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# Trigonometric Functions

Trigonometric functions represent the relationship between sides and angles in right triangles.

There are three main “trig” functions: sine, cosine, and tangent, and a mnemonic often used to remember what they represent is SohCahToa:

• • The SINE of an angle is the ratio of the lengths of the OPPOSITE side and the HYPOTENUSE.
• • The COSINE of an angle is the ratio of the lengths of the ADJACENT side and the HYPOTENUSE.
• • The TANGENT of an angle is the ratio of the lengths of the OPPOSITE side and the ADJACENT side.

For example, in the triangle below, we have $\sin \theta = \frac{4}{5}$, $\cos \theta = \frac{3}{5}$, and $\tan \theta = \frac{4}{3}$.

## Solving for a Side

Trig functions can be used to solve for unknown side lengths in right triangles. For example, if we know that an angle is $25^\circ$, the opposite side has a length of $10$, and we want to find the hypotenuse, we can set up and solve an equation using sine.

\begin{align*} \text{Sine equation} \hspace{.5cm} &\Bigg| \hspace{.5cm} \sin 25^\circ = \frac{10}{c} \\ \text{Solve} \hspace{.5cm} &\Bigg| \hspace{.5cm} c = \frac{10}{\sin 25^\circ} \\ \text{Estimate via calculator} \hspace{.5cm} &\Bigg| \hspace{.5cm} c \approx 23.66 \\ \end{align*}

To find the remaining side, we can use any of three methods: Pythagorean theorem, cosine, or tangent.

No matter which technique we use, we will end up with the same result (though if we use our approximation of $c \approx 23.66$, we might be slightly off due to rounding error).

\begin{align*} \text{Pythagorean theorem} \hspace{.5cm} &\Bigg| \hspace{.5cm} 10^2 + b^2 = 23.66^2 \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} b = \sqrt{ 23.66^2 - 10^2} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} b \approx 21.44 \\ \text{Cosine} \hspace{.5cm} &\Bigg| \hspace{.5cm} \cos 25^\circ = \frac{b}{23.66} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} b = 23.66 \cos 25^\circ \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} b \approx 21.44 \\ \text{Tangent} \hspace{.5cm} &\Bigg| \hspace{.5cm} \tan 25^\circ = \frac{10}{b} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} b = \frac{10}{\tan 25^\circ} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} b \approx 21.45 \\ \end{align*}

## Solving for an Angle

Similarly, using inverse trig functions, we can solve for unknown angles in right triangles.

For example, if we know that the adjacent side is $5$ and the opposite side is $7$, we can set up an equation with tangent and then use inverse tangent to find the angle.

\begin{align*} \text{Tangent equation} \hspace{.5cm} &\Bigg| \hspace{.5cm} \tan A = \frac{7}{5} \\ \text{Inverse tangent} \hspace{.5cm} &\Bigg| \hspace{.5cm} A = \tan^{-1} \left( \frac{7}{5} \right) \\ \text{Evaluate via calculator} \hspace{.5cm} &\Bigg| \hspace{.5cm} A &\approx 54.46^\circ \end{align*}

## The Unit Circle

To find the remaining angle, we can use any of three methods: sum of degrees in a triangle, tangent, or Pythagorean theorem followed by sine or cosine. Regardless of which method we choose, we will end up with the same result.

\begin{align*} \text{Sum of degrees in triangle} \hspace{.5cm} &\Bigg| \hspace{.5cm} 54.46^\circ + B + 90^\circ = 180^\circ \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B \approx 35.54^\circ \\ \text{Inverse tangent} \hspace{.5cm} &\Bigg| \hspace{.5cm} \tan B = \frac{5}{7} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B = \tan^{-1} \left( \frac{5}{7} \right) \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B \approx 35.54^\circ \\ \text{Pythagorean theorem} \hspace{.5cm} &\Bigg| \hspace{.5cm} 7^2 + 5^2 = c^2 \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} c = \sqrt{74} \\ \text{Inverse sine} \hspace{.5cm} &\Bigg| \hspace{.5cm} \sin B = \frac{5}{\sqrt{74}} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B = \sin^{-1} \left( \frac{5}{\sqrt{74}} \right) \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B \approx 35.54^\circ \\ \text{Inverse cosine} \hspace{.5cm} &\Bigg| \hspace{.5cm} \cos B = \frac{7}{\sqrt{74}} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B = \cos^{-1} \left( \frac{7}{\sqrt{74}} \right) \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} B \approx 35.54^\circ \end{align*}

To gain a better understanding of trig functions, we can imagine putting a triangle inside of a circle on the coordinate plane.

The coordinates of the corner point $(x,y)$ on the circle then tell us the other two sides of the triangle: the horizontal side has length $x$ and the vertical side has length $y$. If we make the circle have radius $1$, then the hypotenuse of the triangle is $1$, and we have

\begin{align*} \cos \theta &= \frac{x}{1} = x \\ \sin \theta &= \frac{y}{1} = y \end{align*}

and our point $(x,y)$ can be written as $(\cos \theta, \sin \theta)$.

Immediately, we notice two important things. First, using the Pythagorean theorem on the triangle, we see that

\begin{align*} \sin^2 \theta + \cos^2 \theta = 1. \end{align*}

This is a handy equation that can be useful in simplifying trigonometric expressions. For example, the expression $(\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2$ is actually just equivalent to $2$.

\begin{align*} &(\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2 \\ &= (\sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta) \\ &\hspace{1cm} + (\sin^2 \theta - 2 \sin \theta \cos \theta + \cos^2 \theta) \\ &= 2 \sin^2 \theta + 2 \cos^2 \theta \\ &= 2 \end{align*}

Second, angles repeat every $360^\circ$, since going $360^\circ$ around the circle brings us back to the starting point of $0^\circ$.

That means, for example, that $\sin 390^\circ$ and $\sin -330^\circ$ are both equivalent to $\sin 30^\circ$.

\begin{align*} 390^\circ &= 30^\circ + 360^\circ \\ -330^\circ &= 30^\circ - 360^\circ \end{align*}

## Special Angles

For most angles, a calculator is needed to compute the corresponding trig function values. However, at particular angle measures, the trig functions have simple, exact values:

We can remember which values correspond to which angles and which trig functions by thinking about them visually in the unit circle and mentally pairing $\frac{ \sqrt{3}}{2}$ with $\frac{1}{2}$.

• • At $30^\circ$, the x-coordinate is bigger than the y-coordinate, so the x-coordinate must be $\frac{\sqrt{3}}{2}$ and the y-coordinate must be $\frac{1}{2}$.
• • At $60^\circ$, this is reversed. At $45^\circ$, the x-coordinate and y-coordinate are the same, so they both are $\frac{\sqrt{2}}{2}$.
• • At $0^\circ$, we’re on the x-axis, so the x-coordinate is $1$ and the y-coordinate is $0$.
• • At $90^\circ$, we’re on the y-axis, so the y-coordinate is $1$ and the x-coordinate is $0$.

To get tangent, we can just take the ratio of the y-coordinate to the x-coordinate.

Using symmetry, we can label angles in the other three quadrants of the circle.

## Derivation of Special Angles

You might be wondering where the values $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$ come from in the first place.

To see where $\frac{\sqrt{2}}{2}$ comes from, we can construct a right triangle with a hypotenuse of $1$ and an angle of $45^\circ$.

The other angle must also be $45^\circ$, so the triangle’s two legs must be equal in length, and we can use the Pythagorean theorem to discover that the length of each leg is $\frac{ \sqrt{2}}{2}$.

\begin{align*} \text{Pythagorean theorem} \hspace{.5cm} &\Bigg| \hspace{.5cm} x^2 + x^2 = 1^2 \\ \text{Simplify} \hspace{.5cm} &\Bigg| \hspace{.5cm} 2x^2 = 1^2 \\ \text{Solve} \hspace{.5cm} &\Bigg| \hspace{.5cm} x^2 = \frac{1}{2} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} x = \sqrt{ \frac{1}{2} } \\ \text{Simplify} \hspace{.5cm} &\Bigg| \hspace{.5cm} x = \frac{1}{\sqrt{2} } \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} x = \frac{\sqrt{2} }{2 } \\ \end{align*}

Likewise, to see where $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$ come from, we can construct a right triangle with a hypotenuse of $1$ and an angle of $60^\circ$.

The other angle must be $30^\circ$, which is exactly half – consequently, we can combine two of these triangles to form an equilateral triangle whose side lengths are all equal to the hypotenuse of $1$.

The shortest sides of the two triangles together make up a side of the equilateral triangle, which we know has length $1$, so the shortest sides of the two triangles must each be $\frac{1}{2}$. Using the Pythagorean theorem, we find that the length of the other leg is $\frac{\sqrt{3}}{2}$.

\begin{align*} \text{Pythagorean theorem} \hspace{.5cm} &\Bigg| \hspace{.5cm} \left( \frac{1}{2} \right)^2 + x^2 = 1^2 \\ \text{Simplify} \hspace{.5cm} &\Bigg| \hspace{.5cm} \frac{1}{4} + x^2 = 1 \\ \text{Solve} \hspace{.5cm} &\Bigg| \hspace{.5cm} x^2 = \frac{3}{4} \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} x = \sqrt{ \frac{3}{4} } \\ \text{Simplify} \hspace{.5cm} &\Bigg| \hspace{.5cm} x = \frac{\sqrt{3} }{2} \\ \end{align*}

## Graphs

The graphs of sine, cosine, and tangent are drawn below. They repeat every $360^\circ$, since $360^\circ$ is one full revolution around the unit circle and thus brings us full-circle back to the starting point.

Tangent actually repeats twice every $360^\circ$ (or once every $180^\circ$) because it goes from positive to negative from the first to second quadrant, and again positive to negative from the third to fourth quadrant.

