1. Chapter 29 Atomic Physics

2. Importance of the Hydrogen Atom The hydrogen atom is the only atomic system that can be solved exactly Much of what was learned about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He + and Li 2+

3. More Reasons the Hydrogen Atom is Important The hydrogen atom proved to be an ideal system for performing precision tests of theory against experiment Also for improving our understanding of atomic structure The quantum numbers that are used to characterize the allowed states of hydrogen can also be used to investigate more complex atoms This allows us to understand the periodic table

4. Final Reason for the Importance of the Hydrogen Atom The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids

5. Early Models of the Atom – Newton’s Time The atom was a tiny, hard indestructible sphere It was a particle model that ignored any internal structure The model was a good basis for the kinetic theory of gases

6. Early Models of the Atom – JJ Thomson J. J. Thomson established the charge to mass ratio for electrons His model of the atom A volume of positive charge Electrons embedded throughout the volume

7. Rutherford’s Thin Foil Experiment Experiments done in 1911 A beam of positively charged alpha particles hit and are scattered from a thin foil target Large deflections could not be explained by Thomson’s model

8. Early Models of the Atom – Rutherford Rutherford Planetary model Based on results of thin foil experiments Positive charge is concentrated in the center of the atom, called the nucleus Electrons orbit the nucleus like planets orbit the sun

9. Difficulties with the Rutherford Model Atoms emit certain discrete characteristic frequencies of electromagnetic radiation The Rutherford model is unable to explain this phenomena Rutherford’s electrons are undergoing a centripetal acceleration It should radiate electromagnetic waves of the same frequency The radius should steadily decrease as this radiation is given off The electron should eventually spiral into the nucleus It doesn’t

10. The Bohr Theory of Hydrogen In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory His model includes both classical and non-classical ideas He applied Planck’s ideas of quantized energy levels to orbiting electrons

11. Bohr’s Theory, cont In this model, the electrons are generally confined to stable, nonradiating orbits called stationary states Used Einstein’s concept of the photon to arrive at an expression for the frequency of radiation emitted when the atom makes a transition

12. Problem’s With Bohr’s Model Improved spectroscopic techniques showed many of the single spectral lines were actually a group of closely spaced lines Single spectral lines could be split into three closely spaced lines when the atom was placed in a magnetic field

13. Mathematical Description of Hydrogen The solution to Schrödinger’s equation as applied to the hydrogen atom gives a complete description of the atom’s properties Apply the quantum particle under boundary conditions and determine the allowed wave functions and energies of the atom

14. Mathematical Description, cont For a three-dimensional system, the boundary conditions will generate three quantum numbers A fourth quantum number is needed for spin The potential energy function for the hydrogen atom is

15. Mathematical Description, final The energies of the allowed states becomes a o is the Bohr radius The allowed energy levels depend only on the principle quantum number, n

16. Quantum Numbers, General The imposition of boundary conditions also leads to two new quantum numbers Orbital quantum number, Orbital magnetic quantum number, m

17. Principle Quantum Number The first quantum number is associated with the radial function It is called the principle quantum number It is symbolized by n The potential energy function depends only on the radial coordinate r The energies of the allowed states in the hydrogen atom are the same E n values found from the Bohr theory

18. The orbital quantum number is symbolized by It is associated with the orbital angular momentum of the electron It is an integer The orbital magnetic quantum number is symbolized by m It is also associated with the angular orbital momentum of the electron and is an integer Orbital and Orbital Magnetic Quantum Numbers

19. The values of n can range from 1 to  The values of can range from 0 to n-1 The values of m can range from – to Example: If n = 1, then only = 0 and m = 0 are permitted If n = 2, then = 0 or 1 If = 0 then m = 0 If = 1 then m may be –1, 0, or 1 Quantum Numbers, Summary of Allowed Values

20. Quantum Numbers, Summary Table

21. Shells Historically, all states having the same principle quantum number are said to form a shell Shells are identified by letters K, L, M … All states having the same values of n and are said to form a subshell The letters s, p,d, f, g, h,.. are used to designate the subshells for which = 0, 1, 2, 3, …

22. Shell and Subshell Notation, Summary Table

23. Quantum Numbers, final State with quantum numbers that violate the previous rules cannot exist They would not satisfy the boundary conditions on the wave function of the system

24. Wave Functions for Hydrogen The simplest wave function for hydrogen is the one that describes the 1s state and is designated  1s (r) As  1s (r) approaches zero, r approaches  and is normalized as presented  1s (r) is also spherically symmetric This symmetry exists for all s states

25. Probability Density The probability density for the 1s state is The radial probability density function, P(r), is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr

26. Radial Probability Density A spherical shell of radius r and thickness dr has a volume of 4  r 2 dr The radial probability function is P(r)  4  r 2  2

27. P(r) for 1s State of Hydrogen The radial probability density function for the hydrogen atom in its ground state is The peak indicates the most probable location The peak occurs at the Bohr radius

