Wavelength

Wavelength- and Frequency-Dependent Formulations of Wien ’ s Displacement Law

Ranjan Das*

Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 40005, India

ABSTRACT: Interrelations between the wavelength- and frequency- dependent formulations of Wien’s displacement laws have been derived from the corresponding energy distribution functions of Planck’s blackbody radiation law. Mathematical aspects of the transformation of functions have been illustrated using a simple function. The importance of including the infinitesimal of the independent variable of the distribution function has been highlighted. Some erroneous statements relating to Wien’s displacement law in certain textbooks have been addressed. An iterative method has been described tofind the roots of equations containing exponentials and obtain the values of λmax and νmax, the points at which Planck’s energy distribution functions are maximized. From this, the reason behind the apparent anomaly of the product λmax ×νmax not being equal to the velocity of light has been established.

KEYWORDS: Upper Division Undergraduate, Physical Chemistry, Misconceptions/Discrepant Events, Textbooks/Reference Books, Quantum Chemistry, Continuing Education

T

he spectral characteristics of blackbody radiation, its role in Planck’s proposal of the quantum hypothesis, and his distribution law are almost always the gateway of all undergraduates to the quantum world. The total energy emitted by the blackbody being proportional to the fourth power of the absolute temperature is one experimentally observed property. The other is that the wavelength (λmax) corresponding to the maximum of the emission distribution, plotted as a function of the wavelength of the emitted radiation, is inversely proportional to the absolute temperature. Before Planck’s proposal of the quantum origin of the blackbody spectrum, thermodynamic arguments were advanced to explain these observed characteristics of blackbody radiation. However, with Planck’s formulation of the energy distribution based on the quantum hypothesis, these properties are easily derived from Planck’s law.

The displacement law in terms of wavelength, as originally derived by Wien,¹has the mathematical form

λ_max×T=const₁ (1)

In a recent article, Ball gave an alternative form of the displacement law corresponding to the maximum (νmax) of the energy distribution plotted in terms of frequency²

ν =

T const

max

2 (2)

An important outcome of Ball’s calculations is that, although the wavelength and the frequency of light are related byλ=c/ν, λmax of eq 1 and νmax of eq 2 are not similarly related. The product of the two constants const₁andconst₂is notc, but an unfamiliar number, 1.704 × 10⁸ m/s. The author correctly

states that the difference in the two formulations arises from the way that Planck’s distribution law differs according to whether it is expressed in terms of wavelength or frequency. The author gives the frequency-dependent Planck’s distribution law as

= πν

ν −

⎛

⎝⎜ ⎞

⎠⎟

U h

c e

8 1

h kT 1

3

3 / (3)

and the wavelength-dependent law as

= π λ

λ −

⎛

⎝⎜ ⎞

⎠⎟

U h

c e

8 1

hc kT 1

3

3 / (4)

Ball ends his article by noting that “because both forms of Planck’s law must be maximized numerically, the constant defined byλmax×νmax, 1.704×10⁸m/s, cannot be expressed exactly in terms of fundamental constants,”and concludes that λmax and νmax “occur at seemingly mathematically unrelated points.”² Unfortunately, however, Planck’s energy density distribution as a function of the wavelength is given not by eq 4, but by

π

= λ

λ −

⎛

⎝⎜ ⎞

⎠⎟

U ch

e

8 1

hc kT 1

5 / (5)

In my experience, many elementary students of this subject replace the frequencyνin eq 3 byc/λin an attempt to arrive at eq 5, only to realize the futility of this. This leads many of them to erroneously believe the frequency-dependent relation (eq 3) and the wavelength-dependent relation (eq 5) to be two

pubs.acs.org/jchemeduc

© XXXX American Chemical Society and

Division of Chemical Education, Inc. A DOI: 10.1021/acs.jchemed.5b00116

J. Chem. Educ.XXXX, XXX, XXX−XXX

different formulations of Planck’s law without any internal mathematical relationship between them. Moreover, not many textbooks discuss both formulations of Planck’s law or Wien’s displacement law. In Zettili’s book,³ an expression for λmax

corresponding to the maximum of Planck’s energy distribution has been derived from eq 5

