Chapter 28 Atomic Physics

Importance of Hydrogen Atom Hydrogen is the simplest atom The quantum numbers used to characterize the allowed states of hydrogen can also be used to describe (approximately) the allowed states of more complex atoms This enables us to understand the periodic table

More Reasons the Hydrogen Atom is so Important The hydrogen atom is an ideal system for performing precise comparisons of theory and experiment Also for improving our understanding of atomic structure Much of what we know about the hydrogen atom can be extended to other single-electron ions For example, He+ and Li2+

Early Models of the Atom J.J. Thomson’s model of the atom A volume of positive charge Electrons embedded throughout the volume A change from Newton’s model of the atom as a tiny, hard, indestructible sphere

Early Models of the Atom, 2 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

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 and so 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

Emission Spectra A gas at low pressure has a voltage applied to it A gas emits light characteristic of the gas When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed Each line has a different wavelength and color This series of lines is called an emission spectrum

Examples of Spectra

Emission Spectrum of Hydrogen – Equation The wavelengths of hydrogen’s spectral lines can be found from RH is the Rydberg constant RH = 1.0973732 x 107 m-1 n is an integer, n = 1, 2, 3, … The spectral lines correspond to different values of n

Spectral Lines of Hydrogen The Balmer Series has lines whose wavelengths are given by the preceding equation Examples of spectral lines n = 3, λ = 656.3 nm n = 4, λ = 486.1 nm

Absorption Spectra An element can also absorb light at specific wavelengths An absorption spectrum can be obtained by passing a continuous radiation spectrum through a vapor of the gas The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum

Applications of Absorption Spectrum The continuous spectrum emitted by the Sun passes through the cooler gases of the Sun’s atmosphere The various absorption lines can be used to identify elements in the solar atmosphere Led to the discovery of helium

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 His model included an attempt to explain why the atom was stable

Bohr’s Assumptions for Hydrogen The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction The Coulomb force produces the centripetal acceleration

Bohr’s Assumptions, cont Only certain electron orbits are stable These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state The “jump” cannot be treated classically

Bohr’s Assumptions, final The electron’s “jump,” continued The frequency emitted in the “jump” is related to the change in the atom’s energy It is generally not the same as the frequency of the electron’s orbital motion The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum

Mathematics of Bohr’s Assumptions and Results Electron’s orbital angular momentum me v r = n ħ where n = 1, 2, 3, … The total energy of the atom The energy can also be expressed as

Bohr Radius The radii of the Bohr orbits are quantized This shows that the electron can only exist in certain allowed orbits determined by the integer n When n = 1, the orbit has the smallest radius, called the Bohr radius, ao ao = 0.0529 nm

Radii and Energy of Orbits A general expression for the radius of any orbit in a hydrogen atom is rn = n2 ao The energy of any orbit is En = - 13.6 eV/ n2

Specific Energy Levels The lowest energy state is called the ground state This corresponds to n = 1 Energy is –13.6 eV The next energy level has an energy of –3.40 eV The energies can be compiled in an energy level diagram The ionization energy is the energy needed to completely remove the electron from the atom The ionization energy for hydrogen is 13.6 eV

Energy Level Diagram The value of RH from Bohr’s analysis is in excellent agreement with the experimental value A more generalized equation can be used to find the wavelengths of any spectral lines

Generalized Equation For the Balmer series, nf = 2 For the Lyman series, nf = 1 Whenever an transition occurs between a state, ni to another state, nf (where ni > nf), a photon is emitted The photon has a frequency f = (Ei – Ef)/h and wavelength λ

Bohr’s Correspondence Principle Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small Similar to having Newtonian Mechanics be a special case of relativistic mechanics when v << c

Successes of the Bohr Theory Explained several features of the hydrogen spectrum Accounts for Balmer and other series Predicts a value for RH that agrees with the experimental value Gives an expression for the radius of the atom Predicts energy levels of hydrogen Gives a model of what the atom looks like and how it behaves Can be extended to “hydrogen-like” atoms Those with one electron Ze2 needs to be substituted for e2 in equations Z is the atomic number of the element

