This is it Right one to form M-4.ppt

1
4.0 Elements and Atoms throughout the ages
4.1 The plum pudding / raisin cake atom model of J. J. Thomson
4.2 Rutherford Scattering / Planetary model of the atom?
4.3 The instability of the classic “solar system” model of atoms
3.3 Line spectra of elements, Balmer’s numerology, Rydberg’s equation and
constant
4.4 The Bohr Model of the Hydrogen Atom; Sommerfeld - fine structure
constant
4.5.1 Successes of the Bohr Model
4.5.2 Failures of the Bohr Model
4.6 Shell structure of atoms, characteristic X-Ray Spectra, Moseley’s rule
and its support of Bohr’s model
4.7. There are also Auger electrons, not only characteristic X-rays
4.8. quantum jumps in physics and how the general public misunderstands
the concept
CHAPTER 4
Structure of the Atom / Particle nature of matter
In the present first part of the paper the mechanism of the binding of
electrons by a positive nucleus is discussed in relation to Planck’s theory. It
will be shown that it is possible from the point of view taken to account in a
simple way for the law of the line spectrum of hydrogen. - Niels Bohr, 1913

2
Various atoms and molecules as depicted
in John Dalton's ”A New System of
Chemical Philosophy” (1808).
Law of constant proportion in chemical
reactions, atoms must exist for chemical
book-keeping reasons
2H2 + O2 -> 2H2O
Various Greek (and Indian) philosophers had only a few elements, concept of
an atom that is indivisible smallest unit of any form of matter

3
Faraday, 1838:
Atoms have
“electrical
constituents”

4
John Joseph Thomson, 1897
Electrons are negatively charged particles that can be “pulled out” of any metal by
a strong electric field (the “thingies” pulled out are all identical and have a charge
to mass ratio about 2,000 times larger than hydrogen ions (which we now know to
be protons)
Previous
chapter
So atoms cannot be indivisible, if not
indivisible, they must have some internal
structure that is responsible for the
physical and chemical properties of the
atoms of the various elements

5
There is also radioactivity (from 1896 onwards):
electrons and other particles, e.g. alpha
particles, (we now know nuclei of He, two
protons + two neutrons) come out of the atoms
of certain radioactive elements
end of the second part of this course, nuclear
physics

6
ibid vol. 34, pp. 591-
592, 1911
A. Einstein, “Eine neue
Bestimmung der
Moleküldimensionen”, Annalen der
Physik, vol. 19, pp. 289-306, 1906
Einstein’s PhD thesis and the related 1905
paper, atoms must exist for physical reasons
also start of quantitative
nanoscience

7
Big questions: What is the true nature of atoms?
What is their internal structure that is responsible for the
chemical and physical properties of the atoms of the various
chemical elements, expanded to molecules and condensed
matter (crystals, glasses, liquids, liquid crystals, …)?
Smaller questions (the answers to which will help with the big
questions above:
How come there are characteristic emission and absorption spectra for all
of the elements?
How come there are characteristic X-ray emission spectra for the
elements (there are of course also characteristic X-ray absorption spectra
– not part of this course, there are also characteristic γ-ray absorptions
and emissions, end of the second part of this course, nuclear physics)
How come …

8
Modern Physics
waves or
particles?
How come
they are
characteristic
of atoms?
then applications,
PH 312
the correct theory of matter
at last
J.J. Thompson (1987) electron
How come we can’t
derive this from
Maxwellian waves?
Wave particle duality for p
Angular momentum
is quantized in unit of
h-bar

9
 Thomson’s “plum-pudding” model of the atom had the positive
charges spread uniformly throughout a sphere the size of the
atom, with electrons embedded in the uniform background.
 In Thomson’s view, when the atom was heated, the electrons
could vibrate about their equilibrium positions, thus producing
electromagnetic radiation, but why are there spectral lines for the
elements?
Thomson’s Atomic Model
dough or pudding in the plum pudding, the plums/raisins are the electrons
There needs to be the
same amount of positive
charge, somehow
spread out throughout
the “dough”, which is
collectively tens to
hundreds of thousands
time heavier
The most abandoned
hydrogen ions are
protons, there was no
physical basis for the
periodic table of
elements at that time

