School of Mathematical and Physical Sciences PHYS1220 Sept. 2002 Atoms Slide 1 Quantum Theory of Atoms, Molecules and Solids In the Quantum Mechanics course,

Sept. 2002 Atoms Slide 1

School of Mathematical and Physical SciencesPHYS1220School of Mathematical and Physical SciencesPHYS1220

Quantum Theory of Atoms, Molecules and Solids• In the Quantum Mechanics course, the underlying concepts

of quantum mechanics were discussed and applied to simple systems such as the infinite square well and the finite well.

• While the Bohr model of the atom explains several aspects of the hydrogen spectrum, if does not explain the spectra of complex atoms and the bonding mechanism of atoms in solids and molecules. In this course, quantum mechanics is applied to the ‘real world’ of atoms, molecules and solids to try and explain some of their properties.

• Our starting point is the hydrogen atom (the simplest atom) for which a complete solution of the Schrodinger equation can be obtained. This will not only provide a general understanding of atomic structure, but the results can be extended to more complex atoms.

• We will then look at the structure of simple molecules and the properties of solids as predicted by quantum theory.



The Hydrogen Atom• The hydrogen molecule is the simplest

molecule containing a single electron (charge - e) orbiting a proton (charge +e).

• Solution of the Schrodinger equation requires the potential energy of the electron/proton system.

t

)t,x(i)t,x()x(U

xm2 2

22

Potential energy term (function of x only in this case)

For the hydrogen atom, the potential energy U is due to the electrostatic attraction between the electron and the proton. It is given by

re

41

U2

o



The Hydrogen Atom

• For the hydrogen atom, which is three-dimensional, the Schrodinger equation must be written in three dimensions. It is given by

Er

e

4

1

zyxm2

2

o2

2

2

2

2

22

However, it is easier to solve the Schrodinger equation if it is written in spherical coordinates (r, , ).

z

x

y

r

The wavefunction (r,,) is written in the form

(r,,) = R(r)f()g()

The solution is mathematically complicated and will not be discussed here. However, we will consider the results in some detail.



Quantum Numbers for an Electron in the Hydrogen Atom

Solution of the Schrodinger equation for the hydrogen atom results in three quantum numbers which are:

(a) the principal quantum number n

(b) the orbital quantum number

(c) the magnetic quantum number m

However, four quantum numbers are needed to completely specify the state of the hydrogen atom. The origin and significance of the fourth quantum number are discussed later.



The principal Quantum Number n• The principal quantum number results from solution of

the radial wavefunction R(r).

• Quantum mechanics predicts that the energy E is quantised (as does the Bohr theory) and given by

eVn

6.13E

2n

which is the same expression as that derived from the Bohr theory of the hydrogen atom.

The equation shows that the energy En depends on the quantum number n only.

The negative sign indicates the bound nature of the electron and proton.

n = 1



The Orbital Quantum Numbers The orbital quantum number is related to the magnitude L

of the orbital angular momentum by

In the hydrogen atom, the electron energy depends on the principal quantum number n only. In atoms with more than one electron, the electron energy depends on both n and .

The orbital quantum number has values ranging from 0 to n-1, where n is the principal quantum number. There are n values of for a given value of n.

Examples: if n = 1, =0 only; if n = 3, = 0, 1 and 2.

)1(L



Magnetic Quantum Number m

The magnetic quantum number m is related to the direction of the orbital momentum vector L, and has the values - , - +1, … 0, -1, .

Examples: = 0, m = 0; = 2, m = -2, -1, 0, +1, +2 (2+1 values)

The direction and magnitude of L are both quantised. This is called space quantisation because L can only have certain orientations in space. The directions of the angular momentum that are allowed for =2 are shown in the diagram.

The direction of the angular momentum is specified by giving its component Lz along an arbitrarily chosen z axis.

The component Lz is quantised and given by

mLz

Lz

m = 0

m = 2

m = 1

m = -1

m = -2

0

2

1

1

2 )1(L

)1()(

mCos



Details of the Three Quantum Numbers n, and m

Quantum number

Description Range

n Principal quantum number 1

Orbital quantum number 0,1,2,....., n -1

Number of states

Any number

n

m Orbital magnetic quantumnumber

-, - +1,...,0,

....., -1, 2 + 1



Example

• Calculate (a) the energy and (b) the orbital angular momentum of each state with n=2 for the hydrogen atom.

