text:
# Is it possible to represent max{a,b} as a binary polynomial?

For privacy-preserving comparison, is it possible to represent the function of $$f(a,b)=\max\{a,b\}$$ as a binary polynomial $$p(a,b)$$?

• You have boolean comparator circuits, and maybe you could have another function to just output the larger one determined by the previous circuit. For privacy, you can garble the circuit, other techniques can be used as well. – user2505282 Apr 28 '19 at 11:01

I found the following solution due to Damgard et al.

[1]. Ivan Damgard, Martin Geisler, and Mikkel Kroigaard. Efficient and Secure Comparison for On-Line Auctions. ACISP 2007, LNCS 4586, pp. 416-430, 2007.

Background. Given the binary represetations of $$m=m_{\ell}..m_1$$ and $$x=x_{\ell}..x_1$$, let $$w_i=m_i\oplus x_i\quad\mbox{and}\quad c_i=x_i-m_i+1+\sum_{j=i+1}^{\ell} w_j,$$ then if $$m>x$$ there is exactly one position $$i$$ where $$c_i=0$$, otherwise, no such position exists. (Please see Page 423 of [1].)

A maple code：

with(Bits)：

pcomp := proc (a, b)

local abin, bbin, la, lb, w, v, c, i, f;

abin := convert(a, 'base', 2); la := nops(abin);

bbin := convert(b, 'base', 2); lb := nops(bbin);

if la < lb then

abin := [seq(abin[i], i = 1 .. la), seq(0, i = 1 .. lb-la)]; la := lb;

elif lb < la then

bbin := [seq(bbin[i], i = 1 .. lb), seq(0, i = 1 .. la-lb)]; lb := la;

end if;

w := [seq(Xor(abin[i], bbin[i]), i = 1 .. la)];

c := [seq(bbin[i]-abin[i]+1+sum(w[j], j = i+1 .. la), i = 1 .. la)];

for i to la do

if c[i] = 0 then f := 1; return f; end if;

end do;

f := 0; return f;

end proc;


text:
Obstacle Avoidance and Navigation in the Real World by a Seeing Robot Rover, Hans Moravec, 1980
<-- Previous  Next -->

Chapter 12: Connections

The goal of this research was a program that could drive the cart through realistic environments. I perceived the task to be barely possible, and approached it in an uninhibited frame of mind; no code was too dirty nor method too specialized, so long as it worked. Though I expected to achieve some scientific insights, they would come as byproducts of honest work, not from a direct assault.

Stereo Vision

Computer vision, especially for rovers, is in such a tender and undeveloped state (and so marginally feasible) that very little of the prior progress in the field was of any help in writing my program. Nevertheless, my efforts can be seen to have some relation to the other work. With further development, some of the other methods might even be useful in future cart-like systems.

For understandable reasons, much of the work on extracting three dimensional information from pictures has been done with picture pairs, in imitation of human binocular vision.

Viking Mars Lander picture pairs processed by Gennery \ref{G1} originated from a camera pair. The Mariner 9, and Apollo 15 photograph pairs processed by Quam and Hannah \ref{QH1}, were taken by the same camera at different locations while the spacecraft were orbiting Mars and the Moon. The aerial photograph pairs used by Arnold \ref{A1} were taken successively by a single camera aboard an airplane flying over the scene.

There have been a few efforts besides mine that use more than two pictures. Baker \ref{B4} followed edges between multiple views of an object that rotated in front of a camera, and Baumgart \ref{B5} used the silhouettes in images obtained similarly to build models of objects, except for concavities. Burr and Chien \ref{BC1} use picture triplets. Nevatia \ref{N1} suggested using multiple images, taken successively by a single camera. He forsees using his techniques on orbital, planetary surface, or industrial data. In the latter case, successive images might be available of some mechanical part travelling along a conveyor belt.

Area correlation of small patches, used at times by Quam, Hannah, Nevatia, Gennery and myself is not the only way to find corresponding parts of two images.

Arnold \ref{A1} matches edges, obtained from each image using a standard edge operator, as do Burr and Chien. Quam and Hannah \ref{QH1} match small areas, then grow them into larger areas, to obtain matches between whole shapes in the images. Marr and Poggio \ref{MP1} use a method which examines the whole image and chooses the features matched in parallel with doing the matching.

If a sampling of depths in the image, rather than a dense depth map, is sufficient for an application, then one must select the points to range. A good idea is to pick points in one image which will be easy to find in a second. Pingle and Thomas \ref{PT1} chose windows with high brightness variance, which also got a positive response from a corner finder.

My interest operator is a cheaper version of the same idea, though by depending on local maxima of an “interestingness” measure, it is more tolerant of regions that have low contrast. Yakimovsky and Cunningham \ref{YC1} use an autocorrelation technique that is almost equivalent to my interest measure.

Gennery

Although the current program works quite differently, my initial approach to the obstacle avoider was with a vehicle carrying a single fixed camera. The stereo baseline in that case would be generated by the vehicle motion. Since the motion was known very imprecisely, it would have to be deduced simultaneously with the feature distances. Gennery had written a camera-solving subroutine which tackled this problem. (Fortunately the whole approach proved too error-prone. If it had not, my program would have had a major component that was someone else's prior work. (shudder). It may still be used in future systems).

Gennery's camera solver \ref{G1} is able to determine five of the six parameters which relate the relative position and orientation of two cameras viewing the same scene, from the image plane co-ordinates of five or more features points seen by both cameras. The algorithm determines the azimuth and elevation of the second camera relative to the position of the first, and the pan, tilt and roll of the second relative to the orientation of the first. The sixth parameter, the distance between the cameras, can not be determined, in principle, from image co-ordinates alone.

Gennery has developed another program that may be useful to future carts. This one deduces the position and shape of the ground directly from a stereo pair of pictures of a littered scene \ref{G2}. It assumes the ground surface can be approximated by a two dimensional second degree polynomial (though a first degree fit can be forced a priori if desired). The algorithm (which begins by finding a rough camera model using points found with my interest and correlation operators!) works with a large number of three dimensional positions of points in the scene, found with a high resolution correlator that gives probability ratings to matches, based on a detailed statistical analysis of the errors expected in two noisy images. It separates the points into ground and not ground by successively doing a least squares fit, and then selecting all points within tolerance of the candidate surface or below it for the next fitting. This has the effect of finding lower and lower surfaces, with non-ground clutter above them. Additional heuristics help eliminate spurious points.

Hannah and Quam

Quam and Hannah \ref{QH1} combined techniques they had developed earlier, (Quam \ref{Q1} and Hannah \ref{H1}), to produce contour maps of portions of the Lunar and Martian surfaces from stereo pairs of photographs from the Mariner 9 and Apollo 15 spacecraft. They start their program by hand by indicating a particular planetary surface point in both images. Around this seed, the program grows a region of point pairings whose surrounding windows show a local maximum at the same disparity. When all such contiguous points have been exhausted, a nearby non-matching point is taken, and a local search performed to find a corresponding point which maximizes the correlation of the surrounding windows. If the correlation is sufficiently high, the region growing procedure is repeated. If the program fails to find any suitable new seeds, it appeals to the human operator again. Finally the depth estimates are refined by interpolating the correlation measure for a three by three cell around each point.

An early version of Gennery's camera solver then generates 3D co-ordinates of the surface points, which are used to make contour maps that sometimes look right.

Arnold

Area correlation techniques tend to fail on object boundaries because the edge of an object appears against a different background in each picture. Arnold's stereo vision system \ref{A1} avoids this problem by matching edgels (short edge segments), extracted from the images with the Hueckel operator. He incorporates local context information in the matching process, and eventually determines a depth map of height above the ground, for the edges of the regions.

Edge detectors are unable to operate well in the presence of texture or smooth shading, so Arnold suggests that eventually edge based methods, and area correlation methods should be combined. I say the same thing.

Arnold's system was developed on aerial photographs of airport and parking lot scenes. After finding a camera model and a ground plane with Gennery's solver, the program applies a 3.19 pixel radius circular Hueckel operator to both images, typically producing around 1200 edgels. Each edge has position, angle, and the brightness on each side. The spatial information is normalized using the information from the camera solver and ground finder, starkly limiting the search area for matching edgels. Thresholds on the allowable differences between angles and brightness levels further constrain the plausible matches. Typically there might be 8 possible matches in the right hand image for an edgel in the left. Disparities are calculated for each of these possible matches and attached to the edges in one of the pictures.

Local context further resolves the ambiguity. Two edgels are considered to belong to the same physical edge if they are within 3 pixels and 90° of each other, and the angle of a line connecting their centers lies between their angles and the brightness levels are consistent on at least one side. A voting scheme weighted by edgel distances from each other and by disparity disagreements finds a likely disparity, and hence a height above the ground, for such physical edges.

