Chapter 1basic aspects of atoms

• The compositions of atoms;• The mass and size of the atoms and their experimental methods;• Isotopes;• The periodic system of the elements;• The electrons; • The light/photon;• Rutherford scattering, Rutherford model of the atoms;• Matter waves/the wave-particle duality;• Wave packet, probabilistic interpretation;• Uncertainty principle

Chapter 2

Bohr’s model

The atomic models

Planet model by RutherfordPlum-pudding model by Thomson

Electron cloud model Bohr’s model

Neils Bohr (1885 - 1962)

Bohr was raised in a middle class Danish family and showed no particular talent as a child except for sports. He played soccer at almost a professional level and was an active skier until late in his life. Bohr studied physics as an undergraduate at the University of Copenhagen and, upon graduation, traveled to Cambridge to study under J. J. Thompson. Thompson was particularly busy at this point and was not able to work with Bohr. So Bohr took up an invitation from Rutherford to travel to Manchester and study radioactivity. With Rutherford, Bohr participated in some of the studies on alpha particle scattering by heavy metals, and began to examine the planetary ideas of the atom that had been proposed and rejected by Rutherford. Bohr was very familiar with the new ideas developed by Planck and Einstein and wrote up a preliminary proposal to describe a quantum atom in June of 1912 which he was going to discuss with Rutherford. On talking to a friend he mentioned this talking paper and the friend asked whether Bohr's ideas could account for the spectrum of hydrogen. Curiously, Bohr haven't considered this question and the friend recommended that Bohr look at Balmer's equation to see if the model fit the data. Bohr later related "As soon as I saw Balmer's formula the whole thing was immediately clear to me". Bohr's final model of the planetary atom was published in the Philosophical Magazine in 1913

The colors of firework

Elements in flames

Robert Bunsen's work examined the spectra of elements in flames. It was well known that many salts cause flames to assume brilliant colors. These salts emitted only certain lines of the spectrum and not a continuum of colors as you'd expect from a black body.

Black body and line spectra

Continuous spectrum

Line spectrum

Emission spectra

re dvi

olet

A simple prism spectrometer

Fabry-Perot Interferometer

This interferometer makes use of multiple reflections between two closely spaced partially silvered surfaces. Part of the light is transmitted each time the light reaches the second surface, resulting in multiple offset beams which can interfere with each other. The large number of interfering rays produces an interferometer with extremely high resolution, somewhat like the multiple slits of a diffraction grating increase its resolution.

Solar spectrum with absorption lines

Absorption spectra

Optical spectra

Continuous, band and line spectra:

• Continuous spectra are emitted by radiant solids or high-density gases;• Band spectra consist of groups of large numbers of spectral lines which are very close to on another;• Line spectra are typical of atoms.

Observations of optical spectra:Emission or absorption spectra

The electromagnetic spectrum

ind

igo

Kirchhoff and Bunsen were the first to discover in the mid-19th century that each element possesses its own characteristic spectrum.

Hydrogen is the slightest element, and the Hydrogen atom is the simplest atom, consisting of a proton and an electron. The spectra of the Hydrogen atom have played an important role again and again over the last 90 years in the development of our understanding of the laws of atomic structure and of the structure of the matter.

Balmer series of the Hydrogen atom

Balmer in 1885, the wavelengths of these lines could be extremely well reproduced by a relation of the form:

Gn

n

42

1

21 n1=3,4,…, G: empirical constant

22

1

2

11

nRH

Rydberg constant RH:RH = 109677.5810 cm-1

Balmer series:The series of lines fall closer and closer together in a regular way and approach a short wavelength limit (H)

The optical spectra of Hydrogen atoms

22

11

nnRH '

By Rydberg in 1889:

With n’ < n being integers

We can conclude from observation and inductive reasoning that the frequencies (or wavenumbers) of all the spectral lines can be represented as differences of two terms of the form R/n2. As we shall see in the following, these are just the energy levels of the electron in a hydrogen atom. The spectral lines of the hydrogen atom can be graphically pictured as transitions between the energy levels (terms), leading to a spectral energy level diagram.

Left: Term diagram, right: Grotrian diagram

Bohr’s modelBohr’s model (in 1913) was based on the observations of optical spectroscopy, and following the Rutherford model, he assumed that the electrons move around the nucleus in circular orbits of radius r with velocity, must as the planets move around the sun in the solar system.