(a)

(b) If n = 2, = 0 and 1

eV4.3eV2

6.13eV

n6.13

E 222

0)10(0)1(L,0

Js1049.12)11(1)1(,1 34 xL


School of Mathematical and Physical SciencesPHYS1220School of Mathematical and Physical SciencesPHYS1220Magnetic Effects on Atomic Spectra - Dipole

Moments As early as 1896 it had been observed that the spectral

lines emitted by atoms that are in a magnetic field appear to split. Sometimes the lines split into three lines, a phenomenon known as the normal Zeeman effect, but in some cases more than three lines were observed (anomalous Zeeman effect).

The normal Zeeman effect can be understood by considering an electron orbiting a nucleus as a current loop. Such a current loop would have a magnetic dipole moment given by

where I is the current and A is the loop area. is a vector whose direction is as shown in the diagram.

The current I is given by the charge (q = - e, the electronic charge) divided by the period T (T = 2r/v, r is the radius of the loop and v the electron velocity) for one revolution.

= IA

eL



Magnetic Effects on Atomic Spectra - Dipole Moments (ctd)

L2me

2erv)re)((

ATq

IA2

v/r2

πμ

The magnetic dipole moment can be expressed as

where L is the orbital angular momentum (L = mvr).

Since both and L are vectors,

Lμm2e

The magnetic dipole moments of atoms point in random directions if the atoms are not under the influence of an external magnetic field.

If a magnetic field is present, a magnetic dipole moment experiences a torque = x B which tends to align with B.

z

S

NB

Torque = x B

= B sin()



Magnetic Effects on Atomic Spectra - Dipole Moments (ctd)

The magnetic field establishes a preferred direction in space which is generally taken to be the z-axis.

The z component z of the magnetic dipole moment is therefore given by

mmm2

eL

m2e

Bzz

m2e

Bwhere

.B is called the Bohr magneton.

Work must be done to rotate a magnetic dipole moment away from a magnetic field B. The potential energy in an external field is given by

VB = - .B

(Lz = m)



Magnetic Effects on Atomic Spectra - Dipole Moments (ctd) Since the magnetic field is in the z direction, the potential

energy can be written asVB = - zB = + mBB

Since the magnetic quantum number m is quantised (m = - , - +1, … 0, -1, ; (2+1) values), each energy level with a given value of is split into 2 +1 values in the presence of a magnetic field.

Example1: = 0 and = 1

=0; m=0

B = 0

En

erg

y

=1; m= -1, 0, +1

B = 0

m = +1

m = 0

m = -1

E

E

E = BBm

B = 0 B = 0



Magnetic Effects on Atomic Spectra - Dipole Moments (ctd)Example

(a) What is the value of the Bohr Magneton?

(b) What is the energy difference between the states with m =+1 and m = 0 for the 2p state of hydrogen atoms which are in a magnetic field of 1.5 T?

(a) The Bohr magneton B

12431

3419

B TJ10x27.9)10x11.9(2

)10x055.1)(10x602.1(

m2

e

(b) The energy interval E is given by mBE B

m = 1 - 0 =1, B = 1.5 T E =(9.27x10-24)(1.5)(1) = 1.39x10-23 J = 8.69x10-5 eV

Such an energy difference can be optically measured.



Example of the Zeeman Effect (transition between 1s and 2p states)

1s, = 0

2p, = 1E

nerg

y

B = 0

m = +1

m = 0

m = -1

Single spectral line 3 spectral lines

B = 0

When B = 0, three transitions are possible because the 2p state splits into three states with m = +1, 0 and -1 respectively. Only three transitions are allowed because of the selection rule that must be imposed on m (i.e. m = 0, ± 1). This results in splitting of the single spectral line (for B = 0) into three lines (for B = 0). This phenomenon is known as the Normal Zeeman Effect.



The Zeeman Effect (ctd) The diagram shows an example in

which the upper level has quantum numbers n=3 , = 2 and the lower level n=2, =1.

In a magnetic field, the upper level splits into five and the lower level into three separate levels corresponding to the possible values of m.

Transitions can then occur between the energy levels, subject to the selection rules for m. This results in splitting of the emission line for B =0 into several lines. The figure does not show all the possible transitions.

B = 0

B

Note that the energy of a given state depends not only on n but also on m in the presence of a magnetic field. This is why m is called the magnetic quantum number.

