Ideal chain

In physics, the optical theorem is a general law of wave scattering theory, which relates the forward scattering amplitude to the total cross section of the scatterer. It is usually written in the form

$\sigma _{\mathrm {tot} }={\frac {4\pi }{k}}~\mathrm {Im} \,f(0),$

where f(0) is the scattering amplitude with an angle of zero, that is, the amplitude of the wave scattered to the center of a distant screen, and k is the wave vector in the incident direction. Because the optical theorem is derived using only conservation of energy, or in quantum mechanics from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics, $\sigma _{\mathrm {tot} }$ includes both elastic and inelastic scattering. Note that the above form is for an incident plane wave; a more general form discovered by Werner Heisenberg can be written

$\mathrm {Im} ~f({\mathbf {\hat {k}}}',{\mathbf {\hat {k}}})={\frac {k}{4\pi }}\int f({\mathbf {\hat {k}}}',{\mathbf {\hat {k}}}'')f({\mathbf {\hat {k}}}'',{\mathbf {\hat {k}}})~d{\mathbf {\hat {k}}}''.$

Notice that as a natural consequence of the optical theorem, an object that scatters any light at all ought to have a nonzero forward scattering amplitude. However, the physically observed field in the forward direction is a sum of the scattered and incident fields, which may add to zero.

History

The optical theorem was originally discovered independently by Wolfgang von Sellmeier and Lord Rayleigh in 1871. Lord Rayleigh recognized the forward scattering amplitude in terms of the index of refraction as

$n=1+2\pi {\frac {Nf(0)}{k^{2}}},$

(where N is the number density of scatterers) which he used in a study of the color and polarization of the sky. The equation was later extended to quantum scattering theory by several individuals, and came to be known as the Bohr–Peierls–Placzek relation after a 1939 publication. It was first referred to as the Optical Theorem in print in 1955 by Hans Bethe and Frederic de Hoffmann, after it had been known as a "well known theorem of optics" for some time.

Derivation

The theorem can be derived rather directly from a treatment of a scalar wave. If a plane wave is incident on an object, then the wave amplitude a great distance away from the scatterer is approximately given by

$\psi ({\mathbf {r}})\approx e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}.$

All higher terms, when squared, vanish more quickly than $1/r^{2}$, and so are negligible a great distance away. Notice that for large values of z and small angles the binomial theorem gives us

$r={\sqrt {x^{2}+y^{2}+z^{2}}}\approx z+{\frac {x^{2}+y^{2}}{2z}}.$

We would now like to use the fact that the intensity is proportional to the square of the amplitude $\psi$. Approximating the r in the denominator as z, we have

$|\psi |^{2}=|e^{ikz}+{\frac {f(\theta )}{z}}e^{ikz}e^{ik(x^{2}+y^{2})/2z}|^{2}$

$=1+{\frac {f(\theta )}{z}}e^{ik(x^{2}+y^{2})/2z}+{\frac {f^{*}(\theta )}{z}}e^{-ik(x^{2}+y^{2})/2z}+{\frac {|f(\theta )|^{2}}{z^{2}}}.$

If we drop the $1/z^{2}$ term and use the fact that $A+A^{*}=2~\mathrm {Re} ~A$ we have

$|\psi |^{2}\approx 1+2~\mathrm {Re} \left({\frac {f(\theta )}{z}}e^{ik(x^{2}+y^{2})/2z}\right).$

Now suppose we integrate over a screen in the x-y plane, at a distance which is small enough for the small angle approximations to be appropriate, but large enough that we can integrate the intensity from $-\infty$ to $\infty$ with negligible error. In optics, this is equivalent to including many fringes of the diffraction pattern. To further simplify matters, let's approximate $f(\theta )=f(0)$. We quickly obtain

$\int |\psi |^{2}~da\approx A+2~\mathrm {Re} \left({\frac {f(0)}{z}}\int _{-\infty }^{\infty }e^{ikx^{2}/2z}dx\int _{-\infty }^{\infty }e^{iky^{2}/2z}dy\right)$

where A is the area of the surface integrated over. The exponentials can be treated as Gaussians and so

$\int |\psi |^{2}~da=A+2~\mathrm {Re} \left({\frac {f(0)}{z}}{\frac {i2z\pi }{k}}\right)$

$=A-{\frac {4\pi }{k}}~\mathrm {Im} ~f(0),$

which is just the probability of reaching the screen if none were scattered, lessened by an amount $(4\pi /k)~\mathrm {Im} ~f(0)$, which is therefore the effective scattering cross section of the scatterer.

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534