The fundamental difficulties of Rutherford model with classical physics:• From classical physics, a charge traveling in a circular path should lose energy by emitting electromagnetic radiation • If the "orbiting" electron loses energy, it should end up spiraling into the nucleus (which it does not). Therefore, classical physical laws either don't apply or are inadequate to explain the inner workings of the atom.

The fundamental difficulties of Rutherford model with classical physics

Classical electron orbit

The orbital frequency: /2, where

202

0

2

4

rm

r

eF lcentripeta

r

erd

r

eE

r

pot0

2

20

2

44

r

ZeET pot

0

2

8

Bohr’s three postulates—to be an extremely important step towards

quantum mechanics

To avoid the discrepancy with the laws of classical physics, Bohr proposed three postulates to describe the deviations from classical behavior for electron in an atom.

Bohr’s three postulates1. Quantised orbits. The classical equations of motion are valid for electro

ns in atoms. However, only certain discrete orbits with the energy En are allowed. These are the energy levels of the atom.

2. The motion of the electron in these quantised orbits is radiationless. An electron can be transferred from an orbit to another orbit. Emission: transferring from an orbit with lower binding energy En to an orbit with higher binding energy En’; absorption: higher to lower binding energy levels.

3. Corresponding principle. The orbital frequency is comparison with the frequency of emission or absorption. For large n, one can calculate the Rydberg constant RH from atomic quantities.

In general, the corresponding principle: with increasing orbital radius r, the laws of quantum atomic physics become identical with those of classical physics.

Schematic representation of Bohr’s model

orbits

Bohr’s model

For the orbit, what is the energy level, radii, and R?

The frequency of the emitted radiation:

The energy term of the Hydrogen:

2n

RhcEn

)11

()(1

22 nnRcEE

h nn

We consider the emission between neighboring orbits, n - n’ = = 1, for large n:

nn 21)1( 2

The frequency of the emitted radiation:

332222

221

)(

1)

11(

n

RC

nRC

nnRc

nnRc

According to the correspondence principle, for large n, then = the classical orbital frequency /2. The energy level En of electron in Bohr’s orbit is equal to that of the total energy Etot in classical orbit: En = Etot

2n

RhcEn

3/120

43/2

00

2

)()4(2

1

8

me

r

eEtot

3

2

2 n

Rc

The Rydberg constant: ch

emRH 32

0

40

8

R = (109737.318 ± 0.012) cm-1

The radius rn of the nth orbital:0

20

22 4

me

nrn

The quantum number n is called the principal quantum number

The energy term:

2n

RhcEn

For n = 1: The smallest orbital radius r1 = 0.529Å = Bohr radius = a0

The lowest energy state E1 = -13.59eV. Then for H atom:

02anrn

eVn

En 2

159.13

Optical spectra of Hydrogen atom

orbits

Energy level diagram, term diagram

ultraviolet

visible infrared

Ground state and excited state

At normal temperature only the lowest energy term, n = 1, is occupied, and only Lyman series in hydrogen is observable in absorption. Ground state: the lowest energy stateExcite state:

At higher temperature, the excited state, n = 2, 3, …, can be occupied according to the Boltzmann distribution. For example, in the sun, with a surface temperature of 6000K, only 10-8 of the hydrogen atoms in the solar atmosphere are in the n = 2 state.

Bohr model of the atom—emission of light

If we have a hydrogen atom with its electron in an excited state (either by light absorption or by heating) the electron may fall down to a lower orbit by emission of light. The electron may fall into any lower orbit, and the energy it loses will be exactly equal to the energy difference between the two orbits. In the above example, the electron falling from the third to the first orbit emits green light. Were it to go to the second orbit, the difference in energy would be less and the atom only emits red light.The Balmer equation fit the energies of electrons falling from higher orbits into the second orbit. In real life (and not the simple example above) electrons falling into the first orbit emit ultraviolet light.

Bohr model of the atom — absorption of light

If we assume that a hydrogen atom starts out with its electron in the lowest possible orbit, the electron may move to other obits if and only if absorbs a photon with exactly the right amount of energy.In the illustration above, the transition from the first orbit to the third orbit requires exactly the amount of energy supplied by green light. The spectrum of light leaving the hydrogen atom would be missing a line in the green.

Bohr's model allowed a simple understanding of the absorption or emission of light by atoms that explained line spectra. But it make no statements about processes. The classical orbital concept is abandoned. The electron’s behavior as a function of time is not investigated, but only its stationary initial and final states.

