured total zero-point entropy
$S_{0}^{\mathrm{tot}}$ have to include both contributions, the zero-point
entropy of spin jam $S_{0}^{\mathrm{SJ}}$ and that of spin glass
$S_{0}^{\mathrm{SG}}$,

$\displaystyle
S_{0}^{\mathrm{tot}}(p)=f(p)S_{0}^{\mathrm{SJ}}(p)+[1-f(p)]S_{0}^{\mathrm{SG}}(p).$
(4)

This equation together with Eq. 3, however, cannot give us a set of unique
solutions, because $S_{0}^{\mathrm{SJ}}(p)$ and $S_{0}^{\mathrm{SG}}(p)$ can,
in general, vary with $p$, and the analysis of $\Delta S$(0.5 K, 50 K) as a
function of $p$ [shown in Fig. 4(b)] to estimate $S_{0}^{\mathrm{SJ}}(p)$ and
$S_{0}^{\mathrm{SG}}(p)$ becomes an underconstrained problem.

To overcome this problem, we performed the entropy analysis by imposing two
assumptions. The first assumption is based on the $p$-dependence of the
correlation length, $\xi(p)$, reported by a previous neutron scattering study
of SCGO [18] in which $\xi(p)$ remains constant for $1.0>p>0.8$, i.e., it is
robust against small nonmagnetic doping. At the same time, it linearly and
gradually decreases with further decreasing $p$ below $p=0.8$. Since $\xi(p)$
is directly proportional to the spin-jam domain size, we assume that the spin-
jam domain size is constant for $1.0>p>0.8$, and gradually changes in the same
way as $\xi(p)$ does with decreasing $p$ for $p<0.8$. The second assumption is
that the zero-point entropy of spin jam scales with the perimeter of the spin
jam domains as predicted by the spin-jam theory [16].

Figure 4: The $p$ dependence of (a) $T_{\textrm{f},C_{\mathrm{mag}}}(p)$ and
(b) $\Delta S(p)$ between 0.5 and 50 K. The horizontal dashed line is
$S_{\textrm{max}}=R\ln 4=11.53$ JmolCr-1K-1. The vertical dashed line
indicates the percolation threshold $p_{\textrm{c}}=0.5$. The solid lines are
guides to the eyes. The gradient color represents the crossover of the system
from spin-jam to spin-glass state. (c) The p dependence of the zero-point
entropy of the spin jam $S_{0}^{\mathrm{SJ}}(p)$ and that of the spin glass
$S_{0}^{\mathrm{SG}}(p)$ are calculated using Eq. 4, where
$S_{0}^{\textrm{tot}}(p)$ is obtained from
$S_{0}^{\textrm{tot}}(p)=S_{\textrm{max}}-\Delta S(p)$.

Now let us recall that our $C_{\mathrm{mag}}$ data yields the total zero-point
entropy for $p=0.98$ to be $S_{0}^{\mathrm{tot}}=6.34$
Jmol${}_{\mathrm{Cr}}^{-1}$K-1. The zero-point entropy of the spin-glass state
for $p=0.98$ is likely to be very close to $S_{\mathrm{max}}=R\ln(2s+1)=11.53$
Jmol${}_{\mathrm{Cr}}^{-1}$K-1 because, when the vacancy density is low, the
spin glass is made of almost uncorrelated orphan spins that fluctuate nearly
freely, resulting in $S_{0}^{\mathrm{SG}}(0.98)\approx S_{\mathrm{max}}$.
Using Eq. 4, we obtained $S_{0}^{\mathrm{SJ}}(0.98)=5.92$
Jmol${}_{\mathrm{Cr}}^{-1}$K-1, which is close to $S_{0}^{\mathrm{tot}}(0.98)$
as expected, as the magnetic glassy state for $p=0.98$ is predominantly a spin
jam. For other $p$ values, we first estimated the perimeter of the spin jam
domains with the experimental correlation length $\xi(p)$ of SCGO($p$) and
scaled $S_{0}^{\mathrm{SJ}}(p)$ to $S_{0}^{\mathrm{SJ}}(p=0.98)$ according to
the change in the perimeter of the magnetic domain (see Section IIA in [36]
for details). Once $S_{0}^{\mathrm{SJ}}(p)$ is obtained,
$S_{0}^{\mathrm{SG}}(p)$ for each $p$ was calculated using Eq. 4. Figure 4(c)
shows the resulting $S_{0}^{\mathrm{SJ}}(p)$ and $S_{0}^{\mathrm{SG}}(p)$ for
all $p>p_{\mathrm{c}}$. It is interesting to note that
$S_{0}^{\mathrm{SJ}}(p)$ and $S_{0}^{\mathrm{SG}}(p)$ exhibit strikingly
different behaviors with $p$. For $1.0>p>0.8$, the spin-jam zero-point entropy
$S_{0}^{\mathrm{SJ}}(p)$ is much lower than $S_{\mathrm{max}}=11.53$
Jmol${}_{\mathrm{Cr}}^{-1}$K-1. For $p<0.8$, as $p$ decreases, on the other
hand, $S_{0}^{\mathrm{SJ}}(p)$ gradually increases up to
$0.81S_{\mathrm{max}}$ for $p=0.51$, which is expected since the interaction-
driven magnetic constraints become weaker as the domain size decreases. In
contrast, as $p$ decreases after $p\sim 0.8$, $S_{0}^{\mathrm{SG}}(p)$ rapidly
decreases down to $0.25S_{\mathrm{max}}$ for $p=0.51$. The rapid decrease of
$S_{0}^{\mathrm{SG}}(p)$ suggests that as the vacancy density in the magnetic
lattice increases, the orphan spins begin to correlate with one another
resulting in a smaller degeneracy as expected for canonical spin glasses [3].
The value of $p\sim 0.8$, below which the orphan spins become strongly
correlated, is also consistent with the neutron experimental results of SCGO
[18].

To reaffirm the validity of our analysis, we estimated the typical perimeter
of the spin-jam domain from the obtained value of $S_{0}^{\mathrm{SJ}}(0.98)$
based on the spin-jam theory [16]. For $S_{0}^{\mathrm{SJ}}(0.98)\approx 5.9$
Jmol${}_{\mathrm{Cr}}^{-1}$K-1, the estimated number of bipyramids, denoted as
$N_{\mathrm{p}}$, on the domain perimeter is approximately 1.5(1) (see Section
IIB in [36] for detailed calculations). To put this value into context, we
compared it with the magnetic domain size derived from the correlation length
$\xi$ of a related system. In a single-crystal sample of SCGO(0.67), the
measured $\xi$ using neutron scattering is $4.6(2)\mathrm{\AA}$ [30]. By
considering the $p$ dependence of $\xi$ [18], we estimated $\xi$ to be 5.5(4)
$\mathrm{\AA}$ for $p\sim 1$. This value closely matches the distance between
the two centers of the nearest neighboring bipyramids of 5.85 Å.
$\xi=5.5(4)~{}\mathrm{\AA}$ aligns with $N_{\mathrm{p}}\approx 1.5$ estimated
from the spin jam theory, considering that $\xi$ is defined by the distance at
which the spin correlation reduces to $e^{-1}$, while the spin jam theory
using the transfer matrix formalism [16] assumes the spin correlation being 1
within the magnetic domain.

