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**Atomic Physics & Quantum Effects**

AP Physics B

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**Blackbody radiation A blackbody absorbs all incident light rays.**

All bodies, no matter how hot or cold, emit electromagnetic waves. We can see the waves emitted by very hot objects because they are within the visible spectrum (light bulb filament; red-hot metal). At lower temps we can’t see the waves but they are still there. For example, the human body emits waves in the infrared range. This is why we can use infrared detecting devices to “see” in the dark. The distribution of energy in blackbody radiation is independent of the material from which the blackbody is constructed – it depends only on the temperature.

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The diagrams below show the intensity of various frequencies of EM radiation emitted by blackbodies of various temperatures. Note that as the temperature increases, the energy emitted (area under curve) increases and the peak in the radiation shifts to higher frequencies.

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This is important because classical physics predicts a completely different curve that increases to infinite intensity in the ultraviolet region. (thus called the Ultraviolet Catastrophe). The only way to make sense of this finding is by saying energy is quantized (Planck’s quantum hypothesis)

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Quantum? Quantum mechanics is the study of processes which occur at the atomic scale. The word "quantum" is derived From Latin to mean BUNDLE. Therefore, we are studying the motion of objects that come in small bundles called quanta. These tiny bundles that we are referring to are electrons traveling around the nucleus.

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**“Newton, forgive me..”, Albert Einstein**

At the atomic scale Newtonian Mechanics cannot seem to describe the motion of particles. An electron trajectory between two points for example IS NOT a perfect parabolic trajectory as Newton's Laws predicts. Where Newton's Laws end Quantum Mechanics takes over.....IN A BIG WAY! One of the most popular concepts concerning Quantum Mechanics is called , “The Photoelectric Effect”. In 1905, Albert Einstein published this theory for which he won the Nobel Prize in 1921.

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**What is the Photoelectric Effect?**

In very basic terms, it is when electrons are released from a certain type of metal upon receiving enough energy from incident light. So basically, light comes down and strikes the metal. If the energy of the light wave is sufficient, the electron will then shoot out of the metal with some velocity and kinetic energy.

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**The Electron-Volt = ENERGY**

Before we begin to discuss the photoelectric effect, we must introduce a new type of unit. Recall: This is a very useful unit as it shortens our calculations and allows us to stray away from using exponents.

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**The Photoelectric Effect**

"When light strikes a material, electrons are emitted. The radiant energy supplies the work necessary to free the electrons from the surface."

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Photoelectric Fact #1 The LIGHT ENERGY (E) is in the form of quanta called PHOTONS. Since light is an electromagnetic wave it has an oscillating electric field. The more intense the light the more the field oscillates. In other words, its frequency is greater.

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Light Review

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**More on Fact #1 h hc 6.63x10-34 Js 1.99x10-25 Jm 4.14x10-15 eVs**

Make sure you USE the correct constant! h hc 6.63x10-34 Js 1.99x10-25 Jm 4.14x10-15 eVs 1.24x103 eVnm Planck’s Constant is the SLOPE of an Energy vs. Frequency graph!

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Photoelectric Fact #2 The frequency of radiation must be above a certain value before the energy is enough. This minimum frequency required by the source of electromagnetic radiation to just liberate electrons from the metal is known as threshold frequency, f0. The threshold frequency is the X-intercept of the Energy vs. Frequency graph!

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Photoelectric Fact #3 Work function, f, is defined as the least energy that must be supplied to remove a free electron from the surface of the metal, against the attractive forces of surrounding positive ions. Shown here is a PHOTOCELL. When incident light of appropriate frequency strikes the metal (cathode), the light supplies energy to the electron. The energy need to remove the electron from the surface is the WORK! Not ALL of the energy goes into work! As you can see the electron then MOVES across the GAP to the anode with a certain speed and kinetic energy.

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Photoelectric Fact #4 The MAXIMUM KINETIC ENERGY is the energy difference between the MINIMUM AMOUNT of energy needed (ie. the work function) and the LIGHT ENERGY of the incident photon. THE BOTTOM LINE: Energy Conservation must still hold true! The energy NOT used to do work goes into KINETIC ENERGY as the electron LEAVES the surface. Light Energy, E WORK done to remove the electron

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**Putting it all together**

KINETIC ENERGY can be plotted on the y axis and FREQUENCY on the x-axis. The WORK FUNCTION is the y – intercept as the THRESHOLD FREQUNECY is the x intercept. PLANCK‘S CONSTANT is the slope of the graph.

