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Talk:Introduction to Philosophy/Empiricism

"Tabula Rasa" basically translates as "blank slate." Empiricists believe that we are born with a "blank slate" and the environment "stamps" us with experiences.

Rationalists believe that we're born with some "innate ideas"--every one of us is born with them imprinted on our souls or minds. Locke argued that we are not born with innate ideas, but that we learn absolutely everything we ever know from our experience with the environment. Sensation (using the senses to learn) and Reflection (using our minds to assemble our experiences into new knowledge) were the 2 methods of gaining knowledge in Locke's empiricism.


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The Once and Future King/Major Characters

The very essence of the wonderfully weird wizard that dominates the King Arthur legends is rooted in mystery. Merlyn is a powerful teacher and a kind friend, though his absentmindedness leads Arthur into trouble. He’s already lived in the future and is fully aware of what will occur as he continues to grow younger over the years. He is the main majestic force behind Arthur’s ideals and power. This character has graced countless novels and bedtime stories, enlivening each with his quirkiness.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Nanotechnology/Optical Methods

=Optical Microscopy=

The Abbe diffraction limit
Observation of sub-wavelength structures with microscopes is difficult because of the Abbe diffraction limit. Ernst Abbe found in 1873 that light with wavelength λ,travelling in a medium with refractive index n and converging to a spot with angle φ will make a spot with radius

 * $$d=\frac{0.61 \lambda}{n sin \phi}$$

The denominator nsinφ is called the numerical aperture (NA) and can reach about 1.4 in modern optics, hence the Abbe limit is roughly d=λ/2. With green light around 500nm the Abbe limit is 250nm which is large compared to most nanostructures or biological cells with sizes on the order of 1μm and their internal organelles being much smaller. To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes. These techniques offer splendid resolution but are expensive, suffer from lack of contrast in biological samples, and also tend to damage the sample.

Resources

 * Microscope optics
 * Microscopy Primer
 * Olympus Interactive Java Tutorials on Microscopy
 * Nikon MicroscopyU with interactive tutorials on microscopy techniques

Bright Field
The light is sent to the sample in the same directions as you are looking - most things will look bright unless they absorb the light.

Dark Field
Light is sent towards the sample at an angle to your viewing direction and you only see light that is scattered. This makes most images appear dark and only edges and curved surfaces will light up.

DIC vs H
=Laser Scanning Confocal Microscopy (LSCM)= Confocal laser scanning microscopy is a technique that allows a much better resolution from optical microscopes and three dimensional imaging. A review can be found in Paddock, Biotechniques 1999

Using a high NA objective also gives a very shallow depth of focus and hence the image will be blurred by structures above or below the focus point in a classical microscope. A way to circumvent this problem is the confocal microscope, or even better the Laser Scanning Confocal Microscope (LSCM). Using a laser as the light source gives better control of the illumintaion, especially when using fluorescent markers in the sample. The theoretical resolution using a 1.4 NA objective can reach 140nm laterally and 230nm vertically while the resolution quoted in ref is 0.5×0.5×1μm. The image in the LSCM is made by scanning the sample in 2D or 3D and recordning the signal for each point in space on a PC which then generates the image.

=X-ray microscopy= X-ray microscopy uses X-rays to image with much shorter wavelength than optical light, and hence can provide much higher spatial resolution and use different contrast mechanisms. X-ray microscopy allows the characterization of materials with submicron resolution approaching the 10's of nanometers. X-ray microscopes can use both laboratory x-ray sources and synchrotron radiation from electron accelerators. X-ray microscopes using synchrotron radiation provide the greatest sensitivity and power, but are unfortunately rather large and expensive. X-ray microscopy is usually divided into two overlapping ranges, referred to as soft x-ray microscopy (100eV - 2keV) and hard x-ray microscopy (1keV-40keV). All x-rays penetrate materials, more for higher energy x-rays. Hence, soft x-ray microscopy provides the best contrast for small samples. Hard x-rays do have the ability to pass nearly unhindered through objects like your body, and hence also give rather poor contrast in many of the biological samples you would like to observe with the x-ray microscope. Nevertheless, hard x-ray microscopy allows imaging by phase contrast, or using scanning probe x-ray microscopy, by using detection of fluorescent or scattered x-rays. Despite its limitations, X-ray microscopy is a powerful technique and in some cases can provide characterization of materials or samples that cannot be done by any other means.

=UV/VIS spectrometry=

=Infrared spectrometry (FTIR)=

vIdentification of the functional groups present in a nanomaterial is a frequent requirement in nanoscience and nanotechnology research. Among other tools, FT-IR has found much popularity among researches due to its versatility, relative ease of use and ability to use as a quantification tool.

Atoms in a chemical bonds constantly vibrate. This vibration can be analogue to a system with two masses attached to a spring. The vibration frequency depend upon the weight of the masses and the spring constant of the connecting spring. In the same way, depending on the masses of the atoms that contributes to a bond and cohesiveness of the bond, frequency differ. Since bonds have atoms with different shapes and sizes and different strength, each combination of atoms in an each type of bond has a unique harmonic frequency. This natural frequency lies in the range of infrared region and therefore a spectroscopic method that use IR can be devised to analyze bond vibrations.

When the IR radiation with the same harmonic frequency of the bond shines upon the bond. The bond vibration is amplified by increased transfer of energy from the IR radiation. When range of IR frequencies given to the material, it only absorb IR frequencies that corresponds to the natural frequencies of the bonds that exist in the sample. Others are not absorbed and can be analyzed using an Infrared spectrometer, which tells you the frequencies that are absorbed by the sample. This provides important information about the functional groups present in the sample. This is exactly what FT-IR does.

As FT-IR can be used to get information about functional groups present in nanomaterials. This is particularly useful in cases such as when one attempts to surface modify nanomaterials to increase affinity, reactivity or compatibility. Analyzing the FT-IR of a nanomaterial would tell you what groups present and then appropriate surface modification strategy be decided based on the groups present. Further, it can also be useful in characterizing the surface modification has taken place, as new groups should emerge if the reaction is successful.Identification of the functional groups present in a nanomaterial is a frequent requirement in nanoscience and nanotechnology research. Among other tools, FT-IR has found much popularity among researches due to its versatility, relative ease of use and ability to use as a quantification tool.

Atoms in a chemical bonds constantly vibrate. This vibration can be analogue to a system with two masses attached to a spring. The vibration frequency depend upon the weight of the masses and the spring constant of the connecting spring. In the same way, depending on the masses of the atoms that contributes to a bond and cohesiveness of the bond, frequency differ. Since bonds have atoms with different shapes and sizes and different strength, each combination of atoms in an each type of bond has a unique harmonic frequency. This natural frequency lies in the range of infrared region and therefore a spectroscopic method that use IR can be devised to analyze bond vibrations.

When the IR radiation with the same harmonic frequency of the bond shines upon the bond. The bond vibration is amplified by increased transfer of energy from the IR radiation. When range of IR frequencies given to the material, it only absorb IR frequencies that corresponds to the natural frequencies of the bonds that exist in the sample. Others are not absorbed and can be analyzed using an Infrared spectrometer, which tells you the frequencies that are absorbed by the sample. This provides important information about the functional groups present in the sample. This is exactly what FT-IR does.

As FT-IR can be used to get information about functional groups present in nanomaterials. This is particularly useful in cases such as when one attempts to surface modify nanomaterials to increase affinity, reactivity or compatibility. Analyzing the FT-IR of a nanomaterial would tell you what groups present and then appropriate surface modification strategy be decided based on the groups present. Further, it can also be useful in characterizing the surface modification has taken place, as new groups should emerge if the reaction is successful.

=Terahertz Spectroscopy=

=Raman Spectroscopy=

 * Raman scattering
 * Resonance Raman spectroscopy

Surface Enhanced Raman Spectroscopy (SERS)

 * Surface Enhanced Raman Spectroscopy (SERS)
 * Surface-enhanced Raman spectroscopy: a brief retrospective

=References=

See also notes on editing this book Nanotechnology/About.

title: Nanotechnology/Optical Methods


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Nanotechnology/Electron microscopy

=Electron microscopy= An overview:

Electron microscopes uses electrons instead of photons, because electrons have a much shorter wavelength than photons and so allows you to observe matter with atomic resolution.

There are two general types of electron microscopes: the Scanning Electron Microscope (SEM) that scans an electron beam over the surface of an object and measures how many electrons are scattered back, and the Transmission Electron Microscope (TEM) that shoots electrons through the sample and measures how the electron beam changes because it is scattered in the sample.

Using electron beams however requires working in a vacuum environment, and this makes the instruments considerably larger and expensive. All electron microscope work under at least low pressures and usually in high vacuum chambers to avoid scattering the electrons in the gas. In environmental electron microscopes, differential pumping systems are used to actually have gasses present by the sample together with the electron beam.

