Madelung定数の計算

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以前の記事で、結晶の格子エネルギーにおける静電エネルギー項の計算について説明しました。この記事では、この項を計算する方法について詳しく説明し、また単純なイオン系におけるこの項の値を提供します。

晶体の総クーロン相互作用エネルギーは、個々の相互作用項の合計として与えられます。

\displaystyle V_{AB} = \frac{e^2}{4\pi \epsilon_0} \frac{Z_AZ_B}{r_{AB}} \hfill (1)

固体の結晶構造では、電荷qAとqBを持つイオン対の間の距離rABによって生じるクーロン相互作用力で構成されます。この和は、固体中のすべてのイオン対にわたって計算されます。

最初の近傍は負の重要な寄与を提供するため、この和は非常に遅く収束します。2番目の近傍原子はわずかに弱い正の項を生成します。このプロセスは無限遠まで続き、交互の符号でますます小さい値を通じて行われます。このようにして、陽イオンと陰イオンの間の引力が主導し、固体のエネルギーに有益な負の寄与を提供します。

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Unix C-Shell Programming Notes: Part II

This is the second part of this series of notes on the advanced use of the Unix shell. The first part can be accessed by following the link. In this second part, I will show some advanced examples I wrote for my research activity.

1. CHANGE FILE NAMES SCRIPT

I wrote this script to change the default names of the files generated during a molecular dynamics simulation by the program mdrun in the early version of the GROMACS package for MD simulation. GROMACS is a widely used software package for simulating the behavior of molecules and molecular systems. Although it is now possible to define the root name of the files upon running the simulation program, this script can still be helpful. The script is written in the C Shell (csh) .

 #! /bin/csh -ef
#
# Change the names of the default  output Files
# generated by the program GROMACS mdrun program
#
# (c) Danilo Roccatano
#######
#
#Define the radix of the files name
#
setenv RADIX  'EH_min'

if (-e md.log) then
   if (  -e $RADIX'.log')  then 
     echo Warning $RADIX'.log' exist
   else
     mv md.log            $RADIX'.log'
   endif
else
   echo The md.log file does not exist!
endif

if (-e ener.ene) then
  if (  -e 'e'$RADIX'.ene') then 
   echo Warning 'e'$RADIX'.ene' exist
  else
    mv ener.ene       'e'$RADIX'.ene'
  endif
else
  echo The energy file does not exist
endif

if (-e ctraj.xtc ) then
  if ( -e 'r'$RADIX'.xtc') then 
    echo Warning 'r'$RADIX'.xtc' exist
  else
    mv ctraj.xtc      'r'$RADIX'.xtc'
  endif
else
  echo The compressed trajectory file does not exist
endif

if (-e confout.gro) then
  if ( -e 'x'$RADIX'.gro') then 
    echo Warning 'x'$RADIX'.gro' exist
  else
    mv confout.gro    'x'$RADIX'.gro'
  endif
else
  echo The final configuration file does not exit.
  echo Probably your simulation crashed before the end 
endif
 
exit
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The shell of a snail and its 3D Digitization with the Structure-from-Motion algorithm

To grass, or leaf, or fruit, or wall,
The snail sticks close, nor fears to fall,
As if he grew there, house and all
                                                 Together.
Within that house secure he hides,
When danger imminent betides
Of storm, or other harm besides
                                                Of weather.
Give but his horns the slightest touch,
His self-collecting power is such,
He shrinks into his house, with much
                                                Displeasure.
Where’er he dwells, he dwells alone,
Except himself has chattels none,
Well satisfied to be his own
                                             Whole treasure.
Thus, hermit-like, his life he leads,
Nor partner of his banquet needs,
And if he meets one, only feeds
                                                The faster.
Who seeks him must be worse than blind,
(He and his house are so combin’d)
If, finding it, he fails to find
                                                Its master.

The Snail by William Cowper (1731-1800)

Introduzione

The beautiful poetry of Cowper expresses the pleasant charm that this small inhabitant of our gardens instills. I have always been fascinated by this gastropod, to the point that it was one of my favorite invertebrates for my amateur naturalistic observations. Furthermore, I still recall with pleasure and nostalgia the collection of those called ‘ciammaruchelle‘ in the Ciociaro dialect, which are small snails. These were gathered by the handful in the wheat fields after the harvest. It was one of the various culinary traditions that involved my entire family every year and were carried out with constant devotion. The collection was organized with careful timing, locations, and weather conditions to increase the likelihood of success. Usually, we would return home with a rich and tasty haul, but not without difficulties, as the little snails would climb onto the thistle plants (Cynara cardunculus L., 1753) where they would hide among the thorns to protect themselves from predators. Unfortunately for them, the predator Homo Sapiens Sapiens Frusinenses, equipped with keen eyesight and great tenacity, did not easily give up its prey!

