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# Math Help - Non continuous function

1. ## Non continuous function

Hey everyone,
How would you prove that this function is not continuous in every x != 1:
x-1 +((x-1)^2)*D(x)

D(x) is Dirichlet function.
if x rational -> D(x) = 0
if not -> D(x) = 1

Thanks a lot,
Michael Engslter

2. ## Re: Non continuous function

There are several different "Dirichlet functions" but I assume you mean the most common, D(x)= 1 if x is rational, 0 otherwise. Given any x, there exist a sequence of rational numbers that converges to it and there exist a sequence of irrational numbers that converges to it. What is the limit of this function as you approach x along either sequence?

3. ## Re: Non continuous function

Hey I just wrote which Dirichlet function I meant.

4. ## Re: Non continuous function

Okay, so what I said still applies. Take the limits as you approach x along a sequence of rational numbers and along a sequence of irrational numbers.

5. ## Re: Non continuous function

Or look at the interval $(x-\delta,x+\delta)$ (using any value of $\delta$). Can you get $|f(x)-f(x_0)|<\epsilon$?

- Hollywood

format: plain


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## LaTeX: How to add bibliography and references?

There are two ways to add bibliography (references) to your document. One is for small document with few bibliography items using thebibliography environment. The other is for large documents where there are many bibliography items using packages like biblatex.

1) BIBLIOGRAPHY ON SMALL DOCUMENTS

If you have few bibliography items, which is a general case for small few pages documents, then you can use thebibliography environment to create bibliography on your document. \bibitem command is used to add bibliography items. \cite command is used to cite the bibliography item in the document chapter or section.

Below is an example of creating bibliography using “thebibliography” environment. The number “10” at the beginning of “thebibliography” indicates the number of bibliography entries that can be added in the document.

Bibliography will have the title as “References” if the document class is “article” and it will have title “Bibliography” if the document class is “book” or “report”.

Below is the pdf file generated after you compile the above LaTeX source:

2) BIBLIOGRAPHY ON LARGE DOCUMENTS

For large documents like large report or book, you need to specify .bib file. Bibliographic references are kept in this file. \cite command is used to cite the bibliography items in the document.

Here is an example of the BibTeX database stored in .bib file. Let us name the file as references.bib.

I have used biblatex package which uses bibtex to generate bibliographies. However, now biblatex uses biber as the default backend instead of bibtex.

If you have not installed biber, you need to install it first. Here is the command to install biber on Ubuntu:

Here is the sample LaTeX document that uses references.bib as the bibliography database. In the document chapter, citation is done using \cite command. Package defination and bibliographic resource file is defined in the preamble of the document. \printbibliography command is used to print the bibliography at our desired position.

Suppose that the tex file is named as myfile.tex. To compile the tex and bib file, you need to run the following commands:

Now, myfile.pdf is generated with the bibliography items.

Below is the pdf file generated after you compile the above LaTeX source:

If you are using document class as ‘report’ or ‘book’ then the title will be “Bibliography” and if you are using document class as ‘article’ then the bibliography title will be “References“.

More detail on Bibliography Management

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Question #2f3e8

To understand better the reaction, the ammonia is generally written as $N {H}_{4} O H$ that is ammonium hydroxide,where $N {H}_{4}$ is like a metallic cation . So you can obtain salt and water.
$2 N {H}_{4} O H + {H}_{2} S {O}_{4} = {\left(N {H}_{4}\right)}_{2} S {O}_{4} + 2 {H}_{2} O$
But it is less right, because the ammonia is present in water as gas in solution and doesn't exist the molecula $N {H}_{4} O H$


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# Question on speed of light

Tags:
1. Feb 1, 2014

### Arjun Kumar

As per Maxwell's equations the speed of light is

c = 1/√(permeability X permittivity)

I find that both permeability and permittivity have pi in their calculations. Since spacetime is not flat, the value of pi would change due to the curvature of space near large masses. Does this mean that the speed of light would also be altered by the curvature of spacetime?

2. Feb 1, 2014

### mathman

π is constant! It is a mathematical constant, unaffected by the physics of spacetime or anything else.

3. Feb 1, 2014

### Bill_K

The so-called permeability of free space is also a mathematical constant, an artifact of the SI system of units, having nothing at all to do with physics. It is exactly 4π x 10-7 by definition. (Next, someone will say the same thing about c.)

4. Feb 1, 2014

### Arjun Kumar

Pi is constant only in Euclidian geometry / space. Pi is defined as the number of times the diameter of a circle divides into the circumference. If flat Euclidean space that is 3.14 and remains constant. However, space time is not a flat Euclidean space. It can be curved (by the presence of mass/energy). In such cases pi will not equal 3.14. For example if you draw a circle on the surface of a sphere, the diameter of the circle (while remaining on the surface of the sphere) would be larger than in flat space and so pi would be less. (for the equator, on the surface of the earth, pi would be 2!). So Pi is not always constant. Thus I am still looking for an answer to my question.

5. Feb 1, 2014

### Staff: Mentor

The value of pi we use is the one from Euclidean geometry. As pointed out, it's a constant. While the ratio of circumference to diameter of a circle may well change in different geometries, we wouldn't call such ratios pi.

6. Feb 1, 2014

7. Feb 1, 2014

### pervect

Staff Emeritus
Even on a curved manifold, the value of the circumference of a circle to its radius will approach pi locally for a small enough circle, because the tangent space to the manifold is flat.

As others point out , "Pi" really is defined as a constant, but even if you look at the geometry, the ratio of circumference to diameter approaches a constant even for non-euclidean geometries.

(There are a few pathological exceptions, such as if the circle encloses a singularity, but they don't change the argument).

Nope

8. Feb 2, 2014

### Arjun Kumar

I agree that the value of the circumference of a circle to is radius will approach pi locally for a small enough circle. While this is true and that every 'point' in space is flat, beyond a 'point' space is not flat and can take different forms. (Every point on the surface on earth is flat, but we know that the surface forms a sphere - both views are correct!)
My confusion was that if pi, (being in my belief a ratio and not a constant) can change, then by also being a part of the equations for the speed of light, the speed of light would also be subject to change.

9. Feb 2, 2014

### vanhees71

The choice of the values for $\epsilon_0$ and $\mu_0$ is some men made convention to get convenient quantities for currents, voltages, and charges in electrical engineering, something Nature couldn't care less about.

The only fundamental constant that enters the theory of electromagnetism compared to Newtonian mechanics is the universal speed of light in the vacuum. As it has come out through Einstein's SRT paper of 1905, this is a much more universal speed than just the speed of light in vacuum.

More recent analyses of the special relativity principle has shown that there are, up to equivalence, only two space-time manifolds that obey the special relativity principle (i.e., the indistinguishablitiy of all inertial frames, i.e., the impossibility to measure an absolute constant velocity) are either Galilei-Newton space-time (a fibre bundle, i.e., you simply pin 3D Euclidean spaces at each point along the time axis) or Einstein-Minkowski space-time (a pseudo-Euclidean affine manifold with a fundamental form of signature (1,3) or equivivalently (3,1), depending on which sign convention you prefer). The latter implies a universal speed $c$, which to the best of our knowledge is the phase velocity of electromagnetic waves in the vacuum. This is because empirically there is no hint for a non-zero photon mass (to put it in modern QFT languague).

This confusion can be avoided by using the more natural Gaussian or the rationalized Gaussian (Heaviside-Lorentz) system of units, where no artificial constants due to the choice of an arbitrary fourth independent unit for electric current (in the SI the unit Ampere) or charge (in the SI the unit Coulomb=Ampere times second) is introduced. That's why not long ago usually textbooks on theoretical classical electrodynamics were written using the Gaussian units and why in theoretical high-energy physics one usually uses the Heaviside-Lorentz system of units. Nowadays, unfortunately the textbooks on classical electromagnetism are written using the SI units, spoiling somewhat the beauty of Maxwell theory as a relativistic classical field theory.

In my opinion, the best compromise is found in the third edition of Jackson's textbook: He starts using the SI units in order to be compatible with the use of this units in experimental physics and engineering and then switching to Gaussian units in the chapters where the electrodynamics is treated in its true form as a relativistically covariant field theory, writing "SI" or "G" in the header line of the chapters :-).

10. Feb 2, 2014

### TrickyDicky

There is no confusion on your part, but there seems to have been some misunderstanding in the replies as they got stuck in a "constants discussion" that is unrelated to the OP.

The very first paper on GR by Einstein in 1911 "On the Influence of Gravitation on the Propagation of Light" addresses this issue. This is the one where he got the deflection of light wrong by a half since he didn't have the EFE yet, but the equivalence principle was there already.

