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The '''Sakuma–Hattori equation''' is a mathematical model for predicting the amount of [[thermal radiation]], radiometric flux or radiometric power emitted from a perfect [[blackbody]] or received by a thermal radiation detector.
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== History ==
The Sakuma–Hattori was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1987.<ref name=Sakuma1/> In 1996 a study investigated the usefulness of various forms of the Sakuma–Hattori equation. This study showed the Planckian form to provide the best fit for most applications.<ref name=Sakuma2/> This study was done for 10 different forms of the Sakuma–Hattori equation containing not more than three fitting variables. In 2008, BIPM CCT-WG5 recommended its use for radiation thermometry uncertainty budgets below 960 °C.<ref name=Fischer/>
 
== General form ==
The Sakuma–Hattori equation gives the [[electromagnetic radiation|electromagnetic signal]] from thermal radiation based on an object's [[temperature]]. The signal can be electromagnetic [[flux]] or signal produced by a detector measuring this radiation. It has been suggested that below the silver point{{Cref2|A}}, a method using the Sakuma–Hattori equation be used.<ref name=Sakuma1>F Sakuma, S Hattori, "Establishing a practical temperature standard by using a narrow-band radiation thermometer with a silicon detector", in ''Temperature: Its Measurement and Control in Science and Industry'', vol. 5, edited by J F Schooley, New York, AIP, 421–427 (1982).</ref> In its general form it looks like:<ref name=Fischer>J. Fischer, P. Saunders, M. Sadli, M. Battuello, C. W. Park, Y. Zundong, H. Yoon, W. Li, E. van der Ham, F. Sakuma, Y. Yamada, M. Ballico, G. Machin, N. Fox, J. Hollandt, M. Matveyev, P. Bloembergen and S. Ugur, "[http://www.bipm.org/wg/CCT/CCT-WG5/Allowed/Miscellaneous/Low_T_Uncertainty_Paper_Version_1.71.pdf Uncertainty budgets for calibration of radiation thermometers below the silver point]" (pdf), CCT-WG5 on Radiation Thermometry, BIPM, Sèvres, France (2008).</ref>
:<math>S(T) = \frac{C}{\exp\left(\frac{c_2}{\lambda _x T}\right)-1}</math>
where:
{| class="wikitable"
| <math>C</math>
| Scalar coefficient
|-
| <math>c_2</math>
| Second Radiation Constant (0.014387752 m⋅K<ref>{{cite web |publisher=National Institute of Standards and Technology (NIST) |title=2006 CODATA recommended values |url=http://physics.nist.gov/cuu/index.html | date=Dec 2003 |accessdate=Apr 27, 2010}}</ref>)
|-
| <math>\lambda _x</math>
| Temperature dependent effective wavelength in meters
|-
| <math>T</math>
| Temperature in Kelvin
|-
|}
 
== Planckian form ==
 
=== Derivation ===
 
The Planckian form is realized by the following substitution:
:<math>\lambda _x = A + \frac{B}{T}</math>
 
Making this substitution renders the following the Sakuma–Hattori equation in the Planckian form.
 
{| class="wikitable"
|-
| Sakuma–Hattori equation (Planckian form)
| <math>S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1}</math>
|-
| Inverse equation <ref name=MSLNZ/>
| <math>T = \frac{c_2}{A \ln \left(\frac{C}{S} + 1\right)} - \frac{B}{A}</math>
|-
| First derivative <ref>''ASTM Standard E2758-10 – Standard Guide for Selection and Use of Wideband, Low Temperature Infrared Thermometers'', ASTM International, West Conshohocken, PA, (2010).</ref>
| <math>\frac {dS}{dT} = \left[S(T)\right]^2 \frac{A c_2}{C\left(AT + B\right)^2}\exp\left(\frac{c_2}{AT + B}\right)</math>
|}
 
=== Discussion ===
 
The Planckian form is recommended for use in calculating uncertainty budgets for [[radiation thermometry]]<ref name=Fischer/> and [[infrared thermometry]].<ref name=MSLNZ>''[http://msl.irl.cri.nz/sites/all/files/training-manuals/tg22-july-2009v2.pdf MSL Technical Guide 22 – Calibration of Low Temperature Infrared Thermometers]'' (pdf), Measurement Standards Laboratory of New Zealand (2008).</ref> It is also recommended for use in calibration of radiation thermometers below the silver point.<ref name=Fischer/>
 
The Planckian form resembles [[Planck's law|Planck's Law]].
 
:<math>S(T) = \frac{c_1}{\lambda^5\left[\exp\left(\frac{c_2}{\lambda T}\right)-1\right]}</math>
 
However the Sakuma–Hattori equation becomes very useful when considering low-temperature, wide-band radiation thermometry. To use Planck's Law over a wide spectral band, an [[integral]] like the following would have to be considered:
 
:<math>S(T) = \int_{\lambda _1}^{\lambda _2}\frac{c_1}{\lambda^5\left[\exp\left(\frac{c_2}{\lambda T}\right)-1\right]} d\lambda</math>
 
This integral yields an [[incomplete polylogarithm]] function, which can make its use very cumbersome.
The standard numerical treatment expands the incomplete integral in a geometric series of the exponential
:<math>\int_0^{\lambda_2} \frac{c_1}{\lambda^5[\exp(\frac{c_2}{\lambda T})-1]}d\lambda
=c_1(\frac{T}{c_2})^4\int_{c_2/(\lambda_2 T)}^{\infty}
\frac{x^3}{\exp(x)-1}dx
</math>
after substituting <math>\lambda = c_2/(xT)</math>, <math>d\lambda = -c_2/(x^2T
)dx</math>. Then
:<math>J(c)\equiv \int_c^\infty \frac{x^3}{\exp x-1}dx
=\int_c^\infty \frac{x^3 \exp(-x)}{1-\exp(- x)}dx
=\int_c^\infty \sum_{n\ge 1}x^3 \exp(-nx)dx
</math>
:<math>
=\sum_{n\ge 1} \exp(-nc)\frac{(nc)^3+3(nc)^2+6nc+6}{n^4}
</math>
provides an approximation if the sum is truncated at some order.
 
The Sakuma–Hattori equation shown above was found to provide the best curve-fit for interpolation of scales for radiation thermometers among a number of alternatives investigated.<ref name=Sakuma2>Sakuma F, Kobayashi M., "Interpolation equations of scales of radiation thermometers", ''Proceedings of TEMPMEKO 1996'', pp. 305–310 (1996).</ref>
 
The inverse Sakuma–Hattori function can be used without iterative calculation. This is an addition advantage over integration of Planck's Law.
 
== Other forms ==
 
The 1996 paper investigated 10 different forms. They are listed in the chart below in order of quality of curve-fit to actual radiometric data.<ref name=Sakuma2/>
 
{| class="wikitable"
|-
! Name
! Equation
! Bandwidth
! Planckian
|-
| Sakuma–Hattori Planck III
| <math>S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1}</math>
| narrow
| yes
|-
| Sakuma–Hattori Planck IV
| <math>S(T) = \frac{C}{\exp\left(\frac{A}{T^2} + \frac{B}{2T}\right)-1}</math>
| narrow
| yes
|-
| Sakuma–Hattori – Wien's II
| <math>S(T) = C \exp\left(\frac{-c_2}{AT + B}\right)</math>
| narrow
| no
|-
| Sakuma–Hattori Planck II
| <math>S(T) = \frac{C T^A}{\exp\left(\frac{B}{T}\right)-1}</math>
| broad and narrow
| yes
|-
| Sakuma–Hattori – Wien's I
| <math>S(T) = C T^A {\exp\left(\frac{-B}{T}\right)}</math>
| broad and narrow
| no
|-
| Sakuma–Hattori Planck I
| <math>S(T) = \frac{C}{\exp\left(\frac{c_2}{AT}\right)-1}</math>
| monochromatic
| yes
|-
| New
| <math>S(T) = C \left(1 + \frac{A}{T}\right) - B</math>
| narrow
| no
|-
| Wien's
| <math>S(T) = C \exp\left(\frac{-c_2}{A T}\right)</math>
| monochromatic
| no
|-
| Effective Wavelength – Wien's
| <math>S(T) = C \exp\left(\frac{-A}{T}+\frac{B}{T^2}\right)</math>
| narrow
| no
|-
| Exponent
| <math>S(T) = C T^A</math>
| broad
| no
|}
 
== See also ==
 
*[[Stefan–Boltzmann law]]
*[[Planck's law]]
*[[Rayleigh–Jeans law]]
*[[Wien approximation]]
*[[Wien's displacement law]]
*[[Kirchhoff's law of thermal radiation]]
 
*[[Infrared thermometer]]
*[[Pyrometer]]
*[[Thin filament pyrometry]]
*[[Thermography]]
 
*[[Black body]]
*[[Thermal radiation]]
*[[Radiance]]
*[[Emissivity]]
 
* [[ASTM Subcommittee E20.02 on Radiation Thermometry]]
 
== Notes ==
{{Cnote2|A
| Silver point, the melting point of silver 962°C [(961.961 ± 0.017)°C<ref>
{{Cite journal
  |author=J Tapping and V N Ojha
  | title = Measurement of the Silver Point with a Simple, High-Precision Pyrometer
  | journal = Metrologia
  | volume = 26
  | issue = 2
  | pages = 133
  | year = 1989
  | url = http://iopscience.iop.org/0026-1394/26/2/008
  | doi = 10.1088/0026-1394/26/2/008
  | accessdate = 2010-07-26}}
</ref>
] used as a calibration point in some temperature scales.<ref>
{{Cite web
  | title = Definition of Silver Point - 962°C, the melting point of silver
  | url = http://www.eudict.com/?word=silver+point+melting&lang=engchi
  | accessdate = 2010-07-26}}
</ref>
It is used to calibrate IR thermometers because it is stable and easy to reproduce.
}}
 
== References ==
<references/>
 
{{DEFAULTSORT:Sakuma-Hattori equation}}
[[Category:Statistical mechanics]]
[[Category:Equations]]

Latest revision as of 20:49, 14 May 2014

Self defense and survival are two issues that should by no means be taken without any consideration. You should not wait until the day if you end up mugged by robbers so to be taught the significance of being prepared. It is vitally unlucky that tragedies usually come after we least anticipate. For this reason you need to all the time have the most effective pocket knife with you the place ever you go. It is not essentially for self defense. The best knife can serve you in several occasions, while you unfastened your means in the forest, and also if you end up about to be attacked by wild animals or even by hostile people.



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