B = 0



Electron Spin - The Stern-Gerlach Experiment

The first evidence of space quantisation came to light in an

experiment performed by O. Stern and W. Gerlach in 1922. A schematic diagram of what is now known as the Stern-Gerlach experiment is shown in the figure.

A narrow beam of silver atoms was produced by heating silver in an oven and collimating the atoms that emerged from the oven.

The beam passed through an inhomogeneous magnetic field resulting in a force on the atomic magnetic moments.



The Stern-Gerlach Experiment (ctd.) A magnetic dipole moment may be represented as a

magnet as shown in the diagram. In an inhomogeneous magnetic field the magnet will experience a net force as well as a net torque. The direction of the net force depends on the orientation of the dipole moment in the inhomogeneous magnetic field B.

The potential energy U is given by

If the magnetic field is in the z-direction, the force is given by

dz

dB

dz

dUF z

zz

U = - .B

South Pole

North

N

S

B greater

B lower

Net force

Net force

N

S

North Pole

z



The Stern-Gerlach Experiment (ctd.) In the Stern-Gerlach experiment, the silver atoms are

deflected either up or down depending on the value of for each atom.

If the space quantisation was due to orbital angular momentum, the expected number of lines would be an odd number because the possible values of m for a given value of is (2 +1), an odd number (e.g. for = 1; m = -1, 0, +1 (3 possible values))

Classically a continuous distribution along the z-axis would be expected, but only two lines were observed by Stern and Gerlach for silver. In 1927 the experiment was repeated by Phipps and Taylor using hydrogen and, once again, only two lines were observed.

While this was clear evidence of space quantisation, the result was a bit of a mystery at the time and was only explained a few years later.

dz

dBm

dz

dB

dz

dUF z

Bz

zz



Spin Angular Momentum S

The observed pattern of two lines was eventually explained by Goudsmit and Uhlenbeck using the concept of electron spin angular momentum S. They proposed that the electron must have a spin quantum number s where s = 1/2.

The magnitude S of the spin angular momentum S is given by

)1s(sS

In a magnetic field the spinning electron reacts in the same way as an orbiting electron and the magnetic spin quantum number has only two values namely ms = -1/2, +1/2.

For example, if = 1, m = -1, 0, +1 and three lines would be expected.



Spin Angular Momentum S

• The two states of the silver atoms observed in the Stern Gerlach experiment are due to the spin of their single valence electron.

The effects of the orbital momentum are not observed because the angular momentum of the single s valence electron is 0 (i.e. = 0 and m = 0)

A similar argument applies to the results for hydrogen.



The Stern-Gerlach Experiment (ctd.) For a spin of 1/2, we would expect the z-component of

the dipole moment to be given by an expression which is similar to that for the orbital angular momentum, namely

m2e

21

msBz

However, z is about twice the expected value and is given by

sBz mgwhere g is called the g-factor or gyromagnetic ratio which has been measured to be 2.0023 for a free electron. In the case of the orbital angular momentum, g =1.

The unexpected value of about 2 for g indicates that electron spin should not be seen as arising from a classical picture of angular momentum. In other words, the electron should not be viewed as a spinning charge.



Details of the Four Quantum Numbers n, , m and ms needed to specify a given state

Quantum number

Description Range

n Principal quantum number 1

Orbital quantum number 0,1,2,....., n -1

ms Spin quantum number +1/2, -1/2

Number of states

Any number

n

m Orbital magnetic quantumnumber

-, - +1,...,0,

....., -1, 2 + 1

2



Shells and Subshells

• For historical reasons, electrons in states that have the same value of the quantum number, n, are said to be in the same shell.

• The shells are given the letters K, L, M, … for states with quantum numbers n = 1, 2, 3, … respectively.

• Electrons in states that have the save values of, n, and, , are said to be in the same subshell.

• The subshells are given the letters s, p, d, f,… to specify the states for which = 0, 1, 2, 3, … respectively.

The letters s, p, d and f are abbreviations of the words “sharp”, “principal”, “diffuse” and “fundamental” which refer to experimentally observed spectra.



Shells and Subshells (ctd)

n Value Shell Value Subshell

1 K 0 s

2 L 0 s

L 1 p

3 M 0 s

M 1 p

M 2 d

4 N NNN

01

23

sp

df



Hydrogen Atom Wavefunctions• The wavefunctions (solutions of Schrodinger’s equation) are

written as n, and m are as defined previously.