##  IV Conclusion

In summary, our thermodynamic studies reveal that the low-temperature glassy
state of BSZCGO is a mixture of the spin-jam and spin-glass states,
characterized by the Halperin-Saslow modes and the localized two-level
systems, respectively. The population ratio of the spin-glass state to the
spin-jam state increases as $p$ decreases down to the percolation threshold,
$p_{\mathrm{c}}=0.5$. Furthermore, by quantitatively analyzing the magnetic
zero-point entropy, we found that, as $p$ decreases, the zero-point entropy of
the spin jam $S_{0}^{\mathrm{SJ}}(p)$ gradually increases, whereas that of the
spin glass $S_{0}^{\mathrm{SG}}(p)$ rapidly decreases below $p\sim 0.8$. This
work elucidates the coexistence of the two glassy states in the frustrated
quantum magnetism.

###### Acknowledgements.

J.Y. and S.-H.L. thank Dr. Matthias Thede and Dr. Andrey Zheludev for their
help during some of our DC susceptibility measurements performed at
Eidgenössische Technische Hochschule (ETH) Zurich. C.P. thanks Prof. Satoshi
Kameoka for access to his X-ray diffractometer at Tohoku University. C.P. was
supported by the DPST scholarship from the Institute for the Promotion of
Teaching Science and Technology. Work at Mahidol University was supported in
part by the National Research Council of Thailand Grant N41A640158 and the
Thailand Center of Excellence in Physics. A.T. and S.-H.L. were supported by
the US Department of Energy, Office of Science, Office of Basic Energy
Sciences Award DE-SC0016144. W.-T.C. thanks the support by NSTC-Taiwan with
project number 108-2112-M-002-025-MY3, TCECM project number
110-2124-M-002-019, and Academia Sinica iMATE grant number AS-iMATE-111-12.
T.J.S. was supported by Grants-in-Aids for Scientific Research (JP22H00101,
19KK0069, 19H01834, 19K21839, 19H05824) from MEXT of Japan.

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Supplemental Material
Zero-point entropies of spin-jam and spin-glass states in a frustrated magnet

##  I Temperature-dependent molar heat capacity

Figure S1 shows the temperature dependence of the total heat capacity $C(T)$
measured up to room temperature for $p=0$ and $p=0.98$ samples. From the
figure, above $\sim$50 K, the heat capacity of both samples coincides with one
another indicative of vanishing magnetic contribution to the magnetic sample
($p=0.98$) above such temperature. Figure S2 depicts the temperature
dependence of $C_{\mathrm{mag}}/T$, obtained from subtracting the phonon
contribution measured on the non-magnetic sample ($p=0$) from the magnetic
$p=0.98$ sample in Fig. S1, showing that $C_{\mathrm{mag}}/T$ of the $p=0.98$
sample tends to zero above $\sim$50 K. Hence, we conclude that there is no
further significant increase in the magnetic entropy up to the room
temperature.

Figure S3 shows $C(T)$ of all samples measured from 0.5 K to 50 K. To obtain
magnetic heat capacity, the magnetic $p\neq 0$ sample’s heat capacity is
subtracted by the heat capacity of the non-magnetic $p=0$ sample, where the
non-magnetic data are interpolated before the subtraction. After the
subtraction, the data in the overlapping region taken with He-3 and He-4
options are averaged, weighted by errors as shown in Fig. 3(a).

##  II Calculation of spin-jam zero-point entropy, $S_{0}^{\mathrm{SJ}}$

###  II.1 The $p$ dependence of $S_{0}^{\mathrm{SJ}}$

The calculation of the $p$-dependent zero-point entropy of spin jam
$S_{0}^{\mathrm{SJ}}(p)$ is performed based on the spin jam theory of
triangular network of bipyramids and the previously measured spin correlation
lengths of the related compound SrCr9pGa12-9pO19 (SCGO($p$)) with a similar
structure as a function of $p$. Then, using Eq. (4) in the main text, the $p$
dependence of spin glass’s zero-point entropy $S_{0}^{\mathrm{SG}}(p)$ is
obtained from $S_{0}^{\mathrm{SJ}}(p)$ and $S_{0}^{\mathrm{tot}}(p)$.

The $p$-dependent normalized perimeter length is obtained from the
$p$-dependent normalized correlation length measured from SCGO($p$). The
correlation length is found to remain constant for $0.8<p<1.0$ and it linearly
decreases with $p$ below $p\sim 0.8$. To calculate the normalized perimeter
length for each domain size, we suppose that the correlation length
corresponds to the radius of the domain. We start with a circle of radius 1
unit. Then, we fill this circle with smaller circles representing smaller
domains in a close-packed pattern. The total perimeter length of all circles
with the same radius within the largest circle is, then, calculated. As a
result, we can determine the total perimeter length of domains with different
sizes covering the same total area. Hence, the total perimeter length of a
larger domain, normalized with the total area, is smaller. According to the
theory, the ground-state configurational entropy of spin jam scales with the
normalized perimeter of the domain. Thus, the $p$ dependence of the spin jam’s
zero-point entropy $S_{0}^{\mathrm{SJ}}(p)$ can be determined from that of the
normalized domain perimeter.

###  II.2 Perimeter scaling of $S_{0}^{\mathrm{SJ}}$

$S_{0}^{\mathrm{SJ}}$ can be calculated using the formula
$S^{\mathrm{SJ}}_{0}=k\ln\Omega/n=R\ln\Omega/N$, where $\Omega$ is the number
of all possible ground-state spin configurations, $n$ the number of moles of
spins, and $N$ the number of spins. As detailed in Ref. [37], to obtain a
ground-state spin configuration, a sign state must be imposed with a color
configuration. As there are 6 different color configurations, the number of
all spin configurations, $\Omega$, is hence equal to
$6\cdot\Omega_{\mathrm{sign}}$, where $\Omega_{\mathrm{sign}}$ is the total
number of sign states. $\Omega_{\mathrm{sign}}$ is numerically calculated in
Ref. [37], where $\log_{10}\Omega_{\mathrm{sign}}$ increases linearly with the
number of bipyramids positioned on the perimeter $N_{\mathrm{p}}$, i.e.,
$\log_{10}\Omega_{\mathrm{sign}}=mN_{\mathrm{p}}+c$, as shown in the inset of
Fig. 2(b) in Ref. [37].

