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**Max Karl Ernst Ludwig Planck (1858-1947)**

CHAPTER 3 Planck’s Constant 3-1 Atoms and Radiation in Equilibrium 3-2 Thermal Radiation Spectrum 3-3 Quantization of Electromagnetic Radiation 3-4 Atomic Spectra and the Bohr Model Max Karl Ernst Ludwig Planck ( ) Photo: We have no right to assume that any physical laws exist, or if they have existed up until now, or that they will continue to exist in a similar manner in the future. An important scientific innovation rarely makes its way by gradually winning over and converting its opponents. What does happen is that the opponents gradually die out. - Max Planck

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**Blackbody Radiation Why is Black Body Radiation important?**

When matter is heated, it emits radiation. A blackbody is a cavity with a material that only emits thermal radiation. Incoming radiation is absorbed in the cavity. Blackbody radiation is theoretically interesting because the radiation properties of the blackbody are independent of the particular material. Physicists can study the properties of intensity versus wavelength at fixed temperatures.

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**Rayleigh-Jeans Formula**

Lord Rayleigh used the classical theories of electromagnetism and thermodynamics to show that the blackbody spectral distribution should be: It approaches the data at longer wavelengths, but it deviates badly at short wavelengths. This problem for small wavelengths became known as the ultraviolet catastrophe and was one of the outstanding exceptions that classical physics could not explain.

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**Planck’s Radiation Law**

Planck assumed that the radiation in the cavity was emitted (and absorbed) by some sort of “oscillators.” He used Boltzman’s statistical methods to arrive at the following formula that fit the blackbody radiation data. Planck’s radiation law Planck made two modifications to the classical theory: The oscillators (of electromagnetic origin) can only have certain discrete energies, En = nhf, where n is an integer, n is the frequency, and h is called Planck’s constant: h = × 10−34 J·s. The oscillators can absorb or emit energy in discrete multiples of the fundamental quantum of energy given by:

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**Planck’s Radiation Law**

Exercise 3-1: Derive Planck’s radiation law. Application: The Big Bang theory predicts black body radiation. This radiation was discovered in 1965 by A. Penzias & R. Wilson. Cosmic Background Explorer (COBE) and Wilkinson Microwave Anisotropy Probe (WMAP) detected this radiation field at 2.725± °K This data supports the Big Bang Theory

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Stefan-Boltzmann Law The total power radiated increases with the temperature: This is known as the Stefan-Boltzmann law, with the constant σ experimentally measured to be × 10−8 W / (m2 · K4). The emissivity є (є = 1 for an idealized blackbody) is simply the ratio of the emissive power of an object to that of an ideal blackbody and is always less than 1.

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**Wien’s Displacement Law**

The spectral intensity R(l,T ) is the total power radiated per unit area per unit wavelength at a given temperature. Wien’s displacement law: The maximum of the spectrum shifts to smaller wavelengths as the temperature is increased.

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**Planck’s Radiation Law**

Exercise 3-2 : Show that Stefan’s Boltzmann law, Wein’s displacement law and Rayleigh-Jean’s law can be derived from Planck’s law Exercise 3-3: What is the average energy of an oscillator that has a frequency given by hf=kT according to Planck’s calculations? Exercise 3-4: How Hot is a Star? Measurement of the wavelength at which spectral distribution R(λ) from the Sun is maximum is found to be at 500nm, how hot is the surface of the Sun? Exercise 3-5: How Big is a Star? Measurement of the wavelength at which spectral distribution R(λ) from a certain star is maximum indicates that the star’s surface temperature is 3000K. If the star is also found to radiate 100 times the power Psun radiated by the Sun, how big is the star? Take the Sun’s surface temperature as 5800 K.

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What is a Photon? Planck introduced the idea of a photon or quanta. A cavity emits radiation by way of quanta. How does the radiation travel in space? We think that radiation is a wave phenomenon however the energy content is delivered to atoms in concentrated groups of waves (quanta).

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Properties of a Photon If a photon is to be considered as a particle we must be able to describe its mass, momentum, energy, statistics etc. Energy of a photon limiting value 1/2hf otherwise integral multiple of hf Interaction of photons with matter Complete absorption or partial absorption with the photon adjusting its frequency to remain as particle Intensity of photon intensity has nothing to do with the energy of photons

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Properties of Photons Constant h of photon h defines the smallest quantum angular momentum of a particle Exercise 6 Show that h has units of angular momentum Mass and momentum of photon photons move with velocity v=c Photons have no rest mass m0 Photon is not a material particle since rest mass is zero, it is a wave structure that behaves like a particle

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Properties of Photons Charge of a photon photons do not carry charge, however they can eject charge particles from matter when they impinge on atoms Photon Statistics Consider the radiation as a gas of photon. Photons move randomly like molecules in a gas and have wide range of energies but same velocity Statistics of photons is described by Bose-Einstein. We can talk about intensity and temperature in the same way as density and temperature.