Burr and Chien

Burr's and Chien's system \ref{BC1} is given piece-wise linear wire frame models of objects to be found in scenes. It assumes three fixed cameras, one in the center, another rotated 20° east, and a third rotated 10° north. The program compares center and east pictures when matching vertical edges, and center and north for horizontal edges. Depth maps are made using the region matching algorithms of Hannah \ref{H1}, then an edge operator is used on the center image. Edgels are linked into chains of near neighbors with consistent directionality. Then the depth maps guide a local search in one of the two other images for a matching edge. The 3D edges so obtained are matched against the wire frame models to locate objects in the scene.

Mori et al.

Mori et al. \ref{MKA1} try to make extremely accurate depth maps from aerial photographs. The camera model is determined from accurately known ground points. A fixed size correlation window is inadequate for the desired accuracy because a patch of sloped surface can vary significantly in size between two views. They make an initial approximation using a standard feature correlator, then correct one of the images for distortions that the estimated ground surface gradients would have caused. The process is iterated, and eventually an accurate map is produced.

Yakimovsky and Cunningham

Yakimovsky's and Cunningham's \ref{YC1} stereo program was developed for the JPL Robotics Research Vehicle, a testbed for a Mars rover and remote processing systems (Levine et al. \ref{LOY1} describe an earlier stereo system using the same hardware). The RRV has a pair of rigidly attached cameras, with highly linear lenses, so the search space is a very narrow line parallel to the projection of the stereo axis. A small correlation window is chosen in one image and expanded until its correlation with itself shifted one pixel is sufficiently poor, showing it has adequate information content, or until it gets too large (this is essentially an interest operator step). Windows that pass this test are correlated with the other image.

Marr and Poggio

Marr and Poggio \ref{MP1} have a different approach. Their matching pattern has no precise edges, it is fuzzy and implicit from a cooperative (or relaxation) algorithm in which independent measurements made at nearby parts of the image inhibit or reinforce each other. They start with two general principles; Each image point may assigned at most one disparity and, except for a few boundaries, disparity varies smoothly almost everywhere.

For each pixel in one image and for a range of disparities there is a node which represents the likelihood that the corresponding pixel in the other image represents the same physical object. The nodes are initialized by a simple pixel comparison. An iterative process then repeatedly updates each pixel/disparity node with a normalized measure of how surrounding nodes of the same disparity agree.

Marr's and Poggio's program works well on random-dot stereograms. (Julesz \ref{J1}), taking about 15 iterations to produce a solid representation of the hidden figures.

In 1979 Marr and Poggio \ref{MP2} published and Grimson and Marr \ref{GM1} implemented a different approach that involves low pass filtering the images to various resolutions, then correlating on the zero crossings in the filtered images. A low frequency version provides a coarse positioning, and the less filtered images are then used to refine the positions. In some ways the process is similar to the binary search correlation method described in this thesis.

Path Planning

Plausible or minimum cost path finding is a simpler, and in an abstract sense, less interesting problem than stereo vision. A number of approaches have been published, some heuristic, many depending on exhaustive search. Some of these depend intimately on a “Manhattan distance” (runs exactly in X or Y directions exclusively) geometry, which are good for circuit board layout, but uninteresting to us.

Many of the more general approaches, including the one implemented in the cart program, translate the problem into a connected graph with arc costs. The best general solution for finding the shortest distance in such a graph is still the $O(n^2)$ algorithm published by Dijkstra \ref{D1}. My program uses a variant of it.

A number of heuristic methods have been presented. SRI's Shakey system used one such. A generalization is described by Hart, Nilsson and Raphael \ref{HNR1}. They call it the A* algorithm.

Recently Lozano-Perez and Wesley \ref{LW1} published an obstacle avoidance algorithm whose core is nearly identical to mine. Instead of circular obstacles, their's avoids polygons. Instead of circle to circle tangents, they consider vertex to vertex paths (which is a little easier, because there are fewer vertices than tangents). Their program, like mine, maps the possible path space into a graph, and solves with an efficient graph algorithm.

<-- Previous  Next -->

format: plain


text:
# Divided by three

Algebra Level 3

$\large \sqrt{3} + \frac{1}{ \sqrt{3} } + \frac{1}{3 \sqrt{3} } + \frac{1}{ 9 \sqrt{3} } + ......... = \frac{a \sqrt{b} }{ c }$

Where a and c are positive coprime integers and b is a square free integer.

Find $a + b + c$.

###### This problem is a part of the sets - 1's & 2's & QuEsTiOnS.
×

Problem Loading...

Note Loading...

Set Loading...


text:
+0

0
112
1

Hi, I'm really struggling to solve this problem, I've looked and looked for help, and I can't seem to find anything... here's the problem:

The equation of the line passing through (1,11) and (4,5) can be expressed in the form

$$\frac{x}{a} + \frac{y}{b} = 1.$$
Find $$a$$.

Oct 23, 2020

#1
+29148
+1

First find the equation of the line        slope = 6/-3 = -2

point slope form of the line

y-5 = -2(x-4)

y-5 = 8 - 2x

y+2x = 13      divide by 13

y /13  + x / (13/2) = 1

Oct 23, 2020
edited by ElectricPavlov  Oct 23, 2020


text:
View Single Post

Recognitions:
Gold Member
Homework Help
 Quote by tom.stoer I think my example with S³ was missleading; I'll try it again: The dimension of the vector space of C-functions on S³ is infinite-dimensional. But if we look at the Lie group SU(2) we can write its elements as $$U[f] = \exp\left\{i t^a f^a(x)\right\}$$ The functions fa are members of an infinite dimensional vector space, but nevertheless we talk about a finite dimensional group b/c we have finitly many generators ta.
Diffeomorphisms which are infinitesimally close to the identity look like gauge transformations. The problem is that infinitesimal diffeomorphisms do not generate the whole group.

 Looking at the Ashtekar formulation of gravity there are the Gauss-law generators Ga(x) and the spacelike diffeomorphism generators = vector-constraints Va(x). The transformations are generated via functionals like $$G[f] = \int d^3x\,G^a(x)\,f(x);\quad V[f] = \int d^3x\,V^a(x)\,f(x);\quad$$ Again we have finitly many generators even if the functions f belong to an infinite dimensional vector space.
If there are a finite number of generators, we are not talking about the full diffeomorphism group, only the infinitesimal diffeomorphisms.


text:
Notebook

# The Basal Ganglia

The basal ganglia according to Stewart 2010 is an action selector that chooses whatever action has the best "salience" or "goodness". Its really interesting behaviour manifests itself when it interacts with the thalamus and other components of the brain, but in this example we will only show the basal ganglia's basic behaviour. It will choose between three actions that we'll pretend are "eating", "sleeping" and "playing".

In [ ]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

import nengo

<IPython.core.display.Javascript at 0x7fd1e8682610>

## Step 1: Create the Network

Here we create the basal ganglia and the action input node.

In [2]:
model = nengo.Network(label='Basal Ganglia')
with model:
basal_ganglia = nengo.networks.BasalGanglia(dimensions=3)

class ActionIterator(object):
def __init__(self, dimensions):
self.actions = np.ones(dimensions) * 0.1

def step(self, t):
# one action at time dominates
dominate = int(t % 3)
self.actions[:] = 0.1
self.actions[dominate] = 0.8
return self.actions

action_iterator = ActionIterator(dimensions=3)

with model:
actions = nengo.Node(action_iterator.step, label="actions")

## Step 2: Connect the Network

Connect the input to the basal ganglia and connect the probes

In [3]:
with model:
nengo.Connection(actions, basal_ganglia.input, synapse=None)
selected_action = nengo.Probe(basal_ganglia.output, synapse=0.01)
input_actions = nengo.Probe(actions, synapse=0.01)

## Step 3: Simulate the Network and Plot the Results

In [4]:
with nengo.Simulator(model) as sim:
# This will take a while
sim.run(6)

In [5]:
plt.figure()
plt.subplot(2, 1, 1)
plt.plot(sim.trange(), sim.data[input_actions].argmax(axis=1))
plt.ylim(-0.1, 2.1)
plt.xlabel('time [s]')
plt.title("Index of actual max value")
plt.subplot(2, 1, 2)
plt.plot(sim.trange(), sim.data[selected_action].argmax(axis=1))
plt.ylim(-0.1, 2.1)
plt.xlabel('time [s]')
plt.title("Basal ganglia selected max value")
plt.tight_layout()

As expected, the maximum index is found at 0, then 1, then 2 or "eating", "sleeping", then "playing". Note that if you zoom in enough on the basal ganglia values, you'll be able to see a bit of a delay between finding max values. If you read the aforementioned paper, you'll see that this is expected and matches previous experiments.

Download basal_ganglia as an IPython notebook or Python script.


text:
# Math Help - is this set closed?

1. ## is this set closed?

this is a review problem that I am not sure about.

is the set {(x,y)∈R^(2 ) ∶ 3x^2+ 5y^2=1} a closed set?

my initial thought is yes however I think I am missing something important.