So to make it short, just saying the speed of light is subject to change can be misunderstood. As you can see since c is referred to the local measure of c (where you have admitted as explained by others that pi is a constant) this speed doens't vary. However it is alright to say(with qualifiers) that in the presence of gravitational fields it does, and that's what we see as bending or deflection of light, gravitational lensing etc. In the words of Einstein "From the proposition which has just been proved, that the velocity of light in the gravitational field is a function of the location...".
So the conclusion you draw in the OP is right, but you have to be careful about what you mean. The speed of light c is a local constant just by the same reason pi is a local constant.

Last edited: Feb 2, 2014
11. Feb 2, 2014

### Staff: Mentor

As Bill_K mentioned, c is a defined constant. It is like π in that its value is set by definition, but it is different from π in that its value depends on the system of units used. In natural units c=1 by definition. In SI units c=299792458 m/s by definition.

Now, whether or not the speed of light is equal to c is a slightly more interesting question. Again, the answer depends on your system of units. In SI units the speed of light is c also by definition. In other units the question is not a tautology, but nevertheless the answer tells you about the units rather than about the physics. A closely related physical question is whether or not the photon has mass, if it has mass then light of different wavelengths will travel at different speeds under any consistent system of units (and units that assume otherwise will be inconsistent).

Now, you asked about c in curved spacetime, where things get a little more complicated. You can talk about the coordinate speed of light, either locally or non-locally. Locally this can tell you some information about your coordinate system. Non-locally, I am not sure what you can learn from the coordinate speed of light. I suspect that it is still mostly information about your coordinates, however by combining local and non-local measurements maybe you could learn something about the curvature of spacetime along your non-local measurement also.

Instead of using coordinates you can use a reference frame (aka frame field or vierbein). However, a vierbein is defined as being orthonormal, and the normalization makes it so that c is defined once again. So even in curved spacetime the physical question remains whether or not the photon has mass.

12. Feb 2, 2014

### mathman

You are right in that the ratio depends on the geometry. However, π itself is a defined constant. The ratio in non-Euclidean geometry doesn't have a particular name.

As for the speed of light - I believe it remains unchanged.

Last edited: Feb 2, 2014
13. Feb 2, 2014

### pervect

Staff Emeritus
To use your example, if you reject the idea of taking the limit, what would you say the ratio of the circumference of a circle to its radius on the Earth's surface (or, if you prefer, a perfect sphere the same size as the Earth) is?


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# Monotonicity condition for the $\theta$-scheme for diffusion equations

1 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : We derive the necessary and sufficient condition for the $L^{\infty}-$monotonicity of finite difference $\theta$-scheme for a diffusion equation. We confirm that the discretization ratio $\Delta t = O(\Delta x^2)$ is necessary for the monotonicity except for the implicit scheme. In case of the heat equation, we get an explicit formula, which is weaker than the classical CFL condition.
Document type :
Reports

Cited literature [2 references]

https://hal.inria.fr/inria-00634417
Contributor : J. Frederic Bonnans Connect in order to contact the contributor
Submitted on : Friday, October 21, 2011 - 9:49:21 AM
Last modification on : Saturday, June 25, 2022 - 7:39:54 PM
Long-term archiving on: : Thursday, November 15, 2012 - 10:15:33 AM

### File

RR-7778.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : inria-00634417, version 1

### Citation

J. Frederic Bonnans, Xiaolu Tan. Monotonicity condition for the $\theta$-scheme for diffusion equations. [Research Report] RR-7778, INRIA. 2011, pp.6. ⟨inria-00634417⟩

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# Hand winches with worm gear

The hand winches with worm gears are known for their robust construction. The cable winch can be used either as a hoisting winch or a pulling winch. The high quality manual winches are equipped with a load pressure brake. This brake keeps the load at any required height during lifting and lowering. These manual winches have one of the best braking systems in which 50% of the braking is done by friction in the worm gear. This greatly reduces the load on the pressure brake. The load capacities are calculated on the basis of the first rope layer on the drum. You can operate the winches manually and thus use them in all environments such as the offshore and marine industries.


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Categories

# ISI MStat 2015 PSA Problem 17 | Basic Inequality

This is a problem from ISI MStat 2015 PSA Problem 17. First, try the problem yourself, then go through the sequential hints we provide.

This is a problem from ISI MStat 2015 PSA Problem 17. First, try the problem yourself, then go through the sequential hints we provide.

## Basic Inequality – ISI MStat Year 2015 PSA Question 17

Let $X=\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}$. Then,

• x<1
• $x>\frac{3}{2}$
• $1<x< \frac{3}{2}$
• None of these

### Key Concepts

Basic Inequality

Answer: is $1<x< \frac{3}{2}$

ISI MStat 2015 PSA Problem 17

Precollege Mathematics

## Try with Hints

Take it easy. Group things up. Use $\frac{1}{n+k} < \frac{1}{n}$ for all natural (k)

$\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{1999} < 1000 \times \frac{1}{1000}$
$\frac{1}{2000}+\frac{1}{2001}+\frac{1}{2002}+\dots+\frac{1}{3001} < 1000 \times \frac{1}{2000}$

$\Rightarrow x < 1+ \frac{1}{2}= \frac{3}{2}$

Again see that we can write , x=$(\frac{1}{1001}+\frac{1}{3001}) + (\frac{1}{1002}+\frac{1}{3000})+ \dots + (\frac{1}{2000}+\frac{1}{2002}) + \frac{1}{2001}$ > $\frac{2}{2001} +\frac{2}{2001} + \dots + \frac{2}{2001} +\frac{1}{2001} > \frac{2001}{2001}=1$

Hence , $1<x< \frac{3}{2}$.

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# Homework Help: Parameterization of a Circle Question

1. Sep 15, 2011

### royblaze

This isn't really a HW question, it's just something that's been confusing me in my Calc class.

We recently went over how to find curvatures of curves in 3D space. In lecture, the professor went over a simple example: a circle of radius 3 at any given point.

Maybe it's because I don't remember my Calc II class that well, but he parameterized the equation of a circle,

x2 + y2 = 9

to become r(t) = 3cos(t)i + 3sin(t)j .

So from:

x2 + y2 = r2

If x = cos(t) and y = sin(t), the identity cos2(t) + sin2(t) = 1 still holds true.

x2 + y2 = 32

3(x2 + y2) = 9(12)

9cos2(t) + 9sin2(t) = 9(12)

9(cos2(t) + sin2(t)) = 9(12)

is how I understand it. If there's an easier way to picture it, please let me know!

So then r(t) = 3cos(t)i + 3sin(t)j describes the circle when "traced."

My question: why does the "x" part and the "y" part of the parameterization have to be the respective cos, sin trig functions? In class, one student said that x was sin; but the prof said that no, it's cos. Why does x have to be cos, rather than sin? Does it have to do something with the direction the curve "goes in?" How can I determine the "direction" the curve goes in? Or did I just hear my professor wrong...?

2. Sep 15, 2011

### jncarter

Try drawing a picture. Do you remember your unit circle? For a right-handed coordinate system, x = rcos$\theta$ and y the sine function. Positive theta corresponds to counter-clockwise motion from the positive x-axis.
Now when you parametrize the equation $\theta$ becomes $\dot{\theta}$t, where $\dot{\theta}$ is how quickly you are going around the circle. In your case this is simply one.
Strictly speaking, x doesn't HAVE to be the cosine. It's just convention to use a right handed coordinate system. If you measured $\theta$ from a different axis or you switched positive x being in the vertical, your sines and cosines may change.

3. Sep 15, 2011

### LCKurtz

Yes, it has to do with the orientation of the curve and where you want it to be when t = 0. Parameterizations are in general not unique. If the curve is specified to start at (1,0) and go counterclockwise then R(t) = < cos(t), sin(t) > is a natural choice. If it went clockwise you might use <cos(t), -sin(t)>. If nothing is given about the orientation of the curve then you could use <sin(t), cos(t)> or other choices; they are all equally correct.

4. Sep 15, 2011

### HallsofIvy

They don't have to be. There exist an infinite number of parametric equations describing any curve. $x= 3sin(t)$, $y= 3cos(t)$ are perfectly good parametric equations and, in fact, have the same "direction" as $x= 3cos(t)$ $y= 3sin(t)$: with the first, as t goes from 0 to $\pi/2$, (x, y) goes from (0, 3) to (3, 0), counter-clockwise around the circle and the second goes from (3, 0) to (0, 3), also counterclockwise.

But before you go telling your teacher that we say he/she is wrong, note that they do have different points for specific values of t. The parametric equations $x= cos(t)$, $y= sin(t)$ "start" (t= 0) at (1, 0) and then go around the circle. Perhaps for some reason, that is important.

5. Sep 15, 2011

### royblaze

Wow, thank you all for the absolutely fantastic replies. It REALLY cleared it up for me, big time.

Thanks again!


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Limited access

Let $A$ be a $3\times3$ diagonal matrix.

Which one of the following statements is necessarily TRUE?