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**Photoelectric Effect Methods of electron emission:**

Thermionic emission: Applying heat allows electrons to gain enough energy to escape. Secondary emission: The electron gains enough energy by transfer from another high-speed particle that strikes the material from outside. Field emission: A strong external electric field pulls the electron out of the material. Photoelectric effect: Incident light (electromagnetic radiation) shining on the material transfers energy to the electrons, allowing them to escape. We call the ejected electrons photoelectrons. Image adapted from

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**Photo-electric Effect Experimental Setup**

Image adapted from

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**Photo-electric effect observations**

The kinetic energy of the photoelectrons is independent of the light intensity. The kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light. Classically, the kinetic energy of the photoelectrons should increase with the light intensity and not depend on the frequency. Electron kinetic energy Picture from

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**Photo-electric effect observations**

Electron kinetic energy There was a threshold frequency of the light, below which no photoelectrons were ejected (related to the work function f of the emitter material). Picture from The existence of a threshold frequency is completely inexplicable in classical theory.

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**Photo-electric effect observations**

(number of electrons) When photoelectrons are produced, their number is proportional to the intensity of light. Also, the photoelectrons are emitted almost instantly following illumination of the photocathode, independent of the intensity of the light. Classical theory predicted that, for extremely low light intensities, a long time would elapse before any one electron could obtain sufficient energy to escape. We observe, however, that the photoelectrons are ejected almost immediately. Picture from

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**Einstein’s Theory: Photons**

Einstein suggested that the electro-magnetic radiation field is quantized into particles called photons. Each photon has the energy quantum: where f is the frequency of the light and h is Planck’s constant. Alternatively, where:

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Einstein’s Theory Conservation of energy yields: where f is the work function of the metal (potential energy to be overcome before an electron could escape). In reality, the data were a bit more complex. Because the electron’s energy can be reduced by the emitter material, consider fmax (not f):

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**Example – Photoelectric Effect**

Exercise 3-6: An experiment shows that when electromagnetic radiation of wavelength 270 nm falls on an aluminum surface, photoelectrons are emitted. The most energetic of these are stopped by a potential difference of 0.46 volts. Use this information to calculate the work function of aluminum in electron volts. Exercise 3-7: The threshold wavelength of potassium is 558 nm. What is the work function for potassium? What is the stopping potential when light of 400 nm is incident on potassium? Exercise 3-8 Light of wavelength 400 nm and intensity 10-2 W/m2 is incident on potassium. Estimate the time lag expected classically.

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**Atomic Spectra Newton discovers the dispersion of light**

Invention of Spectroscopy

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**Spectra Three Kinds of spectra**

Solid, liquid or a dense gas excited to emit a continuous spectrum Light passing through low density gas excites atoms to produce emission spectra Light passing through cool low density gas results in absorption spectra From Stars and Galaxies by Seeds/Thomson Brookscole

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Line Spectra Chemical elements were observed to produce unique wavelengths of light when burned or excited in an electrical discharge.

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Balmer Series In 1885, Johann Balmer found an empirical formula for the wavelength of the visible hydrogen line spectra in nm: Image from Thornton and Rex

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Rydberg-Ritz Formula As more scientists discovered emission lines at infrared and ultraviolet wavelengths, the Balmer series equation was extended to the Rydberg equation: From Stars and Galaxies by Seeds

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**The Classical Atomic Model**

Consider an atom as a planetary system. The Newton’s 2nd Law force of attraction on the electron by the nucleus is: where v is the tangential velocity of the electron: The total energy is then: This is negative, so the system is bound, which is good.

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**The Classical Atomic Model**

Exercise 7: Show that in the classical model the frequency of radiation for an accelerating electron is and the Energy is

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**The Planetary Model is Doomed**

From classical E&M theory, an accelerated electric charge radiates energy (electromagnetic radiation), which means the total energy must decrease. So the radius r must decrease!! Electron crashes into the nucleus!? Electron does not crash in the Bohr model Physics had reached a turning point in 1900 with Planck’s hypothesis of the quantum behavior of radiation, so a radical solution would be considered possible.

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**The Bohr Model of the Hydrogen Atom**

Bohr’s general assumptions: 1. Stationary states, in which orbiting electrons do not radiate energy, exist in atoms and have well-defined energies, E. Transitions can occur between them, yielding light of energy: Bohr frequency condition E = Ei − Ef = hf 2. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere. n = 1 n = 3 n = 2 Angular momentum is quantized! 3. The angular momentum of the nth state is: where n is called the Principal Quantum Number.