A linear fit to $\log_{10}\Omega_{\mathrm{sign}}$ vs. $N_{\mathrm{p}}$ yields
$m\approx 0.26$ and $c\approx-0.78$ as shown in Fig. S5. Therefore, we obtain
$\ln\Omega=\ln(6\cdot\Omega_{\mathrm{sign}})=\ln(6\cdot
10^{mN_{\mathrm{p}}+c})\approx\alpha N_{\mathrm{p}}$ where $\alpha=m\ln
10\approx 0.60$. With this result, we obtained $S^{\mathrm{SJ}}_{0}\approx
R(\alpha N_{\mathrm{p}})/N$. For a circular-shape domain, the number of
bipyramids in the domain can be approximated by $N_{\mathrm{p}}^{2}/4\pi$. As
a bipyramid contains 7 spins, the number of spins $N$ in the typical domain
is, therefore, $N\approx 7N_{\mathrm{p}}^{2}/4\pi$, and hence
$S^{\mathrm{SJ}}_{0}=4\pi\alpha R/7N_{\mathrm{p}}$. Using the experimental
value of $S^{\mathrm{SJ}}_{0}=5.9(5)$ Jmol${}_{\mathrm{Cr}}^{-1}$K-1 for
$p=0.98$, we obtain $N_{\mathrm{p}}=1.5(1)$ for the magnetic domain.

##  III Structural refinements

The X-ray diffraction results are analyzed using Rietveld refinements. To
demonstrate the capability of X-ray diffraction in determining the occupancy
of Cr3+ ions in a magnetic sample, we compare the diffraction peaks of three
different samples within the range of $63.0^{\circ}<2\theta<64.5^{\circ}$,
normalizing them to have the same highest peak count, as shown in Figure S6.
In the figure, besides varying peak positions, each sample also exhibits
distinct intensities within this range. These differences in peak positions
are likely attributed to variations in the lattice parameters of each sample.
Conversely, simulations based on similar lattice and atomic parameters but
differing Cr3+ occupancies, as illustrated in Figure S7, indicate that the
disparities in intensities observed in Figure S6 are a result of varying Cr3+
occupancies.

The initial crystal-structure parameters used in the refinements are taken
from Ref. [51]. Figures S9-S19 show the results of Rietveld refinements for
each sample along with corresponding optimal atomic parameters in Tables
S1-S11. The refined lattice parameters, $a=b$ and $c$, are plotted against $p$
in Fig. S8. To minimize errors of the occupancies of the Cr3+ ions in the
magnetic samples, other atomic and lattice parameters were fixed in the final
refinement. The $p$ value for each sample is obtained by averaging the
occupancies of the two Cr3+ sites weighted by the corresponding site
multiplicity.

Figure S1: Temperature dependence of the total heat capacity $C(T)$ for
$p=0.98$ and $p=0$ samples up to room temperature. Error bars represent one
standard deviation. Figure S2: Temperature dependence of $C_{\mathrm{mag}}/T$
for the $p=0.98$ sample obtained from the results in Fig. S1. Error bars
represent one standard deviation. Figure S3: Temperature dependence of the
total heat capacity $C(T)$ for all samples up to 50 K. Figure S4: Low-
temperature region of $C_{\mathrm{mag}}/T$ for all samples. Figure S5: Scaling
of $\log_{10}\Omega_{\textrm{sign}}$ with the domain perimeter
$N_{\textrm{p}}$ in a unit of bipyramids. $\Omega_{\textrm{sign}}$ is the
number of sign states as described in Ref. [37]. This plot is reproduced from
the inset of Fig. 2(b) in Ref. [37]. Figure S6: Comparison of diffraction
peaks of 3 different samples in $63.0^{\circ}<2\theta<64.5^{\circ}$ region.
The $p$ values are obtained from the Rietveld refinements. Figure S7:
Comparison of simulated diffraction peaks in the same region as in Fig. S6
based on two models with similar lattice and atomic parameters but with
distinct Cr3+ occupancies.