10
What to do? Rutherford’s idea: shoot small but highly
energetic streams of particles at atoms, see what happens,
Still done in particle physics today, also Rutherford managed to transform one
type of atom into another by shooting at them with α-particles
That is what physicists do, take things apart, figure out what is its “internal
structure” is, how it works on the basis of this structure
Rutherford himself: “In Science there is only physics, anything else is
stamp collecting.” Various versions: ”The only real science is physics,
everything else is stamp collecting.”
2 positive
charges

12
Experiments of Geiger and Marsden
 Rutherford conceived a new technique
for investigating the structure of matter
by scattering a particles from atoms.
 Marsden and Geiger showed that a few
a particles were scattered from thin gold-
leaf targets at backward angles greater
than 90°.
not the Bragg angle

13
No experiments for Z = 60
(Neodymium) as it is rare and
cannot easily be beaten into
very thin foils as the crystal
structure is hexagonal close-
packed, but Ag (Z = 47) and Au,
Cu, and Al all can, because they
are cubic densest packed
Experimental
rule, change
only one thing
at a time
This will be
important for
Max Born’s
interpretation
of the wave
function
relationships of
form y = axk
appear as straight
lines in a log–log
graph, slope = k
Log-
values of y
to cover
many
orders of
magnitude

14
 Scattering experiments help us study matter too small to be
observed directly.
 There is a relationship between the impact parameter b and the
scattering angle θ.
When b is small,
r gets small.
Coulomb force gets large.
θ can be large and the particle can be repelled backward.
4.2: Rutherford Scattering
In modern accelerators, kinetic energy K needs to treated relativistically
Z1 and Z2 are the
repulsive charges in units
of the elemental charge e

15
 Impact parameter b gets expanded into a circular area πb0
2 which is called
cross section.
 cross section σ = πb2 enters into relation for probability for a particle being
scattered by a nucleus
 Anything within a cross section area will be scattered to angles larger than
θ0 .
Rutherford Scattering
with n number of scattering
nuclei per unit area and t
thickness of the target foil

16
 In actual experiment a detector is positioned from θ to θ + dθ that
corresponds to incident particles between b and b + db.
 The number of particles scattered per unit area is
Rutherford Scattering Equation
Where Ni is total number of
incident particles, K again
kinetic energy, r the angular
widths of the detector
4.16
no h in these
equations, so
there are just
the correct
classical
treatment of
particles, we
are looking at
collapsed
wavefunctions
De Broglie
wavelength of α-
particles very
small, so we can
ignore it, no h in
these equations

17
with k = 1/4π ε0
Shooting with the fastest
bullets at the weakest
target, Rutherford’s
scattering theory broke
down, this means the
underlying assumptions are
no longer valid, from that
one can determine the size
of the nucleus, still done
today for other particles in
particle accelerators
Order of magnitude correct ! Atoms are on the order of 1 to 3 Å (10-10 m)

18
1908 Nobel prize in chemistry to Ernest Rutherford (later
Lord Rutherford of Nelson after the town where he was
born in New Zealand) and the physical chemist Frederic
Soddy, both at McGill University in Canada
for the radioactive transmutation of chemical elements
"The energy produced by the breaking down of the atom is a very
poor kind of thing. Anyone who expects a source of power from the
transformation of these atoms is talking moonshine." – 1933
"All science is either physics or stamp collecting"
"If your experiment needs statistics, you ought to
have done a better experiment."
"An alleged scientific discovery has no merit unless it can be explained to
a barmaid."
the alchemists’ dreams realized about 100 years ago, Glenn
Seaborg has produced several thousand atoms of gold from
bismuth, but at a big net financial loss.
e.g. 14N + α → 17O + p. 1917-1919 but that stable
isotope of oxygen was not identified until the 1930s