28. P(r) for 1s State of Hydrogen, cont The average value of r for the ground state of hydrogen is 3/2 a o The graph shows asymmetry, with much more area to the right of the peak According to quantum mechanics, the atom has no sharply defined boundary as suggested by the Bohr theory

29. Electron Clouds The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud This shows the probability density as a function of position in the xy plane The darkest area, r = a o, corresponds to the most probable region

30. Electron Clouds, cont The electron cloud model is quite different from the Bohr model The electron cloud structure remains the same, on the average, over time The atom does not radiate when it is in one particular quantum state This removes the problem of the Rutherford model Radiation occurs when a transition is made, causing the structure to change in time

31. The next-simplest wave function for hydrogen is for the 2s state n = 2; = 0 The wave function is  2s depends only on r and is spherically symmetric Wave Function of the 2s state

32. Comparison of 1s and 2s States The plot of the radial probability density for the 2s state has two peaks The highest value of P corresponds to the most probable value In this case, r  5a o

33. Physical Interpretation of The magnitude of the angular momentum of an electron moving in a circle of radius r is L = m e v r The direction of is perpendicular to the plane of the circle In the Bohr model, the angular momentum of the electron is restricted to integer multiples of

34. Physical Interpretation of , cont According to quantum mechanics, an atom in a state whose principle quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum: That L can equal zero causes great difficulty when attempting to apply classical mechanics to this system In the quantum mechanical interpretation, the electron cloud for the L = 0 state is spherically symmetrical with no fundamental axis of rotation

35. Physical Interpretation of m The atom possesses an orbital angular momentum Because angular momentum is a vector, its direction must be specified An orbiting electron can be considered an effective current loop with a corresponding magnetic moment

36. Physical Interpretation of m , 2 The direction of the angular momentum vector relative to an axis is quantized Once an axis is specified, the angular momentum vector can only point in certain directions with respect to this axis Therefore, L z, the projection of along the z axis, can have only discrete values

37. Physical Interpretation of m , 3 The orbital magnetic quantum number m specifies the allowed values of the z component of orbital angular momentum L z = m The quantization of the possible orientations of with respect to an external magnetic field is often referred to as space quantization

38. Physical Interpretation of m , 4 does not point in a specific direction Even though its z-component is fixed Knowing all the components is inconsistent with the Uncertainty Principle Image that must lie anywhere on the surface of a cone that makes an angle  with the z axis

39. Physical Interpretation of m , final  is also quantized Its values are specified through m is never greater than, therefore  can never be zero can never be parallel to the z-axis

40. Spin Quantum Number, m s Electron spin does not come from the Schrödinger equation Additional quantum states can be explained by requiring a fourth quantum number for each state This fourth quantum number is the spin magnetic quantum number, m s

41. Electron Spins Only two directions exist for electron spins The electron can have spin up (a) or spin down (b) In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases

42. Electron Spins, cont The concept of a spinning electron is conceptually useful The electron is a point particle, without any spatial extent Therefore the electron cannot be considered to be actually spinning The experimental evidence supports the electron having some intrinsic angular momentum that can be described by m s Sommerfeld and Dirac showed this results from the relativistic properties of the electron

43. Spin Angular Momentum The total angular momentum of a particular electron state contains both an orbital contribution and a spin contribution Electron spin can be described by a single quantum number s, whose value can only be s = 1 / 2 The spin angular momentum of the electron never changes

44. Spin Angular Momentum, cont The magnitude of the spin angular momentum is The spin angular momentum can have two orientations relative to a z axis, specified by the spin quantum number m s  1 / 2 m s  1 / 2 corresponds to the spin up case m s  - 1 / 2 corresponds to the spin down case

45. Spin Angular Momentum, final The allowed values of the z component of the spin angular momentum is S z  m s   1 / 2 Spin angular moment S is quantized

46. The spin magnetic moment is related to the spin angular momentum by The z component of the spin magnetic moment can have values Spin Magnetic Moment

47. Bohr Magneton The quantity e  /2m e is called the Bohr magneton It has a numerical value of 9.274 x 10 -24 J/T

48. Stern-Gerlach Experiment A beam of neutral silver atoms is split into two components by a nonuniform magnetic field The atoms experienced a force due to their magnetic moments The beam had two distinct components in contrast to the classical prediction

49. Stern-Gerlach Experimental Results The experiment provided two important results: It verified the concept of space quanitization It showed that spin angular momentum exists even though the property was not recognized until long after the experiments were performed

50. Wolfgang Pauli 1900 – 1958 Important review article on relativity At age 21 Discovery of the exclusion principle Explanation of the connection between particle spin and statistics Relativistic quantum electrodynamics Neutrino hypothesis Hypotheses of nuclear spin

51. The Exclusion Principle The four quantum numbers discussed so far can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure The Exclusion Principle states that no two electrons in an atom can ever be in the same quantum state Therefore, no two electrons in the same atom can have the same set of quantum numbers

