# Atomic And Nuclear Physics Engineering Essay

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Blackbody Radiation A black body is a theoretical object that absorbs 100% of the radiation that hits it. Therefore it reflects no radiation and appears perfectly black. In practice no material has been found to absorb all incoming radiation, but carbon in its graphite form absorbs all but about 3%. It is also a perfect emitter of radiation. At a particular temperature the black body would emit the maximum amount of energy possible for that temperature. This value is known as the black body radiation. It would emit at every wavelength of light as it must be able to absorb every wavelength to be sure of absorbing all incoming radiation. The maximum wavelength emitted by a black body radiator is infinite. It also emits a definite amount of energy at each wavelength for a particular temperature, so standard blackbody radiation curves can be drawn for each temperature, showing the energy radiated at each wavelength. All objects emit radiation above absolute zero. [1]

"Blackbody radiation" or "cavity radiation" refers to an object or system which absorbs all radiation incident upon it and re-radiates energy which is characteristic of this radiating system only, not dependent upon the type of radiation which is incident upon it. The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating. [2]

The amount of radiation given off in a particular frequency band should be proportional to the amount of modes that is in that range. Classical physics suggested that all modes have an equal chance of been produced and that the number of modes goes up proportional to the square of the frequency

http://hyperphysics.phy-astr.gsu.edu/hbase/imgmod/bb2b.gifFigure (1)

But the continual increase in predicted radiated energy with frequency called the (ultraviolet catastrophe) did not happen.

Modes per unit frequency per unit volume

Probability of occupying modes

Average energy per mode

Classical

Equal for all modes

kT

Quantum

Quantized modes;

Require hÎ½ energy to excite upper modes, less probable

Table (1)

## Experimental set-up

DetectorFigure (2)

A simple experiment set up to detect blackbody radiations to have an object (T1) maintained at a constant temperature and the radiation that is been emitted by the light source will be detected by the apparatus that is sensitive at that particular wavelength following this a prism is used to separate the wavelength (Î») at a variety of different angles (Î¸).The wavelengths that will be detected will be quiet large so to make this easier we will measure the dÎ» difference in wavelengths around Î» .This will be repeated for a range of different angles and so in turn it will be done for a range of values of Î»

The quantity that will be measured by the detector for this will be

The results that will be obtained will be for a series of values R dÎ» for each of the different values of Î» that are measured .we repeat this with a variety of different temperatures and the data that is gathered can be plotted on a graph of radiant intensity R against Î» with the outcome looking as follows

This result of this experiment tells us two important results the first been

That the total radiant intensity (R) increases over all the wavelengths as the fourth power of temperature (T) this is equal to the total intensity is the area under the curve which gives us

ÏƒT4

R = radiant intensity R

Ïƒ = Stefan-Boltzmann constant = 5.6703 x 10-8 W/m2 K4

T = Temperature (Kelvin)

This is known as Stefan's Law and is also written in the form of

P = ÏƒT4 Â· A

The second thing that was learned from this experiment was that the wavelength with the most intensity Î»max (in meters) decreases in inverse proportion as the temperature increases so therefore

Î»max

So the constant of proportionality was determined by experimentation so

Î»max T = 2.898 x 10-3 mÂ·K

m = meters K = temperature

This experiment can be simulated on the link below on the following site to help better understand these concepts

Figure (4)

To recap what we know so far as the temperature increases the total energy increases and the peak of the spectrum shifts to smaller wavelengths towards the blue side of the spectrum.

Figure (5)

The dependence of R on Î» that is described by Stefan's and Wien's Laws is an empirical result that is achieved we need to explain these results based on the theories of thermodynamics and electromagnetism

To solve this problem we construct a cavity in which to trap the radiation which is perfectly absorbing and emitting as shown in Figure (2) it should be remembered that it is the hole and not the box that acts as a blackbody. Radiation outside strikes the hole and gets trapped inside the cavity with a very small chance of escaping again then it is assumed there will be no reflections from the blackbody.

If our blackbody box is filled with electromagnetic standing waves and the walls are made up of a metallic material then the electromagnetic radiation reflected back and forth with a mode of an electric field at each wall

The formula used to test this was to find out the number of standing waves N with wavelengths that are between Î» and dÎ» is

N (Î») dÎ» = dÎ»

http://hyperphysics.phy-astr.gsu.edu/hbase/imgmod/ewav1.gifFigure (6)

For the Classical thermodynamic result it was taught that each wave contributes energy kT to the radiation in the box. The radiation in the box is at thermal equal with the wall temperature of the box. In that the radiation is reflected from the walls and then it is absorbed and re-emitted then quickly by the atoms The atoms are said to oscillate at the same frequency of that of the radiation and the average thermal KE (Kinetic Energy) of the atom is equal to half the kT (Boltzmann constant and the absolute temperature) a result which is similar for ideal gas. So from the SHM (Simple Harmonic Motion) we can say that the mean KE is equal to the mean PE .Therefore the mean of the total energy was equal to kT

What was done after this was to find the R (radiant intensity) from the energy density and multiply this by

Is a factor from geometry consideration of energy in a volume that is linked to Stefan's law

