1568:
Kubelka–Munk (K–M) theory has limitations. The term "failure of the
Kubelka–Munk theory" has been applied because it does not "remain valid in strongly absorbing materials". There have been many attempts to explain the limitations and amend the K–M equation. In literature related to diffuse-reflection infrared Fourier-transform (DRIFT) spectra, "particularly specular reflection" is often identified as a culprit. In some corners, there is the working assumption that the problem is that the K–M theory is a two-flux theory, and that introducing additional directions will solve the problem. In particular, two continuous theories, "diffusion theory" and the "equation of radiation transfer" (ERT), have their advocates. Some of the advocates of the ERT have called to our attention the failure of the ERT to predict the desired linear absorption coefficient as particle size gets large, and blamed it on the hidden mass effect. In 2003, Donald and Kevin Dahm illustrated the degree to which the continuous theories all suffer from the fundamental limitation of trying to model a discontinuous sample as a continuum and suggested that as long as the effect of this limitation is unexplored, there is little reason to search for other reasons for "failure".
1282:: that of an opaque (infinitely thick) coating, which can be applied to a sample modeled as an infinite number of infinitesimal layers. The two-stream approximation was embraced by the early practitioners. There were far more mathematics to choose from, but the name Kubelka–Munk became widely regarded as synonymous with any technique that modeled diffuse radiation moving through layers of infinitesimal size. This was aided by the popular assumption that the Kubelka–Munk function (above) was analogous to the absorbance function in transmission spectroscopy.
2932:
238:
simplicity and its acceptable prediction accuracy in many industrial applications, the
Kubelka–Munk model remains very popular. However, in almost every application area, the limitations of the model have required improvements. Sometimes these improvements are touted as extensions of Kubelka–Munk theory, sometimes as embracing more general mathematics of which the Kubelka–Munk equation is a special case, and sometimes as an alternate approach.
2182:
our assumptions, half is remitted and half is transmitted. The portion that is transmitted proceeds to the other surface undiminished. There it is again split where half (1/4 of the original incident intensity) is transmitted and half is remitted. The amounts that remitted from the first surface can be totaled as can the amount that are transmitted through the second. The total remission is 2/3 ≈ 0.667. The total transmission is 1/3 ≈ 0.333.
34:, devised by Paul Kubelka and Franz Munk, is a fundamental approach to modelling the appearance of paint films. As published in 1931, the theory addresses "the question of how the color of a substrate is changed by the application of a coat of paint of specified composition and thickness, and especially the thickness of paint needed to obscure the substrate". The mathematical relationship involves just two paint-dependent constants.
4683:
3611:
2357:
655:
pulp and paper industry. If the optical properties (e.g., reflectance and opacity) of each pulp, filler, and dye used in paper-making are known, then the optical properties of a paper made with any combination of the materials can be predicted. If the contrast ratio and reflectivity of a paper are known, the changes in these properties with a change in basis weight can be predicted.
4142:
3097:
1246:
2927:{\displaystyle {\begin{aligned}T_{xy}&={\frac {T_{x}T_{y}}{1-R_{(-x)}R_{y}}}={\frac {(1/2)(1/2)}{1-(1/2)(1/2)}}={\frac {1/4}{3/4}}=1/3,\\R_{xy}&=R_{x}+{\frac {T_{x}^{2}R_{y}}{1-R_{(-x)}R_{y}}}=1/2+{\frac {(1/2)(1/2)(1/2)}{1-(1/2)(1/2)}}=2/3,\\A_{xy}&=1-T_{xy}-R_{xy}=1-1/2-1/2=0.\end{aligned}}}
2181:
The sketch shows two surfaces bounding a slab of a non-absorbing medium. Notice that the assembly would appear identical regardless of which side was being entered. Apart from the surfaces, the medium has no spectroscopic properties. A beam of light of unit intensity reaches the front surface, and by
801:
Early practitioners, especially D. R. Duncan, assumed that in a mixture of pigments, the colors produced in any given medium may be deduced from formulae involving two constants for each pigment. These constants, which vary with the wavelength of the incident light, measure respectively the absorbing
49:
from a coating surface is the summation of: 1) the reflectance of the coating surface; 2) the remission from the interior of the coating; and 3) the remission from the surface of the substrate. The intensity considered in the latter two parts is modified by the absorption of the coating material. The
654:
The
Kubelka–Munk theory is also used in the paper industry to predict optical properties of paper, avoiding a labor-intensive trial-and-error approach. The theory is relatively simple in terms of the number of constants involved, works very well for many papers, and is well documented for use by the
2947:
Here we will assume that again the surfaces will remit and transmit 1/2 the amount striking it, this time we will assume that half of the intensity will be absorbed a trip across the slab. A beam of light of unit intensity reaches the front surface, and by our assumptions, half is remitted and half
658:
While the
Kubelka–Munk coefficients are assumed to be linear and independent quantities, the relationship fails in regions of strong absorption, such as in the case of dyed paper. Several theories were proposed to explain the non-linear behavior of the coefficients, attributing the non-linearity to
645:
In the original article, there is a solution for remission from a coating of finite thickness. Kubelka derived many additional formulas for a variety of other cases, which were published in the post-war years. Whereas the 1931 theory assumed that light flows in one dimension (two fluxes, upward and
237:
While numerous early authors had developed similar two-constant equations, the mathematics of most of these was found to be consistent with the
Kubelka–Munk treatment. Others added additional constants to produce more accurate models, but these generally did not find wide acceptance. Due to its
1567:
Continuous models are widely used to model diffuse reflection from particulate samples. They are embodied in various theories, including diffusion theory, the equation of radiation transfer, as well as
Kubelka–Munk. In spite of its widespread use, there has long been an understanding that the
1571:
Spectroscopists have a desire to determine the same absorption coefficient quantity from diffuse reflectance measurements as they would from a transmission measurement on a non-scattering sample of the same material. The
Bouguer–Lambert law describes the attenuation of transmitted light as
4678:{\displaystyle {\begin{aligned}T_{xy}&={\frac {T_{x}T_{y}}{1-R_{x}R_{y}}}={\frac {(1/2)(1/4)}{1-(1/8)(1/2)}}=2/15,\\R_{xy}&=R_{x}+{\frac {T_{x}^{2}R_{y}}{1-R_{(-x)}R_{y}}}=1/2+{\frac {(1/4)(1/4)(1/2)}{1-(1/8)(1/2)}}=8/15,\\A_{xy}&=1-T_{xy}-R_{xy}=1-2/15-8/15=1/3.\end{aligned}}}
1572:
exponential falloff in intensity of a direct beam of light as it passes through a medium. The cause of the attenuation may be absorption or scatter. A coefficient unaffected by scatter is desired by absorption spectroscopists. Mathematically, the
Bouguer–Lambert law may be expressed as
802:
power of the pigment for light and its scattering power. The work of
Kubelka and Munk was seen as yielding a useful systematic approach to color mixing and matching. By resolving the Kubelka–Munk equation for the ratio of absorption to scatter, one can obtain a "remission function":
2165:
A coating layer is not the same as the substrate it covers. As
Kubelka was interested in coatings, he was of course very interested in the handling of what he called "inhomogeneous layers". A set of equations, one of which was believed to apply to the case, had been published by
3606:{\displaystyle {\begin{aligned}T_{xy}&={\frac {T_{x}T_{y}}{1-R_{(-x)}R_{y}}}={\frac {(1/2)(1/2)}{1-(1/2)(0)}}=1/4,\\R_{xy}&=R_{x}+{\frac {T_{x}^{2}R_{y}}{1-R_{(-x)}R_{y}}}=1/2+{\frac {(1/2)(1/2)0}{1-(1/2)0}}=1/2,\\A_{xy}&=1-T_{xy}-R_{xy}=1-1/4-1/2=1/4.\end{aligned}}}
2948:
is transmitted, but this time half of the transmitted light, or 1/4 is absorbed before another 1/4 reaches the second surface, and 1/4 is remitted back across the slab to face half of it being absorbed. The sketch shows that the calculated values from the equations should be
1293:(KBr). This led to a situation analogous to the described in the section just above for pigments, where the analyte had little effect on the scatter, which was dominated by the KBr. In this case, the assumption of the function being linear with concentration was reasonable.
1031:
2966:
Now the sketch has three layers, labeled 1, 2, and 3. Layers 1 and 3 remit and transmit 1/2 and absorb nothing. Layer 2 absorbs half and transmits half, but remits nothing. We can build the assembly by first combining layers 1 and 2, and then combining that result as the
2049:
2170:
in 1946 for the case of two light streams through plane parallel layers. However, it did not handle it successfully. Kubelka solved the problem, and we illustrate the solution here. First, a case to which the equation of Benford may be straightforwardly applied.
1300:, the samples are generally measured in their natural (often particulate) state, and deviations from linearity at higher absorption levels were routinely observed. The remission function (also called the Kubelka–Munk function) was almost abandoned in favor of "log(1/
663:, the coefficients were shown to be dependent quantities related to the real and imaginary part of the refractive index. By accounting for this dependency, the anomalous behavior of the Kubelka–Munk coefficients in regions of strong absorption were fully explained.
