Niels Bohr in England

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30 Οκτ 2013 (πριν από 4 χρόνια και 14 μέρες)

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Niels Bohr in England

In 1911 Niels Bohr went to Cambridge to study with J. J.
Thomson. Unfortunately, by the time Bohr arrived,
Thomson had no spare time to think about electrons.
Bohr kept himself busy writing a paper on electrons in
metals, reading Dickens to improve his English, and
playing soccer.

In December, Rutherford came down from Manchester for the
annual Cavendish dinner. Bohr later said that he was deeply
impressed by Rutherford's charm, his force of personality, and
his patience to listen to every young man who might have an
idea
--
certainly a refreshing change after J. J.!

At Manchester, he talked a lot with Charles Galton
Darwin
-
"grandson of the real Darwin". Darwin had just
completed a theoretical analysis of the loss of energy of
those


-
particles that do not scatter from nuclei while
going through matter (i.e. the bulk of them).

Such

's gradually lose energy at a rate that depends on
how many electrons they encounter. In particular, Bohr
concluded, after reviewing and improving on Darwin's
work, that the hydrogen atom almost certainly had a
single

electron outside the nucleus.

Niels Bohr in England

Can atoms exist at all?

A big problem with the nuclear hydrogen atom was:
how could it exist at all? Classical electrodynamics
predicts that accelerating charges radiate energy (em
waves). As the electrons lose energy they should slow
down and their obits should decay…

Nuclear atom size?

Another problem with the nuclear hydrogen atom was:
what determined its size? Classical mechanics gives a
simple dynamical equation for circular orbits:

However,

any
elliptic orbit centered at the nucleus, however
large or small will satisfy the above equation. There is no hint
here that the atom in its "natural" ground state should have
any particular radius. We're missing
something
. But what?


Stationary States

Since the black body oscillators (harmonically oscillating
atoms or molecules in the oven walls) were of the same
general size as atoms, Bohr (and others) thought Planck's
constant must play a role in determining orbit size as it did in
restricting allowed oscillations for black bodies.

The classical picture of how an oscillating charge radiated
couldn't be right at the atomic level. Bohr concluded that
in an atom in its natural rest state, the electron must be in
a special orbit, he called it a "stationary state" to which
the usual rules of electromagnetic radiation didn't apply.
In this orbit, which determined the size of the atom, the
electron, mysteriously, didn't radiate

Bohr’s prediction of atom size

The parallel approach for the hydrogen atom was to
identify the frequency
f
with the circular frequency of the
electron in its orbit. However, in contrast to the simple
harmonic oscillator this hydrogen atom frequency
varied

with the size of the orbit. Using the expression above:

Therefore, if the electron has a kinetic energy
E
, the potential
energy was
-
2
E

and the total energy
-
E
.

K.E.

P.E. x (
-
½)

Bohr’s prediction of atom size

If a hydrogen nucleus captured a passing electron into
its ground state, and emitted one quantum of
electromagnetic radiation, that quantum would have
energy
E
, the same as the electron kinetic energy in
the natural stationary state (called the
ground state
).

H nucleus

Free electron
Potential
Energy = 0

Captured
electron:
Potential
Energy =
-
E

Quantum of light with
Energy = hf = E to
offset energy lost by
electron upon
capture.

The Bohr atom

Bohr suggested in a note to Rutherford in the summer of
1912 that requiring this energy be some constant (~1)
multiplied by
hf

would fix the size of the atom. In
February
1913, Bohr was surprised to find out about
Balmer’s

formula for predicting visible lines of hydrogen in a casual
conversation with the
spectroscopist

H. R. Hansen. They
had very likely seen this in class together years before, but,
given Bohr's opinion that such spectra were pretty but not
important, he hadn't paid attention.


Bohr said later: "As soon as I saw
Balmer's

formula, the whole
thing was immediately clear to me."

The Bohr atom


The set of allowed
frequencies
(proportional to
inverse wavelengths) emitted by the hydrogen atom
could all be expressed as
differences
.


