# Measurement of the way

## Introduction

The path is determined according to its length (l) measured. The SI unit for the length is the meter (m), it is defined as follows:

definition
One meter is 1,650,763.73 times the wavelength of atoms of the nuclide${}^{86}\text{Kr}$ when transitioning from state 5d5 to state 2 p10emitted radiation that propagates in a vacuum.

## Measurement errors and graphical representations

An important task in physics is to determine relationships between two quantities, whereby these two quantities are measured and are therefore prone to errors.
As an example, consider a distance and time measurement for a movement that can be viewed as uniform. The measured values ​​shown in the following table were determined.

Figure 4 shows the corresponding graphical representation of the measured values ​​in one s-t-Diagram.

It would not be appropriate for the physical situation to connect the dots with one another. Since all measured values ​​are faulty, a compensation curve is drawn.
The main question now is whether it is possible to determine more precisely how this regression curve runs. We assume that the time measurement error can be neglected compared to the distance measurement error.

Measured values ​​and compensation curve

A first possibility for more precise determination of the course of the regression curve would be to mark the greatest error of the path for each measuring point in the form of an error bar (Fig. 5). The regression curve then runs through the error bars. In the example we have assumed an error of Δ s = ± 2.5 m for the path.
The function curve of the regression curve can also be calculated. Let us assume that for them the equation

applies, then you can a respectively. b calculate as follows:

a = (∑ i = 1 n x i ⋅ y i) - n ⋅ x ¯ ⋅ y ¯ (∑ i = 1 n x i 2) - n ⋅ x ¯ 2

b = y ¯ - a ⋅ x ¯

The equation for the best-fit line is thus:

If the situation is non-linear and there is a presumption about the type of dependency, a non-linear relationship can be reduced to a linear relationship by a clever choice of the axis sizes.

Consideration of errors before and after measurements
Considerations of errors are important both before measurements are carried out and in the evaluation of measurements.
If the aim of the measurements is to measure a physical quantity as precisely as possible, then one must before the measurement i.a. answer the following questions:

• Which measurement method do I choose?
• What can cause measurement errors?
• Are there ways of correcting, compensating or minimizing errors?
• Which variables have to be measured particularly precisely because their error has a particularly strong influence on the error of the overall result?
• Does it make sense to carry out a test measurement or a control measurement?
• Is a single measurement sufficient or is it advisable to repeat the measurements so often that a statistical evaluation (error calculation) can be made?

Note:
The accuracy of measurements can usually only be influenced before or during the measurement. Therefore, one should carry out error considerations before starting the measurements.

After taking measurements you can usually only estimate how accurately you have measured, i.e. answer the following questions:

• What random and systematic errors have actually occurred?
• How big are the individual errors and the largest errors in the individual measured quantities?
• How do these errors affect the measurement result? How big is the error of the size to be determined?

Example: the acceleration due to gravity
As an example, consider an experiment to determine the acceleration due to gravity G. This natural constant can be used, for. B. using a thread pendulum using the equation

determine by looking for a pendulum's length l the period of oscillation T measures and the gravitational acceleration with the equation

calculated. With such a determination of the gravitational acceleration, the measurement error depends on the accuracy of the length measurement and the accuracy of the time measurement, the time measurement error having a factor of 2. If the absolute error of the length measurement of the pendulum is Δ l and the absolute error of the measurement of the period of oscillation is Δ T, the resulting error is the acceleration due to gravity G:

at l = 70.0 cm with Δ l = ± 1 cm and T = 1.64 s with Δ T = ± 0.2 s one obtains:

Δ g g = ± 1 cm 70.0 cm + 2 ⋅ ± 0.2 s 1.64 s

Δ g g = ± 0.26 or 26%

Another way of determining the acceleration due to gravity is the free fall of a body (Figure 1), which is shown in detail below. The laws of free fall are used.
The following applies to free fall:

This means that the value for the acceleration due to gravity can be determined experimentally. For this purpose, the above equation is used after the acceleration due to gravity G changed over:

The distance and time for a body to free fall must be measured. To get the most accurate value possible for G the fall time to be measured and thus the fall distance should not be too short. The fall time is measured several times and an average value is then calculated. The free fall is triggered with a magnetic switch, the time measurement with an electronic clock (Fig. 1).

