1AMQ-Atoms, Molecules and Quanta Spring Semester, 2010

1. 1AMQ-Atoms, Molecules and Quanta Spring Semester, 2010 -Lecturer: Zsolt Podolyák [Office06BC04] e-mail: z.podolyak@surrey.ac.uk • Course provides an introduction to Modern Physics • It provides the basis for many advanced courses including • -Quantum Mechanics, Solid State, Semiconductor • and Nuclear Physics • It begins by introducing phenomena which led to the need • for a new mechanics, followed by the introduction of • Quantum Mechanics and how it can be used to describe • the properties of atoms. Books:- Krane, Modern Physics, 2nd Edition, Wiley Halliday, Resnick and Walker, Fundamentals of Physics, 4th Edition,Wiley(Chapters 40,41,43-45) Eisberg and Resnick, Quantum Physics,Wiley Lectures/tutorials: slides plus notes on board. All pictures used in class, as well as the solutions to the class tutorials will be posted on the physics intranet www.ph.surrey.ac.uk/~phs1zp/1amq.html Assessment-Week 7-multiple choice test(Worth % of total, exam = %).

2. Outline of Course 1AMQ Quanta 1. Introduction; Microscopic World-Sizes and Units 2. Quanta and Electromagnetic waves 3. Blackbody Radiation 4. Photoelectric Effect 5. Compton Effect Quantum Mechanics 6. Wave-particle duality; Uncertainty principle 7. Schrodinger’s wave equation 8. Simple cases: free electrons, electrons in a box, quantum nos. The Simplest Atom: Hydrogen 9. Spectral series for Hydrogen 10. Bohr’s Theory for the Hydrogen atom 11. Hydrogen atom in Quantum Mechanics 12. Spatial quantisation and electron spin 13. Fine Structure and Zeeman splitting Multi-electron Atoms 14. Spectroscopic notation, Pauli principle, level ordering 15. Electron screening, shell and sub-shell structure 16. Characteristic X-rays and selection rules 17. Optical spectra of atoms and selection rules 18. Adding angular momenta for two electrons [He atom] Molecules 19. Hydrogen molecular ion 20. Hydrogen molecule and covalent bonding

3. INTRODUCTION • Classical Physics vs. Modern Physics ( approx. before and after ~ 1900 ) ● End of 19th Century-physicists thought they had a good grasp of the physical world with Newton’s Laws and Maxwell’s equations for electromagnetism. ●Suddenly all this was changed by a series of discoveries: radioactivity, X-rays, discovery of and measurements on electrons. They also found it impossible to explain the spectra from blackbodies. •Reason for new paradigms: ability to make better measurements New phenomena observed, hence new theories •Main Result:- The exploration of 3 extremes of Nature - Very fast: special relativity replaces Newtonian mechanics - Very small: quantum mechanics replaces Newtonian mechanics - Very large: general relativity replaces Newtonian mechanics •These new theories of modern physics are refinements of the old ideas but are quite radical in conception. •The old theories work perfectly well at everyday velocities and scales.

4. New Experiments  New Theories  New Concepts RELATIVITY   Measurements of New concepts of speed of light Space & Time (Einstein) QUANTUM MECHANICS   Spectrum of LightNew ideas about measurement and determinism a) from hot glowing objects b) from electrical breakdown in gases Key experiments: -to do with light (very fast, c =3 x 108 ms) -to do with atoms (very small: 10-10 m) Here we are concerned with atoms and the theories needed to describe their properties.We will not be concerned with relativity.

5. ● To give a precise value for any quantity we need at least three things No.x order-of-magnitude x units (=/- error)

6. Voyage from Infinity to Zero Telescopes 1022 m 10-15 m 10-14 m 10-10 m 1019 m 10-9 m 1012 m 10-6 m 107 m 10-5 m Microscopes

8. Our World View As we have seen our World view is of a Universe with a series of layers: each layer containing objects on a particular length scale. Universe-------------?????? Galaxy clusters----6x1022 m Galaxies-------------1019-1020 m Solar system--------6x1012 m Earth-----------------12.7x106 m Crystals/humans---10-2 – 10 m Atoms----------------10-10 m Nuclei----------------10-14 m Nucleons------------ 10-15 m Quarks---------------?????? Is there a significance to this picture? At each scale there is a dominant force. Thus gravity dictates the motion of the planets in the solar system and the nuclear force dictates the size of nuclei. Questions: How do we know these sizes? What units are appropriate?