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**Consequences of the Bohr Model**

The angular momentum is: a0 So the velocity is: So: Solving for r: a0 is called the Bohr radius.

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Bohr Radius The Bohr radius, is the radius of the unexcited hydrogen atom. The “ground” state Hydrogen atom diameter is:

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Energy of an electron Exercise 3-9: Show that the energy of an electron in any atom at orbit n is quantized and that it gives the ground state energy of Hydrogen atom to be -13.6eV.

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Rydberg-Ritz formula Exercise 3-10: Derive the Ryderberg-Ritz formula

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**Transitions in the Hydrogen Atom**

The atom will remain in the excited state for a short time before emitting a photon and returning to a lower stationary state. In equilibrium, all hydrogen atoms exist in n = 1.

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**Successes and Failures of the Bohr Model**

Why should the nucleus of the atom be kept fixed? The electron and hydrogen nucleus actually revolve about their mutual center of mass. Conservation of momentum require that the momenta of nucleus and electron equal in magnitude. The total kinetic energy is then The electron mass is replaced by its reduced mass:

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**Limitations of the Bohr Model**

The Bohr model was a great step in the new quantum theory, but it had its limitations. Failures: Works only for single-electron (“hydrogenic”) atoms. Could not account for the intensities or the fine structure of the spectral lines (for example, in magnetic fields). Could not explain the binding of atoms into molecules.

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**Rydberg-Ritz Formula -Example**

Exercise 3-11 Use the Rydberg-Ritz formula to calculate the first line of Balmer, Lyman and Paschen series for the Hydrogen atom. From Stars and Galaxies by Seeds

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**The Correspondence Principle**

When energy levels are very close quantization should have little effect Bohr’s correspondence principle is rather obvious: In the limits where classical and quantum theories should agree, the quantum theory must reduce the classical result.

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**The Correspondence Principle**

Exercise 3-12: Show that in the limit of large quantum number the Bohr frequency is the same as the classical frequency.

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**Fine Structure Constant**

In Bohr’s theory we know that transitions can occur for ∆n≥1, for small n values. If we allow this for large n and calculate the classical and Bohr frequencies as in previous exercise, we will find that they do not agree. To avoid this disagreement, A. Sommerfeld introduced special relativity and elliptical orbits. From Bohr orbit in hydrogen for n=1, we have We will skip the mathematical treatment of A. Sommerfeld work α is called the fine structure constant

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**Fine Structure Constant**

The fine structure constant can be understood in the following way. For each circular orbit rn and energy En a set of n elliptical orbits exist whose major axis are the same but they have different eccentricities and thus different velocities and Energies. Electron transitions depend on the eccentricities of the initial and final orbits and on the major axes, thus resulting in splitting of energy levels of n called fine-structure splitting. Fine structure constant leads to the notion of electron spin

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Example Exercise 3-13: Show that the energy levels of oscillators in simple harmonic motion are quantized. Do it yourself exercise: solve the differential equation

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Example Exercise 3-14: Derive the Bohr quantum condition from Wilson-Sommerfeld quantization rule

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**X-Ray Production: Theory**

An energetic electron passing through matter will radiate photons and lose kinetic energy, called bremsstrahlung. Since momentum is conserved, the nucleus absorbs very little energy, and it can be ignored. The final energy of the electron is determined from the conservation of energy to be:

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**X-Ray Production: Theory**

If photons can transfer energy to electrons, can part or all of the kinetic energy of electron be converted into photons? “The Inverse photoelectric effect” This was discovered before the work of Planck and Einstein

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Observation of X Rays 1895: Wilhelm Röntgen studied the effects of cathode rays passing through various materials. He noticed that a phosphorescent screen near the tube glowed during some of these experiments. These new rays were unaffected by magnetic fields and penetrated materials more than cathode rays. He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube. physics/rontgen/ Wilhelm Röntgen

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**X-Ray Production: Experiment**

Current passing through a filament produces copious numbers of electrons by thermionic emission. If one focuses these electrons by a cathode structure into a beam and accelerates them by potential differences of thousands of volts until they impinge on a metal anode surface, they produce x rays by bremsstrahlung as they stop in the anode material.

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**Electromagnetic theory Predicts X-Ray**

Accelerated charges produce electromagnetic waves, when fast moving electrons are brought to rest, they are certainly accelerated 1906: Barkla found that X-Ray show Polarization, this establishing that X-Rays are waves. X-Rays have wavelength range 0.1 nm – 100 nm Even though classical theory predicts x-ray’s, the experimental data is not explainable.