I know that this is an ellipse but it is not in standard form. I had mathlab graph it and it complained that there
complex numubers that it was going to ignore, so I suspect that while the graph appeared to be a solid curve it is actually
full of spaces between the real solutions.

any thoughts?

2. ## Re: is this set closed?

Do you mean a closed set in the usual topology on $R^2$? If so, is its complement open? That is, if you take a point (x, y) NOT satisfying that equation, must there be a (possibly very small) neighborhood about it that is also not in the set.

If (x,y) is not on the ellipse, draw a line from (x,y) perpendicular to the ellipse. That is the shortest distance from the point to the ellipse. What about a disk centered on (x, y) with radius half that distance?

I don't use mathlab but if you asked it to graph $y= \frac{1}{5}\sqrt{1- 3x^2}$ and $y= -\frac{1}{5}\sqrt{1- 3x^2}$ then it was "complaining" about x values larger than $\sqrt{1/3}$ and x values less than $-\sqrt{1/3}$. That is, outside the ellipse.

Yes, this is an ellipse- its "standard form" would be $\frac{x^2}{\left(\frac{1}{\sqrt{3}}\right)^2}+ \frac{y^2}{\left(\frac{1}{\sqrt{5}}\right)^2}= 1$


text:
Department of Mathematics

## BÝLKENT TOPOLOGY SEMINAR

"Automorphisms and Extensions of Fusion Systems"

By

BOB OLIVER
(Université Paris 13)

Abstract: Fix a prime $p$. A fusion system $\mathcal{F}$ is \emph{tamely realized} by a finite group $G$ if $\mathcal{F}\cong\mathcal{F}_S(G)$ (for $S\in\textup{Syll_p(G)$) and the natural map from $\textup{Out}(G)$ to $\textup{Out}(BG^\wedge_p)$ is split surjective. Here, $\textup{Out}(X)$(for a space $X$) means the group of homotopy classes of self homotopy equivalences, and $BG^\wedge_p$ is the $p$-completion of the classifying space of $G$. We say that a fusion system $\mathcal{F}$ is \emph{tame} if it is tamely realized by some finite group.

Tameness plays an important role when studying extensions of fusion systems, and through that when using fusion systems to classify certain classes of finite groups. It is also interesting in its own right as a means of describing self equivalences of the space $BG^\wedge_p$ in terms of $\textup{Out}(G)$. The goal of this talk is to explain these connections, and also to describe recent progress in proving tameness forfusion systems of  finite simple groups.

Date:  Monday, May 2, 2016

Time: 13.40-14.30

Place: Mathematics Seminar Room, SA-141

All are most cordially invited. Tea and cookies will be served after the talk.


text:
If you have a pilot friend then during xmas Xmas, then folks at The Point Guy so if you want to get them something amazing.

The two coolest things are a desk made out of a wing if you happen to have a cool $2.5k handy. Or wander over to Etsy and find a restore airline galley cart. Expensive but cool for$800. Perfect to put your cool stuff in your garage.

For more ordinary mortals there is another great list:

• Crayola Mess Free Markers. These won’t draw on anything but the special paper so very useful on a flight.
• Sleeping masks. The cheap ones are yucky. The key is a silk and you can get it for \$10.


text:
Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Very sharp diffraction peak in nonglass-forming liquid with the formation of distorted tetraclusters

## Abstract

Understanding the liquid structure provides information that is crucial to uncovering the nature of the glass-liquid transition. We apply an aerodynamic levitation technique and high-energy X-rays to liquid (l)-Er2O3 to discover its structure. The sample densities are measured by electrostatic levitation at the International Space Station. Liquid Er2O3 displays a very sharp diffraction peak (principal peak). Applying a combined reverse Monte Carlo – molecular dynamics approach, the simulations produce an Er–O coordination number of 6.1, which is comparable to that of another nonglass-forming liquid, l-ZrO2. The atomic structure of l-Er2O3 comprises distorted OEr4 tetraclusters in nearly linear arrangements, as manifested by a prominent peak observed at ~180° in the Er–O–Er bond angle distribution. This structural feature gives rise to long periodicity corresponding to the sharp principal peak in the X-ray diffraction data. A persistent homology analysis suggests that l-Er2O3 is homologically similar to the crystalline phase. Moreover, electronic structure calculations show that l-Er2O3 has a modest band gap of 0.6 eV that is significantly reduced from the crystalline phase due to the tetracluster distortions. The estimated viscosity is very low above the melting point for l-ZrO2, and the material can be described as an extremely fragile liquid.

## Introduction

Determining the liquid structure is the first step in understanding the nature of glass-liquid transitions. However, a diffraction measurement of liquid provides very limited structural information because the liquid structure lacks long-range periodicity, and a Fourier transform of the diffraction data provides only pairwise correlations. Moreover, high-quality measurements are difficult to obtain at high temperatures.

Since glasses play an important role in technology, glass formation has been studied extensively. Zachariasen1 and Sun2 proposed the basic concepts of glass formation by classifying constituents into glass formers, glass modifiers, and intermediates. Furthermore, Angell3 introduced the concept of “fragility” in glass-forming liquids (GFLs). He interpreted the strong and fragile behavior of liquids in terms of topological differences in potential energy hypersurfaces of the configuration space. Typical strong liquids are SiO2, GeO2, and B2O3. Their networks are covalently bonded, and the viscosities show an Arrhenius temperature dependence. In contrast, typical fragile liquids are chalcogenides and iron phosphates, the networks of which are mostly ionic and the viscosities of which deviate significantly from the Arrhenius behavior. Many experimental and theoretical structural studies of liquids and glasses have been performed, and with the advent of advanced synchrotron and neutron sources and the development of high-performance computers, they have led to great progress in our understanding of liquid and glass structures4,5. The structural analysis of liquids with high melting points has advanced significantly with the advent of the levitation technique6, especially in combination with diffraction techniques6. The structure of a typical non-GFL, liquid (l-) Al2O3, and its undercooled liquid have been studied extensively by X-ray diffraction7,8,9,10, neutron diffraction9,10,11, and molecular dynamics (MD) simulations9,10,11,12,13. In addition to the l-Al2O3 structure, several structures of molten pure oxides with high melting points (Tm) have been studied recently. For example, structures of UO214 and compounds in the UO2–ZrO2 system15 have been investigated for nuclear reactor accidents. The structures of ZrO216,17,18, HfO216, and lanthanide oxides17 have also been investigated to understand the fundamental properties of high-temperature liquids. Although these investigations are very important not only for materials science but also for preventing severe accidents, the research methods and data are still limited by the high melting points of the materials in question.

Er2O3 is a representative nonglass former that is commonly used as a refractory material and dopant for luminescent materials. Because Er2O3 has an extremely high melting point (Tm = 2686 K), the difficulties in handling the liquid lead to problems in selecting suitable container materials that do not contaminate the sample. To avoid contact with other materials, levitation furnaces have been developed that enable us to measure precise synchrotron X-ray diffraction and thermophysical properties for liquids at extremely high temperatures6.

This article presents the results of accurate high-energy X-ray diffraction and density measurements on containerless levitated l-Er2O3 using an electrostatic levitation furnace (ELF) at the International Space Station (ISS)19, as it is impossible to measure density data on the ground. We also perform reverse Monte Carlo – molecular dynamics simulations and obtain persistence diagrams from topological analyses to demonstrate liquid properties at the atomic level, comparing l-Er2O3 with other non-GFLs and a typical GFL, l-SiO2. Furthermore, a sample of l-Er2O3 is simulated for a short period with the density functional – molecular dynamics method to investigate the electronic structure and to obtain a realistic estimate of the viscosity above the melting point. The combination of an experiment and a simulation allows trends in single-component nonglass-forming liquid oxides to be identified, with a focus on atomic ordering and topology. Furthermore, the article compares the features of single-component nonglass-forming oxide liquids with those of other systems.

## Materials and methods

### Density measurement

The density of liquid (l-) Er2O3 was measured with an ELF at the ISS. A sample of 2 mm in diameter was prepared by melting Er2O3 powder with a purity of 99.99% and solidifying it in an aerodynamic levitator. It was charged by friction or contact with other materials in the ISS-ELF and then levitated to the center between six electrodes that applied a Coulomb force. The sample position was stabilized by tuning the voltages between electrodes at 1000 Hz and monitoring the image of the sample backlit by a He–Ne laser. The levitated sample was heated and melted by four 40 W semiconductor lasers (980 nm) under 2 atm of dry air. The temperature of the sample was measured by a pyrometer (1.45–1.8 μm). It was calibrated using an emissivity calculated from the plateau temperature at recalescence and the reference value of the melting point (2686 K). After melting, the nonspherical sample became spherical upon cooling after shutting off the lasers. During cooling, the sample image was observed by an ultraviolet back light and a CCD camera. The pixel size was calibrated against an image of 2.0 mm stainless steel spheres, which were recorded under the same conditions as the sample. The sample volume was calculated from its diameter, obtained from the image. Then, the density was calculated from the volume and weight.