A

$A$ commutes with any $3\times3$ symmetric matrix

B

$A$ is invertible

C

$A$ commutes with the matrix $\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}$

D

$A$ commutes with the standard matrix of any rotation of the plane spanned by the first two standard basis vectors of $\mathbb{R}^3$

E

$A$ commutes with any other diagonal $3\times3$ matrix

Select an assignment template


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## Abstract

In previous work on galvanized reinforcements in contact with Ca-containing highly alkaline media the authors have reported the existence of a threshold pH below which the Zn in contact with such a medium is passivated, but at pH values above this it corrodes continuously until it totally disappears. In this paper, after a brief literature review concerning the identification of the corrosion products in these media, they report that the passivation detectedfrom the data of the polarisation technique agreed with the results of an SEM study. This showed that below the threshold pH value the surface is covered with a continuous layer of calcium hydr.oxyzinccite, Ca ${\displaystyle (Zn(OH)_{3})_{2}\cdot 2H_{2}O}$, and above this pH this layer is increasingly composed ofnon-passivating ZnO, to the detriment of the calcium hydroxyzincate content. They have also established that it is the (${\displaystyle Ca^{2+}}$) ions that control the amount of calcium hydroxyzincate formed, and therefore the likelihood of the formation of a continuous passivating layer. Finally, they report that if the calcium hydroxyzincate isperfectly developed, a later increase of thepH does not affect the stability.

### Document information

Published on 01/01/1984

DOI: 10.1179/000705984798273524


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Beta distribution

Probability density function:

$f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathcal{B}(\alpha,\beta)}$ with $$\alpha$$ and $$\beta$$ two shape parameters and $$\mathcal B$$ beta function.

Cumulative distribution function:

$F(x) = \frac{\int_{0}^{x} y^{\alpha-1}(1-y)^{\beta-1}dy} {\mathcal{B}(\alpha,\beta)} =\mathcal{B}(x; \alpha,\beta)$ with $$\mathcal B (x; \alpha,\beta)$$ incomplete beta function.

Log-likelihood function:

$L(\alpha,\beta;X)=\sum_i\left[ (\alpha-1)\ln(x)+(\beta-1)\ln(1-x)-\ln \mathcal{B}(\alpha,\beta) \right]$

Score function vector:

$V(\mu,\sigma;X) =\left( \begin{array}{c} \frac{\partial L}{\partial \alpha} \\ \frac{\partial L}{\partial \beta} \end{array} \right) =\sum_i \left( \begin{array}{c} \psi^{(0)}(\alpha+\beta)-\psi^{(0)}(\alpha)+\ln(x) \\ \psi^{(0)}(\alpha+\beta)-\psi^{(0)}(\beta)+\ln(x) \end{array} \right)$ with $$\psi^{(0)}$$ being log-gamma function.

Observed information matrix:

$\mathcal J (\mu,\sigma;X)= \left( \begin{array}{cc} \psi^{(1)}(\alpha)-\psi^{(1)}(\alpha+\beta) & -\psi^{(1)}(\alpha+\beta) \\ -\psi^{(1)}(\alpha+\beta) & \psi^{(1)}(\beta)-\psi^{(1)}(\alpha+\beta) \end{array} \right)$ with $$\psi^{(1)}$$ being digamma function.


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# Total force exerted by the fluid on the surface is the sum of normal forces

In https://en.wikipedia.org/wiki/Pressure#Formula, the normal force per area $dA$ is $d\vec{F}_n=-p\,\vec{n}\,dA$, where $p$ is the pressure. It is stated that, for any surface $S$ in contact with the fluid, the total force exerted by the fluid on that surface is the surface integral over $S$ of the right-hand side of the equation: $\vec{F}=\int_Sd\vec{F}_n=\int_S -p\,\vec{n}\,dA$. My question is why $\vec{F}=\int_Sd\vec{F}_n$, and not $\vec{F}=\int_S d\vec{F}$. I would like to understand why we sum the normal forces $d\vec{F}_n$ at each point, and not the whole forces $d\vec{F}$ at each point of $S$. Is it because the parallel component of each infinitesimal force $d\vec{F}$ is $0$ ?


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Language:   Search:   Contact
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Zbl 1229.46035
Nguetseng, Gabriel; Svanstedt, Nils
$\Sigma$-convergence.
(English)
[J] Banach J. Math. Anal. 5, No. 1, 101-135, electronic only (2011). ISSN 1735-8787/e

The usual approach to the weak convergence of a product of functions is to require one of the sequences to converge strongly. For example, if $u_n \to u$ in $L^p(\Omega)$, $v_n \rightharpoonup v$ in $L^{p^{\prime}}(\Omega)$, then $u_n v_n \rightharpoonup uv$ in $L^1(\Omega)$, where $\rightharpoonup$ means weak convergence. The authors generalize this by considering weak $\Sigma$-convergence, $\Sigma$-convergence, homogenization algebras and the Gelfand transform. The details are rather technical (see the paper). It is a generalization of the two scale convergence of [{\it G. Nguetseng}, SIAM J. Math. Anal. 20, No. 3, 608--623 (1989; Zbl 0688.35007)] to the non-periodic setting, using et al the Gelfand transform. The authors illustrate the techniques by applying the ideas to a few homogenization problems.
[Raymond Johnson (Houston)]
MSC 2000:
*46J10 Banach algebras of continuous functions
35B27 Homogenization, etc.
28A20 Measurable and nonmeasurable functions

Keywords: homogenization; homogenization algebras; $\Sigma$-convergence; Gelfand transformation

Citations: Zbl 0688.35007

Cited in: Zbl pre06099665

Highlights
Master Server


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## Introduction

Angle-resolved photoemission spectroscopy (ARPES) is the most direct experimental technique to measure the momentum-resolved electronic eigenvalues (energy bands) of crystalline solids1,2. The technique is based on the photoelectric effect3,4, in which electrons inside solids absorb a photon with energy larger than the work function and escape into the vacuum. A key aspect of ARPES is that the energy (E) and momentum (k)-resolved photoemission signal is proportional to the single-particle spectral function A(k, E)5, a fundamental quantity containing key information about many-body interactions inside solids.

However, more than two decades ago, it was realized that the direct relationship between ARPES spectral lineshapes and the spectral function was obstructed by additional intensity modulation coming from the transition dipole matrix element connecting the initial Bloch wave and the final photoelectron states6. Because of its semimetallic 2D nature and its speculated textbook Fermi-Liquid behavior, the layered transition metal dichalcogenide (TMDC) 1T-TiTe2 was the central material around the debate on directly linking the spectral function and the ARPES lineshape7,8,9,10. Indeed, 1T-TiTe2 served as a benchmark material to demonstrate the signatures of many-body effects in ARPES lineshape11, the possibility to retrieve the photohole lifetime12, as well as the potential to dissect the role of the different quasiparticle scattering processes from ARPES measurements10.

On the other hand, very little attention has been devoted to the other key ingredient governing ARPES spectra of 1T-TiTe2: the transition dipole matrix element. The effect of the transition dipole matrix element is often viewed as detrimental, since it breaks the quasi-one-to-one correspondence between the ARPES spectrum and the occupied electronic band structure, by imprinting a complex and often strong energy-dependent and momentum-dependent modulation onto the photoemission intensity13,14. Matrix element effects can sometimes be confused with signatures of many-body interactions6,15 in ARPES spectra, and are often interpreted by hand-waving arguments. While computational schemes to remove matrix element effects from measured ARPES spectra are an attractive route to focus on the band structure16, it has become clear that additional information on the electronic wavefunction can be obtained by understanding matrix element effects. While the matrix element has some extrinsic contributions coming from the experimental geometry (e.g. photon momentum, light polarization, and crystal orientation), it also encodes intrinsic information on the nature of the initial state from which the electrons are emitted. Therefore, disentangling extrinsic and intrinsic contributions is of capital importance to access the additional information encoded in the photoemission matrix elements. Important examples include gaining insights into the orbital character of the electronic band structure in solids17,18,19,20,21, the chirality of charge carriers22,23, quantum geometric properties, such as orbital pseudospin texture24,25 as well as Berry curvature26,27,28,29. The orbital texture in TMDCs is very sensitive to the lattice structure and collective effects such as charge density waves (CDW)30,31,32,33,34 and the debated excitonic insulator state35,36,37. As demonstrated recently, CDW phases and orbital texture can be intertwined38. Tracing the orbital texture from matrix element effects in photoemission will also greatly benefit discerning structural and collective phases in TMDCs. More specifically, 1T-TiTe2 has been recognized to be a very interesting prototypical 2D material: it exhibits a strong thickness33,39 and substrate-dependent34 emergence of CDW and pseudogap phases33.