λ = hc = × ⁻

k T T

4.9663

1 2898.9 10 m K

max

6

(6) From this, the expression for the maximum of the frequency distribution (eq 3) is written as

ν = λc =

h kT 4.9663

max

max (7)

which is not correct, however, as shown in Ball’s derivation.² Another book⁴ gives an approximate expression for the maximum of the frequency-dependent Wien’s law (eq 3) as

ν ≈

h kT

( )_peak 2.82 ₍₈₎

but then goes on to state:“[A]ccordingly,λmT=hc/2.82k, and on comparing with the experimentally determinedλmT= 0.29 cm K, gives us the valueh/k≈0.27×10⁻¹⁰K s.”This value is wrong, however, as can be seen by inserting the values of Planck’s constant and the Boltzmann constant. The error arises by equatingνpeakof eq 8 withc/λm.

Transforming a function by changing its independent variable to another independent variable is a common exercise in high school mathematics. For example,f(x) may be changed to g(z), where x is a function of z, say x = h(z). In such transformations, the properties of fand the locations of the extrema of f in particularare transformed to those of g. At first glance, it may appear surprising, and perhaps confusing, that νmax and λmax in the two formulations of Wien’s displacement law are not related byνmax =c/λmaxeven though ν = c/λ. In this article, I attempt to explain this apparent anomaly, which arises from Planck’s law being a distribution function rather than an ordinary function. Using a simple function as an example, I first show how the positions of its extrema change when the function is transformed by changing its independent variable. I then show how the above result changes when the function is interpreted as a distribution function. This allows derivation of the relationship between the Planck distribution functions in terms of wavelength and frequency, from which the two formulations of Wien’s displacement law are seen to emerge naturally.

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TRANSFORMATION OF A FUNCTION BY CHANGING ITS INDEPENDENT VARIABLE

Consider the functionU(x) = x² exp(−x); 0 <x<∞. If the independent variable x is substituted by 1/z, U(x) is transformed to U(z) = (1/z²) exp(−1/z); 0 < z < ∞. The functions U are shown in Figure 1. The extrema of the functions can be located by differentiating them with respect to the independent variable. Thus, the solutions of

= = U x

x

U z z d ( )

d 0 d ( )

d ⁽⁹⁾

will give the locations of their extrema, including those of the maxima. Thus,

= − − =

U x

x x x x

d ( )

d (2 ) exp( ) 0

(10)

givesx_max= 2, and

= ^⎜⎛− + ^⎟=

⎝

⎞

⎠ U z

z

U z

z z

d ( ) d

( ) 2 1

0 (11)

gives z_max = 1/2. It is seen that z_max = 1/x_max, which is in accordance with the relation z = 1/x between the two independent variables.

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BEHAVIOR OF A DISTRIBUTION FUNCTION WHEN THE INDEPENDENT VARIABLE IS CHANGED In this section, we use the letter P to denote a probability distribution function. IfU(x) is a distribution function, possibly un-normalized, of xin the range 0 < x< ∞, the probability, P(x)dx, thatxlies in the rangextox+ dxwill be proportional to U(x)dx. Without loss of generality, we take the proportionality constant to be unity and write

=

P x( ) dx U x( ) dx (12)

Ifxis substituted with 1/zin the functionU(x) =x²exp(−x) of the previous section, the new distribution function should describe the probabilityP(z)dzthatzlies in the rangeztoz+ dz. As noted above,U(z) = (1/z²) exp(−1/z). Becausex= 1/z, the infinitesimal dxbecomes

= −^⎜⎛ ^⎟

⎝

⎞ x ⎠

z z

d 1

2 d (13)

The negative sign above indicates that the infinitesimals dxand dzchange in the opposite sense: an increase in dx leads to a decrease in dz, and vice versa. Thus, for the probability, we use only their magnitude and ignore the negative sign. The probability distribution in terms ofzis then given by