• The Wavefunction for the ground state (E1 = -13.6eV) depends on r only and is spherically symmetric.

ro = 0.0529 nm (same as first Bohr radius)

n = 1, = 0 and m = 0(ms = +1/2 or -1/2)

1002

mn

dV is the probability of finding the electron in a volume dV about a given point in space.

2

1dV2100 (wavefunction is normalised)

Electron must be somewhere in space.

ror

3o

100 er

1



Hydrogen Atom Wavefunctions (ctd)

• It is convenient to define the radial probability P(r) such that P(r)dr is the probability of finding the electron at a radial distance in the interval from r to r+ dr.

• P(r)dr is essentially the probability of finding the electron within a shell of thickness dr at a radius r.

Most probable radius

Ground state

o100

r

r2

30

2

r er

r4P




First excited state. n=2, =0, m=0

Radial probability distribution. Electrons are on average

further away from the nucleus that for the ground state.

or2

r

o3o

200 err

2r32

1

or

r2

o3o

2

200r err

2rr

81

P




States with n=2, =1, m=1,0,-1i.e. three states

Distribution is still spherical due to the combination of all three.

Radial probability distribution

or2

r

5o

210 er32

z

or2

r

5o

211 er64

iyx

or2

r

5o

121 er64

iyx



Probability Distributions

• It is important to note that we cannot accurately predict the path of the electron.

• According to the Uncertainty Principle (xp /2), if we know the position of the electron at some point in time, its momentum is uncertain and we, therefore, cannot predict its position at a later time.

• We can only determine the probability of finding the electron at some point in space. This is given by the probability distribution.

• In other words, if a large number of measurements of the electron’s position are made, we are more likely to find it where the probability is high (dense regions of dots on the probability plots).



Quantum Mechanics of Complex atoms

Unlike the hydrogen atom, complex atoms contain more than one electron.

The electron energies are not the same as for the hydrogen atom because the electrons interact with one another and with the nucleus.

The allowed energies depend on the principal quantum number, n, and to a lesser extent on the orbital quantum number .

Each electron in the atom occupies a state which is identified by the principal quantum number, n, orbital quantum number, , magnetic quantum number, m and spin quantum number ms for that state.



If the Pauli principle were not valid, every electron would end up in the lowest energy state and the chemical behaviour of the elements would be greatly changed.

The electronic structure of complex atoms can be viewed as a succession of filled subshells of increasing energy, with the outermost electrons (valence electrons) being responsible for the chemical properties of a particular element.

The Pauli Exclusion Principle The arrangement of electrons in complex atoms is

governed by the Pauli Exclusion Principle which can be summarised as follows:

No two electrons in an atom can be in the same quantum state. In other words, no two electrons can have the same value of the quantum numbers n, , m and ms.



As a general rule, after a given subshell has been filled the next electron goes into the vacant subshell which has the next lowest energy.

The number of electrons (and of protons) in neutral atoms is called the atomic number, Z.

The atomic number therefore determines the properties of an atom. Atoms with different atomic numbers have different properties because the arrangement of the electrons and the quantum numbers are different.

The Pauli Exclusion Principle underlies not only the understanding of complex atoms but also of molecules and bonding phenomena.

The Pauli Exclusion Principle (ctd)



States with Principal Quantum Number n =1, 2 and 3

The above procedure can be extended to determine the quantum numbers for states with principal quantum number n = 4, 5,… etc.

n 1 2 3

0 0 1 0 1 2

m 0 0 -1 0 1 0 -1 0 1 -2 -1 0 1 2

ms

N 2 8 18

N : Number of allowed states for n =1, 2 and 3

Spin quantum number ms = +1/2

Spin quantum number ms = -1/2

How many electron can reside in the K (n=1), L (n=2) and M (n=3) shells of an atom?



