††thanks<EMAIL_ADDRESS>
# First Simultaneous K-p $\rightarrow(\Sigma^{0}/\Lambda)\,\pi^{0}$ Cross
Sections Measurements at 98 MeV/c

Kristian Piscicchiaa,b, Magdalena Skurzok∗,c,d, Michael Cargnellie,b, Raffaele
Del Grandef,b, Laura Fabbiettig,f, Johann Martone,b, Pawel Moskalc,d,
Alessandro Scordob, Àngels Ramosh, Diana Laura Sirghib,i, Oton Vazquez Doceb,
Johann Zmeskale,b, Sławomir Wycechj, Paolo Branchinik, Filippo Ceradinil,k,
Eryk Czerwińskic,d, Erika De Luciab, Salvatore Fiorem,n, Andrzej Kupsco,j,
Giuseppe Mandagliop,q, Matteo Martinib,r, Antonio Passerik, Vincenzo
Pateras,n, Elena Perez Del Rioc,d, Andrea Selcem,b, Michał Silarskic,d and
Catalina Curceanub
a Centro Ricerche Enrico Fermi - Museo Storico della Fisica e Centro Studi e
Ricerche “Enrico Fermi”, Roma, Italy, EU
b Laboratori Nazionali di Frascati INFN, Frascati (Rome), Italy, EU
c Institute of Physics, Jagiellonian University, Kracow, Poland, EU
dCenter for Theranostics, Jagiellonian University, Krakow, Poland, EU
e Stefan-Meyer-Institute for subatomic physics, Austrian Academy of Science,
Austria, EU
f Physik Department E62, Technische Universität München, Garching, Germany, EU
g Excellence Cluster “Origin and Structure of the Universe”, Garching,
Germany, EU
h Facultat de Física, Universitat de Barcelona, Barcelona, Spain, EU
i IFIN-HH, Institutul National pentru Fizica si Inginerie Nucleara Horia
Hulubei, Romania, EU
j National Centre for Nuclear Research, Warsaw, Poland, EU
k INFN Sezione di Roma Tre, Roma, Italy
l Dipartimento di Matematica e Fisica dell’Università “Roma Tre”, Roma, Italy
m ENEA, Department of Fusion and Technology for Nuclear Safety and Security,
Frascati (Rome), Italy
n INFN Sezione di Roma, Roma, Italy
oDepartment of Physics and Astronomy, Uppsala University, Uppsala, Sweden
p Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e
Scienze della Terra dell’Università di Messina, Messina, Italy
q INFN Sezione di Catania, Catania, Italy
r Dipartimento di Scienze e Tecnologie applicate, Università “Guglielmo
Marconi”, Roma, Italy
s Dipartimento di Scienze di Base ed Applicate per l’Ingegneria
dell’Università “Sapienza”, Roma, Italy

###### Abstract

We report the first simultaneous and independent measurements of the K-p
$\rightarrow\Sigma^{0}\,\pi^{0}$ and K-p $\rightarrow\Lambda\,\pi^{0}$ cross
sections around 100 MeV/c kaon momentum. The kaon beam delivered by the
DA$\Phi$NE collider was exploited to detect K- absorptions on Hydrogen atoms,
populating the gas mixture of the KLOE drift chamber. The precision of the
measurements ($\sigma_{K^{-}p\rightarrow\Sigma^{0}\pi^{0}}=42.8\pm
1.5(stat.)^{+2.4}_{-2.0}(syst.)\ \mathrm{mb}$ and
$\sigma_{K^{-}p\rightarrow\Lambda\pi^{0}}=31.0\pm
0.5(stat.)^{+1.2}_{-1.2}(syst.)\ \mathrm{mb}\,$) is the highest yet obtained
in the low kaon momentum regime.

††preprint: IPM/P-2012/009

The experimental investigation of the $\overline{\mathrm{K}}$N low energy
interaction aims at providing constrains to the non-perturbative QCD models in
the strangeness sector. Chiral perturbation theory cannot be extended from
SU(2) to SU(3) when a strangeness $S=-1$ quark is introduced in the theory.
This is due to the existence of broad resonances ($\Lambda(1405)$ and
$\Sigma(1385)$) just below the $\overline{\mathrm{K}}$N threshold. The
$\Lambda(1405)$ is an isospin $I=0$ state whose existence was first predicted
by Dalitz and Tuan DT1 ; DT2  as a resonant structure in a coupled-channel
formulation of the $\overline{\mathrm{K}}$N interaction. Since its first
experimental observation expl1405  the interpretation of this quantum state is
still controversial, although the inability of the quark models to predict its
mass correctly has strengthened the original idea of the $\Lambda(1405)$ being
a meson-baryon quasi-bound state DT1 ; DT2 , an interpretation that was later
formulated in terms of modern chiral lagrangians kaiser ; oset-ramos . The
spectral shape of the resonance is observed to depend on both the adopted
production mechanism and the measured decay channel exper ; hemingway ; zychor
; moriya ; agakishiev , a fact that is naturally explained in terms of a two-
poles structure for the $\Lambda(1405)$, first pointed out in oller  but found
in all chiral unitary models, of which the most recent versions IHW ; Guo ;
feijoo ; lu  (see also the review papers of Hyodo  and Mai:2020ltx ) have
already been constrained by the precise measurement of the energy shift and
width of the 1s kaonic-hydrogen state from the SIDDHARTA collaboration SIDD .
While all recent chiral models provide a rather similar description of the K-p
scattering length, the subthreshold elastic and inelastic amplitudes, which
are necessary to describe kaon nuclear absorption processes, may differ by
factors of more than three. Since the extrapolation of the scattering
amplitude below the threshold is so challenging, accurate experimental input
on the K-p$\rightarrow\Sigma^{0}\pi^{0}$ dynamics above the threshold is
strongly needed. The $\Sigma^{0}\pi^{0}$ channel, being almost purely isospin
$I=0$ (with a small $I=2$ component), gives much cleaner information with
respect to the charged $\Sigma^{\pm}\pi^{\mp}$ channels piscicchiaepjweb2018 .

Presently, the available data for the inelastic K-p
$\rightarrow\Sigma^{0}\pi^{0}$ cross section close to threshold are obtained
by means of indirect extrapolations, in bubble chamber experiments (s0pi01 ;
martin ; ferrari  and references therein), from the measurement of the
$\Lambda\pi^{0}$ channel, by exploiting the isospin symmetry. Three cross
sections are given at mean kaon momenta $p_{\mathrm{{K^{-}}}}=120,160$ and 200
MeV/c, which are affected by sizable relative errors (as large as 50% at
$p_{\mathrm{{K^{-}}}}=120$ MeV/c), and where the $p_{\mathrm{{K^{-}}}}$
momentum intervals around the central values are 25 MeV/c wide.

This work takes advantage of the low momentum negatively charged kaon beam
produced at the DA$\Phi$NE e+e- collider Gallo , which has been designed to
work at the center of mass energy of the $\phi$ meson, thereby delivering
almost monochromatic charged kaons with $p_{\mathrm{{K^{-}}}}\sim$127 MeV/c.
Utilizing the KLOE KLOEcite  spectrometer as an active detector
$\Sigma^{0}\pi^{0}$ and $\Lambda\pi^{0}$ events are reconstructed. The K-
absorption in-flight on Hydrogen atoms, from the gas mixture of the KLOE Drift
Chamber, is disentangled to independently obtain the K-p
$\rightarrow(\Sigma^{0}/\Lambda)\,\pi^{0}$ cross sections at
$p_{\mathrm{{K^{-}}}}=98\pm 10$ MeV/c.

In this work a total integrated luminosity of 1.74 fb-1, corresponding to the
2004/2005 KLOE data taking campaign, was analyzed by the AMADEUS
collaboration. The KLOE detector is centered around the DA$\Phi$NE interaction
region and consists of a large cylindrical Drift Chamber (DC) KLOEdc , for
tracking analysis, and a fine sampling lead-scintillating fibers calorimeter
KLOEemc , used in this study to determine position, time and energy of the
ionization deposits in the fibers caused by particles hitting the calorimeter
(clusters), all immersed in the axial magnetic field of a superconducting
solenoid (0.52 T). The DC has an inner radius of 0.25 m, an outer radius of 2
m and a length of 3.3 m and is filled with a mixture of Helium and Isobutane
C4H10 (90% of Helium and 10% of Isobutane in volume). For more details on the
DC and calorimeter performances, resolutions for the tracks and clusters
reconstruction, we refer to Refs. KLOEdc ; KLOEemc . The strategy for the
measurement of the K-p $\rightarrow\Sigma^{0}\pi^{0}$ and K-p
$\rightarrow\Lambda\pi^{0}$ cross sections consists in the identification of
K- absorptions in-flight on the Hydrogen atoms of the Isobutane molecules.

The identification of a $\Lambda(1116)$ hyperon, through its decay into a
proton and a negatively charged pion (BR${}_{\Lambda\rightarrow p\pi^{-}}$ =
(63.9 $\pm$ 0.5) %) PDG , represents the signature of a hadronic interaction
and the starting point of this analysis. Protons and pions are selected by
combining $dE/dx$ (truncated mean of the ADC collected counts due to
ionization in the DC gas) versus momentum information provided by the DC, with
the measurement of the clusters energies, as described in Ref. acta ; oton2016
. K- nuclear absorptions in the gas filling the DC volume are sorted by
requiring the radial distance of the $\Lambda$ decay vertex from the
DA$\Phi$NE beam pipe axis ($\rho_{\Lambda}$) to be greater than $30$ cm. The
cut is optimized based on Monte Carlo (MC) simulations (performed by means of
the standard KLOE GEANT digitization (GEANFI recons )), in order to minimize
the contamination with K- absorptions in the DC entrance wall (aluminated
carbon fiber), which amounts to less then 3% for the selected sample.

The events selection proceeds through the photons identification by time of
flight. Following the convention $\pi^{0}\rightarrow\gamma_{1}\,\gamma_{2}$
and $\Sigma^{0}\rightarrow\Lambda\gamma_{3}$, for the two channels under
study, the two (or three) neutral clusters in the calorimeter (characterized
by energy $E>20$ MeV, no associated track and not produced in a K+ decay) are
first selected by minimizing a time of flight based $\chi_{t}^{2}$ variable.
Further, the photons are associated to the $\pi^{0}$ (and to the $\Sigma^{0}$)
by a second $\chi_{m}^{2}$ analysis, using the $m_{\gamma_{1}\gamma_{2}}$ or
the combined $m_{\gamma_{1}\gamma_{2}}$ and $m_{\Lambda\gamma_{3}}$ invariant
masses information respectively. The constraints $\chi_{t}^{2}<20$,
$\chi^{2}_{m_{\gamma_{1}\gamma_{2}}}<5$ and
$\chi^{2}_{m_{\Lambda\gamma_{3}}}<4$ are tuned based on MC studies. A check
against cluster splitting (single clusters in the calorimeter erroneously
recognized as two clusters) is performed by analyzing the clusters distance as
a function of their energy acta . The selected sample is not significantly
affected by splitting. The algorithm’s efficiency for the photons clusters
recognition is $0.98\pm 0.01$.

