The inelastic mean free path ( IMFP ) is an index of how far an electron on average travels through a solid before losing energy.
31-459: If a monochromatic , primary beam of electrons is incident on a solid surface, the majority of incident electrons lose their energy because they interact strongly with matter , leading to plasmon excitation, electron-hole pair formation, and vibrational excitation. The intensity of the primary electrons, I 0 , is damped as a function of the distance, d , into the solid. The intensity decay can be expressed as follows: where I ( d )
62-666: A x {\displaystyle \omega _{\mathrm {max} }} ), the dielectric function ϵ {\displaystyle \epsilon } , the energy loss function (ELF) I m ( − 1 ϵ ( k , ω ) ) {\displaystyle \mathrm {Im} ({\frac {-1}{\epsilon (k,\omega )}})} and the smallest and largest momentum transfer k ± = 2 E ± 2 ( E − ω ) {\displaystyle k_{\pm }={\sqrt {2E}}\pm {\sqrt {2(E-\omega )}}} . In general, solving this integral
93-477: A transmission electron microscope , provided the sample thickness is known. Such measurements reveal that IMFP in elemental solids is not a smooth, but an oscillatory function of the atomic number . For energies below 100 eV, IMFP can be evaluated in high-energy secondary electron yield (SEY) experiments. Therefore, the SEY for an arbitrary incident energy between 0.1 keV-10 keV is analyzed. According to these experiments,
124-549: A Monte Carlo model can be used to simulate the SEYs and determine the IMFP below 100 eV. Using the dielectric formalism, the IMFP λ − 1 {\displaystyle \lambda ^{-1}} can be calculated by solving the following integral: with the minimum (maximum) energy loss ω m i n {\displaystyle \omega _{\mathrm {min} }} ( ω m
155-425: A desired artistic effect; if only the red channel is selected by the weighting then the effect will be similar to that of using a red filter on panchromatic film . If the red channel is eliminated and the green and blue combined then the effect will be similar to that of orthochromatic film or the use of a cyan filter on panchromatic film. The selection of weighting so provides a wide variety of artistic expression in
186-518: A monochromatic color provides a strong sense of visual cohesion and can help support communication objectives through the use of connotative color. The relative absence of hue contrast can be offset by variations in tone and the addition of texture. Monochromatic in science means consisting of a single wavelength of light or other radiation (lasers, for example, usually produce monochromatic light), or having or appearing to have only one color (in comparison to polychromatic). That means according to science
217-400: A monochromatic image. In computing, monochrome has two meanings: A monochrome computer display is able to display only a single color, often green, amber , red or white, and often also shades of that color. In film photography, monochrome is typically the use of black-and-white film . Originally, all photography was done in monochrome . Although color photography was possible even in
248-551: A narrow band of wavelengths, which is a distinct concept. Of an image , the term monochrome is usually taken to mean the same as black and white or, more likely, grayscale , but may also be used to refer to other combinations containing only tones of a single color, such as green -and-white or green-and-red. It may also refer to sepia displaying tones from light tan to dark brown or cyanotype ("blueprint") images, and early photographic methods such as daguerreotypes , ambrotypes , and tintypes , each of which may be used to produce
279-456: A single hue . Tints are achieved by adding white, thereby increasing lightness ; Shades are achieved by adding black, thereby decreasing lightness; Tones are achieved by adding gray, thereby decreasing colorfulness . Monochromatic color schemes provide opportunities in art and visual communications design as they allow for a greater range of contrasting tones that can be used to attract attention, create focus and support legibility. The use of
310-453: A universal curve that is the same for all materials. The knowledge of the IMFP is indispensable for several electron spectroscopy and microscopy measurements. Following, the IMFP is employed to calculate the effective attenuation length (EAL), the mean escape depth (MED) and the information depth (ID). Besides, one can utilize the IMFP to make matrix corrections for the relative sensitivity factor in quantitative surface analysis. Moreover,
341-443: Is quite challenging and only applies for energies above 100 eV. Thus, (semi)empirical formulas were introduced to determine the IMFP. A first approach is to calculate the IMFP by an approximate form of the relativistic Bethe equation for inelastic scattering of electrons in matter. Equation 2 holds for energies between 50 eV and 200 keV: with and and the electron energy E {\displaystyle E} in eV above
SECTION 10
#1732884574726372-487: Is required for conductors. If non-conducting materials are considered, one also needs to know either E g {\displaystyle E_{g}} or H {\displaystyle H} . An analytical formula for calculating the IMFP down to 50 eV was proposed in 2021. Therefore, an exponential term was added to an analytical formula already derived from 1 that was applicible for energies down to 500 eV: For relativistic electrons it holds: with
403-536: Is the heat of formation of a compound in eV per atom) and the average atomic spacing a {\displaystyle a} : with the Avogadro constant N A {\displaystyle N_{A}} and the stoichiometric coefficients g {\displaystyle g} and h {\displaystyle h} describing binary compounds G g H h {\displaystyle G_{g}H_{h}} . In this case,
434-483: Is the intensity after the primary electron beam has traveled through the solid to a distance d . The parameter λ( E ) , termed the inelastic mean free path (IMFP), is defined as the distance an electron beam can travel before its intensity decays to 1/ e of its initial value. (Note that this is equation is closely related to the Beer–Lambert law .) The inelastic mean free path of electrons can roughly be described by
465-497: The Fermi level (conductors) or above the bottom of the conduction band (non-conductors). m e {\displaystyle m_{e}} is the electron mass, c {\displaystyle c} the vacuum velocity of light, N ν {\displaystyle N_{\nu }} is the number of valence electrons per atom or molecule, ρ {\displaystyle \rho } describes