Introduction to Electron Microscopy
For imaging of nanoscale objects, optical microscopy has limited resolution since the objects are often much smaller than the wavelength of light. The achievable resolution for a wavelength $$\lambda $$ is often given by the diffraction limit $$d$$ as

 * $$d=0.61 \frac {\lambda}{(NA)}$$ (r.g., diffraction limit)

with numerical aperture $$NA$$, which can be approximated by the largest angle of incidence $$\alpha$$ of the wavefront towards the sample, $$NA \approx \alpha$$.

Since $$\alpha \ll 1$$ for the present purposes, we can approximate $$\sin(\alpha)\approx \tan(\alpha)\approx \alpha$$ and hence $$NA \approx \alpha \approx r_{a}/r_{wd}$$ where $$r_{a}$$ is the radius of the objective lens aperture and $$r_{wd}$$ the working distance.

Optical microscopes can often reach a resolution of about $$d=200$$ nm. For nanoscale resolution this is unfortunately not sufficient to distinguish for instance a single nanotube from two adhering to each other, since they have diameters of less than 100 nm.

The figure below gives an overview typical magnifications achievable by the different electron microscopes compared to a light microscope.

Electron optical systems use electrical and magnetic fields to control the electron beam. Although the law of refraction in optics is exchanged with the Lorentz force in electrodynamics, the electron optical system has similar diffraction limits as optical systems, since they depend on the wave nature of the electron beam.

One can achieve a considerable improvement in resolution with instruments such as the transmission electron microscope and the scanning electron microscope that use electrons with De Broglie wavelength much smaller than that of visible light. The De Broglie wavelength λ of an electron with momentum p is

 * $$\lambda = \frac{h}{p} = \frac {h}{\sqrt{2m_{e}E_{b}}}, $$ (Eq. De Broglie wavelength)

where $$h$$ is Plancks constant. The electron has rest mass $$m_{e}$$ and energy $$E_{e}=m_{e}c^2=511 keV$$.

If an electron with charge $$q_{e}$$ is accelerated from rest by an electrical potential $$U$$, to the electron beam energy $$E_{b}=q_{e}U$$, it will have a wavelength of 1 nm at 1 eV decreasing to 1 pm at 100 keV where it will be travelling with 50% the speed of light.

This chapter will briefly review fundamental issues for electron microscopy that are similar for SEM and TEM: the limitations imposed by the electron optical beam system in the microscope column; the interaction of the electron beam with the sample; the standard image formation method in SEM and TEM. These issues are essential to understand the results and limitations reached in SEM and TEM microscopy.

For further details, please refer to reviews of electron microscopes and their applications, such as Goldstein et al. that contains a thorough review of SEM, while Goodhew and Humphreys is a more general introduction to both SEM and TEM.

=The Electron Optical System=

For high resolution imaging, a well focused beam is required, just as in optical microscopy. Due to the short wavelength of electron beams with keV energies, as given by the, the properties of the electron optical system and the electron emitter mainly defines the limits on the achievable beam diameter. The current density in the electron beam can be approximated by a Gaussian distribution of current density j [A/m²] as function of radius, r, from the beam center

 * $$j(r)=j_0 e^{- \frac{r^2}{r_{0}^2}}, $$

with radius determined by $$r_0$$, giving a the full width half maximum $$FWHM=2v(ln2)r0$$. Integrating $$I_{b}=\int \int j(r)rdrd\theta$$ gives the total beam current

 * $$I_{b}=j_0 \pi r_{0}^2.$$

The electron optics impose a limit on the achievable beam current density and radius by the brightness of the electron emitter $$\beta_{e}$$, which is conserved throughout the system .

Brightness, ß, is a measure of the current per area normal to the beam direction and per element of solid angle. At the center of the Gaussian beam,

 * $$\beta=\frac{j_0}{\pi \alpha^2} $$

and the brightness is related to the current density in eq SEM Gaussian beam profile. The emitter brightness $$\beta_{e}$$ is determined by the type of electron emitter and the beam energy $$E_{b}$$

$$\beta_{e}=\frac{j_{e}E_{b}}{\pi \Delta E}$$

with emission current density for W-filament sources about $$j_{e}$$~3 A/cm², for LaB6 sources about 100 A/cm², while field emission guns (FEG) can reach 105A/cm². The energy spread of the electrons from the sources are about ΔE~1 eV and slightly lower for FEGs. Due to conservation of the brightness in the system, the beam diameter depends on current as

$$r_{0}=\frac{1}{\pi}\frac{r_{wd}}{r_{a}}\sqrt{\frac{I_{b}}{\beta_{e}}}. $$

The ideal beam probe size determined by the conservation of brightness cannot be obtained in a real system. Effects such as aberration will make the minimum achievable beam diameter larger. Equation however seem to adequately describe the beam diameter for the present discussion. Apart from the additional beam widening contributions, the image detection method imposes limits on useful values for the parameters in Eq. SEM beam diameter which differ for SEM and TEM.

=Electron Range=

The electron optical system sets limitations to the achievable primary beam current and radius. The expected image resolution set by the primary beam cannot be reached if the signal detected for imaging is caused by electrons scattered far in the sample. The trajectory of an electron penetrating a bulk solid is a complex trajectory due to multiple elastic and inelastic collision events. As the primary electron (PE) penetrates into the sample it will gradually change direction and loose energy in collisions. The mean free path due to elastic and inelastic collisions, $$\lambda_{mfp}$$, depends on the atomic number of the material and the PE energy. At 100 keV $$\lambda_{mfp}=150 nm$$ for carbon and 5 nm for gold . For samples thinner than $$\lambda_{mfp}$$ the main part of the PE will pass relatively unaffected through the sample, which is the basis for TEM.

SEM can be used for thicker specimens. The electrons that escape from the sample in a new direction compared to the PE due to elastic collisions are called backscattered electrons (BSE).

For samples thicker than $$\lambda_{mfp}$$, the volume interacting with the scattered PE defines the range of the electrons in the material, and this is considerably larger than the minimum achievable primary beam diameters.

The electron range is about 1 µm at 10 keV for carbon, decreasing with higher atomic number for the material. Both the high energy PE and BSE generate secondary electrons (SE) by inelastic scattering events. The SE are generally defined as having energy below 50 eV while the BSE have energies up to the PE energy. The range of SE is typically 1 nm for metals and about 10 nm for insulators.

The short range of the SE make the yield of SE highly dependent on the energy lost by the PE within the SE range from the surface, and this makes high Z substances efficient generators of SE. The main emission of SE takes place in the region where the PE strikes the surface and within the SE escape depth from this region.

=Scanning electron microscopy (SEM)=
 * Wikipedia: Scanning electron microscopy (SEM)

In a scanning electron microscope a beam is scanned over the sample surface in a raster pattern while a signal is recorded from electron detectors for SE or BSE. The PE energy is kept relatively low (1-30 keV) to limit the interaction volume in the specimen that will contribute to the detected signal. Especially low energy PE will provide high sensitivity to surface composition as they cannot penetrate far into the sample.

The figure above showed the effect of PE penetration depth of a carbonaceous nanostructure with a gold core, where only the surface is visible at low PE energies, while the carbon becomes increasingly transparent and the core visible at high PE energies.

The low energy SE can easily be attracted and collected by a positively charged detector and are hence an efficient source for an image signal. The standard SE detector is an Everhart-Thornley (ET) detector where a positively charged grid attracts the SE and accelerates them to sufficiently high energies to create a light pulse when striking a scintillator. The light pulse is then amplified by a photomultiplier. Despite the complex construction, the ET detector is remarkably efficient, but requires large $$r_{wd}$$ for effective collection of the SE by the charged grid.

Another SEM detector is the in-lens detector, where SE passing through the column aperture are accelerated towards a solid state detector. The in-lens detector complements the ET by being more efficient at short $$r_{wd}$$.

Environmental SEM (ESEM)

The ESEM makes it possible to use various gasses in the sample chamber of the microscope since there are narrow apertures between the sample chamber and the gun column, and a region in between that is connected to a differential pumping system. Pressures up to about 10 Torr are normally possible in the sample chamber.

The standard Everly-Thornhart SE detector would not work under such conditions since it would create a discharge in the low pressure gas. Instead a "gaseous secondary electron detector (GSD)" is used, as shown in the figure below. The GSD measures the current of a weak cascade discharge in the gas, which is seeded by the emission of electrons from the sample.

In the ESEM one can work with for instance water vapour or argon as the environmental gas, and it is possible to have liquid samples in the chamber if the sample stage is cooled sufficiently to condense water.

=Transmission electron microscopy (TEM)=
 * Transmission electron microscopy (TEM)
 * High Resolution Transmission electron microscopy (HRTEM)

When the specimen thickness is about the mean free path, $$\lambda_{mfp}$$, TEM can be used to achieve high resolution images such as the image above where the atomic lattice of a gold nanocrystal is visible. Since the detected electrons are transmitted PE where the energy can be in the 100 keV range, the resolution is not limited by the issues regarding secondary electrons. The electron beam optics can be optimized for higher current densities (Eq. ) at higher energies compared to SEM.

To achieve optimal imaging conditions for the thin TEM samples, the working distance has been made short. In most TEMs, the space for the sample holder is only about (5 mm)³ between the two objective lenses for the incoming and transmitted beam. Before reaching a CCD camera, the transmitted beam is sent through several magnification lenses to achieve the high magnification (500.000X is not unusual).

