# Quantitative nanostructure characterization using atomic pair distribution
functions obtained from laboratory electron microscopes

Milinda Abeykoon  Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory  Christos D. Malliakas  Department of
Chemistry, Northwestern University  Pavol Juhás  Department of Applied Physics
and Applied Mathematics, Columbia University  Emil S. Božin  Condensed Matter
Physics and Materials Science Department, Brookhaven National Laboratory
Mercouri G. Kanatzidis  Department of Chemistry, Northwestern University
Chemistry Department, Argonne National Laboratory  Simon J. L. Billinge
Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory  Department of Applied Physics and Applied Mathematics, Columbia
University

###### Abstract

Quantitatively reliable atomic pair distribution functions (PDFs) have been
obtained from nanomaterials in a straightforward way from a standard
laboratory transmission electron microscope (TEM). The approach looks very
promising for making electron-derived PDFs (ePDFs) a routine step in the
characterization of nanomaterials because of the ubiquity of such TEMs in
chemistry and materials laboratories. No special attachments such as energy
filters were required on the microscope. The methodology for obtaining the
ePDFs is described as well as some opportunities and limitations of the
method.

##  I Introduction

One of the great challenges of nanoscience is to obtain the quantitative
structures of nanoparticles billi;s07 ; jadzi;s07 . The atomic pair
distribution function (PDF) method has recently emerged as a powerful tool for
doing this egami;b;utbp03 ; juhas;n06 ; billi;jssc08 ; petko;cm06 ;
page;chpl04 ; kodam;aca06 ; neder;pssc07 ; masad;prb07 ; petko;jpcc07 , but
obtaining the required high quality diffraction data to high momentum transfer
with good statistics generally requires synchrotron x-ray or spallation
neutron data from a national user facility. Here we show that data of
sufficient quality for quantitative analysis of nanoparticle structure using
the PDF can be obtained from transmission electron microscopes (TEM) available
at many research institutions. Quantitative structural models were applied to
PDFs of several nanoparticle systems showing that electron PDFs can be modeled
with the powerful emerging modeling tools for studying PDFs in general
farro;jpcm07 ; neder;b;dsadss08 ; tucke;jpcm07 ; cerve;jac10 . This approach
complements medium and high resolution imaging methods for studying
nanoparticles in the TEM. The ease of data collection and ubiquity of TEMs
will make this an important tool in the characterization of nanostructured
materials.

A challenge when using electrons as a probe is that they scatter strongly
cowle;mic04 ; cowle;b;edt92  and not according to the weak scattering
kinematical scattering equations on which the PDF analysis is based debye;ap15
; warre;b;xd90 . This would appear to rule out electrons as a source of
diffraction data for PDFs except in the cases of very dilute, such as gas-
phase schoo;nl05 , samples. However, kinematical, or nearly kinematical,
scattering is obtained from electrons when sample volumes are sufficiently
small that multiple scattering events are not of high probability before the
electrons exit the sample (typically a few nm of thickness), or when the
scattering from the samples is highly incoherent, for example the scattering
from amorphous materials and away from zone axes in a crystal
weiri;b;ecnafsdonm06 . In these latter cases there is still significant
multiple scattering, but it is sufficiently incoherent that it can be treated
as a background and subtracted and the resulting coherent signal can be
treated kinematically. This has been discussed in detail in a number of
publications cocka;armr07 ; ansti;u88 ; ankel;zna05 ; karle;pnas77 . This is
used in the rapidly growing field of electron crystallography
weiri;b;ecnafsdonm06 , and has been demonstrated in previous work of electron
diffraction (ED) from glasses and amorphous materials moss;prl69 ; cocka;aca88
; cocka;armr07 ; hirot;mcp03 ; noren;jac99 , though little has been done in
the way of quantitative modeling in those studies. In these respects, the
study of small nanoparticles is particularly favorable. The samples are
inherently thin, limited to the diameter of the nanoparticles when they are
dispersed as a sub-mono-layer on a holey carbon support, and the structure is
typically less coherent than from crystals because of the finite size effects
that significantly broaden Bragg peaks and the often lower symmetries of
nanoparticle structures due to surface and bulk relaxations. In fact
fortuitously, the scattering is most kinematical precisely for the small
nanoparticles ($<10$ nm) that are most beneficially studied using PDF methods
egami;b;utbp03 ; juhas;n06 ; billi;jssc08 ; petko;cm06 ; page;chpl04 ;
kodam;aca06 ; neder;pssc07 ; masad;prb07 ; petko;jpcc07 .

Here we show how to obtain PDFs from a normal transmission electron microscope
(TEM) found in many research labs. We find that the resulting electron PDFs
(ePDFs) can be modeled to extract quantitative structural information about
the local structure using PDF refinement programs such as PDFgui farro;jpcm07
. This opens the door to broader application of PDF methods for nanostructure
characterization since TEM is already a routine part of the nanoparticle
characterization process wang;jpcb00 ; won;jap06 . With this development, as
well as obtaining low and high resolution TEM _images_ of nanoparticles,
quantitative structural information, similar to that normally obtained from a
Rietveld refinement rietv;jac69 ; young;b;trm93  in bulk materials, is also
available from nanoparticles with little additional effort. This approach also
complements high resolution TEM by getting an average signal from a large
number of nanoparticles rather than giving information from a small part of
the sample that may not be representative. The fact that the real-space images
and the diffraction data suitable for structural analysis can be obtained at
the same time and from the same region of material is also a large advantage,
resulting in more complete information for the characterization of the sample.
In some cases, the small quantity of material required for ePDF, compared to
x-ray and neutron PDF measurements (xPDFs and nPDFs, respectively), may also
be a major advantage, as well as the ability to study thin films.