Execution:
The fall distance is s = 0.95 m.

Evaluation:
The following mean value can be calculated from the series of measurements for the time:

For the acceleration due to gravity one can thus calculate:

The experimentally determined value agrees relatively well with the table value (9.81 m / s 2).

The measured value can be used if the exact value known in this case lies in the error interval.

We provide the following considerations Measurement accuracy and Error interval The considerations are also shown in the sequence of steps preparation - implementation - evaluation.

Preparation:
The acceleration due to gravity is after G = 2 s / t 2 determined, so the fall distance and fall time are to be measured.
The fall distance is measured with a ruler with cm graduation.
It should also be noted that the distance is measured from the bottom edge of the body to the point of contact. Parallax errors can occur. Overall, random errors dominate the path measurement. A multiple measurement with averaging would be useful.
The fall time must be measured particularly precisely because its error is entered with a factor of 2. The accuracy of the time measurement cannot be influenced in the experiment arrangement and is essentially determined by the accuracy class of the measuring device. However, since random influences (e.g. air movement) can occur in this case, a series of measurements with subsequent averaging is advisable.

Execution:
It is particularly important to ensure that the set fall path does not change during all measurements of the time. If there are obvious “outliers”, these can be dropped.

Evaluation:
The largest errors in the measurement of the fall distance and fall time can be estimated and the largest error in the acceleration due to gravity can be determined.
The possible reading error is decisive for the fall path.
With a cm division it is:

Assuming a device error of ± 1% for the ruler, this means an error of:

The greatest error is therefore:

The accuracy class of the measuring device is decisive for electronic time measurement. It is 1.0 with a measuring range of 1 s, so the absolute error is 0.01 s.If one assumes a random error of 0.01 s for the timer, which is accurate to a hundredth, then:

the end G = 2 s / t 2 one obtains according to the rules of error propagation for the absolute error of the acceleration due to gravity:

The values ​​given in the table above result in:

The relative error is 11%, the absolute error ± 1.1 m / s 2.

The result could be stated as follows:

Annotation:
With the stated values, the error of the time measurement dominates with approx. 9%. Even assuming a measurement error in the time measurement of 0.01 s, the relative error would be G still be 4.5% or ± 0.44 m / s 2. Even then, the table value would be in the error interval.

The external photoelectric effect was discovered experimentally in 1887 by Heinrich Hertz [1] and Wilhelm Hallwachs [2] and later explained by Albert Einstein [3] (Nobel Prize in Physics 1921).

Hallwax realized that it is not the intensity of the light but its frequency that decides whether electrons can be released from the surface of a photocathode. Einstein introduced the term light quantum (photon) and showed that its energy, which - as Max Planck had previously discovered for thermal radiation - results directly from the light frequency ν, must be at least as large as the work function Φ of the solid surface. Its photoelectric equation gives the kinetic energy of a photoelectron E.kincreated by a photon of energy E.photon from a state with the binding energy E.B. is stimulated.

Photoelectron spectroscopy was systematically developed from 1960 by Kai Siegbahn in Uppsala into an important experimental investigation method in surface and solid-state physics, for which he received the Nobel Prize in 1981.

The underlying idea was to convert the energy distribution of the electrons in the solid body into an intensity distribution by photoemission excitation I.(E.kin) of photoelectrons of a certain energy E.kin to convict. The kinetic energy of the photoelectrons can then be measured (spectroscopically) using suitable magnetic or electrostatic analyzers.

To excite the photoelectrons, he used two different types of light sources, which are still common in laboratories today, the gas discharge lamp and the X-ray tube. The radiation generated in these sources and used for PES is in the hard ultraviolet range or in the soft X-ray range. According to the energy of the radiation used, a distinction is made between photoelectron spectroscopy to this day UPS (Ultraviolet Photoelectron Spectroscopy) and XPS, after the English name X-ray for X-rays. The energy resolution of the first instruments used was typically between 1 and 2 eV in the XPS range and 100 meV or less in the UPS range.

A major discovery by Siegbahn was that the spectra of the core electrons depend on the chemical environment of the system under investigation. In the XPS spectra of the same element, depending on the chemical form in which it is present, there are differences in the binding energy of a core electron of up to a few electron volts, and in many cases the form of the spectra can also provide information about the valence state of an element. The second name of XPS is based on these observations and the resulting possible applications. ESCA (Electron Spectroscopy for Chemical Analysis).

