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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 atoms

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 Earths

atmosphere

Balmer Series

- In 1885, Johann Balmer found an empirical

equation that correctly predicted the four

visible emission lines of hydrogen - Ha is red, ? 656.3 nm
- Hß is green, ? 486.1 nm
- H? is blue, ? 434.1 nm
- Hd is violet, ? 410.2 nm

Emission Spectrum of Hydrogen Equation

- The wavelengths of hydrogens 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

Other Hydrogen Series

- Other series were also discovered and their

wavelengths can be calculated - Lyman series
- Paschen series
- Brackett series

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, Thomsons

- 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

Rutherfords 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

Thomsons model

Early Models of the Atom, Rutherfords

- 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 - Rutherfords 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 doesnt

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 Plancks ideas of quantized energy

levels to orbiting electrons

Bohrs 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

Bohrs 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

Bohrs 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 electrons motion - This representation claims the centripetally

accelerated electron does not emit energy and

therefore does not eventually spiral into the

nucleus

Bohrs 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 atoms energy - The frequency is independent of frequency of the

electrons 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

Bohrs Assumptions, 4

- The size of the allowed electron orbits is

determined by a condition imposed on the

electrons orbital angular momentum - The allowed orbits are those for which the

electrons orbital angular momentum about the

nucleus is quantized and equal to an integral

multiple of h

Mathematics of Bohrs Assumptions and Results

- Electrons orbital angular momentum
- mevr nh 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

Bohr Radius

- The radii of the Bohr orbits are quantized
- This shows that the radii of the allowed orbits

have discrete valuesthey are quantized - When n 1, the orbit has the smallest radius,

called the Bohr radius, ao - ao 0.052 9 nm

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

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

PLAY ACTIVE 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

Wavelength of Emitted Photons

- The wavelengths are found by
- The value of RH from Bohrs analysis is in

excellent agreement with the experimental value

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

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

Bohrs Correspondence Principle

- Bohrs 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 ltlt 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

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

- ?(x, y, z) is converted to ?(r, ?, f)
- Then, the space variables can be separated
- ?(r, ?, f) R(r), ƒ(?), g(f)
- 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

?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

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

Radial Probability Density

- A spherical shell of radius r and thickness dr

has a volume of 4pr2 dr - The radial probability function is
- P(r) 4pr2 ?2

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

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
- ?2s depends only on r and is spherically symmetric

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

PLAY ACTIVE 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 ?

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

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

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

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

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 ? with the z axis

Physical Interpretation of ml, final

- ? is also quantized
- Its values are specified through
- ml is never greater than l, therefore ? can never

be zero

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

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

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

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

Hunds Rule

- Hunds 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
- ?l 1
- ?ml 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

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 - ? is the wavelength of the Ka line of each

element - The Ka 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

PLAY ACTIVE 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

PLAY ACTIVE 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