19
 Rutherford proposed that an atom has a very small positively
charged core (nucleus) surrounded by the negative electrons.
Rutherford’s Atomic Model
Something three–dimensional where
there is a particle nucleus and particle
electrons in certain orbits, not correct
There is no positively charged dough or pudding in the plum pudding/raisin
cake, only raisins, empty space, and a tiny core that possess all of the positive
charges
A smeared out
positive charge
would only result in
small scattering
angles

20
4.3: The classical/solar system Atomic
Model is doomed
where v is the tangential velocity of the electron.
Circular motion is accelerated, accelerated charges need to radiate
energy off according to Maxwell, loosing kinetic energy
For the centripetal
force in a circular orbit
Let’s consider atoms as a quasi sun/planet model (only one planet so that
it is just a two body problem.
The force balance of circular orbits for an electron “going around” a
stationary nucleolus

21
Other weaknesses:
(1) gravity in the solar system is attractive, if there is
more than one electron in an atom, they will repel
each other.
(2) most importantly: if two solar system atom models
were to “collide”, they would form a “molecule model”,
but when that “molecule model” is broken up, there is
no physics that would ensure that the two original
“solar system atom models” would be obtained again.

22
Balmer Series – hydrogen is the simplest
atom, so theoretical modeling starts there
 In 1885, Johann Balmer found an empirical formula for wavelengths of
the emission spectrum for hydrogen in nm within the visible range
nm (where k = 3,4,5…)
A good physical theory for H atom needs to make sense of this empirical result
There is a minimum wavelength corresponding to
a maximum frequency and energy of photons

23
Rydberg Equation
 As more scientists discovered emission lines at infrared and ultraviolet
wavelengths, the Balmer series equation was extended to the Rydberg
equation, actually on the basis of this equation, people went out looking
for more lines :
Can be applied to isotopes of hydrogen by modifying R slightly

 R
m
R
e
H

1
7
2
_
10
09737
.
1
2





m
h
c
m
R e
a
with μ as reduced
mass of the elecron,
α as fine structure
constant, and M
mass of the nucleus
M
m
M
m
e
e




Aside:
Empirical constant, needs to be explained

24
4.4: The Bohr Model of the Hydrogen Atom
Niels Bohr’s general assumptions:
1) “Stationary states” (orbiting electron does not radiate energy) exist
in the hydrogen atom.
2) E = Einitial − Efinal = hf
3) Classical laws of physics do not apply to transitions between
stationary states, the electron just “jumps” makes a “quantum leap”
4) Angular momentum is quantized in units of h/2π , in the future simply
called h-bar
5) Correspondence principle: at very high quantum numbers (very
large subscripts in (2)), binding energies become so low that
transitions between stationary states can be achieved without us
noticing their discrete quantum nature, energy changes seem to be
continuous again as the object is then nearly a classical object
Nota Bene: All of these assumptions are consistent with the assumption that
the electron is a classical particle, as supported by J. J. Thomson’s
experiments and the rest of classical physics except for the Maxwell
equations which forbids (1) and obviously anything that contains h


25
without k
new physics







r
r r
d
F
U
U


bound system
r
m
n
v
e

 quantized condition for tangential
velocity and radius of orbit
r
m
e
v
e
0
4

r
m
e
r
m
n
e
e 0
4


has to be the same as
this r from classical
force balance, allows
us to resolve for
quantized energy
levels, perhaps some later
homework or quiz question
just classical physics,
not good enough

26
radius of the hydrogen atom for stationary
states, n = 1, 2, 3, …
The smallest diameter of the hydrogen atom is
two times the Bohr radius ≈ 1 Å = 0.1 nm
 n = 1 gives its lowest energy state (called the
“ground” state), there is formally no E0
since quantum numbers start with 1 and
are positive integers
But some books may call it E0 nevertheless, E1 is also called 1 Rydberg ≈ 13.6
eV, needs to be divided by h and c in order to get the Rydberg constant in m-1
Note that
this is a
“flat
model”,
apparently
2D