In Fig. 1 $m_{\Lambda\gamma 3}$ is represented as function of $m_{\gamma
1\gamma 2}$. Gaussian fits to the invariant mass distributions yield the
resolutions $\sigma_{\gamma_{1}\gamma_{2}}\sim$ 20 MeV/c and
$\sigma_{\Lambda\gamma_{3}}\sim$ 15 MeV/c respectively. Accordingly, the
following additional cuts are applied for the two channels:
$\left(m_{\Lambda\gamma_{3}}-m_{\Sigma^{0}}\right)^{2}/(3\sigma_{\Lambda\gamma_{3}})^{2}+\left(m_{\gamma_{1}\gamma_{2}}-m_{\pi^{0}}\right)^{2}/(3\sigma_{\gamma_{1}\gamma_{2}})^{2}\,<\,1$
and
$\left|m_{\gamma_{1}\gamma_{2}}-m_{\pi^{0}}\right|\,<\,3\sigma_{\gamma_{1}\gamma_{2}}$.

Figure 1: The plot shows the $m_{\Lambda\gamma_{3}}$ distribution as function
of $m_{\gamma_{1}\gamma_{2}}$. The black ellipse represents the applied
selection in the two invariant masses space.

The optimization of the following phase space selections is aimed to
disentangle the signal processes (K-H
$\rightarrow(\Sigma^{0}/\Lambda)\,\pi^{0}$ absorptions in-flight) from the
competing background reactions, taking advantage of the signal’s distinctive
kinematics. This requires simulations of the involved reactions, which also
serve for the data fit.

When a negatively charged kaon is absorbed in the DC gas, the final state
$\Sigma^{0}\pi^{0}$ can be produced in K- \+ H $\rightarrow\Sigma^{0}+\pi^{0}$
at-rest ($ar$) or in-flight ($if$), or because the K- is absorbed on a bound
proton in 4He or 12C, $ar$ or $if$. Analogous reactions also give rise to the
$\Lambda\pi^{0}$ production both in direct processes (K-p
$\rightarrow\Lambda\pi^{0}$) or as a result of a $\Sigma^{0}$ decay (K-p
$\rightarrow\Sigma^{0}\pi^{0}\rightarrow\Lambda\gamma\pi^{0}$). Besides the
direct processes and $\Sigma^{0}$ decay, in the nucleus also elastic or
inelastic Final State Interactions (FSI) can end up with the observed hyperon
($Y\,=\,\Sigma^{0}\,,\,\Lambda$) and $\pi^{0}$. After phase space selections
the contribution of the FSI is minor, and will be considered in the evaluation
of the systematic uncertainties.

The case of K- absorption on H is straightforward, since the kinematics is
determined by energy-momentum conservation, for both $ar$ and $if$.
Calculation of the $if$ reaction requires as input the negatively charged kaon
momentum, which is sampled according to the true MC (i.e. not passed for the
events reconstruction) momentum distribution of the kaons inside the DC
volume. The simulations of the K- nuclear absorption processes are performed
according to a phenomelogical model io ; PisWycCur ; lpi ; lproton  which
provides, for each reaction, the total hyperon-pion momentum probability
distribution P($p_{Y\pi^{0}}$). The kinematics of an event is completely
defined by the momentum vectors ($\textbf{{p}}_{Y},\textbf{{p}}_{\pi^{0}}$)
(the residual(s) nucleus track(s) is not detected), which are generated by
sampling the P($p_{Y\pi^{0}}$) distribution, and applying energy and momentum
conservation. For each event the calculated
($\textbf{{p}}_{Y},\textbf{{p}}_{\pi^{0}}$) pairs represent the input for the
KLOE GEANT digitization, followed by the event reconstruction.

The signal K-H $\rightarrow(\Sigma^{0}/\Lambda)\,\pi^{0}$ $if$ is
characterized by an almost back-to-back production of the emerging $Y-\pi^{0}$
pairs. We use this feature to enhance the signal over background ratio. In
Figs. 2 (top) and (bottom) the reconstructed MC $p_{\pi^{0}}$ versus $p_{Y}$
distributions are shown, for signal events in the $\Sigma^{0}\pi^{0}$ and
$\Lambda\pi^{0}$ channels, respectively. The simulations are used to optimize
the two-dimensional cuts, which are represented in the Figs. 2 as black
contours.

Figure 2: The plot shows reconstructed MC $p_{\pi^{0}}$ vs. $p_{Y}$
distributions for the K- H $\rightarrow\Sigma^{0}\,\pi^{0}$ $if$ reaction
(top) and K- H $\rightarrow\Lambda\,\pi^{0}$ $if$ reaction (bottom). The phase
space selections are represented as black contours.

Additionally, thanks to the good resolution in the $\Lambda$ momentum
($\sigma_{p_{\Lambda}}\sim 1.9$ MeV/c for $\Lambda$s produced in the DC gas
acta ), K-H $\rightarrow\Lambda\pi^{0}$ ($ar$ and $if$) events can be
effectively sampled as they are characterized by the sharp angular correlation
$\cos\theta_{\Lambda\pi^{0}}<$ -0.85 (where $\theta_{\Lambda\pi^{0}}$ is the
angle between the two particles momentum vectors). Hence, this additional
condition is set for the $\Lambda\pi^{0}$ sample.

Figure 3:  From top to bottom the figure shows the result of the simultaneous
fit of p${}_{\Sigma^{0}\pi^{0}}$, m${}_{\Sigma^{0}\pi^{0}}$ and
$cos\theta_{\Sigma^{0}\pi^{0}}$. The experimental data and the corresponding
statistical errors are represented by black crosses, the systematic errors are
light blue boxes. The contributions of the various physical processes are
shown as colored histograms, according to the color code shown in the caption.
The light and dark bands correspond to systematic and statistical errors,
respectively. The gray distribution reproduces the global fit function.

Panels a) in Figs. 3 and 4 exemplify the core of the final analysis step. Due
to momentum conservation, the total hyperon-pion momentum ($p_{Y\pi^{0}}$)
distributions of the K-H $\rightarrow Y\pi^{0}$ $if$ samples reflect the
original K- momentum spectrum, as demonstrated by the pre-eminent peaks
centered at $p_{Y\pi^{0}}\sim 98$ MeV/c. This hallmark allows to efficiently
disentangle the remaining background due to the competing K- absorption
processes.

At this stage of the events selection 6.5% of the $\Sigma^{0}\pi^{0}$ events
lie in the region $p_{\Sigma^{0}\pi^{0}}>300$ MeV/c, a phase space sector far
away from the region of interest, mainly populated by FSI processes lpi . A
final phase space selection $p_{\Sigma^{0}\pi^{0}}<300$ MeV/c is then applied.

The selected $\Sigma^{0}\pi^{0}$ and $\Lambda\pi^{0}$ samples amount to 2130
and 7106 events, respectively.

The data are fitted by minimizing the following $\chi^{2}$ function:

$\chi^{2}=\sum\limits_{q}\sum\limits_{n=1}^{N_{bins}^{q}}\frac{(N_{n}^{q}-\mathcal{F}^{q}(q_{n}))^{2}}{\sigma_{n}^{q2}}\
,$ (1)

in which $q$ ranges over the observables, $n$ over the number of bins of the
$q$-th spectrum and $N_{n}^{q}$ is the measured content of the corresponding
$n$-th bin. The fit function $\mathcal{F}$ contains the following physical
processes (with the color code adopted in Figs. 3 and 4):

  1. 1.

$K^{-}\,H\quad\rightarrow\quad(\Sigma^{0}/\Lambda)\,\pi^{0}$ $if$ (red),

  2. 2.

$K^{-}\,H\quad\rightarrow\quad(\Sigma^{0}/\Lambda)\,\pi^{0}$ $ar$ (blue),

  3. 3.

$K^{-}+\,{}^{4}He/{}^{12}C\quad\rightarrow\quad\Sigma^{0}/\Lambda+\pi^{0}+{}^{3}H/{}^{11}B$
(magenta).

In Ref. lpi  equivalent contributions are measured, within the uncertainties,
for K- absorptions on 4He and 12C nuclei in the DC gas, as well as for the
competing in-flight and at-rest nuclear interactions of the kaon. Accordingly,
component 3 is prepared as an equal weight admixture of the mentioned
reactions and the related uncertainty is considered as a systematic effect.
For the $\Lambda\pi^{0}$ channel, besides the direct processes 1-3,
$\mathcal{F}$ also accounts for the same reactions initiated by
$\Sigma^{0}\pi^{0}$ production and followed by
$\Sigma^{0}\rightarrow\Lambda\gamma$ decay. $\mathcal{F}$ is given by the
linear combination:

$\mathcal{F}^{q}(q_{n})=\sum\limits_{i=1}^{N_{par}}\alpha_{i}\cdot
h_{i}^{q}(q_{n})\ ,$ (2)

with the index $i$ running over the number of free parameters $\alpha_{i}$ and
$h_{i}^{q}$ representing the distribution of the observable $q$, for the
$i$-th physical process, normalized to the data entries. In Eq. (1) the errors
are evaluated as
$\sigma_{n}^{q2}=\sqrt{\sum\limits_{i=1}^{N_{par}}(\alpha_{i}\cdot
h_{i}^{q}(q_{n}))^{2}+N_{n}^{q\ 2}}$ and the $\chi^{2}$ function minimization
is performed by means of the SIMPLEX, MIGRAD and MINOS routines of ROOT MINUIT
.

For the $\Sigma^{0}\pi^{0}$ sample a simultaneous fit is performed over
$q=m_{\Sigma^{0}\pi^{0}},\cos\theta_{\Sigma^{0}\pi^{0}},p_{\Sigma^{0}\pi^{0}}$,
which indicate the $\Sigma^{0}\pi^{0}$ invariant mass, angular correlation and
total momentum, respectively. In the $\Lambda\pi^{0}$ channel the sharp
selection of back-to-back events, renders the $\cos\theta_{\Lambda\pi^{0}}$
variable insensitive to the uncorrelated K- nuclear absorption reactions,
hence the - kinematically equivalent - combination
$q=m_{\Lambda\pi^{0}},p_{\Lambda},p_{\pi^{0}},p_{\Lambda\pi^{0}}$ is chosen.
The fits results are presented in Fig. 3 (for $\Sigma^{0}\pi^{0}$) and Fig. 4
(for $\Lambda\pi^{0}$) and the obtained parameters values are summarized in
Table 1.