To disentangle between extrinsic and intrinsic contributions to dichroism in ARPES, we recently introduced a measurement methodology to extract the modulation of the photoemission intensity upon a crystal rotation by a well-chosen angle θ in the laboratory frame ($${{{{{{\mathcal{R}}}}}}}_{\theta }$$)24. We benchmarked this approach by investigating the modulation of photoemission intensity from 2H-WSe2 upon a 60° azimuthal crystal rotation. In 2H-TMDCs, due to the broken inversion symmetry within each monolayer and the strong spin-orbit splitting, a 60° crystal rotation ($${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}$$) is mimicking a time-reversal operation ($$\hat{T}$$), i.e. $${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}\equiv \hat{T}$$, which is equivalent to swapping valley-pseudospin indices, i.e., K → K′, and vice versa. We have shown that since the extrinsic contribution to dichroism was invariant upon $${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}$$, the antisymmetric components of the dichroism upon $${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}$$ (i.e. the differential signal which undergoes a sign change, called time-reversal dichroism in the photoelectron angular distribution—TRDAD) is a background-free intrinsic dichroic signal sensitive to the orbital pseudospin texture in 2H-WSe224.

In this work, we use the same experimental methodology—i.e. we measure the photoemission intensity modulation upon sample rotation—to unveil the orbital texture of 1T-TiTe2. We introduce the intrinsic linear dichroism in photoelectron angular distributions (iLDAD)—a generalized version of TRDAD, in samples featuring inversion symmetry. Indeed, TRDAD can be seen as a special case of iLDAD, when a sample possesses broken inversion symmetry and when the crystal rotation is thus equivalent to a time-reversal operation. The iLDAD from 1T-TiTe2, extracted from ARPES measurements for two different crystal orientations, is fully reproduced by first-principle simulations based on the Korringa–Kohn–Rostoker (KKR) method. The excellent agreement of experiment and theory allows us to disentangle different contributions to the ARPES signal and to construct a tight-binding (TB) model capturing the orbital texture. Combining the TB model with qualitative modeling of the photoemission matrix elements, we trace the measured dichroism to the tilted real-space character of the relevant orbitals. The intrinsic dichroism is found to be particularly sensitive to the orbital character, which is beyond standard LDAD based on the anisotropy of the photoemission intensity upon rotating the light polarization direction by 90°17,19,20. Hence, iLDAD is a powerful addition to the toolbox of dichroism in ARPES, in the spirit of going beyond band structure mapping and gaining information about energy- and momentum-resolved electronic wavefunction of solids.

## Results

### Band structure mapping

The experimental apparatus is composed of a table-top femtosecond (fs) XUV source (centered around 21.7 eV, spectral bandwidth 110 meV FWHM)40 coupled to a time-of-flight momentum microscope spectrometer (METIS 1000, SPECS GmbH) (see Fig. 1a). This detector allows measuring all electrons emitted above the surface of the sample and recording a stream of individually detected photoelectrons, resolved in the three intrinsic dimensions of the detector: kinetic energy (E) and both parallel momentum components (kx, ky). One of the major advantages of this type of detector is the parallel detection of the 3D photoemission intensity I(E, kx, ky) in a single measurement, without having to rotate the sample in the laboratory frame41. More details about the experimental setup can be found elsewhere40,42 and in the “Methods” section. Giving the energy-dependence of the inelastic mean free path of the outgoing electrons43,44, working in the XUV spectral range ensure that most of the detected electrons are coming from the topmost 1T-TiTe2 layer (first atomic trilayer)24. More details about this statement, as well as its theoretical validation, will be given later.

The experimentally measured 3D photoemission intensities and associated constant energy contours (CECs) at the Fermi energy (EEF = 0 eV) are shown in Fig. 1c and d. Looking at the 3D photoemission intensity (Fig. 1c), we recognize the well-known semimetallic nature of 1T-TiTe2, featuring spectrally overlapping Te-5p hole pockets at Γ and Ti-3d electron pockets around two inequivalent M points (denoted by M/M′)45. Looking at the CECs in Fig. 1c, the most striking observation is the strong photoemission intensity anisotropy between Ti-3d electron pockets located at M and M′. Indeed, while the photoemission intensity from M′ pockets is strong, it is almost completely suppressed for adjacent M pockets. The emergence of this strong linear dichroism in photoemission a priori can originate from extrinsic (e.g. experimental geometry) and/or intrinsic (i.e. orbital/Bloch wavefunction symmetries) effects.

Motivated by the recently introduced approach for disentangling extrinsic and intrinsic contributions to dichroism in ARPES (see ref. 24 and the section “Introduction”), we also performed band structure mapping of 1T-TiTe2 for a second crystal orientation rotated by 60°. In inversion symmetric 1T-TiTe2, this crystal rotation ($${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}$$) is strictly equivalent to a mirror-image symmetry of each layer (see Fig. 1b), i.e. $${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}\equiv z\to -z$$. Although the effect of a 60° crystal rotation in 1T-TMDC ($${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}\equiv z\to -z$$) is different than in 2H-TMDC ($${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}\equiv \hat{T}$$), in both cases it corresponds to a well-defined symmetry operation on the electronic wavefunction of the crystal. Linking a given symmetry operation on the electronic wavefunction to the modulation of the photoemission intensity, complemented by proper theoretical analysis, is a powerful tool to investigate the energy-dependent and momentum-dependent orbital texture, a quantity of fundamental importance but hardly accessible with other techniques.

### Intrinsic linear dichroism in photoelectron angular distributions

In Fig. 2, we show the procedure to extract the antisymmetric components of the dichroism (referred to as iLDAD) from photoemission data obtained from two crystal orientations rotated by 60° with respect to each other. In Fig. 2a, one can see the CEC at the Fermi energy featuring the triangular doughnut-shaped Te-5p hole pockets around Γ, which shows subtle but non-vanishing angular anisotropies. More remarkably, the strongly anisotropic photoemission from Ti-3d electron pockets located around M and M′ is dominating the signal. As already discussed in the context of Fig. 1, while the photoemission intensity is strong for the M′ pockets, it is almost vanishing for the adjacent M pockets. Upon a 60° azimuthal rotation of the crystal (see Fig. 2b), this asymmetric photoemission yield from M and M′ pockets remained mostly unchanged, leading to a 60° rotation of the momentum-resolved photoemission patterns. This is a strong indication that this asymmetric photoemission yield is of intrinsic origin and should encode information about the symmetry properties of the orbitals.

While Fig. 2b shows the raw normalized photoemission intensity difference between the two crystal orientations, using our multidimensional detection scheme with the p-polarized fs-XUV pulses impinging in the x-z plane (along with Γ−M/M′), the normalized intensity differences between the forward Iα(E, kx, ky) and backward Iα(E, − kx, ky) hemisphere, i.e. the linear dichroism asymmetry in the photoelectron angular distribution $${A}_{{\mathrm {LDAD}}}^{\alpha }(E,{k}_{x},{k}_{y})$$, can be extracted (see Eq. (1)), without the need to rearrange the sample geometry nor the light-polarization state46,47.

$${A}_{{{{\mathrm{LDAD}}}}}^{\alpha }=\frac{{I}_{\alpha }(E,{k}_{x},{k}_{y})-{I}_{\alpha }(E,-{k}_{x},{k}_{y})}{{I}_{\alpha }(E,{k}_{x},{k}_{y})+{I}_{\alpha }(E,-{k}_{x},{k}_{y})}$$
(1)

Looking at $${A}_{{{{\mathrm{LDAD}}}}}^{{0}^{\circ }}$$ (Fig. 2d), we notice very strong dichroism (up to ±97%) around M/M′ pockets with an opposite sign for adjacent valleys, and weaker dichroism (up to ±35%) with the same threefold symmetry and sign texture for the Te-5p hole pockets around Γ. The dichroic signal is not perfectly up/down (ky > 0/ky < 0) symmetric, suggesting small experimental imperfection in the crystal orientation, XUV beam alignment, or photoelectron imaging, since the up/down symmetry, should not be broken within this experimental configuration (i.e. p-polarization along Γ−M/M′). Moreover, since we are dealing with normalized signals (see Eq. (1)), we only show the dichroism for momentum regions where the right-left (kx > 0/kx < 0) symmetrized signal is > 0.1Imax, where Imax is the maximum photoemission intensity for this CEC. This thresholding procedure ensures that we do not over-interpret the potentially strong dichroism coming from momentum regions of vanishing photoemission intensity. Figure 2e shows $${A}_{{{{\mathrm{LDAD}}}}}^{6{0}^{\circ }}$$, i.e. the dichroism extracted after rotating the crystal orientation by 60. The measured $${A}_{{{{\mathrm{LDAD}}}}}^{6{0}^{\circ }}$$ dichroic signal is almost perfectly antisymmetric with respect to $${A}_{{{{\mathrm{LDAD}}}}}^{{0}^{\circ }}$$, again indicating its intrinsic origin since an extrinsic (experimental geometry) origin would have led to a dichroic signal invariant upon an operation.

