MATHEMATICS AND PHYSICAL ENGENEERING DEPT.
Physics Lectures
Chapter IIi
ATOMIC LINE SPECTRA
3.1 Wave-Particle Duality
Early in the twentieth century, there was a debate about whether light was composed of particles or waves.Most commonly observed phenomenon goes with light iselectromagnetic waves. However, the blackbody radiation, x-rays, photoelectric effect, and atomic line spectra suggest a particle nature for light, photon theory.
Fig. (3.1) The wave-particle duality for light and electrons
A wave-particle dual nature soon is characteristic of electrons as well. The evidence for the description of light as waves was well established when the photoelectric effect introduced firm evidence of a particle nature as well. On the other hand, the particle properties of electrons were well documented when the DeBroglie hypothesis and the subsequent experiments by Davisson and Germer established the wave nature of the electron.
The frequency available is continuous and has no upper or lower bound, so there is no finite lower limit or upper limit on the possible energy of a photon, the light quantum particle. On the upper side, there are practical limits because you have limited mechanisms for creating really highenergy photons. Low energy photons are bound, but when you get below radio frequencies, the photon energies are so tiny compared to room temperature thermal energy, theyare never sensedas distinct quantized entities, they are swamped in the background. Another way to say this, is that in the low frequency limits, things just blend in with the classical treatment of things and a quantum treatment is not necessary.
Fig.(3.2) The electromagnetic waves spectrum
The Photoelectric Effectoccurs when light falls on a metallic plate, electrons are emitted from it. This metal plate is called the emitter. The emitted electrons are collected at another plate called the collector.Both electrodes are connected to an outer circuit in forward or reverse bias as shown in fig.(3.3).
As soon as the incident light strikes the emitter, the photocurrent begins. The photocurrent is instantaneous and directly proportional to the intensity of the incident light beam, I0. The potential difference between the emitter and the collector, V, is varied and the resulting photoelectric current, i, has two main characteristics:
Fig. (3.3) The photoelectric effect and photodiode
i.i is proportional with the incident light intensity I0
ii.i is equal to zero for incident light with frequency less than the threshold frequency, 0 , even for the most intense beams of light.
Classical theory would conclude that the emitted electrons must acquire their kinetic energy from the light beam. Hence increasing the intensity of the light beam would emit photoelectrons with higher energy values.The energy of the emitted electrons is independent of the light intensity whereas the photocurrent is, as the number of incident photons per second increase.
If the applied voltage is reversed and increased in the reverse bias till i is zero; collector is negative while the emitter is positive. The potential is hence known as the stopping potential, Vs. The total energy possessed by the electrons, in case of their stopping on the emitter, is thus the maximum potential energyis given as: P.Emax =eVs.
For a constant frequency and light intensity I0, the photocurrent,i,decreases with increasing the retarding potential. Vsdepends on the frequency of the light, but is independent of the light intensity and therefore is independent of the photocurrent. The V-i characteristic of the photodiode is shown in fig.(3.4), for two light intensities. It is obvious that higher intensities cause higher saturation current and vice versa. On the other hand both incident intensities give the same stopping potential.
3.1.1.Classical interpretation Failure:
Based on classical properties of electromagnetic waves the following statements must be justified:
1-The absorbed energy is proportional to the intensity of light,the area illuminated,and the time of illumination. This means that at lower light intensities we must wait more timeuntil photo current is emitted which doesn’t happen, as delay measured never exceeded 10-9 second.
2-Vs does not depend on intensity which mean that the maximum kinetic energy, is independent on the total energy absorbed by the surface as shown in fig.(3.4).
3- The existence of threshold frequency, , for a given metal is completely unexplainable on classical basis. Fig.(3.5) shows that of sodium is 4.39*1014HzLikewise, the dependence of the stopping potential on the frequency is unexplained.