This immediately suggested a generalization of the
"stationary state", in which the electron did not radiate.


There must be a
sequence

of these stationary states,
which radiation only taking place as the electron
jumps from one to another emitting a single quantum
of frequency
f, such that

hf = E
n

-

E
m

The Bohr atom

The stationary states in the Rydberg
-
Ritz formula, could
be labeled 1, 2, 3, ... ,
n
, ... and had energies
-
1,
-
1/4,
-
1/9, ...,
-
1/
n
2
, ... in units of
hcR
H

(using
l
f = c).
The
energies are now negative, because these bound states,
have zero energy where the two particles are infinitely
far apart.

Bohr knew that if the energy of the orbit was
-
hcR
H

/
n
2
,
that meant the kinetic energy of the electron, ½mv
2

=
hcR
H
/n
2
, and the potential energy would be



-
(1/4

0
)e
2
/r =
-
2hcR
H
/n
2
.

It follows that the
radius

of the
nth

orbit is proportional to
n
2
, and
the
speed

in that orbit is proportional to 1/
n
. It then follows that
the angular momentum of the nth orbit is just proportional to n
.

Bohr predicts the Rydberg Constant

The Balmer formula led Bohr to the realization that the
angular momentum was quantized:
L

=
Kh
, 2
Kh
, 3
Kh
,
... where
K

is some constant numerical factor ~1. Bohr
gave a very clever argument to find
K

without doing any
experiment

!


Firstly, the magnitude of K affects the
physical
properties of the
hydrogen atom.

For
K

= 1, allowed orbits would be have angular momentum
h
,
2
h
, 3
h
, 4
h
,….

For
K

= 10, allowed orbits would have angular momentum 10
h
,
20
h
, ... . The average
spacing

between spectral lines will be
much
greater

that for
K

= 1. E.g. much fewer lines in visible
spectrum.


Bohr predicts the Rydberg Constant

Next, Bohr did a thought experiment in the spirit of
Einstein. He imagined an
immense
hydrogen atom, an
electron going around a proton in a circle of one meter
radius in the depths of space. For this very large atom,
the electron
moves
slowly. At this scale, we know that
accelerating charges emit radiation according to
Maxwell's equations.

If the electron is going
round at
f

revolutions per
second, it will emit em
radiation at that frequency
f

as the electric dipole, seen
from far away, will oscillate
f

times per second.

+

-

-

Bohr predicts the Rydberg Constant

The angular momentum quantization condition must also
be true for this very large atom and the radiation emitted
must still be given by the difference in energies of
neighboring orbits,




hf = E
n+1

-

E
n
.


But this energy spacing depends on K. Therefore, Bohr
concluded K is fixed by requiring that the frequency of radiation
emitted
in the case of the really large atom

must be the same
as the orbital frequency of the electron.










E
n+1

-

E
n

= hf = h(v/2


r)

Correspondence Principle

Assume that the only allowed orbits are those having
angular momentum integral multiples of Kh, where K is
some constant, so the nth orbit has angular momentum
nKh. We can then use the equation of motion to
determine the radius r
n
, the electron speed v
n

and the
energy E
n

for the nth orbit.

Orbits with r
n

~ 1 m will radiate at the frequency given by the
equations of motion for an electron circling a proton one meter
away. We must find the value of K for which this frequency
matches the frequency given by the energy difference between
neighboring orbits divided by h.

This is called the
Correspondence Principle
: in the limit of
large systems, quantum predictions must correspond to known
classical results.

Bohr predicts the Rydberg Constant

For a large orbit is the
n
th

(
n

~10
5
!) with angular momentum, L

The radii of the allowed
orbits are given by


Bohr predicts the Rydberg Constant

Therefore the allowed energies are:


For
very large n
:


for the appropriate
v
,
r

of this large orbit.


Bohr predicts the Rydberg Constant

Therefore the allowed energies are:


For
very large n
:


for the appropriate
v
,
r

of this large orbit. We need value for K!


