Chapter 42

Chapter 42

Atomic Physics

Importance of the Hydrogen Atom

The hydrogen atom is the only atomic system that can be solved exactly

Much of what was learned in the twentieth century about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+

More Reasons the Hydrogen Atom is Important

The hydrogen atom is 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

Final Reasons 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

The full mathematical solution of the Schrödinger equation applied to the hydrogen atom gives a complete and beautiful description of the atom’s properties

Atomic Spectra

A discrete line spectrum is observed when a low-pressure gas is subjected to an electric discharge

Observation and analysis of these spectral lines is called emission spectroscopy

The simplest line spectrum is that for atomic hydrogen

Emission Spectra Examples

Uniqueness of Atomic Spectra

Other atoms exhibit completely different line spectra

Because no two elements have the same line spectrum, the phenomena represents a practical and sensitive technique for identifying the elements present in unknown samples

Absorption Spectroscopy

An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed

The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source

Absorption Spectrum, Example

A practical example is the continuous spectrum emitted by the sun

The radiation must pass through the cooler gases of the solar atmosphere and through the Earth’s atmosphere

Balmer Series

In 1885, Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen Hα is red, λ = 656.3 nm

Hβ is green, λ = 486.1 nm

Hγ is blue, λ = 434.1 nm

Hδ is violet, λ = 410.2 nm

Emission Spectrum of Hydrogen – Equation

The wavelengths of hydrogen’s spectral lines can be found from

RH is the Rydberg constant RH = 1.097 373 2 x 107 m-1

n is an integer, n = 3, 4, 5,… The spectral lines correspond to different values

of n

H 2 2

1 1 12

Rλ n

Other Hydrogen Series

Other series were also discovered and their wavelengths can be calculated

Lyman series:

Paschen series:

Brackett series:

H 2

1 11 2 3 4, , ,R n

λ n

H 2 2

1 1 14 5 6

3, , ,R n

λ n

H 2 2

1 1 15 6 7

4, , ,R n

λ n

Joseph John Thomson

1856 – 1940 English physicist Received Nobel Prize in

1906 Usually considered the

discoverer of the electron Worked with the deflection

of cathode rays in an electric field Opened up the field of

subatomic particles

Early Models of the Atom, Thomson’s

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

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

Early Models of the Atom, Rutherford’s

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 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

Niels Bohr 1885 – 1962 Danish physicist An active participant in the

early development of quantum mechanics

Headed the Institute for Advanced Studies in Copenhagen

Awarded the 1922 Nobel Prize in physics For structure of atoms and

the radiation emanating from them

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

Bohr’s Theory, cont.

This model is now considered obsolete It has been replaced by a probabilistic

quantum-mechanical theory The model can still be used to develop ideas

of energy quantization and angular momentum quantization as applied to atomic-sized systems

Bohr’s Assumptions for Hydrogen, 1

The electron moves in circular orbits around the proton under the electric force of attraction The Coulomb force

produces the centripetal acceleration

Bohr’s Assumptions, 2

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

This representation claims the centripetally accelerated electron does not emit energy and therefore does not eventually spiral into the nucleus

Bohr’s Assumptions, 3

Radiation is emitted by the atom when the electron makes a transition from a more energetic initial state to a lower-energy orbit The transition cannot be treated classically The frequency emitted in the transition is related to the

change in the atom’s energy The frequency is independent of frequency of the electron’s

orbital motion The frequency of the emitted radiation is given by Ei – Ef = hƒ

If a photon is absorbed, the electron moves to a higher energy level

Bohr’s Assumptions, 4

The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum

The allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of h

Mathematics of Bohr’s Assumptions and Results

Electron’s orbital angular momentummevr = nħ where n = 1, 2, 3,…

The total energy of the atom is

The total energy can also be expressed as

Note, the total energy is negative, indicating a bound electron-proton system

221

2 e e

eE K U m v k

r

2

2ek e

Er

Bohr Radius

The radii of the Bohr orbits are quantized

This shows that the radii of the allowed orbits have discrete values—they are quantized When n = 1, the orbit has the smallest radius, called

the Bohr radius, ao

ao = 0.052 9 nm

2 2

21 2 3, , ,n

e e

nr n

m k e

Radii and Energy of Orbits

A general expression for the radius of any orbit in a hydrogen atom is rn = n2ao

The energy of any orbit is

This becomes

En = - 13.606 eV / n2

2

2

11 2 3,

2, ,e

no

k eE n

a n

Specific Energy Levels

Only energies satisfying the previous equation are allowed

The lowest energy state is called the ground state This corresponds to n = 1 with E = –13.606 eV