The image formation in TEM can be based on several principles, but practically all images used in this work were made by phase contrast imaging, here called High Resolution TEM or HRTEM. At sufficiently high brightness, electron sources can produce coherent electron beams due to the point-like emitter surface area and small energy spread. The coherent electron beam can be considered as a spherical wave propagating from the emitter and out through the electron optical system, much like a laser beam would propagate through an optical system.

The HRTEM images are often based on the interference of the electron wavefront after it has passed through the sample and reaches a CCD detector to give a phase contrast image of the sample. The image will have a resolution determined of course by the wavelength of the electrons (Eq. ) but mainly by the imperfections of the electron optics which also perturbs the wavefront. The optimal imaging condition is for a sample thickness about $$\lambda_{mfp}$$, where the wavefront is only slightly perturbed by passing through the sample. TEM instruments are normally easily capable of resolving individual shells of a carbon nanotubes. The fine-tuning of the electron optical system to the required resolution can be achieved in about 30 min for many microscopes.

=Electron Holography=

In special TEM microscopes, the diffracted beam can be combined with a part of the original electron beam from the electron gun, and the image that is recorded is an interference pattern that depends on how much the phase of the diffracted beam was changed. By recording such images, one can measure how the electron wave function changes as it passes through or nearby a nanostructure - and this allows you to measure the electric and magnetic fields surrounding nanostructures.

=Electron Tomography=

By recording numerous TEM images of an object at many different angles, these images can in a computer be combined to create a three-dimensional model of the object. The technique is time consuming but allows you to see nanostructures in 3D.

title: Nanotechnology/Electron microscopy


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Nanotechnology/Scanning probe microscopy

= Scanning probe microscopy = Scanning probe microscopy covers the methods where a sharp tip is scanned over a surface in a raster pattern and the interaction with the surface is recorded in each pixel to form an image of the interaction. There are a multitude of methods and interactions in SPM. Broadly speaking, there are three main categories:

[[Image:SPM_overview.jpg|thumb|400px| Overview of the main types of Scanning Probe Microscope types:

Scanning Tunneling Microscope (STM) - using the tunneling current I between the outermost atom of a conducting probe within an atomic distance from a substrate to map out the sample topography and electrical properties.

Atomic Force Microscope (AFM) - using the van der Waals forces or contact forces between a tip and the sample to measure the sample topography or mechanical properties.

Scanning Near-field Optical Microscope (SNOM) - using the scattered light through a sub-wavelength aperture to form an image. ]]

 * In scanning tunneling microscopy (STM), one uses an atomically sharp metallic tip and records the minute tunneling current between the tip and the surface, when the tip is hovering so close to the surface that electrons can move between the surface and the tip.
 * In Atomic force microscopy (AFM), a cantilever with a sharp tip - somewhat like the needle of an old record player - is scanned over the surface and the topography or surface softness can be recorded.
 * In Scanning near-field optical microscopy (SNOM) a probe with a smalle aperture is scanned over the surface collecting the light comming from regions much smaller than the wavelength of the light used.

Atomic force microscope (AFM)

 * Upcoming page in this wiki book on AFM

 * Atomic force microscope (AFM)
 * Kelvin force microscopy
 * Magnetic force microscope
 * Scanning voltage microscopy
 * Magnetic resonance force microscope
 * Lateral force microscopy

Scanning tunneling microscopy (STM)

 * Upcoming page in this wiki book on STM
 * Scanning tunneling microscopy (STM)

Scanning Near-field optical microscopy (SNOM)

 * Upcoming page in this wiki book on SNOM

Wiki links:
 * Scanning Near-field optical microscopy (SNOM)
 * Evanescent wave coupling

= Resources =
 * A Practical Guide to Scanning Probe Microscopy by Rebecca Howland and Lisa Benatar
 * homebrew atomic-resolution microscope

= References =

See also notes on editing this book about how to add references Nanotechnology/About.


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Nanotechnology/Additional methods

=Point-Projection Microscopes= Point-Projection Microscopes are a type of field emission microscope, and consists of three components: an electron source, the object to the imaged, and the viewing screen.
 * Field emission microscope
 * Field ion microscope
 * Atom Probe

=Low energy electron diffraction (LEED)= LEED is a technique for imaging surfaces, and has two principle methods of use: qualitative and quantitative. The qualitative method measures relative size and geometric properties, whereas the quantitative method looks at diffracted beams as a way of determining the position of atoms.

 * Low energy electron diffraction (LEED)
 * LEED intro

=Reflection High Energy Electron diffraction= RHEED is similar to LEED but uses higher energies and the electrons are directed to the be reflected on the surface at almost grazing incidence. This way the high energy electrons only penetrates a few atomic layers of the surface.

=X-ray Spectroscopy and Diffraction= X-ray Spectroscopy refers to a collection of techniques including, but not limited to X-ray Absorption Spectroscopy and  X-ray Photoelectron Spectroscopy.

X-rays can be used for X-ray crystallography.

=Auger electron spectroscopy (AES)= Auger Electron Spectroscopy is a technique that takes advantage of the Auger Process to analyze the surface layers of a sample.

=Nuclear Magnetic Resonance (NMR)=
 * Nuclear Magnetic Resonance (NMR) - in a magnetic field the spin of the nuclei of molecules will precess and in strong fields (several tesla) this happens with rf frequencies that can be detected by receiving rf antennas and amplifiers. The precession frequency of an individual nucleus will deviate slightly depending on the its surrounding molecules' electronic structure and hence detecting a spectrum of the radiofrequency precession frequencies in a sample will provide a finger print of the types of molecules in that sample.
 * Nuclear quadrupole resonance is a related technique, based on the internal electrical fields of the molecules to cause a splitting of the nuclear magnetic moments energy levels. The level splitting is detected by rf as in NMR. Its is used mainly for experimental explosives detection.

=Electron Paramagnetic Resonance (EPR) or Electron Spin Resonance (ESR)= Electron Spin Resonance (ESR) measures the microwave frequency of  paramagnetic ions or molecules .

=Mössbauer spectroscopy= Mössbauer spectroscopy detects the hyperfine interactions between the nucleus of an atom, and the ambient environment. The atom must be part of a solid matrix to reduce the recoil affect of a gamma ray emission or absorption.

=Non-contact Nanoscale Temperature Measurements= Heat radiation has infrared wavelengths much longer than 1 µm and hence taking a photo of a nanostructure with e.g. a thermal camera will not provide much information about the temperature distribution within the nanostructure (or microstructure for that sake).

Temperatures can be measured locally by different non-contact methods:
 * Spectroscopy on individual quantum dots.
 * Spectra of laser dyes incorporated in the structure
 * Raman microscopy (the temperature influences the ratio of stokes and anti-stokes lines amplitude, the width of the lines and the position of the lines.)
 * Transmission electron microscopy can also give temperature information by various techniques
 * Special AFM probes with a temperature dependent resistor at the tip can be used for mapping surface temperatures
 * Infrared Near-field Microscopy
 * Confocal raman microscopy can provide 3D thermal maps

=References=

See also notes on editing this book about how to add references Nanotechnology/About.

title: Nanotechnology/Additional methods


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Nanotechnology/Modelling Nanosystems

=Modelling Nanosystems=

=The Schrödinger equation= the Schrödinger equation

 * $$H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{\partial}{\partial t} \left| \psi \left(t\right) \right\rangle$$

where $$\mathrm{i}$$ is the imaginary unit, $$t$$ is time, $$ \partial / \partial t$$ is the partial derivative with respect to $$t$$, $$\hbar$$ is the reduced Planck's constant (Planck's constant divided by $$2\pi$$), $$\psi (t)$$ is the wave function, and $$H\!\left(t\right)$$ is the Hamiltonian operator.

=Hartree-Fock (HF) or self-consistent field (SCF)= In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

The Hartree–Fock method often assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of N spin-orbitals. By invoking the variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system.

Especially in the older literature, the Hartree–Fock method is also called the self-consistent field method (SCF). In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartree–Fock equations also behave as if each particle is subjected to the mean field created by all other particles (see the Fock operator below) and hence the terminology continued. The equations are almost universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method.

The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state.

For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately.

The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case. The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods:

 * Restricted open-shell Hartree–Fock (ROHF)
 * Unrestricted Hartree–Fock (UHF)

history
The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. In 1927 D. R. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B. Lindsay, and himself) set in the old quantum theory of Bohr.

In the Bohr model of the atom, the energy of a state with principal quantum number n is given in atomic units as $$ E = -1 / n^2$$. It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the quantum defect d as an empirical parameter, the energy levels of a generic atom were well approximated by the formula $$ E = -1/(n+d)^2$$, in the sense that one could reproduce fairly well the observed transitions levels observed in the X-ray region (for example, see the empirical discussion and derivation in Moseley's law). The existence of a non-zero quantum defect was attributed to electron-electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial screening of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data.

Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., ab initio. His first proposed method of solution became known as the Hartree method. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions.

In 1930 Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function. The Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of quantum statistics.

It was then shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. Fock's original method relied heavily on group theory and was too abstract for contemporary physicists to understand and implement. In 1935 Hartree reformulated the method more suitably for the purposes of calculation.

The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the central field approximation, to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, solution by hand of the Hartree–Fock equations for a medium sized atom were laborious; small molecules required computational resources far beyond what was available before 1950.

Hartree–Fock algorithm
The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the Born–Oppenheimer approximation. Since there are no known solutions for many-electron systems (hydrogenic atoms and the diatomic hydrogen cation being notable one-electron exceptions), the problem is solved numerically. Due to the nonlinearities introduced by the Hartree–Fock approximation, the equations are solved using a nonlinear method such as iteration, which gives rise to the name "self-consistent field method."

Approximations
The Hartree–Fock method makes five major simplifications in order to deal with this task:

 * The Born–Oppenheimer approximation is inherently assumed. The full molecular wave function is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
 * Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
 * The variational solution is assumed to be a linear combination of a finite number of basis functions, which are usually (but not always) chosen to be orthogonal. The finite basis set is assumed to be approximately complete.
 * Each energy eigenfunction is assumed to be describable by a single Slater determinant, an antisymmetrized product of one-electron wave functions (i.e., orbitals).
 * The mean field approximation is implied. Effects arising from deviations from this assumption, known as electron correlation, are completely neglected for the electrons of opposite spin, but are taken into account for electrons of parallel spin. (Electron correlation should not be confused with electron exchange, which is fully accounted for in the Hartree–Fock method.)

Relaxation of the last two approximations give rise to many so-called post-Hartree–Fock methods.

Variational optimization of orbitals
The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that is greater than or equal to the true ground state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at the Hartree–Fock limit; i.e., the limit of the Hartree–Fock energy as the basis set approaches completeness. (The other is the full-CI limit, where the last two approximations of the Hartree–Fock theory as described above are completely undone. It is only when both limits are attained that the exact solution, up to the Born–Oppenheimer approximation, is obtained.) The Hartree–Fock energy is the minimal energy for a single Slater determinant.

The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as spin-orbitals. For an atomic orbital calculation, these are typically the orbitals for a hydrogenic atom (an atom with only one electron, but the appropriate nuclear charge). For a molecular orbital or crystalline calculation, the initial approximate one-electron wave functions are typically a linear combination of atomic orbitals (LCAO).

The orbitals above only account for the presence of other electrons in an average manner. In the Hartree–Fock method, the effect of other electrons are accounted for in a mean-field theory context. The orbitals are optimized by requiring them to minimize the energy of the respective Slater determinant. The resultant variational conditions on the orbitals lead to a new one-electron operator, the Fock operator. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a unitary transformation between themselves. The Fock operator is an effective one-electron Hamiltonian operator being the sum of two terms. The first is a sum of kinetic energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear-electronic Coulombic attraction terms. The second are Coulombic repulsion terms between electrons in a mean-field theory description; a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge. This is the major simplification inherent in the Hartree–Fock method, and is equivalent to the fifth simplification in the above list.

Since the Fock operator depends on the orbitals used to construct the corresponding Fock matrix, the eigenfunctions of the Fock operator are in turn new orbitals which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals are calculated. The Hartree–Fock electronic wave function is then the Slater determinant constructed out of these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed.

The Fock operator
Because the electron-electron repulsion term of the electronic molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under Hartree–Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below, for closed-shell atoms or molecules (with two electrons in each spatial orbital). The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.

 * $$\hat F[\{\phi_j\}](1) = \hat H^{\text{core}}(1)+\sum_{j=1}^{N/2}[2\hat J_j(1)-\hat K_j(1)]$$

where

 * $$\hat F[\{\phi_j\}](1)$$

is the one-electron Fock operator generated by the orbitals $$\phi_j$$, and

 * $$\hat H^{\text{core}}(1)=-\frac{1}{2}\nabla^2_1 - \sum_{\alpha} \frac{Z_\alpha}{r_{1\alpha}}$$

is the one-electron core Hamiltonian. Also

 * $$\hat J_j(1)$$

is the Coulomb operator, defining the electron-electron repulsion energy due to each of the two electrons in the jth orbital. Finally

 * $$\hat K_j(1)$$

is the exchange operator, defining the electron exchange energy due to the antisymmetry of the total n-electron wave function. This (so called) "exchange energy" operator, K, is simply an artifact of the Slater determinant. Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation:

 * $$\hat F(1)\phi_i(1)=\epsilon_i \phi_i(1)$$

where $$\phi_i\;(1)$$ are a set of one-electron wave functions, called the Hartree–Fock molecular orbitals.

Linear combination of atomic orbitals
Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a linear combination of atomic orbitals. These atomic orbitals are called Slater-type orbitals. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more Gaussian-type orbitals, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.

Various basis sets are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as the Gram–Schmidt process is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving the Roothaan–Hall equations by converting the overlap matrix effectively to an identity matrix. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the generalized eigenvalue problem, of which the Roothaan–Hall equations are an example.

Numerical stability
Numerical stability can be a problem with this procedure and there are various ways of combating this instability. One of the most basic and generally applicable is called F-mixing or damping. With F-mixing, once a single electron wave function is calculated it is not used directly. Instead, some combination of that calculated wave function and the previous wave functions for that electron is used—the most common being a simple linear combination of the calculated and immediately preceding wave function. A clever dodge, employed by Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge. In molecular calculations a similar approach is sometimes used by first calculating the wave function for a positive ion and then to use these orbitals as the starting point for the neutral molecule. Modern molecular Hartree–Fock computer programs use a variety of methods to ensure convergence of the Roothaan–Hall equations.

Weaknesses, extensions, and alternatives
Of the five simplifications outlined in the section "Hartree–Fock algorithm", the fifth is typically the most important. Neglecting electron correlation can lead to large deviations from experimental results. A number of approaches to this weakness, collectively called post-Hartree–Fock methods, have been devised to include electron correlation to the multi-electron wave function. One of these approaches, Møller–Plesset perturbation theory, treats correlation as a perturbation of the Fock operator. Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such as multi-configurational self-consistent field, configuration interaction, quadratic configuration interaction, and complete active space SCF (CASSCF). Still others (such as variational quantum Monte Carlo) modify the Hartree–Fock wave function by multiplying it by a correlation function ("Jastrow" factor), a term which is explicitly a function of multiple electrons that cannot be decomposed into independent single-particle functions.

An alternative to Hartree–Fock calculations used in some cases is density functional theory, which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of the two methods—the popular B3LYP scheme is one such hybrid functional method. Another option is to use modern valence bond methods.

Software packages
For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the list of quantum chemistry and solid state physics software.

Two-particle case
The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen orthogonal wave functions of the individual particles. For the two-particle case with spatial coordinates $$\mathbf{x}_1$$ and $$\mathbf{x}_2$$, we have

\Psi(\mathbf{x}_1,\mathbf{x}_2) = \chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2). $$

This expression is used in the Hartree–Fock method as an ansatz for the many-particle wave function and is known as a Hartree product. However, it is not satisfactory for fermions because the wave function above is not antisymmetric, as it must be for fermions from the Pauli exclusion principle. An antisymmetric wave function can be mathematically described as follows:

\Psi(\mathbf{x}_1,\mathbf{x}_2) = -\Psi(\mathbf{x}_2,\mathbf{x}_1) $$

which does not hold for the Hartree product. Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products

\Psi(\mathbf{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}}\{\chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) - \chi_1(\mathbf{x}_2)\chi_2(\mathbf{x}_1)\} $$

= \frac{1}{\sqrt2}\begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) \end{vmatrix} $$

where the coefficient is the normalization factor. This wave function is now antisymmetric and no longer distinguishes between fermions, that is: one cannot indicate an ordinal number to a specific particle and the indices given are interchangeable. Moreover, it also goes to zero if any two wave functions of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.

Generalizations
The expression can be generalised to any number of fermions by writing it as a determinant. For an N-electron system, the Slater determinant is defined as

\Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \left| \begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{matrix} \right|\equiv \left| \begin{matrix} \chi _1 & \chi _2 & \cdots & \chi _N  \\ \end{matrix} \right|, $$

where in the final expression, a compact notation is introduced: the normalization constant and labels for the fermion coordinates are understood – only the wavefunctions are exhibited. The linear combination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for N = 2. It can be seen that the use of Slater determinants ensures an antisymmetrized function at the outset; symmetric functions are automatically rejected. In the same way, the use of Slater determinants ensures conformity to the Pauli principle. Indeed, the Slater determinant vanishes if the set {χi } is linearly dependent. In particular, this is the case when two (or more) spin orbitals are the same. In chemistry one expresses this fact by stating that no two electrons can occupy the same spin orbital. In general the Slater determinant is evaluated by the Laplace expansion. Mathematically, a Slater determinant is an antisymmetric tensor, also known as a wedge product.

A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree–Fock theory. In more accurate theories (such as configuration interaction and MCSCF), a linear combination of Slater determinants is needed.