27
Transitions in the Hydrogen Atom
Lyman series
The atom will remain in the
excited state for a short time
before emitting a photon and
returning to a lower stationary
state. Most hydrogen atoms exist
in n = 1 at room temperature.
Balmer series (of formula fame)
When sunlight passes through the
atmosphere, hydrogen atoms in
water vapor absorb certain
wavelengths (visible), seen as
dark lines in absorption spectrum.
From slide 24
E = Einitial − Efinal
= hf emitted
-3.4 - -13.6 = 10.2

28
If the material is very dilute there are
discrete spectra: Ångstrom (~ 1860)
There is typically more emission
lines than absorption lines,
because there are many more
possible transitions to go back to a
lower state from a multitude of
excited states
Very dense material, i.e. hot
“black body”, big-bang
background radiation, sun if
one does not look too carefully
Absorption lines are fewer in
number because there are
naturally only a small numbers of
atoms in exited states
1
last chapter, we learned that emission and absorption lines are at the
same wavelength
2
reasonable model of an atom needs to explain these lines


29
Optical Spectrometer and emission
spectra
 Spectroscopy can be done very precisely, in fact so precisely that isotopes
of hydrogen, were discovered that way, Deuterium as early as 1932, later
on also Tritium)
 Much depends on the quality of the grating …spectroscopic
measurements can be very precise, one can distinguish between
isotopes of hydrogen
gasses in tubes are typically
quite dilute, discrete spectra
where d is
“line spacing
on the
diffraction
grating”
Note that
there is a
high
voltage
rather than
a couple of
volts as in
the Frank-
Hertz
experiment
If highly excited due to high tension, there are many different transitions
downwards, so typically more emission lines than absorption lines
λ 1 > λ2

30
Also possible in reflection

31
Absorption spectra, light of certain wavelengths is taken out of the
“quasi black-body spectrum” of the sun, the glass prism provides a limited
dispersion so that the emission lines from the excited atoms in the sun are
not readily visible
It is typically only transition from the ground state to higher
states that are missing in the otherwise continuous spectrum
from the sun, so it is only a few
Phenomenon
is refraction,
refractive
index is
wavelength
dependent
This line
should be
dotted

32
No need for special relativity / Fine Structure Constant
 The electron’s velocity in the Bohr model:
 For any state, n = 1, 2, ..
 v1 = 2.2 × 106 m/s ~ less than 1% of the speed of light.
 In any other orbit, there is less kinetic energy, i.e. the
electron needs to move even slower than that
 There is no need to use special relativity, relativistic
correction one part in 1372 ≈ 18,000
 The ratio of v1 to c is known as fine structure constant.
It is dimension-less, so other
creatures in distant galaxies will
have it as well (will have the very
same numerical value for it) !!
α was actually introduced by Arnold Sommerfeld (nominated 81 times for
Nobel prize, didn’t win but died in a car accident) in his attempt to explain
spectral line doublets by elliptical electron orbits, presumed confirmation
of his theory was just fortuitous, as his physics was as wrong as Bohr’s

33
discrete Hydrogen Atom Energy levels (after all it’s a bound system)
 The energies of the stationary states
where E0 = E1 = -13.6 eV (negative as all binding energies). Ground
state energy, also called one Rydberg of energy
 Emission of light occurs when the atom is in
an excited state and de-excites to a lower
energy state (nu → nℓ).
where f is the frequency of a photon.
R∞ is the Rydberg constant for a
hypothetical nucleolus that is so heavy that it
remains stationary while the electron moves
around it, note that R∞ is a model concept in
accordance to the Bohr’s model
Something negative minus
something even more
negative is positive, u for
upper, l for lower

34
 The electron and hydrogen nucleus actually have to revolve about
their mutual center of mass
(from classical mechanics)
 The electron mass is replaced by its reduced mass.
 The Rydberg constant for infinite nuclear mass is replaced by RH
for hydrogen, RD for deuterium, RT for tritium .
See also earlier slide where we
used α instead
All three hydrogen isotopes have a slightly different R values in
precision experiments, a way to detect these isotopes in the sun by
their “signature” on earth

