Chapter 7

Many-Electron Atoms

6 Quantum Theory of the Hydrogen Atom

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7.1 Many-Electron Atoms: Electron Spin

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• Quantum mechanics explains certain properties of the hydrogen

atom in an accurate, straightforward, and beautiful way.

• However, it cannot approach a complete description of this atom or of

any other without taking into account electron spin and the exclusion

principle.

• Many spectral lines actually consist of two separate lines that are very

close together.

• An example of this fine structure (electron spin and relativistic corrections): first

line of the Balmer series of hydrogen, theoretical prediction is for a single line of

wavelength 656.3 nm while in reality there are two lines 0.14 nm apart.

• Another failure occurs in the (normal) Zeeman effect that the spectral lines of an

atom in a magnetic field should each be split into the three components.

• Round and round it goes forever. two Dutch graduate students,

Samuel Goudsmit and George Uhlenbeck, proposed in 1925 that

Figure 7.1 The normal Zeeman effect.

Every electron has an intrinsic angular momentum,

called spin. Associated with this angular momentum

is a magnetic moment.

7.1 Electron Spin

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• Classical picture of an electron as a charged sphere spinning on its

axis.

• The rotation involves angular momentum, and because the electron is negatively charged, it has a

magnetic moment µ

opposite in direction to its angular momentum vector S.

• Serious objection: observations of the scattering of electrons by other electrons indicate that the

electron must be less than 10

-16

m across, and quite possibly is a point particle. In order to have the

observed angular momentum associated with electron spin, so small an object would have to rotate

with an equatorial velocity many times greater than the velocity of light!

• In 1929 the fundamental nature of electron spin was

confirmed by Paul Dirac's development of relativistic

quantum mechanics. He found that a particle with the mass

and charge of the electron must have the intrinsic angular

momentum and magnetic moment proposed for the electron

by Goudsmit and Uhlenbeck.

Paul Adrien

Maurice Dirac

(1902–1984)

Nobel Prize in

Physics in 1933

(Spin angular momentum)

The quantum number s describes the spin angular momentum of the electron. The only

value s can have is s =1/2, which follows both from Dirac's theory and from spectral

data. The magnitude s of the angular momentum due to electron spin is given in terms

of the spin quantum number s by

7.1 Electron Spin

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Example 7.1

Find the equatorial velocity of an electron under the assumption that it

is a uniform sphere of radius r=5.00x10

-17

m that is rotating about an

axis through its center.

7.1 Electron Spin

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• The space quantization of electron spin is described by

the spin magnetic quantum number m

• Spin angular-momentum vector can have the 2s + 1 = 2

orientations specified by m

=+ 1/2 (“spin up”) and m

= - 1/2

(“spin down”), as in Fig. 7.2.

• The component S

of the spin angular momentum of an electron

along a magnetic field in the z direction is determined by the spin

magnetic quantum number, so that

Figure 7.2 The two

possible orientations

of the spin angular

momentum vector are

“spin up” (m

=+ 1/2 )

and “spin down”

= - 1/2 ).

(z component of spin

angular momentum)

• The gyromagnetic ratio for electron orbital motion is

-e/2m. The gyromagnetic ratio characteristic of electron

spin is almost exactly twice.

• Spin magnetic moment µ

of an electron is related to its

spin angular momentum S by

(Spin magnetic moment)

The possible components of µ

along any axis, say the z axis, are

therefore limited to where µ

is the Bohr

magneton (9.274x10

-24

J/ T = 5.788x10

-5

eV/ T).

(z component of spin

magnetic moment)

7.1 Electron Spin

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The introduction of electron spin into the theory of the atom means that

a total of four quantum numbers, n, l, m

, and m

, is needed to describe

each possible state of an atomic electron. These are listed in Table 7.1.

7.2 Exclusion Principle

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• A different set of quantum numbers for each electron in an atom.

• In a normal hydrogen atom, the electron is in its quantum state of

lowest energy.

• What about more complex atoms? Are all 92 electrons of a uranium atom in the

same quantum state, jammed into a single probability cloud?

• An example is the great difference in chemical behavior shown by certain elements

whose atomic structures differ by only one electron. Thus the elements that have

the atomic numbers 9, 10, and 11 are respectively the chemically active halogen

gas fluorine, the inert gas neon, and the alkali metal sodium.

Since the electron structure of an atom controls how it interacts with other

atoms, it makes no sense that the chemical properties of the elements should

change so sharply with a small change in atomic number if all the electrons

in an atom were in the same quantum state.