2941:
2175:
232:
3907:
919:
4827:
1551:
3615:
Kubelka has shown by theory and experiment that remittance and absorption of a non-homogeneous specimen depend on the direction of illumination, whereas transmittance does not. Consequently, for non-homogeneous layers, the remission
4720:, and then producing the output in RGB. There are additional concerns for dealing with wider gamuts and improving speed. This RGB adaptation makes it easier for digital painting software to integrate the more realistic K–M method.
5677:
Burger, T.; Kuhn, J.; Caps, R.; Fricke, J. (1997-03-01). "Quantitative Determination of the Scattering and Absorption Coefficients from Diffuse Reflectance and Transmittance Measurements: Application to Pharmaceutical Powders".
2936:
Next we will examine the case where the medium is absorbing one. While the total assembly would behave the same in either direction, in order to apply the mathematics, we will need to use an intermediate step where it does not.
1854:, we know that the ART function is constant for all sample thicknesses of the same material. This would include the infinitesimal layer used in the Kubelka–Munk differentiation. Consequently we may equate numerous functions:
1727:
1857:
646:
downward within the layer), in 1948 Kubelka derived the same equations (up to a factor of 2) assuming spherical scatter within the paint layer. Later he generalized the theory to inhomogeneous layers (see below).
80:
as a remission fraction. The authors noted that the remission from an infinite number of these infinitesimal layers is "solely a function of the ratio of the absorption and back-scatter (remission) constants
53:
While Kubelka entered this field through an interest in coatings, his work has influenced workers in other areas as well. In the original article, there is a special case of interest to many fields is "the
3690:
50:
concept is based on the simplified picture of two diffuse light fluxes moving through semi-infinite plane-parallel layers, with one flux proceeding "downward", and the other simultaneously "upward".
3093:
4147:
3102:
2362:
805:
1241:{\displaystyle {\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {a_{0}}{r_{0}}}={\frac {K}{S}}={\frac {\Sigma (c_{i}K_{i})}{\Sigma (c_{i}S_{i})}}\approx {\frac {\Sigma (c_{i}K_{i})}{S}}.}
103:
720:
2350:
97:, but not in any way on the absolute numerical values of these constants". (The equation is presented in the same mathematical form as in the article, but with symbolism modified.)
607:
472:
5837:
250:(an ability to hide the surface of an object). The hiding power of a coating measures its ability to obscure a background of contrasting color. Hiding power is also known as
640:
510:
379:
3652:
from the first layer that occurs in the denominator is the remission when illuminated from the reverse (not the forward) direction, so we will need to know the value for
4073:
4032:
3991:
3950:
1787:
4109:
3650:
1845:
1818:
767:
2224:
543:
3677:
2284:
2254:
2077:
in the ART function, but do not correctly predict the fractions of incident light that are transmitted directly. From this, it can be deduced that the coefficients
1575:
1364:
1337:
1026:
4739:
346:
4137:
1416:
4867:
4847:
2155:
2135:
2115:
2095:
2075:
1767:
1747:
1404:
1384:
1268:
999:
961:
941:
791:
403:
306:
5598:
Pracht, G.; Weckler, B.; Lutz, H. D. (2003-10-01). "Diffuse Reflection Infrared Fourier Transform (DRIFT) Spectra and High-Temperature DRIFT Spectra of β—Ni(IO
5209:
5328:"Color in business, science, and industry. By Deane B. Judd. John Wiley & Sons, Inc., New York, 1952. 401 pp. Illustrated. 15.5 × 24 cm. Price $ 6.50".
5515:
Alcaraz de la Osa, R.; Iparragirre, I.; Ortiz, D.; Saiz, J. M. (March 2020). "The extended Kubelka–Munk theory and its application to spectroscopy".
2981:
246:
In the original article, there are several special cases important to paints that are addressed, along with a mathematical definition of
2167:
17:
1270:
is dominated by the base material and is assumed to be constant. In such a case, the equation is linear in concentration of pigment.
59:
2044:{\displaystyle 2F(R)=A(R,T)={\frac {(2-a-2r)a}{r}}\approx 2{\frac {a_{0}}{r_{0}}}=2{\frac {a_{0}/dx}{r_{0}/dx}}=2{\frac {K}{S}}.}
1851:
1305:
5184:
4967:
46:
5215:
2053:
Using a simple system (albeit rather complex mathematics), it can be shown that continuous models correctly predict the
659:
the non-isotropic structure of paper at both the micro- and macroscopic levels. However, using an analysis based on the
733:
axis. There is a simpler method adapted from the Kubelka–Munk theory, in which the band gap is calculated by plotting,
5200:
Koukoulas, A. A.; Jordan, B. D. (May 1997). "Effect of strong absorption on the Kubelka-Munk scattering coefficient".
227:{\displaystyle R_{\infty }=1+{\frac {a_{0}}{r_{0}}}-{\sqrt {{\frac {a_{0}^{2}}{r_{0}^{2}}}+2{\frac {a_{0}}{r_{0}}}}}.}
5560:
5405:
981:
in the Kubelka–Munk equation above. Then assuming separate additivity of the absorption and coefficients for each of
5721:
Dahm, Donald J.; Dahm, Kevin D. (2003-12-01). "Illustration of Failure of Continuum Models of Diffuse Reflectance".
3902:{\displaystyle R_{yx}=R_{y}+{\frac {T_{y}^{2}R_{x}}{1-R_{(-y)}R_{x}}}=0.0+{\frac {(1/2)(1/2)(1/2)}{1-0(1/2)}}=1/8.}
1279:
62:, which describes the remission from a sample composed of an infinite number of infinitesimal layers, each having
5231:"How To Correctly Determine the Band Gap Energy of Modified Semiconductor Photocatalysts Based on UV–Vis Spectra"
1559:
In other areas of spectroscopy, there are shifts away from the strict use of the Kubelka–Munk treatment as well.
682:
5103:"New Contributions to the Optics of Intensely Light-Scattering Materials. Part II: Nonhomogeneous Layers"
2289:
4696:
by Sochorová and Jamriška in 2021. Their "Mixbox" approach works by converting the inputs into a version of
660:
548:
4705:
408:
261:
is the fraction of incident light that is remitted (reflected) by a coated substrate under consideration,
914:{\displaystyle F(R_{\infty })\equiv {\frac {(1-R_{\infty })^{2}}{2R_{\infty }}}={\frac {a_{0}}{r_{0}}}.}
5584:
5472:
Dahm, Kevin (2013). "Separating the Effects of Scatter and Absorption Using the Representative Layer".
4925:
1297:
5555:. Donald A. Burns, Emil W. Ciurczak (2nd ed., rev. and expanded ed.). New York: M. Dekker. 2001.
5068:
5791:
Sochorová, Šárka; Jamriška, Ondřej (December 2021). "Practical pigment mixing for digital painting".
1308:, was developed, along with a scheme to separate the effects of scatter from absorption in the log(1/
963:
as absorption and back-scattering coefficients, which replace the absorption and remission fractions
5272:"Sımple Method For The Determınatıon of Band Gap of a Nanopowdered Sample Usıng Kubelka Munk Theory"
5102:
5021:
4984:
612:
482:
351:
38:
1790:
729:. Then the band-gap energy can be obtained by extending the straight segment of the graph to the
4037:
3996:
3955:
3914:
2185:
Alternatively, we can use the equation of Benford that applies. For two plane parallel layers,
1772:
5069:"Errata: New Contributions to the Optics of Intensely Light-Scattering Materials. Part I"
4078:
3619:
1847:
may be used to represent the absorption and scattering parts of the extinction coefficients.)
1823:
1796:
736:
4822:{\displaystyle H_{\infty }=1+{\frac {s}{r}}-{\sqrt {{\frac {s^{2}}{r^{2}}}+2{\frac {s}{r}}}}}
2196:
1286:
515:
3655:
2259:
2229:
1546:{\displaystyle A(R,T)\equiv {\frac {(1-R_{n})^{2}-T_{n}^{2}}{R_{n}}}={\frac {(2-a-2r)a}{r}}}
5687:
5635:
5481:
5438:
5368:
5302:
5033:
4900:
1342:
1315:
1004:
4716:) plus a residue (to account for the gamut difference), performing the K–M mixing in that
322:
8:
4114:
5691:
5639:
5485:
5442:
5372:
5306:
5037:
5808:
5738:
5703:
5659:
5578:
5532:
5497:
5454:
5176:
4852:
4832:
2140:
2120:
2100:
2080:
2060:
1752:
1732:
1389:
1369:
1253:
984:
946:
926:
776:
388:
291:
5022:"New Contributions to the Optics of Intensely Light-Scattering Materials. Part I"
5832:
5812:
5773:
5651:
5566:
5556:
5536:
5411:
5401:
5314:
5271:
5252:
5180:
5049:
4963:
1290:
308:
is the fraction of incident light transmitted by the sample under consideration, and
5742:
5707:
5663:
5501:
5458:
4959:
5800:
5765:
5730:
5695:
5643:
5524:
5489:
5446:
5376:
5337:
5310:
5242:
5172:
5147:
5114:
5080:
5041:
4955:
2286:
for the two layers can be calculated from the properties of the individual layers (
4693:
251:
42:
5247:
5230:
5152:
5135:
5647:
4713:
1722:{\displaystyle \mathrm {T} (direct)=\exp(-kd)\exp(-sd)=\exp=\exp(-\epsilon d),}
5528:
5826:
5699:
5450:
5415:
723:
37:
In their article, "fundamental differential equations" are developed using a
5804:
5570:
5429:
Dahm, Donald (1999). "Representative Layer Theory for Diffuse Reflectance".