Modifications of the Bohr Theory – Elliptical Orbits Sommerfeld extended the results to include elliptical orbits Retained the principle quantum number, n Added the orbital quantum number, ℓ ℓ ranges from 0 to n-1 in integer steps All states with the same principle quantum number are said to form a shell The states with given values of n and ℓ are said to form a subshell

Modifications of the Bohr Theory – Zeeman Effect Another modification was needed to account for the Zeeman effect The Zeeman effect is the splitting of spectral lines in a strong magnetic field This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field A new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introduced m ℓ can vary from - ℓ to + ℓ in integer steps

Modifications of the Bohr Theory – Fine Structure High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field This splitting is called fine structure Another quantum number, ms, called the spin magnetic quantum number, was introduced to explain the fine structure

de Broglie Waves One of Bohr’s postulates was the angular momentum of the electron is quantized, but there was no explanation why the restriction occurred de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron wavelengths

de Broglie Waves in the Hydrogen Atom In this example, three complete wavelengths are contained in the circumference of the orbit In general, the circumference must equal some integer number of wavelengths 2  r = n λ n = 1, 2, …

de Broglie Waves in the Hydrogen Atom, cont The expression for the de Broglie wavelength can be included in the circumference calculation me v r = n  This is the same quantization of angular momentum that Bohr imposed in his original theory This was the first convincing argument that the wave nature of matter was at the heart of the behavior of atomic systems Schrödinger’s wave equation was subsequently applied to atomic systems

QUICK QUIZ 28.1 In an analysis relating Bohr's theory to the de Broglie wavelength of electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the wavelength of the electron becomes 9 times as long, or (d) the electron is moving 9 times as fast.

QUICK QUIZ 28.1 ANSWER (b). The circumference of the orbit is n times the de Broglie wavelength (2πr = nλ), so there are three times as many wavelengths in the n = 3 level as in the n = 1 level. Also, by combining Equations 28.4, 28.6 and the defining equation for the de Broglie wavelength (λ = h/mv), one can show that the wavelength in the n = 3 level is three times as long.

Quantum Mechanics and the Hydrogen Atom One of the first great achievements of quantum mechanics was the solution of the wave equation for the hydrogen atom The significance of quantum mechanics is that the quantum numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make the theory agree with experiments

Quantum Number Summary The values of n can range from 1 to  in integer steps The values of ℓ can range from 0 to n-1 in integer steps The values of m ℓ can range from -ℓ to ℓ in integer steps Also see Table 28.2

How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level? QUICK QUIZ 28.2

QUICK QUIZ 28.2 ANSWER The quantum numbers associated with orbital states are n, , and m . For a specified value of n, the allowed values of  range from 0 to n – 1. For each value of , there are (2  + 1) possible values of m. (a) If n = 3, then  = 0, 1, or 2. The number of possible orbital states is then [2(0) + 1] + [2(1) + 1] + [2(2) + 1] = 1 + 3 + 5 = 9. (b) If n = 4, one additional value of  is allowed ( = 3) so the number of possible orbital states is now 9 + [2(3) + 1] = 9 + 7 = 16

When the principal quantum number is n = 5, how many different values of (a)  and (b) m are possible? QUICK QUIZ 28.3

QUICK QUIZ 28.3 ANSWER (a) For n = 5, there are 5 allowed values of , namely  = 0, 1, 2, 3, and 4. (b) Since m ranges from – to + in integer steps, the largest allowed value of  ( = 4 in this case) permits the greatest range of values for m. For n = 5, there are 9 possible values for m: -4, -3, -2, –1, 0, +1, +2, +3, and +4.

Spin Magnetic Quantum Number It is convenient to think of the electron as spinning on its axis The electron is not physically spinning There are two directions for the spin Spin up, ms = ½ Spin down, ms = -½ There is a slight energy difference between the two spins and this accounts for the Zeeman effect

Electron Clouds The graph shows the solution to the wave equation for hydrogen in the ground state The curve peaks at the Bohr radius The electron is not confined to a particular orbital distance from the nucleus The probability of finding the electron at the Bohr radius is a maximum

Electron Clouds, cont The wave function for hydrogen in the ground state is symmetric The electron can be found in a spherical region surrounding the nucleus The result is interpreted by viewing the electron as a cloud surrounding the nucleus The densest regions of the cloud represent the highest probability for finding the electron

The Pauli Exclusion Principle No two electrons in an atom can ever be in the same quantum state In other words, no two electrons in the same atom can have exactly the same values for n, ℓ, m ℓ, and ms This explains the electronic structure of complex atoms as a succession of filled energy levels with different quantum numbers

The Periodic Table The outermost electrons are primarily responsible for the chemical properties of the atom Mendeleev arranged the elements according to their atomic masses and chemical similarities The electronic configuration of the elements explained by quantum numbers and Pauli’s Exclusion Principle explains the configuration

Krypton (atomic number 36) has how many electrons in its next to outer shell (n = 3)? (a) 2 (b) 4 (c) 8 (d) 18 QUICK QUIZ 28.4

QUICK QUIZ 28.4 ANSWER (d). Krypton has a closed configuration consisting of filled n=1, n=2, and n=3 shells as well as filled 4s and 4p subshells. The filled n=3 shell (the next to outer shell in Krypton) has a total of 18 electrons, 2 in the 3s subshell, 6 in the 3p subshell and 10 in the 3d subshell.

Characteristic X-Rays When a metal target is bombarded by high-energy electrons, x-rays are emitted The x-ray spectrum typically consists of a broad continuous spectrum and a series of sharp lines The lines are dependent on the metal The lines are called characteristic x-rays

Explanation of Characteristic X-Rays The details of atomic structure can be used to explain characteristic x-rays A bombarding electron collides with an electron in the target metal that is in an inner shell If there is sufficient energy, the electron is removed from the target atom The vacancy created by the lost electron is filled by an electron falling to the vacancy from a higher energy level The transition is accompanied by the emission of a photon whose energy is equal to the difference between the two levels

Moseley Plot λ is the wavelength of the K line K is the line that is produced by an electron falling from the L shell to the K shell From this plot, Moseley was able to determine the Z values of other elements and produce a periodic chart in excellent agreement with the known chemical properties of the elements

Atomic Transitions – Energy Levels An atom may have many possible energy levels At ordinary temperatures, most of the atoms in a sample are in the ground state Only photons with energies corresponding to differences between energy levels can be absorbed

Atomic Transitions – Stimulated Absorption The blue dots represent electrons When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level These higher levels are called excited states ΔE = hƒ = E2 – E1 In general, ΔE can be the difference between any two energy levels

Atomic Transitions – Spontaneous Emission Once an atom is in an excited state, there is a constant probability that it will jump back to a lower state by emitting a photon This process is called spontaneous emission

Atomic Transitions – Stimulated Emission An atom is in an excited stated and a photon is incident on it The incoming photon increases the probability that the excited atom will return to the ground state There are two emitted photons, the incident one and the emitted one The emitted photon is in exactly in phase with the incident photon

Population Inversion When light is incident on a system of atoms, both stimulated absorption and stimulated emission are equally probable Generally, a net absorption occurs since most atoms are in the ground state If you can cause more atoms to be in excited states, a net emission of photons can result This situation is called a population inversion

Lasers To achieve laser action, three conditions must be met The system must be in a state of population inversion The excited state of the system must be a metastable state Its lifetime must be long compared to the normal lifetime of an excited state The emitted photons must be confined in the system long enough to allow them to stimulate further emission from other excited atoms This is achieved by using reflecting mirrors

Production of a Laser Beam

Laser Beam – He Ne Example The energy level diagram for Ne The mixture of helium and neon is confined to a glass tube sealed at the ends by mirrors A high voltage applied causes electrons to sweep through the tube, producing excited states When the electron falls to E2 in Ne, a 632.8 nm photon is emitted