Figure S8: The lattice parameters $a$ ($a=b$) and $c$ obtained from the refinements. The results are shown as percentage change with respect to that of the $p=0.98$ sample, $a_{0}$ and $c_{0}$. Lines are guides to the eye. Figure S9: Refinement result of the $p=0$ sample. Figure S10: Refinement result of the $p=0.44$ sample. Figure S11: Refinement result of the $p=0.51$ sample. Figure S12: Refinement result of the $p=0.60$ sample. Figure S13: Refinement result of the $p=0.63$ sample. Figure S14: Refinement result of the $p=0.67$ sample. Figure S15: Refinement result of the $p=0.71$ sample. Figure S16: Refinement result of the $p=0.83$ sample. Figure S17: Refinement result of the $p=0.86$ sample. Figure S18: Refinement result of the $p=0.93$ sample. Figure S19: Refinement result of the $p=0.98$ sample. Table S1: Refined atomic parameters of the $p=0$ sample. Refined lattice parameters are $a=b=5.857241(44)$ Å and $c=14.253215(78)$ Å. $R_{\mathrm{wp}}=8.81\%$ and GoF = 2.83. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4256(1) | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6822(1) | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.955(43) | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.955(41) | 0.50
Ga ($2c$) | 0 | 0 | 0.3731(2) | 1.00
Ga ($1a$) | 0 | 0 | 0 | 1.00
Ga ($6i$) | 0.1699(2) | -0.1699(2) | 0.17031(8) | 1.00
O ($2c$) | 0 | 0 | 0.2373(9) | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0968(8) | 1.00
O ($6i$) | 0.1572(7) | -0.1572(7) | 0.9144(5) | 1.00
O ($6i$) | 0.4946(9) | -0.4946(9) | 0.2375(6) | 1.00
O ($6i$) | 0.1711(6) | -0.1711(6) | 0.5892(5) | 1.00
Table S2: Refined atomic parameters of the $p=0.44$ sample. Refined lattice parameters are $a=b=5.85692$ Å and $c=14.25790$ Å. $R_{\mathrm{wp}}=5.16\%$ and GoF = 1.52. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.42463 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.68137 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9548 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9548 | 0.50
Ga ($2c$) | 0 | 0 | 0.3721 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.445(14)
Ga ($1a$) | 0 | 0 | 0 | 0.555(14)
Cr ($6i$) | 0.1690 | -0.1690 | 0.17114 | 0.441(4)
Ga ($6i$) | 0.1690 | -0.1690 | 0.17114 | 0.559(4)
O ($2c$) | 0 | 0 | 0.2380 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0913 | 1.00
O ($6i$) | 0.1527 | -0.1527 | 0.9103 | 1.00
O ($6i$) | 0.4920 | -0.4920 | 0.2403 | 1.00
O ($6i$) | 0.1745 | -0.1745 | 0.5925 | 1.00
Table S3: Refined atomic parameters of the $p=0.51$ sample. Refined lattice parameters are $a=b=5.85382$ Å and $c=14.24923$ Å. $R_{\mathrm{wp}}=6.49\%$ and GoF = 1.76. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4251 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6812 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9553 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9553 | 0.50
Ga ($2c$) | 0 | 0 | 0.3712 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.409(18)
Ga ($1a$) | 0 | 0 | 0 | 0.591(18)
Cr ($6i$) | 0.1689 | -0.1689 | 0.17114 | 0.523(5)
Ga ($6i$) | 0.1689 | -0.1689 | 0.17114 | 0.477(5)
O ($2c$) | 0 | 0 | 0.2476 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0888 | 1.00
O ($6i$) | 0.1592 | -0.1592 | 0.9120 | 1.00
O ($6i$) | 0.4899 | -0.4899 | 0.2407 | 1.00
O ($6i$) | 0.1679 | -0.1679 | 0.5878 | 1.00
Table S4: Refined atomic parameters of the $p=0.60$ sample. Refined lattice parameters are $a=b=5.85502$ Å and $c=14.25203$ Å. $R_{\mathrm{wp}}=6.05\%$ and GoF = 1.68. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.42493 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.68166 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9552 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9552 | 0.50
Ga ($2c$) | 0 | 0 | 0.3717 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.599(15)
Ga ($1a$) | 0 | 0 | 0 | 0.401(15)
Cr ($6i$) | 0.1691 | -0.1691 | 0.17067 | 0.604(4)
Ga ($6i$) | 0.1691 | -0.1691 | 0.17067 | 0.396(4)
O ($2c$) | 0 | 0 | 0.2448 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0913 | 1.00
O ($6i$) | 0.1566 | -0.1566 | 0.9122 | 1.00
O ($6i$) | 0.4927 | -0.4927 | 0.2400 | 1.00
O ($6i$) | 0.1693 | -0.1693 | 0.5889 | 1.00
Table S5: Refined atomic parameters of the $p=0.63$ sample. Refined lattice parameters are $a=b=5.85676$ Å and $c=14.25397$ Å. $R_{\mathrm{wp}}=5.06\%$ and GoF = 1.52. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.42444 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6815 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9548 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9548 | 0.50
Ga ($2c$) | 0 | 0 | 0.3732 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.625(16)
Ga ($1a$) | 0 | 0 | 0 | 0.375(16)
Cr ($6i$) | 0.1687 | -0.1687 | 0.17032 | 0.631(4)
Ga ($6i$) | 0.1687 | -0.1687 | 0.17032 | 0.369(4)
O ($2c$) | 0 | 0 | 0.2390 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0945 | 1.00
O ($6i$) | 0.1545 | -0.1545 | 0.9123 | 1.00
O ($6i$) | 0.4923 | -0.4923 | 0.2367 | 1.00
O ($6i$) | 0.1754 | -0.1754 | 0.59183 | 1.00
Table S6: Refined atomic parameters of the $p=0.67$ sample. Refined lattice parameters are $a=b=5.85758$ Å and $c=14.25553$ Å. $R_{\mathrm{wp}}=5.45\%$ and GoF = 1.61. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4240 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6814 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9546 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9546 | 0.50
Ga ($2c$) | 0 | 0 | 0.3732 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.621(17)
Ga ($1a$) | 0 | 0 | 0 | 0.379(17)
Cr ($6i$) | 0.1689 | -0.1689 | 0.17019 | 0.672(4)
Ga ($6i$) | 0.1689 | -0.1689 | 0.17019 | 0.328(4)
O ($2c$) | 0 | 0 | 0.2358 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0966 | 1.00
O ($6i$) | 0.1558 | -0.1558 | 0.9119 | 1.00
O ($6i$) | 0.4925 | -0.4925 | 0.2363 | 1.00
O ($6i$) | 0.1740 | -0.1740 | 0.5935 | 1.00
Table S7: Refined atomic parameters of the $p=0.71$ sample. Refined lattice parameters are $a=b=5.85367$ Å and $c=14.24537$ Å. $R_{\mathrm{wp}}=6.14\%$ and GoF = 1.69. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4245 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6817 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9551 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9551 | 0.50
Ga ($2c$) | 0 | 0 | 0.3722 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.703(16)
Ga ($1a$) | 0 | 0 | 0 | 0.297(16)
Cr ($6i$) | 0.1690 | -0.1690 | 0.17036 | 0.715(4)
Ga ($6i$) | 0.1690 | -0.1690 | 0.17036 | 0.285(4)
O ($2c$) | 0 | 0 | 0.2425 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0938 | 1.00
O ($6i$) | 0.1548 | -0.1548 | 0.9126 | 1.00
O ($6i$) | 0.4895 | -0.4895 | 0.2392 | 1.00
O ($6i$) | 0.1732 | -0.1732 | 0.5906 | 1.00
Table S8: Refined atomic parameters of the $p=0.83$ sample. Refined lattice parameters are $a=b=5.85338$ Å and $c=14.2452$ Å. $R_{\mathrm{wp}}=6.24\%$ and GoF = 1.81. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4247 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6817 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9550 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9550 | 0.50
Ga ($2c$) | 0 | 0 | 0.3734 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.823(19)
Ga ($1a$) | 0 | 0 | 0 | 0.177(19)
Cr ($6i$) | 0.1685 | -0.1685 | 0.1695 | 0.829(5)
Ga ($6i$) | 0.1685 | -0.1685 | 0.1695 | 0.171(5)
O ($2c$) | 0 | 0 | 0.2383 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0969 | 1.00
O ($6i$) | 0.1560 | -0.1560 | 0.9125 | 1.00
O ($6i$) | 0.4944 | -0.4944 | 0.2381 | 1.00
O ($6i$) | 0.1771 | -0.1771 | 0.5927 | 1.00
Table S9: Refined atomic parameters of the $p=0.86$ sample. Refined lattice parameters are $a=b=5.85326$ Å and $c=14.2454$ Å. $R_{\mathrm{wp}}=5.80\%$ and GoF = 1.69. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4236 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6814 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9550 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9550 | 0.50
Ga ($2c$) | 0 | 0 | 0.3735 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.857(21)
Ga ($1a$) | 0 | 0 | 0 | 0.143(21)
Cr ($6i$) | 0.1686 | -0.1686 | 0.1695 | 0.861(6)
Ga ($6i$) | 0.1686 | -0.1686 | 0.1695 | 0.139(6)
O ($2c$) | 0 | 0 | 0.2388 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0961 | 1.00
O ($6i$) | 0.1546 | -0.1546 | 0.9129 | 1.00
O ($6i$) | 0.4931 | -0.4931 | 0.2390 | 1.00
O ($6i$) | 0.1771 | -0.1771 | 0.5945 | 1.00
Table S10: Refined atomic parameters of the $p=0.93$ sample. Refined lattice parameters are $a=b=5.85457$ Å and $c=14.2491$ Å. $R_{\mathrm{wp}}=6.38\%$ and GoF = 1.82. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4237 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6824 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9554 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9554 | 0.50
Ga ($2c$) | 0 | 0 | 0.3734 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 0.937(25)
Ga ($1a$) | 0 | 0 | 0 | 0.063(25)
Cr ($6i$) | 0.1677 | -0.1677 | 0.1686 | 0.933(7)
Ga ($6i$) | 0.1677 | -0.1677 | 0.1686 | 0.067(7)
O ($2c$) | 0 | 0 | 0.245 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.1021 | 1.00
O ($6i$) | 0.156 | -0.156 | 0.9126 | 1.00
O ($6i$) | 0.4889 | -0.4889 | 0.2355 | 1.00
O ($6i$) | 0.1769 | -0.1769 | 0.5878 | 1.00
Table S11: Refined atomic parameters of the $p=0.98$ sample. Refined lattice parameters are $a=b=5.85204$ Å and $c=14.2434$ Å. $R_{\mathrm{wp}}=8.30\%$ and GoF = 2.18. Atom | $x/a$ | $y/b$ | $z/c$ | occupancy
---|---|---|---|---
Ba ($2d$) | 1/3 | 2/3 | 0.4237 | 1.00
Sn ($2d$) | 1/3 | 2/3 | 0.6818 | 1.00
Zn ($2d$) | 1/3 | 2/3 | 0.9560 | 0.50
Ga ($2d$) | 1/3 | 2/3 | 0.9560 | 0.50
Ga ($2c$) | 0 | 0 | 0.3724 | 1.00
Cr ($1a$) | 0 | 0 | 0 | 1.00(2)
Ga ($1a$) | 0 | 0 | 0 | 0.00(2)
Cr ($6i$) | 0.1688 | -0.1688 | 0.1707 | 0.976(4)
Ga ($6i$) | 0.1688 | -0.1688 | 0.1707 | 0.024(4)
O ($2c$) | 0 | 0 | 0.244 | 1.00
O ($2d$) | 1/3 | 2/3 | 0.0929 | 1.00
O ($6i$) | 0.1555 | -0.1555 | 0.9126 | 1.00
O ($6i$) | 0.4853 | -0.4853 | 0.2387 | 1.00
O ($6i$) | 0.1709 | -0.1709 | 0.5895 | 1.00