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**Can we use this idea in a circuit?**

We can then use this photoelectric effect idea to create a circuit using incident light. Of course, we now realize that the frequency of light must be of a minimum frequency for this work. Notice the + and – on the photocell itself. We recognize this as being a POTENTIAL DIFFERENCE or Voltage. This difference in voltage is represented as a GAP that the electron has to jump so that the circuit works What is the GAP or POTENTIAL DIFFERENCE is too large?

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**Photoelectric Fact #5 - Stopping Potential**

If the voltage is TOO LARGE the electrons WILL NOT have enough energy to jump the gap. We call this VOLTAGE point the STOPPING POTENTIAL. If the voltage exceeds this value, no photons will be emitted no matter how intense. Therefore it appears that the voltage has all the control over whether the photon will be emitted and thus has kinetic energy.

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**Importance of photoelectric effect:**

Classical physics predicts that any frequency of light can eject electrons as long as the intensity is high enough. Experimental data shows there is a minimum (cutoff frequency) that the light must have. Classical physics predicts that the kinetic energy of the ejected electrons should increase with the intensity of the light. Again, experimental data shows this is not the case; increasing the intensity of the light only increases the number of electrons emitted, not their kinetic energy. THUS the photoelectric effect is strong evidence for the photon model of light.

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**Wave-Particle Duality**

The results of the photoelectric effect allowed us to look at light completely different. First we have Thomas Young’s Diffraction experiment proving that light behaved as a WAVE due to constructive and destructive interference. Then we have Max Planck who allowed Einstein to build his photoelectric effect idea around the concept that light is composed of PARTICLES called quanta.

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**The momentum of the photon**

Combining E=mc2 and p=mv, you get: p / E = v / c2 The photon travels at the speed of light, so v = c and p / E = 1 / c Therefore the momentum, p, of the photon is p = E / c But we also know that E = hf and λ=c/f, so

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**This led to new questions….**

If light is a WAVE and is ALSO a particle, does that mean ALL MATTER behave as waves? That was the question that Louis de Broglie pondered. He used Einstein's famous equation to answer this question.

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YOU are a matter WAVE! Basically all matter could be said to have a momentum as it moves. The momentum however is inversely proportional to the wavelength. So since your momentum would be large normally, your wavelength would be too small to measure for any practical purposes. An electron, however, due to it’s mass, would have a very small momentum relative to a person and thus a large enough wavelength to measure thus producing measurable results. This led us to start using the Electron Microscopes rather than traditional Light microscopes.

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**The electron microscope**

After the specimen is prepped. It is blasted by a bean of electrons. As the incident electrons strike the surface, electrons are released from the surface of the specimen. The deBroglie wavelength of these released electrons vary in wavelength which can then be converted to a signal by which a 3D picture can then be created based on the signals captured by the detector.

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The Compton Effect: when an X-ray photon strikes an electron in a piece of graphite, the X-ray scatters in one direction, and the electron recoils in another direction after the collision (like two billiard balls colliding on a pool table) the scattered photon has a frequency f ’ that is smaller than the frequency f of the incident photon, thus the photon loses energy in the collision the difference between the two frequencies depends on the angle at which the scattered photon leaves the collision

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**energy of incident photon = energy of scattered photon + KE of e- **

Similar to the analysis for the kinetic energy and work for photoelectric effect, we find: The electron is assumed to be initially at rest and essentially free (not bound to the atoms of the material) According to principle of conservation of energy: hf = hf ’ + K or energy of incident photon = energy of scattered photon + KE of e- For an initially stationary electron, conservation of total linear momentum requires that: Momentum of incident photon = momentum of scattered photon + momentum of electron From this point, these equations are combined with the relativistic equations for energy and momentum to derive the equation for Compton Scattering.

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The difference between the wavelength λ’ of the scattered photon and the wavelength λ of the incident photon is related to the scattering angle θ by: m is the mass of the electron. h/mc is the “Compton wavelength of the electron” and is h/mc = 2.43x10-12 m. It is interesting to note that according to Einstein’s theory of relativity, the rest mass of a photon is zero. However, it is never at rest, it is always moving (at the speed of light!) so it does have a finite momentum (even though p=mv doesn’t work)

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The Davisson-Germer experiment demonstrated the wave nature of the electron, confirming the earlier hypothesis of deBroglie. Davisson and Germer measured the energies of electrons scattering from a metal surface. Electrons from a heated filament were accelerated by a voltage and allowed to strike the surface of nickel metal, which could be rotated to observe angular dependence of the scattered electrons. They found that at certain angles there was a peak in the intensity of the scattered electron beam. In fact, the electron beam was scattered by the surface atoms on the nickel at the exact angles predicted for the diffraction of x-rays according to Bragg's formula nλ=2dsinθ, with a wavelength given by the de Broglie equation, λ=h/p. X-rays are accepted to be wavelike, thus this is evidence for wavelike behavior of electron.

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This is Davisson-Germer’s data relating the intensity of scattered electrons as a function of accelerating voltage for a particular angle.

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X-Rays can be produced when electrons, accelerated through a large potential difference, collide with a metal target (made from molybdenum or platinum for example) contained within an evacuated glass tube a plot of X-ray intensity per unit wavelength versus wavelength consists of sharp peaks or lines superimposed on a broad continuous spectrum X-ray spectrum shown here is produced when a molybdenum target is bombarded with electrons that have been accelerated through a potential difference

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when the energetic electrons impact the target metal, they undergo a rapid deceleration (braking). as the electrons suddenly come to rest they give off high-energy radiation in the form of X-rays over a wide range of wavelengths. This is referred to as “Bremsstrahlung continuum” (bremsstrahlung is German for “braking radiation”) This is the base for the peaks seen in the graph the sharp peaks are called characteristic lines or characteristic X-rays because they are characteristic of the target material. the characteristic lines are marked Kα and Kβ because they involve the n=1 or K shell of a metal atom. (K shell is the innermost electron shell) if an electron with enough energy strikes the target, one of the K-shell electrons can be knocked entirely out of a target atom an electron in one of the outer shells can then fall into the K shell, and an X-ray photon is emitted in the process. Kα is a change from n=2 to n=1; Kβ is a change from n=3 to n=1 there is a cutoff wavelength (as seen in the diagram). an impinging electron cannot give up any more than all of its KE, thus an emitted X-ray photon can have an energy no more than the KE of the impinging electron. the wavelength that corresponds to this is the cutoff wavelength (max frequency-min wavelength). K=eV, E=hf thus eV=hf. f=c/λ, so… V is the potential difference applied across the X-ray tube; e is the charge of an electron

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Line Spectra for a solid object the radiation emitted has a continuous range of wavelengths, some of which are in the visible region of the spectrum. the continuous range of wavelengths is characteristic of the entire collection of atoms that make up the solid. in contrast, individual atoms, free of the strong interactions that are present in a solid, emit only certain specific wavelengths, rather than a continuous range. these wavelengths are characteristic of the atom. a low-pressure gas in a sealed tube can be made to emit EM waves by applying a sufficiently large potential difference between two electrodes located within the tube with a grating spectroscope the individual wavelengths emitted by the gas can be separated and identified as a series of bright fringes or lines. these series of lines are called the LINE SPECTRA.

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**Bohr Model of Hydrogen atom**

Bohr tried to derive the formula that describes the line spectra that was developed by Balmer using trial and error. he used Rutherford’s model of the atom, the quantum ideas of Planck and Einstein, and the traditional description of a particle in uniform circular motion. assumptions of Bohr model: electron in H moves in circular motion only orbits where the angular momentum of the electron is equal to an integer times Planck’s constant divided by 2π are allowed electrons in allowed orbits do not radiate EM waves. Thus the orbits are stable. (if it emitted radiation, it would lose energy and spiral into the nucleus) EM radiation is given off or absorbed only when an electron changes from one allowed orbit to another. ΔE = hf (ΔE is energy difference between orbits and f is frequency of radiation emitted or absorbed) Bohr theorized that a photon is emitted only when the electron changes orbits from a larger one with a higher energy to a smaller one with a lower energy.

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**Bohr Energy levels in electron volts**

the centripetal force that provides the circular orbit is electrostatic force (Coulomb’s Law). setting this equal to centripetal force we get: or note that electrostatic force between an electron and a nucleus with Z protons is ke(Ze)/r2 = kZe2/r2. (Z = # of protons = atomic number) angular momentum is L=rmv . the second assumption listed above states the angular momentum of a given orbit, Ln, equals an integar, n, times h/2π; or Ln=nh/2π. setting these equal and solving for v gives: combining these two equations gives: (note that n is the orbital number) solving for rn gives us Bohr orbital radius

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**energy of electron is sum of kinetic and potential energies:**

E = K + U = ½ mv2 + U the electrostatic potential energy is U = -kZe2/r and using the first equation in this section we get: substituting in the Bohr orbital radius, rn, from above: grouping all of the constants yields: plugging in values for the constants gives us: Z = atomic number; n = energy level (1,2,3,…)

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when an electron in an initial orbit with a larger energy Ei, drops to a lower orbit with energy Ef, the emitted photon has an energy of Ei – Ef , consistent with the law of conservation of energy. combining this with Einstein’s E = hf, we get Ei – Ef = hf To find the wavelengths in hydrogen’s line spectrum (Z=1), we apply that to the equation above: given that E = hc/λ : the value in the first parenthesis is only constants. calculating this value gives the Rydberg Constant (1.097x107) – the exact value that Balmer found

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**de Broglie Waves and Bohr Model**

the angular momentum assumption in Bohr’s model is there because it produces results in agreement with experiment. however, de Broglie matter-wave relationship explains the significance: think of matter (electron) waves as analogous to a wave on a string – except that the string is a circle representing the electrons orbit. the standing wave must fit an integral number of wavelengths into the circumference of the orbit: nλ=2πr. combine this with p=h/λ: rearranging and multiplying both sides of the equation by r gives us angular momentum Thus this condition of the Bohr model is a reflection of the wave nature of matter

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Life and Atoms Every time you breathe you are taking in atoms. Oxygen atoms to be exact. These atoms react with the blood and are carried to every cell in your body for various reactions you need to survive. Likewise, every time you breathe out carbon dioxide atoms are released. The cycle here is interesting. TAKING SOMETHING IN. ALLOWING SOMETHING OUT!

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The Atom As you probably already know an atom is the building block of all matter. It has a nucleus with protons and neutrons and an electron cloud outside of the nucleus where electrons are orbiting and MOVING. Depending on the ELEMENT, the amount of electrons differs as well as the amounts of orbits surrounding the atom.

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**When the atom gets excited or NOT**

To help visualize the atom think of it like a ladder. The bottom of the ladder is called GROUND STATE where all electrons would like to exist. If energy is ABSORBED it moves to a new rung on the ladder or ENERGY LEVEL called an EXCITED STATE. This state is AWAY from the nucleus. As energy is RELEASED the electron can relax by moving to a new energy level or rung down the ladder.

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Energy Levels Yet something interesting happens as the electron travels from energy level to energy level. If an electron is EXCITED, that means energy is ABSORBED and therefore a PHOTON is absorbed. If an electron is DE-EXCITED, that means energy is RELEASED and therefore a photon is released. We call these leaps from energy level to energy level QUANTUM LEAPS. Since a PHOTON is emitted that means that it MUST have a certain wavelength.

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Energy of the Photon We can calculate the ENERGY of the released or absorbed photon provided we know the initial and final state of the electron that jumps energy levels.

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Energy Level Diagrams To represent these transitions we can construct an ENERGY LEVEL DIAGRAM Note: It is very important to understanding that these transitions DO NOT have to occur as a single jump! It might make TWO JUMPS to get back to ground state. If that is the case, TWO photons will be emitted, each with a different wavelength and energy.

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Example An electron releases energy as it moves back to its ground state position. As a result, photons are emitted. Calculate the POSSIBLE wavelengths of the emitted photons. Notice that they give us the energy of each energy level. This will allow us to calculate the CHANGE in ENERGY that goes to the emitted photon. This particular sample will release three different wavelengths, with TWO being the visible range ( RED, VIOLET) and ONE being OUTSIDE the visible range (INFRARED)

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**Energy levels Application: Spectroscopy**

Spectroscopy is an optical technique by which we can IDENTIFY a material based on its emission spectrum. It is heavily used in Astronomy and Remote Sensing. There are too many subcategories to mention here but the one you are probably the most familiar with are flame tests. When an electron gets excited inside a SPECIFIC ELEMENT, the electron releases a photon. This photon’s wavelength corresponds to the energy level jump and can be used to indentify the element.

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**Different Elements = Different Emission Lines**

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Emission Line Spectra So basically you could look at light from any element of which the electrons emit photons. If you look at the light with a diffraction grating the lines will appear as sharp spectral lines occurring at specific energies and specific wavelengths. This phenomenon allows us to analyze the atmosphere of planets or galaxies simply by looking at the light being emitted from them.

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**Line Spectra of Hydrogen Atom**

Lyman series occurs when electrons make transition s from higher energy levels with ni = 2, 3, 4, … to the first energy level where nf = 1. notice when an electron transitions from n=2 to n=1, the longest wavelength photon in the Lyman series is emitted, since the energy change is the smallest possible. when the electron transitions from highest to lowest, the shortest wavelength is emitted.

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Quantum Physics ISAT 241 Analytical Methods III Fall 2003 David J. Lawrence.

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