Structure of Some Simple Atoms in the Ground State

Example 1

Helium ( Z =2, helium has two electrons)

The two electrons are in states which are characterised by the quantum numbers listed in the table below. Note that each state can accommodate only one electron in keeping with the Pauli exclusion principle. Compare the table with that on slide 34.

n m ms

1 0 0 +1/2

1 0 0 -1/2



Structure of Some Simple Atoms in the Ground State (ctd)

Example 2

Lithium ( Z = 3, lithium has three electrons)

The three electrons are in states which are characterised by the quantum numbers listed in the table below. Compare the table with that on slide 34.

n m ms

1 0 0 -1/2

1 0 0 +1/2

2 0 0 -1/2



Structure of Some Simple Atoms in the Ground State (ctd)

Example 3

Sodium (Z = 11, sodium has 11 electrons)

The eleven electrons are in states which are characterised by the quantum numbers listed in the table. Compare the table with that on slide 34.

n m ms

1 0 0 -1/2

1 0 0 +1/2 1s2

2 0 0 -1/2

2 0 0 +1/2 2s2

2 1 1 -1/2

2 1 1 +1/2

2 1 0 -1/2

2 1 0 +1/2

2 1 -1 -1/2

2 1 1 +1/2

2p6

3 0 0 -1/2 3s1



Occupied States for Helium, Lithium and Sodium

n =1, = 0 n =1, = 0 n =1, = 0

n =2, = 0 n =2, = 0n =2, = 0

n =2, = 1 n =2, = 1

n =3, = 0

Helium (Z = 2) Lithium (Z = 3) Sodium (Z= 11)

1s

2s

2p

3s



The Periodic Table

In 1871 Dmitri Mendelev arranged the elements that were known at the time in a table which is now called the Periodic Table.

The elements in the periodic table are arranged so that all those in a vertical column have similar chemical properties.

Since the energy of an electron is determined predominantly by the quantum numbers n and , electron configurations are specified by giving the value of the quantum number n, the appropriate letter for the quantum number and the number of electrons in each subshell as superscript.

As an example, the ground state electronic configuration of sodium is: 1s22s22p63s1

The chemical properties of the elements are governed by their electron distribution. Why is this so?



The Periodic Table (ctd)

An atomic shell or subshell that holds the maximum number of electrons it can accommodate is said to be filled or closed.

A closed s subshell ( = 0) holds 2 electrons, a closed p subshell ( = 1) holds 6 electrons, a closed d subshell ( = 2) holds 10 electrons, etc (consult slide 34).

The noble (inert) gases (He, Ne, Ar, Kr, etc.) in group VIII of the periodic table have filled shells or subshells ( e.g. for neon, electron configuration is 1s22s22p6 ) and the electron distribution is, therefore, spherically symmetric.

The spherical symmetry of the electron distribution results in tightly bound electrons. Consequently the atom does not attract other electrons and its electrons are not readily lost (the atom is not easily ionized). This is why the noble gases are not chemically reactive.



The Periodic Table (ctd)

Group 1 of the periodic table contains the alkali metals. All the alkali metal atoms have a single s electron in the outermost subshell. Such an electron is shielded from the nucleus by electrons in the inner closed shells and subshells.

The outermost electron is relatively far from the nucleus and experiences an effective nuclear charge of +1e rather than +Ze because of the shielding effect of the other electrons.

Consequently, the outermost electron is easily removed and spends a lot of its time around another atom forming a molecule. This is why the alkali metals are highly reactive and have a valency of 1.



Atomic Spectra: Visible Spectral Region An electron in the ground state can absorb electromagnetic

radiation at a specific wavelength and make a transition to a state at higher energy.

The set of wavelengths at which a particular species absorb electromagnetic radiation is known as its absorption spectrum.

In a similar way, atoms emit electromagnetic radiation when an electron in an excited state makes a transition to a state at lower energy.

The set of wavelengths emitted by a particular species is known as its emission spectrum. Emission

Emittedphoton

En

erg

yAbsorption

Incidentphoton

En

erg

y



Atomic Spectra: Visible Spectral Region (ctd) The figure shows the energy level

diagram of hydrogen including the quantum numbers n and for the states.

The diagonal lines show the allowed transitions from a state of higher energy to one of lower energy.

The allowed transitions are those for which changes by 1. In other words, the selection rule for an allowed transition is given by n unrestricted, = ± 1, m = 0, ± 1 Transitions for which = 1 are in principle forbidden (e.g. note that there are no transitions from = 0 to = 2 since this violates the selection rule). However such transitions can actually take place but with a probability that is negligible compared to those of allowed transitions.



Atomic Spectra: X-Ray Spectra Transitions of the outer electrons in

an atom involves energies of only a few electron-volts. Such transitions therefore results in spectral emission in or near the visible spectral region.

Inner electrons of heavier elements are not shielded from the nuclear charge and are very tightly bound.