The method of studying molecules in the gas phase using ultraviolet light was developed by David W. Turner and described in a series of publications from 1962 to 1970. As a light source he used a He gas discharge lamp (E. = 21.22 eV) whose emission is in the ultraviolet range. With this source, Turner's group achieved an energy resolution of approx. 0.02 eV and was thus able to determine the energy of molecular orbitals very precisely and to compare it with theoretical values ​​of the quantum chemistry currently developed at the time. Due to the excitation by means of UV light, this measurement method - based on XPS - was called UPS.

Photoelectron spectroscopy is a measurement method based on the external photo effect. If you irradiate a gas or a solid with light of known energy E.photonso are electrons of kinetic energy E.kin free. With his photoelectric equation, Einstein was able to establish the relationship between the incident photon energy and the kinetic energy of the electrons:

With known photon energy and measured electron energy, statements can be made about the bonding conditions of the electrons in the examined material using this equation. The binding energy E.B. refers to the chemical potential of the solid and the work function Φ (determined when calibrating the spectrometer)spec of the spectrometer. The work function is a characteristic, material or surface-specific variable that can be determined by means of the external photo effect (see Figure 2). In an approximation according to Koopmans, it is assumed that the position of the energy levels of an atom or molecule does not change when it is ionized. As a result, the ionization energy I < displaystyle I> for the highest occupied orbital (HOMO: highest occupied molecular orbital) is equal to the negative orbital energy ε < displaystyle varepsilon>, i.e. the binding energy. With a closer look at this energy in core level electrons, conclusions can be drawn about the type of atom and the chemical composition (stoichiometry) of the sample and, to a certain extent, the chemical bonding in the investigated solid are obtained from the quantitative analysis. In addition, the analysis of the binding energy of the valence band and conduction electrons allows a very detailed investigation of the excitation spectrum of the electron system of crystalline solids.

The additional determination of the angle at which the photoelectrons leave a solid allows a more precise investigation of the valence band structures of crystalline solid bodies, whereby one makes use of the conservation of momentum in the photoemission process. Due to the relationship between the momentum of the photoelectron and the wave vector of a Bloch electron, it is possible to infer the dispersion relations of the valence states from the angle dependence of the spectra. This angle-resolved photoelectron spectroscopy is also called ARPES for short (angular resolved photoelectron spectroscopy). In the case of metals, the electronic dispersion relations contain information about the shape of the Fermi surface, which can also be determined using a number of other methods, such as B. the De Haas van Alphen, Schubnikow de Haas or the anomalous skin effect. The methods mentioned must, however, be carried out on highly pure single crystals at the lowest possible temperatures, whereas ARPES can also be used at room temperature and comparatively defect-rich crystals.

### X-ray Photoelectron Spectroscopy (XPS) edit

X-ray photoelectron spectroscopy (English: X-ray photoelectron spectroscopy , XPS, often too electron spectroscopy for chemical analysis , ESCA) is an established method to non-destructively determine the chemical composition of solids or their surface. You first get an answer to the question of qualitative element analysis, i.e. which chemical elements the solid consists of. Only hydrogen and helium can generally not be detected due to the small cross-sections. [4]

The method uses high-energy X-rays, mostly from an Al-Kα- or Mg-Kα-Source to release electrons from the inner orbitals. The kinetic energy of the photoelectrons can then be used to determine their binding energy E.B. to be determined. It is characteristic of the atom (more precisely even for the atomic orbital) from which the electron originates. The analyzer used for the measurement (usually a hemispherical analyzer) is set using electrostatic lenses and counter voltages in such a way that only electrons of a certain energy can pass through it. For the XPS measurement, the electrons that still arrive at the end of the analyzer are detected by a secondary electron multiplier, so that a spectrum is created, which is usually shown in a graph by plotting the intensity (counting rate) against the kinetic energy of the photoelectrons. The intensity is proportional to the frequency of occurrence of the various elements in the sample. In order to determine the chemical composition of a solid, one has to evaluate the area below the observed lines that are characteristic of the elements. However, there are some measurement-specific features to consider (see main article).