### Theoretical Background

The Fourier transform of X-ray or neutron powder diffraction data yields the
PDF, $G(r)$ according to farro;aca09

$\displaystyle G(r)=(2/\pi)\int_{Q_{\min}}^{Q_{\max}}Q[S(Q)-1]\sin(Qr)dQ,$ (1)

where the structure function, $S(Q)$, is the properly normalized powder
diffraction intensity and $Q$, for elastic scattering, is the magnitude of the
scattering vector, $Q=4\pi\sin(\theta)/\lambda$.warre;b;xd90 ; egami;b;utbp03
The PDF is also related to the atomic structure through

$\displaystyle G(r)=\frac{1}{Nr}\sum_{ij}\frac{f_{i}(0)f_{j}(0)}{\langle
f(0)\rangle^{2}}\delta(r-r_{ij})-4\pi r\rho_{o},$ (2)

where the sum goes over all pairs of atoms _i_ and _j_ separated by $r_{ij}$
in the model. The form factor of atom _i_ is $f_{i}(Q)$ and $\langle
f(Q)\rangle$ is the average over all atoms in the sample. In equation 2, the
scattering factors are evaluated at $Q=0$, which in the case of x-rays is the
atomic number of the atom. The double sums are taken over all atoms in the
sample. For a multicomponent system, $S(Q)$ can be written in terms of the
concentrations, $c_{i}$, of the atomswarre;b;xd90 ; egami;b;utbp03

$S(Q)=1+\frac{I(Q)-\sum c_{i}|f_{i}(Q)|^{2}}{\big{|}\sum
c_{i}f_{i}(Q)\big{|}^{2}}.$ (3)

In the case of electrons as a probe, the equations are the same, providing the
scattering can be treated kinematically cowle;b;edt92 ; however, the form-
factor must be that appropriate for electrons, $f_{e}(Q)$, which is the
Fourier transform of the electronic potential distribution of an atom. Note
that in the electron diffraction literature, it is common to use
$s=2\sin(\theta)/\lambda=Q/2\pi$ instead of $Q$ for the independent variable
in the scattering. The electron form factor, $f_{e}(Q)$, is different to, but
closely related to, the x-ray form factor of the same atom, $f_{x}(Q)$, which
is the Fourier transform of the electron density. A useful relationship
between $f_{e}(Q)$ and $f_{x}(Q)$ is cowle;b;edt92

$f_{e}(Q)=\frac{m_{e}e^{2}}{2\hbar^{2}}\left(\frac{Z-f_{x}(Q)}{Q^{2}}\right),$
(4)

where $m_{e}$ and $e$ are the mass and charge of the electron, respectively,
$\hbar$ is Plank’s constant, and $Z$ the atomic number. This equation does not
give a definite value for $f_{e}(Q)$ at $Q=0$, but $f_{e}$(0) can be
calculated by extrapolation or by using

$f_{e}(0)=4\pi^{2}\frac{me^{2}}{3\hbar^{2}}(Z\langle r_{e}^{2}\rangle),$ (5)

where $\langle r_{e}^{2}\rangle$ is the mean square radius of the electronic
shell of the atom.cowle;b;edt92  Figure 1 shows a comparison between x-ray and
electron form factors, $f_{x}$(Q) and $f_{e}$(Q) of Au.

In the case of single crystal ED, a rule of thumb is that when the crystal
thickness is greater than $\sim 300$-400 Å, data reduction must be done based
on the dynamical diffraction theory which assumes the presence of coherent
multiple scattering components of electrons cowle;b;edt92 . Depending on the
energy of the electrons, this thickness limit may even fall below these
numbers in the presence of heavy elements cowle;b;edt92 , and in the case of
electron powder diffraction, the average thickness of crystallites in the
specimen should also be less than a few hundred Ångströms to avoid dynamical
scattering effects. cowle;b;dp95  Coherent multiple scattering changes the
relative intensities of Bragg peaks from the kinematical structure factor
values, redistributes intensity to the weaker peaks at higher values of $Q$
cocka;aca88  and can allow symmetry disallowed peaks to appear in the pattern.
If $\Gamma$ is the elastic mean free path of the electron, it has been shown
that a PDF determined from a polycrystalline Pt sample does not affect the
positions of the PDF peaks for D/$\Gamma\leq 5$, where D is the particle size,
but it does affect the determination of coordination numbers ansti;u88 . Here
we show that model fits may be good, even in the presence of significant
multiple scattering, while refined thermal factors are underestimated, though
it is desirable to optimize experimental conditions such as to minimize
multiple scattering.

Incoherent multiple scattering can be observed in ED patterns in the form of
increased background cowle;b;edt92  which does not affect the relative
intensities of the Bragg peaks. This is why, in the case of a less coherent
structure, dynamical scattering effects are less important.

In this study, all the specimens used were nanosized samples: thin films,
discrete nanoparticles, or agglomerates of nanoparticles. In this case, the
hope was that multiple scattering would not introduce undue aberrations into
the kinematical diffraction pattern and a reliable PDF will result. We found
this to be largely true with an exception we discuss below.

##  II Experimental

Nanocrystalline thin film, or dispersed nanoparticulate samples, were
distributed on a holey carbon grid and ED data taken with a short camera
length, to give the widest $Q$-range, and a relatively large beam-size (2-5
$\mu$m diameter) on the sample, to obtain the best possible powder average. To
improve the powder average different regions of the sample were illuminated by
translating the sample under the beam. In other respects, the TEM was used in
a standard configuration using a CCD detector and no energy filtering and
operated at 200 keV (wavelength, $\lambda=0.025079$ Å).

All selected area ED experiments were carried out at room temperature on a
Hitachi H8100 200 KeV transmission electron microscope equipped with a Gatan
Orius SC600 CCD Camera (24mm x 24 mm active area). Typical exposure time per
frame was around 0.3s. Formvar coated 300 mesh copper grids (Electron
Microscopy Sciences) stabilized with an evaporated carbon film were used to
support the metallic films and nanoparticles. Deposition of gold on the carbon
coated side of the TEM grid was performed with a Denton Vacuum DeskIII
sputterer and gave a uniform film. The thickness of the film was measured in
real time during the sputtering process with the aid of a thickness monitor
(Maxtek, Inc TM-350). Part of the grid was masked during the deposition and
this masked area was used to extract the diffraction intensity of the support.
No differences were found in the diffraction intensity data of the background
(carbon and polymer films) between different grids. Deposition on the side of
the grid that was coated with the polymer gave Au nanoparticles with a wide
range of sizes up to $\sim 100$ nm. NaCl nanoparticles were deposited on a TEM
grid by a radio-frequency thermal evaporation method.

For comparison, x-ray measurements on Au nanoparticles were carried out in the
rapid acquisition mode (RAPDF) chupa;jac03  using a Perkin Elmer amorphous
silicon 2D detector at X7B beamline of National Synchrotron Light Source
(NSLS) at Brookhaven National Laboratory (BNL). Nanoparticles in ethanol
solution were loaded in a 1 mm diameter kapton tube sealed at both ends, and
mounted perpendicular to the x-ray beam. The data were collected at room
temperature using the x-ray energy of $\sim$38 keV ($\lambda=0.3196$ Å). The
data were collected in a multiple 4 s exposures for a total collection time of
5 min.

To calibrate the conversion from detector coordinates to scattering angle, it
is necessary to measure the ED pattern from a standard of known lattice
parameters. The software for reducing the data to 1D, Fit2D hamme;hpr96  uses
this to optimize the effective sample-detector distance, find the center of
the Scherrer rings on the detector, and correct for aberrations such as any
deviation from orthogonality of the detector and the scattered beam. Typical
standards used by the program are Al2O3, CeO2, LaB6, NaCl and Si. However, for
the ED experiment it is necessary to have a nano-sample standard to obtain a
good powder average. For this gold nanoparticles of diameter $\sim$100 nm were
used and a literature value of 4.0782 Å for the lattice parameter. The
effective sample-detector distance depends on the settings of the magnetic
lenses used in the microscope. We assumed that the energy of the electrons,
200 keV, is well known (resulting in $\lambda=0.025079$ Å), though for the
most accurate results the electron wavelength should be calibrated using
standard methods.Once these calibration quantities are known, they are fixed
and the same values are used to convert the sample data. From this perspective
it is essential that the sample is measured under identical conditions as the
standard, including camera length and focus. We found that even scanning
around a sample to find a different viewing area resulted in a small variation
in the position on the detector of the center of the resulting diffraction
pattern. It was thus necessary to run a separate calibration run on each
diffraction pattern to determine the center of the rings, while keeping the
camera-length from the Au calibration.

##  III Data Analysis

The 2D ED images were read and integrated into 1D powder diffraction patterns,
after masking the missing beam stop region. The data have to be further
processed to obtain the PDF. Corrections were applied to the raw data to
account for experimental effects and properly normalized and divided by
($\langle f_{e}(Q)\rangle^{2}$) egami;b;utbp03 , resulting in the total
scattering structure function, $S(Q)$. The kernel of the Fourier transform is
the reduced structure function, $F(Q)=Q[S(Q)-1]$. We used a home-written
program, PDFgetE, to carry out these steps. The PDF is then straightforwardly
obtained as the Fourier transform of $F(Q)$ according to Eq. 1, which is also
carried out in PDFgetE. Once the PDFs are obtained, they can be modeled using
existing PDF modeling programs. Here we used PDFgui farro;jpcm07 .

##  IV Results

A low resolution TEM image of the 2.7 nm thick Au film is shown in Fig. 2(a).
The film is uniform and featureless in the image, but a region at the edge of
the film was selected for imaging so that the edge of the film gives a visual
cue to its presence. An ED pattern from a position away from the edge of the
film is shown in Fig. 2(b). We can see a series of concentric circles due to
the Scherrer powder diffraction rings in transmission geometry. The resulting
1D ED pattern, obtained by integrating around the rings in the 2D pattern is
shown in Fig. 2(c). Broad diffuse features are observed consistent with the
nanocrystallinity of the sample. Weak features are clearly evident up to
$Q=12$ Å-1 (Fig. 2(c) inset), but less apparent beyond that point.

The $F(Q)$ from the same data after correction is shown in Fig. 3(a) and the
resulting ePDFs in Fig. 3(c), with the calculated PDF from a model of the gold
fcc structure plotted on top in red. For comparison, in Fig. 3(b) and (d) we
show the x-ray derived $F(Q)$ and xPDF, respectively. Unfortunately this is
not a direct comparison between identical samples. We were not able to collect
x-ray data from the same film as the ePDF as it was too thin to get a
sufficient signal in the x-ray measurement.

The structure functions of the electron and x-ray data (Fig. 3(a) and (b),
respectively) are clearly highly similar. Features in the $eF(Q)$ are broader
than the x-ray case but the features are all recognizable and have the correct
relative intensities. Likewise, the e- and xPDFs (Fig. 3(c) and (d),
respectively) are highly similar, with the features in the ePDF of the
nanoscale film being broader. The quality of the fits is comparable for both
the ePDF and xPDF curves, with the ePDF giving a slightly lower (better)
agreement factor. The refined parameters are presented in Table 1. The breadth
of the ePDF peaks are accommodated in the model by giving gold very large
atomic displacement parameters (ADPs), twice as large as those in the x-ray
measured gold nanoparticles that are already large. This indicates the
presence of significant atomic scale disorder in the film and is not coming
from the ePDF measurement itself. This is discussed in greater detail below.

These results clearly demonstrate that quantitatively reliable ePDFs can be
obtained from nanocrystalline materials in a standard laboratory TEM. The
counting statistics from the electron data compare favorably to those from the
x-ray measurements (Fig. 3(a) and (b)), despite the much shorter measurement
time, suggesting that ePDF determination could become a useful general
characterization tool during nanoparticle synthesis. Two effects are clearly
evident in the $Q$-space data: low $Q$-space resolution and the rapid
diminishing of the amplitude of scattered features with increased $Q$. The
latter is likely to reflect real differences in the samples, with the range of
structural coherence being lower in the gold film than the gold nanoparticles
used in the x-ray experiment. The lower $Q$-space resolution could be either a
sample or a measurement effect but this cannot be disentangled without having
a well characterized, kinematically scattering, nanoparticle standard for ED,
which doesn’t currently exist. The sputtered gold film has an fcc gold
structure, like the bulk, but with significantly more disorder and a nanometer
range for the structural coherence.