9. Sizes of Atoms-Dalton’s Atomic Theory 1803-Knowledge of Chemistry was good enough for John Dalton to propose an atomic theory. Basic idea: chemical elements are composed of tiny, indivisible fundamental particles, all identical, they dictate properties of the element Dalton’s Atomic Theory 1. Chemical elements composed of extremely small particles which retain their identities in chemical processes. Atom is smallest mass of an element which can take part in chemical change. 2. Each atom has a definite weight. 3. Each element consists of a particular type of atom which differs in weight from atoms of every other element 4. Atoms combine in simple numerical ratios. Note: Smallest particle of a chemical compound is a molecule e.g. HCl-Dalton’s compound atom. Later Modification:-Discovery of ISOTOPES means that a. An element may have atoms differing in weight. b. It is not ATOMIC WEIGHT which characterises the element c. It is ATOMIC NUMBER= +ve charge on the atomic nucleus -all isotopes of an element have the same atomic number.

10. Avogadro’s Number • Avogadro’s hypothesis: Equal volumes of all gases, under the same conditions of temperature and pressure, contain identical numbers of molecules. • We define 1 Atomic Mass Unit(amu) as being one-twelfth of the mass of the 12 C isotope. [Note: C chemical element, 12 is the number of protons plus neutrons in the nucleus and 6 is the number of protons. • Now consider Avogadro’s hypothesis in terms of mass. Define: Kilogram molecule [kmole] = amount of substance with mass in kg equal numerically with its molecular weight i.e. 1kmole N2 = 28.014 kg N2 • For 1kmole of any substance which has a mass M kg and contains N0 molecules of mass m kg M = N0m or N0 =M/m Since by definition M  m then N0 is the same for all substances. • So N0 is a universal constant = No. of molecules in 1 kmole From experiment N 0 = 6.022 x 1026 kmole -1(Avogadro’s Number) Note: 1 kmole atom contains N0 atoms

11. Atomic Masses and Sizes • For 12 C: mass m = M/N0 = 12/6.022 x 10 26 = 1.99 x 10 -26 kg By definition: 1 amu = mass(12 C)/12 = 1/ N0 = 1.66 x 10-27 kg We find a range of atomic masses up to 4 x 10-25 kg In general masses are well defined. Sizes are less well defined •Let us make an estimate: consider a solid in which the atoms are packed closely together. If an individual atom has a diameter of 2r,where r is the atomic radius, then in 1m we can lay 1/2r atoms side by side. In a cube of 1m side we then have (1/2r)3 atoms Each atom occupies a volume of V=(2r)3 Now in 1kmole we have 6 x 1026 atoms and it occupies N0 x (2r)3m 3 and has a mass of N0 x (2r)3 x =A where A=atomic weight and ρ =density. Thus r = 1/2x(A/N0 )1/3 e.g. Be: A = 9.01,  = 1.84 x 10 3 kg/m 3  rBe = 1.0 x 10 -10 m Now r  (A/ )1/3 which varies only slowly with A so we expect all atoms to have radii which are a few times 10-10m 1m ---------------------------------

12. What about molecules? Assume that they are spherical. V = m /  = M/(N0. ) = (4/3)  r3 where m = mass of a molecule  r = [3M/4N0 ] 1/3 = [3 x 18.015/4 x 6 x 1026 x 10 3] m for water =1.92 x 10 -10 m All determinations of r for small molecules give values of r = 10 -9 - 10 -10 m Our simple estimates tell us that atoms are approx. 10 -10 m in radius and small molecules about r = 10 -9 - 10 -10 m

13. What about Nuclei? •The idea of the nuclear atom comes from experiments in Manchester[1911].Ernest Rutherford suggested that Geiger and Marsden look at large angle scattering of alpha particles(4He nuclei) from metal foils of Au.They observed that a small number were scattered backwards.This is consistent with the current picture of an atom as consisting of a massive,+vely charged,central nucleus surrounded by a cloud of electrons. Overall the atom is electrically neutral. •The same measurement gives an idea of nuclear size.The scattering of charged particles from the nucleus by the Coulomb force alone is called Rutherford Scattering. As we increase the energy of the alpha particles there comes a point where what we observe is not consistent with RS.This is because the particle is close enough to feel the STRONG or Nuclear force.This distance we can define as the nuclear radius. •If we consider scattering at 1800 then (1/2)mv2 = Z1 Z2 e2/40r  r = (Z1 Z2 e2/40) x (2/mv2) For He(Z = 2) on Cu(Z = 29) at 5 MeV energy we get (ε0=8.85x10-12 F/m) r  10-14 m  Z2e  Z2e r

14. Nuclear Radii •One thing we ignored so far is the size of the electron! Interestingly all the evidence we have so far is that it is a genuine particle: an object with a mass and no size, which interacts via the EM force alone. •As a result it is the ideal tool for probing the sizes of nuclei. Thus many experiments have been done in which high energy electrons have been scattered from nuclei and the nuclear radius deduced from the results. The picture summarises the results.The plot shows the Mean square radius plotted against A1/3,where A is the number of nucleons [neutrons plus protons] in the nucleus. This can be summed up as R = R0 A1/3, where R0 is a constant equal to 1.2 x 10-15 m •Note:-This is the charge radius.However other expts. tell us the matter radius is essentially the same. Density = Mass/volume =1.66 x 10-27/1.33x3.1412x(1.2x10-15)3  2.3 x 10 17 kg/m3