# Multiple Photodetachment of Silicon Anions via K-Shell Excitation and
Ionization

A. Perry-Sassmannshausen  I. Physikalisches Institut, Justus-Liebig-
Universität Gießen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany  T. Buhr  I.
Physikalisches Institut, Justus-Liebig-Universität Gießen, Heinrich-Buff-Ring
16, 35392 Giessen, Germany  M. Martins  Institut für Experimentalphysik,
Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany  S.
Reinwardt  Institut für Experimentalphysik, Universität Hamburg, Luruper
Chaussee 149, 22761 Hamburg, Germany  F. Trinter  Institut für Kernphysik,
Goethe-Universität Frankfurt am Main, Max-von-Laue-Straße 1, 60438 Frankfurt
am Main, Germany  Molecular Physics, Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  A. Müller  Institut für
Atom- und Molekülphysik, Justus-Liebig-Universität Gießen, Leihgesterner Weg
217, 35392 Giessen, Germany  S. Fritzsche  Helmholtz-Institut Jena,
Fröbelstieg 3, 07743 Jena, Germany  Theoretisch-Physikalisches Institut,
Friedrich-Schiller-Universität Jena, 07743 Jena, Germany  S. Schippers
<EMAIL_ADDRESS>I. Physikalisches Institut, Justus-
Liebig-Universität Gießen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany

###### Abstract

Experimental cross sections for $m$-fold photodetachment ($m=3-6$) of silicon
anions via $K$-shell excitation and ionization were measured in the photon-
energy range of 1830-1900 eV using the photon-ion merged-beams technique at a
synchrotron light source. All cross sections exhibit a threshold behavior that
is masked by pre-threshold resonances associated with the excitation of a $1s$
electron to higher, either partly occupied or unoccupied atomic subshells.
Results from multi-configuration Dirac-Fock (MCDF) calculations agree with the
experimentally derived cross sections for photo-absorption if small energy
shifts are applied to the calculated resonance positions and detachment
thresholds. Moreover, a systematic approach is applied for modeling the
deexcitation cascades that set in after the initial creation of a $K$-shell
hole. The resulting product charge-state distributions compare well with the
measured ones for direct $K$-shell detachment but less well for resonant
$K$-shell excitation. The present results are potentially useful for
identifying silicon anions in cold plasmas such as interstellar gas clouds.

##  I Introduction

Atomic anions are prototypal quantum systems for studying correlation effects
since the extra electron in theses systems is bound by short-range forces. A
fundamental process that enables one to probe correlation is photodetachment
of atomic anions, i.e., the removal of one or more electrons by absorption of
a single photon. Thorough investigations of this process have repeatedly lead
to a fruitful interplay between experiment and quantum theory, in particular,
for outer-shell photodetachment (see, e.g., [1, 2] and references therein).

The correlation between core and valence electrons can be probed by inner-
shell detachment where the valence electrons are subject to strong many-
electron relaxation effects following the creation of core holes [3, 4]. These
effects are strongest in the case of $K$-shell detachment, which can
experimentally be investigated by the photon-ion merged-beams technique using
soft x-ray beams from synchrotron light sources [5, 6, 7]. So far, $K$-inner-
shell photodetachment has been studied for a number of light ions comprising
Li- [8, 9], B- [10], C- [11, 12, 13], O- [14, 15], and F- [16].

Here, we extend these studies to an atomic anion where the outer shell is the
$M$ shell. We present experimental cross sections $\sigma_{m}$ for $m$-fold
photodetachment of Si- ions with $m=3,4,5,6$ resulting in the production of
multiply positively charged Si(m-1)+ ions. This process can be represented as

$h\nu+\textrm{Si}^{-}\to\textrm{Si}^{(m-1)+}+m\,e^{-}.$ (1)

The experimental photon-energy range of 1830–1900 eV comprises the threshold
for direct photodetachment of a $K$-shell electron. Furthermore, we present
theoretical results for resonant and nonresonant Si- photoabsorption. Using a
recently developed theoretical toolbox [17, 18], we followed in addition the
complex deexcitation cascade that sets in after the initial $K$-hole creation
and gives rise to photon-energy-dependent distributions of product-ion charge
states.

Apart from addressing fundamental questions in many-body physics, the present
experimental and theoretical investigations are also useful for astrophysics
considering the fact that negative ions have been identified, e.g., in the
interstellar medium [19]. In particular, they may help to improve the current
astrophysical models for the x-ray absorption by silicon ions [20].

##  II Experiment

The measurements were carried out using the photon-ion merged-beams technique
[21] at the PIPE facility [22, 23], which is a permanently installed end-
station at the photon beamline P04 [24] of the synchrotron radiation source
PETRA III operated by DESY in Hamburg, Germany. Silicon anions were produced
by a Cs-sputter ion source [25] with a silicon single crystal as a sputter
target and a sputter potential of about 2 kV.  Si- is known to posses long-
lived metastable levels with excitation energies of 1.36 eV and 0.86 eV and
lifetimes of 22 s and more than 5.7 h, respectively [26]. However, Si- ions
produced by a Cs-sputter ion source in an earlier experiment were found to be
exclusively in their $1s^{2}\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{3}\;{}^{4}S_{3/2}$
ground level [27]. We assume that the same holds for the present experiment.
After acceleration to a kinetic energy of 6 keV, the ions were passed through
an analyzing dipole magnet which was adjusted such that 28Si- ions were
selected for further transport to the photon-ion merged-beams interaction
region.

