LLight production metrics of radiation sources
C. Tannous
Laboratoire de Magn´etisme de Bretagne - CNRS FRE 3117UBO, 6, Avenue le Gorgeu C.S.93837 - 29238 Brest Cedex 3 - FRANCE ∗ (Dated: November 15, 2013)Light production by a radiation source is evaluated and reviewed as an important concept ofphysics from the Black-Body point of view. The mechanical equivalent of the lumen, the unit ofperceived light, is explained and evaluated using radiation physics arguments. The existence of anupper limit of luminous efficacy is illustrated for various sources and implications are highlighted. PACS numbers: 42.66.Si,42.72.-g,85.60.JbKeywords: Visual perception,light sources,light-emitting diode
I. INTRODUCTION
Physics students are exposed to Black-Body radia-tion in undergraduate Quantum Physics or in Gradu-ate/Undergraduate Statistical Physics without any clueregarding its significance as to the fundamental role itplayed in the development of light source calibration, de-velopment and of lighting in general.Perhaps, only students with astrophysics or atmo-spheric physics curriculum will be generally more awareof radiation physics in connection with the Black-Bodyfundamentals.Presently, lighting is undergoing a tremendous evolu-tion because of the swift evolution of the light-emission-diode (LED) that is now gaining larger and larger lumi-nous efficacy to a point such that it is now replacing, at avery impressive pace, our traditional home lighting, carheadlights, LCD-monitors backlights, street lighting...Additionally, LED colors are becoming more versa-tile and sharper both in the case of traditional inorganicLED’s or their organic counterpart, the OLED.The underlying basis of lighting progress is the exis-tence of Haitz rule (see fig. 1) that is similar to GordonMoore rule of Electronics evolution.Light production by a radiation source is described bya luminous efficacy ratio η L given by the product of η C the conversion efficacy ratio of number of photons pro-duced (having any wavelength) to input energy (usuallyelectrical but it could also be mechanical, thermal orchemical...) and η P the light perception efficacy ratioor Photometric efficacy ratio (PER). η P is the ratio ofnumber of photons perceived by the human eye (photonwavelength in the visible spectrum) to the total numberof photons. Some authors call it η S the spectral efficacy.This work can be taught as an application chapter ina general course of Statistical physics or in Semiconduc-tor physics at the Undergraduate or Graduate level sincephysicists can contribute readily in improving light pro-duction level of radiation sources or conversion efficacy(through the increase of either η P or η C ) once the basisfor luminous efficacy is explained and illustrated alongsome notions of colorimetry and light calibration.The notions reviewed in this work are primarily con-cerned with η P the perception efficacy and are clearly −4 −3 −2 −1 l u m e n s Year White LED
FIG. 1. (Color online) Haitz rule and evolution of white LED(in cyan) after year 2000 when S. Nakamura introduced Inwithin GaN . Before 2000, evolution was driven by red LEDstarting in the 1970’s with GaAs, GaAsP, GaP, GaAsP:N,GaAlAs and finally GaAlInP. important not only in Physics and technology but alsofor energy savings and efficiency, renewable energy andconsequently for sustainable development of the Planet.This report is organized as follows: In section 2, a re-view of lighting metrology is made, in Section 3 luminousefficacy of radiation sources is explained and derived andin Section 4, we derive the maximum luminous efficacy onthe basis of the colorimetry standard established by theCIE . This standard is briefly explained and reviewed inthe Appendix that comes after Section 5 carrying discus-sions and conclusions. II. METROLOGY OF LIGHTING
