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Search Results (1,902)

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Keywords = quantum physics

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44 pages, 5362 KiB  
Review
Performance of Low-Dimensional Solid Room-Temperature Photodetectors—Critical View
by Antoni Rogalski, Weida Hu, Fang Wang and Piotr Martyniuk
Materials 2024, 17(18), 4522; https://doi.org/10.3390/ma17184522 (registering DOI) - 14 Sep 2024
Viewed by 187
Abstract
In the last twenty years, nanofabrication progress has allowed for the emergence of a new photodetector family, generally called low-dimensional solids (LDSs), among which the most important are two-dimensional (2D) materials, perovskites, and nanowires/quantum dots. They operate in a wide wavelength range from [...] Read more.
In the last twenty years, nanofabrication progress has allowed for the emergence of a new photodetector family, generally called low-dimensional solids (LDSs), among which the most important are two-dimensional (2D) materials, perovskites, and nanowires/quantum dots. They operate in a wide wavelength range from ultraviolet to far-infrared. Current research indicates remarkable advances in increasing the performance of this new generation of photodetectors. The published performance at room temperature is even better than reported for typical photodetectors. Several articles demonstrate detectivity outperforming physical boundaries driven by background radiation and signal fluctuations. This study attempts to explain these peculiarities. In order to achieve this goal, we first clarify the fundamental differences in the photoelectric effects of the new generation of photodetectors compared to the standard designs dominating the commercial market. Photodetectors made of 2D transition metal dichalcogenides (TMDs), quantum dots, topological insulators, and perovskites are mainly considered. Their performance is compared with the fundamental limits estimated by the signal fluctuation limit (in the ultraviolet region) and the background radiation limit (in the infrared region). In the latter case, Law 19 dedicated to HgCdTe photodiodes is used as a standard reference benchmark. The causes for the performance overestimate of the different types of LDS detectors are also explained. Finally, an attempt is made to determine their place in the global market in the long term. Full article
(This article belongs to the Special Issue Recent Advances in Photoelectric Functional Materials and Devices)
50 pages, 3558 KiB  
Article
Dark Atoms of Nuclear Interacting Dark Matter
by Vitaly A. Beylin, Timur E. Bikbaev, Maxim Yu. Khlopov, Andrey G. Mayorov and Danila O. Sopin
Universe 2024, 10(9), 368; https://doi.org/10.3390/universe10090368 - 11 Sep 2024
Viewed by 367
Abstract
The lack of positive evidence for Weakly Interacting Massive Particles (WIMPs) as well as the lack of discovery of supersymmetric (SUSY) particles at the LHC may appeal to a non-supersymmetric solution for the Standard Model problem of the Higgs boson mass divergence, the [...] Read more.
The lack of positive evidence for Weakly Interacting Massive Particles (WIMPs) as well as the lack of discovery of supersymmetric (SUSY) particles at the LHC may appeal to a non-supersymmetric solution for the Standard Model problem of the Higgs boson mass divergence, the origin of the electroweak energy scale and the physical nature of the cosmological dark matter in the approach of composite Higgs boson. If the Higgs boson consists of charged constituents, their binding can lead to stable particles with electroweak charges. Such particles can take part in sphaleron transitions in the early Universe, which balance their excess with baryon asymmetry. Constraints on exotic charged species leave only stable particles with charge 2n possible, which can bind with n nuclei of primordial helium in neutral dark atoms. The predicted ratio of densities of dark atoms and baryonic matter determines the condition for dark atoms to dominate in the cosmological dark matter. To satisfy this condition of the dark-atom nature of the observed dark matter, the mass of new stable 2n charged particles should be within reach of the LHC for their searches. We discuss the possibilities of dark-atom binding in multi-atom systems and present state-of-the-art quantum mechanical descriptions of dark-atom interactions with nuclei. Annual modulations in such interactions with nuclei of underground detectors can explain the positive results of DAMA/NaI and DAMA/LIBRA experiments and the negative results of the underground WIMP searches. Full article
Show Figures

Figure 1

Figure 1
<p>The ratios of densities as a function of charge parameter <span class="html-italic">y</span>.</p>
Full article ">Figure 2
<p>The dependence of the ratio of densities on the total mass (<math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>E</mi> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>m</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>) for different values of <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>L</mi> <mi>B</mi> </mfrac> </mstyle> </semantics></math>.</p>
Full article ">Figure 3
<p>Allowed and forbidden regions for small values of charge parameter <span class="html-italic">y</span>.</p>
Full article ">Figure 4
<p>The ratio of technileptonic DM and baryonic matter densities as a function of mass in the assumption <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow> <mi>N</mi> <mo>/</mo> <mi>E</mi> </mrow> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>General dependence of the density ratios on the mass difference <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>m</mi> <mo>=</mo> <msub> <mi>m</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow> <mi>N</mi> <mo>/</mo> <mi>E</mi> </mrow> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>The forbidden parameter regions for <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mo>&lt;</mo> <msub> <mi>m</mi> <mi>crit</mi> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The ratios of number densities as a function of mass differences. The case <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>L</mi> <mi>B</mi> </mfrac> </mstyle> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>The ratios of number densities as a function of mass differences. The case <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mi>L</mi> <mi>B</mi> </mfrac> </mstyle> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Coefficients of Equation (<a href="#FD32-universe-10-00368" class="html-disp-formula">32</a>) as the total mass functions.</p>
Full article ">Figure 10
<p>The ratio of densities as a function of total mass in assumption <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mrow> <mi>N</mi> <mo>/</mo> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>m</mi> <mrow> <mi>U</mi> <mi>U</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mstyle> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>Qualitative form of the effective potential of the interaction of the <span class="html-italic">X</span>He dark atom with the nucleus of a heavy element [<a href="#B71-universe-10-00368" class="html-bibr">71</a>].</p>
Full article ">Figure 12
<p>Eigenvalues of the Hamiltonian of the helium nucleus (the first 3 energy levels) in the potential of the <span class="html-italic">O</span>He dark atom (blue solid line) and the graphs of the squared modulus of the wave function corresponding to these energy levels (red solid line).</p>
Full article ">Figure 13
<p>Potentials of the Coulomb (red dotted line) and nuclear (green dotted line) interaction between helium and the sodium nucleus, the potential of the Coulomb interaction between helium and the <math display="inline"><semantics> <msup> <mi>O</mi> <mrow> <mo>−</mo> <mo>−</mo> </mrow> </msup> </semantics></math> particle (black dotted line) and the total interaction potential for the helium nucleus (blue dotted line) in the system <span class="html-italic">O</span>He –<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> for fixed <math display="inline"><semantics> <msub> <mover accent="true"> <mi>R</mi> <mo>→</mo> </mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math>. The red circle shows the radius of the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus.</p>
Full article ">Figure 14
<p>Graph of the dependence of the energy values of the helium ground state in a polarized <span class="html-italic">O</span>-helium atom (red stars) on the radius vector of the outer sodium nucleus.</p>
Full article ">Figure 15
<p>The total potential of helium in the <span class="html-italic">O</span>He–<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> system for a fixed position of sodium <math display="inline"><semantics> <msub> <mover accent="true"> <mi>R</mi> <mo>→</mo> </mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math> (blue solid line), graph of the squared modulus of the wave function of the ground state of helium in a polarized dark atom for a fixed <math display="inline"><semantics> <msub> <mover accent="true"> <mi>R</mi> <mo>→</mo> </mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math> (red solid line), the intersection points of the graph of the total potential of helium and the graph of the squared modulus of the wave function of the ground state of helium (black circles).</p>