So after we consider all this information we are left with

Radiant intensity(R) equal number of waves per unit volume multiplied by the energy per wave multiplied by radiant energy per energy density

And we get

R (Î») = = Rayleigh-Jeans Law

R = Radiant intensity

Î» = wavelength

k = Boltzmann constant

T = absolute temperature

c = speed of light

The problem with using the Rayleigh-Jean formula was that at long wavelengths ,radiant intensity (R) predicts the same results as the real data however at shorter wavelengths this does not work and a failure know as the ultraviolet catastrophe happens shown below

http://www.haverford.edu/physics/songs/cavendish/uvcat.jpgFigure (7)

This happened due to there been too much radiant intensity been predicted at small Î» or high frequencies even thou the thermodynamics and the electromagnetism were in good agreement with many other experiments at the time a new theory was required .Plank attempted to find a way to try and reduce the number of high frequency standing waves by reducing the number of frequency oscillators in the cavity wall. Plank came up with the idea that an atom could only absorb and re-emit energy (Îµ) in discrete bundles called quanta. He also assumed that the quanta was proportional to the frequency of the radiation and so the result that Plank came up with was that when the frequencies are large the energy will be large and no individual wave can have more than kT of energy

E = n Îµ

E = energy

n = number of quanta

Îµ = quanta

With the energy of each quanta been determined by

Îµ = h f

h = Plank's constant

f = frequency

When Plank recalculated the spectrum of radiation intensity R he got a result that matched the experimental data exactly

http://hyperphysics.phy-astr.gsu.edu/hbase/imgmod/bb7b.gifFigure (8)

## Blackbody radiation in instrumentation

Infra red thermometers avail of this technology to measure the heat signature of objects using the blackbody principle

Working Principle:

In nature, all objects above absolute zero temperature are always sent to the infrared radiation energy of the surrounding space. Infrared radiation energy of the object size and the distribution by wavelength - and its surface temperature has a very close relationship. Thus, through the infrared energy radiated by objects in their own measurements, can accurately determine its surface temperature, infrared temperature measurement which is based on an objective basis. Infrared thermometerÂ by the optical system, photoelectric detector, amplifier and signal processing, display output and other components. Optical system field of view of its goal of gathering infrared radiation energy. The field size of optical components by the thermometer is to determine its location. Infrared energy is focused on the photoelectric detector and into a corresponding electrical signal. The signal through the amplifier and signal processing circuit, and treatment algorithms in accordance with the instrument and the target emissivity corrected temperature of the target into the measured value. In addition, the thermometer should also consider the target and where the environmental conditions such as temperature, atmosphere, pollution and interfering factors such as the impact on the performance and the correction method. [5]

Typical IR-thermometer

## Photoelectric Effect

The photoelectric effect is when a metal surface is lit up with a beam of light and electrons are emitted from the surface of the metal in question.

http://t1.gstatic.com/images?q=tbn:ANd9GcRdDpGXmoVKeT0jIbkBRUo9Dqbx5wAoUstzc9JL0CbwLgzVzvB9Figure (1)

The Photoelectric effect describes the first (1st) mechanism in which radiation interacts with the atom (the photon gives up all its energy to e-)

The experiment chosen to show the photoelectric effect is performed in a vacuum tube this is to prevent energy loss from electron due to collisions with molecules of air

The emission rate of the electron is measured using an ammeter connected external to a circuit. The KE (Kinetic Energy) of the electrons is determined by applying a Vs (stopping voltage) to the anode so the electrons do not have enough KE to break free and escape.

Most electrons cannot overcome the Vs we can therefore say Vs is equal to the maximum electrons 'Kmax' so we are left with

## Kmax = eVs

Kmax = maximum electron

e = charge

Vs = stopping voltage

A diagram of the experiment is shown in Figure (2) below

01fc46c39a1bc6dabf1088e8d2e83cd4_78928.jpg

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01fc46c39a1bc6dabf1088e8d2e83cd4_78928.jpg

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Figure (2)

01fc46c39a1bc6dabf1088e8d2e83cd4_78928.jpg

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The results we obtain from this experiment are as follows

The emission rate of the electron is dependent on how bright the light is shinning

The' cut-off' wavelength is totally dependent on the metal been used Î»c

The maximum KE of electrons is dependent on Î»

The Photo-current flows as soon as light source is turned on

This experiment can be simulated on the link below on the following site

## Simulation of Photoelectric effect

Figure (3)

Einstein developed a theory based on Plank's idea of quantum energy to explain the observed effects in this he assumed that a quantum of energy was a property of the radiation itself In this theory Einstein assumed that photons have an energy equal to the energy difference between adjacent levels of a blackbody In stating this we must remember that the energy of radiation that is absorbed in discrete packets of energy are called photons and is obtained by the following formula