The SI system of units is based on seven entities: themeter, the kilogram, the second, the ampere, the kelvin,the mole and the candela as the unit of luminous inten-sity (Lumen is candela per unit solid angle).Prior to 1979, the SI system of units defines the candelaas follows:
A pure sample of Platinum at its fusion temperature a r X i v : . [ qu a n t - ph ] N ov (T=2042 K) emits exactly 60 candelas/cm along thevertical direction to the sample and per unit solid angle .The SI system changed the definition during the 16thGeneral Conference on Weights and Measures in October1979 to: The candela is the luminous intensity, in a given direc-tion, of a source that emits monochromatic radiation offrequency 540 × Hz with a radiant intensity, in thatdirection, of 1/683 Watts per unit solid angle .In order to relate these two definitions, despite the ob-vious equivalence of 555 nm wavelength and 540 × Hz frequency, some reminders about Black-Body radia-tion must be made.Black-Body radiation was noticed for the first time byGustav Kirchhoff who used to watch the color changeof cavities present in heated metals worked by black-smiths in his neighborhood. He noticed a systematiccolor change as the metal is being heated along the se-quence of red, orange, yellow, white and finally blue.Cavity color change did not depend on the nature ofthe metal but depended on the size of the cavity.This means that we have a photon gas in the cavityobeying Planck radiation law with the average densityof photons having energy (cid:126) ω of the Bose-Einstein form(the chemical potential being zero since the number ofphotons is not fixed): n ( ω ) = 1 (cid:104) exp( (cid:126) ωk B T ) − (cid:105) (1)The average radiation energy in the [ ω, ω + dω ] is( n ( ω ) + ) (cid:126) ωg ( ω ) with g ( ω ) the density of states . Thevacuum energy (cid:126) ω should not be included since it isindependent of T . The density of states being ω π c , weget: E ( ω ) = ω π c (cid:126) ω (cid:104) exp( (cid:126) ωk B T ) − (cid:105) (2)In order to get E ( λ ) the average energy density of pho-tons having wavelength λ , we use the conservation rulegiven by: E ( ω ) dω = E ( λ ) dλ (3)with the transformation from angular frequency ω tolinear frequency f to wavelength λ : (cid:126) ω = (cid:126) πf = hc/λ .This yields: E ( λ ) = 8 πhcλ (cid:104) exp( hcλk B T ) − (cid:105) (4)This law is derived in many Statistical Physics booksand is usually enough for standard Physics curriculum. It represents the thermal average number of photons perunit volume emitted by a Black-Body at temperature T .Since a Black-Body (absorbs and) emits radiation, weneed a Planck equivalent law for the emitted energy den-sity. By analogy with electrical charge current density J ( J = ρ v with ρ the charge density and v their velocity),we multiply the average photon density by the velocityof light c and divide by 4 π , the whole space solid anglefactor. Thus we obtain the Planck distribution of pho-ton energy emission at wavelength λ , temperature T andunit solid angle: f B ( λ ) = 2 hc λ (cid:104) exp( hcλk B T ) − (cid:105) (5)Light sensed by the human eye is given by the distri-bution of energy average over the eye sensitivity functioncalled V ( λ ) by the CIE .This function depicted in fig. 2 peaks at a 555 nm(Yellow-Green) wavelength that is the maximum sen-sitivity of the human eye. The analytical expressionsof V ( λ ) = 1 .
019 exp( − (cid:2) ( λ ) − . (cid:3) ) (sensitiv-ity in daylight or photopic sensitivity) with λ expressedin nm. The sensitivity in the dark V (cid:48) ( λ ) (scotopic) isshifted with respect to V ( λ ) by 45 nm to shorter wave-lengths and peaks at 510 nm (Purkinje shift). Analyti-cally V (cid:48) ( λ ) = 0 .