The features of Bohr’s model

Electron transitionsThe Bohr model for an electron transition in hydrogen between quantized energy levels with different quantum numbers n yields a photon by emission with qua

ntum energy:

This is often expressed in terms of the inverse wavelength or wave number:

22

21

111

nnRH

Hydrogen level energy plotThe basic structure of the hydrogen energy levels can be calculated from the Schrodinger equation. The energy levels agree with the earlier Bohr model, and agree with experiment within a small fraction of an ele

ctron volt.

Hydrogen spectral tubea hydrogen spectral tube excited by a 5000 volt transformer. The three prominent hydrogen lines are shown at the right of the image through a 600 lines/mm diffraction grating.

An approximate classification of spectral colors:

•Violet (380-435nm) •Blue(435-500 nm) •Cyan (500-520 nm) •Green (520-565 nm) •Yellow (565- 590 nm) •Orange (590-625 nm) •Red (625-740 nm)

Measured hydrogen spectrum

Wavelength Relative Transition Color(nm) Intensity

383.5384 5 9 -> 2 Violet388.9049 6 8 -> 2 Violet397.0072 8 7 -> 2 Violet410.174 15 6 -> 2 Violet434.047 30 5 -> 2 Violet486.133 80 4 -> 2 Bluegreen (cyan)656.272 120 3 -> 2 Red656.2852 180 3 -> 2 Red

Bohr’s explanation of line spectra

Angular momentum quantisation

prl

nrmrvml nnnn 200

With n = 1,2,3,…

From Bohr’s model, an electron has velocity Vn and orbital frequency n in the nth orbit with the orbit radius rn ,the orbital angular momentum is quantised:

Hydrogen-like atoms

Hydrogen-like atoms: atoms with an electron and a nucleus with charge +Ze, such as He+, Li++.

++Ze

-e

Hydrogen-like atoms

nnlcentrifuga r

vm

r

ZeF

20

20

2

4

For the possible orbital radius: 0

2

02

022 4

aZ

n

mZe

nrn

The energy states: eVn

Z

n

meZEn 59.13

32 2

2

220

20

42

Equilibrium between the Coulomb force and the centrifugal force:

The wavenumber of the spectral lines:

2

12

2

2 1159.13

121 nnhc

ZEE

hc nn

Motion of the nucleus

From spectroscopic measurement RH = 109677.5810 cm-1

From theoretical calculation R = 109737.318 ± 0.012 cm-1

= R - RH 60 cm-1

The reason for the difference is the motion of the nucleus during the revolution of the electron, which was neglected in the above model calculation. This calculation was made on the basis of an infinitely massive nucleus; we must now take the finite mass of the nucleus into account.

The motion of the nucleus

The motion of two particles, of masses m1 and m2 and at the distance r from one another, takes place around the common center of gravity. The reduced mass:

21

21

mm

mm

Replace the mass of the orbiting electron, m0 by :

MmRR

/1

1

0

The mass of the orbiting electron m1 = m0, the mass of the nucleus m2 = M.

m2

m1Gravitycenter

Due to the motion of the nucleus, different isotopes of the same element have slightly different spectral lines. The mass ratio M/m0 can be determined by spectroscopic observation. This so-called isotope displacement led to the discovery of heavy hydrogen with the mass number A = 2. It was found that each line in the spectrum of hydrogen was actually double, for H and D.

Mproton/melectron = 1836.15

1

0

1

0

419.109707/1

1

584.109677/1

1

cmMm

RR

cmMm

RR

DD

HH

The difference in wavelengths for corresponding lines in the spectra:

D

HH

H

DHDH R

R11

Spectra of Hydrogen-like atoms

According to Bohr model, the spectra of hydrogen-like atoms, all atoms or ions with only one electron, are the same except for the factor Z2 and the Rydberg number. This has been completely verified experimentally.

Some energy levels of the atoms H, He+, and Li2+

For He+, astronomers found the Fowler series:

22

1

3

14

nRHeF

The Pickering series:

22

1

4

14

nRHeP

Preparation of the hydrogen-like heavy atoms in lab: accelerate the singly-ionised atoms to high energies and pass them through a thin foil, their electrons are stripped off on passing through the foil.

For example, in order to strip all the electrons from a uranium atom and produce the U92+ ions to recapture one electron, one can then obtain the hydrogen-like ion U91+. The corresponding spectral lines are emitted as the captured electron makes transitions from orbits of high n to lower orbits. For U91+, the Lyman series has been observed in the spectral region around 100keV and the Balmer series is in the region between 15 and 35keV.

Muonic atomsMuonic atoms: the electron was replaced by a heavier meson or muon.