Full article ">Figure 16
<p>Graph of the dependence of the dipole moment of polarized <span class="html-italic">O</span>He atom (red stars) on the radius vector of the outer sodium nucleus.</p>
Full article ">Figure 17
<p>Graph of the dependence of the dipole moment of polarized <span class="html-italic">O</span>He atom (red stars) on the radius vector of the outer sodium nucleus. at the moment of repolarization of the dark atom.</p>
Full article ">Figure 18
<p>Graphs of the squared modulus of wave functions (red solid line) of certain values of the energy of the ground state of helium in the total potential of the <span class="html-italic">O</span>He–<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> system (blue solid line), corresponding to the positions of the sodium nucleus <math display="inline"><semantics> <msub> <mover accent="true"> <mi>R</mi> <mo>→</mo> </mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math> at the moment of the beginning of repolarization of the dark atom.</p>
Full article ">Figure 19
<p>Graphs of the squared modulus of wave functions (red solid line) of certain values of the energy of the ground state of helium in the total potential of the system <span class="html-italic">O</span>He–<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> (blue solid line), corresponding to the positions of the sodium nucleus <math display="inline"><semantics> <msub> <mover accent="true"> <mi>R</mi> <mo>→</mo> </mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math> at the moments of the beginning of highly probable tunneling of helium from a repolarized dark atom into the sodium nucleus.</p>
Full article ">Figure 20
<p>Graphs of the nuclear potential of the Woods–Saxon type (green dotted line), <math display="inline"><semantics> <msubsup> <mi>U</mi> <mrow> <mi>X</mi> <mi>H</mi> <mi>e</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math> (blue dotted line), Stark potential (gray dotted line) and the total effective interaction potential <span class="html-italic">O</span>He with the nucleus of the sodium (red dotted line) on the distance between the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus located in the Bohr orbit of the <span class="html-italic">O</span>He atom and the <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nucleus. The red circle shows the sum of the radii of the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nuclei for the interval of the helium radius vector equal to <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mo>|</mo> <mn>1.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mrow> <mspace width="3.33333pt"/> <mi>cm</mi> <mo>|</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 21
<p>Graphs of the nuclear potential of the Woods–Saxon type (green dotted line), <math display="inline"><semantics> <msubsup> <mi>U</mi> <mrow> <mi>X</mi> <mi>H</mi> <mi>e</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math> (blue dotted line), Stark potential (gray dotted line) and the total effective potential of interaction between <span class="html-italic">O</span>He and the nucleus of the sodium (red dotted line) on the distance between the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus located in the Bohr orbit of the <span class="html-italic">O</span>He atom and the <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nucleus for the interval of the helium radius vector equal to <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mo>|</mo> <mn>1.1</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mrow> <mspace width="3.33333pt"/> <mi>cm</mi> <mo>|</mo> </mrow> </mrow> </semantics></math> at the moment of repolarization of the dark atom.</p>
Full article ">Figure 22
<p>Graphs of the nuclear potential of the Woods–Saxon type (green dotted line), <math display="inline"><semantics> <msubsup> <mi>U</mi> <mrow> <mi>X</mi> <mi>H</mi> <mi>e</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math> (blue dotted line), Stark potential (gray dotted line) and the total effective interaction potential of <span class="html-italic">O</span>He with the nucleus of the sodium (red dotted line) on the distance between the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus located in the Bohr orbit of the <span class="html-italic">O</span>He atom and the <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nucleus for the interval of the helium radius vector equal to <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mo>|</mo> <mn>2.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mrow> <mspace width="3.33333pt"/> <mi>cm</mi> <mo>|</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 23
<p>Graphs of the nuclear potential of the Woods–Saxon type (green dotted line), <math display="inline"><semantics> <msubsup> <mi>U</mi> <mrow> <mi>X</mi> <mi>H</mi> <mi>e</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math> (blue dotted line), Stark potential (gray dotted line) and the total effective interaction potential of <span class="html-italic">O</span>He with the nucleus of the sodium (red dotted line) on the distance between the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus located in the Bohr orbit of the <span class="html-italic">O</span>He atom and the <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nucleus for the interval of the helium radius vector equal to <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mo>|</mo> <mn>2.5</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mrow> <mspace width="3.33333pt"/> <mi>cm</mi> <mo>|</mo> </mrow> </mrow> </semantics></math> at the moment of repolarization of the dark atom.</p>
Full article ">Figure 24
<p>Potentials of Coulomb (red dotted line), nuclear (green dotted line) and centrifugal (green solid line) interaction between helium and the nucleus of <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math>, the potential of Coulomb interaction between helium and <math display="inline"><semantics> <msup> <mi>O</mi> <mrow> <mo>−</mo> <mo>−</mo> </mrow> </msup> </semantics></math> particle (black dotted line) and the total interaction potential of the helium nucleus (blue dotted line) in the <span class="html-italic">O</span>He–<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> system at fixed <math display="inline"><semantics> <msub> <mover accent="true"> <mi>R</mi> <mo>→</mo> </mover> <mrow> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math>. The red circle marks the value of the radius of the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus.</p>
Full article ">Figure 25
<p>Graphs of Woods–Saxon nuclear potential (green dotted line), <math display="inline"><semantics> <msubsup> <mi>U</mi> <mrow> <mi>X</mi> <mi>H</mi> <mi>e</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math> (blue dotted line), Stark potential (gray dotted line), centrifugal potential (purple dotted line) and total effective interaction potential of <span class="html-italic">O</span>He with the nucleus of the sodium (red dotted line) on the distance between the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus, located in the Bohr orbit of the <span class="html-italic">O</span>He atom, and the <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nucleus for <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>J</mi> <mo>→</mo> </mover> <mrow> <mo>(</mo> <mi>O</mi> <mi>H</mi> <mi>e</mi> <mo>−</mo> <mi>N</mi> <mi>a</mi> <mo>)</mo> </mrow> </msub> <mo>=</mo> <mover accent="true"> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> <mo>→</mo> </mover> </mrow> </semantics></math>.</p>
Full article ">Figure 26
<p>Graphs of Woods–Saxon nuclear potential (green dotted line), <math display="inline"><semantics> <msubsup> <mi>U</mi> <mrow> <mi>X</mi> <mi>H</mi> <mi>e</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math> (blue dotted line), Stark potential (gray dotted line), centrifugal potential (purple dotted line) and total effective interaction potential of <span class="html-italic">O</span>He with the nucleus of the sodium (red dotted line) on the distance between the <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> </mrow> </semantics></math> nucleus, located in the Bohr orbit of the <span class="html-italic">O</span>He atom, and the <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>a</mi> </mrow> </semantics></math> nucleus for <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>J</mi> <mo>→</mo> </mover> <mrow> <mo>(</mo> <mi>O</mi> <mi>H</mi> <mi>e</mi> <mo>−</mo> <mi>N</mi> <mi>a</mi> <mo>)</mo> </mrow> </msub> <mo>=</mo> <mover accent="true"> <mn>3</mn> <mo>→</mo> </mover> </mrow> </semantics></math>.</p>
Full article ">
13 pages, 2442 KiB  
Article
Critical Assessment of Information Back-Flow in Measurement-Free Teleportation
by Hannah McAleese and Mauro Paternostro
Entropy 2024, 26(9), 780; https://doi.org/10.3390/e26090780 - 11 Sep 2024
Viewed by 198
Abstract
We assess a scheme for measurement-free quantum teleportation from the perspective of the resources underpinning its performance. In particular, we focus on claims recently made about the crucial role played by the degree of non-Markovianity of the dynamics of the information carrier whose [...] Read more.