= −

=

⎜ ⎟⎜ ⎟

⎛

⎝

⎞

⎠

⎛

⎝

⎞ P z z ⎠

z z

z z U z

z z

( ) d 1

exp( 1/ ) 1 d ( ) d

2 2

2 (14)

The functionsP(x) andP(z) are shown in Figure 2. Equations 12 and 14 show that whereasP(x) is equal toU(x),P(z) is not equal toU(z) but rather toU(z)/z². This is a consequence of the relation between the infinitesimals dxand dzgiven by eq 13. Equations 12 and 14 also show that whereas the maxima of P(x) can be determined from those ofU(x) the maxima ofP(z) Figure 1.Plots of ordinary mathematical functionsU(x) =x²exp(−x) andU(z) = exp(−1/z)/z²;z= 1/x.

DOI: 10.1021/acs.jchemed.5b00116 J. Chem. Educ.XXXX, XXX, XXX−XXX B

are rather determined by U(z)/z². Therefore, although the maximum ofP(x) is given by eq 10 and occurs atx_max= 2, the maximum of P(z) is given by

= − +

= − + − +

= − + =

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝⎜ ⎞

⎠⎟⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠ P z

z z U z

z U z

z

z U z z

U z

z z

z U z

z d ( )

d

2 ( ) 1 d ( ) d

2 ( ) 1 ( )

2 1

4 1 ( )

0

3 2

3 2

3 (15)

which occurs atz_max= 1/4. This value ofz_maxis not related to x_max = 2 (from eq 10) or z_max = 1/2 (from eq 11) in any obvious manner.

To see the interrelation between x_max and z_max as obtained above, let us consider a general function U(x) whose independent variable x is changed to z through x = f(z).

Then, while P(x)dxis given by U(x)dx,P(z)dzis given by

=

P z z U z f z

z z

( ) d ( )d ( )

d d

(16) because dx= (df(z)/dz)dz. For the extremum ofP(z),

= × +

P z z

U z z

f z

z U z f z

z d ( )

d

d ( ) d

d ( )

d ( )d ( ) d

2

2 (17)

is equated to zero. Because

= × = ×

U z z

U f z f z

f z z

U x x

f z z d ( )

d

d ( ( )) d ( )

d ( ) d

d ( ) d

d ( )

d ₍₁₈₎

eq 17 can also be written as

= ⎛ +

⎝⎜ ⎞

⎠⎟ P z

z

U x x

f z

z U x f z

z d ( )

d

d ( ) d

d ( )

d ( )d ( )

d

2 2

2 (19)

and the condition for the extremum ofP(z) becomes

+ =

⎛

⎝⎜ ⎞

⎠⎟ U x

x f z

z U x f z

z d ( )

d

d ( )

d ( )d ( )

d 0

2 2

2 (20)

While the extremum ofP(x) is given by dU(x)/dx= 0, eq 20 is an entirely different relation involving dU(x)/dx. Equation 20 also shows that whenf(z) is a linear function ofz, (d²f(z)/dz²)

becomes zero, and the extrema of both P(x) and P(z) are decided by the condition dU(x)/dx = 0 itself. For arbitrary functions x= f(z), d²f(z)/dz²will not, in general, be zero; in that case, dP(x)/dx= 0 and dP(z)/dz= 0 will give different values forxandz. As eqs 17 and 19 are different manifestations of the same relation, either can be used to obtain the conditions for the extremum. For the functionsU(x) =x²exp(−x) andx= 1/z, for example, eq 17 gives

=^⎜⎛ − ^⎟

⎝ ⎞

⎠ P z

z z z z

d ( )

d 4 1 1

exp(1/ )

5 (21)

and eq 19 gives

= − −

P z

z x x x

d ( )

d (4 ) ⁵exp( )

(22) where df(z)/dz=−1/z²=−x²and d²f(z)/dz²= 2x³have been used. The maximum occurs at z_max = 1/4 or, equivalently, at x_max= 4.