The ED data were taken with a standard CCD camera and no filtering of
inelastically scattered electrons. This is the most straightforward protocol
for data collection as it is the standard setup in most laboratory TEMs. It is
expected to result in lower quality PDFs than those measured with energy
filtered electrons because of the higher backgrounds due to inelastically
scattered electrons cocka;aca88 . ED data collected with an image plate
detector are also expected to be higher quality due to the low intrinsic
detector noise and better dynamic range of that detector technology. Thus, the
resulting PDF shown in Fig. 3(c) represents the baseline of what is possible
without specialized instrumentation. The resulting $F(Q)$ shows excellent
signal to noise up to the maximum accessible $Q$-range of 17 Å-1, as evident
in Fig. 3(a).

To explore the size limits for Au NPs to scatter kinematically, we collected
data from larger, 100 nm Au nanoparticles, and the results are also given in
Table 1 and Figures 4 and 5. Comparing the integrated 1D diffraction patterns
of the large NPs and the thin Au film, Figs. 4(c) and 2(c), respectively, we
see similar features, but in the case of the NPs, the amplitudes of the
scattered intensities extend to much higher $Q$ values, as if there is a much
smaller Debye-Waller factor for the data. This can be clearly observed by
comparing the e$F(Q)$ of the large nanoparticles in Fig. 5(a) with those from
x-ray diffraction data, x$F(Q)$ in Fig. 5(b). The enhancement in the high-$Q$
features is large and is almost certainly due to significant coherent multiple
scattering in this sample. The resulting ePDF from the NPs has peaks that are
correspondingly sharp compared to the thin film gold and the xPDFs of gold
NPs. Regardless of the presence of significant multiple scattering, a model
was refined against the ePDF of the 100 nm Au nanoparticles to see the extent
that the refined structural model parameters are affected. The structure
refinement gave fits that were slightly worse but comparable in quality to the
xPDF fits (see Table 1), $R_{w}=0.24$. The refined values were similar also,
except for much smaller atomic displacement parameters (ADPs), due to the
artificially sharpened PDF peaks. It is somewhat remarkable that, in this
case, the dynamical scattering produces features in the $F(Q)$ with
approximately the correct relative amplitude, but extending to much
higher-$Q$. Not only are the PDF peaks in the right position ansti;u88 , but
have the right relative amplitudes. Gold may be a special case because the
structure factors are all either ones or zero’s.

This clearly shows that for a strong scatterer such as Au, 100 nm
nanoparticles already give significant dynamical effects. The resulting PDFs
give useful semi-quantitative and qualitative information but the refined
thermal parameters are not reliable. Indeed, the effect of the multiple
scattering to increase the real-space resolution by boosting the intensities
of the high-$Q$ peaks makes the PDF peaks sharper with the result that bond-
lengths can be extracted with greater precision from the ePDF data in this
case. When accurate PDF peak positions rather than quantitative peak
intensities are desired, this could be a significant advantage of the ePDF
method, for example, when looking for small peak splittings, or resolving peak
overlaps, to aid in structure solution.

A less trivial structure factor is obtained from binary compounds, such as the
NaCl studied here. The TEM image of the sample in Fig. 6(a) shows that it
consists of nanoscale crystallites, some of which have a cubic habit and
others that have no particular morphology. The corresponding ED pattern in
Fig. 6(b) shows clear and fairly uniform rings, with some spottiness from an
imperfect powder average. Fig. 6(c) shows the integrated ED pattern. The
$F(Q)$ and the resulting ePDF obtained from this data set is shown in Figs.
7(a) and (c), respectively. For comparison, an x$F(Q)$ and an xPDF obtained
from a bulk crystalline NaCl sample is also shown in Fig. 7(b) and (d).

The rock-salt structure model fits to the PDFs are shown in Fig. 7(c) and (d)
and the results are presented in Table 2. The e- and xPDFs are qualitatively
highly similar, with all features in the xPDF easily recognizable in the ePDF.
Notably, the relative intensities of adjacent peaks are similar between the e-
and xPDFs. Peaks in the ePDF die out in amplitude with increasing $r$ more
quickly, due to the broader features in the ED pattern. The overall quality of
the fit to the ePDF is worse than the xPDF of bulk NaCl. Refined lattice
constants agree well within the experimental uncertainty. The ePDF refined
thermal parameters are much smaller than those obtained from the x-ray data.
This is unlikely to be a real effect as both the x-ray and electron data were
measured at room temperature, and it is rather implausible that the
nanoparticulate samples have _less_ static structural disorder than bulk NaCl.
We therefore assume that this is the effect of multiple scattering in the
data, similar to that observed for large Au NPs. Clearly, ADPs refined from
ePDFs present a lower bound on actual sample ADPs. They are accurate in the
case where multiple scattering is negligible, but underestimate the thermal
motions and static disorder in the presence of multiple scattering.

##  V Discussion and Conclusions

The Au and NaCl examples establish that quantitatively, or semi-
quantitatively, reliable PDFs can be obtained from nanomaterials using
electron diffraction data obtained on a standard laboratory TEM, without the
use of filtering. Because of the ease and speed of collecting such data and
the ubiquity of such instruments in chemistry and materials laboratories, if
the barriers to data processing could be overcome making the whole process
straightforward, this could become a broadly applicable standard and useful
characterization method for nanoparticles and thin films.