In order to quantitatively isolate the antisymmetric part of the $${A}_{{{{\mathrm{LDAD}}}}}^{\alpha }$$ dichroism upon crystal rotation ($${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}$$), i.e. to remove any spurious contributions from experimental geometry, we introduce the iLDAD:

$${i}{{{\mathrm{LDAD}}}}=\frac{{A}_{{{{\mathrm{LDAD}}}}}^{\alpha }-{A}_{{{{\mathrm{LDAD}}}}}^{\alpha ^{\prime} }}{2}$$
(2)

where the crystal rotation is hereby an angle α$$\alpha ^{\prime}$$ = 60. The resulting iLDAD is shown in Fig. 2f. The strong similarity between $${A}_{{{{\mathrm{LDAD}}}}}^{{0}^{\circ }}$$ and iLDAD is another observation revealing that the dichroism in 1T-TiTe2 is almost exclusively originating from intrinsic origins (orbital symmetry/orientation). We will investigate the microscopic origin of the emergence of this strong intrinsic dichroism and how it can be used to reveal key information about the orbital texture of this layered semimetallic TMDC.

### Comparison with DFT band structure and one-step model of photoemission

To extract quantitative information about orbital texture from such differential (iLDAD) ARPES measurements, a systematic comparison with quantitative theories is highly beneficial. Here, we use inputs from two different theoretical approaches in a synergistic manner. First, we performed one-step model photoemission calculations based on fully relativistic density functional theory (DFT). The one-step model of photoemission is implemented in the fully relativistic KKR method48,49. The calculated photoemission signal includes all matrix element effects such as experimental geometry, photon energy, polarization state, and final state effects (for more details, see the “Methods” section). Second, we constructed a first-principle TB model by computing projective Wannier functions (see the “Methods” section). The TB model leads to an excellent description of the electronic structure of 1T-TiTe2 and provides an intuitive picture of the link between orbital degrees of freedom and dichroism in photoemission.

An important step, before calculating iLDAD, is to ensure that both theoretical approaches can accurately capture the electronic structure of 1T-TiTe2. The experimentally measured photoemission intensity, the simulated KKR photoemission intensity, and the calculated DFT band structure along the M–Γ–M′ cut are shown in Fig. 3. The experimental measured photoemission intensity are shown in Fig. 3a, from which one can clearly recognize the well-known semimetallic nature of 1T-TiTe2, featuring Te-5p hole pockets at Γ and Ti-3d electron pockets around M/M′ points. The experimentally obtained band structure is well captured by the calculated DFT band structure. Within the local density approximation (LDA)—including spin–orbit coupling (SOC)—used for the KKR method (Fig. 3b), the measured band structure near the Fermi level is well reproduced, both at the Γ point and at the M/M′ pockets. At lower energies (EEF ~ −0.9 eV) the measured spectrum shows bands that are absent in the calculated spectrum. We also performed DFT calculations using the Perdew–Burke–Ernzerhof (PBE) functional (Fig. 3c), which captures these bands. The size of the pockets at M/M’ is slightly smaller than in the experiments and within the LDA, which is consistent with calculations including exchange effects39. We have also calculated the band structure of a free-standing monolayer (Fig. 3d) for comparison. Albeit the band structure is generally modified with respect to the bulk (especially near Γ), the electron pockets at M/M′ are still present. Constructing a TB model from projective Wannier functions (red lines in Fig. 3d) provides an excellent fit of PBE band structure. The TB description provides us with a simple model for understanding the measured photoemission features at M/M′, which is discussed below.

By looking more carefully at Fig. 3a, and as seen in Fermi level CECs (Fig. 1c and d), one can notice strong photoemission anisotropies between positive and negative kx (corresponding to Γ−M′ and Γ−M direction, respectively). These photoemission anisotropies, originating from transition dipole matrix element effects, are well-captured by the calculated KKR photoemission intensity, shown in Fig. 3b. As mentioned earlier, the most striking asymmetry in the photoemission signal is coming from Ti-3d electron pockets around M/M’ points, near the Fermi level. Moreover, because the role of orbital texture near the Fermi level is of capital importance in determining materials’ optical and transport properties, we will focus the subsequent analysis on the photoemission anisotropies near the Fermi energy.

### Calculated iLDAD and its photon-energy dependence using KKR

In Fig. 4a and b we compare the iLDAD calculated using the one-step model (KKR) for two different photon energies (18.7 and 21.7 eV, respectively). For both photon energies, the calculated iLDAD at M/M′ pockets is very similar to the experimentally measured iLDAD. However, surprisingly, using 18.7 eV instead of the experimentally used 21.7 eV photon energy captures better the iLDAD features around Γ. This can be interpreted as a limitation of DFT: it is well known that the calculated time-reversed low-energy electron diffraction (TRLEED) final state can be shifted in energy with respect to the experimental value (for more details see Fig. 4). However, the dispersion and the orbital character of the time-reversed LEED state are typically well captured. Moreover, it was pointed out by Strocov et al. 45 that for 1T-TiTe2 the photoemission final state in the UV photon energy regime features dramatic final state effects such as multiband composition and non-free electron like dispersions around the center of the Brillouin zone. In addition, it was shown that the states around Γ at the Fermi level show three-dimensional character due to the interactions between 1T-TiTe2 monolayers (atomic trilayers)45.

To investigate the role of the final states, we performed a detailed theoretical photon-energy-dependent iLDAD scan (kz-scan). In Fig. 4c, we show the photon energy dependence of the iLDAD in the selected regions of interest (marked as 1–4). While the iLDAD of Ti-3d derived electron pockets around M/M′ points (features 1 and 2) are extremely stable against variation of the photon energy, the hole pockets around Γ point (features 3 and 4) exhibits strong modulations including even sign reversal upon modification of the photon energy. We attribute this to the strong 3D character of Te-5p states. A similar trend was observed for the soft-X-ray photoemission, where the final state becomes truly free-electron-like (see SI), ruling out the predominance of final-state effects in the emergence of iLDAD.

Performing such a photon-energy scan of the iLDAD is currently not possible using our HHG-driven XUV source. This is specifically for this reason that we took advantage of our quantitative KKR theoretical framework to investigate the photon-energy-dependence of the iLDAD. In the future, synchrotron light source could be used for photon-energy-dependent iLDAD studies.

### The role of orbital texture in photoemission: tight-binding analysis

While the iLDAD exhibits a pronounced photon energy dependence near the Γ point, the robustness of the iLDAD near the M/M′ pockets gives us confidence that we can link its origin to microscopic properties of the electronic structure of 1T-TiTe2. To find an intuitive link between the orbital texture and this intrinsic dichroic signal, we employ an ab initio TB model. Within this model, the Bloch wavefunction is represented by

$${\psi }_{{{{{{\bf{k}}}}}}n}({{{{{\bf{r}}}}}})=\frac{1}{N}\mathop{\sum}\limits_{{{{{{\bf{R}}}}}}}{e}^{i{{{{{\bf{k}}}}}}\cdot {{{{{\bf{R}}}}}}}\mathop{\sum}\limits_{j}{C}_{jn}({{{{{\bf{k}}}}}}){\phi }_{j}({{{{{\bf{r}}}}}}-{{{{{\bf{R}}}}}}),$$
(3)

where ϕj(r) denote the Wannier functions localized at particular sites R. The coefficients Cjn(k) are the coefficients connection orbital (index j) and band (index n) space; they are the eigenvectors of the TB Hamiltonian $${H}_{j{j}^{\prime}}^{{{{{{\rm{TB}}}}}}}({{{{{\bf{k}}}}}})$$. The Wannier representation (3) provides a direct of real and reciprocal space. For each momentum k the Bloch wavefunction is a coherent superposition of the Wannier functions ϕj(r), whose orbital symmetry is directly reflected in the ARPES signal.

From a layer-resolved KKR calculation, we found that the characteristic intensity variation can be captured by including the top layer TiTe2 only (see SI). Therefore, we constructed the TB model for a free-standing monolayer, including five Ti-d orbitals three p orbitals for each of the two Te chalcogen atoms in Eq. (3). The model provides a systematic way to examine the orbital character of the bands; in particular, we find that the contribution of the Te-p orbitals is negligible near EF around M/M′ pockets (see SI). Figure 5 shows the corresponding orbital-projected band structure for the dominant Ti-d orbitals along the M–Γ–M′ (upper row) and a M′–Γ–M cut rotated by 60° (lower row in Fig. 5), respectively. Focusing on the electron pockets crossing the Fermi level near M/M′, it is clear that the out-of-plane Ti-$${d}_{{z}^{2}}$$ orbitals (Fig. 5b, e) contribute equally to the density of state along both paths. Inspecting the Ti-dxz and Ti-dyz orbital-projected band structure (Fig. 5c, d and f, g, respectively), the situation is drastically different. Along the path parallel to kx the weight of the Ti-dxz orbital is negligible at the M/M’ pockets, while the Ti-dyz orbital contributes significantly. Along the rotated path the band at the M/M′ pocket is comprised of the Ti-$${d}_{{z}^{2}}$$ and the Ti-dxz orbital with almost equal weight, with a smaller contribution from the Ti-dyz orbital. It is important to note that while the band structure possesses six-fold rotational symmetry in momentum space, the orbital texture exhibits a three-fold symmetry corresponding to the lattice symmetry.