### High-energy synchrotron X-ray diffraction measurement

The high-energy X-ray diffraction measurement of l-Er2O3 was performed at the BL04B2 beamline20 of SPring-8 using an aerodynamic levitator21. The energy of the incident X-rays was 113 keV. The 2-mm Er2O3 sample was levitated in dry air and heated by a 200 W CO2 laser. The temperature of the sample was monitored by a two-color pyrometer. The background of the instrument was successfully reduced by shielding the detector and by optimizing a beam stop. The measured X-ray diffraction data were corrected for polarization, absorption, and background, and the contribution of Compton scattering was subtracted using standard analysis procedures22. The corrected data sets were normalized to give the Faber–Ziman23 total structure factor S(Q), and the total correlation function T(r) was obtained by a Fourier transform of S(Q).

### Molecular dynamics – reverse Monte Carlo simulation

To determine the atomic configuration of l-Er2O3, a molecular dynamics – reverse Monte Carlo (MD-RMC) simulation was performed with 5000 particles in a cube to reproduce the X-ray S(Q). The MD simulation was carried out with a Born-Mayer type of pairwise potential with a Coulomb interaction and a repulsive component, given by the following equation:

$$U_{ij}\left( r \right) = \frac{{e^2}}{{4\pi \varepsilon _0}}\frac{{Z_iZ_j}}{r} + B_{ij}\exp \left( { - \frac{r}{{\rho _{ij}}}} \right),$$
(1)

where r is the interatomic distance, Z is the effective charge, B is the repulsion, e is the elementary charge (ZEr = 2.1, ZO = −1.4), ε0 is the permittivity of the vacuum, and ρ is the softness parameter. The parameters used in the MD simulation are summarized in Table 1.

The simulations were carried out for a system of 2000 Er and 3000 O atoms in the unit cell with a random atomic configuration. The cell volume was determined from the number densities of l-Er2O3 at the melting point, which were calculated with the density measured by the ISS-ELF. Periodic boundary conditions were used, and the long-range Coulomb interaction was treated with Ewald’s summation. A time step of 1 fs was used in the Verlet algorithm. First, the temperature of the system was maintained at 4000 K for 20,000 time steps and then cooled to 2923 K over 20,000 steps. The structural model was finally annealed at 2923 K for 150,000 steps. After the MD simulation, RMC refinement was conducted using the RMC++ code24. The benchmark RMC runs were performed using simulation boxes with 250, 500, 1000, and 3000 particles.

### Density functional – molecular dynamics simulation

The simulations based on the density functional theory (DFT) of the electronic structure were performed with the projector augmented wave (PAW) method25, implemented in the VASP software26,27. The PAW potentials supplied within VASP for Er (5p, 5d, and 6s, with 11 4f electrons frozen in the core) and O (2s, 2p) were tested and used (see supplementary information for a comparison between the frozen core approximation and the treating of the Er-4f electrons explicitly, Fig. S1). For a liquid sample of 500 atoms, the energy cutoff of the plane waves was set to 400 eV, with a single Γ-point in the Brillouin zone. In comparison, the bulk crystalline (c)-Er2O3 unit cell was fully relaxed until the forces on all atoms were below 0.01 eV/Å with a Γ-centered 2 × 2 × 2 k-point grid and a plane-wave energy cutoff of 550 eV. The PBE functional28 was used for the geometry optimization and the molecular dynamics simulations, whereas the HSE06 hybrid functional29 was used to obtain the electronic densities of states (DOSs) and their projections to produce more realistic electronic band gaps and test the effect of the 4f electrons in c-Er2O3. The effective charges and atomic volumes were evaluated by Bader analysis30,31,32 using the PBE functional.

The density functional – molecular dynamics (DF-MD) simulations were performed with a Nóse-Hoover thermostat33 and a time step of 2 fs, with an initial atomic configuration given by the benchmark RMC model mentioned above with 500 atoms. The system was simulated at 2923 K (~2650 °C) for a total of 30 ps, where the last 25 ps were used for data collection (Fig. S2). The electron occupancy was described with a Fermi smearing corresponding to the kBT value at the target temperature. The mean-square displacements (MSDs) of the atoms show a liquid (diffusion) behavior where equilibrium is already achieved during the first few picoseconds.

### Topological analysis using a persistent homology

The homology of atomic configurations has been investigated using the persistence diagram D1, which consists of two-dimensional histograms showing a persistent homology. The details of the analysis are described elsewhere34. The persistence diagram D1 of a set of atoms is given by the following thickening process of spheres: (1) place a sphere with a radius r at the center of each atom, (2) increase the radii of the spheres from 0 to a sufficiently large value, and (3) encode the pair of birth and death radii (bi, di) for each ring ci consisting of a set of spheres. The persistence diagram is then constructed by the two-dimensional histogram on the birth and death plane obtained by the pairs for independent ci, i = 1,…, K. Here, the birth (death) radius is defined as the radius of spheres at which the ring ci first appears (disappears). The birth radius has information about the distances between atoms of the ring ci, and the death radius has information about the size of the ring. The persistence diagram provides statistical information on the shapes of all independent rings and thereby provides insight into intermediate ordering in the liquid structure. The rings and cavities detected by this process are recorded for the computation of the persistence diagrams; hence, their geometric shapes can be identified for further analysis. The persistence diagrams were calculated using the HomCloud package35.

## Results and discussion

### Density data

Figure 1 shows the density of l-Er2O3 as a function of temperature, which exhibits a linear temperature dependence. The least-squares fit to the data is given by the following equation:

$$\rho \left( T \right) = {\rho_{\mathrm{m}}}\left[ {1 - \alpha \left( {T - {T_{\mathrm{m}}}} \right)} \right]\left( {{\mathrm{kg}}/{\mathrm{m}}^3} \right),$$
(2)

where ρm is the molten density at Tm (8170 kg/m3) and α (=1/ρm[dρ(T)/dT]) is the thermal expansion coefficient and is assumed to be constant (1.0 × 10−4 K−1) at any temperature of the liquid. The correlation coefficient of this fitting is 0.98. The uncertainty in the measurements is estimated to be 2% from the image resolution (640 × 480 pixels) and from the uncertainty in the mass measurement (±0.1 mg).

The density and the expansion coefficient for l-Er2O3, together with those for l-SiO236 and other non-GFLs18,37, are compared in Table 2. Although the density trends increase with increasing cation atomic number, they do not show a clear relation. On the other hand, the thermal expansion coefficients show a similarity as each value approaches 1 × 10−4 K−1. The thermal expansion coefficient of l-Er2O3 is especially close to those of l-SiO2 and l-Al2O3.

### Structure factors and real-space functions

The Faber-Ziman X-ray total structure factors, S(Q), for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218, together with the results of the MD-RMC simulation for l-Er2O3, are compared in Fig. 2a. It is noted that the scattering vector Q is scaled by multiplying by rA-X (distance between the center and corners of the polyhedron). The experimental S(Q) of l-Er2O3 (solid cyan curve) is well reproduced by the MD-RMC simulation (dotted black curve) using the liquid density measured by the ISS-ELF shown in Fig. 1. A well-defined first sharp diffraction peak (FSDP)39 is observed only for l-SiO2 (GFL) at QrA-X = 2.6, while a principal peak (PP)39 is observed in both the l-ZrO2 and l-Er2O3 data at QrA-X ~ 4.5. On the other hand, l-Al2O3 gives rise to a small peak between the FSDP and PP, suggesting that the structure of l-Al2O3 is intermediate17 between l-SiO2 and l-ZrO2/l-Er2O3. It is well known that the PP reflects the packing of oxygen atoms in neutron diffraction data40, since neutrons are sensitive to oxygen. For the same reason, a PP is not observed in the X-ray S(Q) for l-SiO2 (see Fig. 2a), and the origin of the PP in l-ZrO2 and l-Er2O3 is ascribed to the packing of cations. The X-ray total correlation functions T(r) for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218 are shown in Fig. 2b. The first peak observed at approximately 2.2 Å is assigned to the Er–O correlation, and a tail to ~3 Å implies the formation of distorted ErOn polyhedra in the liquid. The second peak, observed at 3.7 Å, can be assigned mainly to the Er–Er correlation, and the O–O correlation peak is unclear due to its small weighting factor for X-rays. The Er–O correlation length of 2.2 Å, as well as that of Zr–O (2.1 Å), is longer than those of Si–O (~1.63 Å at 2373 K) and Al-O (~1.78 Å at 2400 K) owing to substantial differences between the ionic radii of the elements. The increased cation-oxygen correlation length in the liquid phases of Er–O and Zr–O suggests that the oxygen coordination number around cations is higher than 4 because the Er–O correlation length (2.2 Å) or Zr–O correlation length (2.1 Å) is close to the sum of the ionic radii41 of oxygen (1.35 Å) and six-fold erbium (0.89 Å) or zirconium (0.72 Å), respectively. The structures of l-Er2O3 and l-ZrO2 therefore consist of large interconnected polyhedral units and are very different from those of l-SiO2 and l-Al2O3. This behavior is consistent with the fact that the peaks observed at QrA-X ~ 4.5 in Fig. 2a are not the FSDP, which is typically associated with intermediate-range ordering in oxide glasses and liquids; thus, there is no such ordering in l-Er2O3 and l-ZrO2 due to a very densely packed structure.

