text:
Muting

Anshul:

Have you been to Mars? Where are you, man

Look forward to seeing you too.

Anshul

Muting..... thanks..

I will see you on Monday.

take care

Rachit

Sachin and Praveen.. congratulations and welcome to Baruch..

prashar

Thanks

Thanks Andy for your encouraging remarks

@Anshul and Rachit...Thanks for the wishes.

Looking forward to meeting you guys...

Saurabhagarwal

Congratulations to all....

All those who have got admits, i request them to kindly post their profiles here so that other aspirants like me can gain a lot of understanding on the kind of profiles Baruch is looking out for.

Once again i wish the best of luck to all those who have conquered this uphill task.

alain

Older and Wiser
Congratulations to all....

All those who have got admits, i request them to kindly post their profiles here so that other aspirants like me can gain a lot of understanding on the kind of profiles Baruch is looking out for.

Once again i wish the best of luck to all those who have conquered this uphill task.

The profiles will be posted eventually. I think there are threads already in the forum with a mini-bio of each of the current students. If you do a search you will find them.

Hello Everyone,

i know i am a bit late.
But better late than never

Actually got an admit a few days back...so here i am amongst all of you.

Computer Science - Ahmedabad, Gujarat, India.

bhargav

Hi guys ...

I got admission into Baruch last Thursday after having the 'talk' with professor Dan. I would say I am extremely happy. I have been around here for quite a while but never posted any except specific queries i had. I look forward to interacting with all of you.

Bhargav

thanks andy...

btw...left a private message in your inbox..

amm....i dont know whether this is the right place to say this...
but i am taking my chances

bhargav, sachin, pawan....can i get your email ids..yahoo or gmail....

bhargav

Andy,

When do classes start? August or September (beginning or end).

Thanks

bhargav

Thank you very much Andy

Bhargav

alain

Older and Wiser
But the refreshers usually start in July

bhargav

Alain

I don't think it is feasible for me to attend refreshers as I am international student. But i would love to know from you guys as to what would be covered in these courses. Atleast I can brush up undergrad math and C++. I cannot start this until June as I am preparing for CFA L2.
Any suggestions appreciated

Thanks

dstefan

Baruch MFE Director
But the refreshers usually start in July

The C++ refresher is in June.

Ahh....I appreciate the point made by bhargav....

We would really wish to know what precisely would be covered in the refresher courses....

bhargav

Andy and other seniors

I got the following email fromInternational Student Service Center.

In order to process your SEVIS I-20, we need the followingsupportive documentation:

-Original bank statements translated into English showing a minimum of $29,881.00, not older than 3 months and in U.S. dollar equivalent. An Affidavit of support. -Proof of income from all contributing parties. -Copy of your valid passport (identification page). FAQ in website cites that entire amount to be shown to get I20 is 57,800$. Which one is

Sachin

Hi ,

This is what I did. I showed liquid assets (i.e cash funds) of around 60KUSD. In addition to this ,I sent an affidavit of support from my parents and also my father's annual income on a letter head.

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# Generating And / Or Conditions Automatically from a List

I have the following list:

l1 = {x y, s t, a b}

From this list, I am trying to essentially generate a new list that looks like this:

    l1 = {x==1 && y==1 && s == 0 && t == 0 && a == 0 && b == 0
|| x == 0 && y == 0 && s == 1 && t == 1 && a == 0 && b == 0
|| x == 0 && y == 0 && s == 0 && t == 0 && a == 1 && b == 1}

I.e., only one pair at a time can take on the value of 1 in each OR statement.

l1 = {x y, s t, a b};

res1 = {Or @@
Inner[Sequence @@ Thread[Equal[List @@ #2, #]] &, IdentityMatrix[Length @ l1], l1, And]}

{(x == 1 && y == 1 && s == 0 && t == 0 && a == 0 && b == 0) ||
(x == 0 && y == 0 && s == 1 && t == 1 && a == 0 && b == 0) ||
(x == 0 && y == 0 && s == 0 && t == 0 && a == 1 && b == 1)}

Alternatively,

l2 = Flatten[l1 /. Times -> List]

{x, y, s, t, a, b}

res2 = {Or @@
Inner[Equal[#2, #] &, Riffle[#, #] & /@ IdentityMatrix[Length[l2]/2], l2,  And]}

{(x == 1 && y == 1 && s == 0 && t == 0 && a == 0 && b == 0) ||
(x == 0 && y == 0 && s == 1 && t == 1 && a == 0 && b == 0) ||
(x == 0 && y == 0 && s == 0 && t == 0 && a == 1 && b == 1)}

• Hi @kglr . I tried this solution, and it works great. The only problem I am having is that it doesn't work exactly as expected because your l1 is slightly different then mine. In your l1, you have l1 = {x,y,s,t,a,b}, but I have l1 = {x y, s t, a b}, i.e., each element in the list contains a product. – Thomas Moore Oct 24 '19 at 19:21
• @ThomasMoore, you can use l0 = {x y, s t, a b}; l1 = Flatten[l0 /. Times -> List]. I will post an alternative approach using {x y, s t, a b} directly. – kglr Oct 24 '19 at 19:24
• Hi @kglr Thank you very much. It is much appreciated! – Thomas Moore Oct 24 '19 at 19:24

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Similar Question 1

Determine the total number, \displaystyle n , of compounding periods and the interest rate, \displaystyle i , as a decimal, per compounding period for each scenario.

\displaystyle 6.2 \%  per year, compounded daily for 2 years

Similar Question 2

$100 is put into a bank account that pays interest so that the amount in the account grows according to the expression 100(1.06)n, where n is the number of years. Find the amount in the account after

a) 5 years

b) 10 years
 Similar Question 3
Determine the present value of each future amount for the given conditions.

In 4 years, an investment eaming 6% per year, compounded annually, will have a value of 5800. Find the present value.
 Similar Questions Learning Path L1 Quick Intro to Factoring Trinomial with Leading a L2 Introduction to Factoring ax^2+bx+c L3 Factoring ax^2+bx+c, ex1 Now You Try
Determine the total number, \displaystyle n , of compounding periods and the interest rate, \displaystyle i , as a decimal, per compounding period for each scenario.

\displaystyle 7 \%  per annum, compounded quarterly for 5 years

For each compounding condition, determine the number of compounding periods.

quarterly compounding for 4 years

Determine the total number, \displaystyle n , of compounding periods and the interest rate, \displaystyle i , as a decimal, per compounding period for each scenario.

\displaystyle 8.5 \%  per year, compounded annually for 4 years

For each compounding condition, determine the interest rate per compounding period, expressed as a decimal.

\displaystyle 9 \%  per annum, compounded quarterly

For each compounding condition, determine the number of compounding periods.

semi-annual compounding for 5 years

Determine the total number, \displaystyle n , of compounding periods and the interest rate, \displaystyle i , as a decimal, per compounding period for each scenario.

\displaystyle 6.2 \%  per year, compounded daily for 2 years

Determine the total number, \displaystyle n , of compounding periods and the interest rate, \displaystyle i , as a decimal, per compounding period for each scenario.

\displaystyle 3.6 \%  per year, compounded monthly for 3 years

$100 is put into a bank account that pays interest so that the amount in the account grows according to the expression 100(1.06)n, where n is the number of years. Find the amount in the account after

a) 5 years

b) 10 years

For each compounding condition, determine the number of compounding periods.

annual compounding for 7 years

\displaystyle \$650  is invested for 7 years at \displaystyle 5 \%  interest per year, compounded annually.

a) Determine the amount in the account after 7 years.

b) How much interest was earned?

For each compounding condition, determine the interest rate per compounding period, expressed as a decimal.

\displaystyle 7 \%  per year, compounded monthly

For each compounding condition, determine the interest rate per compounding period, expressed as a decimal.

\displaystyle 8 \%  annual interest, compounded semi-annually

Determine the present value of each future amount for the given conditions.

In 4 years, an investment eaming 6% per year, compounded annually, will have a value of 5800. Find the present value.

For each compounding condition, determine the interest rate per compounding period, expressed as a decimal.

\displaystyle 11 \%  per year, compounded bi-weekly

Determine the total number, \displaystyle n , of compounding periods and the interest rate, \displaystyle i , as a decimal, per compounding period for each scenario.

5.5\% per annum, compounded semi-annually for \displaystyle 6.5  years

For each compounding condition, determine the number of compounding periods.

daily compounding for 3 weeks

For each compounding condition, determine the number of compounding periods.

monthly compounding for \displaystyle \frac{2}{3}  of a year

Wes borrows \displaystyle \$ 850  at a rate of \displaystyle 9.5 \%  interest per year, compounded annually, for 4 years.

a) Determine the amount to be repaid after 4 years.

b) How much interest will Wes have to pay?