++Ze

-e muon

m = 206.8 m0 , Muonic atoms are extremely small in diameter and very close to the nucleus.

Muons behave like heavy electrons, we can simply apply the results of the Bohr model. For the orbital radii:

m

m

Z

na

m

mern

mZer nn

02

002

2

20 )(

4)(

A numerical example: Mg11+:

Electron: mer Ao

121 105.4

12

53.0)(

muon: mer

r 1411 102.2

207

)()(

Muonic atoms are interesting objects of nuclear physics research. Since the muons approach the nucleus very closely, much more than the electrons in an electronic atom, they can be used to study details of the nuclear charge density distribution, the distribution of the nuclear magnetic moment within the nuclear volume and of nuclear quadrupole deformation.

Excitation of quantum jumps by collisions—another experimental support for Bohr’s model

Lenard investigated the ionisation of atoms as early as 1902 using electron collisions.

Experimental arrangement for detecting ionisation process in gases. Only positive ions, which are formed by collisions with electrons, can reach the plate A. The voltage are chosen so that the electrons cannot reach the plate; they pass through the grid and are repelled back to it. When an electron has ionised an atom of the gas in the experimental region, however, the ion is accelerated towards the plate A. Ionisation events are thus detected as a current to the plate.

plate

Franck-Hertz experiment

Franck and Hertz showed for the first time in 1913 that the existence of discrete energy levels in atoms can be demonstrated with the help of electron collision processes independently of optical-spectroscopic results. Inelastic collisions of electrons with atoms can result in the transfer of amounts of energy to the atoms which are smaller than the ionisation energy and serve to excite the atoms without ionising them.

Electrons pass through the grid and are carried by their momenta across a space filled with Hg vapour to an anode A. Between the anode and the grid is a braking voltage of about 0.5eV. Electrons which have lost most of their kinetic energy in inelastic collisions in the gas-filled space can no longer move against this braking potential and fall back to the grid. The anode current is the measured as a function of the grid voltage VG at a constant braking potential VB. At a value of VG 5V the current I is strongly increased; it then increase again up to VG 10V , where the oscillation is repeated.

When the electrons have reached an energy of about 5eV, they can give up their energy to a discrete level of the mercury atoms. They have then lost their energy and can no longer move against the braking potential. If their energy is 10eV, this energy transfer can occur twice, etc.

One can find an intense line in emission and absorption at E = 4.85eV in the optical spectrum of atomic mercury, corresponding to a wavelength of 2537Å. This line was also observed by Franck and Hertz in the optical emission spectrum of Hg vapour after excitation by electron collisions.

electron

Hg

collision

electron

Improved experiment

The resolving power for the energy loss of the electrons may be improved by using an indirectly heated cathode and a field-free collision region. In this way, one obtains a better uniformity of the energies of the electrons. A number of structures can be seen in the current-voltage curve; these correspond to further excitations of the atoms. For example 6.73eV related to 1850Å. But not all the maxima in the current-voltage curve can be correlated with observed spectral lines. For the optically forbidden transitions can, in some cases, be excited by collisions.

Electron collision and optical excitation

1) From the improved Franck-Hertz experiment, the selection rules for collision excitation of atoms are clearly not identical with those for optical excitations.

2) The optical excitation occurs only when the light has exactly the same energy as the quantum energy of atom, for example, Na vapour, yellow line E = 2.11eV. Both smaller and larger quantum energy are ineffective in producing an excitation.

3) The yellow line is emitted whenever the energy of the electrons E >= 2.11eV. Because the kinetic energy of free electrons is not quantised. After excitation of a discrete atomic energy level by electron collision, the exciting electron can retain an arbitrary amount of energy.

These electron collision experiments prove the existence of discrete excitation states in atoms and thus offer an excellent confirmation of the basic assumptions of the Bohr theory.

Electron waves and orbits

Asking why electrons can exist only in some states and not in others is similar to asking how your guitar string knows what pitch to produce when you pluck it. It is a standing wave phenomenon and has to do with resonance.

Electron wave length for different states

Sommerfeld’s extension of the Bohr model

To explain the double or multiplet structure of the spectral lines, Sommerfeld derived an extension of the Bohr model: not only circular orbits, but also elliptical orbits are possible.

According to the Kepler’s Law of Areas, the electron sweep out equal areas between its orbit and the nucleus in equal time. Then, in elliptical orbit, the electron is faster when it closer to the nucleus, more massive. The relativistic mass change of the electron lifted the orbital degeneracy. The Sommerfeld’s calculation:

scorrectionorderhigher

k

n

n

Z

n

ZRhcE kn 4

31

2

22

2

2

,

Sommerfeld’s extension was on one hand of great historical importance in introducing a second quantum number k, but, on the other hand, been made obsolete by the later quantum mechanical treatment.