This work also explores the experimental parameters for obtaining good data
for reliable ePDFs from nanomaterials. Principally, samples should be thin
enough or, for the case of nanoparticles, have a sufficiently small diameter.
What this diameter is depends on the average atomic number of the sample. For
Au, 100 nm diameter NPs gave significant coherent multiple scattering, 2.7 nm
thick films did not. For all materials we expect that 10 nm and smaller
particles will scatter kinematically; and these are precisely in the size-
range of nanoparticles that benefit the most from a PDF analysis masad;prb07 .
Obtaining a good powder average is also a very important part of powder
diffraction regardless of the probing technique, XRD, ND or ED. This can be
easily achieved by using a large sample volume in ND and spinning the sample
in XRD. However, in ED, both of these methods become difficult due to the
limitations of the configuration and careful sample preparation in this regard
is very helpful. Again, for the particular application in nanoparticle
structure characterization, the small size of the particles means that better
powder averages can be obtained even from small sample volumes. However, the
quality of the powder average should be checked by visual inspection of the ED
images from the CCD, which is readily done as is evident in the figures in
this paper. The powder average can be improved by increasing the beam-spot
size on the sample and also by taking multiple images from different regions
of the sample and averaging them. The maximum Qmax attainable is determined by
the operational energy, camera length, dimensions of the detector and the
diameter of the microscope, but in general should be maximized. Our electron
microscope configuration equipped with a CCD camera limited Qmax to
$\sim(17-18)$ Å. The advantage of using a higher Qmax is the better real space
resolution that results in the ePDF. However, standard microscope
configurations naturally give sufficiently high $Q_{\max}$ values for most
applications. Thus there seems to be no impediment to the use of ED from
standard laboratory electron microscopes for quantitative nanoparticle
structural characterization using the PDF.

##  VI Acknowledgements

We would like to acknowledge helpful discussions with Michael Thorpe. We also
thank Jon Hanson for allowing access to the X7B beamline at NSLS, which is
supported by DOE-BES under contract No DE-AC02-98CH10886. Work in the Billinge
group was supported by DOE-BES through account DE-AC02-98CH10886. Work in the
Kanatzidis group was supported by NSF through grant DMR-1104965.

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Figure 1:  A comparison between normalized (to f(0)) x-ray and electron form
factors, $f_{x}$(Q) and $f_{e}$(Q) of Au. Figure 2: (a) A TEM image of the 2.7
nm thick Au film used for ED. (b) A false-color 2D ED pattern collected on
this sample using 200 keV electrons. Lighter colors indicate higher intensity.
The black bar across the middle of the image is the shadow of the beam-stop.
(c) 1D Au electron powder diffraction pattern obtained by integrating around
the rings in 2 (b). The inset shows the high $Q$ region of the ED pattern on
an expanded $y$-scale. The dotted lines are guides to the eye.  Figure 3: (a)
Reduced structure function, $F(Q)$, of Au obtained from the integrated ED
pattern in Fig. 2(c), (b) An $F(Q)$ of Au nanoparticles calculated from an XRD
pattern collected at X7B at the NSLS. (c) Au bulk structure model fit to the
resulting ePDF from 3(a). (d) Au bulk structure model fit to the resulting
xPDF from 3(b). Observed and calculated PDFs are presented with blue circles
and a solid red line respectively. The difference between observed and
calculated is offset below (green solid lines). In both cases used Qmax=15.25
Å.

Figure 4: (a) A TEM image of $\sim$ 100 nm Au nanoparticles used for ED. Black
dots are the nanoparticles on the grid and the large white areas are the holes
in the grid. (b) A background subtracted ED image, collected from the same
region of the sample using 200 keV electrons. (c) The 1D ED pattern obtained
by integrating around the rings in 4(b). The inset shows a magnified region of
the integrated ED pattern as indicated by the dotted lines. This ED pattern
clearly suffers from multiple scattering due to the thickness of the sample.

.

Figure 5: (a) The reduced structure function, $F(Q)$, of Au calculated from
the integrated ED pattern in Fig. 4(c), (b) An F(Q) generated from an
integrated Au NP XRD pattern from a 2D data set collected at X7B at the NSLS.
(c) Au bulk structure model fit to the resulting ePDF from 5 (a). (d) Au bulk
structure model fit to the resulting xPDF from 5 (b). Observed and calculated
PDFs are presented with blue circles and a solid red lines respectively. The
difference between observed and calculated is offset below (green solid
lines). Figure 6: (a) A TEM image of the NaCl film used for ED. (b) A false-
color 2D ED image collected on this sample using 200 keV electrons. Lighter
colors indicate higher intensity. The black bar across the middle of the image
is the shadow of the beam-stop. (c) 1D ED pattern obtained by integrating
around the rings in 6 (b). The inset shows the high $Q$ region of the ED
pattern on an expanded $y$-scale. The dotted lines are guides to the eye.
Figure 7: (a) Reduced structure function, F(Q), of NaCl obtained from the
integrated ED pattern in Fig. 6(c). (b) An F(Q) of NaCl calculated from an
x-ray data set. (c) NaCl bulk structure model fit to the resulting ePDF from 7
(a). (d) NaCl bulk structure model fit to the resulting xPDF from 7 (b).
Observed and calculated PDFs are presented with blue circles and solid red
lines respectively. The difference between observed and calculated is offset
below (green solid lines). In both cases used Qmax=13.6 Å.

Table 1: Refined parameters for 2.7 nm thick nanoparticulate Au film, $\sim
100$ nm diameter nanoparticles (NP) from ePDFs and from a gold nanoparticle
sample from xPDFs. The structure model is the fcc bulk gold structure, space-
group Fm-3m. It was not possible to measure the nanoparticle size from the
ePDFs as we were not able to calibrate the intrinsic $Q$-space resolution of
the ED measurement allowing us to separate the instrumental resolution and
particle size effects in the ePDFs.

| ePDF (film) | ePDF (NP) | xPDF
---|---|---|---
Qmax (Å-1) | 15.25 | 15.25 | 15.25
Fit range (Å) | 1-20 | 1-20 | 1-20
Cell parameter (Å) | 4.075(3) | 4.076(2) | 4.058(1)
Uiso (Å2) | 0.033(4) | 0.006 (3) | 0.014(1)
Diameter (Å) |  $\sim$27∗ |  $\sim$1000∗∗ | 24.51(9)
Q-damp (Å-1) | 0.095(5) | 0.095(5) | 0.047(2)
Rw (%) | 17 | 24 | 20
∗film thickness measured during deposition ∗∗NP diameter estimated directly
from the TEM image