The ionization energy is the energy needed to completely remove the electron from the ground state in the atom The ionization energy for hydrogen is 13.6 eV

Energy Level Diagram

Quantum numbers are given on the left and energies on the right

The uppermost level, E = 0, represents the

state for which the electron is removed from the atom Adding more energy than

this amount ionizes the atom

Active Figures 42.7 and 42.8

Use the active figure to choose initial and final energy levels

Observe the transition in both figures

PLAYACTIVE FIGURE

Frequency of Emitted Photons

The frequency of the photon emitted when the electron makes a transition from an outer orbit to an inner orbit is

It is convenient to look at the wavelength instead

2

2 2

1 1

2ƒ i f e

o f i

E E k e

h a h n n

Wavelength of Emitted Photons

The wavelengths are found by

The value of RH from Bohr’s analysis is in excellent agreement with the experimental value

2

2 2 2 2

1 1 1 1 1

2

ƒ eH

o f i f i

k eR

λ c a hc n n n n

Extension to Other Atoms

Bohr extended his model for hydrogen to other elements in which all but one electron had been removed

Z is the atomic number of the element and is the number of protons in the nucleus

2

2 2

21 2 3

2, , ,

on

en

o

ar n

Z

k e ZE n

a n

Difficulties with the Bohr Model

Improved spectroscopic techniques found that many of the spectral lines of hydrogen were not single lines Each “line” was actually a group of lines spaced

very close together Certain single spectral lines split into three

closely spaced lines when the atoms were placed in a magnetic field

Bohr’s Correspondence Principle

Bohr’s correspondence principle states that quantum physics agrees with classical physics when the differences between quantized levels become vanishingly small Similar to having Newtonian mechanics be a

special case of relativistic mechanics when v << c

The Quantum Model of the Hydrogen Atom

The potential energy function for the hydrogen atom is

ke is the Coulomb constant r is the radial distance from the proton to the

electron The proton is situated at r = 0

2

( ) e

eU r k

r

Quantum Model, cont.

The formal procedure to solve the hydrogen atom is to substitute U(r) into the Schrödinger equation, find the appropriate solutions to the equations, and apply boundary conditions

Because it is a three-dimensional problem, it is easier to solve if the rectangular coordinates are converted to spherical polar coordinates

Quantum Model, final

ψ(x, y, z) is converted to ψ(r, θ, φ)

Then, the space variables can be separated:

ψ(r, θ, φ) = R(r), ƒ(θ), g(φ) When the full set of

boundary conditions are applied, we are led to three different quantum numbers for each allowed state

Quantum Numbers, General

The three different quantum numbers are restricted to integer values

They correspond to three degrees of freedom Three space dimensions

Principal Quantum Number

The first quantum number is associated with the radial function R(r) It is called the principal 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 En values found from the Bohr theory

Orbital and Orbital Magnetic Quantum Numbers

The orbital quantum number is symbolized by ℓ It is associated with the orbital angular

momentum of the electron It is an integer

The orbital magnetic quantum number is symbolized by mℓ It is also associated with the angular orbital

momentum of the electron and is an integer

Quantum Numbers, Summary of Allowed Values

The values of n can range from 1 to The values of ℓ can range from 0 to n - 1 The values of mℓ can range from –ℓ to ℓ Example:

If n = 1, then only ℓ = 0 and mℓ = 0 are permitted If n = 2, then ℓ = 0 or 1

If ℓ = 0 then mℓ = 0

If ℓ = 1 then mℓ may be –1, 0, or 1

Quantum Numbers, Summary Table

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 ℓ are said to form a subshell The letters s, p, d, f, g, h, .. are used to designate

the subshells for which ℓ = 0, 1, 2, 3,…

Shell and Subshell Notation, Summary Table

Wave Functions for Hydrogen

The simplest wave function for hydrogen is the one that describes the 1s state and is designated ψ1s(r)

As ψ1s(r) approaches zero, r approaches and is normalized as presented

ψ1s(r) is also spherically symmetric This symmetry exists for all s states

1 3

1( ) or a

s

o

ψ r eπa

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

2 21 3

1or a

so

ψ eπa

Radial Probability Density

A spherical shell of radius r and thickness dr has a volume of 4πr2 dr

The radial probability function is

P(r) = 4πr2 ψ2

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

22

1 3

4( ) or a

so

rP r e

a

P(r) for 1s State of Hydrogen, cont.

The average value of r for the ground state of hydrogen is 3/2 ao

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

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 = ao, corresponds to the most probable region

Wave Function of the 2s state

The next-simplest wave function for the hydrogen atom is for the 2s state n = 2; ℓ = 0

The normalized wave function is

ψ2s depends only on r and is spherically symmetric

32

22

1 1( ) 2

4 2or a

so o

rψ r e

a aπ

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 5ao

Active Figure 42.12

Use the active figure to choose values of r/ao

Find the probability that the electron is located between two values

PLAYACTIVE FIGURE

Physical Interpretation of ℓ

The magnitude of the angular momentum of an electron moving in a circle of radius r is

L = mevr The direction of is perpendicular to the

plane of the circle The direction is given by the right hand rule

In the Bohr model, the angular momentum of the electron is restricted to multiples of

L

Physical Interpretation of ℓ, 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:

L can equal zero, which causes great difficulty when attempting to apply classical mechanics to this system

H 2 2

1 1 1

2R

λ n

Physical Interpretation of mℓ

The atom possesses an orbital angular momentum

There is a sense of rotation of the electron around the nucleus, so that a magnetic moment is present due to this angular momentum

There are distinct directions allowed for the magnetic moment vector with respect to the magnetic field vector

μ

B

Physical Interpretation of mℓ, 2

Because the magnetic moment of the atom can be related to the angular momentum vector, , the discrete direction of translates into the fact that the direction of is quantized

Therefore, Lz, the projection of along the z axis, can have only discrete values

μ

μ

L

L

L

Physical Interpretation of mℓ, 3

The orbital magnetic quantum number mℓ specifies the allowed values of the z component of orbital angular momentum

Lz = mℓ The quantization of the possible orientations

of with respect to an external magnetic field is often referred to as space quantization

L

Physical Interpretation of mℓ, 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 Imagine that must lie anywhere on the

surface of a cone that makes an angle θ with the z axis

L

L

Physical Interpretation of mℓ, final

θ is also quantized Its values are specified

through

mℓ is never greater than ℓ, therefore θ can never be zero

cos

1zL m

θL

Zeeman Effect

The Zeeman effect is the splitting of spectral lines in a strong magnetic field

In this case the upper level, with ℓ = 1, splits into three different levels corresponding to the three different directions of µ

Spin Quantum Number ms

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 ms

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

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 ms

Dirac showed this results from the relativistic properties of the electron

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 = ½

The spin angular momentum of the electron never changes

L

S

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 ms = ± ½ ms = + ½ corresponds to the spin up case

ms = - ½ corresponds to the spin down case

3( 1)

2S s s

Spin Angular Momentum, final

The z component of spin angular momentum is Sz = msh = ½ h

Spin angular moment is quantized

Spin Magnetic Moment

The spin magnetic moment µspin is related to the spin angular momentum by

The z component of the spin magnetic moment can have values

Sspine

eμ

m

spin 2,ze

eμ

m

Quantum States

There are eight quantum states corresponding to n = 2 These states depend on the addition of the

possible values of ms

Table 42.3 summarizes these states

Quantum Numbers for n = 2 State of Hydrogen

Wolfgang Pauli 1900 – 1958 Austrian physicist 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