Figure 4: From top to bottom the figure shows the result of the simultaneous
fit of p${}_{\Lambda\pi^{0}}$, m${}_{\Lambda\pi^{0}}$, pΛ and p${}_{\pi^{0}}$.
The experimental data and the corresponding statistical errors are represented
by black crosses, the systematic errors are light blue boxes. The
contributions of the various physical processes are shown as colored
histograms, according to the color code shown in the caption. The light and
dark bands correspond to systematic and statistical errors, respectively. The
gray distribution reproduces the global fit function.

The fit is not sensitive to the $\Lambda\pi^{0}$ kinematic distributions
originated in $\Sigma^{0}$ decays, for the K${}^{-}\,H$ interaction ($if$) and
the nuclear K- absorptions; it yields negligibly small contributions, which
are set to zero in the final fit.

The cross sections are calculated in accordance to:

$\sigma=\frac{N^{if}_{K^{-}p\rightarrow Y\pi^{0}}}{N_{K^{-}}\cdot n\cdot L},$
(3)

where the total number of expected signal events $N^{if}_{K^{-}p\rightarrow
Y\pi^{0}}$ is obtained re-scaling for the detection and reconstruction
efficiencies and for the branching ratio of the $\Lambda\rightarrow p\pi^{-}$
decay. Normalization to an absolute number of kaons is performed, taking
advantage of the K+ tagging lproton ; oton2016 . The number of projectiles
$N_{K^{-}}$ corresponds to the number of tagged kaons corrected for the decay,
by means of a MC simulation following the K- tracks (till the decay) in steps
of 1 cm. $n$ is the number density of Hydrogen scattering centers. $L$ is the
effective mean path length, obtained through a MC simulation, accounting for
the K- impinging angle, i.e. the angle between the tangent vector to the
particle’ trajectory at the DC entrance point, and the radial direction.

Table 1: The table summarizes the results obtained from the fits of the $\Sigma^{0}\pi^{0}$ and $\Lambda\pi^{0}$ samples. The values of the reduced chi-squares and of the fit parameters are summarized. $\bm{\Sigma^{0}-\pi^{0}}$ CHANNEL | $\frac{\chi^{2}}{dof}=\frac{92}{54}$ |  |
---|---|---|---
process | fit par. value | $\mathbf{\sigma_{stat.}}$ |
$K^{-}H\rightarrow\Sigma^{0}\pi^{0}\,(if)$ | 0.511 |  $\pm$ 0.018 |
$K^{-}H\rightarrow\Sigma^{0}\pi^{0}\,(ar)$ | 0.017 |  $\pm$ 0.005 |
$K^{-}+^{4}He/^{12}C\rightarrow\Sigma^{0}\pi^{0}$ |  |  |
\+ residual $(ar/if)$ | 0.463 |  $\pm$ 0.018 |
$\bm{\Lambda-\pi^{0}}$ CHANNEL | $\frac{\chi^{2}}{dof}=\frac{165}{57}$ |  |
process | fit par. value | $\mathbf{\sigma_{stat.}}$ |
$K^{-}H\rightarrow\Lambda\pi^{0}\,(if)$ | 0.659 |  $\pm$ 0.011 |
$K^{-}H\rightarrow\Lambda\pi^{0}\,(ar)$ | 0.021 |  $\pm$ 0.003 |
$K^{-}+^{4}He/^{12}C\rightarrow\Lambda\pi^{0}$ |  |  |
\+ residual $(ar/if)$ | 0.298 |  $\pm$ 0.012 |
$K^{-}H\rightarrow\Sigma^{0}\pi^{0}$ |  |  |
$\rightarrow\Lambda\gamma\pi^{0}\,(ar)$ | 0.018 |  $\pm$ 0.006 |

The obtained cross sections

  * •

$\sigma_{K^{-}p\rightarrow\Sigma^{0}\pi^{0}}=42.8\pm
1.5(stat.)^{+2.4}_{-2.0}(syst.)\ \mathrm{mb}$

  * •

$\sigma_{K^{-}p\rightarrow\Lambda\pi^{0}}=31.0\pm
0.5(stat.)^{+1.2}_{-1.2}(syst.)\ \mathrm{mb}\,$,

correspond to a mean kaon momentum $p_{K}=(98\pm 10)$ MeV/c, calculated on the
basis of the true MC information of the kaons momentum distribution inside the
DC volume.

The systematic errors are determined repeating several times the same fit
procedure, by varying independently all the analysis cuts which were optimized
for the $\Sigma^{0}\pi^{0}$ and $\Lambda\pi^{0}$ samples selection. The
systematic error on the $i$-th parameter of the fit, due to a variation of the
$j$-th cut, is defined as:

$\sigma_{syst,i}^{j}=\alpha_{i}^{j}-\alpha_{i}$ (4)

Total, positive and negative systematic errors are obtained by summing in
quadrature the positive and negative systematic fluctuations.

With the exception of those quantities for which the statistical error is
known (e.g. $m_{\gamma_{1}\gamma_{2}}$, $m_{\Lambda\gamma 3}$ and
$\cos\theta_{{\Lambda\pi^{0}}}$), in which case the systematics are evaluated
by applying $\pm 1\sigma$ fluctuations to the corresponding cuts, for the
other selections we chose to change the cuts of the amount necessary to
increase (or decrease) the selected number of events of 15%, with respect to
the standard. This is the case of the constraints on $\rho_{\Lambda}$,
$\chi_{t}^{2}$, $\chi^{2}_{m_{\gamma_{1}\gamma_{2}}}$ and
$\chi^{2}_{m_{\Lambda\gamma_{3}}}$, of the phase space selections in the
$p_{\pi^{0}}-p_{\Sigma^{0}}$ and in the $p_{\pi^{0}}-p_{\Lambda}$ planes. The
systematic uncertainty introduced by setting equal contributions of K-
absorptions on Helium and Carbon, both for the $ar$ and $if$ processes, is set
by performing 15% variations of the relative contribution of each process.

The $p_{\Sigma^{0}\pi^{0}}$ constraint was optimized based on a scan in the
range ($280\div 350$) MeV/c, in steps of 10 MeV/c (compatible with the
resolution $\sigma_{p_{\Sigma^{0}\pi^{0}}}\sim 15$ MeV/c) yielding the minimum
reduced $\chi^{2}$ for $p_{\Sigma^{0}\pi^{0}}=300$ MeV/c. The contribution to
the systematic errors is obtained by the condition $p_{\Sigma^{0}\pi^{0}}<310$
MeV/c.

The systematics introduced by the decay correction in the $N_{K^{-}}$
calculation and by the evaluation of $L$ are estimated by doubling the 1 cm
step length, and diminishing of 15% the number of simulated kaons.

We reported the analysis, by the AMADEUS collaboration, of the 1.74 fb-1
integrated luminosity collected by the KLOE collaboration in the period
2004/2005. Taking advantage of the low momentum K- beam delivered by the
DA$\Phi$NE collider, two independent and high statistics samples of K-
absorptions in-flight on Hydrogen were disentangled, with final states
$\Sigma^{0}\pi^{0}$ and $\Lambda\pi^{0}$, which allowed to perform high
precision measurements of the inelastic K-p
$\rightarrow(\Sigma^{0}/\Lambda)\,\pi^{0}$ cross sections, for a K- momentum
$p_{K}=(98\pm 10)$ MeV/c:

  * •

$\sigma_{K^{-}p\rightarrow\Sigma^{0}\pi^{0}}=42.8\pm
1.5(stat.)^{+2.4}_{-2.0}(syst.)\ \mathrm{mb}$

  * •

$\sigma_{K^{-}p\rightarrow\Lambda\pi^{0}}=31.0\pm
0.5(stat.)^{+1.2}_{-1.2}(syst.)\ \mathrm{mb}\,$.

With respect to previous experiments (s0pi01 ; martin ; ferrari  and
references therein), which extrapolated
$\sigma_{K^{-}p\rightarrow\Sigma^{0}\pi^{0}}$ from the measurement of the K-p
$\rightarrow\Lambda\pi^{0}$ cross section, assuming isospin symmetry, this is
the first direct simultaneous measurement of the cross sections in the isospin
$I=1$, and almost pure $I=0$ channels. Alongside a comparable range for the
kaon momentum ($\pm$10 MeV/c with respect to $\pm$12.5 MeV/c in Refs. s0pi01 ;
martin ; ferrari  and references therein) the precision achieved in the cross
section determination greatly overcomes all the other low momentum
measurements, providing a key input to the theory for the extrapolation
towards and below the $\overline{K}N$ threshold region.

## Acknowledgements

We acknowledge the KLOE/KLOE-2 Collaboration for their support and for having
provided us with the data and the tools to perform the analysis presented in
this paper. Part of this work was supported by the EU STRONG-2020 project
(grant agreement No 824093); Minstero degli Affari Esteri e della Cooperazione
Internazionale, EXOTICA project, PO21MO03; the Polish National Science Center
through grant No. UMO-2016/21/D/ST2/01155. The project is co-financed by the
Polish National Agency for Academic Exchange, grant no PPN/BIT/2021/1/00037.
The authors acknowledge support from the SciMat and qLife Priority Research
Areas budget under the program Excellence Initiative-Research University at
the Jagiellonian University.