This momentum-dependent orbital content of the band structure underlines the important role of the orbital degree of freedom in photoemission from 1T-TiTe2. For exploring the manifestations in the ARPES signal, we combined the TB model with modeling the Wannier functions and the plane-wave (PW) approximation to the final photoelectron states (see the “Methods” section). We calculated the photoemission spectrum within the resulting TB + PW model for both sample orientations (see the “Methods” section) and extracted the iLDAD. Including specific orbitals only then allows for isolating the individual channels giving rise to the measured iLDAD.

First, in Fig. 6a, we show the iLDAD obtained from the TB + PW model including the Ti-$${d}_{{z}^{2}}$$, dxz and dyz orbitals. The calculated iLDAD is in excellent qualitative agreement with the measured signal near the M/M′ pockets, which confirms that these three orbitals are the main ingredients of the iLDAD signal. Moreover, the coherent superposition of these orbitals is important: when incoherently summing $$i{{{{{\rm{LDAD}}}}}}=i{{{{{{\rm{LDAD}}}}}}}_{{z}^{2}}+i{{{{{{\rm{LDAD}}}}}}}_{xz}+i{{{{{{\rm{LDAD}}}}}}}_{yz}$$, iLDAD is extremely weak and does not exhibit the expected symmetry (as experimentally measured), as shown in Fig. 6b. Including only the $${d}_{{z}^{2}}$$ and dyz orbitals (Fig. 6c) captures the full signal along the kx axis, which is consistent with the corresponding orbital weight (see Fig. 5). In contrast, including only the $${d}_{{z}^{2}}$$ and dxz orbitals (Fig. 6d) accounts for most of the iLDAD for the upper and lower two pockets, respectively. Note that the Ti-d orbitals are defined in the laboratory coordinate system here, which is the natural choice when computing ARPES signals. Earlier experiments50,51 attempted to connect the photoemission signal to the symmetry-adapted crystal-field orbitals. The crystal axes defined by the Ti–Te bonds are not strictly orthogonal for the presented experimental geometry, which complicates this picture. In the SI, we present a similar orbital analysis as above in the crystal-field basis for a complementary picture.

The TB + PW model allows for an even more intuitive picture for the characteristic iLDAD by inspecting the real-space dependence of the wavefunction. As explained in the “Methods” section, the photoemission matrix elements can be expressed in terms of the Wannier functions (for the Ti-d orbitals) and eigenvectors Cjn(k). Directly at the six M/M’ points on the edge of the Brillouin zone (labeled by km, m = 1, …, 6; see Fig. 7a), the matrix element within the TB + PW model reduces to

$$M({{{{{{\bf{k}}}}}}}_{m},E)=\int {{{\mathrm{d}}}}{{{{{\bf{r}}}}}}\ {{{{\mathrm{e}}}}}^{-i{{{{{\bf{p}}}}}}\cdot {{{{{\bf{r}}}}}}}{{{{{\bf{e}}}}}}\cdot {{{{{\bf{r}}}}}}{\widetilde{\psi }}_{m}({{{{{\bf{r}}}}}}),$$
(4)

where e denotes the polarization vector, while p = (km,x, km,y, p) is the three-dimensional momentum of the photoelectron. The hybridized initial state $${\widetilde{\psi }}_{m}({{{{{\bf{r}}}}}})$$ is given by the superposition

$${\widetilde{\psi }}_{m}({{{{{\bf{r}}}}}})=\mathop{\sum}\limits_{j}{C}_{jn}({{{{{{\bf{k}}}}}}}_{m}){\phi }_{j}({{{{{\bf{r}}}}}})\,,$$
(5)

where the band index n corresponds to the band near EF. The magnitude of the matrix element (4) is thus determined by the hybrid orbital (5), in particular its orientation with respect to the photoelectrom momentum p. The hydrid orbital (5) corresponds to taking the R = 0 contribution in Eq. (3). Including only the well-localized Ti-d orbitals, we find $${\widetilde{\psi }}_{m}({{{{{\bf{r}}}}}})\approx N{\psi }_{{{{{{\bf{{k}}}}}_{m}}}n}({{{{{\bf{r}}}}}})$$ (up to a trivial phase) in the unit cell; the hybrid orbital (5) is thus a good representation of the Bloch wavefunction.

In Fig. 7b we plot $${\widetilde{\psi }}_{m}({{{{{\bf{r}}}}}})$$ for the α = 0 sample orientation. Inspecting Fig. 7b lead us to the following observation: the electronic wavefunction can be qualitatively described by tilted $${d}_{{z}^{2}}$$ orbital, where the tilt angle is determined by the coefficients Cjn(km) and is thus momentum-dependent. This rotated $${d}_{{z}^{2}}$$ is mainly composed of the $${d}_{{z}^{2}}$$, dxz and dyz orbitals in the laboratory frame. By projecting the hybrid wavefunction at each M/M’ point onto the xz plane and comparing their relative orientation to the outgoing electron momentum p, the origin of the strong iLDAD becomes clear. For m = 1, 3, 5 the lobes of $${\widetilde{\psi }}_{m}({{{{{\bf{r}}}}}})$$ align with p, leading to enhanced photoemission intensity, while for m = 2, 4, 6 the photoelectron momentum is roughly pointing along the nodal direction of the hybrid orbitals. Rotating by 60° or equivalently transforming z → −z inverts this behavior, as the alignment of p with the orientation of the lobes (nodal direction) transforms into alignment with the nodal direction (orientation of the lobes).

The iLDAD is thus sensitive to the orbital orientation. In contrast to regular linear dichroism, the sample rotation $${{{{{{\mathcal{R}}}}}}}_{6{0}^{\circ }}$$ changes the relative orientation of photoelectron momentum and orbitals while keeping the light polarization fixed. Revisiting Eq. (4), the photoemission matrix element (within the PW approximation) in essence is a Fourier transform of the hybrid orbital. From the momentum-space dependece of iLDAD, we gain insights into the real-space orientation of the orbitals. This information is complementary to varying the light polarization, where the matrix element is determined by the selection rules. In particular, orbital tilting as in 1T-TiTe2 is hard to discern directly from standard linear dichroism—when comparing the photoemission intensity modulation upon swapping between s- and p-polarized light.

## Discussion

We have extended the recently introduced differential measurement methodology in multidimensional photoemission spectroscopy to investigate the role of orbital texture in photoemission from a semimetallic layered TMDC (1T-TiTe2). Indeed, we introduced the intrinsic linear dichroism in photoelectron angular distributions (iLDAD)—a generalized version of TRDAD, in samples featuring inversion symmetry. By tracking the photoemission intensity modulation upon specific crystal rotation in the laboratory frame, we experimentally extracted the intrinsic linear dichroism in the photoelectron angular distribution. Comparing our results with two different theoretical calculations based on a quantitative one-step model of photoemission and an intuitive tight-binding model, we provide an intuitive interpretation for the link between orbital texture and the pronounced photoemission intensity anisotropies from momentum regions with equivalent energy eigenvalues but with nonequivalent electronic wavefunctions. Our joint experimental and theoretical work demonstrated the direct relation between momentum-dependent hybridized orbital orientations and photoemission intensities, unambiguously ruling out final state effects as the primary origin of the observed dichroism50,51.

The methodology that we used is general and can be exploited to investigate a plethora of phenomena governed by orbital physics. Indeed, one expects very similar types of orbital texture to be common in governing properties of similar TMDC exhibiting multi-component charge order (e.g. 1T-TiSe252 and 1T-VSe253), so that our methodology can be used to investigate the link between the orbital texture of the emergence of CDW38,54. For example, we envision that performing a temperature-dependent iLDAD investigation for monolayer 1T-TiTe2 across its CDW transition might allows to shed light on the possible intricate nature of the orbital order and the emergence of CDW phases. In particular, the proposed excitonic insulator state55,56 gives rise to the admixture of the orbital character from the Γ point at the M/M’ pockets. Hence, the iLDAD is expected to exhibit pronounced changes, thus serving as a hallmark of the emergence of the excitonic insulator phase. Our approach could potentially also be used to investigate the orbital texture of exotic and stable hidden quantum states (e.g. in 1T-TaS257) which are not accessible using equilibrium tuning parameters.