3.1.2Early Photoelectric Effect Data
Analysis of data from the photoelectric experiment showed that the energy of the ejected electrons was proportional to the frequency of the illuminating light. This showed that whatever was knocking the electrons out had an energy proportional to light frequency. The remarkable fact that the ejection energy was independent of the total energy of illumination showed that the interaction must be like that of a particle which gave all of its energy to the electron! This fits in well with Planck's hypothesis that light in the blackbody radiation experiment could exist only in discrete bundles with energy, (E = hν).
The details of the photoelectric effect were in direct contradiction to the expectations of very well developed classical physics concepts. The explanation marked one of the major steps toward quantum theory.
Fig.(3.4) The V-i characteristic of the photodiode Fig.(3.5): The K.E.max sketched against incident
light frequency for sodium metal
The aspects of the photoelectric effect that contradict classical analysis are:
- The electrons were emitted immediately - no time lag!
- Increasing the intensity of the light increases the number of photoelectrons but not their maximum kinetic energy!
- Red light will not cause the ejection of electrons from a potassium surface, no matter what the intensity!
4. A low intensity violet light will eject only a few electrons, but theirmaximum kinetic energies are greater than those for intense light of longer wavelengths.
Electrons ejected from a sodium metal surface are measured as an electric current. Finding the opposing voltage it took to stop all the electrons,Vs, gave a measure of the maximum kinetic energy of the electrons in eVs.
K.E.max=e Vs …….(3.1)
Fig.(3.5) shows the relation ofeVsagainst frequency. The minimum energy required to eject an electron from the surface is called the photoelectric work function. The threshold for this element corresponds to a wavelength of 683 nm as shown in fig.(3.5). Using this wavelength in Einestien’s equation, eq.3.3, the photon energy is equal to 1.82 eV. This energy is defined as the work functionof sodium, φNa.
3.1.3.Quantum Explanation:
According to the Planck hypothesis, all electromagnetic radiation is quantized and occurs in finite "bundles" of energy which we call photons. The quantum of energy for a photon is not Planck's constant h itself, but the product of h and the frequency. The quantization implies that a photon of blue light of given frequency or wavelength will always have the same size quantum of energy. For example, a photon of blue light of wavelength 450 nm will always have 2.76 eV of energy. It occurs in quantized chunks of 2.76 eV, and you can't have half a photon of blue light - it alwaysoccurs in precisely the same sized energy chunks.
1-Photo-emission occurs with no delay, because it does not take time to hit the electron by the photon and release an electron; energy transfer takes place instantaneously.
2- Increase in the intensity means increasing the number of photons striking the metal surface and so the number of photoelectrons liberated increases, and accordingly the current increases.
3-Since the energy of the photon and the work function are well defined, a well defined maximum kinetic energy of the photoelectrons exists for a given frequency. As a result, a welldefined stopping potential exists regardless of the intensity of the incident radiation.
4- The work function, φ, is also welldefined for a particular metal. If the energy of the photon is less than φ,no photocurrent emerges and so the minimum energy for a photon to liberate an electron from a metallic surface is hν0 , hence,
φ=hν0
λ0=c/ν0 =hc/φ …….(3.2)
3.2Relations and graphs:
Albert Einstein (1905) showed that the experimental results areexplained by what is known as Einestein’s Equation:
…….(3.3)
For forward bias photodiode energy equation per electron is:
K Emax = h - φ KEmax = h - h …….(3.4)
For reverse bias photodiode energy equation per electron is:
eVs=h (-0)=hc(1/λ-1/λ0) …….(3.5)
where KEmax is the maximum kinetic energy possessed by the photoelectrons, is the threshold frequency and φ is the work function of the metal or the minimum energy needed to free an electron from its metal surface.
Knowing the work function for different materials we can plot the following figures, fig.(3.6) for different metallic surfaces. The slopes of these graphs is always constant and equal to Planck’s constant, h, which gives it more validity and allowed Einestein to announce firmly his equation.
Fig.(3.6)
Max KE against frequency for different metals
Example 3.1:
What is the maximum kinetic energy and speed of an electron ejected from a calcium surface whose work function is φ=2.9 eV when illuminated by a light of wavelength a) 410 nm b) 550 nm?
