The electrons interact with atoms in the sample, producing various signals that contain information about the surface topography and composition of the sample. The electron beam is scanned in a raster scan pattern, and the position of the beam is combined with the intensity of the detected signal to produce an image.

In the most common SEM mode, secondary electrons emitted by atoms excited by the electron beam are detected using an Everhart-Thornley detector. The number of secondary electrons that can be detected, and thus the signal intensity, depends, among other things, on specimen topography. SEM can achieve resolution better than 1 nanometer. The cooling is usually achieved with liquid nitrogen.

CryoSEM of biological samples with a high moisture content can be done faster with fewer sample preparation steps than conventional SEM. Serial-section electron microscopy ssEM. One application of TEM is serial-section electron microscopy ssEM , for example in analyzing the connectivity in volumetric samples of brain tissue by imaging many thin sections in sequence.

Reflection electron microscopy REM. In the reflection electron microscope REM as in the TEM, an electron beam is incident on a surface but instead of using the transmission TEM or secondary electrons SEM , the reflected beam of elastically scattered electrons is detected. Another variation is spin-polarized low-energy electron microscopy SPLEEM , which is used for looking at the microstructure of magnetic domains. Any electron that is accelerated by a voltage change its kinetic energy in a conservative way, so. Suppose that initially and ,.

Then, the final kinetic energy reads. In the quantum realm, any particle like the electron has an associated wave and wavelength. It is the de Broglie wavelength. And it reads. Indeed, you can generalize this equation to microscope of X-particles, where X-particles are particles with mass and electric charger as follows:. Now, some numerology. You can use the value of the Planck constant as roughly , and then you can write. If, in particular, the energy is given in electron-volts , then you get. Example 1. Example 2. Example 3. Example 4. Imagine muon microscopy, tau particle microscopy or boson microscopy….

Now, you can enter the special relativistic electron microscope realm. Just as you have for photons or any massless particle and , for any relativistic particle with mass and energy rest mass , the kinetic energy reads , since. Again, for a conservative force set-up,. Taking into account that. From you get. From you obtain algebraically.

Inserting this momentum into the relatistic de Broglie wavelength you finally derive the full relativistic electron microscope equation. They are linked through the following expressions:. With simple scaling rules, you can extended the relativistic electron microscopes to relativistic X-particle microscopes as follows. Definition 1 Electron microscope. Definition 2 X-particle microscope.

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Definition 3 Electron microscope II. Definition 4 X-particle microscope II. Definition 5 Relativistic electron microscope. Definition 6 Relativistic X-particle microscope. Today or not today. That is the point. How to know if a given particle or sytem is special relativistic?

That is a tricky question, since the reality is…Everythin is special relativistic. The question would be when you can use the usual newtonian aproximation. That is the subject today.

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The momentum is relativistic when. If you define , and as above, then. Mass measuremnts are general non-linear in velocity, but invariant mass definition is possible via scalar product in Minkovski spacetime. Particles moving at exactly the speed of light are massless. This is the case of gluons, photons, and gravitons. Massive particles satisfy a more general dispersion relationship, as above. The special case in which you have a massive particle with is called ultrarelativistic case, and then you can approximate.

Here we note the purely relativistic massless case and the ultrarelativistic case easily, and we can also distinguish the massive or almost massless case in the purely relativistic case or the ultrarelativistic case:. Subject today: errors. And we will review formulae to handle them with experimental data. Due to imperfections of measurements or intrinsically random sources.

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Due to the procedures used to measure or uncalibrated apparatus. Accuracy is closeness to the true value of a parameter or magnitude. It is, as you keep this definition, a measure of systematic bias or error. However, sometime accuracy is defined ISO definition as the combination between systematic and random errors, i.

High accuracy would require, in this case, higher trueness and high precision. It is a measure of random errors. They can be reduced with further measurements and they measure statistical variability. Precision also requires repeatability and reproducibility. This is the unbiased variance, when the total population is the sample, a shift must be done from to Bessel correction.

## "Muonium-antimuonium oscillations in an extended minimal supersymmetric" by Boyang Liu

The unbiased formula is correct as far as it is a sample from a larger population. Standard deviation mean squared error, mean quadratic error :. This is the unbiased estimator of the mean quadratic error, or the standard deviation of the sample. The Bessel correction is assumed whenever our sample is lesser in size that than of the total population.

For total population, the standard deviation reads after shifting :. If, instead of the unbiased quadratic mean error we use the total population error, the corrected standar error reads.

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Variance of the mean quadratic error variance of the variance :. Standard error of the mean quadratic error error of the variance :. Here we provide the probability of a random variable distribution X following a normal distribution to have a value inside an interval of width. Let us define and. Furthermore, define the variable. The error in would be:. Then, we have:. If , you obtain essentially the same result:. With , , then you derive. In the case of a several variables function, you apply a generalized Pythagorean theorem to get. Some simple examples are provided:.

When different experiments with measurements are provided, the best estimator for the combined mean is a weighted mean with the variance, i. This is also the maximal likelihood estimator of the mean assuming they are independent AND normally distributed. There, the standard error of the weighted mean would be. Remark: for non homogenous samples, the best estimation of the average is not the arithmetic mean, but the median. Double post today! Let me introduce you to some abstract uncommon representations for spacetime. I will focus on three simple examples:.

Hint: Geometric algebras or Clifford algebras are useful for this problem and the above challenge questions. Hi, there.