We assess a scheme for measurement-free quantum teleportation from the perspective of the resources underpinning its performance. In particular, we focus on claims recently made about the crucial role played by the degree of non-Markovianity of the dynamics of the information carrier whose state we aim to teleport. We prove that any link between the efficiency of teleportation and the back-flow of information depends fundamentally on the way the various operations entailed by the measurement-free teleportation protocol are implemented while—in general—no claim of causal link can be made. Our result reinforces the need for the explicit assessment of the underlying physical platform when assessing the performance and resources for a given quantum protocol and the need for a rigorous quantum resource theory of non-Markovianity. Full article
(This article belongs to the Special Issue Simulation of Open Quantum Systems)
Show Figures

Figure 1

Figure 1
<p>Quantum circuit of the measurement-free teleportation protocol. Each gate into which the circuit is decomposed is labeled as <math display="inline"><semantics> <msub> <mi>G</mi> <mi>i</mi> </msub> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>i</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>8</mn> <mo>}</mo> </mrow> </semantics></math>). They can be grouped into three unitary <span class="html-italic">blocks</span> of operations <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> as defined in Equations (<a href="#FD2-entropy-26-00780" class="html-disp-formula">2</a>)–(<a href="#FD5-entropy-26-00780" class="html-disp-formula">5</a>). Here, <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> are CNOT gates, <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> </semantics></math> are Hadamard transforms, while <math display="inline"><semantics> <msub> <mi>G</mi> <mrow> <mn>5</mn> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> are SWAP gates.</p>
Full article ">Figure 2
<p>Non-Markovianity of the dynamics given by Equation (<a href="#FD24-entropy-26-00780" class="html-disp-formula">24</a>). Results are plotted only for <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>≥</mo> <mn>0.4</mn> </mrow> </semantics></math> as all the measures listed in <a href="#sec3dot1-entropy-26-00780" class="html-sec">Section 3.1</a> are zero for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mn>0.4</mn> </mrow> </semantics></math>. Inset: Non-Markovianity of the effective Hamiltonian in Equation (<a href="#FD25-entropy-26-00780" class="html-disp-formula">25</a>) as measured by the <math display="inline"><semantics> <msub> <mi mathvariant="script">N</mi> <mrow> <mi>B</mi> <mi>L</mi> <mi>P</mi> </mrow> </msub> </semantics></math> measure.</p>
Full article ">Figure 3
<p>Trace distance between the two states of <span class="html-italic">S</span> as they go through the teleportation circuit. Gate <math display="inline"><semantics> <msub> <mi>G</mi> <mi>i</mi> </msub> </semantics></math> acts when <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mi>i</mi> </mrow> </semantics></math>. We plot only <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>8</mn> </mrow> </semantics></math> as the trace distance remains constant at 1 for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>&lt;</mo> <mn>5</mn> </mrow> </semantics></math>.</p>
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<p>Trace distance between the two states of <math display="inline"><semantics> <msub> <mi>E</mi> <mn>2</mn> </msub> </semantics></math> for the original BBC protocol [<a href="#B3-entropy-26-00780" class="html-bibr">3</a>] as time evolves. The dashed lines are boundaries between the gates acting on the system and the environment. Gates <math display="inline"><semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics></math>–<math display="inline"><semantics> <msub> <mi>G</mi> <mn>4</mn> </msub> </semantics></math> correspond to those in <a href="#entropy-26-00780-f001" class="html-fig">Figure 1</a>, while <math display="inline"><semantics> <msub> <mi>G</mi> <mn>5</mn> </msub> </semantics></math> is a CNOT operation on <math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>G</mi> <mn>6</mn> </msub> </semantics></math> is a Hadamard gate on <math display="inline"><semantics> <msub> <mi>E</mi> <mn>2</mn> </msub> </semantics></math>.</p>
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<p>Correlations in the splitting <span class="html-italic">S</span>-vs-<math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, as quantified by (<b>a</b>) logarithmic negativity, (<b>b</b>) discord and (<b>c</b>) classical correlations as the system evolves according to Equation (<a href="#FD24-entropy-26-00780" class="html-disp-formula">24</a>). Time is denoted <span class="html-italic">t</span>, and <span class="html-italic">p</span> determines the Werner state of the environment at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The system is initially in the vacuum state <math display="inline"><semantics> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Correlations in the partition <math display="inline"><semantics> <mrow> <mrow> <mi>S</mi> <mo>|</mo> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> as quantified by (<b>a</b>) logarithmic negativity, (<b>b</b>) discord and (<b>c</b>) classical correlations when the Hamiltonian of the system and environment is given by Equation (<a href="#FD25-entropy-26-00780" class="html-disp-formula">25</a>). We take the initial state of the system to be <math display="inline"><semantics> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> </semantics></math> and the environment <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>. Here <span class="html-italic">t</span> is a dimensionless time. We only show <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>8</mn> </mrow> </semantics></math> as there are no system-environment correlations before <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p>
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<p>Correlations in the partition <math display="inline"><semantics> <mrow> <mrow> <mi>S</mi> <mo>|</mo> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math> as quantified by (<b>a</b>) logarithmic negativity, (<b>b</b>) discord and (<b>c</b>) classical correlations when the Hamiltonian of the system and environment is given by Equation (<a href="#FD25-entropy-26-00780" class="html-disp-formula">25</a>). We take the initial state of the system to be <math display="inline"><semantics> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> </mrow> </semantics></math> and the environment <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math>.</p>
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11 pages, 1230 KiB  
Article
Solving Time-Dependent Schödinger Equation for Some PT-Symmetric Quantum Mechanical Problems
by Tsin-Fu Jiang
Atoms 2024, 12(9), 46; https://doi.org/10.3390/atoms12090046 - 11 Sep 2024
Viewed by 212
Abstract
Using a high-precision code, we generate the eigenstates of a PT-symmetric Hamiltonian. We solve the time-dependent Schrödinger equation (TDSE) of the non-Hermitian system based on the eigenset. Since the formulation is relatively new and the observables are calculated differently than conventional quantum mechanics, [...] Read more.
Using a high-precision code, we generate the eigenstates of a PT-symmetric Hamiltonian. We solve the time-dependent Schrödinger equation (TDSE) of the non-Hermitian system based on the eigenset. Since the formulation is relatively new and the observables are calculated differently than conventional quantum mechanics, we justify it with a paradigmatic case in Hermitian quantum mechanics. We present the harmonic generation spectra on some model PT-Hamiltonians driven by an electric pulse. We discuss the physical differences with the harmonic spectra of a pulse-driven atom. Full article
(This article belongs to the Section Atom Based Quantum Technology)
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Figure 1

Figure 1
<p>(<b>a</b>) The electric field pulse with <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">E</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>ω</mi> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mi>τ</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>-cycle. (<b>b</b>) The corresponding frequency spectrum of the pulse peaked at <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and banded around <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>−</mo> <mi mathvariant="sans-serif">Ω</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>+</mo> <mi mathvariant="sans-serif">Ω</mi> </mrow> </semantics></math>, where <math display="inline"><semantics> <mi>ω</mi> </semantics></math> is the carrier frequency and <math display="inline"><semantics> <mi mathvariant="sans-serif">Ω</mi> </semantics></math> comes from the pulse duration.</p>
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<p>Occupation probability in excited state k after the same pulse of <a href="#atoms-12-00046-f001" class="html-fig">Figure 1</a> with the initial state in the ground state. The Hermitian quantum mechanical (blue-dotted) results and non-Hermitian one with a small value of <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.0001</mn> </mrow> </semantics></math> (red) show negligible differences. The connected line is for visual guidance.</p>
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<p>(<b>a</b>) The emitted light spectra of a pulse-driven harmonic oscillator. The spectra of a system with an additional term <math display="inline"><semantics> <mrow> <mi>i</mi> <mi>g</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>,</mo> <mi>g</mi> <mo>=</mo> <mn>0.0001</mn> </mrow> </semantics></math> under the same pulse show no difference from those of the harmonic oscillator. (<b>b</b>) The time history of dominant coefficients <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the pulse-driven harmonic oscillator with an additional term <math display="inline"><semantics> <mrow> <mi>i</mi> <mi>g</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>,</mo> <mi>g</mi> <mo>=</mo> <mn>0.0001</mn> </mrow> </semantics></math>. At the end of the pulse, the coefficients are in the Poisson’s distribution.</p>
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<p>The harmonic generation spectra of Hamiltonian <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>d</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>i</mi> <mi>g</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>x</mi> <mi>E</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with carrier frequency <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi mathvariant="script">E</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>τ</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>-cycle. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p>The <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> <mo stretchy="false">|</mo> </mrow> </semantics></math> versus time of <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> for Hamiltonian <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>d</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>i</mi> <mi>g</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>x</mi> <mi mathvariant="script">E</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with carrier frequency <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi mathvariant="script">E</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>τ</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>-cycle.</p>
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<p>The harmonic generation spectra of Hamiltonian <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> </mstyle> <mo>+</mo> <mi>i</mi> <mi>g</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>x</mi> <mi mathvariant="script">E</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with carrier frequency <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi mathvariant="script">E</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>τ</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>-cycle. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The <math display="inline"><semantics> <mrow> <mo stretchy="false">|</mo> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> <mo stretchy="false">|</mo> </mrow> </semantics></math> versus time of <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> for Hamiltonian <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <mn>2</mn> </mfrac> </mstyle> <mo>+</mo> <mi>i</mi> <mi>g</mi> <msup> <mi>x</mi> <mn>3</mn> </msup> <mo>+</mo> <mi>x</mi> <mi mathvariant="script">E</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with carrier frequency <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi mathvariant="script">E</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>τ</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>-cycle.</p>
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18 pages, 630 KiB  
Article
Rethinking Electron Statistics Rules
by Andras Kovacs and Giorgio Vassallo
Symmetry 2024, 16(9), 1185; https://doi.org/10.3390/sym16091185 - 10 Sep 2024
Viewed by 435
Abstract
The Fermi–Dirac and Bose–Einstein statistics are considered to be key concepts in quantum mechanics, and they are used to explain the occupancy limit of electron orbitals. We investigate the physical origin of these two statistics and uncover that the key determining factor is [...] Read more.
