The Photoelectric Effect

 

William Reid, Nick Jameson, James Park

 

Salisbury University, Department of Chemistry, Salisbury, Maryland 21801 USA

 

Ø  INTRODUCTION

 

·         The photoelectric effect is occurs when a light ray hits a metal surface, and it produces a current

·         In 1887, the photoelectric effect was discovered by Hertz, and caused much interest.  Results from other experiments led to these facts:

o   When polished metal plates are irradiated, they may emit electrons, not positive ions

o   Whether the plates emit electrons depends on the frequency of the light, in general there will be a threshold that varies from metal to metal.  Only light with greater frequency than the given threshold will produce the photoelectric effect

o   Its observed that the intensity of the light was proportional to the current produced and that the frequency of the light is proportional to the max energy of the photons

·         According to Einstein, beams of light behaved like particles called Photons

 

Ø  CLASSICAL APPROACH

 

·         Intensity of the light was the determining factor of the energy put into the system

·         Scientists believed light was composed of only waves; not particles

·         Frequency did not play a factor in determining the max energy

 

Ø  PROBLEM

 

·         Experiments showed that there is a min frequency at which no current is observed which contradicts the classical approach

·         These finding helped develop the belief that frequency could be a determining factor for energy electrons emitted

 

Ø  QUANTUM EXPLANATION

 

·         Einstein proposed that energy of photon(Ephoton) is equal to the product of Planck’s constant(h) and the frequency of the light(ν)

·         When a photon strikes the metal, it transfers energy in 2 parts, some energy must be expended to separate the electron from the metal, Eionization or Φ, and the rest of the energy from the photon increases the electrons energy by an amount proportional to ½ me•v2

·         The formula was determined by Einstein to be,  Emax = h•ν = ½ me•v2 +  Φ (derived in later section)

 

Ø  EVIDENCE OF SUPERIORITY

 

·         Einstein’s prediction made one universal prediction, that the slope of the graph Emax vs. ν, will be Planck’s constant (h)

·         This is exactly what was observed, which later led to a Nobel Prize for the photoelectric effect

 

Ø  SIGNIFICANCE OF QUANTUM APPROACH

 

·         Compton Effect was discovered using information from the photoelectric effect

·         Led to the idea of wave/particle duality (light behaves as both wave and/or particle)

 

Ø  QUANTUM TO CLASSICAL

 

·         The quantum approach can accurately predetermine the velocity of the electron measured in the classical approach

·         The quantum approach also explains the metals threshold and its dependency on frequency, not intensity

 

Ø  INFORMATION ON ALBERT EINSTEIN

 

Date of Birth: March 14, 1879

 

Place of Birth: Ulm, Württemberg, Germany

 

Family/Heritage: Born to Hermann EINSTEIN and Pauline KOCH; a Non-Observant Jewish family. Family moved around a lot because of failed business ventures. He finally moved to Italy when he was 15. 

 

Service during WW1: German militarism during WW1 made Einstein a pacifist. When he became famous in 1919 he used his celebrity status to promote such pacifist beliefs. In 1914 he signed the Manifesto to Europeans as one of only 4 people to do so. This document protested Germany’s militarism and aggression during WW1. This is said to be his first public political statement. During this time he was a professor at the University of Berlin, where most of his colleagues had the opposite view of Germanys Militarism than Einstein.

 

Education: Einstein entered the Swiss Federal Polytechnic School, located in Zurich, in 1896 to learn to teach physics and mathematics.  He received his diploma and gained Swiss Citizenship in 1901. He received his doctorate in 1905.

 

Age when he proposed quantum approach: In 1905, when Einstein was 25, he claimed that light behaves like discontinuous, individual particles rather than as a continuous wave as was generally the generally accepted belief.

 

Scientists Embrace of Ideas: For over 15 years Einstein’s beliefs on Light Quanta were not accepted by mathematicians and physicist because his theory seemed to be a needless denial of the classical theory of radiation. Many physicists thought that Einstein would withdraw his ideas because of the overwhelming rejection, but he never did, and his theories are the foundation that modern beliefs are based on.

 

Identification of whether this scientist received the Nobel Prize: Einstein received the Nobel Prize in Physics in 1921 “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.”