Solution :
For calcium
a) λ=410nm, hν=hc/λ=1240/410=3.03eV
K.Emax =3.03-2.9=0.13eV ,
v=√2 *0.13*1.6*10-19/9.1*10-31) = m/s
b) λ=550nm, hν=hc/ λ =1240/550=2.25eV
K.Emax =0 , v=0 → No emission as hν ‹ φ .
3.3X-Rays:
An x-ray machine, like that used in adoctor's or a dentist's office, isvery simple. Inside the machine is anx-ray tube. An electron gun inside the tube shoots high energy electrons at a target made of heavyatoms, such astungsten X-rays come out because of atomic processes caused by energetic electrons shot at the target.
3.3.1.Apparatus:
Fig.(3.7)
3.3.2X-Ray production
X-rays are just like any other kind of electromagnetic radiation. They can be produced in parcels of energy called photons, just like light with shorter wavelengths and higher frequencies of range (1017-1019)Hz. Two different atomic processes can produce x-ray photons. One is called,Bremsstrahlung, which is a German name meaning "braking radiation." The other is called K-shell emission. They can both occur in heavy atoms like tungsten. Both ways of making x-rays involve a change in the state of electrons.Bremsstrahlung is easier to understand using the classical idea that radiation is emitted when the velocity of the electron shot at the tungsten changes. This electron slows down after swinging around the nucleus of a tungsten atom and loses energy by radiating x-rays. In the quantum picture, many photons of different wavelengths are produced, but none of the photons has more energy than the electron. After emitting the spectrum of x-ray radiation the original electron is slowed down or stopped.
3.3.3"K-shell" x-rays
Atoms have their electrons arranged in closed "shells" of different energies.
The K-shell corresponds the lowest energy state of an atom. The incoming electron from the electron gun can give a K-shell electron in a tungsten target atom enough energy to knock it out of its energy state. Then, a tungsten electron of higher energy state, from an outer shell, can fall into the K-shell to fill its space. The energy lost by the falling electron is released as an emitted x-ray photon. Meanwhile, higher energy electrons fall into the vacated energy state in the outer shell, and so on.
K-shell emission produces higher-intensity x-rays than Bremsstrahlung, and the x-ray photon comes out at a single wavelength.
3.3.4The X-Ray spectrum
The highest frequency of electromagnetic wave released in this manner is that resulting from the greatest loss of kinetic energy in a single collision with a target atom. Therefore,
hνmax = eVdc …….(3.6)
Lower frequencies are released when the decelerating electrons make multiple collisions losing energy in stages.
Thus the minimum wavelength min emitted by the X-ray tube is given by:
hc/λmin = eVdc …….(3.7)
Spikes in the curve:
In addition to this background radiation, there are also some pronounced spikes seen in the sketch graph of the emission spectrum of Xrays shown in fig.(3.8). These arecalled the CHARACTERISTIC LINES and are generated from re-radiation after excitation of orbiting electrons from lower to higher permitted shells in atoms of the target material.The reason for those peaks was unknown until Bohr postulated that electrons inside atoms have quantized energy states.Inside a metal, the inner shell electrons are tightly bound to their own nuclei and thus have quantized energy states. If the X-ray tube voltage is high enough, an incident electron may collide with one of those inner shell electrons, transferring it to a higher energy state. An electron from another higher energy state quickly fills in the vacancy by emitting a photon.It turns out that there is a very large probability for certain transitions and thus we find a very high intensity of emitted radiation at particular wavelengths (Kα, Kβ).
Why does the innermost electron while one may guess the outermost electron would be the easiest to knock out?
An x-ray photon has a lot of energy in it, and only transitions of the inner electrons release that much energy. Transitions of the outer electrons, which can happen, might be in the infrared or visible part of the spectrum. For the electron energies used in x-ray tubes, it turns out the inner electrons are the most likely to be knocked out so that Lα & Lβ show low intensity on the spectrum.
The above discussion is summarized in fig.(3.8).