The Fermi–Dirac and Bose–Einstein statistics are considered to be key concepts in quantum mechanics, and they are used to explain the occupancy limit of electron orbitals. We investigate the physical origin of these two statistics and uncover that the key determining factor is whether an individual electron spin is measurable or not. Microscopically, a system with individually measurable electron spins corresponds to the presence of Larmor spin precession in electron–electron interactions, while the non-measurability of individual electron spins corresponds to the absence of Larmor spin precession. Both interaction types are possible, and the favored interaction type is thermodynamically determined. The absence of Larmor spin precession is realized in coherent electron states, and coherent electrons therefore obey Bose–Einstein statistics. Full article
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Figure 1
<p>An illustration of two protons’ Larmor spin precession in a hydrogen molecule. Each proton perceives the other proton’s magnetic field (directed red curves) as an externally applied magnetic field and Larmor spin-precesses (cones with arrow) around the external magnetic field line.</p>
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<p>An illustration of electrons’ coherent state. The dotted line represents the shared Zitterbewegung axis, the ellipses represent the 0.386 pm radius Zitterbewegung trajectories, and the blue spheres represent the electron charges. Each electron has the same momentum, and their kinetic speed vectors point along the Zitterbewegung axis.</p>
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<p>The correlation between <math display="inline"><semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics></math> and spin fluctuation temperature <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>S</mi> <mi>F</mi> </mrow> </msub> </semantics></math> in various superconductors. Reproduced from [<a href="#B11-symmetry-16-01185" class="html-bibr">11</a>].</p>
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<p>An illustration of vortices that comprise the Meissner flow. The externally applied magnetic field (B) is represented by the arrow.</p>
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19 pages, 5805 KiB  
Article
Assessment of Warionia saharea Essential Oil as a Green Corrosion Inhibitor for Mild Steel in HCl: Experimental and Computational Studies
by Abdeslam Ansari, Youssef Youssefi, Mohamed Tanghourte, Nazih Ouassou, Nazih Asoufar, Mohamed Znini, Hassane Lgaz, El Houssine Mabrouk, Mohamed Azrour, Han-Seung Lee and Belkheir Hammouti
Coatings 2024, 14(9), 1164; https://doi.org/10.3390/coatings14091164 - 9 Sep 2024
Viewed by 412
Abstract
The objective of this research work is the study of the inhibitory effect of Warionia saharea essential oil (WSEO) on the corrosion of mild steel (MS) in molar HCl solution, employing both experimental and theoretical methods. This inhibitory effect (IE) has been evaluated [...] Read more.
The objective of this research work is the study of the inhibitory effect of Warionia saharea essential oil (WSEO) on the corrosion of mild steel (MS) in molar HCl solution, employing both experimental and theoretical methods. This inhibitory effect (IE) has been evaluated by using a combination of weight loss measurements (LW) and various electrochemical methods, such as open circuit potential (OCP), potentiodynamic polarization (PDP) and electrochemical impedance spectroscopy (EIS) experiments. The LW results indicated that IE increased with inhibitor concentration, reaching 83.34% at 3.00 g/L. The PDP analysis suggested that WSEO functions as a mixed inhibitor, while in the EIS results the Rct values increased with inhibitor concentration to reach 165.8 Ω cm2 at 2.00 g/L, suggesting a defensive film formation by WSEO molecules over the metallic surface. The thermodynamic study demonstrated that the WSEO molecules adsorption on the MS surface followed a Langmuir isotherm, involving mixed physical and chemical (physicochemical) adsorption on the MS surface. Theoretical methods, including density functional theory (DFT) and molecular dynamics (MD) simulations, were employed to elucidate the inhibition mechanisms of the three main components of WSEO. The quantum chemical analysis, using density functional theory (DFT) and molecular dynamics (MD) simulations, showed a low ΔEgap value of 6.30 eV and a low adsorption energy (Eads) value on an Fe (110) substrate of −258 Kcal/mol for (E)-Nerolidol, indicating the significant contribution of this molecule to the overall corrosion inhibition effect of WSEO. The scanning electron microscope (SEM) analysis verified the presence of a protective film formed by the inhibitor on the MS surface. This study highlights the potential of WSEO as a sustainable and green corrosion inhibitor in acidic environments. Full article
(This article belongs to the Special Issue Surface Science of Degradation and Surface Protection)
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<p>Chemical structures of the major components of WSEO.</p>
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<p>OCP–time curves for MS in 1 M HCl without and with various concentrations of WSEO.</p>
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<p>PDP curves of mild steel (MS) in 1.0 M HCl with and without varying concentrations of <span class="html-italic">Warionia saharea</span> essential oil (WSEO) at 25 °C.</p>
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<p>Nyquist diagrams (<b>a</b>), equivalent circuit (<b>b</b>) and Bode plots (<b>c</b>,<b>d</b>) of MS in 1 M HCl without and with the addition of various concentrations of WSEO at 25 °C.</p>
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<p>Isotherm models tested for MS in 1 M HCl in presence of WSEO using WL, PDP and EIS data at 25 °C.</p>
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<p>SEM morphology of MS surface (<b>a</b>) before immersion, (<b>b</b>) after 24 h immersion in blank solution and (<b>c</b>) after 24 h immersion in inhibited solution.</p>
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<p>The HOMO-LUMO iso-surfaces and optimized geometry for β-Eudesmol, (E)-Nerolidol and Linalool that have been evaluated by the DFT/B3LYP with 6-311++ G (d,p) basis set.</p>
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<p>Top (<b>a</b>) and side (<b>b</b>) view of the most equilibrium adsorption configurations of the selected WSEO components on an Fe (110) surface obtained by MD simulations in the aqueous phase.</p>
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<p>MSD vs. time curves and the diffusion behaviors of corrosive species (H<sub>3</sub>O<sup>+</sup> and Cl<sup>−</sup> ions) in three condensed inhibitor films at 25 °C.</p>
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<p>FFV of Cl<sup>−</sup> and H<sub>3</sub>O<sup>+</sup> in the four condensed inhibitor films at 25 °C.</p>
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<p>Simplified schema of the adsorption mechanism of (E)-Nerolidol molecules on the MS surface.</p>
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13 pages, 8911 KiB  
Article
Efficient and Near-Zero Thermal Quenching Cr3+-Doped Garnet-Type Phosphor for High-Performance Near-Infrared Light-Emitting Diode Applications
by Zaifa Yang
Molecules 2024, 29(17), 4253; https://doi.org/10.3390/molecules29174253 - 7 Sep 2024
Viewed by 480
Abstract
In recent years, near-infrared (NIR) phosphors have attracted great research interest due to their unique physical properties and broad application prospects. However, obtaining NIR phosphors with both high quantum efficiency and excellent thermal stability remains a great challenge. In this study, novel NIR [...] Read more.