Table 2:  Refined parameters for nanoparticulate NaCl from ePDFs and for a bulk powder of NaCl from the xPDF. The structure model is the fcc rock-salt structure, space-group Fm-3m. It was not possible to measure the nanoparticle size from the ePDFs as we were not able to calibrate the intrinsic $Q$-space resolution of the ED measurement allowing us to separate the instrumental resolution and particle size effects in the ePDFs.  | ePDF | xPDF
---|---|---
Qmax (Å-1) | 13.6 | 13.6
Fit range (Å) | (0.2-30) | (0.2-30)
Cell parameter (Å) | 5.62(2) | 5.63(1)
$U_{iso}$ \- Na (Å2) | 0.007(5) | 0.027(1)
$U_{iso}$ \- Cl (Å2) | 0.004(4) | 0.016(1)
Q-damp (Å-1) | 0.095(5) | 0.06(1)
Rw % | 33 | 6


The _muX_ project

F. Wauters1$\star$ and A. Knecht2 on behalf of the _muX_
collaboration111https://www.psi.ch/en/ltp/mux

1 PRISMA+ Cluster of Excellence and Institute of Nuclear Physics, Johannes
Gutenberg Universität Mainz, Germany

2 Paul Scherrer Institut, Villigen, Switzerland

* Corresponding<EMAIL_ADDRESS>
Review of Particle Physics at PSI
10.21468/SciPostPhysProc.2

## Abstract

The _muX_ project is conducting a series of muonic X-ray measurements in
medium- and high-Z nuclei at PSI, utilizing a high-purity germanium detector
array, in-beam muon detectors, and a modern digital data-acquisition system. A
novel hydrogen target for muon transfer was developed, enabling measurements
with as little as a few micrograms of target material. First measurements with
radioactive Cm and Ra targets were conducted, aimed at determining their
nuclear charge radii. These serve as important input for upcoming atomic
parity violation experiments. The apparatus is also used to perform a
feasibility study of an atomic parity violation experiment with the $2s-1s$
muonic X-ray transition. In addition, the setup has been made available for a
wider range of nuclear, particle, and solid-state physics measurements.

###  22.1 Introduction

Muonic atoms are exotic atoms that form when negative muons are stopped in a
target and are subsequently captured by a nearby atom in a highly excited
atomic orbital of $n\geqslant$14\. The muons quickly cascade down to the $1s$
orbital, initially predominantly via Auger transitions: at lower $n$ radiative
transitions take over. As the muon mass is about 207 times larger than the
electron mass, the muonic X-rays range in energy from a few tens of keV for
low-Z nuclei to several MeV for heavier nuclei. The capture and cascade
processes occur on (sub)nanosecond timescales. The emitted radiation therefore
appears prompt relative to a muon stopping in the target. Once in the $1s$
orbit, the muon either decays in orbit, or is captured by the nucleus. The
latter is the dominant decay channel for Z=12 and above [1].

Muonic atoms have proven to be a valuable tool to measure nuclear properties
and probe short-range interactions between the muon and the nucleus. With the
Bohr radius of the muon compared to the electron scaling as $m_{e}/m_{\mu}$,
there is substantial overlap between the muon and nuclear wave functions.
Finite size effects are thus highly amplified. In the past, the absolute
nuclear charge radii $<r^{2}>^{1/2}$ of almost all stable nuclei have been
determined with a typical accuracy of $10^{-4}$ \- $10^{-3}$ by measuring the
$2p-1s$ transition energy [2]. More recently, the radii of the lightest nuclei
were measured by the CREMA collaboration (Section 21 [3]) using laser
spectroscopy on muonic atoms [4, 5, 6, 7].

Formerly, this approach was limited to stable isotopes, as a sufficient amount
of target material is needed to stop a $\mu^{-}$ beam with a momentum of
typically 30 MeV/$c$. This excludes many interesting nuclei, such as the
highly-deformed radium isotopes. Radium is a prime candidate for an Atomic
Parity Violation (APV) experiment, using laser spectroscopy on a trapped ion
[8, 9], where the Parity Non-Conserving (PNC) $E1_{PNC}$ atomic $S-D$
transition is proportional to $K_{r}Z^{2}Q_{W}$, with $Q_{W}$ the weak nuclear
charge, and $K_{r}$ a relativistic enhancement factor which depends on the
nuclear charge radius [10]. The _muX_ collaboration aims to determine this
radius by measuring the $2p-1s$ transition energy of 226Ra ($T_{1/2}$=1600
y.). For this we have developed a novel technique, stopping muons in a high-
pressure H2/D2 target, using a sequence of transfer reactions to efficiently
stop muons in a few micrograms of target material. This technique was first
established with gold targets, then applied to 226Ra and 248Cm (see Section
22.3).

With fundamental interactions being our primary physics motivation, the
collaboration is also investigating the possibility of measuring APV directly
in muonic atoms. A neutral parity-violating interaction mixes the $2s_{1/2}$
and $2p_{1/2}$ atomic levels, resulting in an E1 admixture in the otherwise
pure M1 $2s_{1/2}-1s_{1/2}$ transition. Measuring such a parity-odd observable
was first reviewed by Feinberg & Chen [11] and Missimer & Simons [12]. More
recently, the possibility of searching for interactions between the muon and
the nucleus beyond the Standard Model led to revived interest [13, 14]. While
the PNC effect is largest for low-Z atoms, separating the radiative M1/E1
transition from other transitions in the cascade severely complicates the
design of such an experiment [15]. We focus on Z$\simeq$30 nuclei, where the
single-photon $2s-1s$ transition becomes the dominant path depopulating the
$2s$ level. The current goal of the collaboration is to isolate the transition
in the cascade, and to significantly improve the signal-to-background ratio in
the region-of-interest (ROI) in the X-ray spectrum (see Section 22.4.1).