The collected species was a variety of the snail Eobania vermiculata, commonly known as “rigatella,” which is very common in Mediterranean regions. The snails were gathered in woven baskets and, once back home, they were enclosed in circular cages with fine mesh walls for several days to purge their intestines. They were then cooked for a few hours in a tomato base spiced with mint (Clinopodium nepeta), following an ancient recipe. The dish was consumed with fresh or, even better, baked bread to make it crispy. It was a vibrant celebration of scents, flavors, and colors, with the sound of slurping as they tried to empty the succulent contents of their shells. A delicate feast of aromas and flavors: the scent of tomato infused with snail meat and mint, combined with the red color of the snails’ shells adorned with white-brown stripes.

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Unveiling Stardust on Your Rooftop: a Magnetic Microparticle Sorter and Spotter

With the help of my lovely assistant, we have just published on Instructables another STEM project. This time, it was inspired by the reading of the classic Comets by Carl Sagan and the upcoming period of significant meteor shower events of the year:

  1. Perseids: The Perseids is one of the most popular and reliable meteor showers, occurring annually in mid-August. It is associated with the comet Swift-Tuttle and gets its name from the constellation Perseus, from where the meteors appear to radiate. The Perseids are known for their fast and bright meteors, often leaving long-lasting trails in the sky. They are visible from both hemispheres, but the best viewing opportunities are typically in the northern hemisphere. During its peak, the Perseids can produce up to 60 to 100 meteors per hour.
  2. Geminids: The Geminids meteor shower takes place every December and is considered one of the most reliable and prolific meteor showers of the year. The Geminids are associated with the asteroid 3200 Phaethon and appear to radiate from the constellation Gemini, hence their name. Unlike many other meteor showers, the Geminids can be observed from both the northern and southern hemispheres. They are known for producing bright, colorful meteors and can reach a peak rate of 120 to 160 meteors per hour, making them a spectacular astronomical event.
  3. Quadrantids: The Quadrantids meteor shower occurs annually in early January and is known for its brief but intense display of meteors. The radiant point of the Quadrantids lies in the defunct constellation Quadrans Muralis, which is no longer recognized as a constellation. However, the meteors can be seen anywhere in the sky. The Quadrantids are associated with the asteroid 2003 EH1, which is believed to be an extinct comet. The peak of the Quadrantids is relatively short, usually lasting only a few hours, but it can produce a high meteor rate of around 60 to 200 meteors per hour.

Who won’t be thrilled by capturing a piece of outer space right in your hands? However, you might think, “But I don’t have the necessary equipment!” Fear not, as luck would have it, you’ve stumbled upon this instructable that will guide you in creating your own Micrometeorite Separator and Spotter (MSS).

Sure, it’s true that if a magnificent carbonaceous chondrite were to gracefully land right in front of your doorstep, you wouldn’t need any equipment. However, chances are high that tiny fragments of meteorites, resembling specks of falling stars, are scattered on your rooftop, waiting to be discovered. Countless websites and YouTube videos share fascinating tales of those who have embarked on this labor of love, showcasing the patience and dedication required to uncover these elusive cosmic treasures. The meteorites you’re after are microscopic, so your search will focus on distinguishing them from the terrestrial dust and debris that pervade our surroundings.

Fortunately, a small and mighty Neodymium magnet proves invaluable in this pursuit, especially for isolating metallic meteorites. Yet, be prepared for a challenging quest, as finding these little celestial gems is no easy task. However, the rewards of discovering even a single meteorite are genuinely remarkable.

We have recently published an Instructable project presenting a simple yet effective device to aid you in your meteorite discovery journey. This device remains compact and portable that can be used with a digital microscope or fitted under a stereomicroscope and the wonders they reveal.

So, get ready to embark on an extraordinary adventure and immerse yourself in the captivating world of meteorite hunting. With the Micrometeorite Separator and Spotter (MSS) at your disposal, you’ll be equipped to explore the heavens and unveil the hidden secrets within the depths of our vast universe.

Happy hunting, and do not forget to like this post and the Instructable !

Introducing the Mini LED Table: Compact, Affordable, and Enhanced With Computer Vision

How many times have you found yourself looking at your cake decorations, wondering how many pearls are left in the box? Have you ever tried to count how many mustard, basil or pepper plants you can grow with the seeds you have in stock? And what about the urge to know the exact number of millimeter stainless steel balls you recently purchased at your favorite DIY store?
For this purpose, some scrap cardboard, a strip of LEDs, and a Raspberry Pi Zero with its tiny chamber might help satisfy your mathematical desire to enumerate these tiny objects.
In a recent article on Instructables, we detailed the eye-opening experience with all the construction details of a compact, battery-powered, easy-to-use, and affordable mini LED table (MTL). In addition to the hardware, an open source software based on OpenCv is also strengthened for the acquisition and simple processing of the images of the objects on the MTL for possible applications.

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Il Mini Tavolo LED: Illumina l’Universo dei Dettagli Nascosti!