In recent years, near-infrared (NIR) phosphors have attracted great research interest due to their unique physical properties and broad application prospects. However, obtaining NIR phosphors with both high quantum efficiency and excellent thermal stability remains a great challenge. In this study, novel NIR Ca3Mg2ZrGe3O12:Cr3+ phosphors were successfully prepared using a high-temperature solid-phase method, and the structure and luminescent properties of the material were systematically investigated. Ca3Mg2ZrGe3O12:0.01Cr3+ emits NIR light in the range of 600 to 900 nm with a peak at 758 nm and a half-height width of 89 nm under the excitation of 457 nm blue light. NIR luminescence shows considerable quantum efficiency, and the internal quantum efficiency of the optimized sample is up to 68.7%. Remarkably, the Ca3Mg2ZrGe3O12:0.01Cr3+ phosphor exhibits a near-zero thermal quenching behavior, and the luminescence intensity of the sample at 250 °C maintains 92% of its intensity at room temperature. The mechanism of high thermal stability has been elucidated by calculating the Huang Kun factor and activation energy. Finally, NIR pc-LED devices prepared from Ca3Mg2ZrGe3O12:0.01Cr3+ phosphor with commercial blue LED chips have good performance, proving that this Ca3Mg2ZrGe3O12:0.01Cr3+ NIR phosphor has potential applications in night vision and biomedical imaging. Full article
(This article belongs to the Special Issue Organic and Inorganic Luminescent Materials)
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Figure 1
<p>(<b>a</b>) XRD patterns of CMZG:xCr<sup>3+</sup> (0.005 ≤ x ≤ 0.025); Rietveld refinement pattern of (<b>b</b>) CMZG; (<b>c</b>) CMZG:0.01Cr<sup>3+</sup>; and (<b>d</b>) lattice parameters and cell volumes of CMZG:xCr<sup>3+</sup> (0.005 ≤ x ≤ 0.025).</p>
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<p>The (<b>a</b>) SEM and (<b>b</b>) elemental mapping image for the CMZG:0.01Cr<sup>3+</sup> sample.</p>
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<p>The diffuse reflectance spectra of the CMZG host and CMZG:0.025Cr<sup>3+</sup> phosphor.</p>
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<p>(<b>a</b>) The PLE and PL spectra of CMZG:0.01Cr<sup>3+</sup> phosphor; (<b>b</b>) the PL spectra of CMZG:xCr<sup>3+</sup> (0.005 ≤ x ≤ 0.025) phosphors; (<b>c</b>) linear fit of lg(<span class="html-italic">x</span>) vs. lg[<span class="html-italic">I</span>/(<span class="html-italic">x</span>)]; and (<b>d</b>) Tanabe–Sugano energy level diagrams for Cr<sup>3+</sup>.</p>
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<p>(<b>a</b>) Luminescence decay curves of CMZG:xCr<sup>3+</sup> (0.005 ≤ x ≤ 0.025) phosphors; (<b>b</b>) excitation line of BaSO<sub>4</sub> and emission spectrum of CMZG:0.01Cr<sup>3+</sup> phosphor.</p>
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<p>Temperature-dependent PL spectra of (<b>a</b>) CMZG:0.005Cr<sup>3+</sup> (<b>b</b>) CMZG:0.01Cr<sup>3+</sup> and (<b>c</b>) CMZG:0.015Cr<sup>3+</sup> phosphors; (<b>d</b>) temperature-dependent integral emission intensity of these phosphors; and normalized temperature-dependent PL spectra of (<b>e</b>) CMZG:0.005Cr<sup>3+</sup> (<b>f</b>) CMZG:0.01Cr<sup>3+</sup>, and (<b>g</b>) CMZG:0.015Cr<sup>3+</sup> phosphors.</p>
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<p>(<b>a</b>) Variations in the emission peak and FWHM with increasing temperature; (<b>b</b>) fitted Huang–Rhys factor and phonon energy; (<b>c</b>) the dependence of ln[(<span class="html-italic">I</span><sub>0</sub>/<span class="html-italic">I<sub>T</sub></span>) − 1] on 1/<span class="html-italic">kT</span> for the CMZG:0.01Cr<sup>3+</sup> sample; and (<b>d</b>) configurational coordinate diagram of Cr<sup>3+</sup>; (The red dashed line represents the transition process generated by heating and the black dashed line represents the non-radiative transition).</p>
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<p>(<b>a</b>) The EL spectra of fabricated NIR pc-LEDs under various driving currents; (<b>b</b>) photographs of NIR light penetrating a palm; and (<b>c</b>) images taken under fluorescent light and NIR LED light.</p>
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55 pages, 652 KiB  
Article
On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity
by Pierre-Henri Chavanis
Axioms 2024, 13(9), 606; https://doi.org/10.3390/axioms13090606 - 5 Sep 2024
Viewed by 325
Abstract
In this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equations [...] Read more.
In this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equations from microscopic processes. We show their formal equivalence. Specifically, we show that the real and imaginary parts of the complex Lorentz equation in Nottale’s theory are equivalent to the Nelson equations, which are themselves equivalent to the Madelung and de Broglie hydrodynamical representations of the Schrödinger and Klein–Gordon equations, respectively. We discuss the different physical interpretations of the Nelson and Nottale theories and stress their strengths and weaknesses. We mention potential applications of these theories to dark matter. Full article
(This article belongs to the Special Issue Entanglement in Quantum Field Theory and Its Applications)
31 pages, 3091 KiB  
Review
Silicon-28-Tetrafluoride as an Educt of Isotope-Engineered Silicon Compounds and Bulk Materials for Quantum Systems
by Owen C. Ernst, David Uebel, Roman Brendler, Konstantin Kraushaar, Max Steudel, Jörg Acker and Edwin Kroke
Molecules 2024, 29(17), 4222; https://doi.org/10.3390/molecules29174222 - 5 Sep 2024
Viewed by 1317
Abstract
This review provides a summary of the existing literature on a crucial raw material for the production of isotopically pure semiconductors, which are essential for the development of second-generation quantum systems. Silicon-28-tetrafluoride (28SiF4) is used as an educt for [...] Read more.