E =hf

E = Energy

h =Plank's constant

f = frequency

Along with the formula for the momentum of a photon that is given as

p = E/c

p = momentum

E = photon

c = speed of light

With these two equations we can combine them together to get a useful equation that relates to both the wavelength and the momentum of the photon

p = h/Î»

h = Plank's constant

Î» (lambda) = wavelength

Einstein also noted that the Electron is bound in metal with energy W - 'work function 'and that different metals have different work functions copper,nickel,silver,e.t.c An electron is released from the metal surface if the photon energy is equal to the work function so

hf < (less than) W no photoelectric effect

hf > (greater than) an electron is knocked free and the excess energy is KE

So Kmax = h f -W

Kmax = maximum electron

h =Plank's constant

f = frequency

W = work function

So if the photon is exactly equal to W when Kmax = 0 then the above equation becomes

W = h f = hc/Î»c

W = work function

h = Plank's Constant

f = frequency

c = speed of light

hc =planks constant x speed of light

Î»c = cutoff wavelength

Therefore we are left with an equation that is

Î»c = hc/W

Î»c = cutoff frequency

hc = plank's constant x speed of light

W = work function

With these formulas now know it was possible to plot a graph of frequency versus electron kinetic energy and determine the relationship between them is linear

http://mail.rdcrd.ab.ca/~smolesky/FOV1-000293AF/5Matter/FOV1-0002A9D2/FOV1-0002A9E8/PhotoelectricEffectGraph.jpgFigure (4)

We can then take the plot of the KE (Kinetic Energy) versus the frequency and the mean of this slope of data will give us the value of Plank's constant as shown below

http://resources.teachnet.ie/dkeenahan/2005/images/Graph9.jpgFigure (5)

The explanation of the photoelectric effect was a significant breakthrough in physics as it represented the first unequivocal evidence of duality; the phenomenon whereby light can behave as a wave in some situations and as a stream of particles (or quanta of energy) in others. This duality formed a cornerstone of the new quantum theory and was later found to be a universal truth of the micro world -Â  entities known asÂ  'particles' such as electrons and even atoms were in turn found to exhibit wave behavior [3]

## Compton Effect

Compton Effect is the second (2nd) mechanism

The Compton Effect is energetic incident radiation that scatters from loosely bound electrons. Part of the energy of incident radiation is given to the electron-electron this is then released from the atom a portion of energy is re-radiated as a longer wavelength in electromagnetic radiation in a different direction.

http://venables.asu.edu/quant/proj/Compton.gifFigure (1)

The Compton Effect as a wave re-radiated or scatters the electromagnetic wave which is less energetic than the incident radiation (the energy gone into the electron) but has the same wavelength

The Compton Effect as a particle gives us a different prediction for the scattered radiation in that the photon gives up some of its energy to e- and is then scattered and shifted to a longer wavelength (Î»)

Before scattering happens the photon has energy that is obtained by

E = h f = h c/ Î»

E = Energy

h = Plank's constant

c = speed of light

Î» = frequency

And linear momentum p is given by

p = E / c

p = momentum

E = energy

c = speed of light

The electron at rest has energy of me c2

me = Electron rest mass

c = speed of light

http://www.daviddarling.info/images/Compton_effect.gifFigure (2)

After scattering the photon and the electron have energy and momentum as shown in the previous diagram If we want to measure the direction and the energy of the scattered photon we need to apply the conservation of energy and of momentum to achieve the derivation for the formula for Compton scattering equation shown below taken from the Hyper Physics website

## Compton Scattering Equation

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compton2.gif

In his explanation of the Compton scattering experiment, Arthur Compton treated the x-ray photons as particles and applied conservation of energy and conservation of momentum to the collision of a photon with a stationary electron. Using the Planck relationship and the relativistic energy expression, conservation of energy takes the form

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq1.gif

Conservation of momentum requires

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq2.gif

Where p=E/c is used for the photon momentum. Squaring this equation using the scalar product gives

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq3.gif

Again using the Planck relationship and the relativistic energy expression:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq4.gif

The energy conservation expression above can be squared to give

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq5.gif

These two forms can be equated to give

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq6.gif

This can be rearranged to

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/compeq7.gif[4]

And finally to the standard Compton formula:

Î»'- Î» =

Î» = wavelength of incident photon

Î»' =wavelength of the scattered photon

h = Plank's constant

me = electron at rest mass

c = speed of light

Î¸ = scattering angle

The quantity is known as the Compton wavelength of the electron it must be remembered that the Compton wavelength is not a true wavelength but it is a change in the wavelength itself to prove this an experiment can be done to verify what happens by observing peaks in waveforms

At the different incident angle which is greater than zero there will be two peaks that will be observed

The wavelength of one of the peaks does not change this is due to the tightly bound electrons therefore there is no energy lost by the electron to the photon

The wavelength of the second peak will varies depending on the angle that is incident upon it this is predicted by Compton Formula

Figure (3a)

Figure (3b)

âˆ†Î»Figure (3c) Figure(3d)

Î» = wavelength of the incident quantum

Î»' = is the scattered quantum

It showed be noted that the maximum wavelength increase is when âˆ†Î»max = 2Î»c this occurs when the photon is scattered directly backwards. Since this is the max difference that can happen it would be wrong to use this for wavelengths hundreds of times larger than this as the results would be inadmissible (strong UV light). For this reason it is only significant for X-ray and gamma ray scattering

## References

www.egglescliffe.org.uk/physics/astronomy/blackbody/bbody.html