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# Texture tomography, a versatile framework to study crystalline texture in 3D

M.P.K. Frewein  Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel,
Marseille, France  J.K. Mason  University of California, Davis, California  B.
Maier  University of Konstanz, Konstanz, Germany  H. Cölfen  University of
Konstanz, Konstanz, Germany  A.A. Medjahed  European Synchroton Radiation
Facility, Grenoble, France  M. Burghammer  European Synchroton Radiation
Facility, Grenoble, France  M. Allain  Aix Marseille Univ, CNRS, Centrale Med,
Institut Fresnel, Marseille, France  T. A. Grünewald  Aix Marseille Univ,
CNRS, Centrale Med, Institut Fresnel, Marseille, France

###### Abstract

The crystallographic texture is a key organization feature of many technical
and biological materials. In these materials, especially hierarchically
structured ones, the preferential alignment of the nano constituents is
heavily influencing the macroscopic behaviour of the material. In order to
study local crystallographic texture with both high spatial and angular
resolution, we developed Texture tomography (TexTOM). This approach allows to
model the diffraction data of polycrystalline materials by using the full
reciprocal space of the ensemble of crystals and describe the texture in each
voxel via a orientation distribution function. This means, it provides 3D
reconstructions of the local texture by measuring the probabilities of all
crystal orientations. The TexTOM approach addresses limitations associated
with existing models: It correlates the intensities from several Bragg
reflections, thus reduces ambiguities resulting from symmetry. Further, it
yields quantitative probability distributions of local real space crystal
orientations without further assumptions on the sample structure. Finally, its
efficient mathematical formulation enables reconstructions faster than the
time-scale of the experiment. In this manuscript, we present the mathematical
model, the inversion strategy and its current experimental implementation. We
show characterizations of simulated data as well as experimental data obtained
from a synthetic, inorganic model sample, the silica-witherite biomorph. In
conclusion, TexTOM provides a versatile framework to reconstruct 3D
quantitative texture information for polycrystalline samples. In this way, it
opens the door for unprecedented insights into the nanostructural makeup of
natural and technical materials.

##  1 Introduction

The properties of many materials rely on their arrangement on the nano- and
crystal structural level. While this organization has shown to be of great
importance for a wide host of both technical and biological materials such as
concrete, steel or bone, wood, shells and tendons to give but a few examples,
its actual characterization poses a problem up to the current day. The
challenge for a successful characterization during in-situ and in-operando
studies is to enable a high spatial and angular resolution whilst maintaining
a large field of view and ideally providing a non-destructive imaging
modality. While electron microscopy based techniques can boast impressive
spatial resolution and with focused-ion beam tomography supplies the
possibility of 3D characterization, the investigations are restricted to
destructive sampling and in most cases in-vacuum operation. X-rays however
lend themselves to the task as they easily penetrate millimeters, even
centimeters in the case of hard X- rays for most technical materials. Recent
advances in nanofocusing [1] has enabled the routine operation with beam sizes
of 50 nm or less. The advent of 4th generation synchrotron sources such as
MaxIV [2] or ESRF-EBS [3] has further unlocked the potential for in-situ
studies due to the impressive boost in flux that these machines deliver.

X-ray-based tomography methods have a long history of successful materials
characterization. Following the routine implementation of full-field X-ray
tomography and phase tomography [4], X-ray holo tomography [5] and
ptychographic tomography [6] have enabled spatial resolution on a single digit
nanometer scale. The coupling of tomography of X-ray diffraction further
opened the possibility to obtain orientation information, which is of high
interest for polycrystalline materials with a non-random orientation
distribution of crystallites. This property is called the crystallographic
texture [7]. A whole class of 3D techniques exists to obtain local
orientations of crystals from the position of Bragg-peaks such as X-ray Bragg
ptychography [8], Bragg CDI [9], 3D-XRD [10] or DFXM [11]. All these
techniques boast impressive angular resolution given, however, that the
diffraction patterns show clearly distinguishable Bragg reflections.
Hierarchically structured materials such as biominerals (e.g. bone, tendon,
shell, cuticle) but also technical ceramics and some deformed metals can be
composed of a high number of crystallites whose crystal axes are locally
distributed around a common mean orientation. This leads to azimuthally
overlapping diffraction peaks and yields images resembling powder diffraction
patterns with azimuthal variations in intensity, which are no longer possible
to describe by a model that specifically addresses each individual crystallite
or grain.

Another way to approach the problem is to consider first scattering tomography
without orientation information, e.g. diffraction tomography approaches [12,
13] or SAXS tomography [14]. All these approaches assume a random orientation
of the sample crystalline or nanostructural feature. In order to reconstruct
directions based on nanostructural orientation required new ways to probe the
reciprocal space and a parametrization for the orientation information, first
shown by Georgiadis et al. [15] for serial 2D slices This lead to the
development of tensor tomography, for which the seminal papers of Liebi et al
[16, 17, 18] have shown approaches to reconstruct orientation tensors from the
small-angle scattering signal of nanostructures (SASTT), which have since
found a wide host of scientific applications. The technique has also been
extended into the wide-angle regime by Grünewald et al. for investigating
Bragg peaks [19] and used to study material properties of cartilage [20],
tendon [21], nervous tissue [22] and bone [23], to give a non-exhaustive
overview. Recent developments in the field have seen the introduction of more
flexible reconstruction approaches that aim at overcoming some of the isotropy
and sampling assumptions of the original approach as well as improving
reconstruction speed [24, 25]. In particular Nielsen et al. have presented
significant performance improvements as well as an enhanced model flexibility
by introducing band-limited spherical functions for the reconstruction.

All the aforementioned methods aim to reconstruct features in the diffraction
pattern by modelling their intensity directly in reciprocal space. By
reconstructing the position of a Bragg-peak in 3D reciprocal space, for
example, they obtain the preferred direction of a lattice plane or axis of a
crystal. This leaves, however, one rotational degree of freedom to the
orientation of the crystal, which could be obtained by independent
reconstructions on several peaks and subsequent relation to the full crystal
orientation tensor, in analogy to [26]. The approach neglects however the
interconnection between the positions of the Bragg-peaks during the
reconstruction, which is given by the crystal symmetry, which in many cases is
a priori well known. Motivated by classical texture analysis, a full
implementation of the orientation distribution function (ODF) [27] is used to
tackle this problem. This means, we directly model the real space texture by
assigning a probability to each unambiguous orientation of the crystal. The
ODF fully determines the position of all Bragg peaks at once, allowing the
reconstruction of full diffraction patterns with a single model.

Orientations of 3D objects have three degrees of freedom, which are
traditionally described by Euler angles. A framework using a series expansion
of generalized spherical harmonics [28, 29] is often used to build the ODF and
finds it use in state-of-the art software packages [30]. The use of spherical
harmonics ensures a low number of adjustable parameters while providing the
flexibility to model probabilities all possible orientations. Other approaches
building on alternative ODF implementations and direct inversion strategies
[31, 32, 33] exist, but are geared more towards sharp textures usually
obtained in metals and geological samples and are not reviewed here in detail.

There are however shortcomings to the Euler angle parametrization, such as the
degeneracy of orientations [34, 35, 36], a distortion of the metric tensor
[37] and singularities in the equations of motion [35]. To overcome these
problems, we use an 3D harmonic expansion, which uses an axis-angle rotation
parametrization to describe orientations[38]. This framework of hyperspherical
harmonics (HSH) offers additional advantages such as computationally efficient
rotation operations and the possibility of symmetrization according to the
proper point group, which drastically reduces the number of parameters and
opens the possibility to use only the fundamental zone of the orientation
space.

Coupling this versatile model for describing crystallographic textures to
tensor tomography, we are presenting texture tomography (TexTOM) as a
computationally efficient framework to reconstruct full crystallographic
texture information in 3D based on scanning X-ray diffraction patterns. This
lays the foundation for quantifying local texture by ODFs using the full
information available in the diffraction patterns simultaneously. The outlined
method will be described in details in terms of its mathematical underpinning
and the actual implementation of the code in python. Numerical simulations to
benchmark the performance of the method will be shown alongside the first
experimental characterization of a helicoidal silica-witherite biomorph as an
example of a hierarchically textured material.

##  2 Materials and Methods

###  2.1 Biomorph sample

Silica-carbonate biomorphs are a hierarchically structured, polycrystalline
material. They are generated by the precipitation of barium carbonate in
silicate-rich media at elevated pH [39, 40]. During the formation crystalline
witherite (BaCO3) nanorods are formed and embedded in a silica matrix.
Together, the biomorphs can take a variety of complex, curved shapes. The
exact process governing the final shape is not fully understood yet[41]. By
modifying the properties of either the crystalline fraction [42] or
functionalizing the silica matrix [43, 44, 45] new functionalities can be
added, making biomorphs an attractive material system from a material
chemistry point of view. The biomorph sample employed here was produced a one-
pot co-precipitation method[46]. For this, in each field of a 6-field well
plate 5 mL of a 10 mM barium chloride solution were added to 5 mL of a 16.8 mM
sodium meta silicate solution. The starting pH was adjusted to 11 using 1 mM
NaOH. The gradual diffusion of atmospheric CO2 into the solution then yielded
silica-witherite biomorphs of varying shapes at the bottom of the well-plate.
After 15 h the residual solution was removed and the structures were
subsequently washed with water and ethanol before they were carefully detached
from the bottom of the wells using a silicone brush. After transferring the
biomorphs to a centrifuge vial using a pipette they were sedimented in a
centrifuge using 9000 rpm for 10 min. After decanting the supernatant the
structures were dried. For the synchrotron experiments, a $\sim$
$60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long piece was mounted on a thin
($\sim$ $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$) glass capillary with epoxy
glue.

###  2.2 Synchrotron experiments

For the experimental characterization, X-ray diffraction experiments were
carried out at the ESRF-EBS beamline ID13 EH3 nanobranch. A 15.2 keV X-ray
beam was produced by a channel-cut Si(111) monochromator and pre-focused by a
set of compound beryllium lenses onto the final focusing optics, a set of
multi-layer Laue lenses (MLL), producing a beam of 300x300 nm with a flux of
1012 photons/s on the sample position. The sample was mounted on a custom-
designed goniometer [19] based on Smaract actuators and scanned by a piezo
stage. The diffraction signal was recorded with an Eiger X 4M and 157.78 mm
sample- detector distance. The primary beam was blocked by a
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ lead beamstop. The setup gave
access to a usable $q$-range of 0.5 – 32 nm-1 and detector edges extending up
to 40 nm-1. For each projection, the full sample was scanned with a step-size
of 500 nm in a continuous scanning mode and an adaptive field of view that
enabled to catch the full sample (maximum size 90 x
$70\text{\,}\mathrm{\SIUnitSymbolMicro m}$) while avoiding excessive air
regions around the sample. Diffraction patterns were recorded with 2 ms
exposure time. Subsequently, the sample was rotated around the $z^{\prime}$
axis and tilted around the $y^{\prime}$ axis. A total of 260 projections were
collected for 10 tilt angles between 0 and 45°. At the 0° tilt angle, rotation
angles were collected between 0 and 180°, for all other tilt angles between 0
and 360°. The number of rotation angles in every tilt angle was reduced with a
factor $\cos{\varkappa}$. In total, the data acquisition took 6 hours (with
motor movement overheads). The total time of data acquisition was 3.6 h (6.5
Mio. diffraction patterns $\times$ 0.002 s) hours The dose deposited on the
sample was calculated according to Howells et al. [47] as follows:

$d=\frac{\mu N_{0}\epsilon}{\rho}$ (1)

Where $\mu=269.58$ cm-1 is the linear absorption coefficient for BaCO3 at 15.2
keV, $N_{0}=2.3040\times 10^{-15}$ m2 is the incident flux per unit area,
$\epsilon=2.4370\times 10^{-15}$ J is the photon energy and $\rho=4.3$ g cm-3
the mass density. Thus, the dose imparted on the sample during the full scan
is $3.4\times 10^{10}$ J/kg. Under the assumption that each voxel absorbs an
equal amount of radiation, this equates to a dose of 1.3 x $10^{6}$ J/kg per
voxel.

###  2.3 Simulated sample

To test the overall functioning of the analysis, we generated data from a
sample of 20$\times$20$\times$20 voxels and placed a Gaussian ODF in each
voxel. That is, we defined a mean orientation $g_{\mu}$ and calculated the
probability for each orientation the angular distance $dg$ (see appendix Eq.
39) from $g_{\mu}$:

$\rho(g)=\frac{1}{\mathcal{N}}\exp\left(-\frac{dg(g,g_{\mu})^{2}}{2\sigma^{2}}\right)$
(2)

The normalization $\mathcal{N}$ was adjusted so that $\int\rho(g)d\Omega=1$
(volume element $d\Omega$ defined in Eq. 19). Standard deviations $\sigma$
were set to 40°. These distributions were then mimicked by HSHs of order
$n=12$ to facilitate rotations in the laboratory frame. The sample was
generated with stripes of equal distributions in $x$-direction and random
orientations along the other axes. We generated images according to Eq. 26
with a BaCO3 structure factor and produced 108 projections for 4 tilt angles
from 0 to 45° and rotation angles in an equidistant matter as described above.
Data was renormalized to a maximum number of 200 counts per bin, then Poisson
noise was added to simulate the conditions of a typical measurement.

###  2.4 Texture tomography inversion

The data was regrouped into 120 azimuthal and 50 $q$ bins over a range from 10
to 35 nm-1 using the PyFAI package [48]. $q$-regions between the Bragg-peaks
were masked before refinement. To correct for deviations from the true
rotation center of the sample, the individual projections were aligned using
the tomographic self-consistency method [49]. The regular grid of 140x140x180
voxels was constructed, of which 26425 voxels were identified as sample based
on the scattering intensity in the SAXS region ($q$ 0.5 - 1 nm-1). Gaussian
beam intensities were calculated as described in sec. 3.2 and voxels receiving
$<1\%$ of the maximum intensity were excluded from the simulation of the
respective diffraction pattern. BaCO3 single crystal diffraction patterns were
simulated using a published witherite crystal structure [50], using the
lattice parameters $a=5.3072$ Å, $b=8.8928$ Å, $c=6.4245$ Å and space group
Pmcn. Rotation symmetry generators for fundamental zone and HSH symmetrization
were $2_{001}$ and $2_{010}$ as for the 222-point group.

All reconstructions were carried out on a standard compute node, equipped with
a double CPU setup (2x AMD Epyc 7662 64 core) and 2 TB of RAM. The TexTOM
reconstruction code is written in python, using numpy and numba for just in
time compilation and parallelization of the essential parts of the code. No
further code optimization has been carried out and we expect that GPU
portation of the code will enable a further massive speed-up of the
computations, owing to the small memory footprint of the actual inversion
problem. Further information on the reconstruction times are given in Table 1.
A damping factor of $k=2$ was used to ensure positivity of the ODF.

##  3 Texture tomography

###  3.1 Overview

The general idea of Texture tomography is to provide a reconstruction scheme
to extract quantitative, local crystallographic texture information in 3D from
a series of X-ray diffraction patterns of a sample containing polycrystalline
domains in various orientations.

A brief overview of the experimental configuration is given in Fig. 1a. A
sample is mounted on a goniometric stage that enables scanning
($y^{\prime}$/$z^{\prime}$ direction), rotation and tilting
($\phi$/$\varkappa$) of the sample. The sample is raster scanned with a
focused X-ray beam and 2D diffraction patterns are collected at each scan
position. This procedure is repeated for various tilt and rotation angles, in
strict analogy to tensor tomography [16].

Figure 1: Schematics of acquiring experimental and simulated diffraction
patterns. a) The sample is raster-scanned using a focused X-ray beam in the
$y^{\prime}/z^{\prime}$ direction for various rotation ($\phi$) and tilt
($\varkappa$) angles. At each point, a full diffraction pattern is collected,
parametrized by the momentum transfer $q$ and the azimuthal component $\chi$
of diffraction. b) A simulated diffraction pattern originates from an ODF and
a crystal structure. The ODF is parametrized by orientations the three angles
($\omega,\theta,\phi$), which describe axis-angle rotations in the sample
coordinate system ($x,y,z$). The shown ODF is color-coded so that brighter
colors mean higher probability of the respective orientation. Each crystal
orientation yields a different single crystal diffraction pattern and the
resulting image is the sum over all of them weighted by the ODF.

The reconstructed quantity is a 3-dimensional orientation distribution
function (ODF), representing the local arrangement of the crystallites via
probabilities of all unambiguous orientations. Given that the structure of a
single crystal is known, we can simulate the diffraction pattern of a
polycrystalline sample by an ODF-weighted sum over the diffraction patterns of
all crystal orientations. The challenge in this approach is that a faithful
reconstruction of the diffraction pattern requires a high angular resolution,
to ensure not to miss contributions from Bragg reflections from orientations
between the sampling points. Summing over the single crystal patterns for all
orientations can therefore become computationally expensive, when it comes to
parameter optimization of large samples.

We therefore choose to build up our diffraction patterns from a basis set of
elementary images, further labelled diffractlets, which originate in a set of
orthogonal functions used to model the ODF. The basis is given by a series
expansion of hyperspherical harmonics (HSH)[38], similar to a Fourier
expansion in 1D and spherical harmonics in 2D. This model therefore encodes
the ODF in expansion coefficients, which are optimized in an iterative
process. The calculation is roughly divided into the following steps:

  * •

The sample is partitioned into cubic voxels, whose dimensions correspond to
the stepsize of the raster scan.

  * •

Each voxel contains an ODF, given by a set of coefficients of the HSH
expansion.

  * •

We calculate the projected expansion coefficients by summing them over all
voxels, weighted by the respective beam intensity.

  * •

A diffraction pattern is simulated by summing over the diffractlets multiplied
by the projected coefficients.

  * •

The discrepancy between simulation and data is calculated by an error metric,
which is minimized iteratively by optimizing the voxel-specific coefficients.

###  3.2 Sample Translations and Rotations

The sample center is located at $\mathbf{x}^{\prime}_{0}$ in the laboratory
coordinate system (CS) as shown in Fig. 1 and we call
$\mathbf{x}^{\prime}_{i}$ the position of voxel $i$ in laboratory coordinates.
We assign a central voxel around which the rotations are performed, located at
$\mathbf{x}^{\prime}_{0}$, and surround it by a cubic lattice with edge length
$\Delta x$, which is the distance between two neighboring measurements. When
we rotate the sample by the angles $\phi$ and $\varkappa$, we can calculate
the position of voxel $i$ in the sample CS with origin at
$\mathbf{x}^{\prime}_{0}$ by:

$\mathbf{x}_{i}=\mathbf{x}^{\prime}_{0}+\hat{R}~{}(\mathbf{x}^{\prime}_{i}-\mathbf{x}^{\prime}_{0})$
(3)

using the rotation matrix for Euler angle rotations around $z^{\prime}$ and
$y^{\prime}$

$\hat{R}(\phi,\varkappa)=\left[{\begin{array}[]{ccc}\cos{\phi}\cos{\varkappa}&-\sin{\phi}&\cos{\phi}\sin{\varkappa}\\\
\sin{\phi}\cos{\varkappa}&\cos{\phi}&\sin{\phi}\sin{\varkappa}\\\
-\sin{\varkappa}&0&\cos{\varkappa}\\\ \end{array}}\right].$ (4)

Rotating the sample mathematically comprises the challenge of rotating all
voxel positions and interpolating their values on a new grid of coordinates.
This process is slow and prone to produce numerical errors. For calculating
the relative positions of sample and X-ray beam, we therefore keep the sample
CS static and rotate the function describing the beam by the transposed
rotation matrix $\hat{R}^{\intercal}(\phi,\varkappa)$. In addition, we have to
include translations of the sample within the $y^{\prime}$-$z^{\prime}$ plane
(given by displacement indices $t_{y}$, $t_{z}$ and voxel size $\Delta x$). In
the sample CS the translation of the beam is the negative sample displacement
in the laboratory CS.

To calculate the beam path, we define 2 points traversed by the beam in sample
coordinates: $\mathbf{t}$ is the point, where the beam traverses the
$y^{\prime}$-$z^{\prime}$ plane:

$\mathbf{t}(\phi,\varkappa)=-\hat{R}^{\intercal}(\phi,\varkappa)\mathbf{t^{\prime}}=-\Delta
x~{}\hat{R}^{\intercal}(\phi,\varkappa)\left[{\begin{array}[]{@{\kern
0pt}c@{\kern 0pt}}.0pt\lx@intercol\hfil 0\hfil\kern 0.\\\
.0pt\lx@intercol\hfil t_{y}\hfil\kern 0.\\\ .0pt\lx@intercol\hfil
t_{z}\hfil\kern 0.\end{array}}\right]$ (5)

$\mathbf{b}$ is another point on the beam path, resulting from adding the beam
direction unit vector $\mathbf{b^{\prime}}=\hat{x}^{\prime}$:

$\mathbf{b}(\phi,\varkappa)=\hat{R}^{\intercal}(\phi,\varkappa)(\mathbf{b^{\prime}}-\mathbf{t^{\prime}})=\hat{R}^{\intercal}(\phi,\varkappa)\left(\left[\begin{array}[]{@{\kern
0pt}c@{\kern 0pt}}.0pt\lx@intercol\hfil 1\hfil\kern 0.\\\
.0pt\lx@intercol\hfil 0\hfil\kern 0.\\\ .0pt\lx@intercol\hfil 0\hfil\kern
0.\end{array}\right]-\Delta x\left[{\begin{array}[]{@{\kern 0pt}c@{\kern
0pt}}.0pt\lx@intercol\hfil 0\hfil\kern 0.\\\ .0pt\lx@intercol\hfil
t_{y}\hfil\kern 0.\\\ .0pt\lx@intercol\hfil t_{z}\hfil\kern
0.\end{array}}\right]\right)$ (6)

The beam intensity $B(\mathbf{x}_{i})$ at each voxel position is calculated
from the cumulative distribution function of the beam profile (e.g. Gaussian)
in function of the normal distance $w$ between beam axis and voxel center.
Note that any kind of experimentally determined beam profile can be used here.
Using $\Delta x/2-w$ as an argument gives the intensity until the voxel
border, assuming that the beam width is equal or smaller than the voxel. For a
Gaussian beam profile with standard deviation $\sigma$ this gives

$B(\mathbf{x}_{i})=\frac{1}{2}\biggl{[}1-\text{erf}\left(\frac{\Delta
x/2-w(\mathbf{x}_{i})}{\sqrt{2}\sigma}\right)\biggr{]}$ (7)

The distance $w(\mathbf{x}_{i})$ is calculated from the from the normal
distance the line traversing points $\mathbf{t}$ and $\mathbf{b}$ and the
voxel center $\mathbf{x}_{i}$:

$w(\mathbf{x}_{i})=\frac{|(\mathbf{x}_{i}-\mathbf{t})\times(\mathbf{x}_{i}-\mathbf{b})|}{|\mathbf{b}-\mathbf{t}|}.$
(8)

In order to account for the orientation-dependent absorption of the diffracted
beam in the sample, an implementation of the absorption correction outlined in
[23] can be used. Due to the low absorption of the samples employed in this
study, it was currently not implemented.

###  3.3 Single Crystal Diffraction Patterns

The diffraction pattern of a single crystal $I_{sc}(\mathbf{q})$ is calculated
from all the atom positions $\mathbf{R_{j}}$ from the structure factor
$S(\mathbf{q})$.

$I_{sc}(\mathbf{q})\propto
S(\mathbf{q})=\frac{1}{\sum_{i}f_{i}}\sum_{j}\sum_{k}f_{j}(\mathbf{q})f_{k}(\mathbf{q})\text{e}^{i\mathbf{q}(\mathbf{R_{j}}-\mathbf{R_{k}})},$
(9)

We use atomic form factors $f(q)$ as tabulated in [51].

The crystal is created from a known crystal structure and the unit cell is
repeated in 3 dimensions to resemble the expected crystal size. We would like
to note that the interface is open to accept other input sources for
$S(\mathbf{q})$ such as Discus [52] in order to provide a more detailed
modelling of the crystalline parameters in the future. The
$\mathbf{q}$-vectors corresponding to X-ray wavelength $\lambda$ and
experimental setup are calculated using the surface of the Ewald sphere [53].
With the beam oriented along $\hat{x}^{\prime}$ (see Fig. 1), the origin of
the sphere in reciprocal space is at $(-2\pi/\lambda,0,0)$ and its radius is
$2\pi/\lambda$. We express these points in terms of the momentum exchange
$q=\frac{4\pi}{\lambda}\sin{\theta}$ and the angle in the detector plane
$\chi$

$\displaystyle\mathbf{q}=\left({\begin{array}[]{c}q\cos(\theta^{\prime})\\\
-q\sin(\theta^{\prime})\sin(\chi)\\\ q\sin(\theta^{\prime})\cos(\chi)\\\
\end{array}}\right),\qquad\theta^{\prime}=\pi/2+\theta,$ (13)

with $q$ and $\chi$ corresponding to the polar coordinates on the detector,
defined in Fig. 1.

In the construction of textured diffraction patterns, it is necessary to
compute the contribution from each unambiguous orientation of the crystal. The
diffraction pattern $I^{g}_{sc}(q,\chi)$ of a crystal in orientation $g$ is
calculated by rotating all atom positions and using the structure factor (Eq.
9).

###  3.4 Orientation Distribution Functions

An orientation distribution function (ODF) $\rho(g)$ assigns a probability to
a rotation $g$ from a reference orientation and is expedient to describe
properties of an ensemble of objects with different orientations. We use it to
model the scattering intensity $I(q,\chi)$ produced by an ensemble of
crystallites, assuming all of them have the same crystal structure in
different orientations. The ODF connects single crystal diffraction patterns
$I^{g}_{sc}$ to an image resulting from an ensemble by an integration over all
crystallite orientations $g$:

$I(q,\chi)=\int\rho(g)~{}I^{g}_{sc}(q,\chi)~{}d\Omega.$ (14)

####  3.4.1 Hyperspherical Harmonic Expansion

We model the ODF as a series expansion of hyperspherical harmonics (HSH)
$Z^{n}_{lm}(g)$ (see appendix and [54]). These are complex functions that are
naturally written using the axis-angle rotation parametrization. We use these
rotations to describe crystal orientations with respect to the fixed sample
CS. The axis-angle parameterization expresses a 3D rotation using a single
rotation by an angle $\omega\in[0,\pi]$ around a unit vector axis
$\mathbf{\hat{a}}$, which we define by polar angle $\vartheta\in[0,\pi]$ and
azimuthal angle $\varphi\in[0,2\pi)$:

$\displaystyle\mathbf{\hat{a}}=\left({\begin{array}[]{c}\sin(\vartheta)\cos(\varphi)\\\
\sin(\vartheta)\sin(\varphi)\\\ \cos(\vartheta)\\\ \end{array}}\right).$ (18)

The volume element for this parametrization is given by:

$d\Omega=\sin(\omega/2)\sin\vartheta d\omega d\vartheta d\varphi.$ (19)

Since the HSHs form an orthonormal basis for functions of rotations, it is
possible to construct an ODF as a linear combination of HSHs, where
$Z^{0}_{0,0}$ provides the probability mass and higher order HSHs redistribute
this probability mass as a function of $g$. The total ODF depends on a set of
complex complex coefficients $c^{n}_{l,m}$ as:

$\rho(g)=\sum_{n,l,m}c^{n}_{lm}Z^{n}_{lm}(g)$ (20)

Furthermore, the ODF of a rotated sample can also be written in the form of
Eq. 20 with a different set of coefficients. The transformation of the
coefficients depends on the rotation to which the sample is subject. This is
described by the matrix $R^{n/2}_{l^{\prime}m^{\prime}lm}(g_{l},g_{r})$ [55],
which allows one to write the effect of a rotation of a HSH $Z^{n}_{l,m}$ as a
linear combination of other HSHs $Z^{n}_{l^{\prime},m^{\prime}}$ of the same
order $n$ (see appendix A)

$\rho^{\prime}(g_{l}gg_{r})=\sum_{n,l,m}\sum_{l^{\prime}m^{\prime}}c^{n}_{lm}Z^{n}_{l^{\prime}m^{\prime}}(g)\,R^{n/2}_{l^{\prime}m^{\prime}lm}(g_{l},g_{r}).$
(21)

The variable $g$ of the initial ODF defined in Eq. 20 is related to the
variable $g^{\prime}=g_{l}gg_{r}$ of the rotated ODF, where $g_{r}$ is one of
the point symmetry operations of the crystal and $g_{l}$ is a rotation of the
sample in the laboratory frame. This rotation $g_{l}$ will be used to simulate
sample rotations in the course of the tomography experiment. The other
rotation $g_{r}$ will always be the null rotation $g_{0}$.

####  3.4.2 Symmetrized Hyperspherical Harmonics

HSHs, as defined in Eq. 35, are complex valued functions and require certain
combinations of complex coefficients to produce a real-valued ODF. We
therefore use symmetrized hyperspherical harmonics (sHSHs)
$\mathring{Z}^{n}_{\lambda}(g)$, which obey this constraint by definition. The
sHSHs, as functions of rotations, can also be written as linear combinations
of the HSHs of the same order $n$:

$\mathring{Z}^{n}_{\lambda}(g)=\sum_{l,m}Z^{n}_{lm}(g)X^{n}_{lm;\lambda}.$
(22)

The symmetrization procedure to obtain $X^{n}_{lm;\lambda}$ is outlined in
Mason & Schuh [38]. This strategy additionally allows selecting sHSHs with the
same proper point group as the crystal structure. This reduces the number of
adjustable parameters, and the crystal orientations that need to be evaluated
are limited to the fundamental zone of the point group [56]. The matrix for
the effect of a single rotation on the coefficients
$\mathring{c}^{n}_{\lambda,k}$ of the expansion over the sHSHs is then given
by:

$\mathring{R}^{n}_{\lambda^{\prime},\lambda}(g)=\sum_{l^{\prime}m^{\prime}}\sum_{l,m}X^{n\dagger}_{l^{\prime}m^{\prime};\lambda^{\prime}}R^{n/2}_{l^{\prime}m^{\prime}lm}(g,g_{0})X^{n}_{lm;\lambda}.$
(23)

###  3.5 Full forward model

Figure 2:  a) A sample with HSH expansion coefficients in each voxel,
represented by bar diagrams. These are weighted by the beam intensity for a
given configuration and result in a projection of these coefficients. A
summation of diffractlets weighted by these coefficients results in a
diffraction pattern. b) is a selection of diffractlets of order 4 sHSHs with a
BaCO3 structure factor.

