Planck's formula (English)

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3-4: Planck's Formula

On the preceding page,

we studied about

the heat capacity of

a cavity

or a vacuum.

The result said that,

if we naively apply

the law of equipartition

of energy,

the total energy

of a vacuum (cavity)

becomes infinite.

This is completely

different from the reality.

Is the law

of equipartition of energy

inapplicable for the

cavity radiation?

The law is naturally derived

from the classical theory

consisting of

Newtonian mechanics

and Maxwellian electromagnetism.

This seems to suggest

that there must

be something wrong

in the classical theories.

Then, where is the secret hidden?

This is one

of the biggest problems

in physics at the end

of the 19th century.

[Rayleigh-Jeans's Formula]

As discussed

on the preceding page,

Maxwellian electromagnetism

gives the

number of

the normal oscillations

with the frequency

between

and

per unit volume as

If the constant energy kT

is partitioned

to all these normal oscillations

according to the

law of equipartition of energy,

then the energy

of the radiation (or light)

with the frequencies

between

and

per unit volume becomes

This is the Rayleigh-Jeans formula

proposed by

J. W. S. Rayleigh

(UK, 1842 - 1919)

and

J. H. Jeans

(UK, 1877 - 1946).

If we measure the frequencies

and intensities of

lights existing

inside a high-temperature

space like a blast furnace

in an iron industry,

we can obtain

the spectrum

(intensity distribution

for each frequency of light).

The experimental data

are shown in the following

figure.

Experimental spectra of the cavity radiation

The positions of

the hill tops

of the solid curves

on the horizontal axis

denote the frequency

of the brightest light.

As the absolute temperature

T of each curve

becomes high,

the frequency

of the brightest light

shifts gradually to higher.

This means

that the color

of cavity changes

from red

to white,

as the temperature

becomes higher.

The dashed curve

shows the values

of Rayleigh-Jeans's formula

for T = 1646 K.

For example,

when heating up

an iron block,

the color of the ion

is gray

at a low temperature.

It would be bright red

at about

and become white

and dazzling at

This feature

is shown in

the above figure.

In the above figure,

the values of

the Rayleigh-Jeans formula

are shown by a

dashed curve.

It is well fit

to experimental data

at low frequencies,

but becomes worse

at high frequency.

This appears to imply that

the law of equipartition of energy is not valid for high frequencies.

[Wien's Formula]

W. Wien

(Germany, 1864 - 1928)

extended a very ingenious

discussion of generality

to obtain a formula

describing the intensity

distribution

in the cavity radiation

(1896).

Since Wien's idea

is somewhat difficult

to explain simply,

we omit it and represent

only the result here.

According to Wien,

the energy distribution

of the cavity radiation

for a unit volume

is given by

We cannot obtain

the form of the function

F(x)

only from Wien's discussion.

However the most important

is that

the functional value

is determined only by

the ratio of the frequency

to the temperature

T .

This is quite well fit

to experiment.

This law is called

Wien's law

(or Wien's displacement law).

Let the wave length

of the brightest light

in the cavity radiation be .

Differentiating the above

Wien's law by

we can easily obtain

This is often called

Wien's displacement law.

Setting

F(x) = k/x

in Wien's law,

we have

the Rayleigh-Jeans formula.

Wien took

and obtained Wien's formula

If the constant

is adjusted

to an appropriate value,

the values of

Wien's formula

are well fit to

the experimental data

at the high-frequency region.

[Planck's Formula]

As discussed above,

the Rayleigh-Jeans formula

concerning the cavity radiation

gives very good fit

to experiment

in the low frequency region.

On the other hand,

Wien's formula

is very good

in the high frequency region.

Then, M.K.E.L. Planck

(Germany, 1858 - 1947)

proposed an interpolating formula

which can unify

these two formulae

and

brings about very

good fit to experiment over

all regions of frequencies.

This is the famous

Planck's formula

which is

one of the biggest discoveries

at the end of the 19th century.

Planck proposed

that the function F(x)

in the above mentioned

Wien's law

should be taken as

Accordingly, the energy

distribution in the cavity

radiation becomes

This is Planck's formula,

and the constant k

is Boltzmann's constant

that is a universal constant,

Another constant

is adjustable,

which

is usually written ,

where

h is called

Planck's constant

whose value

is given as

You can easily see in

the following figure

how well Planck's formula

reproduces

the experimental data.

Comparison between the experimental spectrum of the cavity radiation and Planck's formula.

The small circles

are experimental data

and the solid

curves denote

the values of

Planck's formula.

Be careful the

abscissa

shows the wave length,

not the frequency.

[Comparison between the Three Formulae]

As discussed above,

Rayleigh-Jeans's formula

shows a good fit

to experiment

in the low-frequency region

and Wien's formula

is good in the

high-frequency region.

Planck's formula

can reproduce the experimental

energy distribution

throughout all frequency regions.

A comparison between

these three formulae

is given

in the following figure.

You can see

from this figure that

Planck's formula

is just an interpolation

of Rayleigh-Jeans's formula

and Wien's formula.

Comparison between Rayleigh-Jeans's, Wien's and Planck's formula

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