To make sense of the shapes of the graphs, try to trace out the trig function values while following around the unit circle.

The standard way to measure angles is actually not in degrees – rather, it is in radians. One radian is equivalent to the angle whose arc is equal to one radius of a circle.

Since the full arc length (circumference) of the circle is $2\pi$ times the radius, a full $360^\circ$ around the circle is equivalent to $2\pi$ radians.

Below is a copy of the unit circle, using radians instead of degrees.

The trig functions are graphed in terms of radians below. Nothing changes, except for the units of the x-axis.

## Reciprocal Trigonometric Functions

There are three other trig functions: secant, cosecant, and cotangent. They are just the reciprocals of cosine, sine, and tangent.

Consequently, they can be understood by thinking about the properties of cosine, sine, and tangent. We will not explore them further, but we include their graphs below.

## Exercises

Use trigonometry to find the missing sides and angles of the triangles. (You can view the solution by clicking on the problem.)

$1)$

Solution:
$a \approx 6.58$
$b \approx 2.39$
$B = 20^\circ$

$2)$

Solution:
$a \approx 5.96$
$c \approx 7.78$
$A = 50^\circ$

$3)$

Solution:
$c = \sqrt{5} \approx 2.24$
$A \approx 63.43^\circ$
$B \approx 26.57^\circ$

$4)$

Solution:
$b = 6$
$A \approx 53.13^\circ$
$B \approx 36.87^\circ$

Use the unit circle to find the exact values of the following trigonometric expressions.

$5) \hspace{.5cm} \cos 0^\circ$
Solution:
$1$

$6) \hspace{.5cm} \sin 270^\circ$
Solution:
$-1$

$7) \hspace{.5cm} \sin 60^\circ$
Solution:
$\frac{\sqrt{3}}{2}$

$8) \hspace{.5cm} \tan 30^\circ$
Solution:
$\frac{ \sqrt{3}}{3}$

$9) \hspace{.5cm} \cos 135^\circ$
Solution:
$-\frac{ \sqrt{2}}{2}$

$10) \hspace{.5cm} \tan 270^\circ$
Solution:
$\text{undefined}$

$11) \hspace{.5cm} \sin 135^\circ$
Solution:
$\frac{ \sqrt{2}}{2}$

$12) \hspace{.5cm} \cos 240^\circ$
Solution:
$-\frac{1}{2}$

$13) \hspace{.5cm} \sin 720^\circ$
Solution:
$0$

$14) \hspace{.5cm} \cos 480^\circ$
Solution:
$-\frac{1}{2}$

$15) \hspace{.5cm} \tan 765^\circ$
Solution:
$1$

$16) \hspace{.5cm} \cos -45^\circ$
Solution:
$\frac{ \sqrt{2}}{2}$

$17) \hspace{.5cm} \sin -150^\circ$
Solution:
$-\frac{1}{2}$

$18) \hspace{.5cm} \cos \frac{\pi}{3}$
Solution:
$\frac{1}{2}$

$19) \hspace{.5cm} \sin \frac{5\pi}{6}$
Solution:
$\frac{1}{2}$

$20) \hspace{.5cm} \tan \frac{2\pi}{3}$
Solution:
$-\sqrt{3}$

$21) \hspace{.5cm} \cos \pi$
Solution:
$-1$

$22) \hspace{.5cm} \tan \frac{7\pi}{6}$
Solution:
$\frac{ \sqrt{3}}{3}$

$23) \hspace{.5cm} \sin 2\pi$
Solution:
$0$

$24) \hspace{.5cm} \cos 5\pi$
Solution:
$-1$

$25) \hspace{.5cm} \tan \frac{9\pi}{4}$
Solution:
$1$

$26) \hspace{.5cm} \cos \frac{5\pi}{2}$
Solution:
$0$

$27) \hspace{.5cm} \sin -\frac{3\pi}{2}$
Solution:
$1$

$28) \hspace{.5cm} \cos -\frac{\pi}{6}$
Solution:
$\frac{ \sqrt{3}}{2}$

$29) \hspace{.5cm} \tan -\frac{\pi}{3}$
Solution:
$-\sqrt{3}$

$30) \hspace{.5cm} \tan -\frac{5\pi}{4}$
Solution:
$-1$

Tags:


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# Problems from having too many interactions in a regression?

Excluding the 'dummy variable trap', are the problems from including too many interaction terms in a regression any different from the problems of including too many continuous or binary variables in a regression?

Furthermore, are the responses to this question conditional on interaction terms being continuous-with-continuous or binary-with-continuous?

If they involve categorical main effects they can lead to unexpectedly high loss of degrees of freedom as people often fail to take into account that the product of the degrees of freedom for the main effects is involved so if you have an interaction between a categorical variable with 5 levels and one with 6 you lose $$(5 - 1) \times (6 -1) = 20$$ more df.

format: plain


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# Goodness of fit tests for quantile regression in R

What goodness of fit tests are usually used for quantile regression? Ideally I need something similar to F-test in linear regression, but something like AIC in logistic regression will suite as well. I use quantreg R package, but found only some Khmaladze test in there. To be fair I hardly understand what is does.

## 1 Answer

quantreg includes several AIC functions: "AIC.nlrq", "AIC.rq", "AIC.rqs" and "AIC.rqss" and similar log likelihood functions.

It also has a vignette at vignette("rq",package="quantreg").

Do these do what you want?


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The ring shear box is analyzed using an isotropic strain hardening/softening model for granular media, permitting an estimate to be made of the stresses developed under conditions of simple shear, at the critical void ratio. Observation of the radial stress on the inner or outer walls of the test chamber would provide a measure of the relative value of the otherwise unknown intermediate principal stress. In a series of five tests on a quartz sand, average pressures exerted by the sand on the outer wall of the test chamber reached well defined, repeatable levels. As interpreted by the theory, the tests showed that the intermediate principal stress was equal to the direct stress on the Coulomb friction planes during simple shear:
$(1−sinφcv)σ1=σ2=(1+sinφcv)σ3$
where σ1 > σ2 > σ3 (compression positive), and φcv is the angle of internal friction at the critical void ratio, ie, during continuing displacement at constant volume. Similar observations are impossible for plane strain in general because displacement cannot be controlled so completely, but it is reasonable to conjecture that the same relationship holds for all such deformations, including those associated with the active and passive Rankine states.
This content is only available via PDF.


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## Computer Science Technical Reports

Technical Report

3-1-2009

TR2009-642

#### Abstract

We consider the approximability of the Unsplittable Flow Problem (UFP) on tree graphs, and give a deterministic quasi-polynomial time approximation scheme for the problem when the number of leaves in the tree graph is at most poly-logarithmic in $n$ (the number of demands), and when all edge capacities and resource requirements are suitably bounded. Our algorithm generalizes a recent technique that obtained the first such approximation scheme for line graphs. Our results show that the problem is not APX-hard for such graphs unless NP \subseteq DTIME(2^{polylog(n)}). Further, a reduction from the Demand Matching Problem shows that UFP is APX-hard when the number of leaves is Omega(n^\epsilon) for any constant \epsilon > 0. Together, the two results give a nearly tight characterization of the approximability of the problem on tree graphs in terms of the number of leaves, and show the structure of the graph that results in hardness of approximation.

COinS


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Time Complexity of the Word Problem for Finite Permutation Groups

Given a finite permutation group, i.e. a subgroup of the symmetric group on $n$ symbols in terms of generators, what is the complexity of the word problem? That is, computing if two words in the generators represent the same group element?

• The obvious algorithm that just computes both permutations explicitly takes $O(n(l_1+l_2))$ time where $l_1$ and $l_2$ are the lengths of the two words. I'd guess that's close to optimal unless you make special assumptions about the generators. – Johannes Hahn Mar 6 '18 at 18:20
• In terms of structural complexity, the problem can be computed in the class DET, as one can reduce it to multiplication of the corresponding sequence of permutation matrices. – Emil Jeřábek Mar 6 '18 at 18:52

1 Answer

As has been pointed out in comments, you cannot hope to do better in general than $O(n(l_1+l_2))$, where $n$ is the degree of the permutation groups and $l_1$, $l_2$ are the lengths of the words.

But from a practical point of view it is important to observe that once a base for $G$ has been computed (see my answer to this question), the word problem can be solved in time $O(k(l_1+l_2)$, where $k$ is the length of the base, because elements of the group are uniquely determined by their actions on the base. This is perhaps the single most important reason why bases play such a fundamental role in computations in permutation groups.

Many groups with which we wish to compute have bases of length much less than $n$, and there are lots of results (by Babai, Shalev, Liebeck, et al) proving the existence of bases with bounded length in certain families of groups. For example primitive groups other than $A_n$ and $S_n$ have bases of length $O(\sqrt{n} \log n)$, which can be improved to $O(\log n)$ by excluding some known families of examples.


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# Problem with natural units in Cosmology

1. Feb 24, 2007

### Magister

1. The problem statement, all variables and given/known data

$$\rho_r = \frac{\pi^2}{30} g_* T^4 = 8.09 * 10^{-34} g/cm^3$$

where $g_*=3.36$ is the degree of freedom of the radiation (equivalent) and $T=2.75 K$ is the CBR temperature today.

The problem is that they dont give me the exact expression and so this relation seems to be dimensionally wrong. I suppose that the full relation must be

$$\rho_r = \frac{\pi^2}{30} g_* \frac{(kT)^4}{(hc)^3}$$

and this way the relation would be dimensionally correct. But when I put the values on it I get

$$\rho_r = 2.88 * 10^{-22} J/cm^3 = 3.20 * 10^{-36} g/cm^3$$

and this is by far wrong. I have spend so much time around this that I am starting to get frustrated!