15. Units •Systems of units must be SELF-CONSISTENT and must be comprehensible. All systems [SI/cgs/Britsh] satisfy the first criterion but many fail the second. •For example: R0 = 1.2 x 10-15 m. Nuclei clearly do not belong on this scale. So we introduce the Fermi (F) = 10-15 m (femto metre) Similarly for atoms the Angstrom (Å)= 10-10 m •The charge on an electron is 1.6 x 10-19 Coulombs. The energy it acquires when it is accelerated through a potential of 1 volt is 1/2 mv 2 = eV So we introduce a unit of energy the electron volt = 1 eV It turns out to be of the right size for energies in atomic systems •Now from Special Relativity E2 = p2c2 + m20c4 For p = 0 E = m0c2 = 511 keV for an electron = 930 GeV for a proton Thus it is natural in Particle Physics to talk about GeV i.e.10 9 eV and in Nuclear Physics it is equally natural to use MeV= 10 6 eV • Atomic Physics eV-keV Nuclear Physics keV-MeV Particle Physics GeV-TeV

16. Units 1 Fermi (1F) =10 -15 m =1 fm 1 Angstrom (1A) = 10 -10 m 1eV = 1.6 x 10 -19 Joules 1 keV = 1.6 x 10 -16 Joules 1 MeV = 1.6 x 10 -13 Joules 1 GeV = 1.6 x 10 -10 Joules 1 amu = 1.66 x 10 -27 kg Atomic sizes = 10 -10 m Nuclear sizes = 10 -14 m Charge on electron(proton) = 1.6 x 10 -19 C

17. Quanta of Light We will now look at a series of experiments related to electromagnetic radiation. a) Diffraction and interference of light. b) Photoelectric effect c) Blackbody radiation d) Compton Effect. In 19th century it seemed that expts. on diffraction and interference of light had settled question of nature of light. Maxwell predicted that light has speed = c and was described as a transverse wave. 1887-Hertz produced and detected such waves. We will find that the question was not really settled and that we now have a different view.

18. The electromagnetic spectrum

19. Electromagnetic Waves •If charges are accelerated an electromagnetic wave is created. The E and B fields vary with both t and r. • Point source - spherical waves - wave fronts are spherical. Picture shows a plane wave travelling in the +ve Z-direction E = E0 sin (kz- t +  ) B = B0 sin (kz- t +  ) , wave number k = 2/, and ang. freq.  = 2 . B0=E0/c Now c =  so we can write c = /k and the angle  is an arbitrary phase angle. • Note:-The wave shown is plane polarised and the energy flux S = E x B/μ0 in the forward direction. (μ0 =4π x 10-7 Tm/A -permeability of free space) •S is called the Poynting vector and has units of energy/time/area I.e. Wm -2 •

20. EM-Waves(contd.) •Intensity  E0 2 (general property of waves) •Intensity fluctuates with time - 2 = 2( / 2) Normally fluctuation is too fast for us to see.For visible light  = 10 15 oscillations per sec. •Principle of Superposition - net effect is sum of individual effects i. e. two waves cause disturbance at a point which is result of combined disturbance and they emerge from the point with all of their properties unchanged. •This leads to Interference and Diffraction a] Constructive Interference b] Destructive Interference - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - -

21. Young’s Double Slit experiment The observation of both interference and diffraction was seen as a triumph for the wave theory of light. One excellent example is Young’s double slit experiment. Here light from a single source falls on two slits. The two slits act as coherent sources and we observe interference on the screen behind. => minima

22. Single Slit Diffraction-A Major Success for the wave Theory •If the size of the slit is comparable to  then we see a diffraction pattern not a sharp image. •We see a central maximum. •At the first minimum we have a sin/2 = /2,i.e. a sin =  (At first minimum)

23. Diffracton at a single slit • Figure shows diffraction at a single slit with a width b. Assuming a wavefront arrives at the aperture any ray passing through can be associated with a ray leaving the aperture a distance b/2 away.If they are /2 out of phase then destructive interference occurs.Then b/2.sin1 = /2 or sin1 = /b • If we divide the aperture into 4 parts then b/4.sin1 = /2 or sin1 = 2/b More generally sin1 = m/b where m = 1,2,3,4,---------- Thus we get darkness on the screen at these points and we get the diffraction pattern shown in the figure.

24. Peacock Interference due to the structure of the feathers