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# High to Low pellet cladding gap heat transfer modeling methodology in an uncertainty quantification framework for a PWR Rod Ejection Accident with best estimate coupling

3 ASCII - Analyse d’interactions stochastiques intelligentes et coopératives
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : High to Low modeling approaches can alleviate the computationally expensive fuel modeling in nuclear reactor’s transient uncertainty quantification. This is especially the case for Rod Ejection Accident (REA) in Pressurized Water Reactors (PWR) were strong multi-physics interactions occur. In this work, we develop and propose a pellet cladding gap heat transfer (Hgap) High to Low modeling methodology for a PWR REA in an uncertainty quantification framework. The methodology involves the calibration of asimplified $Hgap$ model based on high fidelity simulations with the fuel-thermomechanics code ALCYONE1.The calibrated model is then introduced into the CEA developed CORPUS Best Estimate (BE) multi-physicscoupling between APOLLO3R© and FLICA4. This creates an Improved Best Estimate (IBE) coupling that is then used for an uncertainty quantification study. The results indicate that with IBE the distance to boiling crisis uncertainty is decreased from 57% to 42%. This is reflected to the decrease of the sensitivity of $Hgap$. In the BE coupling $Hgap$ was responsible for 50% of the output variance while in IBE it is close to 0. These results show the potential gain of High to Low approachez for $Hgap$ modeling in REA uncertainty analyses.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-03147612
Contributor : Josselin Garnier <>
Submitted on : Friday, April 23, 2021 - 9:29:31 AM
Last modification on : Saturday, May 1, 2021 - 3:46:28 AM

### File

epjn200009.pdf
Files produced by the author(s)

### Citation

Gregory Kyriakos Delipei, Josselin Garnier, Jean-Charles Le Pallec, Benoit Normand. High to Low pellet cladding gap heat transfer modeling methodology in an uncertainty quantification framework for a PWR Rod Ejection Accident with best estimate coupling. EPJ N - Nuclear Sciences & Technologies, EDP Sciences, 2020, 6, pp.56. ⟨10.1051/epjn/2020018⟩. ⟨hal-03147612⟩

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# Simple integral - but a question of methodology

ianb
simple integral -- but a question of methodology

$$\int \frac{(x-1)}{2} dx$$

First, try substitution. Let $$u = x - 1$$, and $$du = dx$$

$$\int \frac{u}{2} du$$

$$\frac{1}{2} \int u \ du$$

$$\frac{1}{2} \times \frac{u^{2}}{2}$$

$$\frac{1}{2} \times \frac{(x-1)^2}{2}$$

$$\frac{(x-1)^{2}}{4} + C$$

Or we could do this:

$$\int \frac{(x-1)}{2} dx$$

$$\int \frac{1}{2}(x-1)dx$$

$$\frac{1}{2} \int(x-1)dx$$

$$\frac{1}{2} (\frac{x^2}{2} - x) + C$$

$$\frac{1}{4} (x^2 - 2x) + C$$

Why should one of them be right?

Last edited:

ZioX
They're both right. Antiderivatives differ by a constant. (Differentiating eliminates the constant term). Show that those two answers differ by a constant.

Homework Helper
...
$$\frac{1}{4} (x^2 - \frac{x}{2}) + C$$

Why should one of them be right?
The last line is wrong. You've pulled out the factor 1 / 4 incorrectly.
$$\frac{1}{4} \left( x ^ 2 - 2x \right)$$
Both are correct, the two results differ from each other by a constant 1/4.
You should note that if F(x) is one anti-derivative of f(x), then F(x) + C is also f(x)'s anti-derivative, where C is any constant.
Proof: Differentiate the LHS with respect to x, we have:
(F(x) + C)' = F'(x) + C' = F'(x) + 0 (the derivative of a constant is just 0)
= F'(x) = f(x)
Can you get this? :)

ianb
Well I can certainly show that, just expand it. But the issue arises when you evaluate the area, say from 0 to 1.7, and using either ways give different answers. In fact, the answers differ by 0.25.

Thoughts?

ianb
Ah, my apologies. I was rushing through latex'ing my equations (still a newbie)--should have double checked it.

Homework Helper
Well I can certainly show that, just expand it. But the issue arises when you evaluate the area, say from 0 to 1.7, and using either ways give different answers. In fact, the answers differ by 0.25.

Thoughts?
Nope, they are identically the same, let F(x) + C1, and F(x) + C2 be two anti-derivatives of f(x), we have:
$$\int_a^b f(x) dx = (F(b) + C_1) - (F(a) + C_1) = F(b) - F(a)$$
$$\int_a^b f(x) dx = (F(b) + C_2) - (F(a) + C_2) = F(b) - F(a)$$, they are still the same.
The constant will automatically vanish when you subtract the two terms. :)

ianb
Hm. What if the lower bound is zero, I don't think it does. Look at the substitution method, and evaluate it from 0 to 1.7. You will still get a difference of 0.25 than if you compare it with the other method. Of course, you might say that the constants do indeed cancel even with 0 set as the lower bound, but that will require me to expand--why? I mean, why would the answers be the same after expansion, and different without it? Does integration by substitution require expansion of factors?

Last edited:
ianb
I'm sorry, I should have been clearer. Message edited accordingly.

Homework Helper
Hm. What if the lower bound is zero, I don't think it does. Look at the substitution method, and evaluate it from 0 to 1. You will still get a difference of 0.25 than if you compare it with the other method. Of course, you might say that the constants do indeed cancel even with 0 set as the lower bound, but that will require me to expand--why? I mean, why would the answers be the same after expansion, and different without it? Does integration by substitution require expansion of factors?
Uhm, well, no, they are not different.
Well, here we go:
$$\int_0 ^ 1 \frac{x - 1}{2} dx = \left. \left( \frac{(x - 1) ^ 2}{4} \right) \right|_0 ^ 1 = \frac{(1 - 1) ^ 2}{4} - \frac{(0 - 1) ^ 2}{4} = 0 - \frac{1}{4} = -\frac{1}{4}$$

$$\int_0 ^ 1 \frac{x - 1}{2} dx = \left. \frac{1}{4} \left( x ^ 2 - 2x \right) \right|_0 ^ 1 = \frac{1}{4} \left( (1 ^ 2 - 2 \times 1) - (0 ^ 2 - 0) \right) = \frac{1}{4} (-1 - 0) = -\frac{1}{4}$$
They are the same. :)

ianb
Wow. I'm embarrassed to say the least. You, sir, rock my world.


text:
anonymous 4 years ago Trigonometric Identities question: http://d.pr/i/FsnP I got my answer to be 3/2 but the mark scheme says -3/2.

1. .Sam.

$3\cos2 \beta+19\cos \beta+13=0$ $Use ~~~~\cos2\beta=2\cos^2\beta-1$ $3(2\cos^2\beta-1)+19\cos \beta+13=0$ $6\cos^2\beta-3+19\cos \beta+13=0$ $6\cos^2\beta+19\cos \beta+10=0$ $(2\cos \beta+5)(3\cos \beta+2)=0$ $\cos \beta=-5/2, \cos \beta =-2/3$

2. anonymous

Thanks. I made a silly mistake.


text:
# Water Quality Portal¶

[1]:

import contextily as ctx
import geopandas as gpd
from pygeohydro import WaterQuality
from shapely.geometry import Point

[2]:

SAVE_KWDS = {"bbox_inches": "tight", "dpi": 300, "facecolor": "w"}

The WaterQuality has a number of convenience methods to retrieve data from the web service. Since there are many parameter combinations that can be used to retrieve data, a general method is provided to retrieve data from any of the valid endpoints. You can use get_json to retrieve stations info as a geopandas.GeoDataFrame or get_csv to retrieve stations data as a pandas.DataFrame. You can construct a dictionary of the parameters and pass it to one of these functions. For more information on the parameters, please consult the Water Quality Data documentation. For example, let’s find all the stations within a bounding box that have Caffeine data:

[3]:

wq = WaterQuality()
stations = wq.station_bybbox((-92.8, 44.2, -88.9, 46.0), {"characteristicName": "Caffeine"})

Or the same criterion but within a 30 mile radius of a point:

[4]:

stations = wq.station_bydistance(-92.8, 44.2, 30, {"characteristicName": "Caffeine"})

[5]:

point = gpd.GeoSeries([Point(-92.8, 44.2)], crs="epsg:4326")
point = point.to_crs(point.estimate_utm_crs())
buff = point.buffer(30 * 1609.344)

ax = buff.plot(color="b", edgecolor="black", alpha=0.1, figsize=(6, 6))
point.plot(ax=ax, color="g", marker="x")
stations.to_crs(buff.crs).plot(ax=ax, color="r", marker="o", facecolor="none")
ax.set_axis_off()
ax.margins(0)
ax.figure.set_dpi(100)
ax.figure.savefig("_static/water_quality.png", **SAVE_KWDS)

Then we can get for al these stations the data like this:

[6]:

sids = stations.MonitoringLocationIdentifier.tolist()
caff = wq.data_bystation(sids, {"characteristicName": "Caffeine"})

[6]:


text:
Hey! I'm David, a security consultant at Cryptography Services, the crypto team of NCC Group . This is my blog about cryptography and security and other related topics that I find interesting.

# Hacking Week

## posted February 2014

A teacher from my uni (and who was teaching Programming last semester) is organizing a Hacking Week next week. Signs up are still possible there : http://hackingweek.fr/contestant/list/

It should be a Capture The Flag kind of contest. It should be interesting, although I'm going to ski with some friends so I won't be able to be really into it...

comment on this story

# Bitpay launches Bitcore

## posted February 2014

Bitcore seems to be an opensource node application that lets you deal with the bitcoin protocole easily (they give as exemple an function to validate a bitcoin address)

comment on this story

# Coinbase organizing a hackaton

## posted January 2014

So this guy owned @N on twitter and got extorted his account by a phishing attack. The story is well written and you should read it here : https://medium.com/p/24eb09e026dd

but for a tl;dr the attacker called his paypal account to ask them for his credit card's last 4 digits. Then he called godaddy to ask them to reset the password. They only asked him for the 2 first digits and the last 4s. The attacker just had to guess the 2 first digits (and he did it on the first try, he could have kept calling and trying otherwise).

Now that he had @N's domain's name, he could now see his emails. Took over @N's facebook account and started mailing him "threats".

It's pretty crazy how easy phishing is.

comment on this story

# Initial Permutations in DES

## posted January 2014

I have to code a whitebox using DES encryption in a class. Which is pretty cool (I would have prefered doing it with AES but the other group got tails and we got heads).

Here is where the Stanford course I passed on Coursera shines. The explanation of DES on it is brilliant. I was wondering about the initial and final permutations that occurs in the algorithm though and Dan Boneh doesn't really talk about it besides saying it's not for cryptographic purposes.

I found a solution on a new sub-stackoverflow dedicated to Cryptography : http://crypto.stackexchange.com/questions/3/what-are-the-benefits-of-the-two-permutation-tables-in-des

comment on this story

# VPN hacked and server used to mine bitcoins

## posted January 2014

That kind of stuff happens and it's always pretty hard to know it happened and how it happened.