The word "detor" was proposed by S. F. Boys to describe the Slater determinant of the general type, but this term is rarely used.

Unlike fermions that are subject to the Pauli exclusion principle, two or more bosons can occupy the same quantum state of a system. Wavefunctions describing systems of identical bosons are symmetric under the exchange of particles and can be expanded in terms of permanents.

Fock matrix
In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors.

It is most often formed in computational chemistry when attempting to solve the Roothaan equations for an atomic or molecular system. The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Importantly, because the Fock operator is a one-electron operator, it does not include the electron correlation energy.

The Fock matrix is defined by the Fock operator. For the restricted case which assumes closed-shell orbitals and single-determinantal wavefunctions, the Fock operator for the i-th electron is given by:

 * $$\hat F(i) = \hat h(i)+\sum_{ j=1 }^{n/2}[2 \hat J_j(i)-\hat

K_j(i)]$$

where:

 * $$\hat F(i)$$ is the Fock operator for the i-th electron in the system,

 * $${\hat h}(i)$$ is the one-electron hamiltonian for the i-th electron,

 * $$n$$ is the number of electrons and $$ \frac{n}{2} $$ is the number of occupied orbitals in the closed-shell system,

 * $$\hat J_j(i)$$ is the Coulomb operator, defining the repulsive force between the j-th and i-th electrons in the system,

 * $$\hat K_j(i)$$ is the exchange operator, defining the quantum effect produced by exchanging two electrons.

The Coulomb operator is multiplied by two since there are two electrons in each occupied orbital. The exchange operator is not multiplied by two since it has a non-zero result only for electrons which have the same spin as the i-th electron.

For systems with unpaired electrons there are many choices of Fock matrices.

Hartree-Fock (HF) or self-consistent field (SCF)

=Density Functional Theory=

Connection to quantum state symmetry
The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as a sum of states in which one particle is in state $$\scriptstyle |x \rangle$$ and the other in state $$\scriptstyle |y\rangle$$:

$$
 * \psi\rangle = \sum_{x,y} A(x,y) |x,y\rangle

and antisymmetry under exchange means that A(x,y) = −A(y,x). This implies that A(x,x) = 0, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantity A(x,y) is not a matrix but an antisymmetric rank-two tensor.

Conversely, if the diagonal quantities A(x,x) are zero in every basis, then the wavefunction component:

A(x,y)=\langle \psi|x,y\rangle = \langle \psi | ( |x\rangle \otimes |y\rangle ) $$

is necessarily antisymmetric. To prove it, consider the matrix element:

\langle\psi| ((|x\rangle + |y\rangle)\otimes(|x\rangle + |y\rangle)) \,$$

This is zero, because the two particles have zero probability to both be in the superposition state $$\scriptstyle |x\rangle + |y\rangle$$. But this is equal to

\langle \psi |x,x\rangle + \langle \psi |x,y\rangle + \langle \psi |y,x\rangle + \langle \psi | y,y \rangle \,$$

The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:

\langle \psi|x,y\rangle + \langle\psi |y,x\rangle = 0 \,$$.

or

A(x,y)=-A(y,x) \,$$

Pauli principle in advanced quantum theory
According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. Since, nonrelativistically, particles can have any statistics and any spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.

In one dimension, bosons, as well as fermions, can obey the exclusion principle. A one-dimensional Bose gas with delta function repulsive interactions of infinite strength is equivalent to a gas of free fermions. The reason for this is that, in one dimension, exchange of particles requires that they pass through each other; for infinitely strong repulsion this cannot happen. This model is described by a quantum nonlinear Schrödinger equation. In momentum space the exclusion principle is valid also for finite repulsion in a Bose gas with delta function interactions, as well as for interacting spins and Hubbard model in one dimension, and for other models solvable by Bethe ansatz. The ground state in models solvable by Bethe ansatz is a Fermi sphere.

Density Functional Theory

=References=

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w:Category:Nanotechnology

title: Nanotechnology/Modelling Nanosystems


text:
Nanotechnology/Physical Chemistry of Surfaces

=Physical Chemistry of Surfaces=
 * Introduction to Surface Chemistry

=Surface Forces=

=Hydrophobic and hydrophilic surfaces=
 * Wetablility
 * wetting
 * Hydrophobe
 * Hydrophile
 * Super hydrophilicity

=Surface Energy=
 * Surface energy

=Surface Diffusion=
 * Diffusion

=References=

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text:
Nanotechnology/Background material

=Dispersion relations= Dispersion relations are essential in describing a physical system and can often be a bit tricky to comprehend.

=References=

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title: Nanotechnology/Background material


text:
Nanotechnology/Nanomechanics

=Nanomechanics= Some of the mechanical aspects of nanotechnology is
 * Extreme Youngs modulus materials
 * High frequency resonances in nanoscale oscillators

NEMS
Nano-electro-mechanical systems

=Mechanics of beams and cantilevers= Cantilevers are essential in many mechanical systems: Nanotubes, nanowires and atomic force microscopes...

=The harmonic oscillator= Fundamental in the description of any oscillating systems is the harmonic oscillator and the quantum mechanical version,  the Quantum harmonic oscillator.

=References=

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title: Nanotechnology/Nanomechanics


text:
Nanotechnology/Overview of Production methods

 * Chemical vapor deposition
 * Metal Organic Vapor Phase Epitaxy (MOVPE)
 * Arc discharge where an electical arc is made between rods of the precursor materials.
 * Physical vapor deposition
 * Molecular beam epitaxy using solid sources and Chemical beam epitaxy using gaseous sources.
 * Epitaxy in general.
 * Atomic layer epitaxy where only a few monolayers are deposited and bond to chemisorption sites on the surface.
 * Self-assembled monolayers
 * metal organic chemical vapor deposition abbreviation as MOCVD
 * atomic layer deposition, for depositing uniform thin films one atomic layer at a time (By [http://www.fmnt.fi/index.pl?id=2913&isa=Category&op=show Dr. Tuomo Suntola and co-workers in Finland.)

=Types of nanometal synthesis= The most common types of nanometal synthesis deal with 'wet' methods in which metal nanoparticles are produced in a colloid with an organic material of some sort.

Gold nanoparticles can be produced by either:

1) Reduction of HAuCl4 in a solution of sodium citrate, then boiling it, causing gold nanoparticles to form in a wine-red solution. 2) Mixing HAuCl4 in water, which produces a solution that is subsequently transferred into toluene using tetraoctylammonium bromide (TOAB), a phase transfer catalyst. Phase transfer catalysts help reactants dissolve in organic (carbon-containing) material where the reactant otherwise couldn't w/o the PTC. Afterwards, the solution is stirred with sodium borohydride, in the presence of certain alkanes, which bind to the gold in the solution, allowing for the formation of gold nanoparticles.

Synthesis of other metal nanoparticles can possibly be achieved by reducing metal salts in organic solvents such as ethanol, or by variations of the above methods which synthesize gold nanoparticles.

=References=

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Nanotechnology/Semiconducting Nanostructures

=Nanotubes= Certain compounds are capable of forming nanotubes where the tube consists of round shell of a single layer of atoms in a cylindrical lattice. Carbon nanotubes is the most famous example, but also other materials can form nanotubes such as Boron-nitride, Molybdenum sulfide and others.

Nanotubes can also be made by etching the core out of an shell structured rod, but such tubes will normally contain many atomic layers in the wall and have crystal facets on the sides.

=Carbon Nanotubes= Carbon nanotubes are fascinating nanostructures. A sheet of graphene as in common graphite, but rolled up in small tubes rather than planar sheets.

Carbon nanotubes have unique mechanical properties such as high strength, high stiffness, and low density and also interesting electronic properties. A single-walled carbon nanotube can be either metallic or semiconducting depending on the atomic arrangement .

This section is a short introduction to carbon nanotubes. For a broader overview the reader is referred to one of the numerous review articles or books on carbon nanotubes

Geometric Structure
The simplest type of carbon nanotube consists of just one layer of graphene rolled up in the form of a seamless cylinder, known as a single-walled carbon nanotube (SWCNT) with a typical diameter of just a few nanometers. Larger diameter nanotube structures are nanotube ropes, consisting of many individual, parallel nanotubes closed-packed into a hexagonal lattice, and multi-walled carbon nanotubes (MWCNTs) consisting of several concentric cylinders nested within each other.

Single walled Carbon Nanotube
The basic configuration is thus the SWCNT. Its structure is most easily illustrated as a cylindrical tube conceptually formed by the wrapping of a single graphene sheet. The hexagonal structure of the 2-dimensional graphene sheet is due to the $$sp^{2}$$ hybridization of the carbon atoms, which causes three directional, in-plane $$\sigma$$ bonds separated by an angle of 120 degrees.