### Ultraviolet Photoelectron Spectroscopy (UPS) Edit

The purpose of the UPS is to determine the valence band structure of solids, surfaces and adsorbates. The density of states is determined Density of States , DOS). For this purpose, ultraviolet light is used in UPS (often also referred to as valence band spectroscopy in the solid state area), which is only capable of triggering valence electrons. These energies are of course also accessible to the XPS measurement, but the kinetic energy of the photoelectrons triggered in this way can be measured with extremely high accuracy through a suitable choice of the light source (generally He gas discharge lamps). UPS can also be used to resolve minimal energy differences between molecular orbitals or the physical environment (e.g. adsorption on surfaces) of the spectroscoped molecule. The chemical structure of bonds, adsorption mechanisms on substrates and vibrational energies of various molecular gases can be examined.

### Two photon photoemission spectroscopy (2PPE) edit

Two-photon photoemission spectroscopy, or 2PPE spectroscopy for short, is a photoelectron spectroscopy technique that is used to investigate the electronic structure and the dynamics of unoccupied states on surfaces. [5] Femtosecond to picosecond laser pulses are used to photoactivate an electron. [6] After a time delay, the excited electron is emitted into a free electron state by a second pulse. [6] The emitted electron is then detected with special detectors with which both the energy and the emission angle and thus the momentum of the electron parallel to the surface can be determined. [5]

### Angle Resolved Measurements (ARPES) Edit

With the help of angle-resolved measurements, ARPES (engl. angle-resolved PES ) or ARUPS ( angle-resolved UPS ), not only the energy of the photoelectrons is measured, but also the angle at which they leave the sample. In this way it is possible to determine the energy-momentum relationship of the electron in the solid, i.e. the representation of the band structure or the visualization of Fermi surfaces.

#### Edit measurement principle

All of the PES methods listed so far detect the photoelectrons regardless of the angle at which they leave the sample. Strictly speaking, for these measurements one generally selects the measuring position of the analyzer in such a way that predominantly electrons with an exit angle perpendicular to the sample surface can be detected. The analyzer settings (more precisely the lens voltages of the electrostatic lenses, the electron optics) are set in such a way that a very wide angle acceptance range of approx. ± 10 ° results. For the measurement method described below, the settings of the analyzer are changed in such a way that photoelectrons are only detected at a significantly smaller angle. Modern analyzers achieve an angular resolution of less than 0.2 ° with a simultaneous energy resolution of 1–2 meV. Originally, high energy and angular resolution were only achieved with low photoelectron energies in the UV range, from which the name ARUPS was derived. Especially in the years 1990-2000, the resolution of the PES analyzers was improved by the combination of microchannel plates, phosphorescent plates and CCD cameras so that even with far higher photoelectron energies (state of the art 2006: E.kin ≈ 10 keV) a determination of the band structure became possible.

Due to the requirement for low photon energies for the analyzer, for example from a He lamp, ARUPS measurements could only determine the band structure of the near-surface areas of a solid. Together with the improvement of the resolution of the analyzers and the use of high-energy (E.kin & gt 500 eV), extremely monochromatic synchrotron light, it has been possible since approx. 2000 to determine the band structure of the volume of crystalline solids. This is one of the reasons why ARPES has developed into one of the most important spectroscopic methods for determining the electronic structure of solids in our time.

#### Qualitative evaluation of the measurement

An essential prerequisite for the validity of the statement that ARPES can determine the band structure of a crystalline solid is the applicability of Bloch's theorem to the electronic states involved, that is, that they are divided by a wavenumber vector k can be clearly characterized and the associated wave function has the general form:

owns, where uk is a lattice periodic function. This requirement cannot be met in the experiment. Due to the transition from the surface of the sample to the vacuum, the system is not translation-invariant in the vertical direction and thus is not k ⊥ < displaystyle _ < bot >> -part of the measured wavenumber vector not a good quantum number. However, it is k ‖ < Displaystyle _ < | >> -part is retained, since both the crystal potential and the vacuum remain lattice-periodic parallel to the surface. Hence, the magnitude of the wavenumber vector in this direction can be given directly:

If the density of states (band structure) of the examined crystal does not vary too much perpendicular to the surface, it is possible to measure the occupied state density directly. In order to allow comparison with theoretical calculations, measurements are usually made in certain directions of high symmetry of the Brillouin zone. For this purpose, the crystal is often oriented perpendicular to the analyzer by means of a LEED, then rotated along one of the directions and an energy spectrum of the photoelectrons is recorded. Using a microchannel plate and a CCD camera, the energy and the angle at which the electrons leave the surface can even be measured at the same time.