35
2
0
0
2
8 h
a
e
R



From Bohr’s
model
1
7
2
_
10
09737
.
1
2





m
h
c
m
R e
a
c
e

0
2
4
a 
2
2
0
0
4
e
m
a
e



BUT Rydberg’s constant from high precision
experiment RH, very close but not exactly the
same, how come?
with μ as reduced electron mass
and M as mass of the nucleus, in
case of standard hydrogen it’s just
a proton, in case of deuterium a
proton and a neutron (beware
binding energy) in case of tritium a
proton and two neutrons (again
beware of binding energy) how
come? Special relativity says
so!)
from Sommerfeld model
To resolve:
M
m
M
m
e
e
e





 R
m
R
e
e
H

mass of nucleus of one atom of
deuterium (2.0141102 u) is less than
an isolated proton (1.007825 u and
an isolated neutron (1.008665 u)
added together (the rest is 2.224 MeV
binding energy with negative sign)

36
Correspondence Principle in general
Need a principle to relate the new modern results with classical
ones.
Classical electrodynamics Bohr’s atomic model
Determine the properties
of radiation
Bohr’s correspondence
principle
In the limits where classical and quantum
theories should agree, the quantum
theory must reduce the classical result.
That limit is quantum leaps between very
lightly bound electron states (very high
quantum numbers)
+
A part of (not all !!!)
some
Spectral lines of H
His 2nd paper in 1913

37
Bohr’s Correspondence Principle specific for
very high orbits of hydrogen atom
 The frequency of the radiation emitted fclassical is equal to the orbital
frequency forb of the electron around the nucleus.
 The frequency of the transition from n + 1 to n is
 For very large n only,
Substitute for E0:
From this idea he actually figured angular momentum needs to be
quantized in units of for the mathematics to work out

Watch this power

38
Limitations of the Bohr Model
The Bohr model was a great step towards a quantum theory of
matter as it includes Planck’s h, explained the spectral lines of
hydrogen (and its three isotopes) but it had its limitations.
1) Works only for single-electron atoms and ions that have a single
electron left. Only modification in all equations Ze instead of e, where Z
is the number of positive charges in the nucleus
2) Could neither account for the intensities nor the fine structure of the
spectral lines (they are actually doublets) for hydrogen when atoms
were put into magnetic fields (Nobel prize to Lorentz and Zeeman 1902)
3) Could not satisfactorily explain the binding of atoms into molecules
4) Contained unexplained quantum jumps – that’s not good physics, just a
step in the right direction, i.e. energy and other physical entities such as
angular momentum in bound systems need to be quantized
The reason for all of this is that the model is build on the basis
of the particle nature of matter.

39
Within the three paper series by Niels Bohr in
1913, idea that electrons in other atoms are
arranged in shells with lots of empty space
between then, essentially correct
Details not
correct









 2
2
2 1
1
1
u
l n
n
R
Z

2
0
0
2
8 h
a
e
R


Because we have e2 in both
E0 and the Rydberg
constant for H, there needs
to be Z2 for the one electron
left ions
Unexpected lines in
the sun’s spectrum
were identified to be
He+ (one electron
isotopes), i.e. different
effective mass
corrections for R
2
0
2
n
E
Z
En 

∞
2 electrons in first shell

40
4.6: Characteristic X-Ray Spectra and Atomic
Number beyond hydrogen and one electron-left ions
 Shells have letter names:
K shell for n = 1
L shell for n = 2 n = 7 at max, 7 periods in the
periodic table of elements
 The atom is most stable in its ground state, most atoms are in
their ground state at ambient conditions.
 When it occurs in a muti-electron atom, radiation is emitted as an
characteristic X ray photon.
 It has the energy E (x-ray) = Eu − Eℓ.
An electron from higher shells will fill an inner-shell vacancy
at lower energy, after that vacancy has been created by any
sufficiently energetic process in the first place

41
Now with a generalization of Bohr’s H-model, we have a “reasonable” theory
of matter (atoms) that can to some extend explain this peculiar behavior !!
Because of its
high intensity, it
makes sense to
use only Kα X-
rays (nearly
monochromatic)
for X-ray
crystallography
Kβ less intense
than Kα, more
electrons jump
down from L
shell compared
to M shell, Kβ
more energetic
Kα and Kβ, slightly different An
Why not Hg?