### Coordination number distributions from the simulation

The coordination number distributions, NA-X and NX-A, for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218 obtained from the simulation are compared in Fig. 3a, b, and their average values are summarized in Table 3. The Er–O coordination number (up to 3.0 Å) is found to be 6.1 from our combined MD-RMC simulation, which is rather close to the crystalline phase43, and the O–Er coordination number can be estimated to be 4.1. It is suggested that the cations are tetrahedrally coordinated in l-SiO2 (GFL), while they are octahedrally coordinated in l-ZrO2 and l-Er2O3 (non-GFLs), and the cation-oxygen coordination number in l-Al2O3 is intermediate17 between GFL and l-ZrO2/l-Er2O3, although l-Al2O3 is a non-GFL. This behavior is consistent with that of the first correlation peaks in experimental real-space functions (see Fig. 2b) and with the fact that the viscosity of l-ZrO2 is approximately one-tenth of that in l-Al2O318. Another interesting behavior is observed for the oxygen-cation coordination numbers. It is demonstrated that oxygen is twofold in l-SiO2, which is a signature of the formation of a sparse network, while triclusters (XA3) are dominant in l-Al2O3 and l-ZrO2. The formation of tetraclusters (XA4) is confirmed in l-Er2O3, suggesting that this behavior is a distinct feature of this liquid. We suggest that the behavior of the coordination numbers in a series of oxide liquids is affected by both the composition and the ionic radii between the constituent anions and cations. For instance, the ionic radii of Si and Al are small, which results in tetrahedral coordination, although the Al-O coordination number is greater than four on average. The tetracluster formation is caused by the ratio of Er and O in Er2O3.

### Very sharp principal peak (PP) in l-Er2O3

As shown in Fig. 2a, the PP of l-Er2O3 is very sharp compared to that of l-ZrO2. The FWHM of the PP in l-Er2O3 is 0.4299, in comparison to 0.7669 in l-ZrO2 (see Fig. 4). A simulation box with 501 particles was used in the previous RMC – density functional (DF) simulation for l-ZrO218, where a good agreement was observed between the experimental data and simulation (see Fig. 4a). However, as can be seen in the inset data of Fig. 4b, a simulation box of 500 particles is insufficient to reproduce the sharp PP in l-Er2O3; larger atomic models are needed to reproduce this feature. Insight into the structure of l-Er2O3, in comparison with those of l-SiO2 and other non-GFLs, can be obtained by calculating the Faber-Ziman partial structure factors, Sij(Q), and the Bhatia-Thornton44 number – number partial structure factor, SNN(Q), which indicates the topological order in a system:

$${S_{\mathrm{NN}}}\left( Q \right) = {c_{\mathrm{A}}^2}{S_{\mathrm{AA}}}\left( Q \right) + {c_{\mathrm{X}}^2}{S_{\mathrm{XX}}}\left( Q \right) + 2c_{\mathrm{A}}c_{\mathrm{X}}S_{\mathrm{AX}}\left( Q \right),$$
(3)

where Sij(Q) is a Faber-Ziman partial structure factor and ci denotes the atomic fraction of chemical species i. Moreover, it is possible to compare data for the four liquids while ignoring the difference in the sensitivity of elements to X-rays because the weighting factors for X-rays are eliminated in SNN(Q). The Sij(Q) values calculated from the simulation models for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218 are shown in Fig. 5a. It is confirmed that a very sharp PP in l-Er2O3 can be assigned to the Er-Er correlation. The SNN(Q) for l-Er2O3 and those for l-SiO2 and other non-GFLs are compared in Fig. 5b. As mentioned above, only l-SiO2 exhibits an FSDP at QrA-X = 2.6. The QFSDP position arises from an underlying periodicity of 2π/QFSDP that originates, for example, from the formation of pseudo-Bragg planes with a finite correlation length of 2π/ΔQFSDP in l-SiO2, while neither l-Al2O3, l-ZrO2, nor l-Er2O3 show an FSDP in SNN(Q), as discussed in Kohara et al.18. Since the Bhatia-Thornton SNN(Q) can eliminate the weighting factors for X-rays, the absence of an FSDP in SNN(Q) is characteristic of a non-GFL. Another important feature in SNN(Q) is that l-SiO2 and l-Al2O3 exhibit a second PP at QrA-X ~ 5, while a PP is not distinct in the l-ZrO2 or l-Er2O3 data.

The absence of an FSDP in the l-ZrO2 and l-Er2O3 data suggests that both cations and oxygen are densely packed. To confirm this in real space for l-Er2O3, the partial pair distribution functions, gij(r), of l-Er2O3 are compared with those of l-SiO2 in Fig. 6a. The atomic distance r is scaled by dividing by rA-X (distance between the center and corners of the polyhedron). It is found that the scaled first A-A and X-X correlation distances of l-Er2O3 are much shorter than those of l-SiO2, demonstrating that l-Er2O3 has a much more densely packed structure, manifested by the formation of the OEr4 tetracluster network shown in Fig. 6b. This network cannot be found in l-Al2O3 nor in l-ZrO2, suggesting that the very sharp PP in l-Er2O3 is a specific signature of the formation of a tetracluster network with long-range periodicity.

### Topology and homology in l-Er2O3

To reveal the origin of the very sharp PP in l-Er2O3, we calculated the bond angle distributions of the liquid and crystal43 and summarized them in Fig. 7. A pronounced difference was found between the liquid and crystal data for the O–Er–O and Er–O–Er distributions. The O–Er–O bond angle distribution exhibits two peaks at 80° and 140°, suggesting that ErO6 polyhedra are highly distorted in the liquid. Another interesting feature is that the Er–O–Er bond angle distribution exhibits a peak at ~180° in addition to the peak at ~90°, which is not observed for the crystal43 nor in l-ZrO218. This two-peak structure in the Er-O-Er bond angle distribution indicates the formation of a distorted OEr4 tetracluster network, whereas tetraclusters are symmetric (comprising regular tetrahedra) in the crystalline phase. This behavior suggests that the coordination of OEr4 tetraclusters is more octahedral-like and hence tolerant of disorder even in the liquid due to the distortion, providing a linear arrangement manifested by a prominent peak observed at 180° in the Er–O–Er bond angle distribution. This is clearly visible in Fig. 6c, where linear atomic arrangements are highlighted by the magenta lines.

To shed light on the similarity in topology between the crystal and liquid phases, we calculated the persistence diagram for l-Er2O3 and compared it with the crystal data in Fig. 8. The figures show the similarity between the crystal43 and liquid phases. In particular, both the Er-centric and O-centric persistence diagrams for l-Er2O3 do not show a vertical profile along the death axis, which is a pronounced feature in a typical GFL, such as l-SiO242. The short lifetime of the profile manifested by the small death value demonstrates that both the crystal and liquid phases exhibit a very densely packed structure associated with the formation of tetraclusters in both phases. We suggest that this similarity is a signature of non-GFL behavior.

### Electronic structure and viscosity of l-Er2O3

As previously mentioned, a 500-atom RMC model of l-Er2O3 appeared to be too small in reproducing the very sharp PP accurately (see Fig. S3). Nevertheless, we used this model as a starting structure in our DF-MD simulations to study the electronic structure and atomic diffusion in the liquid phase. The electronic density of states (DOS) and effective charges were calculated for snapshots of l-Er2O3 atomic structures and a fully DFT-relaxed (0 K) c-Er2O3 unit cell. The DOSs of l- and c-Er2O3 are shown in Fig. 9a, b. The valence band consists mainly of O-2p states, while the conduction band consists mainly of Er-5d states. As a measure of the orbital localization, we also present the inverse participation ratios (IPRs) for l-Er2O3. The IPRs show increased weight at the valence band maximum (VBM), indicating stronger localization, whereas the states around the conduction band minimum (CBM) are delocalized. The obvious broadening of the valence and conduction bands in l-Er2O3 is caused by the distortion of the ErOn polyhedra at elevated temperatures. The DOS for l-Er2O3 reveals a band gap of 0.57 eV in comparison to the substantially large band gap of 5.46 eV in c-Er2O3. Previously, a vanishing band gap was reported for l-ZrO218, but we ascribe this effect to the PBE functional, which is known to underestimate the real value, whereas the hybrid HSE06 functional used here predicts a more correct band gap.