1289:, it was common to prepare solid samples by finely grinding the sample with
5777:
5769:
5655:
5341:
5256:
5118:
5084:
5053:
5045:
4717:
4709:
4701:
247:
5550:
5493:
319:
An ideal white paint reflects all incident light, and absorbs none, or
5734:
1562:
41:
for light diffusing through a coating whose absorption and remission (
5514:
676:
5229:
Makuła, Patrycja; Pacia, Michał; Macyk, Wojciech (6 December 2018).
4692:
The K–M paint-mixing algorithm has been adapted to directly use the
5381:
5356:
5330:
Journal of the American Pharmaceutical Association (Scientific Ed.)
1366:
are the measured remission by and transmission through a sample of
672:
5756:
Benford, Frank (1946-09-01). "Radiation in a Diffusing Medium".
1386:
layers, each layer having absorption and remission fractions of
55:
2940:
5276:
Journal of the Nigerian Association of Mathematical Physics
4697:
2174:
5398:
Reflectance spectroscopy Principles, methods, applications
675:
energy of semiconductors is frequently determined from a
5293:
Duncan, D. R. (1940). "The colour of pigment mixtures".
4950:
Brill, Michael H.; Vik, Michal (2014). "Kubelka, Paul".
5167:
Bajpai, Pratima (2018). "Optical Properties of Paper".
58:
of an infinitely thick coating". This case yielded the
5838:
Scattering, absorption and radiative transfer (optics)
3088:{\displaystyle R_{x}=1/2,T_{x}=1/2,R_{y}=0,T_{y}=1/2,}
1250:
For the case of small amount of pigments, the scatter
4855:
4835:
4742:
4145:
4117:
4081:
4040:
3999:
3958:
3917:
3693:
3658:
3622:
3100:
2984:
2360:
2292:
2262:
2232:
2199:
2143:
2123:
2103:
2083:
2063:
1860:
1826:
1799:
1775:
1755:
1735:
1578:
1419:
1392:
1372:
1345:
1318:
1256:
1034:
1007:
987:
949:
929:
808:
779:
739:
685:
615:
551:
518:
485:
411:
391:
354:
325:
294:
106:
5676:
2160:
288:
is remission fraction of an infinitely thick layer,
270:
is the remission fraction from the substrate alone,
4926:"Kubelka–Munk Theory – an overview"
1563:
Failure of continuous models of diffuse reflectance
4861:
4841:
4821:
4677:
4131:
4103:
4067:
4026:
3985:
3944:
3901:
3671:
3644:
3605:
3087:
2926:
2344:
2278:
2248:
2218:
2149:
2129:
2109:
2089:
2069:
2043:
1839:
1812:
1781:
1761:
1741:
1721:
1545:
1398:
1378:
1358:
1331:
1262:
1240:
1020:
993:
955:
935:
913:
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634:
601:
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504:
466:
397:
373:
340:
300:
226:
5790:
5597:
5824:
5228:
2193:, having different properties, the transmission
1278:One special case has received much attention in
5199:
3679:for the next step, that is when the layer
5004:
4952:Encyclopedia of Color Science and Technology
4870:
479:A coating of an ideal glaze emits no light (
279:is the remission fraction from the coating,
5626:, and Zn(OH)F: Novel Embedding Materials".
4898:
4849:is related to absorption, not scatter. The
649:
27:Modelling the appearance of paint coatings
5380:
5269:
5246:
5235:The Journal of Physical Chemistry Letters
5151:
715:{\displaystyle {\sqrt {F(R_{\infty })E}}}
5720:
5354:
5348:
5005:Wendlandt, Wesley; Hecht, Harry (1966).
4949:
1410:. Note that the so-called ART function
1304:)". A more general equation, called the
5755:
5508:
5389:
5100:
5066:
5019:
4985:"Autobiographical Sketch: Paul Kubelka"
2971:value in combing with layer 3 (as
2345:{\displaystyle T_{x},R_{x},T_{y},R_{y}}
14:
5825:
5422:
5395:
5357:"Radiation through a foggy atmosphere"
5292:
5166:
5130:
5128:
4982:
4901:"An article on optics of paint layers"
1749:is the linear absorption coefficient,
1556:is constant for any sample thickness.
381:For this case, the remission fraction
5723:Journal of Near Infrared Spectroscopy
5474:Journal of Near Infrared Spectroscopy
5169:Biermann's Handbook of Pulp and Paper
4918:
1769:is the back-scatter coefficient, and
602:{\displaystyle R=R_{g}\exp(-2a_{0}X)}
5471:
5428:
5096:
5094:
5000:
4998:
4996:
4994:
4894:
4892:
467:{\displaystyle R=r_{0}X/(r_{0}X+1).}
45:) coefficients are known. The total
5295:Proceedings of the Physical Society
5125:
4899:Kubelka, Paul; Munk, Franz (1931).
24:
5552:Handbook of near-infrared analysis
5465:
5286:
5177:10.1016/B978-0-12-814238-7.00011-8
5160:
4748:
4687:
2939:
2173:
1580:
1200:
1162:
1131:
1077:
1052:
873:
848:
820:
699:
621:
360:
112:
25:
5849:
5202:Journal of Pulp and Paper Science
5091:
4991:
4889:
2161:Treatment of inhomogeneous layers
666:
609:. For an infinitely thick glaze,
3683:is layer 2, and layer
2352:) from the following equations:
1280:diffuse reflectance spectroscopy
385:for a layer of finite thickness
5784:
5749:
5714:
5670:
5591:
5543:
5321:
5263:
5222:
5193:
4960:10.1007/978-3-642-27851-8_300-1
1789:is their sum, often called the
1273:
793:is the absorption coefficient.
241:
5136:"Paper's appearance: A review"
5060:
5013:
4976:
4943:
4736:The original symbolism was:
4730:
4541:
4527:
4524:
4510:
4499:
4485:
4482:
4468:
4465:
4451:
4413:
4404:
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4261:
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2739:
2725:
2722:
2708:
2670:
2661:
2527:
2513:
2510:
2496:
2485:
2471:
2468:
2454:
2430:
2421:
1924:
1903:
1894:
1882:
1873:
1867:
1713:
1701:
1689:
1686:
1680:
1668:
1662:
1650:
1638:
1629:
1617:
1605:
1584:
1531:
1510:
1464:
1444:
1435:
1423:
1226:
1203:
1188:
1165:
1157:
1134:
1058:
1038:
854:
834:
825:
812:
750:
740:
704:
691:
635:{\displaystyle R_{\infty }=0.}
596:
574:
505:{\displaystyle R_{\text{c}}=0}
458:
436:
374:{\displaystyle R_{\infty }=1.}
71:as an absorption fraction and
13:
1:
4882:
3911:The next step is then to set
5793:ACM Transactions on Graphics
5101:Kubelka, Paul (1954-04-01).
5067:Kubelka, Paul (1948-12-01).
5020:Kubelka, Paul (1948-05-01).
2157:in the Bouguer–Lambert law.
2117:and not proportional to the
1001:components of concentration
7:
5248:10.1021/acs.jpclett.8b02892
5153:10.15376/biores.3.2.627-665
2256:, and absorption fractions
10:
5854:
5648:10.1366/000370203769699126
5336:(12): 757. December 1953.
5315:10.1088/0959-5309/52/3/310
1298:near-infrared spectroscopy
312:is the coating thickness.
18:Kubelka-Munk approximation
5529:10.1007/s40828-019-0097-0
5355:Schuster, Aurhur (1905).
5270:Abdullahi, Sabiu (2016).
5009:. New York: Interscience.
4068:{\displaystyle T_{y}=1/2}
4027:{\displaystyle R_{y}=1/2}
3986:{\displaystyle T_{x}=1/4}
3945:{\displaystyle R_{x}=1/2}
1782:{\displaystyle \epsilon }
1312:) data. In the equation,
1296:However, in the field of
796:
512:) and absorbs a fraction
5700:10.1366/0003702971940404
5451:10.1366/0003702991947298
5146:(2): 627–665. May 2008.
5007:Reflectance Spectroscopy
4983:Westin, Stephen (2011).
4723:
4104:{\displaystyle R_{(-x)}}
3645:{\displaystyle R_{(-x)}}
1840:{\displaystyle \mu _{s}}
1813:{\displaystyle \mu _{a}}
762:{\displaystyle (aE)^{2}}
661:Kramers–Kronig relations
650:Paper and paper coatings
39:two-stream approximation
5805:10.1145/3478513.3480549
5396:Kortüm, Gustav (1969).