Transitions of inner electrons result in x-ray emission because of the large energies involved.

X-rays are produced when electrons that are accelerated by a high voltage ( ~ 50kV) strike a metal target inside an X-ray tube.



Atomic Spectra: X-Ray Spectra The X-ray spectrum consists of a

continuous spectrum, with a cut-off wavelength o, on which a number of peaks are superimposed.

The continuous spectrum and the cut-off wavelength move to shorter wavelength when the voltage across the x-ray tube is increased. However the wavelengths of the sharp peaks do not change for a given target material and are, therefore, a characteristic of the material.

When a high energy electron (~50 keV) collides with the inner electron of a metal target atom (molybdenum in the diagram), the inner electron is knocked out of the atom. An electron in a higher energy state may then relax to fill the void resulting in x-ray emission.



Atomic Spectra: X-Ray Spectra The X-ray spectrum consists of a

continuous spectrum, with a cut-off wavelength o, on which a number of peaks are superimposed.

The continuous spectrum and the cut-off wavelength move to shorter wavelength when the voltage across the x-ray tube is increased. However the wavelengths of the sharp peaks do not change for a given target material and are, therefore, a characteristic of the material.

When a high energy electron (~50 keV) collides with the inner electron of a metal target atom (molybdenum in the diagram), the inner electron is knocked out of the atom. An electron in a higher energy state may then relax to fill the void resulting in x-ray emission.



Atomic Spectra: X-Ray Spectra

• The cut-off wavelength lo is due to a collision in which an incident electron loses all its kinetic energy KE. The energy appears as that of a photon with a maximum frequency f and minimum wavelength min given by

minλhchfKE

electronmin KE

hc

Note that the cut-off wavelength depends on the kinetic energy of the incident electron but does not depend on the target material.

The continuous spectrum is due to photons arising from multiple collisions of an incident electron with the target atoms. For each collision the photon energy is less than that of the photon emitted in the first collision.



Atomic Spectra: X-Ray Spectra The inner energy levels of atoms can be determined from

the characteristic x-ray spectra of the atoms.

The K line is emitted when an electron in a complex atom (containing many electrons) makes a transition from a state with n=2 (L shell) to a vacant state with n=1 (K shell).

The L shell electron experiences a nuclear charge (Z-1)e instead of Ze due to the shielding effect of the remaining electron in the K shell.

In 1914 H. Moseley found that a plot of (1/)1/2 against Z produced a straight line. The value of Z for a number of elements have been determined by making use of the Moseley plot.

(1/

)1/2

Z




K

L

M

N

O

K K K K K

L L L L

M M M

N N

Energy levels of a heavy atom showing the origin of x-ray spectra



Atomic Spectra: X-Ray Spectra• Remembering that the energy of an electron in the

hydrogen atom is given by,......,,neV

n

.En 321

6132

the energy of the innermost electron in the complex atom is approximately given by

eVn

)Z(.En 2

21613

The energy of the K line is given by

eV)Z(.)Z(.EEhc

hfE 222

212 1210

1

1

2

11613

The wavelength is given by

CCZ)Z(C

11 2

1(Equation of a straight line)

(C = 2.86x103 m-1)

Note: here h = 4.135x10-15eV s




Example

The wavelength of the K line from iron is 193 pm.

(a) What is the energy difference between the two states of the iron atom that give rise to this transition?

(b) What is the atomic number of iron?

(a)

(b)

eVx.Jx.x

)x)(x.(hcE 315

12

834

104461003110193

103106266

eVx.eV)Z(.E 32 104461210 26Z



Total angular momentum J

• In an atom, each electron has a certain angular momentum L and spin angular momentum S.

• Both L and S contribute to the total angular momentum J of the electron.

• In this discussion, only those atoms whose total angular momentum is due to a single electron outside a closed shell are considered.

• These include atoms of elements in group I of the periodic table (e.g. hydrogen, lithium, sodium, potassium etc., except that hydrogen does not have an inner closed shell).

• They also include ions such as He+, Be+, Mg+, …, B2+, Al2+, …, and so on, which also have one electron outside inner closed shells.



Total angular momentum J (ctd)

• For these atoms and ions, the total angular momentum J of the outer electron is given by the vector sum of the orbital angular momentum L and the spin angular momentum S.