The difference in electronegativities of Er (1.24) and O (3.44) suggests predominantly ionic chemical bonding between the two atoms. This becomes clear from the partial DOS in Fig. 9a, and the calculated atomic charges are summarized in Table 4. The effective charges for l-Er2O3 are +1.96 e and –1.31 e for Er and O, respectively, similar to those in c-Er2O3 (see Table 4). These values are in agreement with previous works on l-ZrO218, glassy MgO-SiO245, and CaO-Al2O346 and are consistent with the nominal charges Er3+ and O2- (systematically scaled down by a factor of $$\sim \frac{2}{3}$$). As for ZrO218, the increased atomic volume of oxygen in the transition from c- to l-Er2O3 compensates for the corresponding decreased oxygen coordination, which results in similar atomic charges for the two phases. Note here also the similarity with the charges used in the classic MD force field (+2.1 and −1.4 e).

A real space visualization of the highest occupied molecular band (HOMO) is shown in Fig. 9c, where the orbital is found to be distributed over a group of atoms. Locally, the shape of the HOMO is similar to that of c-Er2O3 (Fig. S4); however, some deviations occur caused by the aforementioned distorted tetracluster network in l-Er2O3. Figure 9d shows a real space visualization of the lowest unoccupied molecular band (LUMO), where most of the orbital is distributed over nonbonding regions in-between the ErOn polyhedra. The orbital projections for each of the bands reveal that the HOMO band shows mainly an O-2p character, while the LUMO band consists mainly of Er-5d states and some Er-5p character, in agreement with the DOS and IPR in Fig. 9a.

The DF-MD simulations above the melting point at 2650 °C show that the atoms move rapidly, breaking old and forming new Er-O bonds on a picosecond time scale. The MSD analysis (Fig. S2) shows a consistent linear behavior as a function of time, the evaluated self-diffusion constants are 2.40 × 10−5 and 5.84 × 10−5 cm2/s for Er and O, respectively, and the average self-diffusion constant is 4.64 × 10−5 cm2/s. By using the average value and assuming spherical particles in the Stoke-Einstein relation, one can estimate the viscosity, and the obtained value is ~3 × 10−3 Pa s−1 for l-Er2O3 in comparison to the previously reported value for l-ZrO2, i.e., ~2 × 10−3 Pa s−1 at 2800 °C18. Since these values are an order of magnitude smaller than that for l-Al2O3 (fragile liquid) and 9–10 orders of magnitude smaller than that for l-SiO2 (strong liquid), we characterize l-Er2O3 as an “extremely fragile” liquid18.

We combined an aerodynamic levitation technique and a synchrotron high-energy X-ray diffraction and density measurement at the ISS for l-Er2O3 to reveal the structure of A2X3-type non-GFLs. As the main finding, we observed a very sharp PP in the diffraction data. The Er-O coordination number was estimated to be 6.1 from a combined MD-RMC simulation, which is comparable to that of another nonglass-forming liquid, l-ZrO2, and to that of crystalline Er2O3. The formation of distorted OEr4 tetraclusters in the liquid is confirmed, while OZr3 triclusters are dominant in l-ZrO2 and OAl3 triclusters dominate in another A2X3 non-GFL, l-Al2O3. Apparently, the formation of a distorted tetracluster network in l-Er2O3 gives rise to a long periodicity, yielding the sharp principal peak. This long-range periodicity originates from the significantly increased weight at ~180° in the Er-O-Er bond angle distribution, suggesting that the arrangement of distorted OEr4 tetraclusters involves nearly linear connections, which are not observed in other oxide liquids. Furthermore, persistent homology suggests that l-Er2O3 is homologically similar to the crystalline phase and that both phases are very densely packed, in contrast to a typical GFL such as l-SiO2. This similarity is presumed to be the signature of the liquid, and a considerable difference between two A2X3-type oxide liquids, l-Al2O3 and l-Er2O3, is uncovered. The additional DF-MD simulations demonstrate that l-Er2O3 has an electronic band gap of 0.6 eV, which is considerably lower than that of c-Er2O3 due to angular distortions and nearly linear connections within the polyhedral network. The dynamics of atoms show pronounced mobility, as evidenced by the MSDs, resulting in a very low viscosity, and thus place l-Er2O3 within the regime of extremely fragile liquids.

## Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.

## References

1. 1.

Zachariasen, W. H. The atomic arrangement in glass. J. Am. Chem. Soc. 54, 3841–3851 (1932).

2. 2.

Sun, K.-H. Fundamental condition of glass formation. J. Am. Ceram. Soc. 30, 277–281 (1947).

3. 3.

Angell, C. A. Formation of glasses from liquids and biopolymers. Science 267, 1924–1935 (1995).

4. 4.

Greaves, G. N. & Sen, S. Inorganic glasses, glass-forming liquids and amorphizing solids. Adv. Phys. 56, 1–116 (2007).

5. 5.

Salmon, P. S. & Zeidler, A. Identifying and characterising the different structural length scales in liquids and glasses: an experimental approach. Phys. Chem. Chem. Phys. 15, 15286–15308 (2013).

6. 6.

Price, D. L. High-Temperature Levitated Materials (Cambridge University Press, 2010).

7. 7.

Ansell, S. et al. Structure of liquid aluminum oxide. Phys. Rev. Lett. 78, 464–466 (1997).

8. 8.

Landron, C. et al. Liquid alumina: detailed atomic coordination determined from neutron diffraction data using empirical potential structure refinement. Phys. Rev. Lett. 86, 4839–4842 (2001).

9. 9.

Krishnan, S. et al. Structure of normal and supercooled liquid aluminum oxide. Chem. Mater. 17, 2662–2666 (2005).

10. 10.

Shi, C. et al. The structure of amorphous and deeply supercooled liquid alumina. Front. Mater. 6, 38 (2019).

11. 11.

Skinner, L. B. et al. Joint diffraction and modeling approach to the structure of liquid alumina. Phys. Rev. B 87, 024201 (2013).

12. 12.

Jahn, S. & Madden, P. A. Structure and dynamics in liquid alumina: simulations with an ab initio interaction potential. J. Non-Cryst. Solids 353, 3500–3504 (2007).

13. 13.

Vashishta, P., Kalia, R. K., Nakano, A. & Rino, J. P. Interaction potentials for alumina and molecular dynamics simulations of amorphous and liquid alumina. J. Appl. Phys. 103, 083504 (2008).

14. 14.

Skinner, L. B. et al. Molten uranium dioxide structure and dynamics. Science 346, 984–987 (2014).

15. 15.

Alderman, O. L. G. et al. Corium lavas: structure and properties of molten UO2-ZrO2 under meltdown conditions. Sci. Rep. 8, 2434 (2018).

16. 16.

Hong, Q. J. et al. Combined computational and experimental investigation of high temperature thermodynamics and structure of cubic ZrO2 and HfO2. Sci. Rep. 8, 14962 (2018).

17. 17.

Skinner, L. B. et al. Low cation coordination in oxide melts. Phys. Rev. Lett. 112, 157801 (2014).

18. 18.

Kohara, S. et al. Atomic and electronic structures of an extremely fragile liquid. Nat. Commun. 5, 5892 (2014).

19. 19.

Tamaru, H. et al. Status of the electrostatic levitation furnace (ELF) in the ISS-KIBO. Microgravity Sci. Technol. 30, 643–651 (2018).

20. 20.

Kohara, S. et al. Synchrotron X-ray scattering measurements of disordered materials. Z. Phys. Chem. 230, 339–368 (2016).

21. 21.

Kohara, S., Ohara, K., Ishikawa, T., Tamaru, H. & Weber, R. Investigation of structure and dynamics in disordered materials using containerless techniques with in-situ quantum beam and thermophysical property measurements. Quantum Beam Sci. 2, 5 (2018).

22. 22.

Kohara, S. et al. Structural studies of disordered materials using high-energy x-ray diffraction from ambient to extreme conditions. J. Phys. Condens. Matter. 19, 506101 (2007).

23. 23.

Faber, T. E. & Ziman, J. M. A theory of the electrical properties of liquid metals. Philos. Mag. 11, 153–173 (1965).

24. 24.

Greben, O., Jóvári, P., Temleitner, L. & Pusztai, L. A new version of the RMC++ Reverse monte carlo programme, aimed at investigating the structure of covalent glasses. J. Optoelectron. Adv. Mater. 9, 3021–3027 (2007).

25. 25.

Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

26. 26.

Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).

27. 27.

Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

28. 28.

Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

29. 29.

Krukau, A. V., Vydrov, O. A., Izmaylov, A. F. & Scuseria, G. E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys. 125, 224106 (2006).

30. 30.

Henkelman, G., Arnaldsson, A. & Jónsson, H. A fast and robust algorithm for Bader decomposition of charge density. Comput. Mater. Sci. 36, 354–360 (2006).

31. 31.

Sanville, E., Kenny, S. D., Smith, R. & Henkelman, G. Improved grid-based algorithm for Bader charge allocation. J. Comput. Chem. 28, 899–908 (2007).