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^WIEN’S DISPLACEMENT LAW

Planck’s energy distribution function in terms of frequency, U(ν), is given by

ν ν πν

= ν

ν −

⎛

⎝⎜ ⎞

⎠⎟

U h

c e

( ) d 8 1

1 d

h kT 3

3 / (23)

which is eq 3 with the freqency dependence shown explicitly and the infinitesimal frequency dν included. As seen above, when this distribution function is transformed to a function of wavelength, bothU(ν) and dνmust be appropriately modified.

Becauseν=c/λ, then dν=−(c/λ²) dλ. Once again, we ignore the negative sign associated with dν, for the reasons given earlier. Thus, Planck’s distribution function in terms of wavelength becomes

λ λ π

λ λ

= ^λ −

⎛

⎝⎜ ⎞

⎠⎟

U ch

( ) d 8 e 1

1 d

hc kT

5 / (24)

Definingx_ν≡hν/kTandx_λ≡hc/λkT,

ν ∝

−

ν

U νx

( ) e

x 1

3

(25) and

λ ∝

−

λ

U λx

( ) e

x 1

5

(26) The maxima of U(ν) and U(λ) can be obtained by setting dU(ν)/dx_νand dU(λ)/dx_λto zero. This gives the relations

− =

ν ^ν

ν

x e

e 1 3

x

x (27)

and

− =

λ ^λ

λ

x e

e 1 5

x

x (28)

from eqs 25 and 26, respectively. The two relations are almost indentical; the difference of 2 units in their numerical values is the result of replacing dν in eq 23 by (c/λ²)dλto arrive at eq 24. Thus, the two formulations of Wien’s displacement law have no real difference. The reason for their apparently unrelated maxima is nothing more than the difference in the numerical values in eqs 27 and 28.

Figure 2. Plots of distribution functions P(x) = U(x) and P(z) = U(z)/z²;z= 1/x.

DOI: 10.1021/acs.jchemed.5b00116 J. Chem. Educ.XXXX, XXX, XXX−XXX C

Equations 27 and 28 have to be solved forx_νandx_λto obtain the location of the maxima ofU(ν) andU(λ). Ball has solved eq 27 numerically using an Excel spreadsheet in a stepwise manner that gradually improves the accuracy of the root.² Williams later showed how the root could be obtained using the LambertWfunction.⁵ Whereas the spreadsheet procedure involves “hunting” for an improved root, the Lambert W function using complex analysis would be unfamiliar to most students. I propose here the following useful, elegant, and easily grasped iterative procedure.

Equations 27 and 28 are rewritten as

= − −

x N(1 exp( x)) ₍₂₉₎

withN= 3 and 5, respectively. We start with a guessed value of xand insert it on the right side of eq 29 to obtain a new value ofx. This new value ofxis then inserted on the right side of eq 29 to obtain the next new value ofx. This process is repeated until two successive values of xdiffer by less than the desired accuracy. Thus, ifx_iis the value ofxin thei-th step, the next valuex_i+1is obtained from the relation

= − −

x_i+₁ N(1 exp( x_i)) ₍₃₀₎

The initial guess can be any positive number but convergence will be faster if the guessed value is reasonably close to the correct value. With an initial guess for x₀to be N, successive values of the interative solution of eq 29 are listed in Table 1.

For an accuracy better than 1 part in 10⁷, convergence is reached in 10 steps for eq 27 and in 6 steps for eq 28. The roots are found to bex_ν,max= 2.8214393···andx_λ,max = 4.9651142···

(to 8 significant figures). From these, the maxima of the frequency and wavelength distributions of Planck’s law are given by

ν = _ν λ =

λ

⎜ ⎟ ⎜ ⎟

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠ kT

h x hc

and kT x1

max ,max max

,max (31)

and the corresponding formulations of Wien’s displacement law are

ν =^{⎜ ⎟}⎛ _ν = × ^{− −}

⎝

⎞ T ⎠

k

h x 5.87895 10 K s

max

,max 10 1 1

(32) and

λ × = = ×

λ

⎜ ⎟ −

⎛

⎝

⎞ T hc⎠

k x

1 2.89976 10 m K

max

,max

3

(33)