Wolfgang Pauli

(1900–1958)

Nobel Prize in

Physics in 1945

• In 1925 Wolfgang Pauli discovered the fundamental principle

that governs the electronic configurations of atoms having

more than one electron. His exclusion principle states that

No two electrons in an atom can exist in the same quantum state. Each

electron must have a different set of quantum numbers n, l, m

, m

7.4 Periodic Table: Group

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• Organizing the elements

• In 1869 the Russian chemist Dmitri Mendeleev formulated:

When the elements are listed in order of atomic number,

elements with similar chemical and physical properties recur at

regular intervals

Dimitri

Mendeleyev

(1834-1907)

• Elements with similar properties form the groups shown as vertical

columns.

• Group 1 consists of hydrogen plus the alkali metals, which are all soft,

have low melting points, and are very active chemically.

• Hydrogen, although physically a nonmetal, behaves chemically much like

an active metal.

• Group 7 consists of the halogens, volatile non-metals that form diatomic

molecules in the gaseous state.

• Like the alkali metals, the halogens are chemically active, but as oxidizing

agents rather than as reducing agents.

• Group 8 consists of the inert gases. As their name suggests, they are

inactive chemically: they form virtually no compounds with other

elements, and their atoms do not join together into molecules.

7.4 Periodic Table : Period

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• The horizontal rows are called periods.

• The first three periods are broken in order to keep

their members aligned with the most closely related

elements of the long periods below.

• Most of the elements are metals.

• Across each period is a more or less steady transition from an active

metal through less active metals and weakly active nonmetals to

highly active nonmetals and finally to an inert gas.

• A series of transition elements appears in each period after the third

between the group 2 and group 3 elements.

• The transition elements are metals, in general hard and brittle with

high melting points, that have similar chemical behavior.

• Fifteen of the transition elements in period 6 are virtually

indistinguishable in their properties and are known as the lanthanide

elements (or rare earths).

• Another group of closely related metals, the actinide elements, is

found in period 7.

Figure 7.6 How chemical activity

varies in the periodic table.

7.4 Periodic Table

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7.5 Atomic Structures

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• Shells and subshells of electrons

• Two basic principles determine the structures of atoms with more than

one electron:

1.A system of particles is stable when its total energy is a minimum.

2.Only one electron can exist in any particular quantum state in an atom.

• While the various electrons in a complex atom certainly interact

directly with one another, much about atomic structure can be

understood by simply considering each electron as though it exists in

a constant mean electric field.

• Nuclear charge Ze decreased by the partial shielding of those other electrons that

are closer to the nucleus.

• Electrons that have the same principal quantum number n (same

atomic shell) therefore interact with roughly the same electric field

and have similar energies.

(7.13 Atomic shells)

• The energy of an electron in a particular shell also on its orbital

quantum number l, though not as much as on n.

7.5 Atomic Structures

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• In a complex atom the degree to which the full

nuclear charge is shielded from a given electron by

intervening shells of other electrons varies with its

probability-density distribution.

• An electron of small l is more likely to be found

near the nucleus where it is poorly shielded by the

other electrons than is one of higher l (see Fig. 6.ll).

• The result is a lower total energy (that is, higher

binding energy) for the electron.

• The electrons in each shell accordingly increase in

energy with increasing l (see Fig. 7.8).

• Electrons that share a certain value of l in a shell are

said to occupy the same subshell.

• All the electrons in a subshell have almost identical

energies, since the dependence of electron energy upon m

and m

is comparatively minor.

Figure 7.8 The binding

energies of atomic electrons

in rydbergs. (1 Ry=13.6 eV

ground-state energy of H

atom.)

7.6 Explaining the Periodic Table

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• How an atom’s electron structure determines its

chemical behavior.

• An atomic shell or subshell that contains its full

quota of electrons is said to be closed.

• The total orbital and spin angular momenta of the

electrons in a closed subshell are zero, and their

effective charge distributions are perfectly

symmetrical.

Figure 7.9 Schematic representation of

electron shielding in the sodium and

argon atoms.

• The electrons in a closed shell are all very tightly bound, since the

positive nuclear charge is large relative to the negative charge of the

inner shielding electrons (see Fig. 7.9).

• Because an atom with only closed shells has no dipole moment, it does

not attract other electrons, and its electrons cannot be easily detached.

We expect such atoms to be passive chemically, like the inert gases.

7.6 Explaining the Periodic Table

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• An atom of any of the alkali metals of group 1 has a single s electron in

its outer shell.

• Such an electron is relatively far from the nucleus.

• It is also shielded by the inner electrons from all but an effective

nuclear charge of approximately +e rather than +Ze.

• Relatively little work is needed to detach an electron from such an

atom, and the alkali metals accordingly form positive ions of charge

+e readily.

7.6 Explaining the Periodic Table

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Example 7.2

The ionization energy of lithium is 5.39 eV. Use this figure to find the

effective charge that acts on the outer (2s) electron of the lithium atom.