2. Feb 24, 2007

### nrqed

If you use $\hbar$ instead of "h" in your equation, it works out (the result is $(2 \pi)^3$ times larger).

Patrick

3. Feb 24, 2007

### Magister

Humm! You mean:
$$\rho_r = \frac{\pi^2}{30} g_* \frac{(kT)^4}{(\hbar c)^3} (2\pi)^3$$

Yes, in fact it works out but does this make any sense? Is this because that in natural units is the $\hbar$ that is equal to 1 instead of $h$?

Thanks a lot by the way!!!!!

4. Feb 24, 2007

### nrqed

EDIT : No, this is not what I mean You don't have to put in the extra factor of (2 pi)^3. I mean
$$\rho_r = \frac{\pi^2}{30} g_* \frac{(kT)^4}{(\hbar c)^3}$$
This result will be (2 pi)^3 times larger than the equation written with only h.

That's what I mean, yes. Usually, by "natural units", people mean that they set $\hbar$ equal to 1, not "h". Of course, one could also decide to set h to one instead, but this is not what is usually done.

You are very welcome.

Patrick

Last edited: Feb 24, 2007


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# Which lanthanides are exceptions to the Aufbau principle?

Aug 24, 2017

See here for a fuller explanation on the lanthanides and actinides.

The order by atomic number is down the first column, and then down the second column. In $\textcolor{red}{\text{red}}$ are the 'Aufbau exceptions'.

$\textcolor{w h i t e}{\left[\begin{matrix}\textcolor{red}{L a} & \left(\textcolor{red}{\left[X e\right] 6 {s}^{2} 5 {d}^{1}}\right) & \textcolor{b l a c k}{T b} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{9}}\right) \\ \textcolor{red}{C e} & \left(\textcolor{red}{\left[X e\right] 6 {s}^{2} 4 {f}^{1} 5 {d}^{1}}\right) & \textcolor{b l a c k}{D y} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{10}}\right) \\ \textcolor{b l a c k}{P r} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{3}}\right) & \textcolor{b l a c k}{H o} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{11}}\right) \\ \textcolor{b l a c k}{N d} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{4}}\right) & \textcolor{b l a c k}{E r} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{12}}\right) \\ \textcolor{b l a c k}{P m} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{5}}\right) & \textcolor{b l a c k}{T m} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{13}}\right) \\ \textcolor{b l a c k}{S m} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{6}}\right) & \textcolor{b l a c k}{Y b} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{14}}\right) \\ \textcolor{b l a c k}{E u} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{7}}\right) & \textcolor{b l a c k}{L u} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{14} 5 {d}^{1}}\right) \\ \textcolor{red}{G d} & \left(\textcolor{red}{\left[X e\right] 6 {s}^{2} 4 {f}^{7} 5 {d}^{1}}\right) & \text{ & }\end{matrix}\right]}$

Generally the electron configuration exceptions occur

• early in the lanthanide series (lanthanum, cerium)
• after filling the $4 f$ orbitals halfway (gadolinium).

In general, it has to do with the closeness in energy of the $4 f$, $5 d$, and $6 s$ orbitals. The $5 d$ and $6 s$ are nearly the same energy across the entire lanthanide row, and the $4 f$ is fairly close in energy for the first few lanthanides.

That gives rise to the following rationales:

• $\text{La}$ and $\text{Ce}$ have an electron in their $5 d$ instead of $4 f$ because the $4 f$ is more radially compact, and given the small difference in energy between them (about $\text{6.8 eV}$), it is more energetically favorable to choose the $5 d$, which is radially more diffuse.

• $\text{Gd}$ has about a $\text{16 eV}$ $4 f - 5 d$ energy gap, and apparently, the electron repulsion it would generate to add another $4 f$ electron, combined with the radial compactness of the $4 f$ orbitals, is enough that it would rather have that electron in the $5 d$.


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## Effects of spin density wave quantization on the electrical transport in epitaxial Cr thin films    [PDF]

E. Osquiguil, E. E. Kaul, L. Tosi, C. A. Balseiro
We present measurements of the electrical resistivity, $\rho$, in epitaxial Cr films of different thicknesses grown on MgO (100) substrates, as a function of temperature, $T$. The $\rho(T)$ curves display hysteretic behavior in certain temperature range, which is film thickness dependent. The hysteresis are related to the confinement of quantized incommensurate spin density waves (ISDW) in the film thickness. Our important finding is to experimentally show that the temperature $T_{mid}$ where the ISDW changes from $N$ to $N$\,+\,1 nodes {\it decreases} as the film thickness {\it increases}. Identifying $T_{mid}$ with a first order transition between ISDW states with $N$ and $N$\,+\,1 nodes, and using a Landau approach to the free energy of the ISDW together with Monte Carlo simulations, we show that the system at high temperatures explores all available modes for the ISDW, freezing out in one particular mode at a transition temperature that indeed decreases with film thickness, $L$. The detailed dependence of $T_{mid}(L)$ seems to depend rather strongly on the boundary conditions at the Cr film interfaces.
View original: http://arxiv.org/abs/1212.1408


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Typical methods to conduct meta-analysis—pooling effect sizes or analyzing moderating effects with meta-regression—work under the assumption that the effect size estimates are independent. However, primary studies often report multiple estimates of effect sizes. Presence of multiple effect sizes leads to dependence as the estimates within each study are likely correlated (e.g., because the same participants provide multiple outcome scores). The increasingly popular method to handle such dependence, robust variance estimation (RVE), results in inflated Type 1 error rate when the number of studies is small (Hedges, Tipton & Johnson, 2010; Tipton, 2015).

Tipton (2015) and Tipton & Pustejovsky (2015) examined several small sample correction methods. Tipton (2015) recommended CR2 type correction for RVE as well as the use of Satterthwaite degrees of freedom for single coefficient tests. Tipton & Pustejovsky (2015) examined corrections for multiple-contrast hypothesis tests. The authors found that the HTZ test, which is an extension of the CR2 correction method with the Satterthwaite degrees of freedom, controlled Type 1 error rate adequately even when the number of studies was small. However, Joshi, Pustejovsky & Beretvas (2021) showed, through simulations, that the HTZ test can be conservative. The authors examined another method, cluster wild bootstrapping (CWB), that has been studied in the econometrics literature but not in the meta-analytic context. The results of the simulations from Joshi, Pustejovsky & Beretvas (2021) showed that CWB adequately controlled for Type 1 error rate and provided higher power than the HTZ test, especially for multiple-contrast hypothesis tests. hypothesis tests.

The goal of this package is to provide applied meta-analytic researchers a set of functions with which they can conduct single coefficient tests or multiple-contrast hypothesis tests using cluster wild bootstrapping.

## Installation

You can install the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("meghapsimatrix/wildmeta")

## Example

The following example uses the SATCoaching dataset from the clubSandwich package (Pustejovksy, 2020), originally from DerSimonian and Laird (1983). The standardized mean differences represent the effects of SAT coaching on SAT verbal (SATV) and/or SAT math (SATM) scores.

The code below runs cluster wild bootstrapping to test the multiple-contrast hypothesis that the effect of coaching does not differ based on study type. The study_type variable indicates whether groups compared in primary studies were matched, randomized, or non-equivalent. The meta-regression model also controls for hours of coaching provided (hrs) and whether the students took math or verbal test (test). Below, we run a zero-intercept meta-regression model.

Below, we use the robu() function from the robumeta package to fit the full model. The functions in our package work with models fit using robu() from the robumeta package (Fisher, Tipton, & Zhipeng, 2017) and rma.mv() from the metafor package (Viechtbauer, 2010).

library(wildmeta)
library(clubSandwich)
library(robumeta)

set.seed(12102020)

full_model <- robu(d ~ 0 + study_type + hrs + test,
studynum = study,
var.eff.size = V,
small = FALSE,
data = SATcoaching)

C_mat <- constrain_equal(1:3, coefs = full_model\$b.r)

Wald_test_cwb(full_model = full_model,
constraints = C_mat,
R = 12)
#>   Test Adjustment CR_type Statistic  R     p_val
#> 1  CWB        CR0     CR0   Naive-F 12 0.4166667

# Related Work

We want to recognize other packages that provide algorithms to conduct bootstrapping. The fwildclusterboot package runs the fast version of cluster wild bootstrapping proposed by Roodman et al. (2019) and is based on the Stata’s boottest package (Fischer & Roodman, 2021). The package runs bootstrapping for linear regression models and fixed effects models. To the best of our knowledge, meta-analytic models, weighted linear regression models, and multiple contrast hypothesis tests are not supported by the package.

The multiwayvcov package implements cluster robust variance estimation as well as different types of cluster bootstrapping, including pair, residual, and cluster wild bootstrapping (Graham, Arai & Hagströmer, 2016). For cluster wild bootstrapping, Rademacher weights, Mammen weights, and standard normal weights are available. The functions in the package return cluster robust variance-covariance matrices. Our package, on the other hand, outputs p-values from bootstrap hypothesis tests. Further, our package is particularly created for applied meta-analysis whereas the multiwayvcov can be used with any regression analysis involving clusters.

The boot package can be used to generate bootstrap replicates (Canty & Ripley, 2020). The main function in the package, boot() requires a dataset and a function that calculates the estimate of interest. The function returns the bootstrap estimate. The package also contains another function, boot.ci() that can be used to calculate bootstrap confidence intervals. The package allows parallel programming. However, the boot package does not provide functionality for cluster wild bootstrapping.

The lmboot package implements pair, residual and wild bootstrapping for linear models (Heyman, 2019). The package does not provide Rademacher or Mammen weights for wild bootstrapping. The output of the function is a bootstrap sampling distribution. Further, the package does not work with clustered data.

# Acknowledgments

We are extremely thankful to Wolfgang Viechtbauer for helping us solve this issue we had with the update() function. We also thank Mikkel Vembye for testing our package and giving us very helpful feedback.

