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**Photons, Electrons, and Atoms**

Montwood High School Physics R. Casao

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**Wave – Particle Duality**

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**Emission & Absorption of Light**

The internal energy of atoms is quantized; it is emitted and absorbed in particle-like packages of definite energy called quanta. For a given kind of individual atom the energy can’t just have any value; only discrete values called energy levels are possible. Unique sets of wavelengths are emitted and absorbed by gaseous elements. If a light source is a hot solid (like a light bulb filament) or liquid, the electromagnetic (EM) spectrum emitted is continuous; light of all wavelengths are present.

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Continuous Spectrum

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If the light source is a gas carrying an electric discharge (think neon sign) or a salt heated in a flame (think flame test in chemistry), only a few colors appear in the form of isolated sharp parallel lines. Each line corresponds to a definite wavelength and frequency. Each element in its gaseous state has a unique set of wavelengths in its line spectrum. Scientists find the line spectrum to be a valuable tool to identify elements and compounds.

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**Line Emission Spectrum of Hydrogen Atoms**

Line Spectrum Line Emission Spectrum of Hydrogen Atoms

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**Photons: The Quanta of Light**

All electromagnetic radiation is quantized and occurs in finite "bundles" of energy which we call photons. The quantum of energy for a photon is the product of Planck’s constant h and the frequency; E photon = h·f = x J·s ·f Planck’s constant also equals x eV·s 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 always occurs in precisely the same sized energy chunks.

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Electron-Volt The electron-volt, eV, is a unit of energy used in atomic and nuclear processes. The electron-volt is the energy given to an electron by accelerating it through 1 V of electric potential difference. The work done on the charge is given by the charge times the voltage difference (W = q·V), which in this case is: The work done on the charge increases the kinetic energy of the charge;

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**The Photoelectric Effect**

Photoelectric effect: emission of electrons by a substance (generally a metal), when illuminated by electromagnetic radiation (EM radiation). Electrons are ejected when illuminated by the EM radiation. Ejected electrons are called photoelectrons. Subsequent investigations have shown that all substances exhibit photoemission of electrons. Energy contained within the incident light is absorbed by electrons within the metal, giving the electrons sufficient energy to overcome the attraction of the positive ions in the metal and escape into the surrounding space.

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**First Law of Photoelectric Emission**

First Law of Photoelectric Emission: the rate of emission of photoelectrons is directly proportional to the intensity of the incident light. Einstein proposed that the incident light consisted of individual quanta, called photons, that interacted with the electrons in the metal like discrete particles, rather than as continuous waves. For a given frequency, or 'color,' of the incident radiation, each photon carried the energy E = h·f, where h is Planck's constant and f is the frequency. Increasing the intensity of the light increased the number of incident photons per unit time (flux), while the energy of each photon remained the same (as long as the frequency of the radiation was held constant).

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**First Law of Photoelectric Emission**

increasing the intensity of the incident radiation would cause greater numbers of electrons to be ejected, but each electron would carry the same average energy because each incident photon carried the same energy. This assumes that individual photons are being absorbed by the material and result in the ejection of a single electron.

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**Second Law of Photoelectric Emission**

Second Law of Photoelectric Emission: the kinetic energy of photoelectrons is independent of the intensity of the incident light. increasing the frequency f, rather than the intensity, of the incident radiation increases the average kinetic energy of the emitted electrons. The maximum kinetic energy of photoelectrons increases with the frequency of the light illuminating the emitter. For each kind of surface, there is a cut-off or threshold frequency fo below which the photoelectric emission of electrons ceases. In other words, no electrons are emitted if the frequency falls below fo.

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Work Function Work function, : the minimum energy required to remove an electron from the surface of a material. The work required to free the most weakly bound electron is the work function. If an electron acquires less energy than the work function of the metal, the electron cannot be ejected. If an electron acquires more energy than the work function, the electron is emitted from the surface of the metal and the excess energy appears as kinetic energy of the photoelectron.

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Work Function The greater the kinetic energy of the photoelectron, the greater the speed of the photoelectron. Photoelectrons ejected from atom layers below the surface will lose energy thru collisions in reaching the surface and then must give up energy equal to the work function of the metal in escaping thru the surface. Photoelectrons ejected from the surface layer of metals only lose the energy necessary to overcome the surface attractions. In any photoelectric phenomenon, photoelectrons are emitted at various velocities ranging up to a maximum value possessed by electrons having their origin in the surface layer of atoms.

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**Work Function Equations**

h = x J·s = x eV·s fo = cut-off or threshold frequency o = cut-off or threshold wavelength c = 3 x 108 m/s

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Cut-Off Wavelength Cut-off or threshold wavelength – the wavelength for which the energy of the photon is exactly equal to the work function . The cut-off (threshold) wavelength of the incident light is related to the cut-off (threshold) frequency by c = ·fo. The electron is freed but has zero kinetic energy. Nanometer (nm) to meter (m) conversion: nm = 1 x 10-9 m Example: 535 nm = 535 x 10-9 m = 5.35 x10-7 m

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**Cut-Off (Stopping) Potential**

Cut-off or Stopping Potential, Vo: a negative potential on the collector of a photoelectric cell that reduces the photoelectric current to zero. A negative potential on the collector plate repels the photoelectrons, tending to turn them back to the emitter plate. Only those electrons having enough kinetic energy and velocity to overcome the repulsion reach the collector. As the cut-off potential is approached, only those photoelectrons with the highest velocity reach the collector. These electrons have the maximum kinetic energy and are emitted from the surface layer. At the cut-off potential, even these electrons are repelled.