992 exp( − . (cid:2) ( λ ) − . (cid:3) ). V ( l ) , V ’( l ) l (nm) PhotopicScotopic FIG. 2. V ( λ ) function giving the eye sensitivity versus wave-length in daylight (photopic) and V (cid:48) ( λ ) in the dark (scotopic).They are shifted one with respect to the other by 45 nm, thePurkinje shift. The conversion from Radiometry λ ∈ [0 , ∞ ] to Pho-tometry λ ∈ [380nm , (cid:90) ∞ hc λ (cid:104) exp( hcλk B T ) − (cid:105) × V ( λ ) dλ (6)by a conversion factor K m called the ”Mechanicalequivalent of the Lumen” such that:light perceived = K m (cid:90) ∞ hc λ V ( λ ) dλ (cid:104) exp( hcλk B T ) − (cid:105) (7)It is important to notice that we are carrying the in-tegration over all positive frequencies and not the visiblespectrum given by the wavelength interval [380 nm, 780nm] since we are dealing with (unlimited) radiation en-ergy.In order to evaluate K m , we use the old definition ofthe candela and get: K m = 60 cd/cm /sr (cid:82) ∞ hc λ V ( λ ) dλ (cid:104) exp( hcλkBT ) − (cid:105) (8)Transforming to SI units, the numerator becomes6 × lumens/m (since lumen=cd per unit solid angle).The integral of the denominator has Watt/m dimensionsand when it is numerically evaluated we get K m = 679lm/Watt (lumens/Watt) which is close to the value of683 lm/Watt adopted by the SI . III. LUMINOUS EFFICACY OF RADIATIONSOURCESA. Luminous efficacy of Black-Body radiationsources
Having determined K m we are now in the positionof determining the luminous efficacy of any radiationsource.The value of η P for any radiation source characterizedby a power emission spectrum P ( λ ) is given by: η P = K m (cid:82) D λ P ( λ ) V ( λ ) dλ (cid:82) D λ P ( λ ) dλ (9)The wavelength interval D λ is arbitrary and dependson the radiation source.In the particular case of a source of the thermal Black-Body type, we apply the above formula 9 with P ( λ ) = f B ( λ ) and D λ = [0 , ∞ ]: η P = K m (cid:82) ∞ f B ( λ ) V ( λ ) dλ (cid:82) ∞ f B ( λ ) dλ (10)The result is the temperature dependent PER curvedepicted in fig. 3.PER has lm/Watt dimensions, thus we define effi-ciency as a dimensionless ratio (in %) yielding the frac-tion with respect to the ideal efficacy of 683 lm/Watt.It shows that the Sun efficacy is about 93lm/Watt (temperature about 6000K) or an efficiency P E R ( l m / W a tt ) T(K)
FIG. 3. Photometric Efficacy Ratio η P versus temperatureof the Black-Body. Notice that the maximum is 95 lm/Wattand that the temperature is about 7000K. of 93/683=13.6% and that the ordinary Tungsten lightbulb based on incandescence phenomenom is about 15lm/Watt (temperature about 3000K) or 2% only. Fora candle considered as a Black-Body at T=1800K, weget 0.6 lm/Watt which corresponds to an efficiency of0.6/683 or about 0.1%.The consequence in terms of lighting is that when oneacquires a 60 Watts bulb of the Tungsten type, the lightproduced by the bulb is 60 W x 15 lm/Watt= 900 lumensin total and that has tremendous consequences for thequality and cost of the lighting desired. B. Luminous efficacy of White radiation sources
A White source is considered as having a flat poweremission spectrum P ( λ ) over the entire visible interval,nevertheless in practice the interval is limited and one hasto define precisely the wavelength interval over which thisflatness is observed.Two cases are encountered in lighting systems:1. White source as a truncated Black-Body source:This is a Black-Body source taken at a temperatureT=5800K with a spectrum limited by definition to λ min =400 nm and λ max =700 nm. The PER is ob-tained from: η P = K m (cid:82) λ max λ min P ( λ ) V ( λ ) dλ (cid:82) λ max λ min P ( λ ) dλ (11)Using P ( λ ) = hc λ (cid:104) exp( hcλkBT ) − (cid:105) we get a PER ofabout 250 lm/Watt.2. Equal Energy White Source:For instance, an ”Equal Energy White Source” pos-sesses by definition a flat power emission spectrumover the entire visible spectrum. Mathematically P ( λ ) = W for λ ∈ [380nm , λ min =380nm, whereas λ max =780 nm. Thus we get: η P = K m (cid:82) λ max λ min W V ( λ ) dλ (cid:82) λ max λ min W dλ (12)This yields about 179 lm/Watt. This leads to theconclusion that flatness is not enough to increasePER. We need a compromise between flatness andwavelength interval length.