Since 2015 we have been developing an advanced muonic X-ray experimental
setup, combining a high-purity germanium (HPGe) detector array and a modern
data-acquisition system (DAQ) with various target configurations. The setup is
currently also being used for non-destructive elemental analysis, muon-capture
studies probing matrix elements of interest for neutrinoless double $\beta$
decay, and further nuclear-charge radius measurements of various radioactive
elements and rare isotopes.

###  22.2 Experimental setup.

The _muX_ apparatus (Figure 22.2 and Figure 22.2) is located at the $\pi$E1
beam-line of PSI, where a typically 30-40 MeV/c $\mu^{-}$ beam with a momentum
width $\Delta p/p$ of 3 % passes through an electron separator before reaching
the experiment. A custom beam snout houses an in-vacuum set of beam counters,
thin plastic scintillator slabs read out by SiPMs, a lead target mounted away
from the beam axis for calibration purposes, and a port for directly mounting
various targets, thereby minimizing scattering of the low-energy muons.

Figure 22.1: The _muX_ setup, with 1) the $\mu^{-}$ beam passing through 2) a
veto detector with a 18 mm aperture, and 3) a 200 $\mu$m thick muon detector.
The cell 4) with a 600 $\mu$m carbon fibre window supported by a Ti grid holds
5) 100 bar of hydrogen gas, with the 6) target mounted in the back. 7)
Electron veto detectors. 8) Standard and 9) MiniBall cluster HPGe detectors.

Figure 22.2: The MiniBall array with eight cluster detectors complemented by a
70 % coaxial detector and a low-energy planar detector installed at the
$\pi$E1 beamline for the 2019 experimental run, with the _muX_ beam snout. The
target cell is covered by the black electron detectors.

The target itself is surrounded by 5 mm thick plastic scintillators,
efficiently detecting outgoing decay electrons, thus enabling various cuts on
the data such as suppressing Bremsstrahlung background in the HPGe detectors.

The _muX_ HPGe detector array is constructed from various detectors provided
by the collaborating institutions. Early campaigns, such as the 185/187Re
measurement aimed at determining the charge radii and quadruple moments [16],
were conducted with just a few coaxial HPGe detectors. For the 2017 and 2018
campaigns, 7 compact coaxial detectors from the French/UK loan
pool222https://gepool.in2p3.fr/ with relative efficiencies of around 60% and
one Miniball cluster detector were added. In the summer of 2019, the full
MiniBall detector array [17] was installed at the $\pi$E1 beamline (Figure
22.2), operating for a 7 week measurement campaign. The _muX_ automatic
liquid-nitrogen filling system enables extended continuous operation of the
HPGe detectors.

The MIDAS-based DAQ uses SIS3316 250 MSPS
digitizers333https://www.struck.de/sis3316.html which record all detector hits
above threshold. Physics events are reconstructed offline by the analysis
software. A digital filter running on the digitizer module FPGA integrates the
detector signals, in addition, a section of the raw waveform is saved for
offline analysis, where a time resolution of better than 10 ns (FWHM) for the
HPGe detector hits is achieved.

###  22.3 Radioactive target measurements

One of the principal goals of the _muX_ project 444Proposal R-16-01 is to
measure the $2p-1s$ transition energies for 226Ra, a radioactive isotope for
which the maximum allowed quantity in the experimental area is 5 $\mu$g. As
the stopping power of such a low-mass target is insufficient by orders of
magnitude, the _muX_ collaboration has developed a novel method, stopping
muons in a small 100 bar H2 target with a small admixture of D2. Through a
series of transfer reactions the muon is transported to the target material
mounted at the back of the cell (Figure 22.4), hereby exploiting the Ramsauer-
Townsend effect [18, 19, 20, 21], which causes H2 gas to become almost fully
transparent for a $\mu$d atom.

After a first optimization of the target geometry and conditions with Monte-
Carlo simulations, the transfer method was established by mounting a thin gold
target at the back of the cylindrical gas cell. The beam momentum and
deuterium concentration were optimized for the number of gold X-rays per muon,
after which a small 3 nm thick gold target was installed. A total stopping
efficiency per beam muon of 1.2 % was achieved for this 5 $\mu$g target (see
Figure 22.4).

In order to have an efficient transfer target, it is imperative that the
(radioactive) material is deposited as a uniform surface layer. Due to the low
kinetic energy of the $\mu$d atom, an organic surface layer of $>$100 nm acts
as a barrier and significantly reduces the transfer efficiency, rendering
traditional molecular plating techniques inadequate. Several 248Cm and 226Ra
targets were produced at the Institute of Nuclear Chemistry of the Johannes
Gutenberg University Mainz, combining a custom electro-deposition technique
combined with a novel _drop-on-demand_ method where micro-drops of activity in
solution are deposited on glassy carbon disks, the low-Z backing material of
the target [22].

Figure 22.5 shows the muonic X-rays from 248Cm measured during the 2019
campaign with a 15 $\mu$g curium target. After subtracting several background
contributions, the $2p-1s$ transitions are clearly visible. Despite having
nuclear ground state of spin 0, the energy scale of high-Z muonic atoms is
such that the muon spin couples to excited nuclear states with a non-zero spin
[23, 24]. This leads to a complicated dynamic hyperfine structure in the
observed transition energies, which needs to be understood to extract the
nuclear charge radius from the data. The largest uncertainty in the
calculations of the transition energies is caused by the two-photon exchange
nuclear polarization [25, 26].

In addition to the 248Cm target, two 226 Ra targets were used. The data
obtained are currently under analysis to determine whether the X-ray yield is
sufficient to achieve the necessary accuracy on the nuclear charge radius.

Figure 22.3: 1\. After slowing down a $\mu$p atom is formed. 2. In
$\mathcal{O}$(100) ns, the muon transfers to deuterium, gaining 45 eV in
kinetic energy. 3. After scattering down in energy to around 4 eV, the
$\mu$d-H2 scattering cross section becomes negligibly small, and the $\mu$d
atom travels straight until it hits a wall or our target, where 4. the
$\mu^{-}$ transfers to a high-Z atom.