42
Two fitting constants, with a good
theory for atoms, we should be able to
explain them both, right now we only
have a somewhat reasonable theory for
the hydrogen atom and the one
electron left ions
So we will only be partially able to find
meaning for the two fitting constants!
Full quantum mechanics will be needed
eventually.
Strange: R is back for all atoms,
originated with hydrogen

43
Atomic Number Z
L shell to K shell Kα x-ray
M shell to K shell Kβ x-ray
 Atomic number Z = number of protons in the nucleus (6 for
carbon, 7 for nitrogen)
 Moseley found relationships between the frequencies of the
characteristic x-ray and Z.
This holds for the Kα x-ray.
Why is b = 1 for Kα? There are two electrons in the lowest orbit in all elements
except H, one is left and shields the nucleolus if the other is removed

44
Moseley’s Empirical Results
 The most intense Kα X-ray is produced by transition from n = 2 to
n = 1
Moseley’s research clarified the importance of the electron shells for
all the elements (except hydrogen which has only 1 electron) and
beyond Mg (Z = 12) for practical measurement reasons.
For a vacancy in the K shell to be filled by an electron from the L
shell (and assuming that two electrons can occupy the K shell at
most), the electron that “jumps down” sees the nuclear charge
reduced (shielded) by the 1 remaining electron
valid for whole Kα series, R is
again Rydberg’s constant, as this
is a “modified Rydberg equation”,
which can be explained on the
basis of the Bohr-shell-model of
multi-electron atoms and the idea
of shielding by a single remaining
electron in the K shell

45
Empirically for L series, Lα
most prominent
Note that there are 8
electrons in the fully filled L
shell, we will later on
understand how come
Sulfur, Z = 16
Note that the ratio of the radii is not correct
9 if “further generalized” Bohr H model would work
For Kα (and Kβ)
For K and L series

46
There are also other processes, emission of Auger electrons to reduce an exited
atom’s energy, again what we detect is quantized energy levels in bound systems,
these spectra test the energy of the outermost electrons, so are sensitive to the
valence in compounds, ultra clean surfaces, i.e. ultra high vacuum is required

47
R ≈ R0 A0.3333 with R0 = 1.2 10-15 m and A = Z + N
De Broglie wavelength of α-
particles very small, so we can
ignore it, no h in these
equations

48
particularly important for Kα
lines, then b = 1
Better to call it E1
also for He+. Li++
Discussed in last chapter

49
1,000,000,016,640, one trillion, sixteen thousand six hundred
and forty Pennies [one cube measuring 273 x 273 x 273 feet ]
http://www.kokogiak.com/megapenny/fourteen.asp
the “war on terror”
has cost about $1
trillion so far
(2010), but as
wars go that one
accounts only for a
few % of the
annual GDP since
9/11
A real
quantum
jump would
be adding
or taking
away one
penny from
such a
cube
http://www.fas.or
g/sgp/crs/natsec
/RS22926.pdf

50
$1 million
$100,000
other cubic stacks of pennies, always remember what a quantum jump
really is, Francis Scott Fitzgerald to Ernest Hemmingway: “the rich are
different from you and me”.

51
A real
quantum
jump would
be adding
or taking
away one
penny from
such a
cube
http://www.kokogiak.com/megapenny/seventeen.asp
1,000,067,088,384,000, One quadrillion, sixty-seven billion, eighty-eight million,
three hundred and eighty-four thousand Pennies, [ One cube measuring 2,730 x
2,730 x 2,730 feet ]
Sure “quantum
jumps” in
macroeconomic
and “bankstering”
are not based on
the quantum
jumps physicists,
other natural
scientists, and
engineers are
talking about

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