Where the fine structure constant:

137

1

2 0

2

hc

e

The model only worked for hydrogen and could not be modified to fit anything beyond lithium. It was useless in describing bonding, and it relied on a set of arbitrary postulates whose justification flew in the face of known physics. At best, Bohr's model must be seen as a transitional theory, introducing the quantum theory to the understanding of atoms, but not a fully modern theory.

The limitation of Bohr’s model

Failures of the Bohr modelWhile the Bohr model was a major step toward understanding the quantum theory of the atom, it is not in fact a correct description of the nature of electron orbits. Some of the shortcomings of the model are:1. It fails to provide any understanding of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities.2. The Bohr model treats the electron as if it were a miniature planet, with definite radius and momentum. This is in direct violation of the uncertainty principle which dictates that position and momentum cannot be simultaneously determined.

The Bohr model gives us a basic conceptual model of electrons orbits and energies. The precise details of spectra and charge distribution must be left to quantum mechanical calculations, as with the Schrödinger equation.

Rydberg atomsRydberg atoms: atoms in which an electron has been excited to an unusually high energy level illustrate well the logical continuity between the world of classical physics and quantum mechanics.

Extraordinary properties:Gigantic: 10-2 mm in diameter;Extremely long lifetime: 1s, typical lifetime of lower excited states of atoms are about 10-8 s;Strongly polarised: by relatively weak electric fields.

When the outer electron of an atom is excited into a very high energy level, it enters a spatially extended orbit, which is far outside the orbitals of all the other electrons and the nucleus. The core, consisting of the nucleus and the inner electrons, has a charge +e. Rydberg atoms behave in many respects like highly excited hydrogen atoms.

In interstellar space, n up to 350;In the laboratory, n between 10 and 290.

Rydberg atoms

Apparatus for the detection of Rydberg atoms. An atomic beam is crossed by several laser beams. They cause the excitation of the atoms into Rydberg states when the sum of the quantum energies of the laser beams corresponds to the excitation energy of a Ryderg state. The Rydberg atoms are ionised in the electric field of a condenser, and the ions are then detected.An example of the detection of Rydberg states of the lithium atom with n = 28 to 39.

Rydberg excitation states of barium atoms with the principal quantum number n, observed using Doppler-free spectroscopy.

Artificial atoms—Positonium, muonium, and antihydrogen

It is possible to make artificial atoms in which one or both of the atomic components of hydrogen, the proton and the electron, are replaced by other particles. All the conclusions of the Bohr model concerning atomic radii, energy levels, and transition frequencies should also apply to the artificial atoms

Positronium, (e-, e+) an atom consisting of an electron, e-, and a positron, e+, was discovered in 1949 by M. Deutsch. Positron can be obtained from the radioactive decay of nuclei, 22Na. positron atoms are formed when positrons pass through a gas or impinge on solid surfaces, where the positron can capture an electron. The lifetime of positronium is very short (1. 4·10-7s or 1. 25·10-10s), they annihilate each other with the emission of two -quanta. In condensed-matter physics and in modern medicine, positronium atoms are used as probes for structures and dysfunctions, because the emission of their annihilations radiation is dependent on their material surroundings. In medicine, positron emission tomography is used for example to form an image of diseased tissue in the brain.

+e+

e-

++

e-

Muonium, (+, e-) an atom consisting of a muon, +, and a electron, e-. It is formed when positive + enter into a bound state with electrons on passing through a gas or on a solid surface. + particles are unstable, and the lifetime of muonium is correspondingly only 2.2·10-6 s. These atoms have been studied extensively by spectroscopic methods. They are particularly relevant to the refinements of the Bohr model by Dirac’s relativistic quantum mechanics.

-P-

e+

Antihydrogen atom, (p-, e+) an atom consisting of a positron bound to a negatively-charged antiproton. According to the postulates of quantum mechanics, antimatter should behave just like ordinary matter. An experimental test has yet to be performed, since antimatter was not available until very recently. In 1995, the successful preparation of antihydrogen was reported for the first time. One goal is the spectroscopic investigation of the antihydrogen atoms, as a test of the symmetry of the interactions between matter and antimatter.

homework

P122,

8.5, 8.6, 8.8, 8.9, 8.14, 8.15, 8.17