Let us first derive the forward model for a sample made of a single voxel,
located in $\mathbf{x}_{1}$. We can directly use the sHSH model above to
calculate the diffraction patterns expected in that case: let us expand the
ODF in Eq. 14 using sHSHs, the expected scattering intensity in any reciprocal
space coordinates $q$ and $\chi$ reads then

$\bar{I}(q,\chi;\mathbf{c}(\mathbf{x}_{1}))=\sum_{n,\lambda}c^{n}_{\lambda}(\mathbf{x}_{1})\int\mathring{Z}^{n}_{\lambda}(g)~{}I^{g}_{sc}(q,\chi)~{}\text{d}\Omega$
(24)

where
$\mathbf{c}(\mathbf{x}_{1}):=\\{c^{n}_{\lambda}\,|\,(n,\lambda)\in\mathbb{N}\times\mathbb{N}\\}$
is the set of sHSH coefficients associated with the single voxel. Here, we
isolate the contribution to the diffraction pattern by a single sHSH, further
labeled diffractlet (see Fig. 2 b), which reads

$d^{n}_{\lambda}(q,\chi):=\int\mathring{Z}^{n}_{\lambda}(g)~{}I^{g}_{sc}(q,\chi)~{}\text{d}\Omega.$
(25)

Obviously, any realistic sample should be defined over a mesh, with a series
of $P$ voxels located in $\\{\mathbf{x}_{p}\\}_{p=1}^{P}$. It is then natural
to associate any voxel $p$ with its specific series of sHSH coefficients,
hence resulting in a set of parameters in the model that reads

$\mathbf{c}:=\\{c^{n}_{\lambda}(\mathbf{x}_{p})\,|\,p=1,\cdots,P\\}\qquad\text{with}\qquad(n,\lambda)\in\mathbb{N}\times\mathbb{N}.$

In order to account for the effect of the tomographic measurement, we
introduce another index $k=1\cdots N_{\phi}\times N_{\varkappa}\times N_{t}$
that refers to a given sample rotation $\phi$, tilte $\varkappa$ and
translation $\mathbf{t}$. The total diffraction pattern (i.e., the expected
measurement) is a weighted sum of the contribution of each voxel in the
sample, with the weights computed from the local beam intensity
$B_{k}(\mathbf{x}_{p})$, as given by Eq. 7 (illustrated in Fig. 2 a). For a
single measurement $k$, we note that most of the weights are zero as only a
small fraction of the voxels in the sample are actually illuminated by the
beam. The sHSH coefficients $c^{n}_{\lambda}(\mathbf{x}_{p})$ are agnostic to
any rotation and tilt in the sample and we need to explicitly account for it
in the forward model. To that end, we introduce first the sample rotation
$g_{k}$ that is associated with the $k$-th tomographic measurement. Then, we
use the rotation matrix given by (Eq. 23) to define the corresponding rotated
diffractlets

$\delta^{n}_{\lambda,k}(q,\chi):=\sum_{\lambda^{\prime}}d^{n}_{\lambda^{\prime}}(q,\chi)\,\mathring{R}^{n}_{\lambda^{\prime},\lambda}(g_{k}).$

It is then straightforward to define the expected intensity for the $k$-th
tomographic measurement

$\displaystyle\bar{I}_{k}(q_{\ell},\chi_{m};\mathbf{c})$
$\displaystyle=\sum_{p}\sum_{n}\sum_{\lambda}B_{k}(\mathbf{x}_{p})\delta^{n}_{\lambda,k}(q,\chi)c^{n}_{\lambda}(\mathbf{x}_{p})$
(26)

where $(q_{\ell},\chi_{m})$ define the measurement mesh in the reciprocal
space coordinates. Clearly, this relation is a linear model that we write now
in a convenient matrix-vector form: $i$) first, we wrap the measurement-
related indices $k,\ell,m$ into a single index $j=1\cdots N_{k}N_{q}N_{\chi}$,
then $ii)$ we use a single index $\nu$ in place of the 3 model-related
parameters $p,n,\lambda$. The relation Eq. 26 now reads

$\bar{I}_{j}(\mathbf{c})=\sum_{\nu}B_{j,\nu}\delta_{j,\nu}c_{\nu}=\mathbf{a}_{j}^{\dagger}\mathbf{c}$
(27)

and if we stack the expected measurements $\bar{I}_{j}$ into a single vector,
we obtain

$\bar{\mathbf{I}}=\mathbf{A}\mathbf{c}$ (28)

where $\mathbf{A}$ is the texture-tomography matrix that can be pre-computed
before we perform the iterative inversion.

###  3.6 TexTOM reconstruction strategy

We use a standard quadratic metric

$\mathcal{L}(\mathbf{c})=||\,\mathbf{A}\mathbf{c}-{\mathbf{I}}\,||^{2}$ (29)

to compute the discrepancy between the output of the forward model (Eq. 28)
and the texture-tomographic experimental measurements
$\mathbf{I}:=\textbf{vect}(I_{j})$. The reconstruction is defined as the
minimization of the following constrained least-square criterion

$\widehat{\mathbf{c}}=\arg\min_{\mathbf{c}\in\mathbb{R}^{N}}\mathcal{L}(\mathbf{c})\qquad\text{subject
to}\qquad\\{c^{0}_{0}(\mathbf{x}_{p})\geq 0\\}_{p=1}^{P}.$ (30)

We note that the positivity constraint over the zero-order sHSH coefficients
is required to produce a physically meaningful results. This constraint can be
fulfill by projecting any negative $c^{0}_{0}(\mathbf{x}_{p})$ to zero after
each update of the following gradient-base iteration

$\qquad\mathbf{c}^{(n+1)}\leftarrow\mathbf{c}^{(n)}-\gamma^{(n)}\nabla\mathcal{L}(\mathbf{c}^{(n)}),\qquad
n=1\cdots\infty.$ (31)

The gradient of the least-square function $\nabla\mathcal{L}$ is easily
derived from Eq. 29

$\nabla\mathcal{L}(\mathbf{c})=2\mathbf{A}^{\dagger}(\mathbf{A}\mathbf{c}-{\mathbf{I}})$
(32)

The stepsize $\gamma^{(n)}$ is adjusted in each iteration with a backtracking
technique to ensure a strict decrease in the criterion [57, p. 29]. We note
that the fitting-function Eq. 29 is strictly convex whenever $\mathbf{A}$ is a
full-column rank matrix (which is expected to be the case here). In this
context, the solution of the constrained optimization problem Eq. 30 exist and
is unique [57, Prop. 1.1.2], and the convergence of the iteration Eq. 31
toward $\widehat{\mathbf{c}}$ is granted, whatever the initial-guess
$\mathbf{c}^{(0)}$. The convergence of the iteration is monitored via the norm
of the gradient, and we stop the iterative reconstruction when

$\biggl{|}\frac{||\nabla\mathcal{L}(\mathbf{c}^{(n+1)})||-||\nabla\mathcal{L}(\mathbf{c}^{(n)})||}{||\nabla\mathcal{L}(\mathbf{c}^{(0)})||}\biggl{|}<\epsilon$
(33)

where $\epsilon$ is a predefined parameter (typically, it is set to
$10^{-3}$).

###  3.7 Data post-processing

A known limitation of the harmonic expansion approach is that it hard to
ensure positivity of the ODF[30], in particular without explicitly calculating
the ODF in every iteration, which can be computationally extremely costly for
large samples. We therefore post-process our data using a kernel that ensures
non-negativity [58]. This means we modify the obtained coefficients by a
prefactor $K$ that depends on the order $n$, the highest used order $N$ and an
exponent $k$, whose value is empirically chosen to be between 1 and 2
according to the situation.

$K=\left(1-\frac{n}{N+1}\right)^{k}.$ (34)

It was observed however that this kernel slightly spreads out the ODF,
therefore the presented standard deviations are possibly over-estimated.

##  4 Results and Discussion

###  4.1 Reconstruction of simulated data

Figure 3: a) TexTOM reconstruction of the simulated sample for testing the
reconstruction algorithm. The sticks represent the reconstructed preferred
orientation of the crystal c-axis in each voxel, color coded by the angular
deviation $dg$ from the simulation. A corner was cut out for showing the
interior of the sample. b) histogram of $dg$ for the same sample. The
distribution of the standard deviations is shown in c).

The reconstruction of ODFs of the simulated sample was performed in two steps:
First is the retrieval of the mean orientation using a HSH-expansion cut at
the lowest order, second the estimation of the variance by including higher
orders as necessary. Figure 3 presents a summary of the inversion results:
Panel a) shows a cut-off view of the reconstruction. Note that this
representation shows only the orientation of one crystal axis, thus does not
represent the full texture information. The colour scale is the angular
distance metric (see appendix B) between the simulation and the
reconstruction, which is shown as a histogram in panel b). The deviations are
distributed around an average of 2.8° and show no clear spatial distribution
at the interface between differently oriented layers of the sample. The
irregular shape in this histogram is connected to the distribution of sampling
points in orientation space, which was constructed with an angular resolution
of 3°. This part of the reconstruction was done with the expansion truncated
at order 4, where the point group 222 possesses 10 sHSHs, and demonstrates
that the lowest order already suffices for the estimating the most likely
orientations.

An expansion up to order 8 was used to estimate standard deviations $sigma$ of
the ODFs. As discussed in section 3.7, our current model over-estimates the
spread of the distribution as a consequence of ensuring non-negativity of the
ODF. We find $\sigma$ distributed around 47.9° with a standard deviation of
2.6°, as shown in panel c) of Fig 3.

###  4.2 Biomorphs

In order to test the performance of TexTOM on experimental data, a dataset of
silica biomorph was collected. The sample consisted of a helicoidal silica-
witherite biomorph of $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ length and
$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter. An exemplary SEM image
can be seen in Fig. 4a. The arrangement of BaCO3 nanorods of about
20$\times$100 nm embedded in an amorphous silica matrix is shown in the TEM-
image, Fig. 4b. Different morphologies can be obtained by varying the local
synthesis conditions [40, 41]. In the context of this manuscript, no in-depth
analysis of different growth conditions or their texture has been carried out.
Shown data server as a sample to demonstrate the performance of TexTOM.

Figure 4:  a) SEM image of a helicoidal silica biomorph. b) Arrangement of
BaCO3 nanorods (black) in amorphous silica (white) as seen by TEM. For
reference, size of the shown region is given by the red rectangle in a). The
red circle corresponds to the dimension of the X-ray beam. c) TexTOM
reconstructions of a $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long piece of
a helix. Sticks represent the preferred orientation of the indicated crystal
axes. The volume is cut in transverse (c-axis) and longitudinal (a- and
b-axes) directions, thus showing the interior of the sample. Images were
produced by Paraview [59]

A dataset containing 260 projections was collected, with an equal angular
sampling in $\phi$ and $\varkappa$ based on the reasoning of Liebi et al. [18]
to create a ’gold standard’ measurement. From the reconstructions, the most
likely orientation of the a, b and c-axis was determined within in the
fundamental zone and the variance was extracted as a metric for the degree of
orientation/angular dispersion within each voxel and plotted as the colour
scale of Fig. 4c. In order to facilitate the interpretation of the data, cross
sections of the volume are presented in Fig. 4c. The reconstructions show an
angular dispersion consistent with TEM observations. The red circle indicated
the beam size in comparison to the TEM image. In the slice of the c-axis the
two distinct strands of the helix can be identified. An interesting
observation is the presence of a gradient of the variance along the two helix
strands, visible in the cuts of the sample showing a and b-axes of Fig. 4c.
These possibly correspond to a core-skin architecture in the arrangement of
the nanorods. This could be caused by local concentration and pH gradients
during the synthesis. A more detailed study, encompassing an in-depth,
comparative analysis of different morphologies is envisaged for the future.
The reconstruction strategy (described in more detail in the Materials and
Methods section) is split into two major parts, one is the precalculation of
single crystal diffraction patterns and the beam trajectories corresponding to
the scan. The rational here is that these time consuming steps can be re-used
for different reconstructions. The actual reconstructions are carried out with
sequentially increasing order, which allows the selection of an optimal order
via a Minimum Description Length (MDL) criterion [60]. In our case, the MDL
criterion is clearly selecting the order 8 (see Fig. 5). A more detailed
account of the timing is given in Tab. 1.

