Download Bohr`s atomic model: the evolution of a theory

Bohr’s atomic model: the evolution of a theory
Dieuwke Hupkes
5652936
supervisor: Dhr.prof.dr. A.J. Kox
second supervisor: Dhr.prof.dr E.P. Verlinde
7-09-2010
Abstract
For a long time no scientist believed in the existence of atoms. This
changed with experimental discoveries like spectral lines. After Thomson’s plumpudding model and the saturnian model of Rutherford, Bohr
introduced the quantum in his model. He used Planck’s quantum hypothese, imposing a condition on a saturnian system that further abided
the classical mechanic rules. He created a quantized one electron atom
model. Such a system only emitted energy while passing from one state
to another. The energies determined by Bohr’s quantumcondition agreed
with experimental values. Sommerfeld extended Bohr’s theory for more
complex systems with more electrons and more degrees of freedom. The
old quantum theory ended when it became clear that only ad hoc conditions were not enough, but that a new kind of mechanics was needed.
1
Contents
1 Introduction
3
2 Spectral lines and the discovery of the electron
4
3 Models
3.1 Thomson and Rutherford . . . . . . . . . . . . . . . . . . . . . .
6
6
4 Bohr
4.1 Line spectra . . . . . . . . .
4.1.1 Justification 1 . . . .
4.1.2 Justification 2 . . . .
4.1.3 Justification 3 . . . .
4.2 The Rydberg constant . . .
4.3 General acceptation of Bohr
. . . .
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5 Sommerfeld and the old quantum theory
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6 Afterword
13
7 References
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2
1
Introduction
The old-Greek philosophers were the first of whom we know they were thinking
about the construction of matter. Most people will know about the thoughts
of Empodocles (490-430 vc). He thought there were 4 essential elements: air,
fire, water and earth. He held these elements indestructible and unchangeable.
Others didn’t think there were such elements at all, they held matter to be
infinitely divisible and continuous. Some of the old Greek had beliefs closer to
present theories. It was in their writings that the word atom first appeared.
They considered the atom the smallest element of which our physical world
is constituted. However, although some people shone their light on the basic
structure of matter, the real discussion about it was remarkably mute in the early
physics. Most physicists refused to believe that our world could be constituted
of small particles that they could not see. The opinion that matter was infinitely
divisible and continuous prevailed until the nineteenth century. Experimental
discoveries made the former theory on the construction of matter less plausible,
and the belief in the existence of atoms became more common, although a
discussion about the constitution of these smaller elements still had not taken
place. This started to be a more debated topic after the discovery of spectral
lines.
This paper tries to describe the path that led to the older quantum theory.
The next section explains how the discovery of the electron followed from the
observation of spectral lines. After that, the very first atomic models and their
difficulties are discussed in section 3, specifically those of Thomson and Rutherford. The most important section of this paper is section 4. In this section
not only Bohr’s atomic model will be described, but also his justifications and
explanations for the steps taken. The last section briefly adresses Sommerfeld’s
extensions of Bohr’s model.
3
2
Spectral lines and the discovery of the electron
Thomas Melvill was the first to observe discrete spectral lines 1 . He thought he
discovered that kitchen salt emits monochromatic light when held in a flame.
In fact, this light is not monochromatic (no single atomic-spectrum can be
monochromatic), but merely a really intense spectral line in the sodium spectrum. These days this line is known as the D-line, which is actually a doublet,
a pair of lines really close together. The next step forward was of the greatest
importance. Bunsen, Heidelberg and Kirchhoff observed that each chemical element has its own unique spectrum. This knowledge led to the discovery of many
new elements in the following period of time. Now it could also be analyzed
which elements the sun contained, for example. After Ångström measured the
wavelenghts of the four known spectral lines, scientists were looking for a mathematical regularity. Ritz already figured out that by adding up two spectral
lines another could be found. This principle is called the combination principle
of Ritz. In 1895 Balmer published his paper on the spectral lines of hydrogen,
in which he showed that the then known spectral lines of hydrogen could be
expressed as a difference of two terms, both containing an integer.