# References

Canty A. & Ripley B. (2020). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-25. https://CRAN.R-project.org/package=boot

DerSimonian, R., & Laird, N. (1983). Evaluating the effect of coaching on SAT scores: A meta-analysis. Harvard Educational Review, 53(1), 1-1.

Fischer, A. & Roodman, D. (2021). fwildclusterboot: Fast Wild Cluster Bootstrap Inference for Linear Regression Models. Available from https://cran.r-project.org/package=fwildclusterboot.

Fisher, Z., Tipton, E., & Zhipeng, H. (2017). robumeta: Robust variance meta-regression. Retrieved from https://CRAN.R-project.org/package=robumeta

Graham N., Arai M., & Hagströmer, B (2016). multiwayvcov: Multi-Way Standard Error Clustering. R package version 1.2.3. https://CRAN.R-project.org/package=multiwayvcov

Hedges, L. V., Tipton, E., & Johnson, M. C. (2010). Robust variance estimation in meta-regression with dependent effect size estimates. Research Synthesis Methods, 1(1), 39–65. https://doi.org/10.1002/jrsm.5

Heyman, M. (2019). lmboot: Bootstrap in Linear Models. R package version 0.0.1. https://CRAN.R-project.org/package=lmboot

Joshi, M., Pustejovsky, J. E., & Beretvas, S. N. (2021). Cluster wild bootstrapping to handle dependent effect sizes in meta-analysis with small number of studies. https://doi.org/10.31222/osf.io/x6uhk

Pustejovsky, J. E. (2020). clubSandwich: Cluster-robust (sandwich) variance estimators with small-sample corrections [R package version 0.4.2]. R package version 0.4.2. https://CRAN.R-project.org/package=clubSandwich

Tipton, E. (2015). Small sample adjustments for robust variance estimation with meta-regression. Psychological Methods, 20(3), 375–393. https://doi.org/10.1037/met0000011

Tipton, E., & Pustejovsky, J. E. (2015). Small-Sample Adjustments for Tests of Moderators and Model Fit Using Robust Variance Estimation in Meta-Regression. Journal of Educational and Behavioral Statistics (Vol. 40). https://doi.org/10.3102/1076998615606099

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48.


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Popular

# Cauchy problem for hyperbolic operators multiple characteristics, micro-local approach by Karen Yagdjian

Written in English

## Subjects:

• Differential equations, Hyperbolic.,
• Partial differential operators.,
• Cauchy problem.

Edition Notes

Includes bibliographical references (p. [379]-393) and indexes.

## Book details

Classifications The Physical Object Statement Karen Yagdjian. Series Mathematical topics,, v. 12, Mathematical topics (Berlin, Germany) ;, v. 12. LC Classifications QA377 .Y34 1997 Pagination 397 p. ; Number of Pages 397 Open Library OL282719M ISBN 10 3055017390 LC Control Number 97185120 OCLC/WorldCa 37865712

A doubly characteristic point of a diﬀerential operator P of order m (i.e. one where Pm = dPm = 0) is eﬀectively hyperbolic if the Hamilton map FPm has real non-zero eigenvalues. When the characteristics are at most double and every double characteristic is eﬀectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower.

The goal of this book is a construction of the fundamental solution to the Cauchy problem for hyperbolic operators with multiple characteristics. Well-posedness of the problem in various functional spaces as well as a propagation of singularities of the solutions are investigated, by: The correctness of the Cauchy problem in the Gevrey classes for operators with hyperbolic principal part is shown in the first part.

In the second part, the correctness of the Cauchy problem for effectively hyperbolic operators is proved with a precise estimate of the loss of derivatives. The Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved.

Subsequently, an existence result for the Cauchy problem is : Annamaria Barbagallo, Vincenzo Esposito. For noneffectively hyperbolic operators, it was proved in the late of s that for the Cauchy problem to be C ∞ well posed the subprincipal symbol has to be real and bounded, in modulus, by the sum of modulus of pure imaginary eigenvalues of the Hamilton map.

The book takes a look at generalized Hamilton flows and singularities of solutions of the hyperbolic Cauchy problem and analytic and Gevrey well-posedness of the Cauchy problem for second order weakly hyperbolic equations with coefficients irregular in time.

The selection is a dependable reference for researchers interested in hyperbolic equations. The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem. Show less Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems.

This book collects together detailed proofs of many well/ill-posed results related to the Cauchy problem for diﬀerential operators. When the characteristics are at most double and every double characteristic is eﬀectively hyperbolic, the Cauchy problem for P can be solved for arbitrary lower order : Springer International Publishing.

The Strictly Hyperbolic Cauchy Problem. The Basic Calculus Necessary Conditions for Correctness of the Cauchy Problem. Hyperbolic Operators of Principal Type In he was awarded the Fields Medal for his contributions to the general theory of linear partial differential operators.

His book Linear Partial. necessary and sufﬁcient for the Cauchy problem to be C∞well-posed. Introduction It is well-known that the Cauchy problem for a hyperbolic operator with con-stant coefﬁcients is C∞well-posed if and only if the operator is hyperbolic in the sense of Garding (see [5]).

For hyperbolic operators with constant coefﬁ-˚. - weakly hyperbolic Cauchy problems with finite time degeneracy; the precise loss of regularity depending on the spatial variables is determined; the main step is to find the correct class of pseudodifferential symbols and to establish a calculus which contains a symmetrizer.

The correctness of the Cauchy problem in the Gevrey classes for operators with hyperbolic principal part is shown in the first part. In the second part, the correctness of the Cauchy problem for effectively hyperbolic operators is proved with a precise estimate of the loss of Cauchy problem for hyperbolic operators book.

This method can be applied to other (non) hyperbolic. On the Cauchy Problem for Hyperbolic Operators with Double Characteristics. Communications in Partial Differential Equations: Vol. 34, No. 8, pp. How to read online Cauchy Problem for Noneffectively Hyperbolic Operators (MSJ Memoirs) ePub books.

- At a double characteristic point of a differential operator with real characteristics, the linearization of the Hamilton vector field of the principal symbol is called the Hamilton map and according to either the Hamilton map has non zero real eigenvalues or not, the operator is.

This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves.

This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and.

Fourier integral operators with complex-valued phase function and the Cauchy problem for hyperbolic operators.- The effectively hyperbolic Cauchy problem. Series Title: Lecture notes in mathematics (Springer-Verlag), Responsibility: Kunihiko Kajitani, Tatsuo Nishitani.

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators Alessia Ascanellia, Massimo Cicognanib,c,∗ a Dipartimento di Matematica, Università di Ferrara, Via Machiave Ferrara, Italy b Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, Bologna, Italy.

2 Cauchy Problem In this section we formulate the Cauchy problem for a linear diﬀerential operator a x, ∂ ∂x!. To begin with we make a few formal reductions. Let S be a hypersurface in Rn deﬁned by an equation ϕ(x) = 0 where ϕis a suﬃciently often continuously diﬀerentiable function with its gradient ϕx(x0) ≡ ∂ϕ ∂x1 (x0.

Corpus ID: Lectures on Cauchy problem @inproceedings{LecturesOC, title={Lectures on Cauchy problem}, author={茂 溝畑 and M. Murthy and B. Singbal}, year={} }. The cauchy problem for a weakly hyperbolic equation with unbounded and non-Lipschitz-continuous coefficients Ascanelli, Alessia, Differential and Integral Equations, On the Cauchy problem for hyperbolic operators of second order whose coefficients depend only on the time variable WAKABAYASHI, Seiichiro, Journal of the Mathematical Society.

HYPERBOLIC PROBLEMS AND REGULARITY QUESTIONS, MARIAROSARIA PADULA Books, SPRINGER Books, at Meripustak. some ill-posed Cauchy problems for linear PDEs can be generated by unbounded linear operators of those PDEs.

These are those operators for which Carleman estimates are valid, e.g. elliptic, parabolic and hyperbolic operators of the second order. Convergence rates of minimizers are established using Carleman estimates.

Nishitani, The Cauchy problem for e↵ectively hyperbolic operators, in Nonlinear variational problems (Isola d'Elba, ), Res. Notes in Math. (Pitman, ), pp. Nonlinear. (weakly) hyperbolic diﬀerential operators.

As it was proved by Bony-Schapira [4] for a single diﬀerential operator, and by Kashiwara-Schapira [13] for general systems, the hyperbolic Cauchy problem is well posed in the sheaf Bof Sato’s hyperfunctions: Theorem Let f N: N →M be a morphism of real analytic manifolds, and let.

The hyperbolic Cauchy problem by Tatsuo Nishitani Department of Mathematics, College of General Education Osaka University, Toyonaka, OsakaJapan (Soeul National University, February, ) 1. Review on basic facts Hyperbolicity Let P be a diﬀerential operator of order m deﬁned on an open set Ω in IRd+1 and let H be a.

Purchase Scattering Theory for Hyperbolic Operators, Volume 21 - 1st Edition. Print Book & E-Book. ISBN hyperbolic differential operators and related problems lecture notes in pure and applied mathematics Posted By Corín TelladoMedia TEXT ID bded1e Online PDF Ebook Epub Library Derivative And Integral Of Trigonometric And Hyperbolic.

In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first − derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic of the equations of mechanics are hyperbolic, and so the study.

Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v, In this monograph we discuss the ∞ well-posedness of the Cauchy problem for hyperbolic systems.

Usage notes. The hypersurface S is called the Cauchy. We consider the well-posedness of the Cauchy problem in Gevrey spaces for N×N first-order weakly hyperbolic systems. The question is to know whether the general results of Bronštein [1 Bronštein, M.D.