In the interaction region, where the residual-gas pressure was in the mid
$10^{-10}$ mbar range, the ion beam was coaxially merged with the counter-
propagating soft x-ray photon beam over a distance of about 1.7 m. Si(m-1)+
ions resulting from multiple photodetachment (Eq. 1) were separated from the
primary ion beam by a second dipole magnet. Inside this magnet, a Faraday cup
collected the primary ion beam, while the charge-selected product ions were
directed to the detector chamber. Along their flight path, they first passed
through a spherical 180-degree out-of-plane deflector to suppress background
from stray electrons, photons, and ions and then entered a single-particle
detector with nearly 100% detection efficiency [28].

Relative cross sections for $m$-fold photodetachment were obtained by
normalizing the count rates of Si(m-1)+ product ions on the primary Si- ion
current and on the photon flux measured with a calibrated photodiode. The ion
current in the interaction region was up to 40 nA, and the photon flux
amounted to $\sim 1.3\times 10^{13}$ s-1 at a photon energy spread of about
1.75 eV. The systematic uncertainty of the measured cross sections is
estimated to be $\pm$15% at 90% confidence level [22].

The photon-energy scale was calibrated by absorption measurements at the
$2p_{1/2}$ ionization threshold of krypton in a gas-phase electron
spectrometer by using the corresponding threshold energy of 1729.5 eV as
measured by Kato et al. [29] as a reference. The remaining uncertainty of the
present photon-energy scale is estimated to be $\pm$1 eV. A detailed
description of the general calibration procedure has been provided recently
[30, 31].

The experimental photon-energy bandwidth could be adjusted by choosing the
exit-slit width of the photon beamline’s monochromator which houses gratings
with rulings of 400 lines/mm and 1200 lines/mm. We used a 1200 lines/mm
grating and exit-slit widths of 1000 $\mu$m and 50 $\mu$m for low-resolution
and high-resolution measurements, respectively.

##  III Computations

To model the photodetachment of negative ions, such as Si-, detailed
computations of their level structure need to be combined with an efficient
treatment of the subsequent relaxation in order to account for the stepwise
autoionization and photon emission of the ions. This stepwise relaxation can
be formally described by an atomic cascade that includes ions of the same
element but in different charge states that are connected to each other by
different (decay) processes. In this work, all calculations presented below
have been performed by means of the Jac toolbox, the Jena Atomic Calculator
[17], which supports the (relativistic) computation of atomic structures and
processes. This toolbox has now been expanded recently to follow rather long
ionization pathways. Here, we shall recall just a few details about such
cascade computations, while the numerical results will be discussed together
with the measurements in Sec. IV.

Atomic cascades often require an enormous effort to generate, combine, and
simulate all data as needed for modeling a given experiment. To analyze such
cascades more systematically, Jac has recently been expanded for modeling
different cascade schemes, i.e., relaxation scenarios, including the
production of (inner-shell) excited atoms by photoexcitation, photoionization,
or other mechanisms and the stepwise decay of the excited states via photon or
electron emission [18]. Any of these schemes starts from setting up a cascade
tree, i.e., the list of possibly contributing configurations and from
selecting all relevant (decay) steps of the cascade.

In a cascade decay, each ion of a given ensemble therefore follows an
individual relaxation pathway. Obviously, a (very) large number of such
pathways occurs in the deexcitation of deep inner-shell holes owing to the
different decay processes that are possible and that have to be combined. For
the stepwise decay of the $K$-shell ionized Si- ion, for example, we started
from the $1s\,2s^{2}\,2p^{6}\,3s^{2}\,3p^{3}$ configuration and considered all
autoionization and photon emission processes to energetically lower-lying
configurations, until up to five electrons are released from the ion. For the
configuration above, this results in a total of 87 configurations and about
160,000 fine-structure transitions of the Si${}^{q+}\;(q=0,\ldots,5)$
atoms/ions that need to be taken into account.

To formalize the competition of the various Auger and photon emission
processes, we distinguish within the Jac toolbox between (so-called) cascade
computations and simulations [18]. While the cascade computations are
performed with the goal to calculate and compile the associated fine-structure
Auger and photon-emission rates, the simulations then make use of these (pre-
compiled) data in order to derive the ion distributions, or any other
information that may be needed. This distinction has been found helpful to
deal with the quite different experimental setups and observations, e.g.,
photon, electron, or ion distributions, that often can be traced back to the
same (single-step) processes.

Figure 1: Overview of the measured cross sections $\sigma_{m}$ for multiple
($m$-fold, Eq. 1) detachment of Si- ions. The differently colored data points
represent $\sigma_{m}$ for $m$=3 (orange), $m$=4 (magenta), $m=5$ (blue), and
$m=6$ (green). The error bars only account for the statistical experimental
uncertainties. The black circles represent the sum $\sigma_{\Sigma}$ (Eq. 2)
of all measured cross sections $\sigma_{m}$. The dashed grey curve is the
cross section for photoabsorption of neutral silicon atoms from the
compilation of Henke et al. [33]. The width of the monochromator exit slit was
1000 $\mu$m. The distorted resonance line shapes are due to a non-Gaussian
contribution to the photon-energy distribution, that becomes noticeable at
large exit-slit widths for photon energies well above 1000 eV.

Not much needs to be said about the detailed implementation of the cascade
computations and simulations which has been summarized elsewhere [18, 32]. The
major computational effort is spent on the representation of the fine-
structure levels at each step of the cascade as well as on the calculation of
the transition amplitudes. Within Jac, these are the building blocks that
enable us to formulate (and implement) all cascade computations. Because of
the complexity of atomic cascades, the (quantum-mechanical) representation of
the fine-structure levels usually needs to be simplified by _tuning_ the
amount of the inter-electronic interactions accounted for in the calculations.
In the Jac toolbox, this need for simplification is captured by an hierarchy
of cascade approaches for representing all the ionic level energies and state
functions. For Si-, unfortunately, we were able to apply only the simplest of
these approaches, the so-called averaged single-configuration approach, in
which all fine-structure states are approximated by single-configuration state
functions and by just the orbitals from the initial configuration of the
cascades. This approach neglects all configuration mixing between the bound-
state levels and restricts the computations to just the Coulomb interaction
among the electrons as well as a single set of continuum orbitals for each
step of the cascade [34].

Cascade computations have been performed for the photoabsorption and
photoionization cross sections in the region of the $1s\to 3p,\;4p$ inner-
shell excitations and $1s$ ionization. In general, the size of such cascade
computations increases rapidly with (i) the number of electrons, that need to
be replaced, and (ii) the depth of the cascade, i.e., the maximum number of
electrons that can be released and, as outlined above, by the number of fine-
structure transitions. This increase also results in an approximate treatment
of the fine-structure representation of the ionic levels that is likely the
main reason for the discrepancies of the predicted ion distributions, when
compared with the measurements.