The results are usually presented by showing all the angle-dependent spectra in a graph, with the energy and intensity plotted on the coordinate axes. In order to be able to follow the angle dependence, the individual spectra are shifted in intensity so that the dispersion can be observed. An alternative representation is an intensity distribution by means of color coding, in which the angle and energy on the coordinate axes and the intensity are shown as color gradations.

The almost complete spectroscopy of the half-space over a metallic sample according to the above-mentioned method now allows the Fermi area of ​​the electron system of the crystal to be determined from the spectra. By definition, the Fermi surface results from the joining of all points in momentum space at which an electronic band crosses the Fermi energy (as in the case of the dispersion relation, it is sufficient to limit the definition of the Fermi surface to the first Brillouin zone) . In PES measurements with constant photon energy, the points of passage generally correspond to emission directions at which the intensity at the Fermi energy is particularly high in the spectra. Therefore it is often sufficient to determine the intensity distribution E.F. to be determined as a function of the emission angle Θ without having to take into account the exact course of the band.

### Photoelectron Diffraction (XPD) edit

Photoelectron diffraction, often with PED, PhD or XPD (engl. X-ray photoelectron diffraction ) abbreviated, is a method to determine the structure of crystalline surfaces or the spatial position of adsorbates on surfaces. The basis of the measuring process is again photoelectron spectroscopy, whereby the intensity of the photoelectrons is determined depending on the emission angle. However, here, as with the angle-dependent PES, the focus is not on the impulse of the photoelectron, but rather the interference of the wave function of the photoelectron. Depending on the direction of emission and the kinetic energy of the photoelectron, there are differences in intensity, known as modulations. These intensity modulations arise from constructive and destructive interference between the electron wave that reaches the detector directly (reference wave) and those that occur from waves elastically scattered one or more times around the emitting atom (object waves). The path differences and intensities of the individual waves depend on the geometric arrangement and the type of neighboring atoms. With a sufficient number of measured intensities, the geometric structure can be determined from the modulations by comparing the experimentally measured modulations with corresponding simulations.

The simplest applications are based on forward focusing by atoms above the photoionized atom. This can be used to determine whether certain atoms are located directly on the surface or deeper, and in the case of adsorbed molecules, whether there are other atoms (and in which direction) above one type of atom. The XPD can be used to determine the crystallographic structure of metal and semiconductor surfaces. In addition, information is obtained about the spatial position of molecules on surfaces, the bond lengths and bond angles.

### Photoemission electron microscopy (PEEM) edit

Another widespread application of PES is photoemission electron microscopy, or PEEM for short. photo emission electron microscopy ) called. Here electrons are released from the sample by the photoelectric effect, but during the detection it is not the number of electrons of a kinetic energy selected by the analyzer that is measured, but rather the intensity distribution of the photoelectrons in a two-dimensional area of ​​the sample. It is therefore, characteristic of microscopes, an imaging measuring technique.

By installing a microanalyser in the beam path that selects the kinetic energy of the photoelectrons (analogous to normal PES) and by using narrow-band and short-wave excitation light sources, such as. B. synchrotron radiation, it is possible to carry out laterally resolved XPS (XPS microscope). The term μ-ESCA describes the chemical analysis of a micrometer-sized area of ​​the sample. This enables both the determination of the elemental composition of the sample and the investigation of local differences in the electronic properties.

### Coincident Photoelectron Spectroscopy Edit

In addition to the emission of a single electron per incident photon, there is also the possibility that two or more electrons are released. This can take place on the one hand as part of a secondary electron cascade, but also through the coherent emission of two electrons by a photon. The coincident measurement of the emitted electrons allows conclusions to be drawn about the underlying coupling mechanisms. Typically, no electrostatic analyzers are used for experimental detection, but rather time-of-flight spectrometers. Due to the small opening angle of an electrostatic analyzer, only very low coincidence rates can be achieved in this way. Time-of-flight projection systems enable significantly higher detection efficiency. The electrons emitted by a pulsed photon beam are projected onto spatially resolving detectors. The initial impulses or angles and kinetic energy can be determined from the flight time and location.