Here's an article about a guy who doesn't seem to know much about security but does a fine job finding out what happened to him and what he can do to avoid future hacks.

comment on this story


text:
It has a charge of 1 and is written Cl.When an atom of sodium donates its sole valence electron to an atom of chlorine, the result-ing positive and negative charges pull both ions tightly together, forming an ionic bond (Figure 2.4c). Chlorine is a chemical element with the symbol Cl and atomic number 17. How to solve: How many electrons will chlorine gain or lose when it forms an ion? Wiki User Answered . A compound is formed when potassium reacts with chlorine, which forms ions with a negative charge. Use this information to w. Then the atom is called a positive ion. Ca atom loses two electrons and becomes Ca 2+ ion. The electronic configuration for the chloride anion is: d=D. 2010-12-11 15:34:00. it gains an electron. how many electrons does chlorine need to gain to become an ion? Ions. The covalent radius of a chlorine atom, for example, is half the distance between the nuclei of the atoms in a Cl 2 molecule. What kind of ions do a sodium atom and a chlorine atom become when a valence electron is transferred from one to the other. O B. An atom has an equal number of protons (+ charged) and electrons (- charged). Chlorine undergoes reduction when it forms the chloride ion (Cl-). 17 35.45 O A. Ions with a negative charge due to the gain of electrons are called anions. Chlorine has one of the highest electronegativity values of any of the elements which means that its electrons are held very tightly to the nucleus. The ions attach themselves to groups of water molecules. A chlorine atom has an atomic number of 17. Opposite charges attract one another while similar charges repel one another. See Answer. Chlorine is a yellow-green gas at room temperature. The size of an atom can be estimated by measuring the distance between adjacent atoms in a covalent compound. ... (such as chlorine). The sodium ion Na + is a sodium atom less one electron; the chlorine ion Cl − is a chlorine atom with one electron more than normal. Looking at Ions We've talked about ions before. Of the following atoms, ... How many single covalent bonds must a chlorine atom form to have a complete octet in its valence shell? The neutral chlorine atom becomes the chloride ion. • Chloride has achieved the Argon electron configuration, therefore, stable than the chlorine atom. Chlorine does not exist as an ion, perhaps you mean chloride ion. The sodium atom becomes a positive ion,, and the chlorine atom becomes a negative ion. In chemistry an atom of a particular element is said to be reduced when it gains an electron (from another atom). The atom that took the electron or 2 became a negatively charged ion. In the case of physical ionization in a fluid (gas or liquid), "ion pairs" are created by spontaneous molecule collisions, where each generated pair consists of a free electron and a positive ion. [2] The covalent radii of the main group elements are given in … Asked by Wiki User. • Chlorine is more chemically reactive than chloride because it is more unstable. Top Answer. An ion is an atom or a molecule carrying some specific charge due to loss or gain of electrons. Chlorine atoms gain one electron to become a chloride ion (Cl−). 17 35.45 o a. the atom gains 1 electron, to make a total of 21. o b. the atom gains 1 electron, to make a total of 18. o c. the atom loses - e-eduanswers.com c. to become more polar ... read more How Ions Affect Your Mood Does the wind put you in a bad mood. When an atom has an unequal number of protons and electrons it is an ion. Potassium loses one electron when it reacts with chlorine. Chloride is an ion of chlorine. 1. The Chloride Ion. a) gain an electron b) form a covalent bond c) lose an electron d) form a hydrogen bond Answer: a Difficulty: Medium Learning Objective 1: LO 2.1 Describe the structure of an atom and explain how its structure functions in elements and chemical bonds. 1. How does an atom become a positively charged ion? But on the other hand an atom is a neutral species therefore have no charge on them. Simultaneously, the chlorine atom, having gained an extra electron, will take on a negative charge and become a chlorine ion. Become a member and unlock all Study Answers. The formula of this compound would therefore be KCl. Potassium forms ions with a positive charge. Answer to Calculate the charge for a chlorine (CI) ion if a chlorine atom gains an electron to become an ion. Ions can be created by chemical reactions. Do you like rain. The atom gains 1 electron, to make a total of 18. b. to attain a noble-gas electron configuration. How does an atom of chlorine-37 become a chloride ion with a - 1 charge? Chlorine in its free form is very dangerous if you breathe the fumes or come in contact with the gas. Because the sodium ion has a positive charge, and the chlorine ion has a negative charge, they are attracted to each other, and form an ionic bond. Metal atoms lose electrons becoming positively charged ions. Reduction is the direct opposite of a reaction called oxidation where an atom loses or gives up an electron and is thus said to be oxidized. will it become a positively charged cation or a negatively charged anion? This electron is transferred to a chlorine atom to form a chloride ion. Because it has unequal numbers of protons and electrons, it’s now an ion. a. to become ions and attract each other. 8) In order to become an ion, an atom of chlorine must _____. Eg. The second-lightest of the halogens, it appears between fluorine and bromine in the periodic table and its properties are mostly intermediate between them. Na atom loses 1 electron and becomes Na+ ion. Hang on, “chlorine ion” is not correct. Chlorine The electron configuration of a chlorine atom is $[\mathrm{Ne}] … 01:17. Chlorine atom is a neutral species and has no charge on it. The process that changes a chlorine atom into a chloride ion is a reaction known as a reduction reaction. An atom that has more or fewer electrons in orbit than protons in its nucleus is called an ion.Once the electron from its valence shell has been transferred, the sodium atom will be missing an electron; it therefore will have a positive charge and become a sodium ion. • Chloride ion is attracted to positively charged electrodes or other positively charged chemical species, but chlorine does … … O C. The atom loses 1 proton, to make a total of 36. #"An ion is a species with an excess or deficiency of electrons. 75% of chlorine atoms have a mass number of 35, while 25% have a mass number of 37. 1 2 3. "# A given atom, say #"chlorine"# or #"sodium"# has a given number of protons, fundamental positively charged NUCLEAR particles, which is given by #Z#, the atomic number.For #Cl#, #Z=17#; for #Na#, #Z=11#.Chlorine atoms tend to gain electrons to form negatively charged ions with 18 electrons (the difference between … How does a chlorine atom become a chloride ion? Correct answer to the question How does an atom of chlorine-37 become a chloride ion with a -1 charge? Ions form salts, not molecules, but they perform many important functions in nutrition. Now, let's consider chlorine atom, Cl: 1s 2 2s 2 2p 6 3s 2 3p 5. There is a huge difference between a chlorine atom and a chloride ion. When it does, the sodium atom becomes a sodium ion with a charge of positive one (+1). Ions consisting of only a single atom are termed atomic or monatomic ions, while two or more atoms form molecular ions or polyatomic ions. Now consider the chlorine atom in sodium chloride. Under standard conditions, the element is a yellow-green gas. The chloride ion now has eighteen electrons and seventeen protons, so it's become a negative ion. I'm thinking it's either B or D but im not too sure. How many electrons does Li atom lose to become an ion? When an atom loses an electron or 2, it has more protons than electrons, and so more positive charges than negative. And because ions that have a negative charge are called anions, it’s now called the: When an electric field is applied, the sets of ions move in opposite directions, and since they are much more massive than electrons, the conductivity produced is markedly inferior to that in metals. So chlorine becomes an ion with a single negative charge. Or, it can gain electrons and become negatively charged (the number of electrons would be more than the number of protons giving it an overall negative charge). Essentially, when atoms gain or lose electrons, they become ions. The atom gains 1 electron, to make a total of 21. The atomic number of an element, also called a proton number, tells you the number of protons or positive particles in an atom.A normal atom has a neutral charge with equal numbers of positive and negative particles. 1 See answer Answer 5.0 /5 7. calipriester +9 ... 27 minutes ago There are two main isotopes of chlorine. For a chlorine atom to become a chloride ion, the chlorine atom must A. gain one electron B. lose one electron C. share an electron D. lose all its electrons which one is it? Chlorine is a naturally occurring element with a symbol Cl and atomic number 17. have a great day! This makes Chlorine atoms prone to accepting a loose electron to complete its outer shell and therefore form the highly stable Chloride ion, Cl- as shown to the right. At this point, chlorine has 17 protons (17 positive charges) and 18 electrons (18 negative charges). When a chlorine atom gains an electron, its outermost principal energy level achieves an octet. Chlorine, as mentioned above, desperately wants an electron so it can fill its outer electron level. It could be the ions in the air affecting your moods. A chlorine ion contains 17 protons, ... Atom: An atom is the smalled stable structure of matter. The neutral chlorine atom has acquired a negative charge by gaining an electron. An ion cannot remain in a free state, and has to combine with another When chlorine gains an electron to become a chloride ion with a 21 charge, ... Chlorine The electron configuration of a chlorine atom is$[\mathrm{Ne}] 3 \… 00:43. chlorine atom has become an anion (AN-ı¯-on), a negatively charged ion. However, after reaction with sodium, we have sodium chloride formed as the sodium atom gives up an electron to chlorine which accepts the electron to form the chloride anion. An ion is an atom which has an electrical charge, beacause it has either lost some of the electrons or received extra electrons (electrons are negatively charged). Only one more electron is needed to achieve an octet in chlorine’s valence shell. O D. The ionic form of chlorine is called a chloride ion. Sodium atoms have 11 electrons. Cl + e —-> Cl- To become the negatively charged chloride ion, the neutral chlorine atom needs to gain an electron from a reducing substance like sodium metal. Expert Answer 100% (2 ratings) Every atom wants to complete its octet i.e there must be eight electrons in its outermost shell. Now it's time to get down to basics. 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text:
tools:ase

# Atomic Simulation Environment

The Atomistic Simulation Environment (ASE) is a set of tools and Python modules for setting up, manipulating, running, visualizing and analyzing atomistic simulations. The code is freely available under the GNU LGPL license.

The ASE comes with a plugin, a so-called calculator, for running simulations with CP2K. The source code of the calculator is in the file ase/calculators/cp2k.py. It's documentation can be found here .

## Appetizing Example

The ASE provides a very convenient, high level interface to CP2K. It bring about the renown Pythonic productivity. Running a CP2K calculation can be as easy as this:

$export ASE_CP2K_COMMAND="mpirun -n 2 cp2k_shell.popt"$ python
>>> from ase.calculators.cp2k import CP2K
>>> from ase.build import molecule
>>> calc = CP2K()
>>> atoms = molecule('H2O', calculator=calc)
>>> atoms.center(vacuum=2.0)
>>> print(atoms.get_potential_energy())
-467.191035845

For more examples have a look at the test suite in ase/test/cp2k.