52. Filling Subshells Once a subshell is filled, the next electron goes into the lowest-energy vacant state If the atom were not in the lowest-energy state available to it, it would radiate energy until it reached this state

53. Orbitals An orbital is defined as the atomic state characterized by the quantum numbers n,  and m From the Exclusion Principle, it can be seen that only two electrons can be present in any orbital One electron will have m s = 1 / 2 and one will have m s = - 1 / 2

54. Allowed Quantum States, Example The arrows represent spin The n=1 shell can accommodate only two electrons, since only one orbital is allowed In general, each shell can accommodate up to 2n 2 electrons

55. Hund’s Rule Hund’s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins Some exceptions to the rule occur in elements having subshells that are close to being filled or half-filled

56. Periodic Table Dmitri Mendeleev made an early attempt at finding some order among the chemical elements He arranged the elements according to their atomic masses and chemical similarities The first table contained many blank spaces and he stated that the gaps were there only because the elements had not yet been discovered

57. Periodic Table, cont By noting the columns in which some missing elements should be located, he was able to make rough predictions about their chemical properties Within 20 years of the predictions, most of the elements were discovered The elements in the periodic table are arranged so that all those in a column have similar chemical properties

58. Periodic Table, Explained The chemical behavior of an element depends on the outermost shell that contains electrons For example, the inert gases (last column) have filled subshells and a wide energy gap occurs between the filled shell and the next available shell

59. Hydrogen Energy Level Diagram Revisited The allowed values of are separated Transitions in which does not change are very unlikely to occur and are called forbidden transitions Such transitions actually can occur, but their probability is very low compared to allowed transitions

60. Selection Rules The selection rules for allowed transitions are    1  m  =  or  1 The angular momentum of the  atom-photon system must be conserved Therefore, the photon involved in the process must carry angular momentum The photon has angular momentum equivalent to that of a particle with spin 1 A photon has energy, linear momentum and angular momentum

61. Multielectron Atoms For multielectron atoms, the positive nuclear charge Ze is largely shielded by the negative charge of the inner shell electrons The outer electrons interact with a net charge that is smaller than the nuclear charge Allowed energies are Z eff depends on n and

62. X-Ray Spectra These x-rays are a result of the slowing down of high energy electrons as they strike a metal target The kinetic energy lost can be anywhere from 0 to all of the kinetic energy of the electron The continuous spectrum is called bremsstrahlung

63. X-Ray Spectra, cont The discrete lines are called characteristic x-rays These are created when A bombarding electron collides with a target atom The electron removes an inner-shell electron from orbit An electron from a higher orbit drops down to fill the vacancy

64. X-Ray Spectra, final The photon emitted during this transition has an energy equal to the energy difference between the levels Typically, the energy is greater than 1000 eV The emitted photons have wavelengths in the range of 0.01 nm to 1 nm

65. Moseley Plot Henry G. J. Moseley plotted the values of atoms as shown is the wavelength of the K  line of each element The K  line refers to the photon emitted when an electron falls from the L to the K shell From this plot, Moseley developed a periodic chart in agreement with the one based on chemical properties

66. Hydrogen Atoms in Space Hydrogen is the most abundant element in space Its role in astronomy and cosmology is very important Assume a cloud of hydrogen near a hot star The high-energy photons from the star can interact with the hydrogen atoms Either raises them to a high energy state or ionizes them

67. Hydrogen Atoms in Space, cont As the excited electrons fall to lower energy levels, many atoms emit Balmer series of wavelengths These atoms provide red, green, blue, and violet colors to nebulae These correspond to the colors in the hydrogen spectrum

68. Emission Nebulae Emission nebulae are near hot stars The hydrogen atoms are excited by light from the star The light is dominated by discrete emission spectral lines and contains colors

69. Reflection Nebulae Reflection nebulae are near cool stars Most of the light from the nebula is starlight reflected from larger grains of material in the nebula rather than emitted by excited atoms The spectrum of light from the nebula is the same as from the star

70. Reflection Nebulae, cont The spectrum is an absorption spectrum with dark lines corresponding to atoms and ions in the outer regions of the star The light from these nebulae tends to appear white

71. Dark Nebulae Dark nebulae are not near a star Little radiation is available to excite atoms or reflect from grains of dust The material in the nebulae screen out light from stars beyond them They appear as black patches against the brightness of more distant stars

72. Other Gases In addition to hydrogen, other atoms and ions are excited by the photons from stars Some of the prominent colors are Violet (  373 nm) from O + ions Green (  496 nm and 501 nm) from O ++ ions Also contributing are helium and nitrogen

73. 21 cm Radiation States where proton spin and electron spin are parallel are slightly higher than when they are antiparallel Since they are not at the same energy, transitions can occur between them The energy difference is 5.9  eV The wavelength of this photon is 21 cm

74. 21 cm Radiation, cont By looking for this radiation, you can detect hydrogen in space If the wavelength has been Doppler shifted, the shift can be used to measure the relative speed of the source