Fig (3.8) kα is radiated when line results from a change involving the L shell and the Kβ is radiated when the change involving the M shell and so on.
Examples of X-Ray spectrum for different materials:
kα1=hc /E=0.1634nm kβ2=hc /E=0.1479nm
Fig.(3.9.a) The x-ray spectrum for Nickel
kα1=0.1444nm kβ2=0.125nm
Fig.(3.9.b) The x-ray spectrum for Tungesten
The following table illustrates different wavelengths at which x-ray radiation for different metallic targets from spectra as shown in fig.(3.9):
Table.3.1Different wavelengths ofx-ray radiation
Note that as energy states differ for different target materials, so the generated photons will have different frequenciesfor each metallic target as shown in table 3.1. This means that the generated x-ray wavelengths are characteristic of the target material.
3.3.5. Increase in the accelerating voltage:
Increase of the accelerating voltage applied, Vdc, between filament and target is found to increase the penetrating power of the x-rays. Since the maximum loss of kinetic energy at a single collision is now higher, the highest frequency emitted is also higher as expected. Thus the quality of theemitted X rays is altered. These are called ‘hard’ x-rays.
3.3.6.X-ray dose measurement:
To measure x-ray intensity, the scintillation counter is preferable. It gives a reading in the form of n, counts/sec, corresponding to the number of incident photons per sec. n photons/sec possess a total energy/sec of Et (j/s)
Et=n h ν ……. (3.8)
This energy causes a total dose equal of Et multiplied by the time during which the person is subjected to the radiation. It is then produced in rads.
3.4Atomic Electromagnetic Spectrum:
When a a discharge tube is filled with alow-pressure gas then excited by applying a high voltage, the gas glows. The emitted radiation, analyzed by a diffraction grating or a prism, is resolved into different spectral lines. The important issue is that each particular gas shows a different spectrum. The gas spectral lines are unique and discrete as shown in fig. (3.10-3.11). The characteristic spectral lines act like a fingerprint that can be used to identify the gas.This spectrum is quiet different from the continuous spectrum of the sun electromagnetic radiation or from continuous light spectrum, emitting a rainbow of colors. White light has a continuous spectrum and all wavelengths of the visible range, (400-750)nm, are obtained.
The sun hasnot much different Fig. (3.11.a), a continuous spectrum with Franhofer lines, which appear as dark lines on the spectrum. The spectrum of the sun is obtained by photographing a slit in the window blind with a similar grating set-up.These lines signify that some wavelengths are missing. The dark lines are absorbed wavelengthsfrom the continuous spectrum which indicate the presence of some gases that absorb these specific wavelengths.
Fig.(3.10)
Different emission wavelengths from discharge tubes for He and Ne gases
COMPARISON BETWEEN THE EMISSION SPECTRUM OF HYDROGEN AND SPECTRUM OF THE SUN:
In Fig. (3.11.b), the spectrum of a glass tube filled with hydrogen that is excited with a high voltageshows the red line at the left, the blue-green line in the right center, and the violet line is the second from the right. On the extreme right is an ultraviolet line that human eyes cannot see, but is barely detectable by a digital camera. The visible lines in the hydrogen spectrum are called the Balmer lines of hydrogen. Some dark lines, Franhofer lines, on the sun spectrum are coincident with the bright lines of the hydrogen. Notice that there are several other dark lines for elements other than hydrogen.Helium gas, that surrounds the sun causing a part of the absorption spectrum, was discovered on the sun before it was discovered on earth. The dark lines are called absorption lines, they are characteristic of all stars, and are caused by absorption of light by the elements in their atmospheres.
Fig (3.11)
Sets of spectral lines in nm emitted by a particular gas when excited
Accordingly, one can define the emission spectrum as the characteristic set
of spectral lines emitted by a particular gas when excited. The absorption spectrum is the set of spectral lines that are absorbed by a gas when a continuous spectrum passes through it. These absorbed lines appear dark on the rainbow background and correspond exactly to some lines in the absorption spectrum and are known as the gas fingerprints.
3.5. Atomic models: