Chapter 28: Photons, electrons & atoms Albert Einstein

What will we learn in this chapter? Contents: The photoelectric effect Line spectra and energy levels The nuclear atom Bohr’s model (Lasers) X-ray production and scattering Wave nature of particles Wave–particle duality Electron microscopes

smoke detector use the photoelectric effect

What is light? So far: We have established that light is an electromagnetic wave. In this chapter: We show that light has also a particle character. The emission/absorption of light is quantized, i.e., the energy of these particles can only take certain definite values. Light particles are called photons. Furthermore: We show that the energy levels in atoms are quantized. To fully understand these effects quantum mechanics is needed.

The photoelectric effect Experiment by Hertz (1887): Electrons are emitted from a surface of a conductor when light shines on it. What has happened? The (usually confined) electrons absorb the energy of the photons and thus overcome the surface potential barrier to escape. The minimum amount of energy an individual electron has to gain in order to escape from a surface is called the work function φ of the surface. Later experiments by Hallwachs & Lenard (~1900): Studied the photoelectric effect in more detail. Studied the current produced by monochromatic light when the frequency is varied.

Hallwachs & Lenard experiment Two electrodes are enclosed in a high-vacuum tube (to avoid particle-particle collisions). The battery generates an electric field (pink arrows) between the negativelycharged cathode and the positivelycharged anode. Monochromatic light (purple arrows) falls on the cathode causing electrons (blue balls) to be emitted. Because of the electric field, the electrons are pushed to the anode producing a measurable current in the galvanometer G.

Hallwachs & Lenard experiment contd. Experimental observations: No electrons are emitted unless the frequency of the light exceeds a material-dependent threshold frequency f. The required frequencies are in the UV range (200 – 300 nm), except for potassium and cesium oxides, where they are in the visible spectrum (400 – 700 nm).

Conclusions: The results are consistent with the notion that electrons absorb an amount of energy E proportional to the frequency of the light. When f is not big enough, E is not great enough to overcome the potential energy barrier of the surface φ .

Hallwachs & Lenard experiment contd. Further experimental observations: When the field is turned off some electrons are still able to reach the anode, suggesting that the electrons have great speeds. Even when the polarity of the potential difference V is reversed and the field pushes electrons back to the cathode, some electrons still reach the anode. The electron flow stops completely only once the potential energy eV is larger than the maximum kinetic energy of the electrons. Stopping potential: The reversed potential difference required to stop the electron flow completely is called the stopping potential and denoted by V0. It follows 1 2 mvmax = eV0 2

Hallwachs & Lenard experiment contd. all electrons collected Dependence on light intensity: by cathode When the potential of the setup VAC is increased, the photocurrent levels off. The potential where the current levels off is independent of the light intensity. When the light intensity is increased, the maximum current increases, but the stopping potential is the same.

Classical (wrong) predictions: A classical wave theory of light would predict that when the light intensity is increased, the electrons should have a higher energy and thus the stopping potential should be larger. This is not true!

Hallwachs & Lenard experiment contd. Dependency on frequency: When the frequency f of the light is increased, the stopping potential increases linearly. The kinetic energy of an electron depends on the frequency and not on the intensity of the light. Theoretical description (Einstein, 1905): A beam of light consists of small bundles of energy called quanta or photons. The energy of the photons is directly proportional to their frequency.

Photoelectric effect Energy of a photon: The energy E of an individual photon is equal to a constant times the frequency f of the photon: E = hf

h is a universal constant called Planck’s constant with the numerical value h = 6.6260693(11) × 10−34 Js . Note: The energy transfer from a photon to an electron in a conductor is an “all-or-nothing” process. If this energy is greater than the surface potential barrier the electron can escape. Einstein received the Nobel prize for these results in 1921. In terms of electronvolts h = 4.136 × 10−15 eV · s . Typical work functions of elements are between 2 and 5 eV and are very sensitive to surface impurities.

Photoelectric effect contd. The maximum kinetic energy of an electron is thus given by

1 2 mvmax = hf − φ = eV0 2 i.e., the photon energy minus the potential barrier energy. It follows: V0 =

h φ f− e e

If we measure the stopping potential V0 for different values of f, we expect V0 to be a linear function of f. The intercept with the vertical axis gives −φ/e and the slope h/e . Knowing e (Millikan) allows for the determination of the work function φ and h.

V0 (V)

f (1015 Hz)

Visible evidence of light’s particle nature Photographic film contains silver salts which, when exposed to light, undergo a chemical reaction. After development, the salts turn into silver, which prevents the light from passing, creating a (black and white) image. If light is made of photons, these should strike individual silver salt crystals at very low exposures.

increasing number of photons

Only when the number of photons is large, a clear image emerges.

Photon frequency and wavelength Photon frequency and wavelength: The concept of photons is applicable to all regions of the electromagnetic spectrum. Photons always travel with the speed of light c. A photon of frequency f and wavelength λ = c/f has the energy hc E = hf = λ Note: One can show using relativity theory that any particle (even with zero mass) has a momentum given by p = E/c. The momentum of photons is thus h p= λ

So… how do smoke detectors work? Schematic diagram: photoelectric detector

smoke particles light source

light beam

How it works: Light is scattered off the smoke particles and strikes the photosensitive cell. A small current is produced in the cell, triggering an alarm.

Line spectra and energy levels The existence of line spectra has been known for more than 200 years. A diffraction grating can be used to separate the various wavelengths. Two cases: Hot source: ! ! ! ! ! Excited gas: ! ! ! ! !

Continuous spectrum, all wavelengths are present. Example: light bulbs. Line spectrum, only certain colors appear. Example: neon signs, thermally exited salts, …

Line spectra Early knowledge: Early in the 19th century is was discovered that each chemical element has a definite, unchanging set of wavelengths in its line spectrum (@home experiment: sprinkle salt over a flame). Identifying chemical constituents of materials by analyzing spectra became a useful technique. Later theoretical explanation (Niels Bohr 1913): Atoms can only have discrete internal energy levels. If the atoms can emit and absorb photons of a given energy and/ or wavelength (spectra, photoelectric effect), then the atoms themselves can only possess these energy levels. Transitions between these levels are permitted by emission of a photon.

Bohr’s hypothesis Bohr’s hypothesis (emission): If Ei is the initial energy of an atom before a transition from one energy level to another, Ef is the atom’s (smaller) final energy after the transition, and the energy of the emitted photon is hf, then

hf = Ei − Ef Note: For the simplest atom (hydrogen) Bohr pictured these levels in terms of the electron revolving in various orbits around the proton with only certain radii permitted. Bohr’s hypothesis (absorption): In this case the atom’s final energy is greater than the initial energy, we obtain

hf = Ef − Ei

Bohr’s hypothesis & the hydrogen spectrum Hydrogen spectrum:

Hydrogen emits 4 lines in the visible spectrum: Hα (red) – Hδ . By trial-and-error, Balmer discovered in 1885… Balmer series: The formula for the visible hydrogen spectrum is: ! " 1 1 1 ! ! = !R ! !− ! ! Here λ is the wavelength,R = 1.097 × 107 m−1 ! λ! ! ! 22 ! n ! 2 ! is the Rydberg constant and n ≥ 3 is an ! ! ! ! ! ! ! integer. Example: for n = 3 we obtain λ = 656.3nm (which is the Hα line).

Bohr & Balmer contd. We can rewrite Balmer’s equation for the hydrogen spectrum: ! " hc 1 1 hcR hcR = hf = E = hcR − = − 2 λ 22 n2 22 n

We can identify these terms with initial and final energies. This suggests that the hydrogen atom has a series of energy levels given by En = −

hcR n2

n = 2, 3, 4, . . .

For n = ! the electron is separated from the proton. Each wavelength in the Balmer series corresponds to a transition from a state with n " 3 to n = 2. Note: !The numerical value of hcR is 13.6 eV and is a typical unit for ! ! atomic energy levels.

The whole hydrogen spectrum There are other series of spectra named after their discoverers in agreement with Bohr’s predictions. All the spectral series of the hydrogen atom can be understood via: hcR En = − 2 n Lyman series: final state n = 1. Balmer series: final state n = 2. Paschen series: final state n = 3. Brackett series: final state n = 4. Pfund series: final state n = 5.

UV

Infrared

Energy levels and Bohr’s predictions Good news: Atoms or ions with a single electron can be represented by the simple Bohr formula. Bad news: This does not work for more complex elements. Still, it is always possible to analyze more complex spectra of other elements in terms of transitions between energy levels. The numerical values of the levels can be deduced from spectral wavelength measurements. Definitions: Ground state (GS): Lowest energy level an atom can have. Excited state: All energy levels with energies larger than the GS.

Further definitions Levels vs states: There are cases where an atom can be in different states, but with the same energy. Therefore we distinguish between energy states and energy levels because one level can correspond to multiple states. Resonance: When two or more energy levels are very close, the atom can switch easily between these states by absorbing/emitting a photon. Lifetime: Time-span when an atom is excited and then returns to the ground state by emitting a photon (usually around ns).

Continuous spectra Liquids & solids: Line spectra are generally produced by gases where interactions between the atoms can be neglected. blackbody spectrum In liquids and solids interactions shift the spectra of individual atoms. Because there are so many atoms, the shifts overlap and we obtain a continuous emission spectrum. Blackbody radiation: One can show that the total radiation of a liquid or solid is proportional to T4. The radiation consists of a continuous distribution of wavelengths. Blackbody radiation from a hot material is most intense in the vicinity of λmax ∼ 1/T . Therefore, the hotter the material, the “bluer” it is. This is why colors are sometimes given in Kelvin (bulbs, monitors).

The nuclear atom So far: Millikan showed that most of the mass is in the nucleus (electrons are thousands of times lighter than atomic cores). It was known that atoms are of the size of Å. It was known that all atoms except hydrogen contain more than one electron. To be understood: What is the inner structure of an atomic core? What is the “shape” of an atom? Do atoms also show a particle–wave duality like electrons or photons? Right now we are in the early 1910’s…

Rutherford scattering contd. First experiment designed to probe the inner structure of atoms. Idea: Bombard atoms with alpha particles from radioactive sources with speeds of 107 m/s to test the subatomic structure. Setup: A collimated beam of alpha particles targets a gold foil. Why gold? Easy to machine into thin sheets. The scintillation screens spark every time an alpha particle hits them.

Rutherford scattering experiment contd. Measurement & results: Rutherford’s students (Geiger & Marsden) then counted the flashes as a function of the angle. (Note: This is why Geiger invented the Geiger counter…). Scattering angles of even 180º were found! Because the alpha particles pass trough the gold sheet, Rutherford was able to show that… Alpha particles can penetrate atoms. Scattering is only due to interactions with the positive atomic charges. Electrons are 7300 times smaller and their effects negligible due to momentum conservation. But…

Rutherford scattering experiment contd.

Thomson model

Rutherford model

Thomson model: Positive charges are uniformly distributed around the atom’s volume. Rutherford’s results: The scattering angles found showed that positive charges can only be located in a tiny volume and not around the whole atomic volume. The notion of an atomic nucleus was born. Nuclei are ~10-14 m diameter, contain 99.95% of the mass and all the positive charge!

Nuclear size vs possible scattering angles Realistic simulation of a gold nucleus with a radius of 7.0 x 10-15 m, ~105 times smaller than an atom. Large scattering angles are possible.

Computer simulation of a hypothetical nucleus with a radius of 7.0 x 10-14 m. No large-angle scattering is possible.

Rutherford’s postulate Because nuclei are much smaller than atoms, the question arose, what ensured that atoms are stable. Rutherford’s postulate: Electrons revolve on orbits about the nucleus (similar to the solar system). Problems of this postulate: Rotating (accelerating) charges emit radiation. The total electron energy would reduce, as well as the orbit. The atom would be unstable. Furthermore, atoms would emit continuous spectra (not the case!).

The Bohr model Bohr postulated: Electrons in an atom evolve in stable orbits with definite energies without emitting radiation. Only when the electron changes the orbit (transition) does the atom emit/absorb radiation via hf = Ei − Ef . To determine the radii of the permitted orbits, Bohr postulated that the angular momentum is quantized in units of h/2π = ! . Note that the units of [h] = Js = kg m/s2 = [p]. Quantization of the angular momentum in the Bohr model: The allowed values of the angular momentum for an electron in a circular orbit are given by h L = mvn rn = n n = 1, 2, 3, 4, . . . 2π

Mechanical model of the hydrogen atom Model: A negative electron with mass m and charge –e revolves around a proton with charge e. Because the proton is 2000 times heavier than the electron, we assume it does not move.

Mechanical model of hydrogen contd. Equate the electrical force to the centrifugal force (F = ma with a = v2/r): 1 e2 vn2 F = =m 4π"0 rn2 rn Use now the quantization of the angular momentum: h mvn rn = n 2π to obtain… Radius & speed of Bohr orbits: An electron in a state with a quantum number n has orbit radius rn and speed vn given by n 2 h2 1 e2 rn = !0 v = n πme2 !0 2nh Note: h2 The smallest possible orbit is the Bohr radius r1 = !0 πme2 2 All higher orbits rn = r1n , i.e., r2 = 4r1, r3 = 9r1, … −10 n is called the quantum number for the orbit r1 = 0.5293×10 m

Energy levels The expressions for the radius rn and speed vn can be used to obtain the kinetic and potential energies for an electron with the quantum number n: 1 1 me4 2 Kn = mvn = 2 2 2 2 !0 8n h 1 e2 1 me4 Un = − =− 2 2 2 4π"0 rn "0 4n h

The total energy is given by En = Un + Kn = −

1 me4 !20 8n2 h2

Ground state: The atom energy En is least when n = 1. The radius r1 is also least when n = 1.

Rydberg constant revisited We can compare the energy-level equation from spectral analysis hcR n2 to the Bohr expression for the energy 1 me4 En = − 2 2 2 !0 8n h Rydberg constant expressed by me4 It follows: fundamental constants: R = 2 3 8!0 h c En = −

Plugging in the values of the constants, we obtain R = 1.097 × 107 m−1 a major triumph for Bohr’s theory! The ionization energy is given by the energy for a transition from n = 1 to n = !. One finds 13.6 eV in agreement with experiments. Note, it is useful to write: Kn = 13.6eV/n2 En = −13.6eV/n2 Un = −27.2eV/n2

Applicability/limitations of the Bohr model The model works for any one-electron atom. Examples: Single-ionized Helium Double-ionized Lithium … If the nuclear charge is Ze instead of e, one has to simply replace in all equations: e2 → Ze2 The radius decreases by a factor Z The energies are multiplied by Z2. Limitations: Atoms with two or more electrons cannot be studied. More drastic changes to the “classical picture” were needed.

Lasers LASER = Light Amplification by Stimulated Emission of Radiation What is a laser? A light source that produces highly-coherent and nearly monochromatic light. The light is so well collimated, that the earth–moon distance was measured using a laser beam up to 10cm error! Toy model: Assume a transparent container filled with gas. For simplicity the atoms are in the ground state.

Laser contd.

Absorption: Spontaneous emission: Stimulated emission: Atoms can absorb Some (random) time The incoming photon photons of energy E later the atom A* emits causes A* to emit a and be excited to a a photon of energy E photon in the same state A*. and returns to A. direction. The direction of the A correlation occurs photon is random. and all atoms synchronize.

Laser contd. The idea behind the laser can be illustrated best with a ruby laser: High-intensity light excites electrons in the chromium impurities of the ruby crystal. These emit photons of 694 nm which cause a cascade of stimulated emission in the other Cr atoms. The light intensity is amplified. A coherent, monochromatic and very narrow light beam is emitted. Mathematically a population inversion of states (ground state to excited state) occurs: By pumping energy into the material, the number of excited atoms is larger than atoms in the ground state: coherent emission.

Lasers: applications Lasing can be induced from x-ray to microwaves. Applications: High-intensity lasers are used for precision cutting (even diamond). Precision measurements (distances in surveys). Medical science (eye surgery, tumor removal, …). Technology (data reading in CDs and DVDs, printers, scanners). Military (missile guidance). Warning: Be sure to wear eye protection if you see this symbol at a lab door.

X-ray production Production (Röntgen 1895): Idea: Rapidly moving electrons due to a potential difference of 103 – 106 V strike a metal target. Implementation: Place apparatus in vacuum. A heated cathode emits electrons which are accelerated due to a large potential difference to the anode. Electrons strike the metallic anode and emit x-rays. The emitted lights has wavelengths between 0.001 and 1 nm. Note: The principle is inverse to the photoelectric effect. Here the kinetic energy of electrons is used to create photons.

X-ray production contd. Note: In x-ray production we can neglect the initial kinetic energy of the electrons or the work function of the target since the produced radiations is far more energetic. Each element has atomic levels associated to the x-ray part of the spectrum. These are hundreds or thousands of eV above the ground state. The x-ray energy levels are associated with vacancies in the inner electrons and thus are different than than the visible spectra. Recently (Nature), it has been shown that peeling tape off on vacuum produces x-rays (portable devices!).

Compton scattering Discovered in 1924 by A. H. Compton. Direct confirmation of the quantum nature of light. What is it? When x-rays strike matter, some of the light is scattered. Compton discovered that some of the scattered radiation has a longer wavelength than the incident radiation and the increase in wavelength ∆λ depends on the scattering angle. Experimental setup: measure the wave– length of the x-rays after they are scattered.

Compton scattering contd. If λ is the wavelength of the incident x-rays and λ! is the wavelength of the scattered x-rays at an angle φ , then Wavelength increase in Compton scattering: h ∆λ = λ! − λ = (1 − cos φ) mc where m is the electron mass. Note: Classical electromagnetism would predict that the incoming and scattered wave have the same wavelength. The quantity h/mc has units of length, its numerical value being 2.426 pm.

Quantum mechanical explanation Compton scattering can be explained with quantum theory: The scattering occurs between two particles, an electron at rest and an incident photon.

before collision

Part of the photon energy and after collision momentum is transferred to the electron (conservation of momentum). Since the energy changes, so does the wavelength! The wavelength shift equation can be derived from conservation of energy and momentum with pphoton = h/λ .

Applications of x-rays Most materials are opaque to visible light. Therefore x-rays are used to visualize the interiors. Applications: Medical (broken bones). Materials (defects). Security (ask the TSA…). Destruction of cancer cells. Improvement: Computerized Axial Tomography (CAT scans) A thin fan-shaped beam of electrons is rotated around a subject. A computer reconstructs an image which can visualize even tumors (invisible with traditional techniques).

Particle–wave duality? Or wave–particle? So far: Light shows particle-like features. The Bohr model fails to describe multi-electron atoms. Solution: Extend the particle-wave duality of light to particles. This is the main ingredient of quantum mechanics. Note that it is only natural to extend this notion to particles based on symmetry considerations!

de Broglie waves de Brogli’s postulate (1924): If light behaves like a particle, then electrons should behave like a wave. The analogy is simple: Take the expression for the wavelength of the photon, pphoton = h/λ and invert it for the wavelength. de Broglie wavelength of the electron: For an electron with momentum p = mv, (m the electron mass, v its velocity) the de Broglie wavelength is given by h λ= mv Note: Within quantum mechanics, particles are like localized “wavepackets.” Electrons in an atom can thus be visualized as a diffuse cloud.

Davisson-Germer experiment Idea: incident electrons Perform a diffraction experiment on an atomic surface, but use electrons instead of x-rays. The speed of the electrons cab be determined from the accelerating voltage, thus the (de Broglie) wavelength can be computed. Result: The deflection angles found for a given electron wavelength agreed with the deflection angles of x-rays with the same wavelength. Later, experiments using neutrons or alpha particles agreed with the Davisson-Germer results.

Wave–particle duality Classical Newtonian mechanics: We treat particles as points with a definite position and velocity at any instant in time. !

!

!

!

!

!

!

VS

Quantum mechanics: When we look at small enough scales there are fundamental limitations on the precision with which we can describe position and velocity of particles. The particle behavior can only be stated in terms of probabilities.

Single-slit experiment with electrons Setup: Electrons with the same speed are sent trough a single slit.

Result: A diffraction pattern emerges on the screen. Classical mechanics: All electrons should travel on the same path. Clearly this is not the case. Some electrons hit the area of the central peak, others are diffracted to higher maxima.

Single-slit experiment with electrons contd. Quantum mechanics: Although all e- start the same, we cannot follow their path! The best we can do is determine with a given probability, how many electrons will statistically be in a given region. There is no way to determine the final position of an electron. Uncertainties: The uncertainty in the momentum ∆py is directly related to the width of the first maximum. The uncertainty ∆y in the position is directly related to the width of the slit a. Reducing a, increases ∆py , reducing ∆py increases a. It follows ∆py a ≥ const.

Heisenberg uncertainty principle If a coordinate x has a standard deviation ∆x and the x-component of the momentum, px, has a uncertainty ∆px then the two uncertainties are related by the… Heisenberg uncertainty principle for position and momentum: h ∆x∆px ≥ 2π This means neither momentum nor position can be determined with arbitrary precision, as classical physics would predict. Heisenberg uncertainty principle for energy and time: The uncertainty ∆E in the energy of a system in a particular state depends on the time interval ∆t during which the system remains in that state. It follows h ∆E∆t ≥ 2π A system that remains a long time in a state, has a well defined energy.

The electron microscope Recall Rayleigh’s criterion: The limit of resolution is proportional to the wavelength λ θres = 1.22 D Idea: Use the wave character of electrons to generate smaller wavelengths (103 times smaller means 103 better magnification). Same principles as for the light microscope. Optical lenses are replaced by electrostatic lenses: optical

electrostatic