C. Luminous efficacy of fluorescent sources
Fluorescent light sources are known as cold sources(Eco light is also a fluorescent light source) as op-posed to thermal (Black-Body like) or incandescentsources. They need a special circuit called a ballastto stabilize current and accelerate electrons in orderto make them collide inelastically with a gas mix-ture of heavy atoms (typically Mercury, Terbiumand Argon) producing radiation.An example power emission spectrum of the threeband type is displayed in fig. 4. It shows severalpeaks over a finite wavelength interval in sharp con-trast with Black-Body spectrum that is smooth andcontinuous extending over an infinite wavelengthinterval. P ( λ ) , V ( λ ) λ (nm) FIG. 4. (Color online) Relative power emission spectrum ofa three-band type fluorescent bulb compared to eye sensitiv-ity curve (in green). Mercury peak is around 450 nm (Dataadapted from Hoffmann ). Emission by a fluorescent lamp extends over a fi-nite interval [ λ min , λ max ] with λ min =380 nm and λ max =700 nm for this case. The evaluation of thePER is done by spline interpolating the data dis-played in fig. 4. Using the general definition eq. 11 we obtain a PER of 343 lm/Watt which representsan efficiency of 50%. Usually the conversion efficacy η C is about 20% which makes the overall efficacy of68.6 lm/Watt and the total efficiency at 10 %. Suchefficiency is quite interesting, however the problemwith fluorescent light is that it suffers from flicker(fluctuating light intensity) due to the ballast andthe random collision phenomena, besides it relies onMercury which is a highly polluting source of theenvironment. Moreover the ballast circuit mightproduce annoying noise in some cases. D. Luminous efficacy of lasers and LED’s
White light, Black-Body and fluorescent radiators areconsidered as broadband emitters since their radiationspans (at least) the entire visible spectrum.This is not the case of LED’s and lasers since they aresomehow closer to monochromatic (narrowband) sources.The new CIE definition of the mechanical equivalent ofthe lumen is that a monochromatic source that is a poweremission spectrum peaking at λ =555 nm (at the maxi-mum sensitivity of the eye) with power of 1 W producesexactly 683 lumens.This can be understood readily from the general PERdefinition: η P = K m (cid:82) λ max λ min δ ( λ − λ ) V ( λ ) dλ (cid:82) λ max λ min δ ( λ − λ ) dλ (13)where we used: P ( λ ) = aδ ( λ − λ ) with a a constantand the condition that λ ∈ [ λ min , λ max ].The integral gives the result: PER= K m V ( λ ) = K m since V ( λ = 555nm) = 1.As an application, consider a laser emitting at λ =570nm with a power of 50 mW. It has a PER= K m V ( λ )that produces 30 lumens.Turning to lighting with LED’s, one of the main advan-tages of LED is that its lifetime is extremely long (on theorder of 100,000 hours) because it is a rugged solid statedevice. Moreover it does not rely on a ballast or Mer-cury which makes it safer than Fluorescent lamps whoselifetime is about several 1000 hours.The LED power emission spectrum function is usuallyapproximated by a Gaussian or a superposition of severalGaussian functions. In the single Gaussian approxima-tion P ( λ ) = exp (cid:104) − ( λ − λ ) λ (cid:105) can be used to evaluate theefficacy η P from eq. 11.The LED is characterized by an average wavelength λ and a standard deviation ∆ λ . As an example, weconsider a blue LED with λ =450 nm and ∆ λ =20 nm.The PER obtained from eq. 11 is about 39.7 lm/Wattand therefore an efficiency of 6%. The small efficiencyis due to small overlap between the blue LED spectrumand the eye sensibility curve V ( λ ), moreover that numberis further reduced after multiplication by the conversionefficacy η C which is typically about 20% yielding a totalefficiency of 1.2%.This is to be contrasted with the present status ofWhite LED who has a very large PER as illustrated byHaitz law in fig. 1. That might be due to the fact a flatpower emission spectrum enhances the PER as previouslyseen with White sources however the question might beasked more generally in specific terms as explained in thenext section. IV. MAXIMUM LUMINOUS EFFICACY OFRADIATION SOURCES
