3-1 Atoms and Radiation in Equilibrium3-2 Thermal Radiation Spectrum3-3 Quantization of Electromagnetic Radiation3-4 Atomic Spectra and the Bohr Model CHAPTER 3 Plancks ConstantWe 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 PlanckMax Karl Ernst Ludwig Planck (1858-1947)

Blackbody RadiationWhen 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.Why is Black Body Radiation important?

Rayleigh-Jeans FormulaLord 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.

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 Plancks constant: h = 6.6261 1034 Js. The oscillators can absorb or emit energy in discrete multiples of the fundamental quantum of energy given by: Plancks radiation lawPlancks Radiation LawPlanck assumed that the radiation in the cavity was emitted (and absorbed) by some sort of oscillators. He used Boltzmans statistical methods to arrive at the following formula that fit the blackbody radiation data.

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 0.001 K Plancks Radiation LawThis data supports the Big Bang TheoryExercise 3-1: Derive Plancks radiation law.

Stefan-Boltzmann LawThe total power radiated increases with the temperature:

This is known as the Stefan-Boltzmann law, with the constant experimentally measured to be 5.6705 108 W / (m2 K4).

The emissivity ( = 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.

Wiens Displacement LawThe spectral intensity R(l,T ) is the total power radiated per unit area per unit wavelength at a given temperature.Wiens displacement law: The maximum of the spectrum shifts to smaller wavelengths as the temperature is increased.

Plancks Radiation LawExercise 3-2 : Show that Stefans Boltzmann law, Weins displacement law and Rayleigh-Jeans law can be derived from Plancks law

Exercise 3-3: What is the average energy of an oscillator that has a frequency given by hf=kT according to Plancks calculations?

Exercise 3-4: How Hot is a Star? Measurement of the wavelength at which spectral distribution R() 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() from a certain star is maximum indicates that the stars 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 Suns surface temperature as 5800 K.

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).

Properties of a PhotonIf 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 hfInteraction 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

Properties of PhotonsConstant h of photon h defines the smallest quantum angular momentum of a particle Exercise 6 Show that h has units of angular momentumMass 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

Properties of PhotonsCharge 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.

Photoelectric EffectMethods 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.

Photo-electric Effect Experimental Setup

Photo-electric effect observationsThe 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.

Photo-electric effect observationsThere was a threshold frequency of the light, below which no photoelectrons were ejected (related to the work function f of the emitter material). The existence of a threshold frequency is completely inexplicable in classical theory.

Photo-electric effect observationsWhen 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.(number of electrons)

Einsteins Theory: PhotonsEinstein 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 Plancks constant.Alternatively,

where:

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

Example Photoelectric EffectExercise 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.

Newton discovers the dispersion of lightInvention of SpectroscopyAtomic Spectra

Three Kinds of spectra Solid, liquid or a dense gas excited to emit a continuous spectrumLight passing through low density gas excites atoms to produce emission spectraLight passing through cool low density gas results in absorption spectraSpectra

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

Balmer SeriesIn 1885, Johann Balmer found an empirical formula for the wavelength of the visible hydrogen line spectra in nm:

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

The Classical Atomic ModelConsider an atom as a planetary system.The Newtons 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.

The Classical Atomic ModelExercise 7: Show that in the classical model the frequency of radiation for an accelerating electron is

and the Energy is

The Planetary Model is DoomedFrom 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!!Physics had reached a turning point in 1900 with Plancks hypothesis of the quantum behavior of radiation, so a radical solution would be considered possible.Electron crashes into the nucleus!?Electron does not crash in the Bohr model

The Bohr Model of the Hydrogen AtomBohrs 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 = hf2. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere.3. The angular momentum of the nth state is: where n is called the Principal Quantum Number.Angular momentum is quantized!

Consequences of the Bohr ModelThe angular momentum is:So:Solving for r:So the velocity is:a0 is called the Bohr radius.

Bohr RadiusThe Bohr radius,

is the radius of the unexcited hydrogen atom.

The ground state Hydrogen atom diameter is:

Energy of an electronExercise 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.

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

Transitions in the Hydrogen AtomThe 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.

Successes and Failures of the Bohr ModelWhy 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

Success:The electron mass is replaced by its reduced mass:

Limitations of the Bohr ModelWorks 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.Failures:The Bohr model was a great step in the new quantum theory, but it had its limitations.

Rydberg-Ritz Formula -ExampleExercise 3-11 Use the Rydberg-Ritz formula to calculate the first line of Balmer, Lyman and Paschen series for the Hydrogen atom.

The Correspondence PrincipleIn the limits where classical and quantum theories should agree, the quantum theory must reduce the classical result.Bohrs correspondence principle is rather obvious:When energy levels are very close quantization should have little effect

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

Fine Structure ConstantIn Bohrs theory we know that transitions can occur for n1, 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

is called the fine structure constantWe will skip the mathematical treatment of A. Sommerfeld work

Fine Structure ConstantThe 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

ExampleExercise 3-13: Show that the energy levels of oscillators in simple harmonic motion are quantized.

Do it yourself exercise: solve the differential equation

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

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:

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

Observation of X Rays1895: Wilhelm Rntgen 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.

Wilhelm Rntgen

X-Ray Production: ExperimentCurrent 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.

Electromagnetic theory Predicts X-RayAccelerated 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-rays, the experimental data is not explainable.

Two distinctive featuresSome 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 spectrumNo matter what the target , the threshold wavelength depends on the accelerating potential

Inverse Photoelectric EffectSince the work function of the target is of the order of few eV, whereas the accelerating potential is thousand of eV From photoelectric effectDuane-Hunt rule

Shells have letter names:K shell for n = 1L 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 E.X-Ray Spectra An electron from higher shells will fill the inner-shell vacancy at lower energy.Bohr-Rutherford picture of the atom can also be applied to heavy elements

Atomic Number and MoseleyThe x-rays have names:L shell to K shell: K x-rayM shell to K shell: K x-rayetc.

G.J. Moseley studied x-ray emission in 1913.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 K x-ray with b=1 and different An values (from quantum mechanics):

Moseleys Empirical ResultsThe 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-1Moseleys research clarified the importance of Z and the electron shells for all the elements, not just for hydrogen.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.where

Moseleys Empirical ResultsFor the L series of x-ray wavelength the frequencies are Neodymium Z=60 andSamarium Z=62b) Promethium Z=61c) All three elements together

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

Aguer (oh-zhay) EffectIn 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 effectEdN(e)/dE plot: Auger spectrum of CuAl and Al2O3 together note the energy shift in the larges peak due to adjustment in Al electron shell energies

X-Ray ScatteringMax 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.

Braggs LawWilliam 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.

Braggs Law: n = 2d sin (n = integer)

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.The Bragg Spectrometer

ExamplesExercise 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?

4-5: Franck-Hertz ExperimentFranck 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 electronLeads to current increasing

Inelastic collision: E=E2-E1 equal to that of incoming electronE 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

Atomic Excitation by ElectronsAccelerating 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 stateNo 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

*Photo: http://www.spaceandmotion.com/quantum-theory-max-planck-quotes.htm************Image adapted from http://sol.sci.uop.edu/~jfalward/particlesandwaves/particlesandwaves.html*Image adapted from http://www.chem1.com/acad/webtut/atomic/qprimer/#Q8*Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png*Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png*Picture from http://en.wikipedia.org/wiki/Image:Photoelectric_effect.png*****From Stars and Galaxies by Seeds/Thomson Brookscole**Image from Thornton and Rex*From Stars and Galaxies by Seeds************From Stars and Galaxies by Seeds*********www.peoples.ru/science/ physics/rontgen/ ****************