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.

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 intranetwww.ph.surrey.ac.uk/~phs1zp/1amq.html

Assessment-Week 7-multiple choice test(Worth % of total, exam = %).

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

Outline of Course 1AMQ

Quanta1. Introduction; Microscopic World-Sizes and Units2. Quanta and Electromagnetic waves3. Blackbody Radiation4. Photoelectric Effect5. Compton EffectQuantum Mechanics6. Wave-particle duality; Uncertainty principle7. Schrodinger’s wave equation8. Simple cases: free electrons, electrons in a box, quantum nos.The Simplest Atom: Hydrogen 9. Spectral series for Hydrogen10. Bohr’s Theory for the Hydrogen atom11. Hydrogen atom in Quantum Mechanics12. Spatial quantisation and electron spin13. Fine Structure and Zeeman splittingMulti-electron Atoms14. Spectroscopic notation, Pauli principle, level ordering15. Electron screening, shell and sub-shell structure16. Characteristic X-rays and selection rules17. Optical spectra of atoms and selection rules18. Adding angular momenta for two electrons [He atom]Molecules19. Hydrogen molecular ion20. Hydrogen molecule and covalent bonding

INTRODUCTION

• Classical Physics vs. Modern Physics ( approx. before and after ~ 1900 )

•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.

● 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.

New Experiments New Theories New Concepts

RELATIVITY

Measurements of New concepts of speed of light Space & Time

QUANTUM MECHANICS

Spectrum of Light New ideas about measurement and determinisma) 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 neededto describe their properties.We will not be concerned with relativity.

(Einstein)

Sizes, Orders-of- magnitude and Units

•Often important to know roughly how largesomething is!

-Chemical engineers need thermal constantsto 5-fig. Accuracy to predict manufacture

-Mech.engs-factor of two may make bridge falldown.

-Physicists-estimate whether steel is stronger than butter from atomic constitution.

•Mass of proton= 1.66x10-27 kg=No. x Order-of-magn. X unit

•Units-Self-consistent and comprehensibleAll systems are former[SI/CGS/British]not all are latter[see above]

● To give a precise value for any quantity we need at least three things

No.x order-of-magnitude x units (=/- error)

10-15 m

107 m

1012 m

1019 m

1022 m

10-9 m

10-6 m10-5 m

10-10 m

10-14 m

Telescopes

Microscopes

The Microscopic World

•ATOMS--10-10 m

•NUCLEI--10-14 m

•NUCLEONS--10-15 m

•QUARKS--???

Our World View

As we have seen our World view is of a Universe with aseries of layers: each layer containing objects on a particular length scale.

Universe-------------??????Galaxy clusters----6x1022 mGalaxies-------------1019-1020 mSolar system--------6x1012 mEarth-----------------12.7x106 mCrystals/humans---10-2 – 10 mAtoms----------------10-10 mNuclei----------------10-14 mNucleons------------ 10-15 mQuarks---------------??????

Is there a significance to this picture?At each scale there is a dominant force. Thus gravity dictatesthe motion of the planets in the solar system and thenuclear force dictates the size of nuclei.

Questions: How do we know these sizes? What units are appropriate?

Sizes of Atoms-Dalton’s Atomic Theory

Dalton’s Atomic Theory1. 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.

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

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

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)3 m 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---------------------------------

What about molecules?

Assume that they are spherical.

V = m / = M/(N0. ) = (4/3) r3

where m = mass of a molecule

r = [3M/4N0 ] 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

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 e2/40r

r = (Z1 Z2 e2/40) 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

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

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 + m2

0c4

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

Units

1 Fermi (1F) =10 -15 m =1 fm1 Angstrom (1A) = 10 -10 m1eV = 1.6 x 10 -19 Joules1 keV = 1.6 x 10 -16 Joules1 MeV = 1.6 x 10 -13 Joules1 GeV = 1.6 x 10 -10 Joules1 amu = 1.66 x 10 -27 kg

Atomic sizes = 10 -10 mNuclear sizes = 10 -14 m

Charge on electron(proton) = 1.6 x 10 -19 C

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.

The electromagnetic spectrum

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

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 Diffractiona]

ConstructiveInterference

- - - - - - - - - - - - - - - - - - - - - - - - - -- - - - -

Destructive Interference

b]

Young’s Double Slit experiment

The observation of both interference and diffraction was seen as a triumph for the wave theory of light. One excellentexample is Young’s double slit experiment. Here light froma single source falls on two slits. The two slits act as coherentsources and we observe interference on the screen behind.

=> minima

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)

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.sin1 = /2 or sin1 = /b

• If we divide the aperture into 4 parts then b/4.sin1 = /2 or sin1 = 2/b More generally sin1 = 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.

PeacockInterference due to the structure of the feathers