The environment variable $ASE_CP2K_COMMAND must point to the CP2K-shell binary. ## Remote Usage over SSH The ASE calculator communicate with the CP2K-shell solely over stdin/stdout pipes. Therefore, one can launch the CP2K-shell on a remote host via ssh. For this the environment variable $ASE_CP2K_COMMAND has to be setup accordingly:

export ASE_CP2K_COMMAND="ssh your-compute-host.com \"mpirun -np 2 /path/to/cp2k_shell.popt\""


text:
## Introduction

In the search for solid-state electrolytes (SSEs) that could replace the liquid organic electrolytes used today and improve safety in commercial Li-ion batteries1, a significant effort is being directed towards testing and improving ionic conductivity2,3,4,5, chemical and electrochemical stability6,7, and fast transport at the electrodes8,9. However, in order to address the manufacturing and the operando feasibility of all-solid-state batteries (ASSBs), a thorough understanding of the mechanical properties of the SSEs is also needed, and the mechanical stability of the electrolyte is a critical parameter for ASSBs10,11,12. First, the volume changes due to the storage of Li ions in the active materials can not be accommodated as in conventional liquid-electrolyte batteries, resulting in a considerable strain of the SSE and SSE/active-material composite during cycling10,13,14. This can easily turn out into mechanical degradation of the electrolyte, as observed for amorphous Li2S-P2S5 cycled with an Sn-anode15 or for β-Li3PS4 cycled with a Ni-rich NCM cathode16. Second, an SSE with high mechanical resistance would inhibit dendrite propagation and possibly enable the advent of ASSB technologies with superior energy density exploiting Li-metal anodes17.

The fracture toughness18,19,20,21,22, which quantifies the resilience of an SSE to be damaged under tensile stress, has been shown to be related to a high Young’s modulus23,24. On the other hand, the resistance of an SSE to dendrite growth has been related to a high shear modulus25, although quantification of this relationship is under debate26. Knowledge of the elastic behavior (bulk, shear, Young’s modulus, and Poisson’s ratio) of an SSE might thus be ground to predict its mechanical stability even outside the elastic regime. In this respect, softness has been historically regarded as a favorable property for the design of ASSBs technologies: soft materials, in particular sulfides, are easier to deform and thus expected to maintain good conformal contact with the electrodes10,27,28,29, allowing in addition for room-temperature pressure sintering10. However, a low stiffness of the SSE is not necessarily associated with good battery performance, as shown in the above-mentioned experiments on amorphous Li2S-P2S5 and β-Li3PS415,16 and through fracture toughness measurements on amorphous Li2S-P2S523. A theoretical continuum study based on non-linear kinematics models further shows that, perhaps counter-intuitively, compliant SSEs with Young’s moduli in the range of the sulfides (E = 15 GPa) are more prone to micro-cracking than typically brittle materials as oxide SSEs24. Elastic constants and conductivity are also closely related30, and enhanced vibrational31,32 and rotational33,34,35 degrees of freedom for the host lattice (typical of more compliant materials) have been shown to correlate with higher conductivities. Lattice-dynamics descriptors have been recently used to perform high-throughput screening of Li SSEs36,37, and conductivity has proven to increase in rotationally free SSEs, as is the case of the sulfide β-Na3PS438 or some LISICON oxides3. The relation between lattice softness and conductivity is nevertheless not straightforward, since the first usually lowers both activation barriers and jump frequencies11,39,40, which in turn have opposite effects on the latter, giving rise to the Meyer-Neldel rule (also called compensation rule39,40,41,42). Speed-of-sound measurements in halide-doped argyrodites39 and Sn-substituted Li10GeP2S1243 show that, below a certain threshold of optimal lattice softness (or maximum conductivity), stiffer materials are better conductors11.

Routinely, first-principles calculations of elastic tensors and elastic moduli of SSEs exploit static finite-strain methods28,44,45,46,47,48, that fit the total energy49,50,51,52 or the stress53,54 with respect to strain, with energies and stresses obtained from DFT static calculations at different applied strains. Whereas these approaches are a powerful tool for the calculation of the elastic constants of ordered intermetallic alloys49,55, and obviously of single crystals with a defined structure51,52, they are in principle not appropriate for superionic materials, where the dynamical disorder plays a significant role and the standard picture of atoms vibrating around fixed equilibrium positions is not valid anymore56. In this respect, extracting the elastic tensor from the quasi-harmonic vibrational free energy57 under finite-strain deformations58,59 would not improve the picture, as it still assumes the existence of well-defined reference positions around which the atoms vibrate, which is clearly not the case for SSEs56. In a seminal paper, Parrinello and Rahman60 combined constant stress simulations61,62 with the thermodynamical fluctuation theory of the strain63 in order to estimate the elastic stiffness tensor from molecular dynamics. This method, which in the remainder of this paper will be referred to as the “strain-fluctuation method", builds on the knowledge of the dynamics and on the statistical convergence of the strain fluctuations over the simulation time64, and presents the important advantage of considering all the statistically relevant configurations of the atoms at a given temperature and pressure; it is therefore particularly appealing for superionic materials.

It is the purpose of this work to apply the strain-fluctuation method60 to the calculation of the elastic tensors and moduli of SSEs, choosing two benchmark systems, namely Li10GeP2S1244,65,66 and its oxide counterpart Li10GeP2O1267,68,69. For this, extensive and accurate first-principles molecular dynamics simulations are performed in the isothermal-isobaric ensemble at T = 600 K and P = 0, with a Nose-Hoover thermostat70 and a Parrinello-Rahman barostat61,71 using Car-Parrinello molecular dynamics72. The elastic tensors and moduli are extracted from the dynamical covariance of the strain over the trajectory, and a block analysis is provided to estimate statistical errors73. From the knowledge of the errors and the analysis of the elastic moduli convergence over the trajectories, we estimate the simulation length which is needed to produce reliable results. 0K and room-temperature moduli are extracted from NPT simulations at different temperatures using Wachtman’s law74,75,76. Finally, in order to compare the strain-fluctuation method with the static methods alluded to above, we provide additional static calculations, in which we distort fully relaxed snapshots sampled from the molecular dynamics trajectories fitting the Murnaghan equation of state (EOS) or the stress vs strain relation to obtain the elastic constants51.

The paper is organized as follows. In Section Results “Elastic tensors and moduli from the strain fluctuations" we discuss the isobaric-isothermal cell dynamics and we present the results for the elastic tensors and moduli of Li10GeP2S12 and Li10GeP2O12 obtained from the strain-fluctuation method. For comparison, in Section Results “Elastic tensors and moduli from static methods" we present the results for the elastic tensors and moduli obtained from static calculations, using fully relaxed snapshots from the molecular dynamics simulations. The main results of this paper are discussed and summarized in the Discussion.

## Results

### Elastic tensors and moduli from the strain fluctuations

We simulate two superionic materials, Li10GeP2S12 (LGPS) and Li10GeP2O12 (LGPO), each in two phases, namely the quasi-orthorhombic69,77 and tetragonal65 phases for LGPS (that we call o-LGPS and t-LGPS, respectively69), and the orthorhombic67,68 and tetragonal69,78 phases for LGPO (that we call o-LGPO and t-LGPO, respectively69). We use the supercells reported in refs. 69 and 79, having 100 and 50 atoms for the o- and t- structures, respectively (see Supplementary Figure 1 for a description of the structures). We recall here that, while t-LGPS, o-LGPS, and o-LGPO are existing compounds65,68,77, t-LGPO is a hypothetical one69,78,80, that we include in this work following a recent first-principles investigation highlighting its high Li-ion conductivity69. We use Car-Parrinello (CP) molecular dynamics72, based on Kohn-Sham density-functional theory (DFT)81,82 in the plane-wave pseudopotential formalism83,84, as implemented in the cp code of the QUANTUM ESPRESSO distribution85, in the isobaric-isothermal ensemble (NPT) with a Nose-Hoover thermostat86 and a Parrinello-Rahman barostat61,62. We use the Perdew-Burke-Ernzerhof (PBE) generalized-gradient approximation (GGA) functional87 and sample the Brillouin zone at the Γ point, as in refs. 69,78 (for an extensive review on the performance of PBE and other DFT functionals, see, e.g., ref. 88). The NPT CP method and the numerical details of these simulations are described in the Supplementary Methods. We test the reliability of the k-point sampling for the stress tensor, and show that the pressure and the off-diagonal elements of the stress tensor remain within ~10−2 GPa when calculated with an unshifted (2, 2, 2) k-point grid89 compared to Γ-only sampling (see Supplementary Tables 7 and 8). The above test also helps discussing possible finite-size effects in the MD simulations90. For these, we also study (at 600 K, with NPT CP dynamics, for ~70 ps) a 100-atom 2 × 1 × 1 supercell of t-LGPS, also sampled at Γ, and we extract the elastic moduli: comparison with the analogous results for the 1 × 1 × 1 cell shows that increasing the size of the simulation cell does not alter the elastic moduli significantly (see Supplementary Figure 10). In Supplementary Figures 3 and 4, we also report results for molecular dynamics runs with different barostat masses, showing that changing the barostat mass in a range of values around the standard theoretical suggestion70,85 does not change the cell fluctuations. The choice of the thermostat’s mass is already discussed in the Supplemental Material of ref. 69.