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Cut-Off Potential Vo The cut-off potential Vo measures the kinetic energy of the fastest photoelectrons. Work done by the cut-off potential = maximum kinetic energy of electrons: q = charge on electron; q = x C Mass of electron = 9.11 x kg Kinetic energy:

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**Third Law of Photoelectric Emission**

Third Law of Photoelectric Emission: the maximum kinetic energy of photoelectrons varies directly with the difference between the frequency of the incident light and the cut-off frequency. For photoelectric emission from any surface, the incident light radiation must contain frequencies higher than the cut-off frequency characteristic of the surface. The maximum kinetic energy of photoelectrons emitted from any surface can be increased only by raising the frequency of the illuminating light.

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**Third Law of Photoelectric Emission**

Equations:

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**A photon arriving at the surface is absorbed by an electron.**

The energy transfer is an all-or-nothing process; the electron gets all of the photon’s energy or none at all. If the photon energy is greater than the work function , the electron may escape from the surface. Greater intensity at the same frequency means a proportionally greater number of photons per second absorbed and a greater number of electrons emitted per second and a greater current.

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**Experimental Data Supporting Photoelectric Effect**

Photoelectron Kinetic Energy vs. Frequency

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**Experimental Data Supporting Photoelectric Effect**

Threshold frequency fo is the point where the line crosses the x-axis. Work function is the point where the line crosses the y-axis.

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**Experimental Data Supporting Photoelectric Effect**

Photoelectron Kinetic Energy vs. Frequency

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**Frequency and Photoelectrons**

For potassium, the work function = 2 eV. Photon energies less than 2 eV will not produce photoelectrons. Photon energies greater than or equal to 2 eV produce photoelectrons; energy above 2 eV increases the kinetic energy of the photoelectrons.

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**Work Functions for Photoelectric Effect**

Element Work Function (eV) Aluminum 4.08 Copper 4.7 Niobium 4.3 Beryllium 5 Gold 5.1 Potassium 2.3 Cadmium 4.07 Iron 4.5 Platinum 6.35 Calcium 2.9 Lead 4.14 Selenium 5.11 Carbon 4.81 Magnesium 3.68 Silver 4.73 Cesium 2.1 Mercury Sodium 2.28 Cobalt Nickel 5.01 Zinc

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Application Photoelectric cell acts like a switch in an electric circuit in that it produces a current in an external circuit when light of sufficiently high frequency falls on the cell, but it does not allow a current in the dark. Burglar alarms: beams of UV light pass from a source to a photosensitive surface; the current produced is amplified and used to energize an electromagnet that attracts a metal rod. When the light beam is broken, the electromagnet switches off and the spring pulls the iron rod to the right. In this position, a completed electric circuit allows current to pass and activate the alarm system. See figure on next slide. Some garage door systems and fire alarm systems also employ photoelectric sensors as part of their operating system.

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Burglar Alarm System

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Application Photoelectric cell acts like a switch in an electric circuit in that it produces a current in an external circuit when light of sufficiently high frequency falls on the cell, but it does not allow a current in the dark. Burglar alarms: beams of UV light pass from a source to a photosensitive surface; the current produced is amplified and used to energize an electromagnet that attracts a metal rod. When the light beam is broken, the electromagnet switches off and the spring pulls the iron rod to the right. In this position, a completed electric circuit allows current to pass and activate the alarm system. See figure on next slide. Some garage door systems and fire alarm systems also employ photoelectric sensors as part of their operating system.

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**Photoelectric Effect Example**

A sodium surface is illuminated with light of wavelength 300 nm. The work function for sodium is 2.46 eV. Determine the kinetic energy of the ejected photoelectrons.

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**Photoelectric Effect Example**

b. Determine the cut-off wavelength for sodium.

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Photon Momentum A photon of any EM radiation with frequency f and wavelength has energy: Every particle that has energy must also have a momentum, even if it has no rest mass; photons have zero rest mass. A photon with energy E has momentum p given by E = p∙c. Photon momentum:

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**The direction of the photon’s momentum is the direction in which the EM wave is moving.**

De Broglie Waves Nature loves symmetry. Light is dualistic in nature, behaving in some situations like a wave and in others like a particle. If nature is symmetric, the dual nature should also hold for particles. Electrons, which we usually think of as a particle, may in some situations behave like a wave.