Proof that K = 1/2


Using

mv
n
r
n

= nKh,


v
n

= nKh/mr
n
.


Thus

hv/2


r = nKh
2
/2


mr
n
2
.



E
n+1

-

E
n

= hf = hv/2


r = nKh
2
/2


mr
n
2


From before…

From before…

Proof that K = 1/2


Bohr predicts the Rydberg Constant

The allowed energies are now:


The Rydberg

Ritz formula becomes:


Compared to the earlier form…

We get

This formula was found to be correct within the limits of
experimental error in measuring the quantities on the right.


Skepticism for Bohr model

Bohr's interpretation of the Balmer formula didn't make
much mechanical sense. An electron jumping from the
n
th orbit to the
m
th emitted radiation at
f = (E
n
-
E
m
)/h
.

Presumably, it began radiating as soon as it left its
original orbit.

As Rutherford put it in a letter to Bohr, " It seems to me
that you would have to assume that the electron
knows beforehand where it is going to stop."


Another reason for skepticism was the recent discovery
of apparently new spectral lines for hydrogen
corresponding to
half
-
integers in the Balmer formula.

These new lines had been seen in a discharge tube
containing a mixture of hydrogen and helium, and also in
the spectra of a star.

Bohr realized that his formula for the Rydberg constant, should
also apply to the any nucleus with a single electron in orbit around
it, provided
e
2

is replaced by Z
e
2
.

Bohr argued, the “new” hydrogen lines were from He+ because,
for He+, Z
e
2

gives a factor exactly equivalent to replacing the
integers in the denominator of the Balmer formula by half
-
integers.

Skepticism for Bohr model

Confirming Bohr’s ideas

However, the spectral lines were measured to five
significant figures and the ratio of the "Rydberg
constant" for helium to that for hydrogen was
actually 4.0016.


Bohr's then pointed out that he had neglected the finite mass of
the nucleus. He should really have taken the electron to have an
effective mass equal to
mM
/(
m

+
M
). It is easy to verify that if this
correction is made for both hydrogen and helium, the ratio
changes from 4 to 4.0016 (Exercise)!


Einstein said: "This is an
enormous achievement
.
The theory of Bohr must
then be right."

Why does
everyone believe
it when HE says
it?

Thanks
Al!

I vill a little,
think… Ja!
Niels is right!

X
-
ray Spectra

Moseley investigated the x
-
ray spectra of many
elements to pin down their Z values using a cathode ray
tube. He found several different x
-
ray lines for each
element, with very similar patterns for each element,
although gradually shifting to higher frequencies with
increasing
Z
. For example, one particular line, labeled
K


, had a
Z
-
dependent frequency:

X
-
ray Spectra

Bohr's theory was nearly
correct for the inner electrons
of high Z atoms. The theory
worked for these electrons
because they were shielded
form the inter
-
atomic forces.
And because the stronger
coloumbic attraction dominated
the inter
-
electron forces.
However a small amount of
“shielding” occurred (hence Z
-
1 rather than Z).

X
-
ray Spectra

Moseley’s work clearly
separated between
atomic weight and
atomic number and
helped to conform the
validity of the periodic
table of the elements

Franck
-
Hertz Experiment

Electrons of energy
E=qV = eV
0

are used
to excite electrons bound in atoms rather
than light of energy
E=hf
.

Current

n =1

n =2

E < E
2
-
E
1
:
Elastic
Scattering

E = E
2
-
E
1
:
Inelastic
Scattering

Franck
-
Hertz Experiment

As the grid voltage is
increased, the incoming
electrons arrive faster and
the current increases.
Wherever their kinetic energy
corresponds to a level
difference, the target electron
can absorb some of the
incoming electron’s kinetic
energy and the current goes
down because it is only
driven by
D
V
.

This was an independent confirmation of the Bohr Model

Electron Energy Loss Spectra

EELS:

A similar sort
of idea allows
electronic structure
to be probed in
complex materials
(e.g. Solids).

Used in modern
material science.