This review provides a summary of the existing literature on a crucial raw material for the production of isotopically pure semiconductors, which are essential for the development of second-generation quantum systems. Silicon-28-tetrafluoride (28SiF4) is used as an educt for several isotope-engineered chemicals, such as silane-28 (28SiH4) and silicon-28-trichloride (28SiHCl3), which are needed in the pursuit of various quantum technologies. We are exploring the entire chain from the synthesis of 28SiF4 to quantum applications. This includes the chemical properties of SiF4, isotopic enrichment, conversion to silanes, conversion to bulk 28Si and thin films, the physical properties of 28Si (spin neutrality, thermal conductivity, optical properties), and the applications in quantum computing, photonics, and quantum sensing techniques. Full article
(This article belongs to the Special Issue Exclusive Feature Papers in Physical Chemistry, 2nd Edition)
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<p>Summarized reaction pathways of the known synthesis routes to SiF<sub>4</sub>.</p>
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<p>Countercurrent gas centrifuge. Natural SiF<sub>4</sub> is introduced into the centrifuge. The centrifugal force generated by the rotation of the centrifuge exerts a pushing force on the heavier isotopes, causing them to move further outward than the lighter isotopes. With cylinder radii of 5–10 cm and rotation speeds of 750 m/s, the generation of forces in excess of 1,000,000 g is possible. The pressure in the regions of the centrifuge in close proximity to the wall exhibits a notable increase, whereas the pressure in the vicinity of the axis of rotation declines. In a countercurrent centrifuge, an additional flow is introduced, which results in further enrichment. By subjecting the bottom of the centrifuge to a heating process, the heavier isotopes are drawn upward, while the lighter isotopes are drawn further downward. This results in the formation of multiple vertical segments where centrifugal enrichment occurs. The heavier isotopes accumulate in the upper section of the centrifuge, subsequently traversing a perforated plate and being collected. Conversely, the lighter isotopes congregate at the base and flow through an aperture in close proximity to the axis of rotation, ultimately entering a lower chamber where they can be harvested separately.</p>
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<p>Laser separation by <span class="html-small-caps">Silex</span> process. The <span class="html-small-caps">Silex</span> process was developed by <span class="html-small-caps">Silex System Limited</span> and remains under the company’s operational control. A solution comprising SiF<sub>4</sub> and argon is introduced into a chamber or feed pipe under conditions of increased pressure. The concentration of SiF<sub>4</sub> is typically less than 1 mol%. The gas mixture is directed through a fine nozzle into a second chamber. The chamber is maintained at a constant low pressure. Because of the abrupt expansion of the gas as it traverses the nozzle, the temperature of the gas declines rapidly to a value below 100 K. At these low temperatures, the SiF<sub>4</sub> molecules form stable clusters with Ar. The cold gas mixture is subsequently subjected to narrow-band laser light. The wavelength is selected to facilitate greater excitation of the <sup>28</sup>Si isotope in comparison to <sup>29</sup>Si or <sup>30</sup>Si. Because of their excited state, the formed <sup>28</sup>SiF<sub>4</sub>-Ar clusters disintegrate, resulting in the liberation of <sup>28</sup>SiF<sub>4</sub> molecules. Several techniques can be employed to separate the free <sup>28</sup>SiF<sub>4</sub> molecules from the larger <sup>29/30</sup>SiF<sub>4</sub>-Ar clusters. In the simplest case, a molecular sieve can be utilized.</p>
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<p>Two principal routes have been identified. The initial route entails the combustion of <sup>28</sup>SiF<sub>4</sub>, resulting in the production of <sup>28</sup>SiO<sub>2</sub>. This is subsequently reduced with a base metal, such as Al, to generate bulk <sup>28</sup>Si. If thin <sup>28</sup>Si layers are required, for example, for microelectronic applications, physical vapor deposition (PVD) methods may be employed. The second route involves the chemical substitution of fluorine atoms with hydrogen. The resulting <sup>28</sup>SiH<sub>4</sub> can be converted into solid <sup>28</sup>Si via thermal decomposition or chemical vapor deposition (CVD).</p>
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<p>The unique properties of isotopically pure silicon have the potential to facilitate the development of novel applications. (<b>a</b>) The mass of each silicon-28 atom is precisely 28 u. This allows for the precise measurement of physical constants, such as the Avogadro constant, using a well-defined geometry, as demonstrated by the Avogadro project. (<b>b</b>) The highest heat conductivity λ of any dielectric at low temperatures has been observed in silicon-28. This offers the potential for enhanced performance in cryogenic applications, such as in high-energy laser optics in vacuum, where energy can dissipate away rapidly from laser impact, thereby reducing thermal stress or damage. (<b>c</b>) In contrast to the silicon-29 nucleus, the nucleus of silicon-28 is spin-neutral. This implies that the nucleus’ spin does not interact with the spin of electrons, thereby elongating their decoherence time and thus enabling spin quantum computer approaches.</p>
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19 pages, 5780 KiB  
Review
Fluorometric Mercury (II) Detection Using Heteroatom-Doped Carbon and Graphene Quantum Dots
by Mosayeb Chaghazardi, Soheila Kashanian, Maryam Nazari, Kobra Omidfar, Yvonne Joseph and Parvaneh Rahimi
Photonics 2024, 11(9), 841; https://doi.org/10.3390/photonics11090841 - 5 Sep 2024
Viewed by 462
Abstract
Mercury ion (Hg2+) is one of the most toxic pollutants that can exist throughout the environment and be diffused into water, soil, air, and eventually the food chain. Even a very low level of Hg2+ diffused in living organisms can [...] Read more.
Mercury ion (Hg2+) is one of the most toxic pollutants that can exist throughout the environment and be diffused into water, soil, air, and eventually the food chain. Even a very low level of Hg2+ diffused in living organisms can hurt their DNA and cause the permanent damage of the central nervous system and a variety of consequential disorders. Hence, the development of a sensitive and specific method for the detection of Hg2+ at trace ranges is extremely important as well as challenging. Fluorometric detection assays based on graphene quantum dots (GQDs) and carbon quantum dots (CQDs) offer considerable potential for the determination and monitoring of heavy metals due to their fascinating properties. Although the quantum yield of GQDs and CQDs is sufficient for their use as fluorescent probes, doping with heteroatoms can significantly improve their optical properties and selectivity toward specific analytes. This review explores the primary advances of CQDs and GQDs in their great electronic, optical, and physical properties, their synthetic methods, and their use in Hg2+ fluorimetry detection. Full article
(This article belongs to the Special Issue Editorial Board Members’ Collection Series: Photonics Sensors)
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<p>Schematic representation of the synthesis of N,S-GQDs via UV irradiation, illustrating the concentration-dependent properties (<b>left</b>) and the photothermal conversion process (<b>right</b>) [<a href="#B34-photonics-11-00841" class="html-bibr">34</a>]. Reprinted with permission from Elsevier.</p>
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<p>Bottom-up and top-down strategies for synthesis of CQDs.</p>
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<p>Bottom-up and top-down strategies for synthesis of GQDs.</p>
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<p>Nitrogen-doped carbon quantum dots as “turn-off” fluorescent probes for highly selective and sensitive detection of mercury (II) ions [<a href="#B59-photonics-11-00841" class="html-bibr">59</a>]. Reprinted with permission from Wiley.</p>
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<p>Fluorescent sensing method to determine Hg<sup>2+</sup> in aqueous solutions using sulfhydryl functionalized CQDs (HS-CQDs) mediated by Ag<sup>+</sup>. (<b>A</b>) Absorption spectrum of the HS-CDs/Ag precipitate; the insets are corresponding photos of HS-CQDs/Ag under visible light (<b>left</b>) and UV light at 365 nm (<b>right</b>). (<b>B</b>) FL spectra of HS-CDs/Ag and HS-CQDs/Ag + Hg<sup>2+</sup>; the insets are corresponding photos of the HS-CQDs/Ag sensor (<b>left</b>) and the HS-CQDs/Ag <sup>+</sup> Hg<sup>2+</sup> (<b>right</b>) under UV light at 365 nm [<a href="#B88-photonics-11-00841" class="html-bibr">88</a>]. Reprinted with permission from Elsevier.</p>
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<p>The scheme illustration of the synergistic effect of Hg<sup>2+</sup> and NGQDs in accelerating the coordination rate of Mn<sup>2+</sup> and 5,10,15,20-tetrakis(1-methyl-4-pyridinio) porphyrin tetra(p-toluenesulfonate) (TMPyP) [<a href="#B122-photonics-11-00841" class="html-bibr">122</a>]. Reprinted with permission from ACS.</p>
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13 pages, 3186 KiB  
Article
Carbon Dots: A Versatile Platform for Cu2+ Detection, Anti-Counterfeiting, and Bioimaging
by Qian Wang, Xinyi He, Jian Mao, Junxia Wang, Liangliang Wang, Zhongchi Zhang, Yongfei Li, Fenglin Huang, Bin Zhao, Gang Chen and Hua He
Molecules 2024, 29(17), 4211; https://doi.org/10.3390/molecules29174211 - 5 Sep 2024
Viewed by 408
Abstract
Carbon dots (CDs) have garnered extensive interest in basic physical chemistry as well as in biomedical applications due to their low cost, good biocompatibility, and great aqueous solubility. However, the synthesis of multi-functional carbon dots has always been a challenge for researchers. Here, [...] Read more.