().The Cauchy Problem for hyperbolic operators with characteristic of variable multiplicity. Project Euclid - mathematics and statistics online. On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness Bernardi, Enrico and Nishitani, Tatsuo, Kyoto Journal of Mathematics, ; Non-autonomous Ornstein-Uhlenbeck equations in exterior domains Hansel, Tobias and Rhandi, Abdelaziz, Advances in Differential Equations,   If the characteristic Cauchy problem has a solution, then the equation and the second initial condition imply a necessary condition for its solvability: $\nu'(x)=0$, i.e.

the characteristic Cauchy problem may be solvable only if $\nu(x)=\mathrm{const}=\alpha$. hyperbolic differential operators and related problems lecture notes in pure and applied mathematics Posted By Robert LudlumPublic Library TEXT ID bded1e Online PDF Ebook Epub Library hyperbolic differential operators and related problems by vincenzo ancona and jean vaillant abstract presenting research from more than 30 international authorities this reference provides a.

The text then ponders on the Cauchy problem for effectively hyperbolic equations and for uniformly diagonalizable hyperbolic systems in Gevrey classes.

The book takes a look at generalized Hamilton flows and singularities of solutions of the hyperbolic Cauchy problem and analytic and Gevrey well-posedness of the Cauchy problem for second order.

of operators for which the Cauchy problem is weakly well posed. Such results are described in and Taylor on the use of pseudodiﬀerential techniques in nonlinear problems.

Other books on hyperbolic partial diﬀerential equations include those of Hadamard, Leray, G˚arding, and Mizohata The deﬁning properties of hyperbolic problems. operators are unique and that they extend to several spaces of sections.

We argue that Green-hyperbolic operators are not necessarily hyperbolic in any PDE-sense and that they cannot be characterized in general by well-posedness of a Cauchy problem.

The fourth section is devoted to extending the Green’s operators to distributional sections. In this paper, we study the well posed‐ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane.

By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter m ∈(0,1) and. BEHAVIOR OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A HYPERBOLIC EQUATION AS t —» B. VAlNBERG UDC Introduction Suppose that P(id/dt, id/dx), χ - (χ, • •, x), is a homogeneous hyperbolic operator without multiple characteristics, with constant coefficients and of order 2m, This means that for σ.

In particular, symbolic symmetrizers are shown to exist for constantly hyperbolic operators. Subsequently, well-posedness is proved in Sobolev spaces Hs, for any s in the case of infinitely smooth coefficients, and also for s large enough in the case of Hs coefficients.

Uniqueness follows from energy estimates for the direct Cauchy problem. B 2 − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data.

An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0.About this Item: Oxford University Press, United Kingdom, Hardback.

Condition: New. New. Language: English. Brand new Book. This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves.hyperbolic differential operators and related problems lecture notes in pure and applied mathematics Posted By Cao XueqinMedia TEXT ID bded1e Online PDF Ebook Epub Library Hyperbolic Trig Identities List Of Hyperbolic.

61666 views Wednesday, November 18, 2020


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# Forest UCM Ch3 AngMom

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

# Definition of Angular Momentum

The angular momentum of a single particle is defined as

An coordinate must be defined in order to express the vectors for the particles position and momentum. The resulting angular momentum is defined with respect to the origin (rotation point) of the particle.

## Torque

If I take the derivative of angular momentum with respect to time I get

cross product of parallel vectors
Definition of Torque

Newton's second law for angular motion

# Kepler's second Law

Kepler's second Law

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The above law is the result of the conservation of angular momentum when two bodies are attracted to eachother by a central force.

Consider a planet orbiting the Sun such that the sun if Fixed at the origin.

The line segment that joins the planet and the Sun is just the radius vector from the origin of a coordinate system where the sun is located to the position of the planet orbiting the sun.

The displacment vector for the planet is given as

The differential area of the triangle representing a differential element of the orbital motion is given as

as seen at Forest_UCM_NLM#Vector_.28_Cross_.29_product the area of a parallelogram is given by the cross product and the above triangle makes up half of the parallelogram

using the definitionof momentum

Since central forces are conserved forces, the angular momentum is conserved.

Additionally, there is no net Torque for this system.

As a result

constant

Thus

Equal areas are swept out in equal times

# Angular momentum for a system of particles

## Definition

The total angular momentum for a system of particles is defined as

## Conservation of Angular Momentum

Taking the derivative of the above definition

: parallel vectors

For a system of many particles we can write the Total force on one of the particles in terms of a sum of the Internal forces from all the other particles and any external force .

The second term represents the external torques exerted on the system

The first term represents the forces exerted on each particle by every other particle (internal forces)

?

The terms in the above double sum can be regrouped

Newton's Third law

The vector difference is the vector pointing from particle to particle anti-parallel to the central Force.

= vector pointing from particle to particle

If
the NET external torque on a system of particles is zero
Then
the angular momentum is constant

# Moment of Inertia

## Definiitons

for a system of particles
for a Rigid Bodies

## Moment of Inertia of a Uniform disk

Find the center of mass for a disk of radius and area mass density


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# Showing a local webpage in Unity3d using the Prime[31] etcetera plugin.

I’ve been working with Prime31’s Etcetera plugin for iOS development with Unity3d. There are a lot of great things in there but some of the documentation is lacking. Anyways, if you want to show a local web page with the showWebPage command it is a two step process.

Step 1:

Copy the files you wish to access into a folder called StreamingAssets under your Assets directory.

Step 2:

Call EtceteraBinding as follows:

EtceteraBinding.showWebPage(Application.dataPath + "/Raw/FolderWithHTML/index.html");

where FolderWithHTML is the folder you placed in the StreamingAssets folder.


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Hey! I'm David, the author of the Real-World Cryptography book. I'm a crypto engineer at O(1) Labs on the Mina cryptocurrency, previously I was the security lead for Diem (formerly Libra) at Novi (Facebook), and a security consultant for the Cryptography Services of NCC Group. This is my blog about cryptography and security and other related topics that I find interesting.

# Another look on Public key crypto posted January 2016

I was watching this excellent video on the birth of elliptic curves by Dan Boneh, and I felt like the explanation of Diffie-Hellman (DH) felt short. In the video, Dan goes on to explain the typical DH key exchange:

Alice and Bob each choose a public point $g$ and a public modulus $N$.

By the way. If you ever heard of "DH-1024" or some big number associated to Diffie-Hellman, that was probably the bitsize of this public modulus $N$.

The exchange then goes like this:

1. Alice generates her private key $a$ and sends her public key $g^a\pmod{N}$ to Bob.

2. Bob generates his own private key $b$ and sends his public key $g^b\pmod{N}$ to Alice.

3. They can both generate their shared key by doing either $(g^b)^a \pmod{N}$ for Alice, or $(g^a)^b \pmod{N}$ for Bob.

Dan then explains why this is secure: because given $(g, g^a, g^b)$ (the only values an eavesdropper can observe from this exchange) it's very hard to compute $g^{ab}$, and this is called the Computational Diffie-Hellman problem (CDH).

But this doesn't really explain how the scheme works. You could wonder: but why doesn't the attacker do the exact same thing Bob and alice just did? He could just iterate the powers of $g$ until $g^a$ or $g^b$ is found, right?

## A key exchange with hash functions

Let's replace the exponentiation by a hash function. Don't worry I'll explain:

$g$ will be our public input and $h$ will be our hash function (e.g. sha256). One more thing: $h^{3}(g)$ translates to $h(h(h(g)))$.

So now our key exchange looks like this:

1. Alice generates an integer $a$ large enough and compute $a$ iteration of the hash function $h$ over $g$, then sends it to Bob.

2. Bob does the same with an integer $b$ and sends $h^b(g)$ to Alice (exact same thing that Alice did, different phrasing.)

3. They both compute the share private key by doing either $(h^b(g))^a$ for Alice, or $(h^a(g))^b$ for Bob.

So if you understood the last part: Alice and Bob both iterated the hash functions on the starting input $g$ a number of $a+b$ times. If Alice's public key was $h(h(g))$ and Bob's public key was $h(h(h(g)))$ then they both end up computing $h(h(h(h(h(g)))))$.

That seems to work. But how is this scheme secure?

You're right, it is not. The attacker can just hash $g$ over and over until he finds either Alice's or Bob's public key.

So let's ask ourselves this question: how could we make it secure?

If Bob or Alice had a way to compute $h^c(x)$ without computing every single hash ($c$ hash computations) then he or she would take way less time to compute their public key than an attacker would take to retrieve it.

## Back to our discrete logarithm in Finite groups

This makes it easier to understand how the normal DH exchange in finite groups is secure.

The usual assumptions we want for DH to works were nicely summed up in Boneh's talk

The point of view here is that discrete log is difficult AND CDH holds.

Another way to see this, is to see that we have algorithm to quickly calculate $g^c \pmod{n}$ without having to iterate through every integers before $c$.

To be more accurate: the algorithms we have to quickly exponentiate numbers in finite groups are way faster than the ones we have to compute the discrete logarithm of elements of finite groups. Thanks to these shortcuts the good folks can quickly compute their public keys while the bad folks have to do all the work.


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View Single Post
 P: 330 Using snipez90's suggestion for delta, I get $$\delta=min(\frac{1}{2},\frac{\epsilon}{2})$$. I was wondering how the book got the result I said it got, that is why I abandoned my original line of thought.


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Article | Open | Published:

# Picometer polar atomic displacements in strontium titanate determined by resonant X-ray diffraction

## Abstract

Physical properties of crystalline materials often manifest themselves as atomic displacements either away from symmetry positions or driven by external fields. Especially the origin of multiferroic or magnetoelectric effects may be hard to ascertain as the related displacements can reach the detection limit. Here we present a resonant X-ray crystal structure analysis technique that shows enhanced sensitivity to minute atomic displacements. It is applied to a recently found crystalline modification of strontium titanate that forms in single crystals under electric field due to oxygen vacancy migration. The phase has demonstrated unexpected properties, including piezoelectricity and pyroelectricity, which can only exist in non-centrosymmetric crystals. Apart from that, the atomic structure has remained elusive and could not be obtained by standard methods. Using resonant X-ray diffraction, we determine atomic displacements with sub-picometer precision and show that the modified structure of strontium titanate corresponds to that of well-known ferroelectrics such as lead titanate.