In Fig. 1 we report the values of the cell edges a, b, and c of the four structures during a 600 K-NPT trajectory. While for t-LGPS, o-LGPS, and o-LGPO (Fig. 1a, b, d, respectively) these oscillate, for t-LGPO (Fig. 1c) a and b can swap during a fluctuation (as already reported in69). In Supplementary Figures 2 and 5 we also report the cell angles α, β, and γ from the same simulations, and the six components of the Voigt strain vector ϵ (see Equations (4)−(6) in the Methods Section), respectively. We recall here that h and ϵ in Equations (4)−(6) are time-dependent quantities. The diagonal elements of the strain-dynamical-covariance matrix (see also Equation (10) in Methods Section)

$$\left\langle {{\Delta }}{\epsilon }_{i}{{\Delta }}{\epsilon }_{j}\right\rangle =\left\langle {\epsilon }_{i}{\epsilon }_{j}\right\rangle -\left\langle {\epsilon }_{i}\right\rangle \left\langle {\epsilon }_{j}\right\rangle$$
(1)

are reported in Table 1. Larger fluctuations of the cell parameters translate into larger oscillations of the strain components, and in turn into larger values of $$\left\langle {{\Delta }}{\epsilon }_{i}{{\Delta }}{\epsilon }_{i}\right\rangle$$ (Table 1): an example is given by o-LGPS vs o-LGPO.

To show the superionic character of these materials, in Fig. 2 an isosurface of Li-ion probability density (ρ(Li) = 8 × 10−2 Å−3) is displayed for t-LGPS from the 600 K-NPT CP dynamics (we use the implementation in https://github.com/lekah/samos, see also91). In Supplementary Figure 6, we report analogous isodensity plots for o-LGPS, t-LGPO, and o-LGPO.

The strain-dynamical-covariance matrix (Equation (1)) determines the stiffness and compliance elastic tensors from Equations (9) and (10) in the Methods Section, respectively. From the tensors, we obtain the moduli B, G, E, and ν following the Voigt-Reuss-Hill (VRH) approximation (see Methods Section). However, being the tensors statistical quantities, we need first to determine their statistical uncertainties over the trajectory length, and estimate sufficient trajectory lengths to give meaningful results from Equations (9) and (10). We compute the statistical errors on $$\left\langle V\right\rangle$$, $$\left\langle {{{\boldsymbol{\epsilon }}}}\right\rangle$$, and $$\left\langle {{{\boldsymbol{\epsilon \epsilon }}}}\right\rangle$$ in Equations (9) and (10) through a block analysis, i.e., by calculating the variance of the mean of a subset of data (blocks) of the whole set. A careful evaluation of the correct number of blocks is mandatory, so as to avoid error overestimation (too few uncorrelated data) or underestimation (correlated data)73. For each run, we set this number by performing a systematic partitioning of the trajectory in increasing number of blocks up to a maximum number, and by selecting the region where the variance of the mean is stable over the number of blocks69,73. This procedure is illustrated for the 600 K-NPT CP dynamics of t-LGPS in Fig. 3, where we report the relative standard error (square root of the variance of the elastic modulus divided by the value of the mean) for B, G, E, and ν (obtained from error propagation, see Methods Section) as a function of the number of data in block chosen for the block analysis of $$\left\langle V\right\rangle$$, $$\left\langle {{{\boldsymbol{\epsilon }}}}\right\rangle$$ and $$\left\langle {{{\boldsymbol{\epsilon \epsilon }}}}\right\rangle$$. By increasing the number of blocks (going right to left in the plot), the variance oscillates less strongly, and reaches a region of stability (here at ~ 2000 data in each block), after which it decreases monotonically (correlated data). This plateau determines the proper number of data in block73 and thus the error. Analogous plots for the evaluation of the number of blocks for o-LGPS, t-LGPO, and o-LGPO are reported in Supplementary Figure 7. In turn, from these trajectories, we identify 47 uncorrelated blocks for t-LGPS (each block ~4 ps), 41 uncorrelated blocks for o-LGPS (each block ~4 ps), 16 uncorrelated blocks for t-LGPO (each block ~10ps), and 31 uncorrelated blocks for o-LGPO (each block ~5 ps). Next, we study the convergence of the elastic moduli on the simulation time. For each system we perform several calculations of the elastic moduli, each of them using only a part of the whole trajectory simulated, corresponding to n = 2, . . N blocks, N being the number of blocks that we have chosen for the whole trajectory (see above), and each block having the same length as determined from the block analysis on the whole trajectory. Then, for each of these calculations we obtain the standard errors of the moduli from the variance over the blocks, since these blocks are already uncorrelated and there is no need to repeat a block analysis for each of these calculations. The values of the moduli as a function of the simulation time, together with the related standard errors are reported in Fig. 4 for t-LGPS, whereas for the remaining three structures they are reported in Supplementary Figure 8. In Fig. 4, the errors of B, G, E, and ν decrease, and their absolute values converge, while increasing the simulation time. A similar behavior is shown by o-LGPS and o-LGPO (Supplementary Figures 8a, c). For t-LGPO (Supplementary Figure 8b) we have a totally different scenario: the moduli are almost independent of the length of the trajectory, and their relative standard errors are very large, ranging from 10% (Poisson’s ratio) to 60% (Young modulus). In addition, the values of the moduli for t-LGPO are significantly lower than the respective moduli for o-LGPO (Supplementary Figures 8b and 8c). The behavior of t-LGPO is directly related to the cell dynamics of t-LGPO reported in Fig. 1, where a and b oscillate and swap: these fluctuations give rise to a material that would seem more compliant, but they would disappear if larger supercells were used. For this reason, we investigate a 4-times larger 2 x 2 x 1 supercell, already used in69, that we simulate for 100 ps: in Supplementary Figure 9 and Supplementary Table 6 we show that for this larger supercell the oscillations of a and b are considerably diminished, and the moduli higher. However, the statistical uncertainties remain high, and we conclude that for this material, where these two alternate phases can swap, even larger supercells should be used. For the remaining three structures (Fig. 4 and Supplementary Figures 8a, 8c), B, G, E, and ν can be considered reasonably converged from the reported simulations at 600 K for simulation lengths of the order of 150−200 ps.

The converged elastic tensors with errors, obtained from NPT (T = 600 K, P = 0) CP molecular dynamics through the analysis presented above, are reported in Table 2 for t-LGPS, o-LGPS, and o-LGPO. The space-group features of the tensors92 can be deduced: in particular, for both tetragonal and orthorhombic structures, the components in the off-diagonal blocks are  0, apart from c15 in the o-LGPO tensor, suggesting a certain monoclinic degree92 in this structure. For t-LGPS we observe c44c55, c11c22 and c13c23. In Table 2 we also report the moduli (Voigt and Reuss bounds, and VRH average) with their statistical uncertainties. We note that the statistical uncertainties on the moduli are in general within ~2−8%.

Knowledge of the temperature dependence of the elastic moduli can provide room-temperature predictions to be compared with the experimental literature. To this end, we perform NPT CP molecular dynamics simulations at additional temperatures for t-LGPS and o-LGPO, from which we extract the elastic moduli, as described above for the simulations at 600 K. From these simulations we also extract the temperature dependence of the lattice parameters, that we report in Fig. 5, together with the linear fits, for t-LGPS (see Supplementary Figure 11 for o-LGPO). In Fig. 6, the values of B, G, E, and ν of t-LGPS as a function of temperature (T = 400, 500, 600, 700, and 800 K) are reported, together with a fit to the Wachtman’s law74,75,76:

$$M(T)={M}_{0}-\alpha T\exp \left(-\frac{{T}_{0}}{T}\right),$$
(2)

where $$M=\left\{B,G,E,\nu \right\}$$, M0 are the moduli at 0 K, and the meaning of the remaining parameters α and T0 is explained in74,75,76. This equation gives M = M0 at 0 K, approaching this value with a zero slope as required by the third law of thermodynamics74,75,76, and a linear dependence at high temperatures, as $$\exp (-{T}_{0}/T)$$ approaches unity. Although it was originally tested74 for some oxide compounds, it has been shown to be correct also for nonoxide solids76. An analogous plot for o-LGPO (T = 600, 800, 1000 K) is reported in Supplementary Figure 12. In Table 3 we report the extrapolated bulk, shear, Young’s moduli, and Poisson’s ratio at 0 K and 300 K for t-LGPS. For a non-quantitative reference, we also report the experimental room-temperature moduli of related sulfide glassy electrolytes, measured through ultrasonic pulse echo methods93,94. For o-LGPO, we report in Table 3 the values of the moduli at T = 300 K, that can serve as a reference for further studies, since, to date, no experimental reports on the elastic properties of this material are available.

### Elastic tensors and moduli from static methods

As a basis of comparison with the strain-fluctuation method, we provide here results from the static methods (outlined in the Methods Section) aiming to quantify the relevance of the dynamic nature of the elastic response of these materials. We focus on the benchmark material t-LGPS, for which results from computational static methods28,44,45 are also available in the literature, and a relevant number of experimental works investigate the mechanical properties of the class of superionics to which it belongs23,26,93,94,95,96. All the calculations are done both on nine uncorrelated snapshots of the 600 K-NPT CP molecular dynamics, that are previously fully relaxed, and on a global minimum energy structure that was obtained in ref. 78 through an electrostatic energy criterion, that is also previously fully relaxed. In order to compare the dynamics-based strain-fluctuation method with a static method at the same DFT accuracy, we perform here DFT calculations with the same supercell, pseudopotentials, DFT functional, cutoff energy, and k-point sampling as in the 600 K-NPT CP molecular dynamics simulations (see previous section, and the Supplementary Methods), and we relax the internal coordinates following a Broyden-Fletcher-Goldfarb-Shanno algorithm85, with, as convergence criteria, an energy difference between two consecutive steps below 2 × 10−4 a.u. and single components of the forces on the ions below 2 × 10−3 a.u.