7.6 Explaining the Periodic Table: Ionization Energy

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• Inert gases have the highest

ionization energies and the alkali

metals the lowest.

• The larger an atom, the farther the

outer electron is from the nucleus and

the weaker the force is that holds it to

the atom.

• This is why the ionization energy

generally decreases as we go down a

group in the periodic table.

Figure 7.10 Variation of ionization energy with atomic number.

• The increase in ionization energy from left to right across any period is

accounted for by the increase in nuclear charge while the number of

inner shielding electrons stays constant.

• At the other extreme from alkali metal atoms are halogen atoms, whose

imperfectly shielded nuclear charges tend to complete their outer

subshells by picking up an additional electron each.

7.6 Explaining the Periodic Table: Size

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• The periodicity in Fig. 7.11 has a similar

origin in the partial shielding by inner

electrons of the full nuclear charge.

• The greater the shielding, the lower the

binding energy of an outer electron and

the farther it is on the average from the

nucleus.

• The relatively small range of atomic

radii is not surprising in view of the

binding- energy curves of Fig. 7.8.

Figure 7.11 Atomic radii of the elements.

• In contrast to the enormous increase in the binding energies of the

unshielded 1s electrons with Z, the binding energies of the outermost

electrons (whose probability-density distributions are what determine

atomic size) vary through a narrow range.

• The heaviest atoms, with over 90 electrons, have radii only about 3 times that of

the hydrogen atom, and even the cesium atom, the largest in size, has a radius only

4.4 times that of the hydrogen atom.

7.6 Explaining the Periodic Table: Transition Elements

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• The origin of the transition elements lies in the

tighter binding of s electrons than d or f

electrons in complex atoms.

• The first element to exhibit this effect is

potassium, whose outermost electron is in a 4s

instead of a 3d substate.

• The difference in binding energy between 3d

and 4s electrons is not very great, as the

configurations of chromium and copper show.

Figure 7.12 The sequence of quantum

states in an atom. Not to scale.

• In both these elements an additional 3d electron is present

at the expense of a vacancy in the 4s subshell.

• The order in which electron subshells tend to be filled,

together with the maximum occupancy of each subshell,

is usually as follows:

Figure 7.12 illustrates this sequence.

7.6 Explaining the Periodic Table: Transition Elements

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• The remarkable similarities in chemical behavior among the

lanthanides and actinides are easy to understand on the basis of this

sequence.

• All the lanthanides have the same 5s

configurations but have

incomplete 4f subshells. The addition of 4f electrons has almost no

effect on the chemical properties of the lanthanide elements, which

are determined by the outer electrons.

• Similarly, all the actinides have 6s

configurations and differ

only in the numbers of their 5f and 6d electrons.

• These irregularities in the binding energies of atomic electrons are

also responsible for the lack of completely full outer shells in the

heavier inert gases.

7.6 Explaining the Periodic Table: Hund’s Rule

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• In general, the electrons

in a subshell remain

unpaired-that is, have

parallel spins-whenever

possible (see Table 7.5).

Table 7.5 Electron Configurations of Elements from Z=5 to Z=10. The p electrons

have parallel spins whenever possible, in accord with Hund’s rule.

• This principal is called Hund’s rule.

• The ferromagnetism of iron, cobalt, and nickle (Z=26, 27, 28) is in

part a consequence of Hund's rule.

• The 3d subshells of their atoms are only partially occupied, and the

electrons in these subshells do not pair off to permit their spin

magnetic moments to cancel out.

• In iron, for instance, five of the six 3d electrons have parallel spins, so

that each iron atom has a large resultant magnetic moment.

7.6 Explaining the Periodic Table: Hund’s Rule

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• The origin of Hund's rule lies in the mutual repulsion of atomic

electrons.

• Because of this repulsion, the farther apart the electrons in an atom are,

the lower the energy of the atom.

• Electrons in the same subshell with the same spin must have different

values and accordingly are described by wave functions whose

spatial distributions are different.

• Electrons with parallel spins are therefore more separated in space

than they would be if they paired off.

• This arrangement, having less energy, is the more stable one.

7.9 X-Ray Spectra

• They arise from transitions to inner shells.

• We learned that the x-ray spectra of targets bombarded by fast

electrons show narrow spikes at wavelengths characteristic of the

target material.

• The line spectrum comes from electronic transitions within atoms that

have been disturbed by the incident electrons.

• The transitions of the outer electrons of an atom usually involve only a

few electronvolts of energy, and even removing an outer electron

requires at most 24.6 eV (for helium) (in or near the visible part).

• The inner electrons of heavier elements are a quite different matter,

because these electrons are not well shielded from the full nuclear

charge by intervening electron shells and so are very tightly bound.