Figure 5:  Plot of the minimum description length (MDL) for the reconstruction of the helicoidal biomorph in function of the HSH order $n$. Table 1: Duration $t$ of different steps of TexTOM reconstructions and total number of adjustable parameters $N_{par}$. Pre-calculations | t [min] |
---|---|---
Single crystal diffraction patterns | 770 |
Beam trajectories | 5 |
260 projections | t [min] | $N_{par}$
Reconstruction $n=4$ | 35 | 290675
Reconstruction up to $n=8$ | 100 | 1189125
Reconstruction up to $n=12$ | 305 | 3144575
41 projections | t [min] | $N_{par}$
Reconstruction $n=4$ | 5 | 290675
Reconstruction up to $n=8$ | 24 | 1189125
Reconstruction up to $n=12$ | 60 | 3144575

###  4.3 Benchmarking

In order to test the performance of our code and see how under-sampling
affected our results, two further datasets were selected from the existing
data. One being all 41 projections at $\varkappa$=0 and another being 41
projections, equally sampled in $\phi$ and $\varkappa$ orientation space. The
sampling with 41 projections corresponds roughly to a Nyquist-Shannon sampling
[61]. Reconstructions were carried out and compared by calculating the angular
distance between the most likely orientations for each voxel, shown in Fig.
3a. The histogram in Fig. 6b shows amount of voxels for each distance. It is
visible that the 41 equidistant projections reproduce the orientation very
well with a mean angular distance of 22° and a decrease in quality that can be
expected from the sampling reduction. Here, there are no visible regions with
a worse reconstruction quality, it rather adds a general noise on the results.
A similar, but slightly worse trend is visible for the $\varkappa$=0 dataset.
Here, the mean angular distance is 35° and the histogram (blue bars, Fig 6b)
shows a higher tail at larger angles. Furthermore, the large orientation
differences seem to be linked to regions where the crystallites have a wide
ODF (compare red zones in top panel of Fig. 6 a and blue zones in Fig. 4)
hence, a weak texture, whereas voxels with a strong texture are in good
agreement. Together with the very fast reconstruction time (5 min) for n=4 for
the reduced datasets, the strategy of online reconstruction and information-
driven sampling comes into reach. This strategy entails that continuous online
reconstructions are carried out and that projections are added in zones of the
reciprocal space where the fit shows larger deviations between reconstructed
and measured data.

Figure 6: Comparison of reconstructions using all 260 projections and either
41 projections with $\kappa=0$ or 41 projections with an equidistant
distribution over orientation space. a) shows how the angular deviations $dg$
from the full reconstruction are distributed over the interior of the sample,
using the same sectional plane as in Fig. 4. b) shows the distribution of $dg$
for both cases.

##  5 Conclusions and Outlook

In summary, this manuscript presents TexTOM as a new inversion framework to
recover quantitative ODF information in a tomographic fashion from X-ray
diffraction data using hyperspherical harmonics and derived diffractlets. We
present a detailed description of the experiment, the forward model including
the parametrization of the HSH expansion as well as our inversion strategy.We
show the results of an inversion on both simulated and experimental data with
sub $\mu$m resolution and benchmark the reconstruction strategy with reduced
angular sampling.

The presented method presents a large step forward from state-of-the-art
tensor tomography methods [16, 24, 25] by enabling a fully quantitative
reconstruction of the real-space ODF compared to the reconstruction of the
position of a single SAXS or WAXS reflection. Furthermore TexTOM does not rely
on regularization parameters and associated assumptions on local smoothness,
but uses the inherent constraints given by the crystal symmetry. Due to the
nature of the harmonic decomposition, the recovery of mean orientations is
given by the lowest order and therefore allows very fast reconstructions, even
when compared to the actual data acquisition. The estimation of the spread of
the ODF requires the inclusion of higher orders and the use of a damping
kernel also enables to enforce strict positivity of each component of the ODF.
This method currently overestimates the true variance and and we envisage
further tuning if the model to retrieve better estimates.

The joint-optimization of several Bragg-peaks also enables the concurrent
refinement of multiple crystalline phases within one inversion as well as
adding a strain component. For the future, the direct extraction of
crystalline phase information prior to a TexTOM inversion via e.g. a Pawley
extraction can also be envisaged, requiring less prior information on the
crystalline phases present. In this way, TexTOM aims at reconstructing the
full crystalline state tensors for each voxel with high, essentially beam-size
limited resolution. One challenge arising from the wealth of information
contained in the ODF is to find ways to visualize the retrieved data. Further
challenges are arising when trying to push the experimental resolution down
into the range of 100 nm or less. Firstly, the data acquisition becomes more
challenging as both the scanning and rotation need to provide a positioning
accuracy better than the target resolution. Secondly, the alignment of the
data during the pre-processing is subject to equal constraints and factors
such as imperfections in the tilt and rotation axis as well as the coaxiality
of the scanning stages becomes more and more important and might approach the
limit of current mechanical solutions. One way to overcome this challenge
could be the use of nonrigid tomography approaches [62]

Through the efficient use of the collected diffraction information, a
significant speed-up of the experiment can be expected, bringing in-situ
experiments on dynamically changing samples into reach as well as providing an
avenue for measuring radiation sensitive samples with a largely reduced
deposited X-ray dose. Future work will be directed towards accurately
benchmarking the performance in terms of spatial, angular and multi-phase
resolution with specifically designed benchmark samples. A further of
development is implementation of live reconstructions and information-driven
sampling to realize the quickest-possible experiments whilst achieving the
desired angular and spatial resolution.

##  Appendix A Equations related to hyperspherical harmonics

Definition of hyperspherical harmonics [54]:

$\displaystyle\begin{split}Z^{n}_{lm}\Big{(}g(\omega,\vartheta,\varphi)\Big{)}=(-i)^{l}\frac{2^{l+1/2}l!}{2\pi}\sqrt{(2l+1)\frac{(l-m)!(n+1)(n-l)!}{(l+m)!(n+l+1)!}}\\\
\Big{[}\sin(\omega/2)\Big{]}^{l}C^{l+1}_{n-l}\Big{[}\cos(\omega/2)\Big{]}P^{m}_{l}\Big{(}\cos\vartheta\Big{)}\text{e}^{im\varphi}\end{split}$
(35)

The $C^{l+1}_{n-l}$ are the Gegenbauer polynomials and the $P^{m}_{l}$ are the
associated Legendre polynomials with the Condon–Shortley phase. Their integer
indices are restricted to $n\geq 0$, $0\geq l\geq n$, $-l\geq m\geq l$.

The hyperspherical harmonic rotation matrix is given by:

$R^{2l}_{\lambda^{\prime}\mu^{\prime}\lambda\mu}(g_{2},g_{1})=\sum_{m_{2}^{\prime}}\sum_{m_{2}}\sum_{m_{1}^{\prime}}\sum_{m_{1}}C^{\lambda^{\prime}\mu^{\prime}}_{lm_{2}lm_{1}^{\prime}}U^{l*}_{m_{2}^{\prime}m_{2}}(g_{2})U^{l}_{m_{1}^{\prime}m_{1}}(g_{1})C^{\lambda\mu}_{lm_{2}^{\prime}lm_{1}}$
(36)

where $U^{l}_{m,m^{\prime}}$ is the irreproducible representative of $SO(3)$

$U^{l}_{m,m^{\prime}}(g)=\sum_{\lambda}\sum_{\mu}\frac{\sqrt{2(2\lambda+1)}\pi}{2l+1}C^{lm}_{lm^{\prime}\lambda\mu}Z^{2l}_{\lambda\mu}(g)$
(37)

and $C^{lm}_{lm^{\prime}\lambda\mu}$ being the Clebsch-Gordon coefficient for
$SO(3)$.

##  Appendix B Angular distance between orientations

We define the distance between two orientations $g,g^{\prime}$ via the
quaternion formalism. Unit quaternions are related to rotation angle $\omega$
and axis $\mathbf{\hat{a}}$ by:

$\mathbf{q}=q_{0}+q_{1}\mathbf{i}+q_{2}\mathbf{j}+q_{3}\mathbf{k}=\cos\frac{\omega}{2}+\sin\frac{\omega}{2}(a_{x}\mathbf{i}+a_{y}\mathbf{j}+a_{z}\mathbf{k})$
(38)

The angular distance between two unit quaternions is then given by [63]

$dg(g,g^{\prime})=\arccos(2\langle\mathbf{q},\mathbf{q}^{\prime}\rangle-1)$
(39)

Here, $\mathbf{q},\mathbf{q}^{\prime}$ are the quaternions corresponding to
rotations $g,g^{\prime}$, respectively.
$\langle\mathbf{q},\mathbf{q}^{\prime}\rangle$ denotes the inner product of
the quaternions:

$\langle\mathbf{q},\mathbf{q}^{\prime}\rangle=q_{0}q^{\prime}_{0}+q_{1}q^{\prime}_{1}+q_{2}q^{\prime}_{2}+q_{3}q^{\prime}_{3}$
(40)

This however, does not yet take into account the crystal symmetry. To find the
effective minimal distance between two orientations, the therefore rotate
$\mathbf{q}$ by the symmetry generators of the point group and choose the
smallest distance of all possible combinations.

##  Appendix C Preferred orientation and standard deviation of ODFs

We examined the resulting ODFs resulting from the sHSH-expansion and found
that the ODF truncated at the lowest order ($n=4$ for point group ’222’)
generally shows a single maximum within the fundamental zone. We define this
orientation as the mean orientation $g_{\mu}$ of the distribution. To
calculate a standard deviation of an ODF given by a set of coefficients
$\mathbf{c}$, we rotate the distribution by $-g_{\mu}$ using relation 21, thus
obtaining a distribution centered at the center of the fundamental zone (where
$\omega=0$). Then we calculate the standard deviation $\sigma$ in the
conventional way from the rotation angle $\omega$:

$\sigma^{2}=\int\omega^{2}\rho(g)d\Omega$ (41)

##  Appendix D Acknowledgements

We would like to acknowledge Moritz Stammer for helpful discussions and the
ESRF ID13 staff for supplying beamtime and support during the experiment. This
work is funded by the European Union, European Research Council H2020, TexTOM
(grant no. 101041871). Views and opinions expressed are those of the author(s)
only and do not necessarily reflect those of the European Union or the
European Research Council. Neither the European Union nor the granting
authority can be held responsible for them.

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