Carbon dots (CDs) have garnered extensive interest in basic physical chemistry as well as in biomedical applications due to their low cost, good biocompatibility, and great aqueous solubility. However, the synthesis of multi-functional carbon dots has always been a challenge for researchers. Here, we synthesized novel CDs with a high quantum yield of 28.2% through the straightforward hydrothermal method using Diaminomaleonitrile and Boc-D-2, 3-diaminopropionic acid. The size, chemical functional group, and photophysical properties of the CDs were characterized by TEM, FTIR, XPS, UV, and fluorescence. It was demonstrated in this study that the prepared CDs have a high quantum yield, excellent photostability, and low cytotoxicity. Regarding the highly water-soluble property of CDs, they were proven to possess selective and sensitive behavior against Cu2+ ions (linear range = 0–9 μM and limit of detection = 1.34 μM). Moreover, the CDs were utilized in fluorescent ink in anti-counterfeiting measures. Because of their low cytotoxicity and good biocompatibility, the CDs were also successfully utilized in cell imaging. Therefore, the as-prepared CDs have great potential in fluorescence sensing, anti-counterfeiting, and bioimaging. Full article
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<p>The schematic diagram of the synthesis process of the CDs.</p>
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<p>(<b>a</b>) TEM image and particle size distribution measured from TEM images of CDs; (<b>b</b>) XRD profile of CDs; (<b>c</b>) FT-IR spectra of CDs.</p>
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<p>XPS spectra of DBP-CDs. (<b>a</b>) Survey profile, (<b>b</b>) C1s, (<b>c</b>) N1s, and (<b>d</b>) O1s.</p>
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<p>(<b>a</b>) UV–vis absorption and (<b>b</b>) fluorescence spectra of CD solution; (<b>c</b>) normalized fluorescence spectra of CDs solution. (<b>d</b>) Photostability of CDs under 532 nm laser, fluorescence decay curves (black), and fitted curves (red) of CDs.</p>
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<p>The fluorescence intensity of CDs before and after UV lamp irradiation at 365 nm for 40 min.</p>
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<p>(<b>a</b>) Fluorescence responses of CDs in the presence of different metal ions; (<b>b</b>) fluorescence emission spectra of CDs with different concentrations of Cu<sup>2+</sup> (from 10 µM to 100 µM); (<b>c</b>) linear relationship between <span class="html-italic">F</span><sub>0</sub>/<span class="html-italic">F</span> and Cu<sup>2+</sup> concentration within the concentration range of 0–9 µM.</p>
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<p>Fluorescent ink application of CDs. (<b>a</b>) Illustration of CD dispersion as a security ink in anticounterfeiting applications. Inkjet-printed sample images of (<b>b</b>–<b>d</b>) QR code and (<b>e</b>–<b>g</b>) “6699” number in daylight (<b>b</b>,<b>e</b>), 365 nm UV light ON (<b>c</b>,<b>f</b>), and OFF (<b>d</b>,<b>g</b>).</p>
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<p>Cytotoxicity and bioimaging of CDs against HeLa cells. (<b>a</b>) Cell viability of CDs assessed by using MTT assays. Confocal fluorescence images of HeLa cells incubated with CDs: (<b>b</b>) bright field, (<b>c</b>) fluorescent channel, (<b>d</b>) merging of (<b>b</b>,<b>c</b>).</p>
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9 pages, 1878 KiB  
Article
Self-Assembled Hybrid Halide Perovskite Quantum Wire Bundle/Dot for Multiband Applications
by Hee Chang Jeon, Seonghwan Kim and Young-Seong Kim
Nanomaterials 2024, 14(17), 1443; https://doi.org/10.3390/nano14171443 - 4 Sep 2024
Viewed by 390
Abstract
In this study, self-assembled halide perovskite quantum wire bundles (QWBs)/quantum dots (QDs) are fabricated using a room temperature-based formation method. The one-dimensional (1D) perovskite-based QWB structures incorporate zero-dimensional QDs within a composite quantum structure. Transmission electron microscopy reveals that quantum wires with diameters [...] Read more.
In this study, self-assembled halide perovskite quantum wire bundles (QWBs)/quantum dots (QDs) are fabricated using a room temperature-based formation method. The one-dimensional (1D) perovskite-based QWB structures incorporate zero-dimensional QDs within a composite quantum structure. Transmission electron microscopy reveals that quantum wires with diameters ranging from tens of nanometers to approximately 200 nm maintain a single-crystal atomic arrangement in a bundle form. Conversely, QDs are uniformly distributed within the single-phase wire and appear as black dots < 10 nm. Photoluminescence analysis identifies the multiband characteristics of the emissions. The 420–440 nm band is attributed to 1D QWB, whereas the peak appearing in the 530–550 nm range corresponds to lead halide PbBr2 QDs. Thus, the proposed self-assembled 1D QWB/QD composite structure exhibits novel multiband physical properties in the 420–440 and 530–550 nm bands; it offers new opportunities for designing materials with potential applications in optoelectronic devices. Full article
(This article belongs to the Section Nanophotonics Materials and Devices)
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<p>Fabrication method of the self-assembled hybrid halide perovskite QWB/QD. The bundled shape of the wires is evident in the SEM images.</p>
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<p>XRD analysis showing the peaks of QWB and QD, confirming the hybrid structure. The simulated XRD peak results obtained through material modeling have been included alongside the experimental data.</p>
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<p>TEM images of QW and QD: (<b>a</b>,<b>b</b>) TEM images of QW and QD, respectively; (<b>c</b>) HR-TEM image confirming both structures. The region containing QD in QW is demarcated by a yellow dashed line.</p>
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<p>Results of the photoluminescence analysis: (<b>a</b>) temperature dependence of PL, (<b>b</b>) outcomes of a multi-peak analysis of PL conducted at 10 K, calculation of the activation energy using the Arrhenius equation for the wavelength ranges of (<b>c</b>) 420–440 nm and (<b>d</b>) 530–550 nm.</p>
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<p>UV–Vis absorption data represented using a Taut plot confirmed the presence of bandgaps at 2.86 and 3.1 eV.</p>
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<p>XPS spectra of (<b>a</b>) the wide scan spectrum for QWB/QD, (<b>b</b>) Pb 4f, (<b>c</b>) S 2p, and (<b>d</b>) O 1s.</p>
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20 pages, 323 KiB  
Article
Quantum Mechanics and Inclusive Materialism
by Javier Pérez-Jara
Philosophies 2024, 9(5), 140; https://doi.org/10.3390/philosophies9050140 - 3 Sep 2024
Viewed by 991
Abstract
Since its inception, the intricate mathematical formalism of quantum mechanics has empowered physicists to describe and predict specific physical events known as quantum processes. However, this success in probabilistic predictions has been accompanied by a profound challenge in the ontological interpretation of the [...] Read more.