32. 32.

Tang, W., Sanville, E. & Henkelman, G. A grid-based Bader analysis algorithm without lattice bias. J. Phys. Condens. Matter 21, 084204 (2009).

33. 33.

Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511–519 (1984).

34. 34.

Hiraoka, Y. et al. Hierarchical structures of amorphous solids characterized by persistent homology. Proc. Natl Acad. Sci. USA 113, 7035–7040 (2016).

35. 35.
36. 36.

Aksay, I. A., Pask, J. A. & Davis, R. F. Densities of SiO2-Al2O3 melts. J. Am. Ceram. Soc. 62, 332–336 (1979).

37. 37.

Paradis, P. F., Ishikawa, T., Saita, Y. & Yoda, S. Non-contact thermophysical property measurements of liquid and undercooled alumina. Jpn J. Appl. Phys. 43, 1496–1500 (2004).

38. 38.

Mei, Q., Benmore, C. J. & Weber, J. K. R. Structure of liquid SiO2: a measurement by high-energy x-ray diffraction. Phys. Rev. Lett. 98, 057802 (2007).

39. 39.

Salmon, P. S., Martin, R. A., Mason, P. E. & Cuello, G. J. Topological versus chemical ordering in network glasses at intermediate and extended length scales. Nature 435, 75–78 (2005).

40. 40.

Salmon, P. S. Magma under Pressure: Advances in High-Pressure Experiments on Structure and Properties of Melts (Ed. Kono, Y. & Sanloup, C.) pp. 347 (Elsevier, Amsterdam, 2018).

41. 41.

Shannon, R. D. & Prewitt, C. T.Effective ionic radii in oxides and fluorides. Acta Cryst. B25, 925–946 (1969).

42. 42.

Onodera, Y. et al. Understanding diffraction patterns of glassy, liquid and amorphous materials via persistent homology analyses. J. Ceram. Soc. Jpn 127, 853–863 (2019).

43. 43.

Saiki, A., Ishizawa, N., Mizutani, N. & Kato, M. Structural change of C-rare earth sesquioxides Yb2O3 and Er2O3 as a function of temperature. Yogyo Kyokai Shi 93, 649–654 (1985).

44. 44.

Bhatia, A. B. & Thornton, D. E. Structural aspects of the electrical resistivity of binary alloys. Phys. Rev. B 4, 3004–3012 (1971).

45. 45.

Kohara, S. et al. Relationship between topological order and glass forming ability in densely packed enstatite and forsterite composition glasses. Proc. Natl Acad. Sci. USA 108, 14780–14785 (2011).

46. 46.

Akola, J. et al. Network topology for the formation of solvated electrons in binary CaO–Al2O3 composition glasses. Proc. Natl Acad. Sci. USA 110, 10129–10134 (2013).

Download references

## Acknowledgements

The synchrotron radiation experiments were performed at BL04B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2016A0134). This research was supported by JST PRESTO, Japan Grant Numbers JPMPR15N4 (to S.K.); the “Materials Research by Information Integration” initiative (MI2I) project of the Support Program for Starting Up Innovation Hub from JST (to Y.O., S.T., S.K., At.M., and Y.H.); JST CREST 15656429 (to Y.H.); JSPS KAKENHI Grant Number JP17H03121 (to A.M.); and the TIA collaborative research program “Kakehashi”, TK19-004 (to S.K.). D.R.S. and S.M.S. acknowledge the Research Council of Norway (FRINATEK Project No. 275139/F20) for their financial support and UNINETT Sigma2 (Project No. NN9264K) for providing computational resources. The density measurement experiments were supported by the ISS crew members and ground operation staff.

## Author information

Authors

### Contributions

S.K. and C.K. designed this research. The high-energy X-ray diffraction measurements were performed by S.K., C.K, Y.O., At.M., Ak.M., Y.W., K.O., H.T., and J.T.O. The density measurements were conducted by C.K., T.I., Y.W., Y.N., H.T., and H.O. The classic molecular dynamics simulations were performed by S.T. The density functional – molecular dynamics simulations were conducted by D.R.S., S.M.S., and J.A.; S.K., D.R.S., S.M.S., J.A., I.O., Y.H., and O.S. analyzed the data, and S.K., C.K., and J.A. wrote the article.

### Corresponding authors

Correspondence to Chihiro Koyama or Shinji Kohara.

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interest.

## Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

## About this article

### Cite this article

Koyama, C., Tahara, S., Kohara, S. et al. Very sharp diffraction peak in nonglass-forming liquid with the formation of distorted tetraclusters. NPG Asia Mater 12, 43 (2020). https://doi.org/10.1038/s41427-020-0220-0

Download citation

• Received:

• Revised:

• Accepted:

• Published:


text:
# Tag Info

1

So taking your questions one at a time 1) The concave mirror will form a virtual image if the object is placed closer to the mirror than the focal point of the mirror. The formula for the position of the final image is $s'=\frac{sf}{s-f}$ where s is the object-mirror distance and f is the focal length of the mirror. You can see that if $s<f$ this will be ...

1

Two things that might help you. Finding Sellmeier Co-efficients Firstly, although the refractive index formulas you cite are not in the Sellmeier form, they are in the form: $$n(\lambda) = 1+\delta(\lambda) = 1+\sum_j\frac{K_j\,\lambda^2}{\lambda^2 - L_j}$$ where $\delta(\lambda) = \sum_j\frac{K_j\,\lambda^2}{\lambda^2 - L_j}$ is VERY small. So, squaring ...

1

Everything scattering light has a characteristic spectrum. The spectrum is defined by the way wherein the particular thing interacts with light, and this is set by the (1) chemical makeup and (2) texture (at the wavelength-of-visible light scale). You cannot change the spectrum without changing one of these two things, either by e.g. (1) altering the ...

1

The answer to your question depends on what you want to study. But first, let's agree on terms: by "optically conjugate" I mean that the deformable mirror/shack-hartmann wavefront sensor is imaging some plane. In case of astronomical AO, the SH WFS is conjugated with the telescope pupil (or, more precisely, with the primary mirror of the telescope). A good ...

1

You can reach the end of a rainbow, if you make one using a hose pipe on a sunny day. It is possible to locate the ends to within a few cm

2

Why can't we reach the ends of rainbow? The WIkipedia article you linked to contains the explanation The rainbow is not located at a specific distance, but comes from an optical illusion caused by any water droplets viewed from a certain angle relative to a light source. Thus, a rainbow is not an object and cannot be physically approached. Indeed, ...

0

There is nothing wrong with working in a completely enclosed box which has an electric safety switch that turns the power off as soon as it is opened. That is exactly how a professional engineer/physicist will solve this safety problem: with deadlocks like this and measures that prevent the beam intersecting your eye under all circumstances of "normal" use. ...

8

When working with any laser above class 1, you should seek appropriate qualification for dealing with your specific laser system (which includes both its wavelength and its power), and you should inquire with your institution as to any formal safety requirements. Specifically, you should not take laser safety advice from what are essentially random ...

0

Here's an idea. Maybe it's kind of cheating, I don't know, you decide. Take a small diameter clear tube, fill with ordinary, unfiltered tap water and line the laser up to fire down the length of the tube. Seal the far end water tight. If you're firing the beam at a downward angle, you can leave the end closest to the laser open. The impurities in the water ...

1

Videos typically have a frame rate of ~25 frames per second. The 'movie' you see is a result of these snapshots. If you assume the propellers were moving at 9k RPM, that is 150 revolutions per second. If the camera recorded at 150 FPS, the propellers would appear stationary as they would be in the same position at each time a new frame was captured. The ...

1

Yes. There are more complete developments based on electromagnetic theory. Exact solutions are known for a few cases. Almost all applications rely on approximations, many of which are very good approximations under the right conditions. But the more complete explanations end up looking like Huygens Principle, and serve to justify it. Huygens' original ...

0

For a parabolic mirror, the answer is "all rays go through the focus". For a spherical mirror, you can see the answer most easily by looking at the limit as the spherical mirror is almost a hemisphere: Clearly the rays that start further off axis are focused closer to the mirror.

3

As Olin said, for a parabolic mirror lines that start out parallel to the principal axis all converge to a point. This can easily be confirmed with a bit of simple math. For a mirror, angle of incidence = angle of reflection. I show that this results in a parabolic shape in my answer to an earlier question (see the last part of the answer). Now a spherical ...

0

It doesn't. You need a parabolic surface, not a spherical one, to have rays parallel to the axis converge at a focal point. I have no idea where you got your failed math from, but it's just plain wrong.

2

The positions of the diffraction peaks do not depend on the size of the atoms. The peak positions are determined by the spacings of the crystal lattice and it does not matter what the atoms are or how big they are. That's why the atom size does not appear in the Bragg formula. However the intensities of the lines, both absolute and relative, depend very ...