### Measurements in resonance (ResPES) edit

In principle, the course of the photoemission spectrum, in particular that of the valence band, depends on the photon energy used for the excitation. If the photon energy sweeps over the area of ​​an X-ray absorption edge, the changes are generally particularly pronounced. The reason for this are resonance effects that arise from the interaction of two or more different end states, more precisely from continuum states with discrete levels, and thus influence the overall photoemission cross-section. If the photoemission intensity of a selected spectral structure is plotted against the photon energy, one generally obtains asymmetrical excitation profiles, so-called Fano resonances. The shape and intensity of these profiles can provide information about the elementary character of the spectral structure, details of the chemical bond and the interactions between the states involved. This is the case with the resonant photoemission spectroscopy ( resonant photoemission spectroscopy , ResPES).

### Inverse Photoelectron Spectroscopy (IPES) edit

In contrast to PES, with inverse photoelectron spectroscopy (IPS, IPES), often also called inverse photoemission spectroscopy, electrons of known energy are accelerated onto the sample and the photons emitted are detected as bremsstrahlung. The two measurement methods PES and IPES can complement each other very well, as the IPES are good for determining the unoccupied Density of states (above the Fermi energy) is suitable (more details on determining the unoccupied density of states with UPS above). Analogous to the angle-integrated measurements just mentioned, IPES also enables the experimental determination of the band structure above the chemical potential (above the Fermi energy) for angle-resolved measurements. In the same way as ARUPS, IPES uses the UV range to determine the band structure, the k-Information from the direction of incidence of the exciting electrons.

In terms of equipment, an IPES spectrometer consists of a simple electron gun and a photon detector with a bandpass filter or monochromator. For laboratory measurements, the kinetic energy (primary energy) of the electrons is usually varied and the photon energy is kept constant during detection. In this case one speaks of Isochromatic mode, from which the name UNTIL, Bremsstrahlungsisochromaten-Spektroskopie, ableitet. Den häufigsten Einsatz finden für Energien im UV-Bereich Bandpassfilter vom Geiger-Müller-Typ, bei dem ein Erdalkalifluoridfenster als Tiefpass (z. B. CaF2 oder SrF2) und ein geeignetes Zählgas (z. B. I2 oder CS2) als Hochpass kombiniert werden. Detektionsenergie und Bandpassbreite ergeben sich aus der Transmissionsschwelle des Fenstermaterials bzw. aus der molekularen Photoionisationsschranke des Zählgases (etwa 9,5 eV). Die Bandpassbreite bestimmt im Wesentlichen die Spektrometerauflösung. Andere Detektortypen kombinieren das Erdalkalifluoridfenster mit einem geeignet beschichten Kanalelektronenvervielfacher (z. B. mit Natriumchlorid oder Kaliumbromid).

Wegen des geringen Wirkungsquerschnittes des inversen Photoemissionsprozesses ist die typische Zählrate im Vergleich zur Photoelektronenspektroskopie sehr klein. Daher lassen sich bei der IPES auch keine vergleichbaren Energieauflösungen erreichen, da das Signal mit der Bandpassbreite linear abnimmt. Typische Werte für die Auflösung liegen bei wenigen Hundert Millielektronenvolt, sind also zwei Größenordnungen schlechter als bei UPS. Detektoren mit Gittermonochromatoren erreichen prinzipiell deutlich bessere Werte und sind wegen ihrer durchstimmbaren Photonenenergie wesentlich flexibler einsetzbar, sind aber sehr viel teurer und größer als die anderen Detektortypen.

### ZEKE-Spektroskopie Bearbeiten

Bei der ZEKE-Spektroskopie (ZEKE kurz für englisch zero-electron kinetic energy oder auch nur zero kinetic energy ) werden insbesondere Elektronen an der Ionisationsgrenze detektiert [7] . Das zu untersuchende Gas wird mit einem kurzen Laserpuls bestrahlt. Nachdem dieser Laserpuls abgeklungen ist, wird die Zeit τ abgewartet. In dieser Zeit bewegen sich alle Elektronen mit E > E c > aus dem Untersuchungsbereich heraus. Mit Hilfe eines elektrischen Feldes werden nach Ablauf von τ alle restlichen Elektronen abgesaugt und gemessen.