Equation 32 is the same as that derived by Ball,²and eq 33 is the original formulation of Wien’s displacement law.¹ Multi- plying these two relations, we get

ν ×λ = ^ν = ×

λ

⎛ −

⎝⎜⎜ ⎞

⎠⎟⎟

c x

x 1.704757 10 m s

max max

,max ,max

8 1

(34) Equation 34 explains why the product ofνmax andλmax is not equal to the velocity of light, and also shows how the unfamiliar number 1.704757 ×10⁸m s⁻¹arises. The inequality ofx_ν,max and x_λ,max is due to the different numerical values associated with eqs 27 and 28. This difference, in turn, arises when transforming Planck’s distribution function to express it in terms of wavelength rather than frequency, which also requires the infinitesimals of the independent variables to be appropriately transformed.

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^CONCLUSION

The analyses described here show that the interrelation between the two formulations of Wien’s displacement law arises naturally from Planck’s distribution law. The significance of the latter being a distribution function and not just a mathematical function is underlined. This example could be an insightful mathematical exercise for students to study how functions transform with a change of variable. Distribution functions occur in several areas of science. Such distribution functions should always be written in the form U(x)dx (i.e., along with the infinitesimal of the independent variable), so that when the independent variable changes, the distribution function will also change correctly. An exercise of this kind pertaining to kinetic theory of gases has been given in this Journal.⁶ Converting electronic absorption and fluorescence spectra generally recorded as a function of wavelength to a function of wavenumber is another common task in spectroscopic calculations. Angulo et al. critically discuss the possible errors if care is not taken to include the infinitesimal of the variable during the transformation.⁷The numerical exercise for solving eqs 27 or 28 can be used as an illustrative example for iterative calculations. Such calculations are widely used in quantum mechanical self-consistent field calculations, which students will encounter in later years.

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AUTHOR INFORMATION Corresponding Author

*E-mail: ranjan@tifr.res.in.

Notes

The authors declare no competingfinancial interest.

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(1) Wien, W. Temperatur und Entropie der Strahlung.^REFERENCES Ann. Phys.

(Berlin, Ger.)1894,288, 132−165.

(2) Ball, D. W. Wien’s Displacement Law as a Function of Frequency.

J. Chem. Educ.2013,90, 1250−1252.

(3) Zettili, N.Quantum Mechanics: Concepts and Applications; John- Wiley: Chichester, England, 2001; p 9.

(4) Dutta-Roy, B. Elements of Quantum Mechanics; New Age International Publishers: New Delhi, India, 2009; p 21.

(5) Williams, B. W. A Specific Mathematical Form for Wien’s Displacement Law as v_max/T = Constant. J. Chem. Educ. 2014, 91, 623−623.

(6) Marron, M. T. Extraterrestrial Kinetic Theory of Gases.J. Chem.

Educ.1983,60, 526.

Table 1. Iterative Solutions of Eqs 27 and 28 Using the Iteration Method Given by Eq 30 withN = 3 and 5, Respectively

stepi values ofx_i(N= 3) values ofx_i(N= 5)

0 3.0 5.0

1 2.8506 4.9663

2 2.8265 4.9651

3 2.8223 4.965113

4 2.82159 4.9651142

5 2.821466 4.96511423

6 2.821444 4.965114231

7 2.821440

8 2.8214394

9 2.82143938

10 2.82143937

DOI: 10.1021/acs.jchemed.5b00116 J. Chem. Educ.XXXX, XXX, XXX−XXX D

(7) Angulo, G.; Grampp, G.; Rosspeintner, A. Recalling the Appropriate Representation of Electronic Spectra. Spectrochim. Acta, Part A2006,65, 727−731.

DOI: 10.1021/acs.jchemed.5b00116 J. Chem. Educ.XXXX, XXX, XXX−XXX E