3

You have to have a second polarizing device that you can use to analyze the polarization. A common choice is to use a reflection from water or a glossy floor that happens to lie near Brewster's angle, which makes the light strongly polarized in the plane of the reflecting surface. Look through your polarizer at the reflections, and rotate your polarizer ...

0

Polarization effects. For EM radiation you have two types of polarization (TEM, TE and TM) whereas in sound you don't have any polarization (it's a scalar field).

1

I have been recently performing experiments using UVB (and UVA light) light. The personal protection I have been using have been a welder's mask (to protect the face) and sun glass like material for the eyes and sun protective clothing that did not leave any skin exposed (e.g. hands, arms etc). As mentioned in the comments, surround the experiment in a ...

2

You can create wavelets anywhere - the propagation of a wave is always represented by the Huygens construction. You need to keep in mind the phase: it is usually convenient to draw a new wavelet starting at a boundary, and with a known phase, because then it's easy to draw a series of concentric circles with appropriate spacing (wavelength). If you draw ...

1

This is typically because of optical selection rules which forbid certain types of transitions. The most usual case is where the states are, in order of energy, $S$, $D$ and $P$ states, driven by a reasonably-intense laser. In this case, the coupling to the EM field is usually a dipole coupling, which means that the atomic operator that does the transitions ...

1

In addition to @Floris response: You have missed a lot of wavelengths in your list of wavelengths that would experience interference. Take your example of a $6,000,000 \text{ nanometer}$ pane of glass, and consider that 15,000 waves of $400 \text{ nanometer}$ wavelength light exactly fills this space. So, indeed, this light will experience some sort of ...

4

The wikipedia article is correct. The relation between the actual response function and the conductivity is not immediately obvious, however. Let us consider the case of the longitudinal conductivity for example. The susceptibility, $\chi^L(\textbf{q},\omega)$, which is the true response function, is related to the conductivity using the following equations: ...

3

Very simply, when a plate is quite thick, the fringe patterns will be very close together - because a tiny change in angle will result in an additional wavelength's worth of path difference. Different colors will have a different repeat distance (because of different wavelengths); and light will typically arrive at the eye from more than one direction ...

0

A Pockel cell followed by a quarter waveplate will do this. The Pockel cell acts as an electrically controlled waveplate which will give an elliptical state. The waveplate will then convert this to a rotated linear state. Producing a 10MHz driving signal at the several kilovolts required might be a challenge but not impossible.

2

There are "circular-polarization-maintaining fibers" (you may wish to google them). One can get such fibers by introducing circular birefringence, say, by twisting the fiber.

0

Faraday Effect. The plane of polarization in a medium is rotated when exposed to a magnetic field. Solenoid, glass rod.

0

How much is the force of gravity on the $3m$ mass, and how much mass must it accelerate? Similar question for the $2m$ mass? Then carry on as you have...

0

Angular magnification is used to describe the magnification of an afocal system, typically a telescope like a Galilean telescope (which a modern microscope's eyepiece almost always is) or Keplerian Telescope. Indeed your question is a good one insofar that it underlines the meaninglessness of "magnification" (or at least calls into question how worthwhile ...

2

Here are some topics to read about: Frequency doubling, also called second-harmonic generation as Johannes mentions. Here, you put one wave into a medium, and some fraction of it is converted to a wave with a different frequency. By carefully engineering the medium you can get quite a high conversion percentage. Other nonlinear optical processes, not just ...

1

Although normally considered as photon interactions, any inelastic scattering process will result in the alteration of the frequency of the electromagnetic radiation. An obvious example is Compton scattering, where high energy (X-rays+) light scatters from free electrons. The scattered light has lower energy and longer wavelengths than the light incident ...

1

You can place an object anywhere you like with respect to the lens. In one diagram, the author simply chose the separation to be twice the focal length. In this case, the rules for ray tracing tell you the image will also happen to be the same distance from the lens. This doesn't work with any other separation. Also note that the one place you would have ...

1

Although the method illustrated in the pic below can in principle assure an unprecedented level of collinearity of the beams in a rather uncomplicated manner, in practice it is subject to the availability as well as a reasonable level of performance in usage of fiber-optical components in the $10\, \mu$m regime (I had written a long and complete answer, and ...

3

Great setup! I don't envy your task: IR is very tricky and frustrating to align. Do not expect to solve this problem overnight: this is a many month project, unless you can find someone who has done exactly this before, in which case it would have been a many month project for them. Part of the translation is easy: for fine ajustment, you use an angled ...

2

I wanted to post this as a comment, but it grew too long. The final question that was a bit hidden, but that several other users seemed to be also interested in, would be about why the blue LED was maybe so much harder to construct than the red one. Reading through Steve B's link to the nobel scientific background provides me with enough information that I ...

2

Additionally, blue was the last of the primary colours so its invention made the production of white LEDs possible. Ordinary lamps could then be replaced with extremely energy efficient LED alternatives.

0

I'm not sure the other answers are complete. Quarks could be said to have a photopic color under certain conditions. If there is zero interaction (other than elastic scattering) with photons, then quarks could be considrered to be "white," as they have no preferential absorption. If quarks are capable of absorbing photons, they are either "gray" or have a ...

1

For usual way of looking, trying to see an object with size smaller than wavelength you shine at it will give you nothing: you'll basically not notice the object at all: it can't be resolved. But, there's a trick, where instead of using normal "shine and look from a distance" way you watch the object very closely, taking advantage of the properties of ...

1

Roughly speaking, the color of an object is the wavelength which hits your eyes when [typically non-coherent] light interacts with the object. This interaction includes refraction and reflection and the spectrum of incident light. But for objects that are single particles, there aren't really any refraction/reflection properties, there is just stimulated ...

39

The Nobel website scientific background is good. Basically, when you try to make gallium nitride, you usually end up with a material that is (1) chock-full of defects, and (2) n-doped (even when you were trying to p-dope it). So blue LEDs required The invention of MOCVD technology for growing crystals (early 1970s); Finding the right recipe to grow good ...

0

Yes, optical activity can be affected by external electric and magnetic fields. Check out the Faraday effect: https://en.wikipedia.org/wiki/Faraday_effect Explanation of this phenomenon can be attempted with the assumption that the external magnetic field changes the electronic motion inside the molecules in such a way as to modify their effect on the ...

4

The "critical" part was in finding and producing a structure with a large enough bandgap to produce blue photons. The first LEDs produced relatively longwave infrared (IR) photons, which have far less energy than the green or blue photons now available from LEDs. In general, the larger the desired bandgap, the harder it is to manufacture a suitable ...

7

The book The Blue Laser Diode: The Complete Story deals with the issues of p-type doping of GaN. The difficulty of growing high quality GaN crystalline films lies in the problem of finding a suitable substrate material. (...) The link above points to the chapter you may be interested in.

1

As opposed to type II phase matching that produces orthogonally polarized photons in parametric down conversion (PDC), the type I PDC process produces identically polarized photons in the output signal and idler modes (labels $s$ and $i$ below). Normally the output state from type I PDC is not entangled: to get the required phase matching in the nonlinear ...

0

I did some digging and I found a blog that indirectly, but the none the less explicitly, talks about ray tracing chief and marginal rays in thin lens systems: http://taylortechassoc.com/?page_id=617 The equations he presents are the literal mathematical relationships between incoming and out outgoing ray angles (of any type) and it has a spreadsheet linked ...

2

All you need to know about the chief ray is that it will show up in the right place at the focal plane. Any path that gets there is valid. You will see that this is indeed the case for your example - the ray starts up at the top of the image (tip of the arrow) and passes through the tip of the arrow in the "internal image" plane. Any pair of straight lines ...

1

Reflection,refraction and transmission of light are macroscopic manifestation of a phenomenon called scattering.In this incoming photons are absorbed and either the quantum energy level of an atom is raised (as in case of resonance absorption) or the outer electron cloud is set into motion(this is responsible for light around us).Almost instantaneously ...

1

One approach is to try to use the original model as a ground truth and then redo the match to the desired form. I tried to do this in MATLAB: xs = 360.0 : 1.0 : 830.0; ys = 1 + 0.05792105./(238.0185-1.0./(xs.*xs)) + 0.00167917./(57.362-1.0./(xs.*xs)); cftool But, quality software that it is, it reliably crashed. So, I wrote my own nonlinear least ...

0

Neither the antumbra nor pentumbra have a fixed brightness. It varies across the region based on the amount of the solar disk that is visible. Each has regions where it is brighter than places in the other. It does not make sense to call one "brighter" than the other.

0

I'm going do do the annoying teacher thing where I answer a question with a question. Which is darker, a partial solar eclipse, in the penumbra of the moon, or a planetary transit, in the antumbra of an interior planet?

-1

In the penumbral area, part of the disc of the moon obscures a section of the sun. In the antumbral area, the entire disc of the moon obscures a, logically, larger section of the sun. Therefore the antumbra cannot be brighter than the penumbra during the same event.

Top 50 recent answers are included