Quante volte ti sei trovato a guardare le tue decorazioni per i dolci, chiedendoti quante perle rimangono nella scatola? Hai mai provato a contare quante piante di senape, basilico o pepe puoi coltivare con i semi che hai in magazzino? E cosa dire dell’urgenza di conoscere il numero esatto delle sfere in acciaio inossidabile da un millimetro che hai recentemente acquistato nel tuo negozio di fai-da-te preferito?
Per questo scopo, un po’ di cartone di recupero, una striscia di LED e un Raspberry Pi Zero con la sua piccola camera potrebbero aiutare a soddisfare il tuo desiderio matematico di enumerare questi oggetti minuscoli.
In un recente articolo su Instructables, abbiamo descritto l’ illuminante esperienza con tutti i dettagli costruttivi di un mini tavolo LED (MTL) compatto, alimentato a batteria, facile da usare e conveniente. Oltre all’hardware viene fortito anche un software opensource basato su OpenCv per l’acquisizione e la semplice elaborazione delle immagini degli oggetti sul MTL per possibili applicazioni.

Caratteristiche principali del progetto:

  • Un tavolo a LED compatto, portatile ed economico realizzato con un foglio di cartone di imballaggio e una striscia di LED bianchi.
  • Può essere alimentato alimentato da una batteria elettrica e ha un dimmer per regolare l’intensità della luce.
  • È dotato di una camera Raspberry Pi per acquisire le immagini (con la distanza focale modificata per mettere a fuoco oggetti vicini).
  • Il software in Python fornito come OpenSource consente di controllare la camera e di elaborare le immagini per contare gli oggetti sul tavolo.

L’articolo originale (in inglese) del MTL si trova sul mio sito Instructables.

L’articolo contiene le istruzioni dettagliate e i file STL per la stampa 3D dei componenti necessari. Il progetto è stato realizzato in collaborazione con i miei due cari collaboratori, in particolare Leonardo. È un progetto semplice e accessibile che può essere utilizzato anche a scopo educativo per imparare concetti di costruzione, elettronica, ottica e programmazione.

Introduzione alla programmazione in Python (III)

Puntate precedenti:

  1. Introduzione alla programmazione in Python (I)
  2. Introduzione alla programmazione in Python (II)

Siamo giunti alla terza parte di questa serie dove vedremo come usare le istruzion condizionali per il controllo del flusso del programma.

DIAGRAMMI DI FLUSSO

Per una descrizione più semplice e intuitiva dei commandi per il controllo del programma, verranno utilizzati diagrammi di flusso. Un diagramma di flusso è un tipo di diagramma (grafico o simbolico) che rappresenta un algoritmo o un processo. Ogni passo nel processo è rappresentato da un simbolo diverso e contiene una breve descrizione del passo del processo. I simboli del diagramma di flusso sono collegati tra loro con frecce che mostrano la direzione del flusso del processo. Un diagramma di flusso mostra tipicamente il flusso dei dati in un processo, dettagliando le operazioni/passi in un formato pittorico che è più facile da capire rispetto alla lettura di un formato testuale.

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Easter 2023: Superquadric and Squared-Eggs

First there was the Great Cosmic Egg.

Huai-nan Tzu, China 100 BCE 

Another Easter is arriving, and here I am for a new article on old friends: the Eggs. In my blog for Easter 2021, I mentioned squared eggs paraphrasing a sentence attributed to the extraordinary jeweler Peter Carl Fabergé:” This year, your Highness, we will be featuring square eggs.” A hen is unlikely to make a cubic-shaped egg, but we can still transform a cooked egg to get the cuboid shape. If you look for squared eggs on the internet, you will find a tool called the egg cuber that does the magic. Unfortunately, I could not yet try one, but it seems to be doing an excellent job from the reviews. The shape of the egg is not a perfect cube but a cuboid, namely a cube with rounded corners. In my previous article, I anticipated that this shape could be obtained using the superquadric function. Curiouser and curiouser! In this article, I will give a bit of mathematical background and even a source code to play with such a function as my Easter Bunny gift.

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FORTRAN Programming (PART III): NUMERICAL DIFFERENTIATION

In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.

Hugo Rossi. Mathematics Is an Edifice, Not a Toolbox, Notices of the AMS, v. 43, no. 10, October 1996.

Welcome to the third part is the third part of the essential introduction to the FORTRAN programming series. In this part, we will introduce the use of functions and subroutines to structure the programs. Then we use them to make a program to calculate numerical derivatives of mathematical functions.

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Numeric Solutions of the Schrödinger equation

For Schrödinger, the atom was an oscillating system — like the string of a musical instrument — for which there exist a number of modes of oscillation (fundamentals and overtones) which are interpreted as the atom’s energy states.

Armin Hermann. Werner Heisenberg 1901-1976.

This set of slides is from one of my courses in Computational Chemistry. It covers the one-dimensional solution of the Schrödinger equation using Numerov’s method. The material for this article was mainly taken from the excellent Levin textbook in the reference.
If you are interested in more details, please like it and subscribe to my website. This way, I will be motivated to add more of these contents.

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