An important question can now be asked: For a givencolor (chromaticity), is there a maximum PER that canbe realized with any radiation source?Mathematically this can be answered with the follow-ing set of assumptions: Given a source endowed with anormalized power emission spectrum function P ( λ ): (cid:90) λ max λ min P ( λ ) dλ = 1 (14)Is it possible to find the best P ( λ ) such that the PERgiven by: Y = K m (cid:90) λ max λ min P ( λ )¯ y ( λ ) dλ (15)for some given color represented by chromaticity coor-dinates x c , y c (see Appendix) is maximized. Note thatthis stems from the fact ¯ y ( λ ) = V ( λ ) as explained in theAppendix.Thus we have an optimization problem for an unknownfunction P ( λ ) subject to three constraints: normalizationeq. 14 and fixed color ( x c , y c ) constraints eq. A2.Following Ohta et al. suggestion , we transform theproblem into its discrete version by dividing the wave-length interval [ λ min , λ max ] into N values λ i with a step∆ λ such that the objective function to be optimized is:max (cid:90) P ( λ )¯ y ( λ ) dλ → max N (cid:88) i =1 P i ¯ y i (16)Discrete values P i , ¯ x i , ¯ y i , ¯ z i correspond to P ( λ ) , ¯ x ( λ ) , ¯ y ( λ ) , ¯ z ( λ ) taken at λ = λ i .The normalization constraint eq. 14 becomes:∆ λ N (cid:88) i =1 P i = 1 (17)whereas the chromaticity constraints (see Appendix)become: x c = (cid:80) Ni =1 P i ¯ y i (cid:80) Ni =1 P i (¯ x i + ¯ y i + ¯ z i ) y c = (cid:80) Ni =1 P i ¯ y i (cid:80) Ni =1 P i (¯ x i + ¯ y i + ¯ z i ) (18)This can be transformed into: N (cid:88) i =1 P i [ x c (¯ x i + ¯ y i + ¯ z i ) − ¯ x i ] = 0 , N (cid:88) i =1 P i [ y c (¯ x i + ¯ y i + ¯ z i ) − ¯ y i ] = 0 (19)The problem now is expressed in the standard Simplexform (see Numerical Recipes , chapter 10): One oughtto find a set of N values P i such that eq. 16 is maximizedunder three constraints given by eq. 17 and eqs. 19.The Simplex method results are displayed in fig. 5.The constant PER curves or iso-PER curves representedin the CIE diagram (see Appendix) get closer to the CIEcontour as the PER is increased.Low values of PER are in the blue region of the CIEdiagram which explains the result obtained for the bluediode, whereas larger PER values occur as we move to-ward the yellow part of the CIE diagram.We find the largest value of 679 lm/Watt, as before,and the corresponding iso-PER curve in the vicinity ofthe 555 nm region, the area of highest sensibility of theeye, confirming the candela standard once again and theSI metrological data . y x555 nm 650 lm/W600 lm/W550 lm/W500 lm/W450 lm/W400 lm/W350 lm/W300 lm/W250 lm/W200 lm/W150 lm/W100 lm/WCIE-contour CIE-base FIG. 5. (Color online) Constant PER curves in the CIE dia-gram. The maximum value of 679 lm/Watt is attained aroundthe 555 nm region (red spot) as adopted by the SI system.Waviness observed in some curves around borderline are dueto slow convergence of the optimization simplex algorithm.
V. DISCUSSION AND CONCLUSION
Some lighting metrics have been introduced and per-spectives for future development regarding the increaseof lighting intensity and quality were presented.Despite the fact White LED is presently showing tremen-dous potential in terms of quality and PER increase, thediagram presented in fig. 5 indicates that a white sourcemaximum is between 350 and 400 lm/Watt which is yetto be reached by White LED’s.On the other hand, Yellow-Green light sources mayachieve a large increase since their maximum PER isaround the theoretical standard of 683 lm/Watt adoptedby both CIE and SI organizations.The conclusion is that work must be targeted toward in-creasing rather the conversion efficacy η C in the Whitesources case and in particular the White LED case. Appendix A: CIE Chromaticity Coordinates
In 1931, the CIE undertook a series of historical mea-surements called Color Matching experiments in orderto calibrate colorimetry and human color perception. Anumber of ”standard” observers had to make a com-parison between a color of a given wavelength λ and asuperposition of three selected wavelengths called RGBprimaries . The weight of each of the three colors toperform the match was recorded. The observations weredone at a fixed distance of 50 cm with two possibilitiesfor the eye angle opening (defined by the observation di-ameter value) of 2 ◦ and 10 ◦ .This led to the existence of color matching functions(CMF) ¯ r ( λ ) , ¯ g ( λ ) , ¯ b ( λ ) corresponding to the red, green,blue weight coefficients that matched the color λ .The study showed that not all matching weights arepositive and that some values were negative. The alge-braic values of the coefficients meaning that CMF func-tions took positive and negative values originating fromthe overlap versus wavelength between human cone sen-sitivities.This pushed the CIE to perform a linear transfor-mation over ¯ r ( λ ) , ¯ g ( λ ) , ¯ b ( λ ) in order to define threestrictly positive CMF functions ¯ x ( λ ) , ¯ y ( λ ) , ¯ z ( λ ) displayedin fig. 6.The linear transformation is based on equal area of¯ x ( λ ) , ¯ y ( λ ) , ¯ z ( λ ) over the visible spectrum and the choicefor the middle spectrum function ¯ y ( λ ) to be taken equalto V ( λ ), the photopic eye sensitivity.If we have a radiation source characterized by a poweremission spectrum function P ( λ ) its tristimulus coordi-nates are given by: X = K m (cid:90) λ max λ min P ( λ )¯ x ( λ ) dλ,Y = K m (cid:90) λ max λ min P ( λ )¯ y ( λ ) dλ, ¯ x ( λ ) , ¯ y ( λ ) , ¯ z ( λ ) λ (nm) ¯ x ( λ )¯ y ( λ )¯ z ( λ ) FIG. 6. (Color online) Color matching functions of the CIEfor eye opening of 2 ◦ . ¯ x in red, ¯ y in green and ¯ z in blue coverapproximately the corresponding RGB color zones. Z = K m (cid:90) λ max λ min P ( λ )¯ z ( λ ) dλ, (A1) xy D50D65
D93
FIG. 7. (Color online) CIE diagram displaying color of pointswith coordinates ( x, y ) and the Black-Body radiation colorpath as a function of absolute temperature. The various sym-bols D T correspond to Daylight type source (illuminant orsynthetic source) at a given temperature T / is for T=6500K (adapted from Hoffmann ). The color of the P ( λ ) source is given by a point withcoordinates ( x, y ) in the CIE diagram displayed in fig. 7.( x, y ) are called chromaticity coordinates with values ex-plicitly given by: x = XX + Y + Z , y = YX + Y + Z (A2)The CIE diagram (called tongue or horseshoe diagram)shown in fig. 7 displays several interesting characteristics:1. The contour contains pure colors (completely sat-urated or free of any white content) having wave-lengths indicated on the borderline in nanometers.The corresponding wavelengths are called domi-nant since they control the color from pure (on theborder) to White point at the center with coordi-nates x = , y = .2. Colors within the horseshoe diagram are unsatu-rated and as we move forward toward the Whitepoint they become pastel like. This stems from theincrease of white content as we proceed toward thewhite point.3. Black-Body color appears on a path as a functionof absolute temperature. It follows the red, orange, yellow, white and finally blue sequence as tempera-ture is increased. This describes heated metals andagrees with Kirchhoff observations .4. Illuminants (artificial daylight sources) indicatedby D , D and D appear at their correspond-ing colour with index (50,65,93) equal to absolutetemperature divided by 100. Black-Body sourceswith temperatures of 5000K, 6500K and 9300Khave colours close to the White point.5. The CIE contour is closed from below by a straightline (called also the purple line) that does not carryany dominant wavelength. It means that most pur-ple colors cannot be obtained by altering the Whitecontent of some main (dominant) color as done be-fore. This is another consequence of the cone over-lap that resulted in negative CMF weights. ∗ [email protected] J. Ouellette, Phys. Today, December 2007 p. 25. CIE is
Commission Internationale de l’Eclairage or Inter-national Organization for Lighting based in Vienna (Aus-tria) that sets standards for light and colors and is respon-sible of the metrology of lighting like the NIST (NationalInstitute of Standards and Technology) in the US. F. Grum and R. J. Becherer,
Optical radiation measure-ments, Volume 1: Radiometry; Volume 2: Color Measure-ments , Academic Press (New-York, 1979). G. Kirchhoff,
On the relation between the radiating andabsorbing powers of different bodies for light and heat , Phil.Mag. Series 4, Volume 20, 1 (1860) (available on Googlebooks). Photons have the dispersion relation ω ( k ) = c | k | and ac-cording to Kittel the density of states g ( ω ) in d dimen-sions for a system of length L is given by: g ( ω ) = (cid:18) L π (cid:19) d (cid:90) ω<ω ( k ) <ω + dω dS ω v g (A3)where dS ω is the differential area element on the constantenergy surface ω ( k ) = ω and v g the group velocity. Sincethe dispersion is linear in k , v g = c and the constant energysurface is a sphere of radius k = ωc , thus S ω = 4 πk =4 π ( ωc ) . Photons are zero-mass particles with spin S = 1 and a number of polarizations N p = 2. Should their massbe finite, they would have had N p = 2 S + 1 = 3. Thus, thedensity of states in 3D per unit volume L is: g ( ω ) = N p (cid:18) π (cid:19) c πω c = ω π c (A4) Report of the 21st meeting (23-24 February 2012) of theConsultative Committee for Photometry and Radiome-try (CCPR), Bureau International des Poids et Mesures(2012). D. C. Agrawal, H. S. Leff and V. J. Menon, Am. J. Phys. , 649 (1996). G. Hoffmann,
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Second Edition, Cambridge University Press(New-York, 1992). R.D. Edge and R. Howard, Am. J. Phys. , 142 (1979). C. Kittel,