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 can ever be in the same quantum state Therefore, no two electrons in the same atom can have the

same set of quantum numbers

If the exclusion principle was not valid, an atom could radiate energy until every electron was in the lowest possible energy state and the chemical nature of the elements would be modified

Filling Subshells

The electronic structure of complex atoms can be viewed as a succession of filled levels increasing in energy

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

Orbitals

An orbital is defined as the atomic state characterized by the quantum numbers n, ℓ and mℓ

From the exclusion principle, it can be seen that only two electrons can be present in any orbital One electron will have spin up and one spin down

Each orbital is limited to two electrons, the number of electrons that can occupy the various shells is also limited

Allowed Quantum States, Example with n = 3

In general, each shell can accommodate up to 2n2 electrons

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

Configuration of Some Electron States

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

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

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

Hydrogen Energy Level Diagram Revisited

The allowed values of ℓ are separated horizontally

Transitions in which ℓ 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

Selection Rules

The selection rules for allowed transitions are Δℓ = ±1 Δmℓ = 0, ±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

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

Zeff depends on n and ℓ

2eff

2

13 6.n

ZE eV

n

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, the German word for “braking radiation”

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

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

Moseley Plot

Henry G. J. Moseley plotted the values of atoms as shown

λ is the wavelength of the Kα line of each element The Kα line refers to the

photon emitted when an electron falls from the L to the K shell

From this plot, Moseley developed a periodic table in agreement with the one based on chemical properties

Stimulated Absorption

When a photon has energy hƒ equal to the difference in energy levels, it can be absorbed by the atom

This is called stimulated absorption because the photon stimulates the atom to make the upward transition

Active Figure 42.25

Use the active figure to adjust the energy difference between the states

Observe stimulated absorption

PLAYACTIVE FIGURE

Spontaneous Emission

Once an atom is in an excited state, the excited atom can make a transition to a lower energy level

Because this process happens naturally, it is known as spontaneous emission

Stimulated Emission

In addition to spontaneous emission, stimulated emission occurs

Stimulated emission may occur when the excited state is a metastable state

Stimulated Emission, cont.

A metastable state is a state whose lifetime is much longer than the typical 10-8 s

An incident photon can cause the atom to return to the ground state without being absorbed

Therefore, you have two photons with identical energy, the emitted photon and the incident photon They both are in phase and travel in the same direction

Active Figure 42.27

Use the active figure to adjust the energy difference between states

Observe the stimulated emission

PLAYACTIVE FIGURE

Lasers – Properties of Laser Light

Laser light is coherent The individual rays in a laser beam maintain a

fixed phase relationship with each other Laser light is monochromatic

The light has a very narrow range of wavelengths Laser light has a small angle of divergence

The beam spreads out very little, even over long distances

Lasers – Operation

It is equally probable that an incident photon would cause atomic transitions upward or downward Stimulated absorption or stimulated emission

If a situation can be caused where there are more electrons in excited states than in the ground state, a net emission of photons can result This condition is called population inversion

Lasers – Operation, cont.

The photons can stimulate other atoms to emit photons in a chain of similar processes

The many photons produced in this manner are the source of the intense, coherent light in a laser

Conditions for Build-Up of Photons

The system must be in a state of population inversion

The excited state of the system must be a metastable state In this case, the population inversion can be established

and stimulated emission is likely to occur before spontaneous emission

The emitted photons must be confined in the system long enough to enable them to stimulate further emission This is achieved by using reflecting mirrors

Laser Design – Schematic

The tube contains the atoms that are the active medium An external source of energy pumps the atoms to excited

states The mirrors confine the photons to the tube

Mirror 2 is only partially reflective

Energy-Level Diagram for Neon in a Helium-Neon Laser

The atoms emit 632.8-nm photons through stimulated emission

The transition is E3* to E2

* indicates a metastable state

Laser Applications

Applications include: Medical and surgical procedures Precision surveying and length measurements Precision cutting of metals and other materials Telephone communications