• As is the case with the orbital angular momentum L, the total angular momentum J is quantised in both magnitude and direction. The magnitude is given by

• j can have the values

j = + s = + 1/2

or j = - s = - 1/2 but never less than zero.

J = L + S

)1j(jJ



Total angular momentum J (ctd)

• Examples: If = 0, j = 1/2. If = 1, such as for a p state, j = 3/2 or 1/2.

• The component Jz of J in the z-direction is quantised and given by

where mj = j, j - 1,…, -j + 1, -j

• Examples: If j = 1/2, mj = 1/2, or -1/2. If j =3/2, mj = 3/2, 1/2, -1/2, -3/2

• The state of an electron can be specified in terms of the quantum numbers n, , m, ms or n, , j, mj.

jz mJ



Spectroscopic Notation• Recall that lowercase letters are used to specify individual

states with a particular angular momentum. For example, the letter s is used for = 0, p for = 1, and so on.

• A similar scheme is used to specify the state of an atom based on its total angular momentum L, but using capital letters.

• L = 0 1 2 3 4 …...

• Letter = S P D F G …...

• For a single outer electron, the state of the atom is given by a term symbol of the form

2123

0, , :has state

1, 2, :has state

j1nS1

jnP2

21

23

nLj



Fluorescence and Phosphorescence

Atomic or molecular systems may exhibit both fluorescence and phosphorescence.

Fluorescence involves the absorption of a short wavelength (high energy) photon and the immediate relaxation of the excited state to a lower level by the emission of one or two longer wavelength photons.

Fluorescence is typically fast (fs to ns) due to it being highly “allowed”.

WavelengthA

bso

rpti

on a

nd

em

issi

on inte

nsi

ty

Absorbed

radiation

Emittedradiation



Applications of Fluorescence

• Many compounds can be identified from their fluorescence spectra. The compound to be identified is illuminated with UV radiation and the fluorescence spectrum of the compound is then identified from a library of known spectra.

• In a fluorescent lamp, an electric current is passed through a mixture of mercury vapour and an inert gas such as argon. The mixture is contained in a sealed glass tube whose inner surface is coated with a fluorescent material.

• Excitation of the gas mixture results in the emission of ultra-violet light which causes the coating material to fluoresce in the visible spectral region.

• The above process is more efficient than the production of light using an incandescent lamp.



Phosphorescence

• Phosphorescence happens on a much slower timescale because the transitions involved are highly “disallowed”.

• Atoms of phosphorescent materials can be excited by photon absorption to states which have relatively long lifetimes of a few seconds or longer. These states are known as metastable states.

• The radiation emitted in transitions from these states may be emitted minutes and even hours after the initial excitation.

• After exposure of some watch dials to light, the glow that is observed in the dark is due to phosphorescence of materials embedded in the dial.


School of Mathematical and Physical SciencesPHYS1220School of Mathematical and Physical SciencesPHYS1220 Stimulated Absorption and Spontaneous

Emission

• Three types of transitions that involve electromagnetic radiation are possible between two energy levels E1 and E2 of an atom.

• If the atom is initially in the lower state with energy E1, it can be raised to a state with higher energy E2 by absorption of a photon of energy hf where hf = E2 - E1 (h is Planck constant, f the frequency of the incident radiation). This process is known as stimulated (or induced) absorption.

• If the atom is in the state with higher energy E2, it can make a spontaneous transition to the state of lower energy E1 by emitting a photon of energy hf. This process is known as spontaneous emission.

Stimulated absorption

hf

E1

E2

Spontaneous emission

hf

E1

E2


School of Mathematical and Physical SciencesPHYS1220School of Mathematical and Physical SciencesPHYS1220Stimulated

Emission• The third type of transition was first proposed by

Einstein, and is known as stimulated (or induced) emission.

Einstein showed that the probability of stimulated emission is the same as that of stimulated absorption. In other words an incident photon is equally likely to be absorbed as it is of causing stimulated emission.

Stimulated emission

hf

E1

E2

In stimulated emission a photon of energy hf interacts with an atom in the upper state of energy E2, causing a transition to the lower state of energy E1 and the emission of another photon of energy hf. The radiation is emitted in phase with the incident radiation.