We apply a small hydrostatic strain (Equation (14) in the Methods Section) to each given structure of t-LGPS, calculate the energy, and fit to a Murnaghan EOS (Equation (15) in the Methods Section85,97,98), from which we extract V0, B0, and $${B}_{0}^{{\prime} }$$. In Fig. 7 we report the energy-volume relations together with the fits, both for the nine snapshots and for the global minimum energy structure. In all these calculations, all the atoms are let free to relax during the expansion/compression, as explained above. However, in order to quantify the influence of the internal coordinate relaxation, we also perform the energy-volume calculations on the global minimum energy structure by keeping the atoms fixed at their equilibrium positions: we report the corresponding fit from these calculations, compared to the case with internal coordinate relaxation, in Supplementary Figure 13. The bulk modulus from the EOS is reported in the first line of Table 4 (B*(EOS)) for the unrelaxed calculations, and in the second line (B(EOS)) for the relaxed calculations. This comparison reveals that the material is about twice as stiff when the internal coordinates of the atoms are not free to relax. As for the energy-volume curves and bulk moduli obtained from the snapshots configurations (Fig. 7 and Table 4), the nine uncorrelated snapshots from the t-LGPS dynamics reveal a rather wide range of values (~15% of the average value) for the t-LGPS bulk modulus from the EOS, corresponding to rather different energy-volume curves. This result is not unexpected, as superionic materials can have different stable structures, mainly depending on which sites are populated by the mobile Li ions in each of them. Since there are 32 sites for 20 Li ions in one 50-atom supercell99, the number of such structures is huge (~108), and it would not be possible to assign a proper statistical uncertainty to the moduli from the static methods in Table 4 from a block analysis, as done for the strain-fluctuation method in the previous section. However, from the block analysis of the previous section, we know that, by considering 9 uncorrelated snapshots (i.e., separated at least by the length of one block, which is ~4 ps from Fig. 3), we are not underestimating the statistical uncertainty of the static moduli in Table 4. Thus, sampling of the snapshots from the 600 K-NPT CP molecular dynamics is performed here as a way to generate uncorrelated t-LGPS configurations, and not as a rigorous statistical tool, which would be inconsistent as the energy-volume calculations are at 0 K. Finally, we note that the bulk modulus for the global minimum energy structure78 (last column in Table 4) is very similar to the average value obtained from the fully relaxed snapshots.

In order to calculate also G, E, and ν, we apply uniaxial strains to extract the 36 stiffness coefficients of the elastic tensor (Equations (18) and (19) in the Methods Section). For these calculations, we tighten the convergence criteria to energy differences between two consecutive steps below 2 × 10−5 a.u. and single components of the forces on the ions below 2 × 10−4 a.u. In Fig. 8 we report the components c11, c12, c13, c33, c44, and c66 of the stiffness tensor for the nine fully relaxed snapshots considered. The resulting moduli from Equations (20)−(22) are reported in Table 4, together with the moduli obtained from the same stress–strain calculations on the global minimum energy structure of ref. 78, here fully relaxed. In Supplementary Tables 25 we report the full elastic tensor and Voigt-Reuss bounds for each relaxed snapshot configuration, and for the global minimum energy structure. For the snapshot configurations, similar considerations as for the bulk modulus from EOS can be drawn for the moduli from the stress–strain relations: we find values of each elastic modulus distributed over a range of about 15% of the average value. As noted for the calculation of the bulk modulus from the EOS, the moduli obtained from the global minimum energy structure78 (last column in Table 4) do not give additional relevant information, being in the range of the moduli from the fully relaxed snapshots. This result can be understood as a demonstration that the global minimum energy structure and the energy-minimized snapshots are simply different energy-minimized configurations, each with a potentially different Li occupation of the available sites.

## Discussion

This paper provides reference results for the elastic constants and moduli of two benchmark oxide and sulfide SSEs from the strain-fluctuation method60. From Tables 2 and 3, the oxide o-LGPO67,68,100 (whose elastic properties were experimentally and computationally unreported, to date) is predicted to be significantly stiff (B = 44.1 GPa, G = 25.4 GPa, E = 63.9 GPa), and ~3 times stiffer than the corresponding sulfide. This result is compatible with the available results for the garnet Li7La3Zr2O12 (B ~ 100 GPa, G ~ 60 GPa, E ~ 150 GPa47) and the NASICON Li1.2Zr1.9Sr0.1(PO4)3 (E ~ 40 GPa101), showing a superior stiffness of oxide SSEs as compared to sulfide SSEs (see Table 3 for Li3PS493 and LGPS94). However, these results do not univocally determine the relative performances of oxides and sulfides in ASSBs. Even though a large shear modulus has been historically believed to prevent Li penetration through the electrolyte25,102, a SSE that possesses a high G of ~60 GPa, LLZO, has recently proven to suffer from dendrite propagation26. Similarly, despite a low Young’s modulus is usually believed to ensure stress–strain-accommodation ability27,102, a compliant SSE with a low E of ~18 GPa, Li3PS4, has recently appeared to be more prone to micro-cracking than stiffer electrolytes23,24. A major goal for the scientific community would be to determine which balance of low Young’s modulus and high shear modulus leads to the best performance in reducing dendritic growth and interface resistivity. To this end, our results for o-LGPO seem to show a common trend with other oxide superionics, such as LLZO47 and Li2O-ZnO-B2O glasses103, where E ~ 1.5B and B ~ 1.5G (Table 3), differently from sulfide materials, for which this trend is not reported (this work and refs. 28,93,94), and in principle G and E might be more easily tuned to reach such ideal balance. Moreover, from the moduli, we can estimate the ductility, usually related to the ratio B/G104 (also called the Pugh’s ratio105), which quantifies the ability of a material to resist volume changes against shape changes. The B/G ratio for o-LGPO (1.7, Table 3) is in line with the one of another oxide SSE, the garnet LLZO47. The ductility of t- and o-LGPS is considerably higher (>2, Table 3), in line with the above-mentioned experiments for glassy solid-state sulfides93,94,102. A close comparison with the experimental literature (refs. 27,93,94,102 and Table 3) shows a good agreement with ref. 94 for the values of G and E and a fair agreement for the values of B and ν (Table 3). However, we recall that the experimental values of ref. 94, also reported in Table 3, refer to the elastic moduli of glassy Li2S − P2S5 and Li3PS4 − Li4GeS427,93,94, and no experimental investigations of the elastic properties of single-crystal or polycrystalline LGPS are available to date. In addition, ref. 93 shows that the experimental moduli are very sensitive to the molding condition, being higher for higher molding temperature and pressure, and covering a wide range of values (e.g., for the composition 75−30 of Li2S − P2S5, see Table 3, B = 12.5−21.3 GPa, G = 5.9−8.7 GPa, and E = 15−23 GPa93). For the glassy Li3PS4 − Li4GeS4, since the hot-pressed pellets showed higher Li density and ionic conductivity106, the elastic moduli were measured only after hot pressing94, so that ref. 94 (cf. Table 3) reports only the upper limit of the experimental moduli for these glassy LGPS samples. Finally, it would be interesting to clarify the connections between hot- and cold-pressed glass structures, pure crystals, and elastic moduli.

A more theoretical purpose of this work is to compare the strain-fluctuation method exploiting NPT molecular dynamics trajectories (Tables 2 and 3) with static methods applied to fully relaxed snapshots extracted from the same trajectories. We purposely use here the same DFT machinery (DFT functional, pseudopotentials) and choose the same accuracy (k-point sampling, supercell size) in both methods. Moreover, we do not aim here to establish the accuracy of the DFT functionals45,88, but rather to compare the accuracy of static and dynamic methods. Such a comparison, that we perform for t-LGPS (Table 4), is expected to shed light on the role of the many statistically accessible configurations in the determination of the elasticity in superionic materials, and to clarify the need for a dynamical treatment in the description of their elastic response. First, Table 4 shows that the static methods give rather different values for the moduli, depending on the choice of the configuration to be strained. This result confirms that the different stable structures of superionic materials, which are dependent on the sites populated by the mobile Li ions, give rise to statistically different elastic properties, and is in principle not correct to use as a reference only one of these structures44,45,107. Furthermore, we find a marked disagreement between the average moduli from the static calculations (Table 4) and the 0 K extrapolation of the moduli from the strain fluctuations (Table 3), in general, the first exceeding the latter by a considerable amount. The static calculations performed on the global minimum structure78 are in line with those on the fully relaxed snapshots. The overestimation of the moduli from the static methods with respect to the strain-fluctuation method can be explained by the ability of the latter to capture the elastic response of superionic materials, where the non-mobile sublattice responds elastically as in a proper solid, whereas the mobile sublattice behaves inelastically as in a liquid. Conversely, the static methods assume the presence of an elastic response from all the atoms in the material, predicting in turn a stiffer material than it is in reality. For t-LGPS, our results show that such overestimation amounts to ~25–50%, so a proper statistical dynamic treatment of the elastic properties of this material should be desirable. Overall, we show that the elastic response of the material is systematically stiffer when going from a dynamic method (strain-fluctuation), to a static method (EOS or stress–strain) performed by relaxing the internal coordinates, to a static method (EOS) performed at fixed internal coordinates. Thus, relaxing the internal coordinates when performing a static calculation for the elastic moduli is beneficial for these materials (we show that otherwise the bulk modulus would be ~100% stiffer), but is not enough when compared to the correct dynamical treatment (since the moduli are still ~25–50% stiffer than the dynamically obtained ones).

In summary, although superionic conductors are dynamically disordered materials, for which a well-defined microscopic reference configuration does not exist56, computational studies of their elastic properties usually rely on static methods28,44,45,46,47,48, where strains are applied to a chosen ionic configuration (in general, one of the many possible fully relaxed structures starting from molecular dynamics trajectories or educated guesses from the experimental structures), and the resulting energies or stresses are calculated. Such methods neglect the quasi-liquid motion of these materials, which can instead be captured by turning to statistical methods, based on the sampling of the whole configuration space with molecular dynamics. First, we provide a computational study of the elastic moduli of superionic conductors with such a dynamical approach, sampling strain fluctuations60 with first-principles molecular dynamics71,85, followed by an accurate block analysis of the errors. Choosing two benchmark crystalline superionic conductors, LGPS65 and LGPO68, we show that an affordable computational effort is sufficient (~180 ps trajectories) to obtain converged moduli and statistical errors of reasonable size; the calculated moduli agree with the existing experimental literature for similar glassy materials. Second, we compare the strain-fluctuation method with standard static methods for the benchmark t-LGPS. By applying the same static methods to different fully relaxed structures from the molecular dynamics, both with and without internal coordinates relaxation, we note that: (i) static methods predict a material unrealistically stiff when unrelaxing the internal coordinates; (ii) static methods still overestimate the moduli with respect to the correct dynamical treatment by ~25–50% when relaxing the internal coordinates; (iii) static methods have an intrinsic variance up to ~2–3%, and an overall spread of ~15% on the average value. These results argue for the importance of dynamical sampling to address elastic properties of superionic conductors, and provide a computational reference for the community, given that no experimental reports on crystalline LGPS and no experimental or computational reports on LGPO are available to date. Given the growing interest in the mechanical properties of superionic conductors for all-solid-state-battery technologies, justified by the urgency of controlling the manufacturing of these materials and the mechanical phenomena taking place in the electrochemical cell upon cycling (e.g., dendrite propagation through the electrolyte, or formation of interface strains due to volume changes or ionic transport10,13,14,17), we believe that further computational and experimental investigations are warranted. Last, whilst elastic properties, like compliance vs. stiffness or ductility vs. brittleness, are governed by the elastic moduli (bulk, shear, Young’s modulus, and Poisson’s ratio), even mechanical properties outside the elastic regime, such as fracture toughness24,104, fragility104, shear strength108, shear viscosity104 or hardness109, can often be predicted starting from the elastic moduli, underscoring the importance of accurate measurements or predictions.