The nanotube can be described by a chiral vector $$C$$ that can be expressed in terms of the graphene unit vectors $$\textbf{a}_{1}$$ and $$\textbf{a}_{2}$$ as $$\textbf{C}=n\textbf{a}_{1}+m\textbf{a}_{2}$$ with the set of integers $$(n,m)$$ uniquely identifying the nanotube. This chiral vector or 'roll-up' vector describes the nanotube circumference by connecting two crystallographically equivalent positions i.e. the tube is formed by superimposing the two ends of $$\textbf{C}$$.

Based on the chiral angle SWCNTs are defined as zig-zag tubes ($$\theta =0 deg \leftrightarrow m=0$$), armchair tubes ($$\theta =30 deg \leftrightarrow n=m$$), or chiral tubes ($$0 deg < \theta < 30 deg $$).

Multiwalled Carbon Nanotubes
MWCNTs are composed of a number of SWCNTs in a coaxial geometry. Each nested shell has a diameter of $$d=\sqrt{3}a_{C-C}(m^{2}+n^{2}+mn)^{1/2}/\pi$$ where $$a_{C-C}$$ is the length of the carbon-carbon bond which is 1.42 Å. The difference in diameters of the individual shell means that their chiralities are different, and adjacent shell are therefore in general non-commensurate, which causes only a weak intershell interaction.

The intershell spacing in MWCNTs is $$\sim$$ 0.34 nm - quite close to the interlayer spacing in turbostratic graphite

Electronic Structure
The electronic structure of a SWCNT is most easily described by again considering a single graphene sheet. The 2-D, hexagonal-lattice graphene sheet has a 2-D reciprocal space with a hexagonal Brillouin zone (BZ).

The $$\sigma$$ bonds are mainly responsible for the mechanical properties, while the electronic properties are mainly determined by the $$\pi$$ bands. By a tight-binding approach the band structure of these $$\pi$$ bands can be calculated

Graphene is a zero-gap semiconductor with an occupied $$\pi$$ band and an unoccupied $$\pi^{*}$$ band meeting at the Fermi level at six $$K$$ points in the BZ, thus it behaves metallic, a so-called semimetal.

Upon forming the tube by conceptually wrapping the graphene sheet, a periodic boundary condition is imposed that causes only certain electronic states of those of the planar graphene sheet to be allowed. These states are determined by the tube's geometric structure, i.e. by the indices $$(n,m)$$ of the chiral vector. The wave vectors of the allowed states fall on certain lines in the graphene BZ.

Based on this scheme it is possible to estimate whether a particular tube will be metallic or semiconducting. When the allowed states include the $$K$$ point, the system will to a first approximation behave metallic. However, in the points where the $$\pi$$ and the $$\pi^{*}$$ bands meet but are shifted slightly away from the $$K$$ point due to curvature effects, which causes a slight band opening in some cases

This leads to a classification scheme that has three types of nanotubes:

 * Metallic: These are the armchair tubes where the small shift of the degenerate point away from the $$K$$ point does not cause a band opening for symmetry reasons.

 * Small-bandgap semiconducting: These are characterized by $$n-m = 3j$$ with $$j$$ being an integer. Here, the wave vectors of the allowed states cross the $$K$$ point, but due to the slight shift of the degenerate point a small gap will be present, the size of which is inversely proportional to the tube diameter squared with typical values between a few and a few tens meV

 * Semiconducting: In this case $$n-m \neq 3j$$. This causes a larger bandgap, the size of which is inversely proportional to the tube diameter: $$E_{g}=k/d$$ with experimental investigations suggesting a value of $$k$$ of 0.7-0.8 eV/nm

Typically the bandgap of the type 2 nanotubes is so small that they can be considered metallic at room temperature. Based on this it can be inferred that 1/3 of all tubes should behave metallic whereas the remaining 2/3 should be semiconducting. However, it should be noted that due to the inverse proportionality between the bandgap and the diameter of the semiconducting tubes, large-diameter tubes will tend to behave metallic at room temperature. This is especially important in regards to large-diameter MWCNTs.

From a electrical point of view a MWCNT can be seen as a complex structure of many parallel conductors that are only weakly interacting. Since probing the electrical properties typically involves electrodes contacting the outermost shell, this shell will be dominating the transport properties In a simplistic view, this can be compared to a large-diameter SWCNT, which will therefore typically display metallic behavior.

Electrical and Electromechanical Properties
Many studies have focused on SWCNTs for exploring the fundamental properties of nanotubes. Due to their essentially 1-D nature and intriguing electronic structure, SWCNTs exhibit a range of interesting quantum phenomena at low temperature

The discussion here will so far, however, primarily be limited to room temperature properties.

The conductance $$~G~$$ of a 1-dimensional conductor such as a SWCNT is given by the Landauer formula

$$~G=G_{0}\sum_{i}T_{i}~$$,

where $$~G_{0}=2e^{2}/h = (12.9 \rm k\Omega)^{-1}$$; $$~1/G_{0}~$$ is the conductance quantum; and $$~T_{i}~$$ is the transmission coefficient of the contributing channel $$~i~$$.

More information on nanotubes

 * Wikipedia Carbon nanotubes
 * The nanotube site
 * Compiled overview of properties of carbon nanotubes

Commercial suppliers of carbon nanotubes and related products

 * Nanoledge.com nanotubes and related products - fibers, pellets, resins and dispersions...
 * Carbon Solution, Inc. Mainly SWCNT.
 * BuckyUSA Fullerens, SWCNT, MWCNT.
 * CNI HiPco Carbon nanotubes
 * Carbolex SWCNT,sub-gram quantities are sold via sigma aldric
 * Sigma-Aldrich SWCNT

"Buckyball"

Buckminsterfullerene (IUPAC name (C60-Ih)[5,6]fullerene) is the smallest fullerene molecule in which no two pentagons share an edge (which can be destabilizing, as in pentalene). It is also the most common in terms of natural occurrence, as it can often be found in soot.

The structure of C60 is a truncated (T = 3) icosahedron, which resembles a soccer ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge.

The van der Waals diameter of a C60 molecule is about 1 nanometer (nm). The nucleus to nucleus diameter of a C60 molecule is about 0.7 nm.

The C60 molecule has two bond lengths. The 6:6 ring bonds (between two hexagons) can be considered "double bonds" and are shorter than the 6:5 bonds (between a hexagon and a pentagon). Its average bond length is 1.4 angstroms.

Silicon buckyballs have been created around metal ions.

Boron buckyball
A new type of buckyball utilizing boron atoms instead of the usual carbon has been predicted and described by researchers at Rice University. The B-80 structure, with each atom forming 5 or 6 bonds, is predicted to be more stable than the C-60 buckyball. One reason for this given by the researchers is that the B-80 is actually more like the original geodesic dome structure popularized by Buckminster Fuller which utilizes triangles rather than hexagons. However, this work has been subject to much criticism by quantum chemists as it was concluded that the predicted Ih symmetric structure was vibrationally unstable and the resulting cage undergoes a spontaneous symmetry break yielding a puckered cage with rare Th symmetry (symmetry of a volleyball). The number of six atom rings in this molecule is 20 and number of five member rings is 12. There is an additional atom in the center of each six member ring, bonded to each atom surrounding it.

Variations of buckyballs
Another fairly common buckminsterfullerene is C70, but fullerenes with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained.

In mathematical terms, the structure of a fullerene is a trivalent convex polyhedron with pentagonal and hexagonal faces. In graph theory, the term fullerene refers to any 3-regular, planar graph with all faces of size 5 or 6 (including the external face). It follows from Euler's polyhedron formula, |V|-|E|+|F| = 2, (where |V|, |E|, |F| indicate the number of vertices, edges, and faces), that there are exactly 12 pentagons in a fullerene and |V|/2-10 hexagons.

The smallest fullerene is the dodecahedron--the unique C20. There are no fullerenes with 22 vertices. The number of fullerenes C2n grows with increasing n = 12,13,14..., roughly in proportion to n9. For instance, there are 1812 non-isomorphic fullerenes C60. Note that only one form of C60, the buckminsterfullerene alias truncated icosahedron, has no pair of adjacent pentagons (the smallest such fullerene). To further illustrate the growth, there are 214,127,713 non-isomorphic fullerenes C200, 15,655,672 of which have no adjacent pentagons.

Trimetasphere carbon nanomaterials were discovered by researchers at Virginia Tech and licensed exclusively to Luna Innovations. This class of novel molecules comprises 80 carbon atoms (C80) forming a sphere which encloses a complex of three metal atoms and one nitrogen atom. These fullerenes encapsulate metals which puts them in the subset referred to as metallofullerenes. Trimetaspheres have the potential for use in diagnostics (as safe imaging agents), therapeutics and in organic solar cells.

=Semiconducting nanowires= Semiconducting nanowires can be made from most semiconducting materials and with different methods, mainly variations of a chemical vapor deposition process (CVD).

There are many different semiconducting materials, and heterosrtuctures can be made if the lattice constants are not too incompatible. Heterostructures made from combinations of materials such as GaAs-GaP can be used to make barriers and guides for electrons in electrical systems.

Low pressure metal organic vapor phase epitaxy (MOVPE) can be used to grow III-V nanowires epitaxially on suitable crystalline substrates, sucha s III-V materials or silicon with a reasonably matching lattice constant.