Population inversion

In thermal equilibrium, the number of atoms in an ensemble in state n is:

The ratio of the population in two states n and n is given by

kT

En

n Ce

N

kT

EE nn

n

n eNN

For the stimulated emission process to dominate, the population must be driven far from thermal equilibrium. Number

En

erg

y

En

En

Number

En

erg

y

En

En

Equilibriumdistribution

Population inversion



Mechanism for Achieving a Population Inversion A careful selection of an element

whose allowed energy states have the appropriate lifetimes offers a mechanism to achieve population inversion. The mechanism is as follows:

Rapidly excite higher energy levels in an ensemble of atoms.

A cascade of this injected energy populates a particular long lived “metastable” state.

The state to which the metastable state is optically coupled relaxes to its thermal population level very quickly.

There is a greater population in level 3 than level 2 resulting in a population inversion.

Number

En

erg

y

E1

E2

E3

E4

Fast relaxation

Fast relaxation

Laser transitionThermal

distribution

Atoms excited to this state

Metastable level



Population Inversion and Laser Action

• The inverted population means that stimulated emission may dominate over spontaneous emission (that still occurs) from the metastable state, and also over induced absorption from the state of lower energy.

• This would result in amplification of the emitted light, a concept underlying the operation of the laser. The word laser is an acronym for Light Amplification by the Stimulated Emission of Radiation.

• Note that because stimulated emission and absorption are equally likely, it is not possible to achieve a population inversion for a two-level system. The best that can be achieved is equal population of the two levels.



Laser Excitation

• A population inversion can be achieved in a number of ways. The most commonly used methods of excitation include the following:

• Optical excitation using (a) a high intensity flash lamp (e.g. Ruby laser, Neodymium YAG lasers) (b) another laser (e.g. Dye lasers, optical fibre lasers).

• Electron impact (a) a gas discharge ( e.g. Nitrogen laser, helium-neon laser, Argon-ion laser, Excimer lasers (XeF, ArF, KrF, XeCl), to name just a few) (b) an electron beam (e.g. Excimer lasers).

• Chemical reaction (e.g. HF (hydrogen fluoride) laser)



Laser Design The ensemble of atoms is usually placed within an optical

cavity formed by two mirrors, one of which is partially transparent.

The atoms may be excited by an electrical discharge or optically pumped.

A random, spontaneously-emitted photon may just happen to be on the optical axis of the cavity. This photon may stimulate another atom to emit yielding two photons on axis. These two photons stimulate two more atoms to yield 4, followed by 8, 16, 32 and so on.

Random spontaneously-emitted photons



The Helium-Neon Laser

• The Helium-Neon (He-Ne) laser is commonly used in a number of applications. It emits typically a few milliwatts in a continuous (as opposed to pulsed) beam at a wavelength of 632.8nm (red beam).

• The active medium consists of a mixture of helium and neon at low pressure, which gives the laser its name.

• The pumping mechanism is a high-voltage electrical discharge as illustrated in the simplified diagram.

Laser output

Power supply

Anode Cathode

Highly reflectingmirror

Partially reflectingmirror

Glass envelope

Low pressuregas discharge



The Helium-Neon Laser (ctd)• Helium atoms are excited

to the metastable state HE2 in an electrical discharge.

• The excited helium atoms transfer their energy to ground state neon atoms which are in turn excited to the long-lived state NE4.

Helium Neon

Laser emissionat 632.8nm

Relaxation by spontaneous

emission

Radiationlesstransition

Excitation byelectron impact

HE1

HE2

NE1

NE2

NE3

NE4

This creates a population inversion between levels NE4 and NE3. Spontaneous emission starts the stimulated emission process and laser emission occurs at 632.8nm.

Atoms in state NE3 relax by spontaneous emission to NE2, thus maintaining the population inversion.

Energy transfer



Laser Properties A laser beams generally has the following properties:

It is collimated; the beam has low divergence due to the cavity optics.

It is coherent; a phase relationship is maintained across and along the beam.

It is nearly monochromatic; the output has a narrow spectral width.

It is intense; the power is confined to a narrow beam of low divergence.

Lasers may be continuous or pulsed depending on the particular lasing medium and the mode of excitation.

Laser output powers range from milliwatts (e.g. He-Ne lasers), to several kilowatts (e.g. industrial CO2 lasers) or even to terawatts (1012W pulsed lasers used in laser fusion experiments)

Documents

School of Mathematical and Physical Sciences PHYS1220 Sept. 2002 Atoms Slide 1 Quantum Theory of Atoms, Molecules and Solids In the Quantum Mechanics course,