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**If a particle acts like a wave, it should have a wavelength and a frequency.**

De Broglie proposed that a particle with mass m, moving with speed v, should have a wavelength related to its momentum p = m∙v just like a photon: The de Broglie wavelength of a particle is:

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**The electrons in orbit around the nucleus do not radiate energy.**

The electrons can be thought of as a standing wave fitted around a circle in one of the orbits. For the wave to come out even and join onto itself smoothly, the circumference of the circle must include some whole number of wavelengths.

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Compton Scattering When a photon strikes matter, some of the radiation is scattered, just like light falling on a rough surface undergoes diffuse reflection. Some of the scattered radiation has smaller frequency (longer wavelength) than the incident radiation and that the change in wavelength depends on the angle through which the radiation is scattered.

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**Interpretation of Compton Effect**

“Light particle” l1 l2 Before Collision After Collision The Compton Effect describes the collisions of light with electrons perfectly if we treat light as a particle with momentum p and energy E: Let the incident photon energy be referred to as E1=hc/l1. Let the emergent photon energy be referred to as E2=hc/l2. The fact that the emergent photon has a larger wavelength than the incident implies that E2 < E1. In other words, it’s as if the photon transferred some of its energy to something else, and emerged with a lower energy. In fact it did. It struck an electron and gave up some of its energy. Therefore, the photon’s energy must be lower after the collision with the electron. Since the wavelength is inversely proportional to the energy, this means the photon’s wavelength is larger after the collision. The difference in energy between the incident and emergent photon is carried away by the struck electron.

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If the scattered radiation emerges at an angle θ with respect to the incident direction, if is the wavelength of the incident radiation, and is the wavelength of the scattered radiation: Imagine the scattering process as a collision of two particles, the incident photon and an electron that is initially at rest.

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The incident photon disappears, giving part of its energy and momentum to the electron, which recoils as a result of the impact. The remaining energy and momentum goes to a new scattered photon that has less energy, smaller frequency, and longer wavelength than the incident photon. The Compton effect is a quantum phenomenon that cannot be explained by classical physics.

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Compton effect depends on the assumption that while a photon does not possess mass, it does have momentum that can be transferred during a collision. The change in kinetic energy for the electron equals the difference in energy for the incident photon, h∙fi, and the scattered photon,h∙ff.

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The change in wavelength between the incident and scattered photon, the Compton shift, depends on the scattering angle θ (the angle between the incident and scattered photon). If the scattering angle θ = 90°, the Compton shift has a definite, numerical value called the Compton wavelength.

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**Momentum and energy of the incident photon:**

Momentum and energy of the scattered electron before collision:

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** is θ in the other images**

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**Compton Scattering video link Youtube: http://www. youtube. com/watch**

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**Derivation of Compton Scattering Equation**

Momentum and energy of the scattered photon: Momentum and energy of the scattered electron after collision: the energy equation is derived from Einstein’s theory of relativity for particles that have a velocity that approaches the speed of light.

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**Conservation of energy:**

Square both sides to get E2: Substitute E2 = (m·c2)2+(pe·c)2:

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** is θ in the other images**

Conservation of momentum: Apply conservation of vector momentum and the law of cosines: is θ in the other images

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pe2 = p2 + p´2 – 2·p·p´·cos Substitute pe2 into the conservation of energy equation: Squaring the (p·c + m·c2 – p´·c)2 term:

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Combining like terms: Combining both sides of the equation: The m2·c4 terms cancel out:

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**Divide both sides of the equation by c2:**

The p2 and p´2 terms cancel out:

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The 2 cancels out: Combine like terms (p·p´ and m·c) by adding p·p´ to both sides: Factor out p·p´ and m·c:

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**Replace the p’s with the h/λ terms, combine like terms, and find a common denominator:**

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**Factor out from both sides and cancel the term:**

Solve for the change in wavelength, λ´ – λ:

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**Summary of Photons a particle.**

Photons can be thought of as “packets of light” which behave as a particle. To describe interactions of light with matter, one generally has to appeal to the particle (quantum) description of light. A single photon has an energy given by E = h·c/l, where: h = Planck’s constant = 6.6x10-34 [J·s] c = speed of light = 3x108 [m/s] l = wavelength of the light (in m) Photons also carry momentum. The momentum is related to the energy by: p = E/c = h/l

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Matter Waves ? One might ask: “If light waves can behave like a particle, might particles act like waves”? The short answer is YES. The explanation lies in the realm of quantum mechanics, and is beyond the scope of this course. However, you already have been introduced to the answer. Particles also have a wavelength given by: l = h/p = h / mv That is, the wavelength of a particle depends on its momentum, just like a photon! The main difference is that matter particles have mass, and photons don’t !

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Matter Waves Compute the wavelength of a 1 [kg] block moving at 1000 [m/s]. l = h/mv = 6.6x10-34 [J s]/(1 [kg])(1000 [m/s]) = 6.6x10-37 m. This is immeasureably small. So, on a large scale, we cannot observe the wave behavior of matter Compute the wavelength of an electron (m=9.1x10-31 [kg]) moving at 1x107 [m/s]. l = h/mv = 6.6x10-34 [J s]/(9.1x10-31 [kg])(1x107 [m/s]) = 7.3x10-11 m. This is near the wavelength of X-rays

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