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[[Image:Scattering theory illust.png|right|thumb|Top: the [[real part]] of a [[plane wave]] travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of [[index of refraction]] higher than the index of the surrounding medium. This object scatters part of the wave field, although at any individual point, the wave's frequency and wavelength remain intact.]]
Kallang Riverside is about to turn into an exciting addition to the colourful Southern Waterfront of Singapore as a consequence of its wonderful location and provision of a life-style that many would desire. HDB will likely be releasing about three,810 BTO flats in Punggol, Sembawang, Toa Payoh, Woodlands and Yishun in Jul 2014.<br><br>We're glad you are and we now have not forgotten about Jurong New Launches With the most recent preview launch of J Gateway on 28 June 2013, there are strong indication of excessive demand over provide in Jurong from home seekers and buyers alike because the response are over-whelming with greater than 1,four hundred blank cheques placed to specific their curiosity earlier than the preview sale. There are 738 units in the development and this venture is one hundred% absolutely offered on preview day even with prices transacting as excessive as $1,778 psf for a one-bed room unit.<br><br>If you're focused on exploring the Resale Government Condominium market, please send me an enquiry through the hyperlink above. Govt Condominium - Modifications to Executive Condominium Scheme (impacts all EC launches occuring after 12 Jan 2013) Can participating builders severely look into promotions targeting vacationers from top-spending destinations? How about partnering with Singapore Tourism Board for organized purchasing tours with free sightseeing, flight and accommodation bundled with reserving a luxurious residence, with the 15 % foreigner Extra Purchaser Stamp Responsibility absorbed by the builders? Buyers of EC's whose land was bought post eleventh Dec 2013 must pay Resale Levy (for 2nd timers solely) - this doesn't have an effect on 2014 EC launches. Waterbay EC @ Punggol<br><br>Posted by Edison Foo  November 24, 2013 Open for booking NOW !!! Posted by Edison Foo  November 17, 2013 Open for reserving NOW !!! Posted by Edison Foo  November 7, 2013 Open for booking NOW !!! Posted by Edison Foo  October 22, 2013 SCORCHING!!! Promoting Quick!! Posted by  [http://www.mydadhasrights.com/member.php?u=821113-M7498 The Skywoods condo price] Edison Foo  September 29, 2013 HEARTH SALE!!! From solely $9xx,000! Posted by Edison Foo  September 10, 2013 Posted by Edison Foo August 19, 2013 Posted by Edison Foo  August 15, 2013 Posted by Edison Foo  August eleven, 2013 Posted by Edison Foo  July 25, 2013 Located inside minutes to Bishan MRT interchange( Circle & North-South Line). Apartment developments close to MRT have gotten increasingly rare. Bus interchange proper beside. Open for Sale Booking Now! HTC BlinkFeed – Your news direct to your private home screen Reduce EC Cancellation Charge<br><br>Ang Mo Kio Cluster House is launching tentatively in April 2013. Click here to register your curiosity for Ang Mo Kio Cluster House now! Ang Mo Kio Cluster House is a rare freehold strata landed development along Ang Mo Kio Avenue Spottiswoode Residences - Guide now to get pleasure from 3% direct discount + 3% MONEY REBATE upon first cost + three% furnishing CASH SUBSIDY upon TOP! For limited interval only. Selling out quick! Last 15 items! Why wait! E book your unit now! Spottiswoode For more info or Kensington Sq. price, please register with us and we will ship you the latest updates with high precedence. Feel free to contact us directly and we'll reply to you as soon as doable. Reputable developer with many years as Singapores main developer firm Capitaland<br><br>Now that you've all these to consider, it's a good idea to search for for real estate agents that will make it easier to in making the right motion of investing on the right property once you might be ready. A reliable agent will just remember to have the proper funds for a brand new condominium launch investing. Without knowing the exact quantity of how much you might be keen to spend, you may be brief in the long term since you have to make sure that the price range also covers the maintenance bills. These bills will gradually enhance 12 months by yr and if you're investing without a monetary plan, the dream of having your individual non-public residential unit will never be a actuality.<br><br>Marine Blue Rental is a 99-years leasehold new launch rental improvement by Capitaland (Oneof Singapore largest developer) within the very wanted area of Marine Paradein District 15. Marine Blue condo is situated mins away from Parkway Parade andalso walking distance to East Coast Park! With anticipated completion in mid 2018,it contains of 124 items. No. For new Singapore property sales, you'll be able to withdraw at any time before booking the unit, without penalty. At the preview, the agent will let you already know the precise worth for you to resolve whether or not to proceed or not. Solely once you decide to proceed will the agent book the unit for you. Marina One Residences By Malaysia Khazanah & Singapore Temasek. May 14, 2014 The Crest Rental by Wing Tai Preview Launch Quickly<br><br>It's one other measure to safeguard those buying property in Singapore. When you've ever had the experience of paying for an uncompleted property elsewhere only to see the developer vanish midway via development of the mission, you would be glad to know this is not one thing that's likely to happen should you're buying property in Singapore. That is as a result of new Singapore property gross sales are very high regulated. After the seventh spherical of cooling measure in Feb 2013, thedust in property market in Singapore begins to settle. 2ndquarter will see many more new property launches. This is what'scoming up for brand spanking new condominium launch near MRT station! How to pre-register for Singapore property sales previews on my own? Present property launch which are hidden gems New Apartment In 2013 Might
In [[mathematics]] and [[physics]], '''scattering theory''' is a framework for studying and understanding the [[scattering]] of [[waves]] and [[Elementary particle|particles]].  Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance [[sunlight]] scattered by [[rain drop]]s to form a [[rainbow]]. Scattering also includes the interaction of [[billiard balls]] on a table, the [[Rutherford scattering]] (or angle change) of [[alpha particle]]s by [[gold]] [[atomic nucleus|nuclei]], the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and the inelastic scattering of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of [[partial differential equations]], propagating freely "in the distant past", come together and interact with one another or with a [[boundary condition]], and then propagate away "to the distant future".  The '''direct scattering problem''' is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the [[scattering|scatterer]]. The [[inverse scattering problem]] is the problem of determining the characteristics of an object (e.g., its shape, internal constitution) from measurement data of radiation or particles scattered from the object.
 
Since its early statement for [[radar|radiolocation]], the problem has found vast number of applications, such as [[acoustic location|echolocation]], [[geophysical]] survey, [[nondestructive testing]], [[medical imaging]] and [[quantum field theory]], to name just a few.
 
==Conceptual underpinnings==
 
The concepts used in scattering theory go by different names in
different fields. The object of this section is to point the reader
to common threads.
 
===Composite targets and range equations===
 
[[Image:Xsection2.png|288px|thumb|left|Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units.]]
 
When the target is a set of many scattering centers whose relative position varies unpredictably, it is customary to think of a range equation whose arguments take different forms in different application areas.  In the simplest case consider an interaction that removes particles from the "unscattered beam" at a uniform rate that is proportional to the incident flux <math>I</math> of particles per unit area per unit time, i.e. that
 
:<math> \frac{dI}{dx}=-QI \,\!</math>
 
where ''Q'' is an interaction coefficient and ''x'' is the distance traveled in the target.  
 
The above ordinary first-order [[differential equation]] has solutions of the form:
 
: <math>I = I_o e^{-Q \Delta x} = I_o e^{-\frac{\Delta x}{\lambda}} = I_o e^{-\sigma (\eta \Delta x)} = I_o e^{-\frac{\rho \Delta x}{\tau}} ,</math>
 
where ''I''<sub>o</sub> is the initial flux, path length Δx&nbsp;≡&nbsp;''x''&nbsp;&minus;&nbsp;''x''<sub>o</sub>, the second equality defines an interaction [[mean free path]] λ, the third uses the number of targets per unit volume η to define an area [[cross section (physics)|cross-section]] σ, and the last uses the target mass density ρ to define a density mean free path τ.   Hence one converts between these quantities via ''Q'' =&nbsp;1/''&lambda;'' =&nbsp;''&eta;&sigma;'' =&nbsp;''&rho;/&tau;'', as shown in the figure at left.
 
In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm<sup>&minus;1</sup>) is variously called [[opacity (optics)|opacity]], [[absorption coefficient]], and [[attenuation coefficient]]. In nuclear physics, area cross-sections (e.g. σ in [[barn (unit)|barn]]s or units of 10<sup>&minus;24</sup> cm<sup>2</sup>), density mean free path (e.g. τ in grams/cm<sup>2</sup>), and its reciprocal the [[mass attenuation coefficient]] (e.g. in cm<sup>2</sup>/gram) or ''area per nucleon'' are all popular, while in electron microscopy the [[inelastic mean free path]]<ref>R. F. Egerton (1996) ''Electron energy-loss spectroscopy in the electron microscope'' (Second Edition, Plenum Press, NY) ISBN 0-306-45223-5</ref> (e.g. λ in nanometers) is often discussed<ref>Ludwig Reimer (1997) ''Transmission electron microscopy: Physics of image formation and microanalysis'' (Fourth Edition, Springer, Berlin) ISBN 3-540-62568-2</ref> instead.
 
==In theoretical physics==
In [[mathematical physics]], '''scattering theory''' is a framework for studying and understanding the interaction or [[scattering]] of solutions to [[partial differential equation]]s. In [[acoustics]], the differential equation is the [[wave equation]], and scattering studies how its solutions, the [[sound wave]]s, scatter from solid objects or propagate through non-uniform media (such as sound waves, in [[sea water]], coming from a [[submarine]]). In the case of classical [[electrodynamics]], the differential equation is again the wave equation, and the scattering of [[light]] or [[radio wave]]s is studied. In [[particle physics]], the equations are those of [[Quantum electrodynamics]], [[Quantum chromodynamics]] and the [[Standard Model]], the solutions of which correspond to [[fundamental particle]]s.
 
In regular [[quantum mechanics]], which includes [[quantum chemistry]], the relevant equation is the [[Schrödinger equation]], although equivalent formulations, such as the [[Lippmann-Schwinger equation]] and the Faddeev equations, are also largely used. The solutions describe the long-term motion of free atoms, molecules, photons, electrons, and protons. The scenario is that several particles come together from an infinite distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles. The products and unused reagents then fly away to infinity again. (The atoms and molecules are effectively particles for our purposes. Also, under everyday circumstances, only photons are being created and destroyed.) The solutions reveal which directions the products are most likely to fly off to and how quickly. They also reveal the probability of various reactions, creations, and decays occurring. There are two predominant techniques of finding solutions to scattering problems: [[partial wave analysis]], and the [[Born approximation]].
 
==Elastic and inelastic scattering==
 
The term "elastic scattering" implies that the internal states of the scattering particles do not change, and hence they emerge unchanged from the scattering process. In inelastic scattering, by contrast, the particles' internal state is changed, which may amount to exciting some of the electrons of a scattering atom, or the complete annihilation of a scattering particle and the creation of entirely new particles.
 
The example of scattering in [[quantum chemistry]] is particularly instructive, as the theory is reasonably complex while still having a good foundation on which to build an intuitive understanding. When two atoms are scattered off one another, one can understand them as being the [[bound state]] solutions of some differential equation.  Thus, for example, the [[hydrogen atom]] corresponds to a solution to the [[Schrödinger equation]] with a negative inverse-power (i.e., attractive Coulombic) [[central potential]].  The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even [[ionization|ionized]], representing an inelastic scattering process.
 
The term "[[deep inelastic scattering]]" refers to a special kind of scattering experiment in particle physics.
 
==The mathematical framework==
In [[mathematics]], scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a [[differential equation]] is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of [[time]]. One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made  to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future".
 
Solutions to differential equations are often posed on [[manifold]]s. Frequently, the means to the solution requires the study of the [[ Spectrum (functional analysis) | spectrum]] of an [[operator theory|operator]] on the manifold. As a result, the solutions often have a spectrum that can be identified with a [[Hilbert space]], and scattering is described by a certain map, the [[S matrix]], on Hilbert spaces.  Spaces with a [[discrete spectrum]] correspond to [[bound state]]s in quantum mechanics, while a [[continuous spectrum]] is associated with scattering states.  The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.
 
An important, notable development is the [[inverse scattering transform]], central to the solution of many [[exactly solvable model]]s.
==See also==
{{div col}}
*[[Backscattering]]
*[[Brillouin scattering]]
*[[Diffuse sky radiation]]
*[[Extinction (astronomy)|Extinction]]
*[[Linewidth]]
*[[Mie theory]]
*[[Molecular scattering]]
*[[Raman scattering]]
*[[Rayleigh scattering]]
*[[Scattering from rough surfaces]]
*[[Scintillation (physics)]]
{{div col end}}
 
==Footnotes==
{{reflist}}
 
==References==
* [http://www.mif.pg.gda.pl/homepages/slawek/epic/sem.htm Lectures of the European school on theoretical methods for electron and positron induced chemistry, Prague, Feb. 2005]
* [http://fa.its.tudelft.nl/~koelink/dictaat-scattering.pdf E. Koelink, Lectures on scattering theory, Delft the Netherlands 2006]
<!--
{{Author = Koelink, E. | Title = Lectures on scattering theory | Url = http://fa.its.tudelft.nl/~koelink/dictaat-scattering.pdf| Year = 2006 |Place = Delft}}
-->
 
==External links==
*[http://www.opticsinfobase.org/submit/ocis/OCIS_2007.pdf Optics Classification and Indexing Scheme (OCIS)], [[Optical Society of America]], 1997
 
[[Category:Scattering theory| ]]
[[Category:Scattering, absorption and radiative transfer (optics)| ]]
[[Category:Quantum mechanics]]
[[Category:Hilbert space]]

Revision as of 19:01, 11 February 2014

Kallang Riverside is about to turn into an exciting addition to the colourful Southern Waterfront of Singapore as a consequence of its wonderful location and provision of a life-style that many would desire. HDB will likely be releasing about three,810 BTO flats in Punggol, Sembawang, Toa Payoh, Woodlands and Yishun in Jul 2014.

We're glad you are and we now have not forgotten about Jurong New Launches With the most recent preview launch of J Gateway on 28 June 2013, there are strong indication of excessive demand over provide in Jurong from home seekers and buyers alike because the response are over-whelming with greater than 1,four hundred blank cheques placed to specific their curiosity earlier than the preview sale. There are 738 units in the development and this venture is one hundred% absolutely offered on preview day even with prices transacting as excessive as $1,778 psf for a one-bed room unit.

If you're focused on exploring the Resale Government Condominium market, please send me an enquiry through the hyperlink above. Govt Condominium - Modifications to Executive Condominium Scheme (impacts all EC launches occuring after 12 Jan 2013) Can participating builders severely look into promotions targeting vacationers from top-spending destinations? How about partnering with Singapore Tourism Board for organized purchasing tours with free sightseeing, flight and accommodation bundled with reserving a luxurious residence, with the 15 % foreigner Extra Purchaser Stamp Responsibility absorbed by the builders? Buyers of EC's whose land was bought post eleventh Dec 2013 must pay Resale Levy (for 2nd timers solely) - this doesn't have an effect on 2014 EC launches. Waterbay EC @ Punggol

Posted by Edison Foo November 24, 2013 Open for booking NOW !!! Posted by Edison Foo November 17, 2013 Open for reserving NOW !!! Posted by Edison Foo November 7, 2013 Open for booking NOW !!! Posted by Edison Foo October 22, 2013 SCORCHING!!! Promoting Quick!! Posted by The Skywoods condo price Edison Foo September 29, 2013 HEARTH SALE!!! From solely $9xx,000! Posted by Edison Foo September 10, 2013 Posted by Edison Foo August 19, 2013 Posted by Edison Foo August 15, 2013 Posted by Edison Foo August eleven, 2013 Posted by Edison Foo July 25, 2013 Located inside minutes to Bishan MRT interchange( Circle & North-South Line). Apartment developments close to MRT have gotten increasingly rare. Bus interchange proper beside. Open for Sale Booking Now! HTC BlinkFeed – Your news direct to your private home screen Reduce EC Cancellation Charge

Ang Mo Kio Cluster House is launching tentatively in April 2013. Click here to register your curiosity for Ang Mo Kio Cluster House now! Ang Mo Kio Cluster House is a rare freehold strata landed development along Ang Mo Kio Avenue Spottiswoode Residences - Guide now to get pleasure from 3% direct discount + 3% MONEY REBATE upon first cost + three% furnishing CASH SUBSIDY upon TOP! For limited interval only. Selling out quick! Last 15 items! Why wait! E book your unit now! Spottiswoode For more info or Kensington Sq. price, please register with us and we will ship you the latest updates with high precedence. Feel free to contact us directly and we'll reply to you as soon as doable. Reputable developer with many years as Singapores main developer firm Capitaland

Now that you've all these to consider, it's a good idea to search for for real estate agents that will make it easier to in making the right motion of investing on the right property once you might be ready. A reliable agent will just remember to have the proper funds for a brand new condominium launch investing. Without knowing the exact quantity of how much you might be keen to spend, you may be brief in the long term since you have to make sure that the price range also covers the maintenance bills. These bills will gradually enhance 12 months by yr and if you're investing without a monetary plan, the dream of having your individual non-public residential unit will never be a actuality.

Marine Blue Rental is a 99-years leasehold new launch rental improvement by Capitaland (Oneof Singapore largest developer) within the very wanted area of Marine Paradein District 15. Marine Blue condo is situated mins away from Parkway Parade andalso walking distance to East Coast Park! With anticipated completion in mid 2018,it contains of 124 items. No. For new Singapore property sales, you'll be able to withdraw at any time before booking the unit, without penalty. At the preview, the agent will let you already know the precise worth for you to resolve whether or not to proceed or not. Solely once you decide to proceed will the agent book the unit for you. Marina One Residences By Malaysia Khazanah & Singapore Temasek. May 14, 2014 The Crest Rental by Wing Tai Preview Launch Quickly

It's one other measure to safeguard those buying property in Singapore. When you've ever had the experience of paying for an uncompleted property elsewhere only to see the developer vanish midway via development of the mission, you would be glad to know this is not one thing that's likely to happen should you're buying property in Singapore. That is as a result of new Singapore property gross sales are very high regulated. After the seventh spherical of cooling measure in Feb 2013, thedust in property market in Singapore begins to settle. 2ndquarter will see many more new property launches. This is what'scoming up for brand spanking new condominium launch near MRT station! How to pre-register for Singapore property sales previews on my own? Present property launch which are hidden gems New Apartment In 2013 Might

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