## Methods

### Elastic tensors from strain fluctuations, and statistical errors

We present here the formalism to calculate elastic tensors from the molecular dynamics simulations according to the strain-fluctuation method60, and the derivation of the statistical errors from the dynamics. The strain-stress relation can be recast in terms of the fluctuations of the strain in a constant stress ensemble60:

$${S}_{\alpha \beta \mu \nu }={\left(\frac{\partial {\epsilon }_{\alpha \beta }}{\partial {\sigma }_{\mu \nu }}\right)}_{T}=\frac{\left\langle V\right\rangle }{{k}_{{{{\rm{B}}}}}T}{\left\langle {{\Delta }}{\epsilon }_{\alpha \beta }{{\Delta }}{\epsilon }_{\mu \nu }\right\rangle }_{N,\sigma ,T}.$$
(3)

In Equation (3), Sαβμν is the αβμν component of the isothermal compliance tensor60 (inverse of the isothermal stiffness tensor {Cαβμν}), where ϵαβ and σμν are the strain and stress tensors, respectively, and the Greek indices αβμν cover the cartesian coordinates in three dimensions. T and $$\left\langle V\right\rangle$$ are the temperature and average volume of the system in the constant stress and constant-temperature ensemble (NσT), $$\left\langle ..\right\rangle$$ is an ensemble average, and Δ is a deviation from the mean value, i.e., $${{\Delta }}{\epsilon }_{\alpha \beta }={\epsilon }_{\alpha \beta }-\left\langle {\epsilon }_{\alpha \beta }\right\rangle$$. Equation (3) can be derived from statistical thermodynamics through the theory of fluctuations in various ensembles63,110,111. The strain tensor in Equation (3) can be calculated from the instantaneous and average cell matrices via the expression60:

$${{{\boldsymbol{\epsilon }}}}=\frac{1}{2}({\left\langle {{{\bf{h}}}}\right\rangle }^{T,-1}{{{\mathcal{G}}}}{\left\langle {{{\bf{h}}}}\right\rangle }^{-1}-1),$$
(4)

where $${{{\mathcal{G}}}}={{{{\bf{h}}}}}^{T}{{{\bf{h}}}}$$ is the metric tensor (cf. Supplementary Methods) and h the instantaneous cell matrix in the triangular superior form112:

$${{{\bf{h}}}}=\left(\begin{array}{ccc}\parallel {{{\bf{a}}}}\parallel &\parallel {{{\bf{b}}}}\parallel \cos \gamma &\parallel {{{\bf{c}}}}\parallel \cos \beta \\ 0&\parallel {{{\bf{b}}}}\parallel \sin \gamma &({{{\bf{b}}}}\cdot {{{\bf{c}}}}-{b}_{x}\cdot {c}_{x})/{b}_{y}\\ 0&0&{(\parallel {{{\bf{c}}}}{\parallel }^{2}-{c}_{x}^{2}-{c}_{y}^{2})}^{\frac{1}{2}}\end{array}\right)$$
(5)

where a, b, c and α, β, γ are the cell edges and angles, respectively. Equations (3)−(5) can be used to calculate the isothermal stiffness coefficients from molecular dynamics or Monte Carlo simulations from the strain fluctuations at fixed stress64,112,113,114,115.

In the Voigt notation92,116, thanks to symmetry, the stress and strain tensors can be represented as one-dimensional arrays with six components. The strain is:

$${\epsilon }_{1}={\epsilon }_{xx},{\epsilon }_{2}={\epsilon }_{yy},{\epsilon }_{3}={\epsilon }_{zz},{\epsilon }_{4}=2{\epsilon }_{yz},{\epsilon }_{5}=2{\epsilon }_{zx},{\epsilon }_{6}=2{\epsilon }_{xy},$$
(6)

and the stress–strain relation is simplified, with the stiffness tensor becoming a (6 × 6) matrix117:

$${\sigma }_{i}={c}_{ij}{\epsilon }_{j},$$
(7)

so that Equation (3) becomes:

$${c}_{ij}={\left(\frac{\partial {\sigma }_{i}}{\partial {\epsilon }_{j}}\right)}_{T}=\frac{{k}_{{{{\rm{B}}}}}T}{\left\langle V\right\rangle }{\left\langle {{\Delta }}{{{\boldsymbol{\epsilon }}}}{{\Delta }}{{{\boldsymbol{\epsilon }}}}\right\rangle }_{ij}^{-1},$$
(8)

where $$\left\langle {{\Delta }}{{{\boldsymbol{\epsilon }}}}{{\Delta }}{{{\boldsymbol{\epsilon }}}}\right\rangle$$ is the dynamical covariance of ϵ, and the ensemble average is intended to be in the (NσT) ensemble. Equation (8) can also be written as:

$${c}_{ij}=\frac{{k}_{{{{\rm{B}}}}}T}{\left\langle V\right\rangle }{\left[\left\langle {{{\boldsymbol{\epsilon }}}}{{{\boldsymbol{\epsilon }}}}\right\rangle -\left\langle {{{\boldsymbol{\epsilon }}}}\right\rangle \left\langle {{{\boldsymbol{\epsilon }}}}\right\rangle \right]}_{ij}^{-1}.$$
(9)

In the Voigt notation, adopted throughout the remainder of this paper, the compliance reads:

$${s}_{ij}=\frac{\left\langle V\right\rangle }{{k}_{{{{\rm{B}}}}}T}\left[\left\langle {\epsilon }_{i}{\epsilon }_{j}\right\rangle -\left\langle {\epsilon }_{i}\right\rangle \left\langle {\epsilon }_{j}\right\rangle \right].$$
(10)

Next, we calculate the errors on the statistical quantities $$\left\langle V\right\rangle$$, $$\left\langle {{{\boldsymbol{\epsilon }}}}\right\rangle$$, and $$\left\langle {{{\boldsymbol{\epsilon \epsilon }}}}\right\rangle$$ by performing a block analysis, as reported in Section Results “Elastic tensors and moduli from the strain fluctuations". Then, we propagate these errors (Var(V), Var(ϵ) and Var(ϵϵ)) to the compliance of Equation (10):

$$\begin{array}{l}{{{\rm{Var}}}}({s}_{ij})={\left(\frac{\left\langle V\right\rangle }{{k}_{{{{\rm{B}}}}}T}\right)}^{2}\left[{{{\rm{Var}}}}({\epsilon }_{i}{\epsilon }_{j})+\left\langle {\epsilon }_{i}\right\rangle {{{\rm{Var}}}}({\epsilon }_{j})\right.\\\qquad\qquad \quad\,\,+\,\left.\left\langle {\epsilon }_{j}\right\rangle {{{\rm{Var}}}}({\epsilon }_{i})\right]+{\left(\frac{{s}_{ij}}{\left\langle V\right\rangle }\right)}^{2}{{{\rm{Var}}}}(V)\end{array}$$
(11)

and we obtain the error on the stiffness, Var(c), by the following:

$$\begin{array}{ll}{{{\rm{Var}}}}({{{\bf{c}}}})&={{{\rm{Var}}}}({{{{\bf{s}}}}}^{-1})={d}^{2}\left({{{{\bf{s}}}}}^{-1}\right)={\left({{{{\bf{s}}}}}^{-1}d{{{\bf{s}}}}{{{{\bf{s}}}}}^{-1}\right)}^{2}={\left\{{{{{\bf{s}}}}}_{ij}^{-1}d{s}_{jk}{{{{\boldsymbol{s}}}}}_{km}^{-1}\right\}}^{2}\\& =\left\{{{{{\bf{s}}}}}_{ij}^{-1}d{s}_{jk}{{{{\bf{s}}}}}_{km}^{-1}{{{{\bf{s}}}}}_{il}^{-1}d{s}_{ln}{{{{\bf{s}}}}}_{nm}^{-1}\right\}\\ &=\left\{{{{{\bf{s}}}}}_{ij}^{-1}{{{{\bf{s}}}}}_{ij}^{-1}d{s}_{jk}d{s}_{jk}{{{{\bf{s}}}}}_{km}^{-1}{{{{\bf{s}}}}}_{km}^{-1}\right\}+{\left\{o\left(d{s}_{jk}d{s}_{ln}\right)\right\}}_{j\ne l,k\ne n}\\ &=\left\{{({{{{\bf{s}}}}}_{ij}^{-1})}^{2}{{{\rm{Var}}}}({s}_{jk}){({{{{\bf{s}}}}}_{km}^{-1})}^{2}\right\},\end{array}$$
(12)

where, in the last passage, we exclude the term $${\left\{o\left(d{s}_{jk}d{s}_{ln}\right)\right\}}_{j\ne l,k\ne n}$$ as we consider sij to be statistically decorrelated. We note that the compliance and stiffness tensors computed with this procedure are independent of the number of blocks chosen for the error analysis.