 

 

This information and more can be found at:

http://www.amnh.org/exhibitions/einstein/peace/ww1.php

http://nobelprize.org/nobel_prizes/physics/laureates/1921/einstein-bio.html

http://www.aip.org/history/einstein/essay-photoelectric.htm

http://www.springerlink.com/content/vp611310h107q756/

http://nobelprize.org/nobel_prizes/physics/laureates/1921/

 

 

Ø  THE PHOTOELECTRIC EFFECT (Derivation)

 

When a beam of light (photons) hits a metal surface it transfers all of its energy to an atom in the metal plate.  So, Ephoton = Egained by atom.  When the electrons are ejected from the metal, those metal atoms become ions, meaning that they now have a positive charge.  Normally, and in this case, there is ionization energy, Eionization, needed for the electrons to escape from the nucleus and head into infinity creating an ion.  If Ephoton is greater than Eionization then there is some left over energy.  Through the conservation of energy we know that the energy of a photon must be conserved and entirely used.  So, the rest of the energy in converted into kinetic energy, Ekinetic, for the electron. So, because of the conservation of energy we can come up with the theoretical formula of:

 

Ephoton = Egained = Eionization + Ekinetic

 

Kinetic energy can be measured with the equation:

 

Ekinetic = ½ m•v2

 

And Einstein proposed that the energy of a photon can be measured with the equation:

 

Ephoton = h•ν

 

Eionization can be symbolized with the Greek letter Φ, so we write the equation as:

 

h•ν = Φ + ½ m•v2

 

This equation may be moved around algebraically to isolate whichever bit of information is needed.  For exercise 8.13 b it is best rearranged to:

 

            Φ = h•ν - ½ me v2

 

 

 

Ø  EXERCISE 8.13 B 1

 

In an X-ray photoelectron experiment, a photon wavelength of 121 pm ejects an electron from the inner shell of an atom and it emerges with a speed of 56.9 mM/s.  Calculate the binding energy of the electron.

 

The equation for the photoelectric effect is:

 

Φ = h•ν - ½ me v2

 

Given by the problem:

 

λ = 121 pm

v = 56.9 Mm/s

 

Constants:

 

me = 9.11•10-31 kg       (Mass of an Electron)

h = 6.626•10-34 J•s       (Planck’s Constant)

c = 2.99•108 m/s          (Speed of Light)

 

Unknown:

 

Φ, Eionization= ?

 

First we should convert all values to SI units:

 

λ = 121 pm (1 m/ 1012 pm) = 1.21•10-10 m

v = 56.9 Mm/s (1 m/ 10-6 Mm) = 5.69•107 m/s

 

Knowing that ν = c/λ we can substitute that in the equation for ν changing the equation to:

 

Φ = h•c/λ - ½ me v2

 

Then by inputting all of the values we get:

 

Φ = (6.626•10-34 J•s) (2.99•108 m/s)/(1.21•10-10 m) - ½ (9.11•10-31 kg) (5.69•107 m/s)2

 

All of the units except J cancel and the answer is:

 

Φ = 1.67•10-16 J

 

This means that 1.67•10-16 J of energy is needed to release an electron from this metal.

 

 

Ø  EQUATION AND MAPLE GRAPH RELATING VELOCITY OF ELECTRON AND WAVELENTGH FOR 8.13 B

We rearranged the equation using algebra to isolate velocity (v) and plotted in the graph v vs.

 

The velocity equation is:

 

            ve = √[2/me(h•c/λ – Φ)]

 

The constants needed are:

 

me = 9.11•10-31 kg       (Mass of an Electron)

h = 6.626•10-34 J•s       (Planck’s Constant)

c = 2.99•108 m/s          (Speed of Light)

 

Since this is from exercise 8.13 b we also use that Φ, Eionization

 

Φ = 1.67•10-16 J

 

Maple Commands:

 

            First we defined the constants and Φ:

 

Then we typed the velocity equation (determined above) and solved for ve:

 

Next we plotted the data over an appropriate interval so we can view the key points of the graph:

 

 

This plot shows that when a light has higher wavelengths, the velocity of the electron will be slower. Without looking at the plot, one could see this from the equation because the only variable in the equation is , or wavelength, and it is the denominator in a fraction where the numerator is a constant value. As  increases, that fraction gets smaller and smaller, making the entire value inside the square root smaller and smaller. As the number inside the square root gets smaller, the actual square root gets smaller and smaller. This plot supports that trend. At smaller values of , the electron velocity is huge, and at bigger values of , the electron velocity is small.

 

 

 

 

Ø  REFERENCES

 

1Atkins, Peter. Atkins Physic Chemistry, eighth edition. New York: W. H. Freeman and Company, 2006. 249-252, 274.

 

2Gasiorowicz, Stephen. Quantum Physics. Canada: John Wiley & Sons, Ltd, 1974. 8-9.

 

3Scherrer, Robert. Quantum Mechanics: An Accessible Introduction. San Francisco: Pearson Education, 2006. 10-13.