4869:is for the German word
2219:{\displaystyle T_{xy},}
538:{\displaystyle a_{0}X.}
5770:10.1364/JOSA.36.000524
5583:: CS1 maint: others (
5342:10.1002/jps.3030421221
5119:10.1364/JOSA.44.000330
5085:10.1364/JOSA.38.001067
5046:10.1364/JOSA.38.000448
4871:
4863:
4843:
4823:
4679:
4133:
4111:in the denominator as
4105:
4069:
4028:
3987:
3946:
3903:
3673:
3672:{\displaystyle R_{21}}
3646:
3607:
3089:
2944:
2928:
2346:
2280:
2279:{\displaystyle A_{xy}}
2250:
2249:{\displaystyle R_{xy}}
2220:
2178:
2151:
2131:
2111:
2091:
2071:
2045:
1850:Through work with the
1841:
1814:
1791:extinction coefficient
1783:
1763:
1743:
1723:
1547:
1400:
1380:
1360:
1333:
1264:
1242:
1022:
995:
957:
937:
915:
787:
763:
716:
636:
603:
539:
506:
468:
399:
375:
342:
302:
228:
5361:Astrophysical Journal
4930:www.sciencedirect.com
4864:
4844:
4824:
4680:
4134:
4106:
4070:
4029:
3988:
3947:
3904:
3674:
3647:
3608:
3090:
2943:
2929:
2347:
2281:
2251:
2221:
2177:
2152:
2132:
2112:
2092:
2072:
2046:
1842:
1815:
1784:
1764:
1744:
1724:
1548:
1401:
1381:
1361:
1359:{\displaystyle T_{n}}
1334:
1332:{\displaystyle R_{n}}
1287:infrared spectroscopy
1265:
1243:
1023:
1021:{\displaystyle c_{i}}
996:
958:
938:
916:
788:
764:
717:
679:, where the quantity
637:
604:
540:
507:
469:
400:
376:
343:
303:
229:
60:Kubelka–Munk equation
5680:Applied Spectroscopy
5628:Applied Spectroscopy
5431:Applied Spectroscopy
5400:. Berlin: Springer.
5171:. pp. 237–271.
4853:
4833:
4829:, where confusingly
4740:
4706:quinacridone magenta
4143:
4115:
4079:
4038:
3997:
3956:
3915:
3691:
3656:
3620:
3098:
2982:
2358:
2290:
2260:
2230:
2197:
2141:
2121:
2101:
2081:
2061:
1858:
1824:
1797:
1773:
1753:
1733:
1576:
1417:
1390:
1370:
1343:
1316:
1254:
1032:
1005:
985:
947:
927:
806:
777:
737:
683:
613:
549:
516:
483:
409:
389:
352:
341:{\displaystyle a=0,}
323:
292:
104:
5692:1997ApSpe..51..309B
5640:2003ApSpe..57.1254P
5486:2013JNIS...21..351D
5443:1999ApSpe..53..647D
5373:1905ApJ....21....1S
5307:1940PPS....52..390D
5038:1948JOSA...38..448K
4380:
4132:{\displaystyle 1/8}
3740:
3336:
2637:
1490:
722:is plotted against
254:or covering power.
186:
171:
32:Kubelka-Munk theory
5494:10.1255/jnirs.1062
4859:
4839:
4819:
4675:
4673:
4366:
4129:
4101:
4065:
4024:
3983:
3942:
3899:
3726:
3669:
3642:
3603:
3601:
3322:
3085:
2945:
2924:
2922:
2623:
2342:
2276:
2246:
2216:
2179:
2147:
2127:
2107:
2087:
2067:
2041:
1837:
1810:
1779:
1759:
1739:
1719:
1543:
1476:
1396:
1376:
1356:
1329:
1260:
1238:
1018:
991:
953:
933:
911:
783:
759:
712:
632:
599:
535:
502:
464:
395:
371:
338:
298:
257:In the following,
224:
172:
157:
5735:10.1255/jnirs.398
5634:(10): 1254–1259.
5241:(23): 6814–6817.
5186:978-0-12-814238-7
4969:978-3-642-27851-8
4862:{\displaystyle H}
4842:{\displaystyle s}
4817:
4815:
4799:
4770:
4545:
4429:
4307:
4222:
3883:
3789:
3687:is layer 1:
3473:
3385:
3263:
3186:
2802:
2686:
2564:
2531:
2446:
2150:{\displaystyle s}
2130:{\displaystyle k}
2110:{\displaystyle S}
2090:{\displaystyle K}
2070:{\displaystyle T}
2036:
2020:
1964:
1934:
1762:{\displaystyle s}
1742:{\displaystyle k}
1541:
1502:
1399:{\displaystyle a}
1379:{\displaystyle n}
1291:potassium bromide
1263:{\displaystyle S}
1233:
1192:
1123:
1110:
1083:
994:{\displaystyle i}
956:{\displaystyle S}
936:{\displaystyle K}
906:
879:
786:{\displaystyle a}
710:
493:
398:{\displaystyle X}
316:Ideal white paint
301:{\displaystyle T}
219:
217:
187:
148:
16:(Redirected from
5845:
5817:
5816:
5788:
5782:
5781:
5753:
5747:
5746:
5718:
5712:
5711:
5674:
5668:
5667:
5595:
5589:
5588:
5582:
5574:
5547:
5541:
5540:
5512:
5506:
5505:
5469:
5463:
5462:
5426:
5420:
5419:
5393:
5387:
5386:
5384:
5352:
5346:
5345:
5325:
5319:
5318:
5290:
5284:
5283:
5267:
5261:
5260:
5250:
5226:
5220:
5219:
5214:
5197:
5191:
5190:
5164:
5158:
5157:
5155:
5132:
5123:
5122:
5098:
5089:
5088:
5064:
5058:
5057:
5017:
5011:
5010:
5002:
4989:
4988:
4980:
4974:
4973:
4954:. pp. 1–3.
4947:
4941:
4940:
4938:
4936:
4922:
4916:
4915:
4905:
4896:
4876:
4874:
4868:
4866:
4865:
4860:
4848:
4846:
4845:
4840:
4828:
4826:
4825:
4820:
4818:
4816:
4808:
4800:
4798:
4797:
4788:
4787:
4778:
4776:
4771:
4763:
4752:
4751:
4734:
4684:
4682:
4681:
4676:
4674:
4667:
4653:
4639:
4622:
4621:
4606:
4605:
4580:
4579:
4557:
4546:
4544:
4537:
4520:
4502:
4495:
4478:
4461:
4449:
4441:
4430:
4428:
4427:
4426:
4417:
4416:
4391:
4390:
4389:
4379:
4374:
4364:
4359:
4358:
4342:
4341:
4319:
4308:
4306:
4299:
4282:
4264:
4257:
4240:
4228:
4223:
4221:
4220:
4219:
4210:
4209:
4193:
4192:
4191:
4182:
4181:
4171:
4162:
4161:
4138:
4136:
4135:
4130:
4125:
4110:
4108:
4107:
4102:
4100:
4099:
4074:
4072:
4071:
4066:
4061:
4050:
4049:
4033:
4031:
4030:
4025:
4020:
4009:
4008:
3992:
3990:
3989:
3984:
3979:
3968:
3967:
3951:
3949:
3948:
3943:
3938:
3927:
3926:
3908:
3906:
3905:
3900:
3895:
3884:
3882:
3875:
3854:
3847:
3830:
3813:
3801:
3790:
3788:
3787:
3786:
3777:
3776:
3751:
3750:
3749:
3739:
3734:
3724:
3719:
3718:
3706:
3705:
3678:
3676:
3675:
3670:
3668:
3667:
3651:
3649:
3648:
3643:
3641:
3640:
3612:
3610:
3609:
3604:
3602:
3595:
3581:
3567:
3550:
3549:
3534:
3533:
3508:
3507:
3485:
3474:
3472:
3462:
3444:
3434:
3417:
3405:
3397:
3386:
3384:
3383:
3382:
3373:
3372:
3347:
3346:
3345:
3335:
3330:
3320:
3315:
3314:
3298:
3297:
3275:
3264:
3262:
3246:
3228:
3221:
3204:
3192:
3187:
3185:
3184:
3183:
3174:
3173:
3148:
3147:
3146:
3137:
3136:
3126:
3117:
3116:
3094:
3092:
3091:
3086:
3078:
3067:
3066:
3048:
3047:
3032:
3021:
3020:
3005:
2994:
2993:
2962:
2957:= 2/15 ≈ 0.133,
2953:= 8/15 ≈ 0.533,
2933:
2931:
2930:
2925:
2923:
2910:
2896:
2879:
2878:
2863:
2862:
2837:
2836:
2814:
2803:
2801:
2794:
2777:
2759:
2752:
2735:
2718:
2706:
2698:
2687:
2685:
2684:
2683:
2674:
2673:
2648:
2647:
2646:
2636:
2631:
2621:
2616:
2615:
2599:
2598:
2576:
2565:
2563:
2559:
2550:
2546:
2537:
2532:
2530:
2523:
2506:
2488:
2481:
2464:
2452:
2447:
2445:
2444:
2443:
2434:
2433:
2408:
2407:
2406:
2397:
2396:
2386:
2377:
2376:
2351:
2349:
2348:
2343:
2341:
2340:
2328:
2327:
2315:
2314:
2302:
2301:
2285:
2283:
2282:
2277:
2275:
2274:
2255:
2253:
2252:
2247:
2245:
2244:
2225:
2223:
2222:
2217:
2212:
2211:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2116:
2114:
2113:
2108:
2096:
2094:
2093:
2088:
2076:
2074:
2073:
2068:
2056:
2050:
2048:
2047:
2042:
2037:
2029:
2021:
2019:
2012:
2007:
2006:
1996:
1989:
1984:
1983:
1973:
1965:
1963:
1962:
1953:
1952:
1943:
1935:
1930:
1901:
1846:
1844:
1843:
1838:
1836:
1835:
1819:
1817:
1816:
1811:
1809:
1808:
1788:
1786:
1785:
1780:
1768:
1766:
1765:
1760:
1748:
1746:
1745:
1740:
1728:
1726:
1725:
1720:
1583:
1552:
1550:
1549:
1544:
1542:
1537:
1508:
1503:
1501:
1500:
1491:
1489:
1484:
1472:
1471:
1462:
1461:
1442:
1409:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1365:
1363:
1362:
1357:
1355:
1354:
1338:
1336:
1335:
1330:
1328:
1327:
1285:In the field of
1269:
1267:
1266:
1261:
1247:
1245:
1244:
1239:
1234:
1229:
1225:
1224:
1215:
1214:
1198:
1193:
1191:
1187:
1186:
1177:
1176:
1160:
1156:
1155:
1146:
1145:
1129:
1124:
1116:
1111:
1109:
1108:
1099:
1098:
1089:
1084:
1082:
1081:
1080:
1067:
1066:
1065:
1056:
1055:
1036:
1027:
1025:
1024:
1019:
1017:
1016:
1000:
998:
997:
992:
980:
971:
962:
960:
959:
954:
942:
940:
939:
934:
920:
918:
917:
912:
907:
905:
904:
895:
894:
885:
880:
878:
877:
876:
863:
862:
861:
852:
851:
832:
824:
823:
792:
790:
789:
784:
772:
768:
766:
765:
760:
758:
757:
732:
728:
721:
719:
718:
713:
711:
703:
702:
687:
641:
639:
638:
633:
625:
624:
608:
606:
605:
600:
592:
591:
567:
566:
544:
542:
541:
536:
528:
527:
511:
509:
508:
503:
495:
494:
491:
473:
471:
470:
465:
448:
447:
435:
427:
426:
404:
402:
401:
396:
384:
380:
378:
377:
372:
364:
363:
347:
345:
344:
339:
307:
305:
304:
299:
287:
278:
269:
260:
233:
231:
230:
225:
220:
218:
216:
215:
206:
205:
196:
188:
185:
180:
170:
165:
156:
154:
149:
147:
146:
137:
136:
127:
116:
115:
96:
79:
70:
21:
5853:
5852:
5848:
5847:
5846:
5844:
5843:
5842:
5823:
5822:
5821:
5820:
5789:
5785:
5754:
5750:
5719:
5715:
5675:
5671:
5625:
5621:
5617:
5613:
5609:
5605:
5601:
5596:
5592:
5576:
5575:
5563:
5549:
5548:
5544:
5513:
5509:
5470:
5466:
5427:
5423:
5408:
5394:
5390:
5353:
5349:
5327:
5326:
5322:
5291:
5287:
5268:
5264:
5227:
5223:
5212:
5198:
5194:
5187:
5165:
5161:
5134:
5133:
5126:
5099:
5092:
5065:
5061:
5018:
5014:
5003:
4992:
4981:
4977:
4970:
4948:
4944:
4934:
4932:
4924:
4923:
4919:
4903:
4897:
4890:
4885:
4880:
4879:
4875:, "brightness".
4854:
4851:
4850:
4834:
4831:
4830:
4807:
4793:
4789:
4783:
4779:
4777:
4775:
4762:
4747:
4743:
4741:
4738:
4737:
4735:
4731:
4726:
4694:RGB color model
4690:
4688:In computer art
4672:
4671:
4663:
4649:
4635:
4614:
4610:
4598:
4594:
4581:
4572:
4568:
4565:
4564:
4553:
4533:
4516:
4503:
4491:
4474:
4457:
4450:
4448:
4437:
4422:
4418:
4403:
4399:
4392:
4385:
4381:
4375:
4370:
4365:
4363:
4354:
4350:
4343:
4334:
4330:
4327:
4326:
4315:
4295:
4278:
4265:
4253:
4236:
4229:
4227:
4215:
4211:
4205:
4201:
4194:
4187:
4183:
4177:
4173:
4172:
4170:
4163:
4154:
4150:
4146:
4144:
4141:
4140:
4121:
4116:
4113:
4112:
4086:
4082:
4080:
4077:
4076:
4057:
4045:
4041:
4039:
4036:
4035:
4016:
4004:
4000:
3998:
3995:
3994:
3975:
3963:
3959:
3957:
3954:
3953:
3934:
3922:
3918:
3916:
3913:
3912:
3891:
3871:
3855:
3843:
3826:
3809:
3802:
3800:
3782:
3778:
3763:
3759:
3752:
3745:
3741:
3735:
3730:
3725:
3723:
3714:
3710:
3698:
3694:
3692:
3689:
3688:
3663:
3659:
3657:
3654:
3653:
3627:
3623:
3621:
3618:
3617:
3600:
3599:
3591:
3577:
3563:
3542:
3538:
3526:
3522:
3509:
3500:
3496:
3493:
3492:
3481:
3458:
3445:
3430:
3413:
3406:
3404:
3393:
3378:
3374:
3359:
3355:
3348:
3341:
3337:
3331:
3326:
3321:
3319:
3310:
3306:
3299:
3290:
3286:
3283:
3282:
3271:
3242:
3229:
3217:
3200:
3193:
3191:
3179:
3175:
3160:
3156:
3149:
3142:
3138:
3132:
3128:
3127:
3125:
3118:
3109:
3105:
3101:
3099:
3096:
3095:
3074:
3062:
3058:
3043:
3039:
3028:
3016:
3012:
3001:
2989:
2985:
2983:
2980:
2979:
2978:So for step 1:
2949:
2921:
2920:
2906:
2892:
2871:
2867:
2855:
2851:
2838:
2829:
2825:
2822:
2821:
2810:
2790:
2773:
2760:
2748:
2731:
2714:
2707:
2705:
2694:
2679:
2675:
2660:
2656:
2649:
2642:
2638:
2632:
2627:
2622:
2620:
2611:
2607:
2600:
2591:
2587:
2584:
2583:
2572:
2555:
2551:
2542:
2538:
2536:
2519:
2502:
2489:
2477:
2460:
2453:
2451:
2439:
2435:
2420:
2416:
2409:
2402:
2398:
2392:
2388:
2387:
2385:
2378:
2369:
2365:
2361:
2359:
2356:
2355:
2336:
2332:
2323:
2319:
2310:
2306:
2297:
2293:
2291:
2288:
2287:
2267:
2263:
2261:
2258:
2257:
2237:
2233:
2231:
2228:
2227:
2204:
2200:
2198:
2195:
2194:
2163:
2142:
2139:
2138:
2122:
2119:
2118:
2102:
2099:
2098:
2082:
2079:
2078:
2062:
2059:
2058:
2054:
2028:
2008:
2002:
1998:
1997:
1985:
1979:
1975:
1974:
1972:
1958:
1954:
1948:
1944:
1942:
1902:
1900:
1859:
1856:
1855:
1831:
1827:
1825:
1822:
1821:
1804:
1800:
1798:
1795:
1794:
1793:. (The symbols
1774:
1771:
1770:
1754:
1751:
1750:
1734:
1731:
1730:
1579:
1577:
1574:
1573:
1565:
1509:
1507:
1496:
1492:
1485:
1480:
1467:
1463:
1457:
1453:
1443:
1441:
1418:
1415:
1414:
1407:
1391:
1388:
1387:
1371:
1368:
1367:
1350:
1346:
1344:
1341:
1340:
1323:
1319:
1317:
1314:
1313:
1276:
1255:
1252:
1251:
1220:
1216:
1210:
1206:
1199:
1197:
1182:
1178:
1172:
1168:
1161:
1151:
1147:
1141:
1137:
1130:
1128:
1115:
1104:
1100:
1094:
1090:
1088:
1076:
1072:
1068:
1061:
1057:
1051:
1047:
1037:
1035:
1033:
1030:
1029:
1012:
1008:
1006:
1003:
1002:
986:
983:
982:
979:
973:
970:
964:
948:
945:
944:
928:
925:
924:
900:
896:
890:
886:
884:
872:
868:
864:
857:
853:
847:
843:
833:
831:
819:
815:
807:
804:
803:
799:
778:
775:
774:
770:
753:
749:
738:
735:
734:
730:
726:
698:
694:
686:
684:
681:
680:
669:
652:
620:
616:
614:
611:
610:
587:
583:
562:
558:
550:
547:
546:
545:For this case,
523:
519:
517:
514:
513:
490:
486:
484:
481:
480:
443:
439:
431:
422:
418:
410:
407:
406:
390:
387:
386:
382:
359:
355:
353:
350:
349:
324:
321:
320:
293:
290:
289:
286:
280:
277:
271:
268:
262:
258:
244:
211:
207:
201:
197:
195:
181:
176:
166:
161:
155:
153:
142:
138:
132:
128:
126:
111:
107:
105:
102:
101:
95:
88:
82:
78:
72:
69:
63:
43:back-scattering
28:
23:
22:
15:
12:
11:
5:
5851:
5841:
5840:
5835:
5819:
5818:
5783:
5764:(9): 524–554.
5748:
5729:(6): 479–485.
5713:
5686:(3): 309–317.
5669:
5623:
5619:
5615:
5611:
5607:
5603:
5599:
5590:
5561:
5542:
5507:
5480:(5): 351–357.
5464:
5437:(6): 647–654.
5421:
5406:
5388:
5382:10.1086/141186
5347:
5320:
5285:
5262:
5221:
5192:
5185:
5159:
5124:
5113:(4): 330–335.
5090:
5059:
5032:(5): 448–457.
5012:
4990:
4975:
4968:
4942:
4917:
4887:
4886:
4884:
4881:
4878:
4877:
4858:
4838:
4814:
4811:
4806:
4803:
4796:
4792:
4786:
4782:
4774:
4769:
4766:
4761:
4758:
4755:
4750:
4746:
4728:
4727:
4725:
4722:
4714:titanium white
4689:
4686:
4670:
4666:
4662:
4659:
4656:
4652:
4648:
4645:
4642:
4638:
4634:
4631:
4628:
4625:
4620:
4617:
4613:
4609:
4604:
4601:
4597:
4593:
4590:
4587:
4584:
4582:
4578:
4575:
4571:
4567:
4566:
4563:
4560:
4556:
4552:
4549:
4543:
4540:
4536:
4532:
4529:
4526:
4523:
4519:
4515:
4512:
4509:
4506:
4501:
4498:
4494:
4490:
4487:
4484:
4481:
4477:
4473:
4470:
4467:
4464:
4460:
4456:
4453:
4447:
4444:
4440:
4436:
4433:
4425:
4421:
4415:
4412:
4409:
4406:
4402:
4398:
4395:
4388:
4384:
4378:
4373:
4369:
4362:
4357:
4353:
4349:
4346:
4344:
4340:
4337:
4333:
4329:
4328:
4325:
4322:
4318:
4314:
4311:
4305:
4302:
4298:
4294:
4291:
4288:
4285:
4281:
4277:
4274:
4271:
4268:
4263:
4260:
4256:
4252:
4249:
4246:
4243:
4239:
4235:
4232:
4226:
4218:
4214:
4208:
4204:
4200:
4197:
4190:
4186:
4180:
4176:
4169:
4166:
4164:
4160:
4157:
4153:
4149:
4148:
4128:
4124:
4120:
4098:
4095:
4092:
4089:
4085:
4064:
4060:
4056:
4053:
4048:
4044:
4023:
4019:
4015:
4012:
4007:
4003:
3982:
3978:
3974:
3971:
3966:
3962:
3941:
3937:
3933:
3930:
3925:
3921:
3898:
3894:
3890:
3887:
3881:
3878:
3874:
3870:
3867:
3864:
3861:
3858:
3853:
3850:
3846:
3842:
3839:
3836:
3833:
3829:
3825:
3822:
3819:
3816:
3812:
3808:
3805:
3799:
3796:
3793:
3785:
3781:
3775:
3772:
3769:
3766:
3762:
3758:
3755:
3748:
3744:
3738:
3733:
3729:
3722:
3717:
3713:
3709:
3704:
3701:
3697:
3666:
3662:
3639:
3636:
3633:
3630:
3626:
3598:
3594:
3590:
3587:
3584:
3580:
3576:
3573:
3570:
3566:
3562:
3559:
3556:
3553:
3548:
3545:
3541:
3537:
3532:
3529:
3525:
3521:
3518:
3515:
3512:
3510:
3506:
3503:
3499:
3495:
3494:
3491:
3488:
3484:
3480:
3477:
3471:
3468:
3465:
3461:
3457:
3454:
3451:
3448:
3443:
3440:
3437:
3433:
3429:
3426:
3423:
3420:
3416:
3412:
3409:
3403:
3400:
3396:
3392:
3389:
3381:
3377:
3371:
3368:
3365:
3362:
3358:
3354:
3351:
3344:
3340:
3334:
3329:
3325:
3318:
3313:
3309:
3305:
3302:
3300:
3296:
3293:
3289:
3285:
3284:
3281:
3278:
3274:
3270:
3267:
3261:
3258:
3255:
3252:
3249:
3245:
3241:
3238:
3235:
3232:
3227:
3224:
3220:
3216:
3213:
3210:
3207:
3203:
3199:
3196:
3190:
3182:
3178:
3172:
3169:
3166:
3163:
3159:
3155:
3152:
3145:
3141:
3135:
3131:
3124:
3121:
3119:
3115:
3112:
3108:
3104:
3103:
3084:
3081:
3077:
3073:
3070:
3065:
3061:
3057:
3054:
3051:
3046:
3042:
3038:
3035:
3031:
3027:
3024:
3019:
3015:
3011:
3008:
3004:
3000:
2997:
2992:
2988:
2919:
2916:
2913:
2909:
2905:
2902:
2899:
2895:
2891:
2888:
2885:
2882:
2877:
2874:
2870:
2866:
2861:
2858:
2854:
2850:
2847:
2844:
2841:
2839:
2835:
2832:
2828:
2824:
2823:
2820:
2817:
2813:
2809:
2806:
2800:
2797:
2793:
2789:
2786:
2783:
2780:
2776:
2772:
2769:
2766:
2763:
2758:
2755:
2751:
2747:
2744:
2741:
2738:
2734:
2730:
2727:
2724:
2721:
2717:
2713:
2710:
2704:
2701:
2697:
2693:
2690:
2682:
2678:
2672:
2669:
2666:
2663:
2659:
2655:
2652:
2645:
2641:
2635:
2630:
2626:
2619:
2614:
2610:
2606:
2603:
2601:
2597:
2594:
2590:
2586:
2585:
2582:
2579:
2575:
2571:
2568:
2562:
2558:
2554:
2549:
2545:
2541:
2535:
2529:
2526:
2522:
2518:
2515:
2512:
2509:
2505:
2501:
2498:
2495:
2492:
2487:
2484:
2480:
2476:
2473:
2470:
2467:
2463:
2459:
2456:
2450:
2442:
2438:
2432:
2429:
2426:
2423:
2419:
2415:
2412:
2405:
2401:
2395:
2391:
2384:
2381:
2379:
2375:
2372:
2368:
2364:
2363:
2339:
2335:
2331:
2326:
2322:
2318:
2313:
2309:
2305:
2300:
2296:
2273:
2270:
2266:
2243:
2240:
2236:
2215:
2210:
2207:
2203:
2162:
2159:
2146:
2126:
2106:
2086:
2066:
2040:
2035:
2032:
2027:
2024:
2018:
2015:
2011:
2005:
2001:
1995:
1992:
1988:
1982:
1978:
1971:
1968:
1961:
1957:
1951:
1947:
1941:
1938:
1933:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1834:
1830:
1807:
1803:
1778:
1758:
1738:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1582:
1564:
1561:
1554:
1553:
1540:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1506:
1499:
1495:
1488:
1483:
1479:
1475:
1470:
1466:
1460:
1456:
1452:
1449:
1446:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1395:
1375:
1353:
1349:
1326:
1322:
1275:
1272:
1259:
1237:
1232:
1228:
1223:
1219:
1213:
1209:
1205:
1202:
1196:
1190:
1185:
1181:
1175:
1171:
1167:
1164:
1159:
1154:
1150:
1144:
1140:
1136:
1133:
1127:
1122:
1119:
1114:
1107:
1103:
1097:
1093:
1087:
1079:
1075:
1071:
1064:
1060:
1054:
1050:
1046:
1043:
1040:
1015:
1011:
990:
977:
968:
952:
932:
923:We may define
910:
903:
899:
893:
889:
883:
875:
871:
867:
860:
856:
850:
846:
842:
839:
836:
830:
827:
822:
818:
814:
811:
798:
795:
782:
756:
752:
748:
745:
742:
709:
706:
701:
697:
693:
690:
668:
667:Semiconductors
665:
651:
648:
643:
642:
631:
628:
623:
619:
598:
595:
590:
586:
582:
579:
576:
573:
570:
565:
561:
557:
554:
534:
531:
526:
522:
501:
498:
489:
477:
474:
463:
460:
457:
454:
451:
446:
442:
438:
434:
430:
425:
421:
417:
414:
394:
370:
367:
362:
358:
337:
334:
331:
328:
317:
297:
284:
275:
266:
243:
240:
235:
234:
223:
214:
210:
204:
200:
194:
191:
184:
179:
175:
169:
164:
160:
152:
145:
141:
135:
131:
125:
122:
119:
114:
110:
93:
86:
76:
67:
26:
9:
6:
4:
3:
2:
5850:
5839:
5836:
5834:
5831:
5830:
5828:
5814:
5810:
5806:
5802:
5798:
5794:
5787:
5779:
5775:
5771:
5767:
5763:
5759:
5752:
5744:
5740:
5736:
5732:
5728:
5724:
5717:
5709:
5705:
5701:
5697:
5693:
5689:
5685:
5681:
5673:
5665:
5661:
5657:
5653:
5649:
5645:
5641:
5637:
5633:
5629:
5594:
5586:
5580:
5572:
5568:
5564:
5562:0-8247-0534-3
5558:
5554:
5553:
5546:
5538:
5534:
5530:
5526:
5522:
5518:
5511:
5503:
5499:
5495:
5491:
5487:
5483:
5479:
5475:
5468:
5460:
5456:
5452:
5448:
5444:
5440:
5436:
5432:
5425:
5417:
5413:
5409:
5407:9783642880711
5403:
5399:
5392:
5383:
5378:
5374:
5370:
5366:
5362:
5358:
5351:
5343:
5339:
5335:
5331:
5324:
5316:
5312:
5308:
5304:
5300:
5296:
5289:
5281:
5277:
5273:
5266:
5258:
5254:
5249:
5244:
5240:
5236:
5232:
5225:
5217:
5211:
5208:: J224–J232.
5207:
5203:
5196:
5188:
5182:
5178:
5174:
5170:
5163:
5154:
5149:
5145:
5141:
5137:
5131:
5129:
5120:
5116:
5112:
5108:
5104:
5097:
5095:
5086:
5082:
5078:
5074:
5070:
5063:
5055:
5051:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5016:
5008:
5001:
4999:
4997:
4995:
4986:
4979:
4971:
4965:
4961:
4957:
4953:
4946:
4931:
4927:
4921:
4913:
4909:
4908:Z. Tech. Phys
4902:
4895:
4893:
4888:
4873:
4856:
4836:
4812:
4809:
4804:
4801:
4794:
4790:
4784:
4780:
4772:
4767:
4764:
4759:
4756:
4753:
4744:
4733:
4729:
4721:
4719:
4715:
4711:
4707:
4703:
4699:
4695:
4685:
4668:
4664:
4660:
4657:
4654:
4650:
4646:
4643:
4640:
4636:
4632:
4629:
4626:
4623:
4618:
4615:
4611:
4607:
4602:
4599:
4595:
4591:
4588:
4585:
4583:
4576:
4573:
4569:
4561:
4558:
4554:
4550:
4547:
4538:
4534:
4530:
4521:
4517:
4513:
4507:
4504:
4496:
4492:
4488:
4479:
4475:
4471:
4462:
4458:
4454:
4445:
4442:
4438:
4434:
4431:
4423:
4419:
4410:
4407:
4400:
4396:
4393:
4386:
4382:
4376:
4371:
4367:
4360:
4355:
4351:
4347:
4345:
4338:
4335:
4331:
4323:
4320:
4316:
4312:
4309:
4300:
4296:
4292:
4283:
4279:
4275:
4269:
4266:
4258:
4254:
4250:
4241:
4237:
4233:
4224:
4216:
4212:
4206:
4202:
4198:
4195:
4188:
4184:
4178:
4174:
4167:
4165:
4158:
4155:
4151:
4126:
4122:
4118:
4093:
4090:
4083:
4062:
4058:
4054:
4051:
4046:
4042:
4021:
4017:
4013:
4010:
4005:
4001:
3980:
3976:
3972:
3969:
3964:
3960:
3939:
3935:
3931:
3928:
3923:
3919:
3909:
3896:
3892:
3888:
3885:
3876:
3872:
3868:
3862:
3859:
3856:
3848:
3844:
3840:
3831:
3827:
3823:
3814:
3810:
3806:
3797:
3794:
3791:
3783:
3779:
3770:
3767:
3760:
3756:
3753:
3746:
3742:
3736:
3731:
3727:
3720:
3715:
3711:
3707:
3702:
3699:
3695:
3686:
3682:
3664:
3660:
3634:
3631:
3624:
3613:
3596:
3592:
3588:
3585:
3582:
3578:
3574:
3571:
3568:
3564:
3560:
3557:
3554:
3551:
3546:
3543:
3539:
3535:
3530:
3527:
3523:
3519:
3516:
3513:
3511:
3504:
3501:
3497:
3489:
3486:
3482:
3478:
3475:
3469:
3463:
3459:
3455:
3449:
3446:
3441:
3435:
3431:
3427:
3418:
3414:
3410:
3401:
3398:
3394:
3390:
3387:
3379:
3375:
3366:
3363:
3356:
3352:
3349:
3342:
3338:
3332:
3327:
3323:
3316:
3311:
3307:
3303:
3301:
3294:
3291:
3287:
3279:
3276:
3272:
3268:
3265:
3256:
3247:
3243:
3239:
3233:
3230:
3222:
3218:
3214:
3205:
3201:
3197:
3188:
3180:
3176:
3167:
3164:
3157:
3153:
3150:
3143:
3139:
3133:
3129:
3122:
3120:
3113:
3110:
3106:
3082:
3079:
3075:
3071:
3068:
3063:
3059:
3055:
3052:
3049:
3044:
3040:
3036:
3033:
3029:
3025:
3022:
3017:
3013:
3009:
3006:
3002:
2998:
2995:
2990:
2986:
2976:
2974:
2970:
2964:
2961:= 1/3 ≈ 0.333
2960:
2956:
2952:
2942:
2938:
2934:
2917:
2914:
2911:
2907:
2903:
2900:
2897:
2893:
2889:
2886:
2883:
2880:
2875:
2872:
2868:
2864:
2859:
2856:
2852:
2848:
2845:
2842:
2840:
2833:
2830:
2826:
2818:
2815:
2811:
2807:
2804:
2795:
2791:
2787:
2778:
2774:
2770:
2764:
2761:
2753:
2749:
2745:
2736:
2732:
2728:
2719:
2715:
2711:
2702:
2699:
2695:
2691:
2688:
2680:
2676:
2667:
2664:
2657:
2653:
2650:
2643:
2639:
2633:
2628:
2624:
2617:
2612:
2608:
2604:
2602:
2595:
2592:
2588:
2580:
2577:
2573:
2569:
2566:
2560:
2556:
2552:
2547:
2543:
2539:
2533:
2524:
2520:
2516:
2507:
2503:
2499:
2493:
2490:
2482:
2478:
2474:
2465:
2461:
2457:
2448:
2440:
2436:
2427:
2424:
2417:
2413:
2410:
2403:
2399:
2393:
2389:
2382:
2380:
2373:
2370:
2366:
2353:
2337:
2333:
2329:
2324:
2320:
2316:
2311:
2307:
2303:
2298:
2294:
2271:
2268:
2264:
2241:
2238:
2234:
2213:
2208:
2205:
2201:
2192:
2188:
2183:
2176:
2172:
2169:
2168:Frank Benford
2158:
2144:
2124:
2104:
2084:
2064:
2051:
2038:
2033:
2030:
2025:
2022:
2016:
2013:
2009:
2003:
1999:
1993:
1990:
1986:
1980:
1976:
1969:
1966:
1959:
1955:
1949:
1945:
1939:
1936:
1931:
1927:
1921:
1918:
1915:
1912:
1909:
1906:
1897:
1891:
1888:
1885:
1879:
1876:
1870:
1864:
1861:
1853:
1852:Dahm equation
1848:
1832:
1828:
1805:
1801:
1792:
1776:
1756:
1736:
1716:
1710:
1707:
1704:
1698:
1695:
1692:
1683:
1677:
1674:
1671:
1665:
1659:
1656:
1653:
1647:
1644:
1641:
1635:
1632:
1626:
1623:
1620:
1614:
1611:
1608:
1602:
1599:
1596:
1593:
1590:
1587:
1569:
1560:
1557:
1538:
1534:
1528:
1525:
1522:
1519:
1516:
1513:
1504:
1497:
1493:
1486:
1481:
1477:
1473:
1468:
1458:
1454:
1450:
1447:
1438:
1432:
1429:
1426:
1420:
1413:
1412:
1411:
1393:
1373:
1351:
1347:
1324:
1320:
1311:
1307:
1306:Dahm equation
1303:
1299:
1294:
1292:
1288:
1283:
1281:
1271:
1257:
1248:
1235:
1230:
1221:
1217:
1211:
1207:
1194:
1183:
1179:
1173:
1169:
1152:
1148:
1142:
1138:
1125:
1120:
1117:
1112:
1105:
1101:
1095:
1091:
1085:
1073:
1069:
1062:
1048:
1044:
1041:
1013:
1009:
988:
976:
967:
950:
930:
921:
908:
901:
897:
891:
887:
881:
869:
865:
858:
844:
840:
837:
828:
816:
809:
794:
780:
754:
746:
743:
725:
724:photon energy
707:
695:
688:
678:
674:
664:
662:
656:
647:
629:
626:
617:
593:
588:
584:
580:
577:
571:
568:
563:
559:
555:
552:
532:
529:
524:
520:
499:
496:
487:
478:
475:
461:
455:
452:
449:
444:
440:
432:
428:
423:
419:
415:
412:
392:
368:
365:
356:
335:
332:
329:
326:
318:
315:
314:
313:
311:
295:
283:
274:
265:
255:
253:
249:
239:
221:
212:
208:
202:
198:
192:
189:
182:
177:
173:
167:
162:
158:
150:
143:
139:
133:
129:
123:
120:
117:
108:
100:
99:
98:
92:
85:
75:
66:
61:
57:
51:
48:
44:
40:
35:
33:
19:
5796:
5792:
5786:
5761:
5757:
5751:
5726:
5722:
5716:
5683:
5679:
5672:
5631:
5627:
5593:
5551:
5545:
5520:
5516:
5510:
5477:
5473:
5467:
5434:
5430:
5424:
5397:
5391:
5364:
5360:
5350:
5333:
5329:
5323:
5298:
5294:
5288:
5279:
5275:
5265:
5238:
5234:
5224:
5205:
5201:
5195:
5168:
5162:
5143:
5140:BioResources
5139:
5110:
5106:
5079:(12): 1067.
5076:
5072:
5062:
5029:
5025:
5015:
5006:
4978:
4951:
4945:
4933:. Retrieved
4929:
4920:
4911:
4907:
4732:
4718:latent space
4710:Hansa yellow
4702:phthalo blue
4691:
3910:
3684:
3680:
3614:
2977:
2972:
2968:
2965:
2958:
2954:
2950:
2946:
2935:
2354:
2190:
2186:
2184:
2180:
2164:
2052:
1849:
1570:
1566:
1558:
1555:
1309:
1301:
1295:
1284:
1277:
1274:Spectroscopy
1249:
974:
965:
922:
800:
670:
657:
653:
644:
309:
281:
272:
263:
256:
248:hiding power
245:
242:Paint colors
236:
90:
83:
73:
64:
52:
36:
31:
29:
5799:(6): 1–11.
5367:(1): 1–22.
4935:22 February
476:Ideal glaze
5827:Categories
5301:(3): 380.
5282:: 241–246.
4914:: 593–601.
4883:References
4872:Helligkeit
2226:remission
5813:245013978
5622:O, Mg(OH)
5579:cite book
5537:212642389
5517:ChemTexts
5416:714802320
4773:−
4749:∞
4644:−
4630:−
4608:−
4592:−
4508:−
4408:−
4397:−
4270:−
4199:−
4091:−
3860:−
3768:−
3757:−
3632:−
3572:−
3558:−
3536:−
3520:−
3450:−
3364:−
3353:−
3234:−
3165:−
3154:−
2901:−
2887:−
2865:−
2849:−
2765:−
2665:−
2654:−
2494:−
2425:−
2414:−
1937:≈
1916:−
1910:−
1829:μ
1802:μ
1777:ϵ
1708:ϵ
1705:−
1699:
1666:−
1660:
1642:−
1636:
1621:−
1615:
1523:−
1517:−
1474:−
1451:−
1439:≡
1201:Σ
1195:≈
1163:Σ
1132:Σ
1078:∞
1053:∞
1045:−
874:∞
849:∞
841:−
829:≡
821:∞
700:∞
677:Tauc plot
622:∞
578:−
572:
361:∞
151:−
113:∞
47:remission
5833:Lighting
5778:21002043
5743:93926306
5708:98159628
5664:42585664
5656:14639754
5610:O, Ni(IO
5571:46829414
5523:(1): 2.
5502:98416407
5459:96885077
5257:30990726
5054:18916891
773:, where
673:band-gap
5688:Bibcode
5636:Bibcode
5482:Bibcode
5439:Bibcode
5369:Bibcode
5303:Bibcode
5216:2690187
5034:Bibcode
4075:, with
3993:, with
769:versus
252:opacity
5811:
5776:
5741:
5706:
5662:
5654:
5569:
5559:
5535:
5500:
5457:
5414:
5404:
5255:
5213:
5183:
5052:
4966:
4712:, and
1729:where
797:Colors
56:albedo
5809:S2CID
5739:S2CID
5704:S2CID
5660:S2CID
5533:S2CID
5498:S2CID
5455:S2CID
5210:INIST
4904:(PDF)
4724:Notes
5774:PMID
5758:JOSA
5652:PMID
5585:link
5567:OCLC
5557:ISBN
5412:OCLC
5402:ISBN
5253:PMID
5181:ISBN
5107:JOSA
5073:JOSA
5050:PMID
5026:JOSA
4964:ISBN
4937:2021
4698:CMYK
4034:and
3952:and
2189:and
2137:and
2097:and
2057:and
1820:and
1406:and
1339:and
972:and
943:and
671:The
348:and
30:The
5801:doi
5766:doi
5731:doi
5696:doi
5644:doi
5618:·2H
5606:·4H
5525:doi
5490:doi
5447:doi
5377:doi
5338:doi
5311:doi
5243:doi
5173:doi
5148:doi
5115:doi
5081:doi
5042:doi
4956:doi
3795:0.0
2975:).
1696:exp
1657:exp
1633:exp
1612:exp
569:exp
405:is
5829::
5807:.
5797:40
5795:.
5772:.
5762:36
5760:.
5737:.
5727:11
5725:.
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5694:.
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5682:.
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5577:{{
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5519:.
5496:.
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5280:35
5278:.
5274:.
5251:.
5237:.
5233:.
5206:23
5204:.
5179:.
5142:.
5138:.
5127:^
5111:44
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5105:.
5093:^
5077:38
5075:.
5071:.
5048:.
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4962:.
4928:.
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4891:^
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3502:x
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3072:1
3069:=
3064:y
3060:T
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3053:0
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3037:,
3034:2
3030:/
3026:1
3023:=
3018:x
3014:T
3010:,
3007:2
3003:/
2999:1
2996:=
2991:x
2987:R
2973:y
2969:x
2959:A
2955:T
2951:R
2915:=
2912:2
2908:/
2904:1
2898:2
2894:/
2890:1
2884:1
2881:=
2876:y
2873:x
2869:R
2860:y
2857:x
2853:T
2846:1
2843:=
2834:y
2831:x
2827:A
2819:,
2816:3
2812:/
2808:2
2805:=
2799:)
2796:2
2792:/
2788:1
2785:(
2782:)
2779:2
2775:/
2771:1
2768:(
2762:1
2757:)
2754:2
2750:/
2746:1
2743:(
2740:)
2737:2
2733:/
2729:1
2726:(
2723:)
2720:2
2716:/
2712:1
2709:(
2703:+
2700:2
2696:/
2692:1
2689:=
2681:y
2677:R
2671:)
2668:x
2662:(
2658:R
2651:1
2644:y
2640:R
2634:2
2629:x
2625:T
2618:+
2613:x
2609:R
2605:=
2596:y
2593:x
2589:R
2581:,
2578:3
2574:/
2570:1
2567:=
2561:4
2557:/
2553:3
2548:4
2544:/
2540:1
2534:=
2528:)
2525:2
2521:/
2517:1
2514:(
2511:)
2508:2
2504:/
2500:1
2497:(
2491:1
2486:)
2483:2
2479:/
2475:1
2472:(
2469:)
2466:2
2462:/
2458:1
2455:(
2449:=
2441:y
2437:R
2431:)
2428:x
2422:(
2418:R
2411:1
2404:y
2400:T
2394:x
2390:T
2383:=
2374:y
2371:x
2367:T
2338:y
2334:R
2330:,
2325:y
2321:T
2317:,
2312:x
2308:R
2304:,
2299:x
2295:T
2272:y
2269:x
2265:A
2242:y
2239:x
2235:R
2214:,
2209:y
2206:x
2202:T
2191:y
2187:x
2145:s
2125:k
2105:S
2085:K
2065:T
2055:R
2039:.
2034:S
2031:K
2026:2
2023:=
2017:x
2014:d
2010:/
2004:0
2000:r
1994:x
1991:d
1987:/
1981:0
1977:a
1970:2
1967:=
1960:0
1956:r
1950:0
1946:a
1940:2
1932:r
1928:a
1925:)
1922:r
1919:2
1913:a
1907:2
1904:(
1898:=
1895:)
1892:T
1889:,
1886:R
1883:(
1880:A
1877:=
1874:)
1871:R
1868:(
1865:F
1862:2
1833:s
1806:a
1757:s
1737:k
1717:,
1714:)
1711:d
1702:(
1693:=
1690:]
1687:)
1684:d
1681:)
1678:s
1675:+
1672:k
1669:(
1663:[
1654:=
1651:)
1648:d
1645:s
1639:(
1630:)
1627:d
1624:k
1618:(
1609:=
1606:)
1603:t
1600:c
1597:e
1594:r
1591:i
1588:d
1585:(
1581:T
1539:r
1535:a
1532:)
1529:r
1526:2
1520:a
1514:2
1511:(
1505:=
1498:n
1494:R
1487:2
1482:n
1478:T
1469:2
1465:)
1459:n
1455:R
1448:1
1445:(
1436:)
1433:T
1430:,
1427:R
1424:(
1421:A
1408:R
1394:a
1374:n
1352:n
1348:T
1325:n
1321:R
1310:R
1302:R
1258:S
1236:.
1231:S
1227:)
1222:i
1218:K
1212:i
1208:c
1204:(
1189:)
1184:i
1180:S
1174:i
1170:c
1166:(
1158:)
1153:i
1149:K
1143:i
1139:c
1135:(
1126:=
1121:S
1118:K
1113:=
1106:0
1102:r
1096:0
1092:a
1086:=
1074:R
1070:2
1063:2
1059:)
1049:R
1042:1
1039:(
1014:i
1010:c
989:i
978:0
975:r
969:0
966:a
951:S
931:K
909:.
902:0
898:r
892:0
888:a
882:=
870:R
866:2
859:2
855:)
845:R
838:1
835:(
826:)
817:R
813:(
810:F
781:a
771:E
755:2
751:)
747:E
744:a
741:(
731:E
727:E
708:E
705:)
696:R
692:(
689:F
627:=
618:R
597:)
594:X
589:0
585:a
581:2
575:(
564:g
560:R
556:=
553:R
533:.
530:X
525:0
521:a
500:0
497:=
492:c
488:R
462:.
459:)
456:1
453:+
450:X
445:0
441:r
437:(
433:/
429:X
424:0
420:r
416:=
413:R
393:X
383:R
366:=
357:R
336:,
333:0
330:=
327:a
310:X
296:T
285:∞
282:R
276:c
273:R
267:g
264:R
259:R
222:.
213:0
209:r
203:0
199:a
193:2
190:+
183:2
178:0
174:r
168:2
163:0
159:a
144:0
140:r
134:0
130:a
124:+
121:1
118:=
109:R
94:0
91:r
89:/
87:0
84:a
77:0
74:r
68:0
65:a
20:)
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