##  IV Results and Discussion

The measured low-resolution cross sections for multiple detachment of Si- are
displayed in Fig. 1. All cross sections exhibit a rise at a photon energy of
about 1835 eV which is associated with the resonant excitation of a $1s$
electron to higher subshells. Two rather broad resonance features at $\sim$
1840 and $\sim$1843 eV can be discerned in all four cross-section curves. At
higher energies, where the cross sections are dominated by nonresonant direct
removal of a $K$-shell electron, they are structureless. The corresponding
detachment thresholds are masked by the below-threshold resonances.

The $K$-shell hole, that is created either by resonant excitation or by
nonresonant detachment, is filled by a cascade of radiative and Auger
processes, which results in the observed energy-dependent distributions over
the various product charge states analogous to what we have observed
previously for C- [13], where C(m-1)+ product charge states were detected for
$2\leq m\leq 6$. As compared to the C- cross sections, the present Si- cross
sections are almost an order of magnitude smaller. This follows the general
trend of decreasing photoionization cross sections as the nuclear charge
increases. Another significant difference between C- and Si- concerns the
dominant detachment channel which is $\sigma_{2}$ for C- and $\sigma_{4}$ for
Si-.  This can be attributed to the higher number of electrons outside the $K$
shell in the Si- ion which supports many more deexcitation channels as
compared to C-. In particular, the number of possibilities for autoionization
increases when going from C- to Si-.

The measured relative cross sections were put on an absolute scale by matching
the sum cross section

$\sigma_{\Sigma}=\sum_{m=3}^{6}\sigma_{m}$ (2)

with the absorption cross section of neutral silicon atoms [33] which is also
displayed in Fig. 1. Apparently, this cross section represents the energy
dependence of the experimental cross section very well except for the energies
where resonances occur. In order to bring $\sigma_{\Sigma}$ to the same scale,
all cross sections $\sigma_{m}$ were multiplied by the same appropriate
factor. As shown explicitly for C- [7], where independently _absolute_ cross
sections were experimentally determined [13], this procedure is valid if one
can safely assume that $\sigma_{\Sigma}$ represents all significant absorption
channels. This assumption appears justified since the behavior of the cross
sections $\sigma_{m}$ suggests that the unmeasured single and double
detachment channels and the higher detachment channels with $m>6$ are
negligible. The single-detachment channel could not be measured since our
experimental setup is only capable of measuring charged product ions.
Moreover, the measurement of the (weak) double-detachment channel was severely
hampered by a large background from collisions of Si- with the residual gas
such that meaningful results could not be obtained within a reasonable amount
of time. Measurements of the multiple-detachment cross sections for $m>6$ were
not attempted.

Figure 2: High-resolution cross sections $\sigma_{m}$ for $m$-fold ($m$=3-6)
photodetachment of Si- ions in the energy region of the near-threshold
resonances. The width of the monochromator exit slit was 50 $\mu$m.

The nonzero cross sections below 1835 eV are due to direct detachment of an
$L$-shell electron and the subsequent (multiple) autoionization. In this
energy range, the sum cross section $\sigma_{\Sigma}$ agrees, as mentioned
above, well with the cross section for photoabsorption of neutral silicon
atoms. This is also the case for energies above 1855 eV which are beyond the
threshold for the direct detachment of a $K$-shell electron. For neutral
silicon this threshold has been predicted to be at 1838.9 eV [33] (dashed line
in Fig. 1). For Si-, the corresponding step-like cross-section features are
masked by the resonances that occur immediately below these thresholds.

In order to obtain more detailed experimental information on the
photodetachment resonances we performed additional cross-section measurements
with higher resolution as compared to the measurements shown in Fig. 1. These
high-resolution measurements are displayed in Fig. 2. The statistical
uncertainties of $\sigma_{3}$ are comparatively large because a substantial
background contributed to the measured signal in this channel. Such a
background was largely absent in the other scrutinized channels. In all
measured cross sections, the strongest feature is a comparatively narrow
resonance at $\sim$1838.4 eV with an experimental width of 0.55 eV. A second,
broader resonance feature at about 1842.4 eV with a width of 2–3 eV is visible
in the cross sections for fourfold, fivefold, and sixfold detachment. The
relative strength of this feature, when compared to the narrower one,
increases with increasing product-ion charge state.

Figure 3: Comparison between the high-resolution experimental sum cross
section $\sigma_{\Sigma}$ (sum of the cross sections displayed in Fig. 2) with
the present theoretical absorption cross section (full line). The short-dashed
line represents the contribution by direct detachment of a single $L$-shell or
$K$-shell electron to the absorption cross section. As in Fig. 1, the long-
dashed curve is the cross section for photoabsorption of neutral silicon atoms
[33].

A double-peak structure was similarly observed in the multiple photodetachment
of C- [13], where the first resonance was attributed to a $1s\to 2p$
excitation to the $1s\,2s^{2}\,2p^{4}\;{}^{4}P$ term and the second, broader
resonance peak was attributed to higher excitations to ${}^{4}P$ terms mainly
of the $1s\,2s^{2}\,2p^{3}\,3p$ configuration. In analogy, one might therefore
assign the $1s^{-1}\,3p^{4}\;{}^{4}P$ term to the first $K$-shell excited
resonance of Si- and a blend of several ${}^{4}P$ terms associated with $1s\to
4p$ excitations to the second resonance peak. Our calculations confirm this
expectation, though one should keep in mind that they only provide a coarse
orientation on the positions. For any negative ion, the calculated levels
result from strong configuration interactions.

Figure 3 shows the comparison between the sum of the experimental cross
sections from Fig. 2 and our theoretical absorption cross section. The
individual theoretical resonances are represented by Voigt line profiles with
a Gaussian width of 0.05 eV that accounts for the experimental photon-energy
spread. The calculated Lorentzian width of virtually all these resonances
amounts to 0.57 eV. It is determined mainly by the KLL Auger transition that
fills the core hole and it is about 15% larger than the $K$-level width of
neutral silicon [35]. An energy shift of $-1.3$ eV was applied to the
theoretical resonance positions in order to line these up with the
experimental peak positions. This shift is within the uncertainty of the
theoretical resonance energies. In order to achieve the best possible
agreement between theory and experiment, a shift of similar magnitude was
applied to the calculated threshold energies for direct $K$-shell detachment
resulting in the values 1842.5 and 1843.3 eV for the
$1s^{-1}\,3p^{3}\,(^{4}S)\;^{5}S_{2}$ and
$1s^{-1}\,3p^{3}\,(^{4}S)\;^{3}S_{1}$ levels in neutral silicon, respectively.
Apart from some obviously not quite correct relative line strengths, the
general agreement between experiment and theory is very satisfying considering
the complexity of the problem under study. Furthermore, we like to point out
that, at energies above $\sim$1846 eV, our theoretical cross section for
photoabsorption of Si- agrees excellently with the absorption cross section
for photoabsorption of neutral silicon [33]. This supports our earlier
assumption that the presence of the extra electron in Si- does not influence
the cross section for direct inner-shell detachment.