Nanowire growth is catalyzed by various nanoparticles, which are deposited on the substrate surface, typically gold nanoparticles with a diameter of 20-100nm.

To grow for instance GaP wires, the sample is typically annealed at 650C in the heated reactor chamber to form an eutectic with between the gold catalyst and the underlying substrate.

Then growth is done at a lower temperature around 500C in the presence of the precursor gasses trimethyl gallium and phosphine. By changing the precursor gasses during growth, nanowire heterostructures with varying composition can be made

Resources

 * wikipedia Semiconducting nanowires
 * IOFFE semiconductor physical properties

=Nanoparticles=
 * Nanoparticles
 * Quantum dots

Catalytic particles

 * Catalyst
 * Catalysis
 * Haber process

Commercial suppliers of nanoparticles

 * Reade has a wide selection of nanoparticles
 * sigma-aldrich has dispersions of nanoparticles
 * nanophase

=Contributors and Acknowledgements=
 * Jakob Kjelstrup Hansen

=References=

See also notes on editing this book about how to add references Nanotechnology/About.


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Nanotechnology/Nano-optics

Nanophotonics and nanooptics
With the increasing demand for smaller, faster, and more highly integrated optical and electronic devices; as well as extremely sensitive detectors for biomedical and environmental applications; a field called nano-optics or nano-photonics is emerging - studying the many promising optical properties of nanostructures.

Like nanotechnology itself, it is a rapidly evolving and changing field – but because of strong research activity in optical communication and related devices, combined with the intensive work on nanotechnology, nano-optics appears to be a field with a promising future.

Nanophotonics is seen as a crucial technology for extending Moore's Law into the next few decades. In the past few years nanophotonics researchers worldwide have developed, On Chip Silicon Lasers, Gigahertz Silicon Electro Optic Switches, and Low Loss Highly Integratable Compact Nanowires (With waveguides of 100’s of nanometers' width).

Nanophotonics is mainly expected to play a complementary role to micro/nano electronics on chip and extend the capacity of telecommunication networks into the Terabit/s regime. One of the major emphasis’s in the last few years has been developing on-chip interconnects to break the bottle neck for higher data rates within integrated chips.

In conjugation with Nanofluidics, Nanophotonics is also finding applications in biomedical sensors, Medical diagnosis, etc.

Nanophotonic components such as Microcavities with ultra high life time of trapped photons are expected to find applications in fundamental experimental physics such as gravitational wave detection.

Intel, IBM, Lucent, and Luxtera have highly functional and well funded nanophotonic research groups. A number of universities in: US, UK, Japan, Italy, China, Belgium, etc. have been actively pursuing nanophotonics. Apart from a growing number of hits on the word in publication databases like "Web of Science", which shows it is already getting increased attention, it is also increasingly mentioned in the aims of the funding agencies, which will surely add to the activity in the field as increased economical support becomes available.

Electrooptic modulators
Electro-optic modulators are devices used to modulate, or modify a beam of light. Currently they are mainly used in the information technology and telecommunications industries (e.g. fiber-optic cables). EOM’s have good potential in nanophotonics. Nanoscale optical communication devices will have increased speed and efficiency, once they can be engineered and used. Nano-size electrooptic modulators will be an integral part of a nanoscale communications network.

Photodetector
Photodetectors respond to radiant energy. They are basically sensors of light or other electromagnetic energy. A sensor is a electronic device that converts one type of energy to another for various reasons. Nanoscale size photodetectors will be an integral part of a theoretical nanoscale optical information network.

Electrooptic switches
Electrooptic switches change signals in optical fibers to electrical signals. Typically semiconductor-based, their function depends on the change of refractive index with electric field. This feature makes them high-speed devices with low power consumption. Neither the electro-optic nor thermo-optic optical switches can match the insertion loss, back reflection, and long-term stability of opto-mechanical optical switches. The latest technology combines all-optical switches that can cross-connect fibers without translating the signal into the electrical domain. This greatly increases switching speed, allowing today's telcos and networks to increase data rates. However, this technology is only now in development, and deployed systems cost much more than systems that use traditional opto-mechanical switches.

Photonic crystals
"Photonic crystals are composed of periodic dielectric or metallo-dielectric nanostructures that are designed to affect the propagation of electromagnetic waves (EM) in the same way as the periodic potential in a semiconductor crystal affects the electron motion by defining allowed and forbidden electronic energy bands. Simply put, photonic crystals contain regularly repeating internal regions of high and low dielectric constant." Photonic crystals are used to modify or control the flow of light. Photonic crystals may have a novel use in optical data transmission but are not extremely prominent. They may be used to filter for interference in a fiber optic cable, or increase the quality of the transmission. In addition, they can be used to divide different wavelengths of light. Photonic crystals can already be manufactured at close to the nanoscale.

Sensors
Nanotechnology creates many new, interesting fields and applications for photonic sensors. Existing uses, like digital cameras, can be enhanced because more ‘pixels’ can be placed on a sensor than with existing technology. In addition, sensors can be fabricated on the nano-scale so that they will be of higher quality, and possibly defect free. The end result would be that photos would be larger, and more accurate. As part of a communication network, photonic sensors will be used to convert optical data (photons) into electricity (electrons). Nanoscale photonic sensors will be more efficient and basically receive similar advantages to other materials constructed under the nanoscale.

Multiplexers
A multiplexer is a device for converting many data streams into one single data stream, which is then divided into the separate data streams on the other side with a demultiplexer. The main benefit is cost savings, since only one physical link will be needed, instead of many physical links. In nano-optics, multiplexers will have many applications. They can be used as part of a communication network, as well as utilized on a smaller scale for various modern scientific instruments.

Vanadium dioxide
Vanadium dioxide has the interesting property of changing from a transparent state to a reflective, mirror-like state in less than 100 femtoseconds (1/10 of a trillionth of a second). Vanderbilt University discovered the transition at 68 degrees celsius. The temperature that the transition happens can be changed by adding small amounts of impurities, and it is possible to lower the temperature by as much as 35 degrees celsius. However, there is a size limit, the change will not occur in particles that are smaller than 20 atoms across, or 10 nanometers. This property has many applications. Possibilities are a 'solar shade' window that changes from letting light in, to reflecting light back automatically when the temperature starts rising. Also, nanosensors could be created which could measure the temperature at different locations in human cells. However, most importantly, this transition can be utilized in creating an 'ultrafast' optical switch which could be used in communications or computing. Currently, researchers are seeing if they can put a layer of vanadium dioxide nanoparticles on the end of an optical fiber to create a very high speed link.

Quantum dots
Quantum dots have several applications. One of the first applications found was their ability to emit very specific wavelengths of light. This is different from other light emitting bulbs since quantum dots could be tuned across the visible and ultraviolet spectrums very precisely. Researchers have found that if they put about 2,000 quantum dots together, they would have a finely tuned LED. Researchers have tried for an extremely long time to get these dots to emit light. In the 1990’s someone was able to get a dark red light. Since then other researchers have been able to tune the dots to a higher frequency, thus gaining blue and green light. The applications for this would be beneficial so that we could make full color screens and monitors.

Resources

 * Near and far field - near and far field radiation can to some extent be compared to listening to a walkmans earphones; the one carrying the earphones can hear the sound perfectly even though the bass sound wavelength is much larger than the earphone. If you are not wearing the earphones, the high frequency sounds will be much higher than the bass. The bass can only be heard in the near field.
 * plasmonics
 * Rochester Nano Optics

title: Nanotechnology/Nano-optics


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Nanotechnology/Nanoelectronics

=Nanoelectronics= Nanoelectronics is expected to be cheaper to fabricate than silicon / gallium / arsenic based electronics. It also might be small enough to not require a power source as it is possible to abstract a small amount of energy from the surrounding heat by a molecular level energy scavenging system

=Diffusive and Ballistic Electron Transport=

=Double barrier systems= Coulomb Blockade
 * Coulomb Blockade
 * Tunnel Junctions
 * Quantum Tunneling

=Moletronics / Molecular Electronics=

Using molecules for electronics, often called moletronics or molecular electronics , is a new technology which is still in its infancy, but also brings hope for truly atomic scale electronic systems in the future.

One of the more promising applications of molecular electronics was proposed by the IBM researcher Ari Aviram and the theoretical chemist Mark Ratner in their 1974 and 1988 papers Molecules for Memory, Logic and Amplification, (see Unimolecular rectifier ) . This is one of many possible ways in which a molecular level diode / transistor might be synthesized by organic chemistry. A model system was proposed with a spiro carbon structure giving a molecular diode about half a nanometre across which could be connected by polythiophene molecular wires. Theoretical calculations showed the design to be sound in principle and there is still hope that such a system can be made to work.

However one researcher, experimentalist Jan Hendrik Schön, could not wait for the necessary technical progress and at a time when he was publishing one scientific paper a week and winning scholarships, heading for the top in nanotechnology, it was discovered he had fabricated both the experiment where such a device worked and several other potentially important milestones in the field. This incident is discussed by David Goodstein in Physics World . However it seems only a matter of time before something like this proposed elegant solution to the problem demonstrates the behavior of a diode.