Moreover, applying stress to the material investigated here (1T-TiTe2) has been predicted to drive strain-induced topological phase transitions58 associated with a change of the orbital texture. Strain-dependent measurement of the iLDAD could thus allow providing energy- and momentum-resolved view on the modification of orbital texture underlying topological phase transitions. This approach could also be used to investigate the role of orbital texture in the emergence of spontaneous electronic nematic ordering giving rise to the momentum-anisotropy of superconducting gap, for example in FeSe59.

One additional route that we envision is the ultrafast time-resolved investigation of dynamical modification of orbital texture upon impulsive photoexcitation of solids. Angle-resolved photoemission spectroscopy in general, and more specifically the experimental setup used in this study, is compatible with time-resolved studies with femtosecond temporal resolution40,42,60. Thus, combining this differential observable with pump-probe schemes could allow following in real-time modification of orbital texture during non-equilibrium topological phase transitions61,62, photoinduced orbital order63 and coherent phonon excitation breaking fundamental symmetries of the materials64.

## Methods

### Angle-resolved photoemission spectroscopy

The experimental apparatus features a table-top femtosecond XUV beamline coupled to a photoemission end-station. Briefly, a home-built optical parametric chirped-pulse amplifier (OPCPA) delivering 15 W (800 nm, 30 fs) at 500 kHz repetition rate65 is used to drive high-order harmonic generation (HHG) by tightly focusing the second harmonic of the laser pulses (400 nm) onto a thin and dense Argon gas jet. The nonperturbative nonlinear interaction between the laser pulses and the Argon atoms leads to the generation of a comb of odd harmonics of the driving laser, extending up to the 11th order. A single harmonic (7th order, 21.7 eV) is isolated by reflection on a focusing multilayer XUV mirror and propagation through a 400 nm-thick Sn metallic filter. A photon flux of up to 2 × 1011 photons/s at the sample position is obtained (110 meV FWHM). The bulk 1T-TiTe2 samples (HQ Graphene) were cleaved at room temperature and base pressure of 5 × 10−11 mbar, and handled by a six-axis manipulator (SPECS GmbH). The photoemission data is acquired using a time-of-flight momentum microscope (METIS1000, SPECS GmbH). This detector allows for simultaneous detection of the full surface Brillouin zone, over an extended binding energy range, without the need to rearrange the sample geometry41. For the data post-processing, we use a recently developed open-source workflow66 to efficiently convert the raw single-event-based datasets into binned calibrated data hypervolumes of the desired dimension, including axes calibration and artifact corrections (including symmetry distortion corrections67). The resulting 3D photoemission intensity data have the coordinates I(E, kx, ky).

### DFT and tight-binding model

We have performed DFT calculations for both the bulk and the monolayer 1T-TiTe2 using the QUANTUM ESPRESSO code68. All calculations were performed using the PBE functional and norm-conserving pseudopotentials from the PSEUDODOJO project69. Self-consistent calculations were converged on a 15 × 15 Monkhorst–Pack grid for the Brillouin zone. For the monolayer, we constructed projective Wannier on the same grid with the WANNIER90 code70. We included the Ti-d orbitals and the p orbitals on each of the two Te atoms, yielding an 11-band TB model. This procedure yields the TB Hamiltonian $${H}_{j{j}^{\prime}}^{{{{{{\rm{TB}}}}}}}({{{{{\bf{k}}}}}})$$ and thus the coefficients defining the Bloch wavefunction in Eq. (3). The real-space dependence of the Wannier functions is approximated by

$${\phi }_{j}({{{{{\bf{r}}}}}})={R}_{j}(r){X}_{{\ell }_{j}{m}_{j}}(\hat{{{{{{\bf{r}}}}}}})\,,$$
(6)

where $${X}_{\ell m}(\hat{{{{{{\bf{r}}}}}}})$$ denotes the real spherical harmonics. Due to the small weight of the p orbitals we only construct the Ti d orbitals explicitly; the radial dependence entering Rj(r) is taken from the $${d}_{{z}^{2}}$$ Wannier orbital and fixed for all d orbitals. We use atomic units in what follows.

From the Wannier functions (6) and the eigenvectors Cjn(k), we construct the Bloch states via Eq. (3), which allows us to calculate the ARPES intensity according to Fermi’s Golden rule

$$I({{{{{\bf{k}}}}}},E)=\mathop{\sum}\limits_{n}{f}_{n}({{{{{\bf{k}}}}}})| {M}_{n}({{{{{\bf{k}}}}}},E){| }^{2}\delta ({\varepsilon }_{n}({{{{{\bf{k}}}}}})+\omega -E)\,,$$
(7)
$${M}_{n}({{{{{\bf{k}}}}}},E)=\langle {{{{{\bf{k}}}}}},E| {{{{{\bf{e}}}}}}\cdot \hat{{{{{{\bf{D}}}}}}}| {\psi }_{{{{{{\bf{k}}}}}}n}\rangle \,.$$
(8)

Here, fn(k) is the occupation of band n with energy εn(k), ω is the photon energy, and E denotes the energy of the final state $$\left|{{{{{\bf{k}}}}}},E\right\rangle$$. The band structure is shifted to reproduce the experimentally measured work function. The dipole operator is evaluate in the dipole gauge ($$\hat{{{{{{\bf{D}}}}}}}={{{{{\bf{r}}}}}}$$), which is defined in terms of the Berry connection. We ignore nonlocal dipole contributions since photoemission at the M/M’ pockets is predominantly originating from the Ti d orbitals. The final states are approximated by plane waves. In this set-up, the dipole gauge has proven to be more accurate than the velocity gauge ($$\hat{{{{{{\bf{D}}}}}}}=-i\nabla$$) for a number of related systems14,24,71. Within this TB + PW model, the photoemission matrix element is then given by

$${M}_{n}({{{{{\bf{k}}}}}},E)=\mathop{\sum}\limits_{j}{C}_{jn}({{{{{\bf{k}}}}}})\int {{{\mathrm{d}}}}{{{{{\bf{r}}}}}}{{{{\mathrm{e}}}}}^{-i{{{{{\bf{p}}}}}}\cdot {{{{{\bf{r}}}}}}}{{{{{\bf{e}}}}}}\cdot {{{{{\bf{r}}}}}}{\phi }_{j}({{{{{\bf{r}}}}}})\,.$$
(9)

Here, p = (kx, ky, p) is the three-dimensional momentum of the photoelectron; the out-of-plane component p is fixed by the kinetic energy E = p2/2. The orbital-resolved photoemission analysis in the section “The role of orbital texture in photoemission: tight-binding analysis” is performed by including a subset of $$j\in \{{d}_{{z}^{2}},{d}_{xz},{d}_{yz},{d}_{{x}^{2}-{y}^{2}},{d}_{xy}\}$$ orbitals in Eq. (9).

### One-step model of photoemission: KKR method

The one-step model of photoemission is implemented in the spin-polarized relativistic Korringa–Kohn–Rostoker (SPR-KKR) scheme of the Munich bandstructure software package, based on Green’s function and multiple scattering spin-density matrix formalisms48,49. Relativistic phenomena such as SOC are treated by the Dirac equation. The LDA has been chosen to approximate the exchange-correlation part of the potential. The bulk potential is converged within the atomic sphere approximation (ASA) geometry where the empty spheres were used. We have used a lattice constant of 3.779 Å which corresponds to the interlayer Te–Te distance of 4.015 Å. During the tuning procedure of the theoretical model, it was necessary to modify the Wigner–Seitz radius of individual atomic types to the following ratio: Ti = 1.03, Te = 1.08 and the vacuum type = 1. We used an angular momentum expansion up to a lmax = 3. After the self-consistency was reached, the ARPES calculations have been performed and are based on the one-step model of photoemission in its spin density matrix formulation using the same geometry as used in the experiments. This theory accounts for effects induced by the light polarization, matrix-element effects, final state effects and surface effects. The final states are described by TRLEED states. These states are calculated by multiple scattering formalism and it are considered to give the best single particle description of final states in photoemission. In particular, this approach goes far beyond free-electron-like approximation for the final state. Lifetime effects in the final states have been simulated via a constant imaginary part V0f = 1 eV in the inner potential and the lifetime of the initial state was simulated by an imaginary part of V0i = 0.01 eV. The photoemission signal factors in all the matrix element effects such as experimental geometry, photon energy, polarization state, and surface termination or final state effects.


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# Is the Lagrangian a kind of Energy

1. Jan 9, 2007

### Kevin_spencer2

If we have that Lagrangian is equal to L=T-V and Hamiltonian is just H=T+V my question is if for every lagrangian we could consider is just the energy of a particle with a potential -V(x) instead of V(x) what would happen if V(-x)=-V(x)?

In fact if we substitute velocities by momenta in the Lagrangian teh Euler-Lagrange equation for a particle is just (putting -V(x) instead of V(x)) the Hamilton equation $$\dot p =-\frac{dH}{dp}$$.

then it occurred to me if we could use this fact to quantizy the dynamics of a particle using Lagrangians as if they were the pseudo-energy of the system to get energy levels.