496-473: The IMFP is an important parameter in Monte Carlo simulations of photoelectron transport in matter. Calculations of the IMFP are mostly based on the algorithm (full Penn algorithm, FPA) developed by Penn, experimental optical constants or calculated optical data (for compounds). The FPA considers an inelastic scattering event and the dependence of the energy-loss function (EFL) on momentum transfer which describes
527-702: The Simulation of Electron Spectra for Surface Analysis (SESSA). The data contains IMFPs determined by EPES for energies below 2 keV. Otherwise, IMFPs can be determined from the TPP-2M or the S1 formula. Monochromatic A monochrome or monochromatic image, object or palette is composed of one color (or values of one color). Images using only shades of grey are called grayscale (typically digital) or black-and-white (typically analog). In physics, monochromatic light refers to electromagnetic radiation that contains
558-577: The TTP-2M equations show precise agreement with the measurements. Another approach based on Equation 2 to determine the IMFP is the S1 formula. This formula can be applied for energies between 100 eV and 10 keV: with the atomic number Z {\displaystyle Z} (average atomic number for a compound), W = 0.06 H {\displaystyle W=0.06H} or W = 0.02 E g {\displaystyle W=0.02E_{g}} ( H {\displaystyle H}
589-576: The TTP-2M formula one needs to know M {\displaystyle M} , ρ {\displaystyle \rho } and N ν {\displaystyle N_{\nu }} for conducting materials (and also E g {\displaystyle E_{g}} for non-conductors). Employing S1 formula, knowledge of the atomic number Z {\displaystyle Z} (average atomic number for compounds), M {\displaystyle M} and ρ {\displaystyle \rho }
620-422: The atomic number becomes with the atomic numbers Z g {\displaystyle Z_{g}} and Z h {\displaystyle Z_{h}} of the two constituents. This S1 formula shows higher agreement with measurements compared to Equation 2 . Calculating the IMFP with either the TTP-2M formula or the S1 formula requires different knowledge of some parameters. Applying
651-430: The bandgap energy E g {\displaystyle E_{g}} is given in eV. Equation 2 an 3 are also known as the TTP-2M equations and are in general applicable for energies between 50 eV and 200 keV. Neglecting a few materials (diamond, graphite, Cs, cubic-BN and hexagonal BN) that are not following these equations (due to deviations in β {\displaystyle \beta } ),
SECTION 20
#1732884574726682-455: The density (in g c m 3 {\displaystyle \mathrm {\frac {g}{cm^{3}}} } ), M {\displaystyle M} is the atomic or molecular weight and β {\displaystyle \beta } , γ {\displaystyle \gamma } , C {\displaystyle C} and D {\displaystyle D} are parameters determined in
713-451: The electron velocity v {\displaystyle v} , v 2 = c 2 τ ( τ + 2 ) / ( τ + 1 ) 2 {\displaystyle v^{2}=c^{2}\tau (\tau +2)/(\tau +1)^{2}} and τ = E / c 2 {\displaystyle \tau =E/c^{2}} . c {\displaystyle c} denotes
744-492: The final monochrome image. For production of an anaglyph image the original color stereogram source may first be reduced to monochrome in order to simplify the rendering of the image. This is sometimes required in cases where a color image would render in a confusing manner given the colors and patterns present in the source image and the selection filters used (typically red and its complement , cyan ). A monochromatic color scheme comprises ( tones, tints, and shades ) of
775-455: The following. Equation 2 calculates the IMFP and its dependence on the electron energy in condensed matter. Equation 2 was further developed to find the relations for the parameters β {\displaystyle \beta } , γ {\displaystyle \gamma } , C {\displaystyle C} and D {\displaystyle D} for energies between 50 eV and 2 keV: Here,
806-430: The late 19th century, easily used color films, such as Kodachrome , were not available until the mid-1930s. In digital photography , monochrome images use only the data for brightness captured by the sensor, or by post-processing a color image to present only the perceived brightness by combining the values of multiple channels (usually red, blue, and green). The weighting of individual channels may be selected to achieve
837-408: The probability for inelastic scattering as a function of momentum transfer. To measure the IMFP, one well known method is elastic-peak electron spectroscopy (EPES). This method measures the intensity of elastically backscattered electrons with a certain energy from a sample material in a certain direction. Applying a similar technique to materials whose IMFP is known, the measurements are compared with
868-475: The results from the Monte Carlo simulations under the same conditions. Thus, one obtains the IMFP of a certain material in a certain energy spectrum. EPES measurements show a root-mean-square (RMS) difference between 12% and 17% from the theoretical expected values. Calculated and experimental results show higher agreement for higher energies. For electron energies in the range 30 keV – 1 MeV, IMFP can be directly measured by electron energy loss spectroscopy inside
899-481: The strictly scientific meaning of the word. In fact, monochrome in the art world can be as complicated or even more complicated than other polychromatic art. In physics, monochromatic light is electromagnetic radiation of a single wavelength . While no source of electromagnetic radiation is purely monochromatic, in practice, it is usually used to describe very narrowband sources such as monochromated or laser light. The degree of monochromaticity can be defined by
930-410: The true monochromatic images can be strictly created only of shades of one color fading to black. However, monochromatic also has another meaning similar to “boring” or “colorless” which sometimes leads to creating a design composed from true monochromatic color shades (one hue fading to black), and the colors created from the one hue but faded to all wavelengths (to white). This is not monochromatic in
961-516: The velocity of light. λ {\displaystyle \lambda } and a 0 {\displaystyle a_{0}} are given in nanometers. The constants in 4 and 5 are defined as following: IMFP data can be collected from the National Institute of Standards and Technology (NIST) Electron Inelastic-Mean-Free-Path Database or the NIST Database for