=Quantum Computing= A quantum computer would be incredibly fast compared to microelectronics. It would also be able to use the properties of quantum mechanics to be in fuzzy states which could represent many numbers at once allowing a massive density of memory. How such devices would be nano-fabricated is however way beyond current technology.

The first working 2-qubit quantum computer was demonstrated in 1998.

In 2006, the first working 12 qubit quantum computer was demonstrated.

title: Nanotechnology/Nanoelectronics


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Nanotechnology/Nanofluidics

Flow on the large scale, in Microsystems and in nanosystems show very different behaviour.

In nanofluidic systems you willl have
 * Small sample volume
 * High area to volume ratio
 * Domination of surface forces
 * Short diffusion times
 * Enhanced reaction kinetics due to short diffusion times
 * Relatively large electric double layer

 * Microfluidics
 * Brownian motion of nanoparticles
 * Diffusion
 * Fluid dynamics

Nanoscale flow can be enhanced considerably compared to what is predicted by macroscale Knudsen flow or continuum hydrodynamics models - see Science Vol. 312. no. 5776, pp. 1034 - 1037 DOI-link (Fast Mass Transport Through Sub-2-Nanometer Carbon Nanotubes)

=FERROFLUIDS= Ferrofluids are a form of colloidal suspensions. They are tiny iron particles covered with a liquid coating, also surfactant, that are then added to water or oil, which gives them their liquid properties. Ferrofluids are colloidal suspensions. Those are materials with properties of more than one state of matter. In this case, the two states of matter are the solid metal and liquid it is in. This ability to change phases with the application of a magnetic field allows them to be used as seals, lubricants, and may open up further applications in future nanoelectromechanical systems.

=Resources=
 * a wikibook on Microfluidics

 * Microfluidics: Fluid physics at the nanoliter scale REVIEWS OF MODERN PHYSICS, VOLUME 77, JULY 2005 by Squires and Quake
 * The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture Journal of Fluids Engineering 1999, Vol. 121 / 5

=References=

See also notes on editing this book about how to add references Nanotechnology/About.

title: Nanotechnology/Nanofluidics


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User:ARMueller

hey this is villorenterg. thanks for your response on my article. i really wanted to spice my article up by adding some images, but the only ones you can use are embedded images. so Wikipedia has a Commons area where you can pick from hundreds of pictures on just about anything to post on your page. wikipedia's pictures are the only ones that will work with the code that they have. If youre really interested in adding some pictures but need help, please dont hesitate to ask. thanks again!

title: User:ARMueller


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Intermediate Algebra/Absolute Value Inequalities

Compound Inequalities
In the previous section we dealt with inequalities with one specific constraint - that is, only one solution set. However, there is a good chance that real-world situations will require multiple constraints. For example, computer manufacturers will want their products to be sold within certain ranges so that customers will continue to buy their computers.

Such situations are also bound to happen in algebra, and can be represented using compound inequalities. A compound inequality is a statement that puts two constraints on a constant, variable, or expression. For example, the inequality -4 < x < 3 indicates that x has two constraints: it must be (a) greater than -4 and (b) less than 3. These compound inequalities are often referred to as "and" inequalities because they require that the expression in the center meets the two restraints.

Using the graphing methods covered in the last lesson, we can show the solution set of -4 < x < 3 on the number line:

OO                              <--|--|--|--|--|--|--|--|--|--|--|--> -5 -4 -3 -2 -1 0  1  2  3  4  5

Notice that the solution is bounded by the two values.

On the other hand, a compound inequality can have one out of two constraints. These inequalities are called "or" inequalities - the expression can satisfy either one of the conditions. If, for example, x < 3 or x $$\ge$$ 7, then x is equal to any number less than 3 or greater than or equal to 7. Any number between 3 and 7 will not satisfy either condition.

Represented graphically, the inequality x < 3 or x $$\ge$$ 7 looks like this:

 <--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->                       -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Notice that, as opposed to the inequality in the previous example, the two constraints travel in opposite directions. This is a key principle of compound inequalities, and one that should be kept in mind when graphing solutions.

Solving Compound Inequalities
Because compound inequalities are two separate constraints combined into one, all operations must be performed on all sides of the inequality.

Example 1: Solve for x: -7 < 2x + 4 $$\le$$ 12.

Stop for a second and pretend that the statement above was two separate inequalities. The logical first step would be to subtract 4 from both sides. Because the two are now put into one, however, we must subtract 4 from all sides. Therefore:

-7 < 2x + 4 $$\le$$ 12

-7 - 4 < 2x + 4 - 4 $$\le$$ 12 - 4

-11 < 2x $$\le$$ 8

Now divide all sides by 2.

-11 < 2x $$\le$$ 8

$$\frac {-11}{2} < \frac {2x}{2} \le \frac {8}{2}$$

-5.5 < x $$\le$$ 4

Example 2: Solve for x: -3 > -x + 7 or 2x - 6 $$\le$$ -4.

Because the two inequalities act as separate entities, solve each one separately. Start by subtracting 7 from both sides in the first inequality, and add 6 to both sides in the second.

-3 > -x + 7                 or                    2x - 6 $$\le$$ -4 -10 > -x                                               2x $$\le$$ 2

Finish by dividing by respective coefficients (-1 and 2).

-10 > -x                                             2x $$\le$$ 2

x > 10                     or                       x $$\le$$ 1


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XForms/Range

NOTE
These examples do not appear to work with the FireFox 3 XForms addition. Sliders are hidden with XSLTForms (r406).

Motivation
You want to use a range control or "slider" to set a numeric value in your form.

Selecting from a continuous range of values
The range is a control that is part of the XForms specification. It allows the user to select a numeric value from a range of values without using the keyboard. This is ideal for instructors that use interactive learning environments such as a SmartBoard.

The XForms range has four attributes:
 * start - the lower bound of the range
 * end - the upper bound of the range
 * step - the increment of output change
 * incremental - true if the form should send continuous values to the model or false otherwise. The default is false.

These attributes MUST be bound to an element that has a numeric datatype. To do this you must add the following bind statements to your model:

User Interface
There are three range controls in this application. The value of the range is on the right side.

Link to XForms Application
Basic Range

Ranges Using Binds

Sample User Interface
      Data1:     Value=

 Data2:  </xf:range> <xf:output ref="data2"> <xf:label> Value= </xf:label> </xf:output>

<xf:range ref="data3" start="1" end="100" step="5" incremental="true"> <xf:label>Data3: </xf:label> </xf:range> <xf:output ref="data3"> <xf:label> Value= </xf:label> </xf:output>

Testing
Select one of the three range controls with the mouse. Drag the value to the right or left. You should notice that the values at the right also are updated.

The first control just selects decimals from 1 to 5. The second uses the step to return only integers. The last has a range of 1 to 100 in increments of 5. Note that the value is 66 and you cannot set a value from 62 to 65.

Discussion
The upper and lower values of the range control can also be set directly in the model. XML Schema provides a minValue and a maxValue that can be used to adjust the range control limits.

W3C Specification
Here is the specification from the W3C web site: W3C range specification

title: XForms/Range


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Nanotechnology/Top-down and bottom-up approaches

=Top-down and bottom-up approaches= There are two types of approaches for the synthesis of nanomaterial and fabrication of nanostructure.
 * top-Down approaches refers to slicing or successive cutting of bulk material to get nano-sized particles.there are two types *attrition ,* milling

 * Bottom-up refers to method where devices 'create themselves' by self-assembly. Chemical synthesis is a good example. Bottom-up should, broadly speaking, be able to produce devices in parallel and much cheaper than top-down methods, but getting control over the methods is difficult when things become larger and bulkier than what is normally made by chemical synthesis. Of course, nature has had time to evolve and optimize self-assembly processes that can do wonders.

=Microfabrication made smaller= Not much of nanotechnology is based on methods from Microfabrication.

=References=

See also notes on editing this book about how to add references Nanotechnology/About.


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Nanotechnology/Self assembly

 * Self assembly
 * Self assembled monolayers
 * Molecular nanotechnology
 * Molecular engineering

=References=

See also notes on editing this book about how to add references Nanotechnology/About.

title: Nanotechnology/Self assembly


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Nanotechnology/Lithography

=Lithography=
 * Nanolithography

=Electron beam lithography= Electron beam lithography (EBL)

=Nano imprint lithography (NIL)= Nanoimprint lithography (NIL)

=Focused Ion Beam Techniques= Focused Ion Beam Techniques

=Electron Beam Induced Deposition (EBID or EBD)=

The highly focused electron beam in a SEM is used for imaging nanostructures, but it can also be used to make nanoscale deposits. In the presence of carbonaceous or organometallic gasses in the SEM chamber, electron beam induced deposition (EBID or electron beam deposition (EBD)) can be used to construct three-dimensional nanostructures or solder/glue nanostructures.

There is module in this handbook dedicated to EBID

=References=

See also notes on editing this book about how to add references Nanotechnology/About.

title: Nanotechnology/Lithography