Figure 4: Experimental (small filled circles with error bars) and theoretical
(large stars) Siq+ product-ion fractions (Eq. 3, panel a) and mean charge
state $\overline{q}=\sum_{q}qf_{q}$ (panel b).

In Fig. 4(a), the outcome of our cascade calculations, i.e., the distribution
of product-ion charge states $q=m-1$ resulting from $m$-fold photodetachment
(stars in Fig. 4), is compared to the corresponding experimental values

$f_{q}=\sigma_{q+1}/\sigma_{\Sigma}.$ (3)

The behavior of these charge-state fractions can be qualitatively understood
as follows. The cross sections below $\sim$1837 eV, i.e., below the first
$K$-shell excitation energy are mainly due to $L$-shell detachment and the
subsequent autoionization that fills the $L$-shell hole. Over the narrow
displayed energy range below 1837 eV, the branching ratios for the production
of the various final charge states are virtually constant. Therefore, all
partial cross sections $\sigma_{q+1}$ have a similar progression as functions
of photon energy and, thus, also the fractions $f_{q}$. This changes when a
$K$-shell hole is created by excitation or direct detachment. The subsequent
Auger processes that fill the $K$-shell hole significantly influence the
branching ratios giving higher weights to the higher charge states. This is
apparent for the $1s\to 3p$ resonance at about 1838 eV where the fraction
$f_{3}$ rises to above the fraction $f_{2}$. A similar behavior also occurs
(less drastically) at the $K$-shell detachment-threshold energy of $\sim$1841
eV.

At energies above $\sim$1880 eV, the distribution results  almost exclusively
from direct detachment of a $K$-shell electron and the subsequent Auger
cascade that produces the various product-ion charge states. The calculated
fractions are of the correct orders of magnitude. For individual charge states
the theoretical fractions deviate from the corresponding experimental values
by factors of up to 2. However, these uncertainties average out when the mean
product-ion charge state is considered [Fig. 4(b)].

We attribute the discrepancies between the experimental and theoretical
$f_{g}$ values to the approximations that we had to apply for making the
cascade computations tractable (see also Sec. III). The product-ion
distributions in our computations are quite strongly affected by the
thresholds for the subsequent Auger processes. Because we could only
accommodate an averaged single-configuration approach, the fine structure is
poorly taken into account and does not describe all the ionization channels
properly. Very likely, this is the main reason that the theoretical
predictions for $f_{2}$ and $f_{4}$ are underrated. In the future, we hope to
advance our cascade simulations, although this will require some major
developments and tests for some simpler system with, say, $Z<10$. In the
present cascade model, the ion fractions $f_{0}$ and $f_{1}$ are very tiny and
cannot be compared directly with the experimental data. Moreover, the current
cascade model does not predict any ions in the $q\geq 6$ channels. More
accurate predictions for the higher product-ion charge states will also
require the incorporation of simultaneous double-detachment processes, either
during the initial detachment and/or the subsequent decay (see, e.g., [36,
37]).

Below the threshold for $K$-shell detachment, the distribution of product
charge states depends on the individual $K$-shell excited core-hole levels.
Computationally, this is even more demanding than the calculation of the
cascades that follow the direct $K$-shell detachment, since the core-excited
configurations then contain one more electron as compared to the core-ionized
configurations. We have performed one such calculation for the
$1s^{-1}\,3p^{4}$ configuration. The resulting product-ion fractions (stars at
$\sim$1839 eV in Fig. 4) compare less well with the experimental ones. The
fractions $f_{3}$ and $f_{4}$ are underestimated by more than a factor of 2
resulting in a significantly too small mean charge state.

##  V Summary and Conclusions

The present joint experimental and theoretical study of multiple
photodetachment of the silicon anion via $K$-shell excitation and ionization
extends our previous work with light ions (see [7] and references therein) to
an atomic system with a partly occupied $M$ shell. This additional valence
shell challenges experiment and theory alike. The experimental challenge
arises from the fact that $K$-shell detachment cross sections decrease with
increasing nuclear charge. The cross section for the direct detachment of a
$1s$ electron is an order of magnitude smaller for Si- as compared to C-. The
photon-ion merged-beams method as implemented at the PETRA III synchrotron
successfully proved capable of meeting this challenge, with the high photon
flux from the XUV beamline P04 and the high product selectivity of the PIPE
setup being the key assets [23] .

The present theoretical calculations were mainly challenged by the strong
correlation effects that govern all anions and by the complexity of the
deexcitation cascades following after the initial creation of a $K$-shell
vacancy. In particular, the presence of the $M$-shell places additional
demands on the cascade calculations as compared to our earlier studies with
light negative ions that have been summarized in Ref. [7]. The large amount of
levels and transitions to be followed in such calculations requires a
systematic and consistent approach as now offered by the recently developed
Jac toolbox [17, 18].

The measured cross sections for net triple to sixfold photodetachment exhibit
a threshold for direct photodetachment of one $K$-shell electron. This
threshold is masked by resonances associated with $1s\to 3p$ and $1s\to 4p$
excitations in the language of single-particle excitations. The calculated
resonance positions agree with the values from a high-resolution measurement
by less than 2 eV. While the calculated relative resonance strengths agree
reasonably well with our measurements, they leave room for improved
theoretical approaches. The same is true for the fractions of the product ions
as obtained from the cascade computations. These turn out to be more reliable
for the deexcitation cascade that follows direct $K$-shell detachment than for
the one that evolves after $1s\to 3p$ excitation.

The statistical quality of the present data is somewhat lower as compared to a
recently published and similarly comprehensive study on multiple
photodetachment of C- ions [13]. This is partly due to the already mentioned
reduced cross section but also due to a lower photon flux which varies with
photon energy such that it rapidly decreases at energies above 1000 eV. Since
C-([He] $2s^{2}\,2p^{3}$) and Si-([Ne] $3s^{2}\,3p^{3}$) have analogue open-
shell structures, one might expect similarities in the progression of the
detachment cross sections. Indeed, the double-resonance structure at the
$K$-shell detachment threshold is common to both species. However, additional
small resonances at higher energies, that were discovered in the C- triple-
detachment channel, could not be detected in any of the measured Si-
photodetachment channels within the present level of statistical uncertainty.
Finally, we would like to mention that the experimentally derived and
calculated cross sections for Si- photoabsorption might be useful for
identifying silicon anions in cold cosmic gas clouds by their characteristic
x-ray absorption pattern.

###### Acknowledgements.

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association
HGF, for the provision of experimental facilities. Parts of this research were
carried out at PETRA III and we would like to thank Kai Bagschik, Frank
Scholz, Jörn Seltmann, and Moritz Hoesch for assistance in using beamline P04.
We are grateful for support from Bundesministerium für Bildung und Forschung
within the “Verbundforschung” funding scheme (grant nos. 05K19GU3 and
05K19RG3) and from Deutsche Forschungsgemeinschaft (DFG, project no.
389115454).

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