• In sodium, for example, only 5.13 eV is needed to remove the outermost 3s

electron, whereas the corresponding figures for the inner ones are 31 eV for each

2p electron, 63 eV for each 2s electron, and 1041 eV for each 1s electron.

• Transitions that involve the inner electrons in an atom are what give

rise to x-ray line spectra because of the high photon energies involved.

7.9 X-Ray Spectra

Figure 7.20 The origin of x-ray spectra.

Figure 7.20 shows the energy levels

(not to scale) of a heavy atom.

• Let us look at what happens when an

energetic electron strikes the atom and

knocks out one of the K-shell electrons.

• An atom with a missing K electron gives up

most of its considerable excitation energy in

the form of an x-ray photon when an electron

from an outer shell drops into the “hole” in

the K shell.

• As indicated in Fig. 7.20, the K series of

lines in the x-ray spectrum of an element

consists of wavelengths arising in transitions

from the L, M, N, . . . levels to the K level.

• It is easy to find an approximate relationship

between the frequency of the K

x-ray line

of an element and its atomic number Z.

7.9 X-Ray Spectra

• K

photon is emitted when an L (n = 2) electron undergoes a transition to a vacant K

(n= 1) state.

• The L electron experiences a nuclear charge of Ze that is reduced to an effective

charge in the neighborhood of (Z - 1)e by the shielding effect of the remaining K

electron. Thus

(7.21 K

x-rays)

• The energy of a K

x-ray photon is given in electronvolts in terms of (Z - 1) by the

formula

(7.22)

• In the operation of this x-ray spectrometer, a stream of fast electrons is

directed at a sample of unknown composition.

• Some of the electrons knock out inner electrons in the target atoms,

and when outer electrons replace them, x-ray are emitted whose

wavelengths are characteristic of the elements present.

• The identity and relative amounts of the elements in the sample can be

found in this way.

7.9 X-Ray Spectra

Example 7.3

Which element has a K

x-ray line whose wavelength is 0.180 nm?

7.9 X-Ray Spectra: Auger Effect

• An atom with a missing inner electron can also lose excitation energy by the Auger

effect without emitting an x-ray photon.

• In this effect, an outer-shell electron is ejected from the atom at the same time that

another outer-shell electron drops to the incomplete inner shell.

• Thus the ejected electron carries off the atom's excitation energy instead of a photon

doing this (see Fig. 7.21).

• In a sense the Auger effect represents an internal photoelectric effect, although the

photon never actually comes into being within the atom.

Figure 7.21 When an electron from an outer shell of an

atom with a missing inner electron drops to fill the

vacant state, the excitation energy can be carried off by

an x-ray photon or by another outer electron. The latter

process is called the Auger effect.

• The Auger process is competitive with x-ray

emission in most atoms, but the resulting

electrons are usually absorbed in the target

material while the x-rays emerge to be detected.

• Those Auger electrons that do emerge come

either from atoms on the surface of the material

or just below the surface.

• Because the energy levels of an atom are

affected by its participation in a chemical bond,

the energies of Auger electrons provide insight

into the chemical environment of the atoms

involved.

7 Solved Problems

1. (a) How many different sets of quantum numbers (n, l,m

) are

possible for an electron in the 4f level? (b) Suppose a certain atom

has three electrons in the 4f level. What is the maximum possible

value of the total m

of the three electrons? (c) What is the maximum

possible total m

of three 4f electrons? (d) Suppose an atom has ten

electrons in the 4f level. What is the maximum possible value of the

total m

of the ten 4f electrons? (e) What is the maximum possible

total m

of ten 4f electrons?

7 Solved Problems

2. Copper has the electronic configuration [Ar]4s

in its ground

state. By adding a small amount of energy (about 1 eV) to a copper

atom, it is possible to move one of the 3d electrons to the 4s level

and change the configuration to [Ar]4s

. By adding still more

energy (about 5 eV), one of the 3d electrons can be moved to the 4p

level so that the configuration becomes [Ar]4s

. For each of

these configurations, determine the maximum value of the total m

the electrons.

7 Solved Problems

3. (a) The ionization energy of sodium is 5.14 eV. What is the effective

charge seen by the outer electron? (b) If the 3s electron of a sodium

atom is moved to the 4f state, the measured binding energy is 0.85

eV. What is the effective charge seen by an electron in this state?

7 Solved Problems

4. Use Hund’s rules to find the ground-state quantum numbers of

nitrogen.

7 Solved Problems

5. Compute the energy of the K

X ray of sodium (Z = 11).

7 Solved Problems

6. Some measured X-ray energies in silver (Z = 47) are E(K

)= 21.990

keV and E(K

)= 25.145 keV. The binding energy of the K electron

in silver is E

(K)=25.514 keV. From these data, find: (a) the energy

of the L

X ray, and (b) the binding energy of the L electron.

5 Quantum Mechanics

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