Holography Holography is the production of three-dimensional images of an object Light from a laser is split at B One beam reflects off the object and onto a photographic plate The other beam is diverged by Lens 2 and reflected by the mirrors before striking the film

Holography, cont The two beams form a complex interference pattern on the photographic film It can be produced only if the phase relationship of the two waves remains constant This is accomplished by using a laser The hologram records the intensity of the light and the phase difference between the reference beam and the scattered beam The image formed has a three-dimensional perspective

Energy Bands in Solids In solids, the discrete energy levels of isolated atoms broaden into allowed energy bands separated by forbidden gaps The separation and the electron population of the highest bands determine whether the solid is a conductor, an insulator, or a semiconductor

Energy Bands, Detail Sodium example Blue represents energy bands occupied by the sodium electrons when the atoms are in their ground states Gold represents energy bands that are empty White represents energy gaps Electrons can have any energy within the allowed bands Electrons cannot have energies in the gaps

Energy Level Definitions The valence band is the highest filled band The conduction band is the next higher empty band The energy gap has an energy, Eg, equal to the difference in energy between the top of the valence band and the bottom of the conduction band

Conductors When a voltage is applied to a conductor, the electrons accelerate and gain energy In quantum terms, electron energies increase if there are a high number of unoccupied energy levels for the electron to jump to For example, it takes very little energy for electrons to jump from the partially filled to one of the nearby empty states

Insulators The valence band is completely full of electrons A large band gap separates the valence and conduction bands A large amount of energy is needed for an electron to be able to jump from the valence to the conduction band The minimum required energy is Eg

Semiconductors A semiconductor has a small energy gap Thermally excited electrons have enough energy to cross the band gap The resistivity of semiconductors decreases with increases in temperature The white area in the valence band represents holes

Semiconductors, cont Holes are empty states in the valence band created by electrons that have jumped to the conduction band Some electrons in the valence band move to fill the holes and therefore also carry current The valence electrons that fill the holes leave behind other holes It is common to view the conduction process in the valence band as a flow of positive holes toward the negative electrode applied to the semiconductor

Current Process in Semiconductors An external voltage is supplied Electrons move toward the positive electrode Holes move toward the negative electrode There is a symmetrical current process in a semiconductor

Doping in Semiconductors Doping is the adding of impurities to a semiconductor Generally about 1 impurity atom per 107 semiconductor atoms Doping results in both the band structure and the resistivity being changed

n-type Semiconductors Donor atoms are doping materials that contain one more electron than the semiconductor material This creates an essentially free electron with an energy level in the energy gap, just below the conduction band Only a small amount of thermal energy is needed to cause this electron to move into the conduction band

p-type Semiconductors Acceptor atoms are doping materials that contain one less electron than the semiconductor material A hole is left where the missing electron would be The energy level of the hole lies in the energy gap, just above the valence band An electron from the valence band has enough thermal energy to fill this impurity level, leaving behind a hole in the valence band

A p-n Junction A p-n junction is formed when a p-type semiconductor is joined to an n-type Three distinct regions exist A p region An n region A depletion region

The Depletion Region Mobile donor electrons from the n side nearest the junction diffuse to the p side, leaving behind immobile positive ions At the same time, holes from the p side nearest the junction diffuse to the n side and leave behind a region of fixed negative ions The resulting depletion region is depleted of mobile charge carriers There is also an electric field in this region that sweeps out mobile charge carriers to keep the region truly depleted

Diode Action The p-n junction has the ability to pass current in only one direction When the p-side is connected to a positive terminal, the device is forward biased and current flows When the n-side is connected to the positive terminal, the device is reverse biased and a very small reverse current results

Applications of Semiconductor Diodes Rectifiers Change AC voltage to DC voltage A half-wave rectifier allows current to flow during half the AC cycle A full-wave rectifier rectifies both halves of the AC cycle Transistors May be used to amplify small signals Integrated circuit A collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece of silicon