### Elastic tensors from static approaches

We provide here a concise overview of the static methods to evaluate elastic constants and moduli. The expression for the cell matrix under the strain ϵ is:

$$\left(\begin{array}{rcl}{{{{\bf{a}}}}}_{{{{\bf{1}}}}}^{{\prime} }&{{{{\bf{a}}}}}_{{{{\bf{2}}}}}^{{\prime} }&{{{{\bf{a}}}}}_{{{{\bf{3}}}}}^{{\prime} }\end{array}\right)=(I+{{{\boldsymbol{\epsilon }}}})\left(\begin{array}{rcl}{{{{\bf{a}}}}}_{{{{\bf{1}}}}}&{{{{\bf{a}}}}}_{{{{\bf{2}}}}}&{{{{\bf{a}}}}}_{{{{\bf{3}}}}}\end{array}\right)=\left(\begin{array}{rcl}1+{\epsilon }_{1}&{\epsilon }_{6}/2&{\epsilon }_{5}/2\\ {\epsilon }_{6}/2&1+{\epsilon }_{2}&{\epsilon }_{4}/2\\ {\epsilon }_{5}/2&{\epsilon }_{4}/2&1+{\epsilon }_{3}\end{array}\right)\left(\begin{array}{rcl}{{{{\bf{a}}}}}_{{{{\bf{1}}}}}&{{{{\bf{a}}}}}_{{{{\bf{2}}}}}&{{{{\bf{a}}}}}_{{{{\bf{3}}}}}\end{array}\right)$$
(13)

where $$\left(\begin{array}{rcl}{{{{\bf{a}}}}}_{{{{\bf{1}}}}}^{{\prime} }&{{{{\bf{a}}}}}_{{{{\bf{2}}}}}^{{\prime} }&{{{{\bf{a}}}}}_{{{{\bf{3}}}}}^{{\prime} }\end{array}\right)$$ and $$\left(\begin{array}{rcl}{{{{\bf{a}}}}}_{{{{\bf{1}}}}}&{{{{\bf{a}}}}}_{{{{\bf{2}}}}}&{{{{\bf{a}}}}}_{{{{\bf{3}}}}}\end{array}\right)$$ are the strained and unstrained cells, respectively. Under a hydrostatic strain

$$\begin{array}{lll}{\epsilon }_{i}&\!\!\!=x,i&\!\!\!=1,2,3;\\ {\epsilon }_{i}&\!\!\!=0,i&\!\!\!=4,5,6,\end{array}$$
(14)

only the volume V of the cell is changed, and the bulk modulus (or isothermal incompressibility, i.e., the stiffness of the material under the effect of isotropic compression) can be calculated fitting an EOS, such as Murnaghan’s118,119, Birch’s119,120, Keane’s97,121, or Stacey’s122, where P and V are the variables, and volume, bulk modulus, and bulk modulus derivatives at the minimum (V0, B0, $${B}_{0}^{{\prime} }$$, $${B}_{0}^{{\prime\prime} }$$.) are the fitting parameters. We use here a simple Murnaghan’s EOS, equivalent to Keane’s EOS97,98,121 with $${B}_{0}^{{\prime\prime} }=0$$85,98, where the energy-volume relation is:

$$E=-\frac{{B}_{0}{V}_{0}}{{B}_{0}^{{\prime} }-1}+\frac{V{B}_{0}}{{B}_{0}^{{\prime} }}\left[{\left(\frac{{V}_{0}}{V}\right)}^{{B}_{0}^{{\prime} }}\frac{1}{({B}_{0}^{{\prime} }-1)}+1\right]$$
(15)

and the pressure

$$P=\frac{{B}_{0}}{{B}_{0}^{{\prime} }}\left[{\left(\frac{V}{{V}_{0}}\right)}^{-{B}_{0}^{{\prime} }}-1\right].$$
(16)

While the bulk modulus can be obtained directly from the EOS (Equations (15) and (16)), the remaining moduli (shear, Young’s and Poisson’s ratio) can only be derived from the full elastic tensor (Equation (7)), with 21 independent components92. Symmetry reduces this number to, e.g., 13 for monoclinic, 9 for orthorhombic, and 6 for tetragonal space groups92. The full elastic tensor can be calculated by an “energy-strain" approach53, i.e., by expanding the energy over the strain up to the second order50,123, or by a “stress–strain" approach, based on the stress resulting from an applied strain51,54. In the stress–strain approach, the elastic tensor is calculated from the change in stress Δσ associated with the strain ϵ applied to a reference configuration ϵ051:

$${{\Delta }}{{{\boldsymbol{\sigma }}}}={{{\boldsymbol{\sigma }}}}({{{{\boldsymbol{\epsilon }}}}}_{{{{\bf{0}}}}}+{{{\boldsymbol{\epsilon }}}})-{{{\boldsymbol{\sigma }}}}({{{{\boldsymbol{\epsilon }}}}}_{0})={{{\bf{c}}}}({{{{\boldsymbol{\epsilon }}}}}_{{{{\bf{0}}}}}){{{\boldsymbol{\epsilon }}}}.$$
(17)

We recall that in Equation (17) σ and ϵ are six-dimensional vectors and c(ϵ0) is a (6 × 6) matrix (Voigt notation116). The whole tensor c(ϵ0) can be obtained from Equation (17) by imposing the 6 simple uniaxial strains

$$\begin{array}{l}{{{{\boldsymbol{\epsilon }}}}}_{m}=\{{\epsilon }_{i}\},\\ {\epsilon }_{i}={\delta }_{im}x,m=1,\ldots 6.\end{array}$$
(18)

From Equations (17) and (18), each column m of c(ϵ0) is extracted from the differences between the stress calculated at ϵ0 + ϵm and ϵ051:

$${{\Delta }}{{{{\boldsymbol{\sigma }}}}}_{m}={{{\boldsymbol{\sigma }}}}({{{{\boldsymbol{\epsilon }}}}}_{{{{\bf{0}}}}}+{{{{\boldsymbol{\epsilon }}}}}_{m})-{{{\boldsymbol{\sigma }}}}({{{{\boldsymbol{\epsilon }}}}}_{0})=\left({c}_{1m},{c}_{2m},{c}_{3m},{c}_{4m},{c}_{5m},{c}_{6m}\right)x.$$
(19)

This is a simple and general way to extract the elastic tensor from the stress–strain relations. The 36 stiffness coefficients can be obtained independently, although only 21 are necessary (as c(ϵ0) is a symmetric matrix). The method holds for the most general case of a triclinic phase51, i.e., the independent matrix elements can be in principle all different. Of course, c(ϵ0) should display the symmetries of the space group to which the materials belong, which can also become a useful checkpoint for the convergence of the calculations.

### From the elastic tensors to the elastic moduli: Voigt-Reuss-Hill approximation

The effective elastic moduli of an arbitrarily shaped isotropic polycrystalline aggregate can be obtained from the elastic stiffness tensor {cij} of the single crystal, by assuming homogeneous strain (Voigt approximation116):

$$\begin{array}{ll}9{B}_{{{{\rm{V}}}}}&={c}_{11}+{c}_{22}+{c}_{33}+2({c}_{12}+{c}_{23}+{c}_{31})\\ 15{G}_{{{{\rm{V}}}}}&={c}_{11}+{c}_{22}+{c}_{33}-({c}_{12}+{c}_{23}+{c}_{31})\\ &\quad\,+\,3({c}_{44}+{c}_{55}+{c}_{66}),\end{array}$$
(20)

or from the elastic compliance tensor {sij}, by assuming homogeneous stress (Reuss approximation124):

$$\begin{array}{ll}1/{B}_{{{{\rm{R}}}}}&={s}_{11}+{s}_{22}+{s}_{33}+2({s}_{12}+{s}_{23}+{s}_{31})\\ 15/{G}_{{{{\rm{R}}}}}&=4({s}_{11}+{s}_{22}+{s}_{33})-4({s}_{12}+{s}_{23}+{s}_{31})\\ &\quad\,\,+\,3({s}_{44}+{s}_{55}+{s}_{66}).\end{array}$$
(21)

Equations (20) and (21) constitute the upper and lower bounds to the expected values of the moduli, respectively125,126, and their arithmetic or geometric means are considered a reasonable estimate, which goes under the name of Voigt-Reuss-Hill (VRH) approximation126:

$$\begin{array}{lll}B&\!\!\!=\frac{1}{2}({B}_{{{{\rm{V}}}}}+{B}_{{{{\rm{R}}}}})\,{{{\rm{or}}}}\,\,B&\!\!\!=\sqrt{{B}_{{{{\rm{V}}}}}{B}_{{{{\rm{R}}}}}},\\ G&\!\!\!=\frac{1}{2}({G}_{{{{\rm{V}}}}}+{G}_{{{{\rm{R}}}}})\,{{{\rm{or}}}}\,\,G&\!\!\!=\sqrt{{G}_{{{{\rm{V}}}}}{G}_{{{{\rm{R}}}}}}.\end{array}$$
(22)

In Equations (20) and (21) only the 9 independent components of the elastic tensors of orthorhombic single crystals appear92. A more sophisticated approach would be to relax the isotropy and homogeneity assumptions, using the variational principle that holds for arbitrary crystal shapes127. The resulting bounds, which go under the name of Hashin-Shtrikman bounds, have been derived for cubic128, orthorhombic129, tetragonal130, and monoclinic131 symmetries, and prove to give more accurate results than the VRH approximation128,129,130,131, though still lying within the Voigt and the Reuss bounds128,131. Thus, for simplicity, and considering the scope of this work, we will restrict ourselves here to the VRH approximation. Young’s modulus E and Poisson’s ratio ν are calculated from B and G through the relations126:

$$\frac{1}{E}=\frac{1}{3G}+\frac{1}{9B},\quad \nu =\frac{1}{2}\left[1-\frac{3G}{3B+G}\right].$$
(23)

For the strain-fluctuation method, where a statistical error is provided for the elastic tensors (see above), the errors on the elastic moduli B, G, E, and Poisson’s ratio are obtained by propagating the errors on the compliance (Equation (11)) and the stiffness (Equation (12)) to the expressions of the moduli in the VRH approximation (Equations (20) and (23))116,124,126.