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**Two distinctive features**

Some targets have enhanced peaks. For example Molybdenum shows two peaks at specific wavelengths. This is due to rearrangement of electrons of the target material after bombardment. X-ray ‘s have a continuous spectrum No matter what the target , the threshold wavelength depends on the accelerating potential

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**Inverse Photoelectric Effect**

Since the work function of the target is of the order of few eV, whereas the accelerating potential is thousand of eV From photoelectric effect Duane-Hunt rule

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X-Ray Spectra Bohr-Rutherford picture of the atom can also be applied to heavy elements Shells have letter names: K shell for n = 1 L shell for n = 2 The atom is most stable in its ground state. When it occurs in a heavy atom, the radiation emitted is an x-ray. It has the energy E (x ray) = Eu − Eℓ. An electron from higher shells will fill the inner-shell vacancy at lower energy.

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**Atomic Number and Moseley**

The x-rays have names: L shell to K shell: Kα x-ray M shell to K shell: Kβ x-ray etc. G.J. Moseley studied x-ray emission in Atomic number Z = number of protons in the nucleus. Moseley found a relationship between the frequencies of the characteristic x-ray and Z. Moseley found this relation holds for the Kα x-ray with b=1 and different An values (from quantum mechanics):

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**Moseley’s Empirical Results**

The Ka x-ray is produced from the n = 2 to n = 1 transition. In general, the K series of x-ray frequencies are: Form the Bohr model with Z=Z-1 where We use Z-1 instead of Z because one electron is already present in the K-shell and so shields the other's from the nucleus’ charge. Moseley’s research clarified the importance of Z and the electron shells for all the elements, not just for hydrogen.

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**Moseley’s Empirical Results**

For the L series of x-ray wavelength the frequencies are Neodymium Z=60 and Samarium Z=62 b) Promethium Z=61 c) All three elements together

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Example X-Ray Spectra Exercise 3-15: Calculate the wavelength of Kα line of molybdenum, Z=42, and compare with the value λ=0.0721nm measured by Moseley.

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**Aguer (oh-zhay) Effect**

In 1923 Pierre Auger discovered an alternative to X-ray emission. The atom may eject a third electron from a higher-energy outer shell via radiationless process called Auger effect EdN(e)/dE plot: Auger spectrum of Cu Al and Al2O3 together note the energy shift in the larges peak due to adjustment in Al electron shell energies

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X-Ray Scattering Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed. Crystals act as three-dimensional gratings, scattering the waves and producing observable interference effects.

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Bragg’s Law William Lawrence Bragg interpreted the x-ray scattering as the reflection of the incident x-ray beam from a unique set of planes of atoms within the crystal. There are two conditions for constructive interference of the scattered x rays: The angle of incidence must equal the angle of reflection of the outgoing wave. The difference in path lengths must be an integral number of wavelengths. Bragg’s Law: nλ = 2d sin θ (n = integer)

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**The Bragg Spectrometer**

A Bragg spectrometer scatters x rays from crystals. The intensity of the diffracted beam is determined as a function of scattering angle by rotating the crystal and the detector. When a beam of x rays passes through a powdered crystal, the dots become a series of rings.

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Examples Exercise 3-16: What is the shortest wavelength present in a radiation if the electrons are accelerated to 50,000 volts? Exercise 3-16:The spacing of one set of crystals planes in common salt is d=0.282nm. A monochromatic beam of X-rays produces a Bragg maximum when its glancing angle is 7 degrees. Assuming that this is the first order maximum (n=1), find the wavelength of the X-rays, what is the minimum possible accelerating voltage Vo that produced the X-rays?

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**4-5: Franck-Hertz Experiment**

Franck and Hertz studied the phenomenon of ionization. Proof to show the quantization of atomic energies. Elastic collision: ∆E=E2-E1=10.2eV is not equal to that of incoming electron Leads to current increasing Inelastic collision: ∆E=E2-E1 equal to that of incoming electron ∆E is absorbed by the electron in the atom and excitation occurs. The excited electron returns to ground state quickly emitting a photon Leads to sudden fall in current

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**Atomic Excitation by Electrons**

Accelerating voltage is below 5 V: electrons did not lose energy. Accelerating voltage is above 5 V: sudden drop in the current. Hg has an excitation energy of 4.88 eV in the first excited state No energy can be transferred to Hg below 4.88 eV because not enough energy is available to excite an electron to the next energy level Above 4.88 eV, the current drops because scattered electrons no longer reach the collector until the accelerating voltage reaches 9.8 eV and so on. Franck –Hertz experiment is gratifying because we show quantization of energy levels by measurements with voltmeters and ammeters

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28.3 THE BOHR THEORY OF HYDROGEN At the beginning of the 20th century, scientists were puzzled by the failure of classical physics to explain the characteristics.

28.3 THE BOHR THEORY OF HYDROGEN At the beginning of the 20th century, scientists were puzzled by the failure of classical physics to explain the characteristics.

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