## Introduction

Next to high beam intensity, low emittance, and short pulse durations, one of the main benefits of synchrotron light sources is the wide range of accessible photon energies from visible light to hard X-rays. This degree of freedom allows to perform spectroscopy in the X-ray regime (XAS) but also to enhance traditional X-ray scattering methods. This is based on the variation of the complex atomic scattering for photon energies close to electronic transitions in a certain chemical species and therefore referred to as anomalous or resonant X-ray scattering.

Today, the field of resonant X-ray diffraction (RXD) spans a set of techniques based on the different dependencies in the resonant atomic scattering amplitude1, 2. The variation in its real and imaginary part is used for contrast enhancement of atoms with similar3 or even equal4 number of electrons, or to solve the phase problem in crystallography5, 6. Small oscillations of the atomic scattering amplitude beyond absorption edges are used for site-selective study of coordination geometry7. Other methods rely on the dependence of the scattering amplitude on the X-ray polarization and wave vector. These make the resonant atom an anisotropic scatterer, which can result in additional forbidden reflections8 and allows to study different origins of reduced symmetry such as spin, charge order, orbital order, or point defects9,10,11,12.

In this work, we describe a way to enhance the sensitivity of RXD to atomic displacements in the picoscale. In contrast to those above, this our approach employs the energy dependence of atomic scattering amplitudes to vary the structure factor aiming at a suppression of a selected Bragg reflection through the destructive interference of waves scattered by different chemical species. As a result, the structure factor is highly sensitive to a slight dynamic or static displacement of atoms. Moreover, it allows to refine the positions of atoms by analyzing the energy dependence of a few of these reflections. This way, normalization problems are avoided and a structure determination becomes possible in many cases where conventional X-ray crystallography cannot yield this information, e.g., for thin films or heterogeneous samples.

We apply this method to strontium titanate (SrTiO3, STO) both in cubic and polar state. STO receives a lot of attention as a basis for a wide variety of complex oxide electronics. Despite being a wide-bandgap semiconductor with centrosymmetric crystal structure under normal conditions, it can acquire magnetic or ferroelectric properties, become conducting and even superconducting via doping, strain or defect manipulation13,14,15,16,17. Furthermore, it was found that single crystals of centrosymmetric STO undergo a structural transition under static external electric field18. This phase transition is quite unusual as it is triggered by a redistribution of oxygen vacancies in the crystal. It leads to a distortion of unit cells which is stabilized by the electric field and eventually results in the formation of a transformed layer at the anode side. This modified structure is therefore referred to as migration-induced, field-stabilized, polar (MFP) phase19 and is connected to the substrate by a strain gradient. A loss of centrosymmetry in the MFP layer was concluded due to the appearance of additional Raman modes as well as piezo- and pyroelectricity19,20,21. On that basis, a transition from space group $Pm 3 ̄ m$ to P4mm was predicted20, but no information on the atomic positions for the MFP phase could be obtained so far. A structure analysis using conventional methods is not feasible due to the restricted geometry and the overlap of reflections from different layers. This problem is avoided using the outlined resonant RXD method. To test it, we characterize the initial, cubic state of STO and obtain the anisotropic mean square atomic displacements (Debye–Waller factors). Based on that, a refinement of atomic positions corresponding to polar displacements in the MFP phase is performed.

## Results

### Resonant RXD near destructive interference

Although the underlying physics are presented in the Methods section and are common with related RXD techniques, the way we use the resonance in our measurements is different and should be introduced. The central feature is the dependence of the Bragg peak intensity on the photon energy near an absorption edge. The intensity is determined by the norm of the complex structure factor (according to Eqs 1 and 3 in Methods) that runs through a loop in the complex plane when the energy passes the edge (see Fig. 1). This behaviour of the structure factor has been utilized in various anomalous diffraction methods (see Discussion). Beyond that, the Bragg intensity will show dramatic variations in case this loop is close to the complex zero. This leads to a strongly pronounced energy dependence of the intensity which will be particularly prone to structural changes and allow detailed conclusion about absolute values for mean and variance of the atomic positions in a crystal.

In Fig. 1, this situation is sketched for the specific example of the STO 007 reflection near the strontium edge considering two types of positional variation of the titanium atom: a static shift as well as a mean square deviation (due to thermal motion). Only for this illustration, the calculation of the shown curves is based on a simplified model, neglecting the resonant fine structure. As the curves are close to the origin of the complex plane for a certain energy, the measured Bragg intensity as a function of energy will undergo a pronounced minimum. One can see that both considered structural modifications result in orthogonal responses of the progression of the structure factor which can be understood by developing the structure factor for small displacements (see Eq. 5 in Methods). Using the intensity minimum (distance from origin) and its energy position, it is already possible to assess both structural parameters. In a real experiment, the minimum position in energy can be easily determined. The absolute intensity at this position is difficult to measure, but a comparison with several points of the energy dependence can serve as normalization. In addition, the complete curve carries further information, e.g., about the direction of atomic displacement.

However, the obtained results and the high precision would be based on the knowledge of all remaining internal structural parameters. In most cases, there is an uncertainty in more than one atomic position. Even for known structures, the given error margins for position or mean square displacement (Debye–Waller factors) are often too large to give a reliable prediction of the spectral shape near destructive interference. To reach the aspired resolution in atomic positions below 1 pm, this sensitivity and its correlations with static atomic displacements is thus discussed in detail and improved anisotropic atomic displacement parameters (ADPs) are determined before the refinement of atomic positions. Other factors influencing the results, such as site occupancy and valence state, are usually known but can also be disentangled based on the scattering vector dependence of the Debye–Waller factors.

In this work, using several reflections, we vary the weighting of the different atomic scattering amplitudes to gain additional information allowing us to refine all free parameters. We found suitable conditions to perform such measurements for Bragg reflections where two of the Miller indices are even and the other is odd with respect to the pseudocubic setting. If atomic displacement is neglected, the structure factor for this set of reflections is composed of the simple amplitude difference F ≈ fSr − fO − fTi. Usually, these reflections show a substantial intensity since strontium is the dominating scatterer. Near the strontium K absorption edge, however, the real part of the strontium scattering amplitude is strongly reduced, causing the structure factor to approach zero and leading to the pronounced minimum in intensity.

### Characterization of dynamic thermal displacement

In order to test the method but also to get a good starting model for the characterization of structural modifications, we characterized the thermally induced atomic displacement for the initial (cubic) state of SrTiO3 where all average atomic positions are known. The experiments involve a measurement of the Bragg intensity as a function of energy while the crystal is kept in the maximum of the Bragg condition by adjusting the scattering angles resulting in a fixed momentum transfer on the photons.

A high sensitivity of these RXD spectra to thermal displacement is evident in Fig. 2 where the measurement is performed near the Sr-K edge and for different temperatures. A clear trend can be seen and reflects that the Debye–Waller factors of the different atoms do not scale proportionally with temperature. This fairly corresponds to the changes initially calculated (Fig. 1, bottom) for increasing mean square displacement. The simulated curves in Fig. 2 are based on Eq. 1 and the refinement of the anisotropic ADPs Uij for each atom that determine the Debye–Waller factors (see Methods). For energies beyond the edge at ≈ 16,100 eV, it was necessary to include an absorption correction and the resonant fine structure.

In cubic SrTiO3, oxygen occupies a noncubic Wyckoff position leading to two different oxygen ADPs necessary to describe the measured intensity correctly—one parallel and one perpendicular to the Ti–O bond. The results of least-squares refinement displayed in Fig. 2 were based on the assumption that the ADPs show a temperature dependence according to the Debye model22, 23 and taking into account a decomposition of the Debye temperature Θ for oxygen into a parallel and a perpendicular part following Potzel et al.24. For this reason, it was necessary to record spectra of additional reflections to ensure an unambiguous refinement of the ADPs. Table 1 summarizes the results for anisotropic ADPs and Debye temperatures obtained by including room temperature measurements of the reflections 005, 205, 405, 504, and 016 in a similar energy region as for the 007 reflection (see Supplementary Fig. 3).

Despite its lower mass, titanium shows a smaller thermal vibration than strontium and the oxygen oscillations along the Ti–O bond are much smaller than perpendicular to it. Both observations are in good agreement with literature results25 and the absolute values are within the error margins. This is remarkable as the results were obtained by different methods. The errors shown in Table 1 seem relatively high even though the spectra are very sensitive to small variations in the results. A detailed error analysis reveals that the confidence volumes are comparatively small and that the large uncertainties originate from a strong correlation between the parameters. There is a much higher certainty in relative values of the parameters as the confidence regions are only elongated along a direction, which corresponds to proportional change of the Debye temperatures (illustrated in Supplementary Fig. 4). Therefore, we assume that the ratios of values given in Table 1 are more accurate than those taken from literature. To get more precise absolute values, the correlation of fit parameters can in general be further reduced by including spectra of additional reflections.