Since its inception, the intricate mathematical formalism of quantum mechanics has empowered physicists to describe and predict specific physical events known as quantum processes. However, this success in probabilistic predictions has been accompanied by a profound challenge in the ontological interpretation of the theory. This interpretative complexity stems from two key aspects. Firstly, quantum mechanics is a fundamental theory that, so far, is not derivable from any more basic scientific theory. Secondly, it delves into a realm of invisible phenomena that often contradicts our intuitive and commonsensical notions of matter and causality. Despite its notorious difficulties of interpretation, the most widely accepted set of views of quantum phenomena has been known as the Copenhagen interpretation since the beginning of quantum mechanics. According to these views, the correct ontological interpretation of quantum mechanics is incompatible with ontological realism in general and with philosophical materialism in particular. Anti-realist and anti-materialist interpretations of quantum matter have survived until today. This paper discusses these perspectives, arguing that materialistic interpretations of quantum mechanics are compatible with its mathematical formalism, while anti-realist and anti-materialist views are based on wrong philosophical assumptions. However, although physicalism provides a better explanation for quantum phenomena than idealism, its downward reductionism prevents it from accounting for more complex forms of matter, such as biological or sociocultural systems. Thus, the paper argues that neither physicalism nor idealism can explain the universe. I propose then a non-reductionistic form of materialism called inclusive materialism. The conclusion is that the acknowledgment of the qualitative irreducibility of ontological emergent levels above the purely physical one does not deny philosophical materialism but enriches it. Full article
(This article belongs to the Special Issue Philosophy and Quantum Mechanics)
11 pages, 245 KiB  
Article
The Particle of Haag’s Local Quantum Physics: A Critical Assessment
by Gregg Jaeger
Entropy 2024, 26(9), 748; https://doi.org/10.3390/e26090748 - 1 Sep 2024
Viewed by 325
Abstract
Rudolf Haag’s Local Quantum Physics (LQP) is an alternative framework to conventional relativistic quantum field theory for combining special relativity and quantum theory based on first principles, making it of great interest for the purposes of conceptual analysis despite currently being relatively limited [...] Read more.
Rudolf Haag’s Local Quantum Physics (LQP) is an alternative framework to conventional relativistic quantum field theory for combining special relativity and quantum theory based on first principles, making it of great interest for the purposes of conceptual analysis despite currently being relatively limited as a tool for making experimental predictions. In LQP, the elementary particles are defined as species of causal link between interaction events, together with which they comprise its most fundamental entities. This notion of particle has yet to be independently assessed as such. Here, it is captured via a set of propositions specifying particle characteristics and then compared to previous particle notions. Haag’s particle differs decisively with respect to mechanical intuitions about particles by lacking, among other things, even an approximate independent space–time location. This notion is thus found to differ greatly even from those of relativistic quantum mechanics and quantum field theory, which have been applied to the known elementary particles. Full article
32 pages, 3170 KiB  
Article
Inequality in the Distribution of Wealth and Income as a Natural Consequence of the Equal Opportunity of All Members in the Economic System Represented by a Scale-Free Network
by John G. Ingersoll
Economies 2024, 12(9), 232; https://doi.org/10.3390/economies12090232 - 29 Aug 2024
Viewed by 312
Abstract
The purpose of this work is to examine the nature of the historically observed and empirically described by the Pareto law inequality in the distribution of wealth and income in an economic system. This inequality is presumed to be the result of unequal [...] Read more.
The purpose of this work is to examine the nature of the historically observed and empirically described by the Pareto law inequality in the distribution of wealth and income in an economic system. This inequality is presumed to be the result of unequal opportunity by its members. An analytical model of the economic system consisting of a large number of actors, all having equal access to its total wealth (or income) has been developed that is formally represented by a scale-free network comprised of nodes (actors) and links (states of wealth or income). The dynamic evolution of the complex network can be mapped in turn, as is known, into a system of quantum particles (links) distributed among various energy levels (nodes) in thermodynamic equilibrium. The distribution of quantum particles (photons) at different energy levels in the physical system is then derived based on statistical thermodynamics with the attainment of maximal entropy for the system to be in a dynamic equilibrium. The resulting Planck-type distribution of the physical system mapped into a scale-free network leads naturally into the Pareto law distribution of the economic system. The conclusions of the scale-free complex network model leading to the analytical derivation of the empirical Pareto law are multifold. First, any complex economic system behaves akin to a scale-free complex network. Second, equal access or opportunity leads to unequal outcomes. Third, the optimal value for the Pareto index is obtained that ensures the optimal, albeit unequal, outcome of wealth and income distribution. Fourth, the optimal value for the Gini coefficient can then be calculated and be compared to the empirical values of that coefficient for wealth and income to ascertain how close an economic system is to its optimal distribution of income and wealth among its members. Fifth, in an economic system with equal opportunity for all its members there should be no difference between the resulting income and wealth distributions. Examination of the wealth and income distributions described by the Gini coefficient of national economies suggests that income and particularly wealth are far off from their optimal value. We conclude that the equality of opportunity should be the fundamental guiding principle of any economic system for the optimal distribution of wealth and income. The practical application of this conclusion is that societies ought to shift focus from policies such as taxation and payment transfers purporting to produce equal outcomes for all, a goal which is unattainable and wasteful, to policies advancing among others education, health care, and affordable housing for all as well as the re-evaluation of rules and institutions such that all members in the economic system have equal opportunity for the optimal utilization of resources and the distribution of wealth and income. Future research efforts should develop the scale-free complex network model of the economy as a complement to the current standard models. Full article
(This article belongs to the Special Issue Innovation, Reallocation and Economy Growth)
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<p>Schematic comparison of the distribution of links (k) in random networks where most nodes have the same number of links and scale-free networks where a few nodes can have a large number of links, resulting in the formation of hubs (<a href="#B4-economies-12-00232" class="html-bibr">Barabási 2014</a>).</p>
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<p>Schematic illustration of the mapping between an economic system comprised of actors and wealth elements and represented as a network comprised of nodes (actors) and links (wealth elements) and a statistical thermodynamic system of indistinguishable and non-interacting gas of photons (links—wealth elements) distributed in different energy states (nodes—actors)—a node (actor) is characterized by a fitness η and has a varying number of wealth elements (photons—links) occupying the different energy states ε (actors—nodes) of the physical system (complex network—economic system). The dashed node and links signify an actor addition to the network system (<a href="#B4-economies-12-00232" class="html-bibr">Barabási 2014</a>).</p>
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<p>Schematic of the mapping of the economic system into a scale-free complex network and its modeling as a Planck photon gas physical system.</p>
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<p>Exponential distribution of the random variable x with rate λ: (<b>a</b>) Probability density function (PDF); and (<b>b</b>) Cumulative density function (CDF).</p>
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<p>Pareto distribution of the random variable x with index α and a minimum value of the random variable of x<sub>m</sub> = 1: (<b>a</b>) Probability density function (PDF); (<b>b</b>) Cumulative density function (CDF).</p>
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<p>Lorenz function for several Pareto distributions with index α. The case α = ∞ (45-degree line) corresponds to a perfectly equal distribution (<span class="html-italic">G</span> = 0) and the case α = 1 (horizontal line) corresponds to complete inequality (<span class="html-italic">G</span> = 1). The case α = 2 corresponds to the optimal allocation of wealth and income based on the Planck distribution derived herein.</p>
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<p>Map of the income Gini coefficient by country for the time period mostly between 2015 and 2022 based on data from the World Bank and other sources—Created by Alice Hunter (<a href="#B53-economies-12-00232" class="html-bibr">World Bank Group 2022</a>).</p>
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<p>Map of the wealth Gini coefficient by country showing values ranging from a high of 0.850–0.899 (dark red) to a low of 0.550–0.599 (light green) based on data derived from the Wealth Handbook by Credit Suisse in 2019 (<a href="#B18-economies-12-00232" class="html-bibr">Hechler-Fayd’herbe 2019</a>).</p>
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<p>Graphical representation of the exponential integral function E<sub>a</sub>(nβh) with a = δ and with x = nβh per Equations (19) and (20).</p>
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