Last edited: Jan 9, 2007
2. Jan 9, 2007

### dextercioby

Nope, it doesn't work that way. Lagrangians are not observables, they cannot be measured. There's nothing "pseudo" about them. Their unit is energy, just like for a Hamiltonian, but nothing more. They're just a tool. Very useful in quantum physics, i must say.

Daniel.

3. Jan 9, 2007

### Kevin_spencer2

then, what would we get if we put $$L(p,q)|\Phi >$$ where our $$\Phi$$ is just the wave function of a quantum system

4. Jan 9, 2007

### StatMechGuy

I'm not sure why you'd want the eigenstates of the lagrangian. It's certainly observable, but I think that's beyond the point of this thread (being in classical mechanics). I personally have never seen it used, which doesn't mean that it's not of value, it's just that lagrangians enter quantum mechanics in a way that is a little independent of state vectors.

You can measure the lagrangian by just measuring the kinetic and potential energy, certainly. But the physical quantity of most interest is the action, which is just $$S = \int_t L(q, \dot{q},t)$$, and which is the basis for a lot of least-action principles, etc.

5. Jan 9, 2007

### Kevin_spencer2

- the question is without prior knowledge of the system if they give you a quantity $$F(\dot q , q)$$ how could you know if you've got the energy of the whole system or just the Lagrangian?, in fact you could write the minimum-principle in the form $$S=\int_{a}^{b}dt \mathcal H(q,\dot q, t)$$ where H has -V(x) instead of V(x), the question is that it's easier to work with Lagrangians rather than Hamiltonians, and that for example the Lagrangian for GR is just $$(-g)^{1/2}R$$, however the Hamiltonian is just 0, which arises to lots of problems.

6. Jan 9, 2007

### StatMechGuy

The hamiltonian need not equal T + V; it depends on the form of the lagrangian. The lagrangian is also not, generally, T - V -- look to magnetic phenomena to confirm this.

7. Jan 10, 2007

### dextercioby

The Hamiltonian is not 0 for GR, search for ADM Hamiltonian formulation of general relativity.

Daniel.


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×

# How to Solve It?

How can we prove that-

$$n \times (n^{2} + 20)$$ is divisible by 48.

[Given : n is a positive even number]

Note by Vatsalya Tandon
2 years, 1 month ago

Sort by:

Since n is even, Let n=2k for some k.

Then, The Expression becomes,

$$2k({(2k)}^{2}+20)$$ = $$8k({k}^{2}+5)$$

Thus, 8 is a factor. Now, we know that one of k or $${k}^{2}+5$$ has to be a multiple of 2. Hence We get that 2 is also a factor.

let $$n({n}^{2}+20)= {n}^{3}+20n$$.

One can see that, $${n}^{3}=n (mod 3)$$. So, $${n}^{3}+20n$$ = n+20n =21n Which is congruent to 0 (mod 3)

Hence, We find that 3 is also a factor. So, The number is completely divisible by $$8*2*3=48$$.

Hence, Proved :D

This was my first Proof. If you spot any mistakes, Kindly inform it to me. I will clean it up. · 2 years, 1 month ago

Cheers!!! xD · 2 years, 1 month ago

LOL, Cheers! xD · 2 years, 1 month ago

Good! · 2 years, 1 month ago

Thanks! :D · 2 years, 1 month ago

Let n be 2k :

Then, $$2k(4k^{2} +20)$$

$$8k^{3}+40k$$

$$8k(k^{2}+5)$$

$$8k(k^{2}-1+6)$$

$$8k((k+1)(k-1)+6)$$

$$8(k-1)(k)(k+1)+8k \times 6$$

$$8(k-1)(k)(k+1)+48k$$

Now, $$48k$$ is clearly divisible by 48 and,

$$(k-1)(k)(k+1)$$ are three consecutive numbers so they will be divisible by $$3! = 6$$ AND $$(k-1)(k)(k+1)$$ is multiplied by 8 so the product will be divisible by $$8 \times 6$$ which is $$48$$.

It is proved now that $$8(k-1)(k)(k+1)$$ and $$48k$$ are both divisible by 48 and when you add two numbers which are divisible by 48, obviously the sum will also be divisible by 48. · 2 years, 1 month ago

Yeah even i did the same way · 1 year ago

Ok I Finally Get This-

Let us assume $$n$$ to be an even number $$2k$$. Now, the equation becomes- $$2k(4k^{2}+20)$$ and then $$8k^{3}+40k$$. Taking $$8k$$ to be common like this- $$8k(k^{2} + 5)$$, we can split "5" in the equation to form this- $$8k(k^{2} + 6 - 1)$$. And factorizing $$k^{2} + 1$$ gives us, $$8k((k+1)(k-1) + 6)$$. Here we would be using an identity that the product of $$n$$ consecutive numbers is divisible by $$n!$$. Thus in this case we have 3 consecutive numbers- $$k, k+1, k-1$$, which means that this part is divisible by $$6$$. So, this part- $$8 * (k) * (k+1) * (k-1)$$ is divisible by $$48$$ (as we have 8 multiplied to it and 6 divides it). Now the remaining part is- $$8k*6$$ which in itself is divisible by 48. Thus the sum of 2 numbers divisible by 48 will give a number divisible by 48.

Q.E.D (Hope if you Like it!). And thanks to @MehulArora for his initial solution. Cheers! · 2 years, 1 month ago

My Pleasure @Vatsalya Tandon · 2 years, 1 month ago

Sorry Guys! Updated the Values! Sorry for the Inconvenience! · 2 years, 1 month ago

Comment deleted Apr 19, 2015

Comment deleted Apr 18, 2015

1 isn't even plz see read the question more carefully. · 2 years, 1 month ago

Yeah. Sorry :P :P :P xD · 2 years, 1 month ago

Delete it @Siddhartha Srivastava :) -_- · 2 years, 1 month ago

Taking $$n=2x$$ and simplifying a bit gives,$$8(x^3+5x)$$.Now,we just need to prove that $$x^3+5x\equiv 0\pmod{6}$$.For doing that we first write $x(x^2+5)\\ =x(x^2+6-1)\\ =x^3+6x-x\\ =x^3-x+6x\\ x(x^2-1)+6x\\ (x-1)(x)(x+1)+6x$.Now,we just need to prove that $$(x-1)(x)(x+1)\equiv 0\pmod{6}$$ which is obvious.Hence proved.@Vatsalya Tandon · 2 years, 1 month ago


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# Questions tagged [noise]

In signal processing noise can be considered random unwanted data without meaning.

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### Does sum of random variables follows gaussian distribution although each have different mean and variances?

I understand Central limit theorem, but cannot understand the result when it applies to diverse situation. As far as I understand, Central limit theorem states follows: No matter what random variable ...
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### Stockwell S-Transform and multitapering

I want to estimate just the spectral amplitudes at different time points using S-transform given a noisy signal. Would I benefit (in terms of noise reduction) from an approach used in multi taper ...
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### Signal-to-Noise ratio of multivariate stochastic process from Correlation Matrix

I'm not in signal processing, I'm from an another discipline. I've derived a simple result which I presume must be well known in SP and I'd like to know whether there's a paper or textbook that has it ...
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### Concept of energy threshold

I am analyzing some program responsible for detecting the correct signal in the channel on the receiver side. It seems to me that such an implementation allows for the detection of the correct packets ...
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### Is it possible to remove the non-random noise in the signal by averaging?

Ex: We're measuring vibration Frequency response of a structure, and at the same time there is a constant source of vibration (noise) from a pump. The pump is exciting the structure by sinusoid ...
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### Recursive Least Square For Filtering

I just started doing research on Recursive Least Square for filtering noises such as sensors and dc motors noises. The only thing I've seen on the internet was Theoretical information about RLS but ...
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### Extracting ridges in automatically in image

I am trying to extract ridges from an image, calculated using the eigenvalues of the Hessian matrix (see answer in Best way of segmenting veins in leaves?). In this image I have zeroed out the ...
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### What kind of effect is this and what is it caused by?

In the screenshot below I have two audio signals. The upper one is the audio signal that is created with python and then played, whilst the lower one is the "same" signal but recorded via ...
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### What is the advantage of Wiener filter for noise reduction of a 1D signal?

I have a 1D signal, acquired by an accelerator sensor that measure the vibration of a structure. What is the advantage of using a Wiener filter for noise reduction compared to the other (more classic) ...
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### Wavelet denoising. Simple explanation

I am studying wavelet transform. In the matlab in the "wavemenu" package, I use "SWT denoising 1-D". I loaded my signal, performed a Haar 5 wavelet squelch. I got a good result. I ...
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### How can I add any kind of noise to image for a given SNR?

What is the best way to add any kind of noise (not only Gaussian noise) to 2D signal (image) for a given SNR?