Figure 22.4: Muonic X-ray energies versus their time relative to an incoming
muon. X-rays from direct stops appear at 0 ns. The Au X-rays appear over
$\mathcal{O}$(100) ns, the typical timescale for the transfer processes. The
background mainly consists of decay electrons, and neutrons emitted after
nuclear muon capture.

Figure 22.5: The 248Cm muonic X-ray spectrum from the hydrogen transfer cell
after subtracting the lead calibration lines and the $\gamma$ background from
muon capture on 16O.

###  22.4 Extended experimental program

####  22.4.1 2s-1s measurements

With an expected branching ratio of $\mathcal{O}$(10-4) for the single-photon
$2s-1s$ muonic X-ray transition in the cascade of Z$\simeq$30 atoms, a
possible APV experiment with a PNC observable using this transition is
severely hampered by an overwhelming background in the energy region of
interest (ROI) from scattered $(n\geqslant 3)p-1s$ X-rays, Bremsstrahlung from
decay electrons, and neutrons from muon capture. For this reason this
transition has never been observed. The goal of the _muX_ project is to
observe this transition, significantly improve the signal-to-background in the
ROI, and determine the reach of a possible APV experiment.

The initial average orbital quantum number $l$ after $\mu H\rightarrow\mu Z$
transfer is lower than the initial $l$ for direct atomic capture [27]. We have
observed that as a consequence, the $2s$ population in the cascade of Ar, Kr,
and Xe is increased by a factor of 3-4, thus increasing the branching ratio of
the $2s-1s$ transition. A 7 day measurement with a 100 bar H2 target and an
0.1 % Kr addition was performed. After subtracting the nuclear capture
background from muon stops in the surrounding materials, the $2s-1s$ full
energy peak is clearly visible, achieving a signal to background of about 1/10
(Figure 22.7).

Figure 22.6: The $2s-1s$ full energy peak of muonic Kr clearly visible at 2.22
MeV above the Compton background of $(n>2)p-1s$ transitions after subtracting
background $\gamma$’s from nuclear muon capture processes.

Figure 22.7: Partial muon capture rates of selected isotopes provide access to
the transition strengths via virtual states in double $\beta$-decays.

To further reduce the background in the ROI, the transitions feeding the $2s$
level were used to tag events of interest. While sacrificing efficiency, this
approach significantly reduces the background: the continuous Compton
background from e.g. $3p-1s$ photons is fully eliminated, and the accidental
background from neutrons and decay electrons is at the same level as the
signal yield, which can be further reduced by improving the time resolution.
The only remaining challenging background is the satellite peaks introduced in
the spectra by Compton scattered photons with energy depositions in the region
of the $2s$ feeding transitions. This background needs to be controlled by
optimizing the detector geometry. During the 2019 campaign, one week of data
was taken with such an optimized geometry, collecting over 1011 muon stops on
an isotopically pure 64Zn target.

####  22.4.2 Other measurements

To fully benefit from the availability of the MiniBall detector array, the
_muX_ experimental program was expanded in 2019. The partial ordinary muon
capture rates on enriched 130Xe, 82Kr, and 24Mg to specific excited states in
the daughter nucleus were measured. Such measurements provide valuable
information to determine the nuclear matrix elements in neutrinoless double
$\beta$-decay [28, 29], as these states act as intermediate virtual states in
the double $\beta$-decay (Figure 22.7) of isotopes such as 130Te [30, 31] and
82Se [32].

In addition, the _muX_ apparatus was made available to perform elemental
analysis on a series of cultural heritage samples, 17th century Japanese coins
and an ancient Chinese mirror, significantly improving the sensitivity of
previous J-PARC measurements [33], and a number of coins and recently found
artifacts from the Roman Augusta Raurica site, nearby PSI. The intense muon
beam and efficient detector setup permitted a narrowly collimated beam,
probing different areas of a sample. Muonic X-ray spectroscopy provides
information about the bulk material compared to the surface sensitivity of
traditional fluorescence X-ray analysis. Furthermore, for high Z-elements such
as lead the isotopic composition can be extracted.

###  22.5 Conclusions and Outlook

The _muX_ efforts have resulted in a revived muonic X-ray program at the Paul
Scherrer Institut. A new versatile experimental setup allows us to efficiently
take data for extended periods of time.

The new hydrogen transfer target we have developed enables muonic X-ray
measurements with a very small amount of target material. First measurements
were performed with a Cm and Ra targets, with the purpose of extracting the
nuclear charge radius, providing valuable input for upcoming APV experiments.
The radioactive program will be extended to other elements, aiming to measure
the third of three isotopes of odd Z-elements needed to calibrate the vast
amount of isotope shift data available from laser spectroscopy on radioactive
elements [34].

The single photon $2s-1s$ transition in the muonic X-ray cascade was observed
for the first time, and significant progress was made in reducing the
backgrounds. This opens up the possibility for an APV experiment with a
sensitivity of $\mathcal{O}$(1) of the Standard Model amplitude, i.e., such a
measurement would act as a new physics search.

The two additional measurements of the 2019 campaign, the OMC capture
measurements and the elemental analysis, will continue as separate projects
with the support of the _muX_ collaboration.

#### Acknowledgements

This work was supported by the PSI through the Career Return Program, by the
Swiss National Science Foundation through the Marie Heim-Vögtlin grant No.
164515 and the project grant No. 200021165569, by the Cluster of Excellence
"Precision Physics, Fundamental Interactions, and Structure of Matter" (PRISMA
EXC 1098 and PRISMA+ EXC2118/1) funded by the German Research Foundation (DFG)
within the German Excellence Strategy (ProjectID 39083149), and by the DFG
under Project WA 4157/1. We wish to thank all collaborating institutions
providing essential instrumentation such as high-purity germanium detectors.

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