### Static displacement and the structure of the MFP phase

Based on the refined ADPs of cubic SrTiO3, it is possible to characterize the structural changes in the strained MFP layer. The MFP phase is formed under influence of a [001] oriented, static electric field of about 1 MV m−1 in a few micrometer thick layer on the anode side of the crystal (see sketch in Fig. 3) where the oxygen vacancies are depleted. It can be observed by an increase of the lattice parameter along the field direction of up to 1%. A tetragonal distortion with the unique axis c in field direction was found with a Poisson’s ratio of approximately $ν=- Δ a Δ c =0.2$19 which conforms to literature values26. Although the change of lattice parameters alone does not modify the structure factor, the associated atomic displacements should have a clear effect on the energy dependence of the intensity. The formation of the MFP phase is observed as an increased current through the sample, which reaches its maximum after a few hours and is attributed to the migration of oxygen vacancies. It is accompanied by a broadening of Bragg peaks and formation of pronounced shoulders on their lower-angle sides. Both the current and the broadening converge after a period of about 8 h depending on sample and ambient conditions. Following symmetry considerations, a displacement of atoms perpendicular to the field direction would result either in an orthorhombic structure and a consequential twinning/peak splitting or in a two-fold superlattice in ab direction27 causing additional superlattice reflections. We recorded reciprocal space maps and ruled out both cases concluding that all atomic displacement must be parallel to the field. In addition, we defined the position of the strontium atom as the origin of the unit cell and assumed that the MFP phase exhibits the same ADPs, as they were found above for cubic SrTiO3, which is justified due to the very small static displacements in the structure. Therefore, only three free parameters are left which are the z positions of titanium and the two oxygen sites that are not anymore linked by symmetry: for one the Ti–O bond is parallel and for the other it is nearly perpendicular to the electric field vector.

As its strain in equilibrium depends on the applied electric field strength, crystal thickness, as well as sample history, there is no prototype of the MFP phase. Furthermore, there is a continuous decrease of strain towards the cubic bulk SrTiO3, which can be seen in the form of a plateau between the two corresponding Bragg peaks. To study differently strained volumes in an isolated manner, we keep the momentum transfer Q fixed at the corresponding position. In particular, in the conditions of our experiments, we observed a distinct maximum in the diffraction profile towards smaller momentum transfer (see Fig. 3b) corresponding to a value of $- Δ Q z Q z = Δ c c =0.15%$. This maximum is related to a region of constant strain associated with the MFP phase near the crystal surface. Therefore, we select the MFP by choosing accordingly the value of Q z (l) for each studied reflection hkl. At these positions of reciprocal space, we then measured the energy dependence of Bragg intensity for several reflections in a similar region as before the electroformation. The resulting data along with calculations are shown in Fig. 4.

As it is on the order of the penetration depth of the X-rays, the refinement also includes the thickness of the probed, strained region using Eq. 2 from the Methods part. It does not affect the main feature (minimum intensity and position) but is necessary to obtain a good fit for energies beyond the edge. For small thicknesses, the influence of a covering layer on the absorption correction is weaker and was not taken into account. Depending on the reflection, the kinematic penetration depth varies from 18 to 43 μm below and from 5 to 10 μm beyond the absorption edge. Extinction effects have also been neglected, as the measured reflections are typically weak. Furthermore, we assumed that the fine structure does not change significantly during phase transition to the MFP phase. This approximation seems to be valid as the fine structure oscillations (see Fig. 4 beyond 16,100 eV) are well modeled using spectra obtained from bulk SrTiO3. The validity is further discussed with aid of absorption spectroscopy measurements in Supplementary Note 1.

The results of the refinement of internal structure parameters are summarized in Table 2. They confirm that the MFP structure is polar. It belongs to the same structure type as ferroelectric, tetragonal BaTiO3 and PbTiO3 with space group P4mm. In comparison, the absolute displacements of atoms from high symmetry positions are significantly smaller in the present case of the MFP phase. However, similar to PbTiO3, the displacement of titanium is roughly a factor of three smaller than those of oxygen. The relative uncertainties in the refined results are much smaller than in the case of Debye temperatures. Even though the refinement was based on iterating through all combinations of ADPs within the confidence volume, the error did not propagate proportionally indicating a stronger sensitivity to the rather small static displacement. Figure 5 illustrates the pairwise correlations of all displacement parameters. The derived error margins correspond to a proportional movement of the considered atoms in z-direction and away from their original high-symmetry positions meaning that the ratio of the shifts is again known more precisely than one would estimate based on Table 2. The layer thickness, in turn, shows minor correlation and a less well defined confidence region. It should also be pointed out that, using the described method, the accuracy in the determination of average atomic positions is at the sub-picometer level.

### Strain dependence of atomic displacement

The MFP phase is connected with the bulk SrTiO3 crystal via a strain gradient. An open question is, how the displacement evolves with strain or whether a polar phase is only formed in the thin layer of constant strain closest to the anode which is associated with the additional peak in the diffraction profile. In the latter case, the region of continuous strain gradient, which is seen as a plateau between the peaks (see Figs 3 and 6), would be assumed to be tetragonally distorted but to remain centrosymmetric like the substrate. To answer this question, we performed similar energy scans for several fixed values of the momentum transfer Q selecting the MFP phase, the substrate and also the transition region but restricting ourselves to the 007 and 205 Bragg reflections. Although significant uncertainties are expected in a refinement of atomic positions that relies on only two reflections, a loss of centrosymmetry would have an unequivocal effect on the measured spectra. Assuming a proportional movement of all atoms, we further reduced the amount of parameters and can give an estimate of all atomic positions based on the former results in Table 2.

Regions of the sample having different strain in c direction were selected using four positions of the Q z scans (see Fig. 6) near the bulk reflections 007 and 205. One should note that the model of a strain gradient predicts that there is no region of constant strain with a finite width. From the experimental conditions, one can estimate how thick the layers are that contribute to a certain position in Q z scans in reciprocal space. This assessment is based on the value of the strain gradient. In an exponential strain profile model for our case, we obtained an upper limit for the strain gradient of 1.5 · 10−3 μm−1 (near the surface) corresponding to a value of 5.9 · 10−7 per unit cell. For the 007 reflection, this would result in a $π 2$ phase shift after 350 unit cells and therefore in a thickness of 140 nm of the contributing layer. Based on this thickness, the calculated peak width for the 007 reflection is ΔQ z  ≈ 0.0035 r.l.u. (reciprocal lattice units). This estimation has been done for the position of highest strain. For lower strain, the broadening of Bragg peaks will decrease and the depth resolution of the measurement will improve. Therefore, the sampling we used in this work was always below the limits of resolution.

The fixed Q energy scans corresponding to the four selected strain values are presented in Fig. 7 and clearly show that also the transition region exhibits a polar structure. Already for the smallest selected value of ΔQ z /Q z  = −0.04% (at Q z values 6.997 and 4.998 r.l.u.), the energy dependence shows an enormous response which is enhanced with larger strain. The changes in RXD curves saturate when further increasing the strain and this saturation is reached earlier for the 205 reflection. The legend in Fig. 7 lists the titanium displacements that result from the fit with the condition of a proportional movement of the oxygen atoms according to Table 2. These movements (enhanced by a factor of 10) are illustrated in Fig. 6 together with the Q z -scans near the 007 reflection for two selected energies.

The strong response seen in Fig. 7 already for small strain or small displacements suggests that even much smaller perturbations in the average structure can be detected this way. A clear trend of larger atomic displacement and therefore stronger electric polarization with higher strain is observed. Moreover, this dependence is not linear. The obtained relationship between Ti-displacements and the lattice strain suggests that the displacement plays the role of the primary order parameter in the MFP phase. The lattice strain must be summed up from purely microscopic and purely macroscopic contributions. The microscopic contribution expresses the need to optimize the unit cell size to the new positions of atoms, whereas the macroscopic contribution accounts for the need to connect the strained phase to the nonpolar bulk. We have found that the second contribution has negligible effect on the positions of the Ti-atom. The same relationship between atomic positions and lattice parameters was documented in the piezoelectric crystals under quickly rising electric fields28, 29, where the lattice strain was caused by atomic displacements due to electric field and elastic vibrations due to the short rising time.

### Comparison with density functional theory calculations

Electronic structure calculations using density functional theory (DFT) were performed to validate the relation between strain and polar displacements, as well as to estimate the macroscopic polarization resulting from these displacements. The effects caused by the external electric field have been emulated by imposing relative atomic displacements Δz/c of titanium and oxygen in field direction as they were experimentally found for the MFP phase according to Table 2. This was followed by geometrical relaxation of the MFP unit cell size. Calculated changes in lattice parameters as well as strain parameters and polarization Δptot are given in Table 3. The projector-augmented wave method30 in Perdew-Burke-Ernzerhof (PBE) parameterization31 was employed as implemented in the VASP code32. Total energies have converged better than 10−10 eV with a maximum kinetic energy of 450 eV for the plane-wave basis set and Γ-centered 12 × 12 × 12 Monkhorst–Pack k-point meshes with spacings < 0.02 × 2πÅ−1. For the evaluation of strain, we kept relative atomic positions fixed and relaxed the cell geometry within the space group to forces less than 10−5 eV Å−1. The change in polarization has been assessed within the framework of a Berry-phase treatment33.

During the measurements on the structure of the MFP phase, we selected a strain value in c-direction of $Δ c c =0.15%$. This value is in reasonable agreement with the result obtained here by DFT calculations $Δ c c =0.21%$. The Poisson’s ratio even better matches previously observed values for the MFP phase19.

## Discussion

In this study we obtained highly accurate values for atomic positions of the recently discovered MFP phase of SrTiO3 and presented a resonant RXD method to obtain these even for inhomogeneous samples. The method involves adjusting the X-ray wavelength—and thus the scattering amplitude of the resonant atoms—to tune destructive interference of the partial waves from different scatterers. To which extent this can be achieved strongly depends on the atomic positions, giving an access to reveal slight displacements of the atoms. In the present case, a sensitivity to changes of average positions at the level of 1 pm and below has been demonstrated. Prerequisite for a careful analysis of the data was a refinement of temperature induced atomic displacement parameters of cubic SrTiO3. We could confirm directly that SrTiO3 forms a polar structure under electric field, which is of the same type as ferroelectric BaTiO3 at room temperature. This is remarkable, as, in contrast to BaTiO3 and PbTiO3, SrTiO3 does not form this phase when cooling to low temperatures: the beginning of a paraelectric to ferroelectric phase transition was observed at 40 K but remains incomplete down to 0 K34. Yet, it is known that other external influences, such as strain, can stabilize this ferroelectric phase even at room temperature15. Using DFT calculations, we could validate the relation between strain and atomic displacement that was experimentally found for the MFP structure and furthermore give an estimate for the resulting electric polarization of the unit cell. We have shown that the magnitude of polar displacement exhibits a nonlinear relation to the strain. The recently found indications for an electrostrictive origin of piezoelectricity in the MFP phase21 would suggest a quadratic dependence of the strain on polarization. In our case, however, the polarization dependence of strain appears to follow an even higher power law (see legend in Fig. 7), which can be attributed to the elastic interaction of MFP layer and substrate. Therefore, this data adds to a better understanding of the mechanism of formation and the role of the substrate.

The employed resonant suppression of Bragg reflections can be universally combined with all diffraction based X-ray techniques as long as the photon energy of the X-ray source is tunable. There are similar attempts to extract structural information reported in literature, all subject to substantial limitations compared to our development. To avoid ambiguities in the refinement, fixed structural models were presupposed or only one structural parameter was assumed to be unknown35, 36. Often, the analysis required inverting (switching) the polarity of the structure37, 38 with an electric field in order to measure the Friedel pair contrast. This is not always feasible or may alter the material as in the present case. Furthermore, in all cases the impact of Debye-Waller factors was neglected and the obtained resolution was much lower than in our approach, which exploits destructive interference.

In Supplementary Note 2, we briefly compare the method to the variety of other techniques and show that it can be useful for a large group of materials especially in a restricted geometry where only few reflections can be accessed or their absolute intensities cannot be compared, and therefore standard diffraction methods cannot be applied. It also allows to study dynamics in crystals in situ, such as switching in ferroelectric films or polarization changes in pyro- and piezoelectrics and represents a uniquely sensitive tool for cross-checking existing structure models or to reveal pseudosymmetries. Advantages are that the measurements can be performed in the pre-edge region allowing straightforward calculations and that the kinematic approximation of diffraction, neglecting extinction effects, is valid for this method, as it is based on weak reflections.

## Methods

### Theory

Within the kinematic theory of diffraction from crystals, in the absence of extinction effects, the dependence of the diffracted intensity of a Bragg reflection on the photon energy E is given by39

$I kin ∝A ( E ) F ( E , Q ) 2$
(1)

where F denotes the structure factor and A describes absorption. Further dependencies on E, such as in Lorentz and polarization correction, are usually weak and can be approximated as linear. The momentum transfer vector Q defines the Bragg reflection. In Bragg geometry, the absorption factor can be written as40

$A= 1 μ 1 + sin α sin β 1 - e - 1 sin α + 1 sin β μ t$
(2)

with t being the thickness of the probed crystalline layer, μ the linear absorption coefficient, and α and β respectively referring to the angle of the incidence and exit of the X-ray beam with respect to the crystal surface. In the independent-atom model, the structure factor is expressed as a weighted sum of the scattering amplitudes f j of all atoms present in the crystallographic unit cell. Real crystals exhibit a certain degree of disorder due to lattice vibrations and defects. Therefore, an averaging over all unit cells is necessary, which is commonly parameterized with the occupancy of the crystallographic site o j and the Debye-Waller factor $e - M j$ taking account of the reduced scattering amplitude due to the uncertainty in the position r of atom j:

$F ( E , Q ) = ∑ j ∈ u .c . o j f j E , Q e - M j e i Qr j .$
(3)

The Debye–Waller factor is a result of harmonic approximation of lattice vibrations due to temperature and is related to the mean square of the atomic displacement u j of the atom j in the direction of Q via $M j = 1 2 ( Q ⋅ u j ) 2$. The anisotropic mean square displacement is commonly expressed in form of the ADP tensor Uij41. Neglecting the polarization and wave-vector dependence of the dispersion corrections f′ and f″, the scattering amplitude can be written as

$f ( E , Q ) = f 0 ( Q ) +f′ ( E ) +if′′ ( E ) .$
(4)

In resonant diffraction, it is common to investigate Bragg reflections where the structure factor is zero due to destructive interference of scattered waves coming from atoms of the same species. Under resonant conditions, these atoms may have different scattering amplitudes due to their different chemical or magnetic environment lifting the destructive interference and resulting in non-zero intensity and therefore new, previously forbidden reflections. In this work, we went the opposite way and tuned the scattering amplitude of distinct elements to cancel out each other. This can only be true in a small range of photon energy, since the scattering amplitude of each element shows a different energy dependence. It also strongly depends on any contribution to the structure factor such as occupancy, shifts in the atomic positions, temperature-induced dynamic displacement, defects, or valence state.

Let u be the small displacement of a selected atom j from the ideal position ($r j → r j 0 +u$; u = u j to improve legibility). To point out the contributions of displacement, the structure factor can be approximated, assuming it is close to zero, as

$F = F u = 0 ⏟ → 0 + ∂ F ∂ u k u = 0 u k + 1 2 ∂ 2 F ∂ u k ∂ u l u = 0 u k u l + ⋯ ≈ o j f j e - M j e i Qr j 0 i Q k u k - 1 2 Q k Q l u k u l .$
(5)

One can see that the term of the mean displacement carries the imaginary unit resulting in the orthogonal displacement of the complex structure factor for the two contributions that was seen in Fig. 1. It explains why, in the present case, the intensity minimum is shifted in energy due to thermal vibration but only less pronounced in the case of directed displacement of titanium. Furthermore, the measured intensity will be proportional to the square of mean displacement as well as of mean square displacement of the atom, weighted with first and second power of the momentum transfer, respectively. This illustrates an enhanced sensitivity of higher order reflections.

### Experiment and data treatment

We found that the Bragg intensity has a minimum in the pre-edge region of the Sr-K edge (at 16,150 eV) for a set of reflections where exactly one Miller index is odd. On the other hand, the post-edge region contains valuable information about the direction of a considered atomic displacement since a significant imaginary part is typically added to the scattering amplitude. Owing to variations in the Q-dependent scattering amplitudes, the minimum is located at different energies for different reflections which defines the energy range for each measurement. Energy-dependent RXD data have been collected in quasi-symmetric geometry (i.e., α = β) at the beamlines BM2042 and BM2843 of the European Synchrotron Radiation Facility (ESRF) using the diffractometers in four-circle geometry. The preparation of samples (crystal plus electrodes) has been described in detail elsewhere19. To facilitate the formation of the MFP phase, the photon flux had to be reduced such that the leakage current was about 1 μA or lower depending on the formation state. For the study of influences from thermal displacement, a continuous-flow cryostat was used at BM20 to cool the crystals down to a minimum temperature just above the cubic to tetragonal phase transition at 105 K44.

The fit of the energy dependent Bragg intensities was based on Eq. 1 multiplied by a linear device function where a free scale parameter was included for each reflection. Therefore, only relative intensity changes were relevant for the fit results. The dispersion terms f′, f″ far off the edge have been taken from the Sasaki database45, whereas the near edge fine structure in f′, f″ has been obtained for strontium by means of XAS measurements on SrTiO3 powder at the Sr-K edge and subsequent application of Kramers-Kronig relations following Lucarini et al.46. The nonresonant atomic scattering factors f0(Q) have been taken from the International Tables for Crystallography47. Additional peaks appearing in the energy dependent RXD scans are attributed to Umweganregung48 and have been filtered out through multiple scans for small rotations about the Bragg vector Q where feasible.

### Data availability

Raw data were generated at the ESRF and can be accessed from the Open Science Framework49. Software routines used for data modeling are available online50.

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## Acknowledgements

We thank the teams of beamlines BM20 (ROBL) and BM28 (XMaS) of the ESRF, particularly S.D. Brown and C. Bähtz, for their outstanding support during the measurements. Furthermore, we are very grateful for advice given by V.E. Dmitrienko regarding the first principles calculations and for the help of H. Choe and B. Khanbabaee during one of the beamtimes. Financial support by the German Federal Ministry of Education and Research (BMBF) within the projects 05K13OF1 (C.R.), 03EK3029A (M.Z., E.M., J.H., and M.N.), and 05K13PSA (S.G.) is greatly acknowledged.

## Author information

### Affiliations

1. #### Institute of Experimental Physics, TU Bergakademie Freiberg, Leipziger Strasse 23, 09596, Freiberg, Germany

• Carsten Richter
• , Matthias Zschornak
• , Erik Mehner
• , Melanie Nentwich
• , Juliane Hanzig
•  & Dirk C. Meyer
2. #### Photon Science, Deutsches Elektronen-Synchrotron DESY, Hamburg, 22607, Germany

• Carsten Richter
•  & Dmitri Novikov
3. #### Department of Materials Science and Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

• Semën Gorfman

### Contributions

C.R. developed the technique, led the measurements, analyzed the data, performed the database survey, and prepared the initial manuscript. All authors contributed significantly to the manuscript. M.Z. carried out the first principles calculations. C.R., D.N., E.M., M.N., M.Z., and S.G. participated in the experiments. J.H. prepared and pre-characterized the samples. E.M. designed the sample holder with cabling for high-voltage experiments. S.G. contributed to crystallographic considerations prior to the measurements. D.C.M. and M.Z. suggested an investigation of the MFP phase using resonant XRD.

### Competing interests

The authors declare no competing financial interests.

### Corresponding author

Correspondence to Carsten Richter.

## Electronic supplementary material

### DOI

https://doi.org/10.1038/s41467-017-02599-6