The formula he found for the wavelengths (λ) of the spectral lines (in mm)
was:
λ=h
m2
m2 − 22
(1)
With h = 3645.6 x 10−7 and m = 3, 4, 5, 6, this formula perfectly matched
the known hydrogen lines. What he did was amazing, because only four hydrogen lines were known, and this formula proved to be correct! Shortly after
Balmer found his formula, a colleague and friend told him that twelve more
lines were found. Balmer checked them in his formula and these too (with m =
7, 8 etc) fitted.
More spectral lines could be found by subtracting numbers larger than 22 .
The generalization of Balmers formula reads 2
1
1
ν=R
−
(2)
n2
m2
The constant in this formula, now known as the Rydberg constant, became
a popular subject of experiments.
The discovery of the spectral lines was so important for the atomic theory
because this proved that an atom in fact had to have an internal structure.
This internal structure should be able to explain the spectral lines. However,
the physicists were still in the dark, because they did not have the slightest idea
what this internal structure should look like.
1 more
detailed information can be found in Pais (1995)
(1985) p.43
2 Heilbron
4
This changed when Zeeman discovered that the spectral lines of an atom
split when the atom is held in a magnetic field. Lorentz could explain this with
the force he introduced in 1892. This so-called Lorentz force acts on moving
charged particles in a static magnetic field. From Zeemans measurements he
concluded that an atom contained negatively charged vibrating particles with
a very small mass. The measurements of Zeeman could be used to find the
e/m-ratio of these particles, this ratio was a hot topic in the last years of the
19th century. In 1897 Thomson showed that the unknown negative particles in
cathode rays had the same e/m-ratio as Lorentz found. The electron was born.
5
3
Models
Although the discovery of the electron was a huge step forward, it still left many
questions unanswered. Known was, for example, that an atom was neutral in
charge, so what was responsible for the positive charge? Another important
question was the number of electrons in an atom. Electrons were assumed to
be the cause of a huge number of spectral lines. Because every spectral line
should correspond with an oscillating electron, the number of electrons was at
first assumed to be huge (in retrospect), about a 1000 times the atomic weight.
The third difficulty was the stability of the system, why should a system of
oscillating and therefore radiating electrons not collapse? A lot of physicists
were working on the atomic theory, but because their models often differed only
a little bit, only 3 main models will be discussed in this paper. In this section
the work of Thomson and Rutherford will be addressed. Because the main goal
of this paper is to make clear which process Bohr went through, his work wil be
elaborately reviewed in the next paragraph.
3.1
Thomson and Rutherford
Thomson’s work meant a great deal to the atomic theory, mostly because he
”initiated a promising research program” 3 . His model of the atom, called the
plumpudding model, was not very precise. A diffuse sphere without any mass
was responsible for the positive charge. A great many electrons oscillate in
this sphere around a dynamical equilibrium. Because of the huge number of
electrons, the radiation loss is negligible in this situation. Thomson’s atom is
therefore stable. Scattering experiments by Barkla, Rutherford and Thomson
himself produced evidence that the number of electrons in an atom is of the order
of its atomic weight. Thomson deduced the number in his model. Consequently,
several problems arose. First of all, the small number of electrons (of which the
mass was known to be very small) led to the assumption that the weight of
the atom lay in the positive component. Thomson had to reject his massless
positive sphere. The second problem was, that now not enough electrons were
available to account for all the spectral lines. Finally, with this little electrons,
the radiation loss is not negligible at all. This makes the atom collapsing instead
of stable.
Rutherford decided to take another path. The experiments on alpha-scattering
proved to him that an atom was nearly completely empty. He turned back to
a saturnian model with a massive nucleus and electrons circling around it. In
such a system he could calculate energies, frequencies and diameters by use of
classical formulas.
3 Heilbron
(1977) p.53
6
4
Bohr
Bohr was the man who eventually made huge progress in the search for the
model of atoms. In July 1913 he published his first paper on the hydrogen
atom. There was already evidence that a hydrogen atom contained only one
electron. Inspired by Rutherford, Bohr started with calculations on a saturnian
model. The nucleus was assumed to be very small and of charge +e, the electron
orbiting around it of charge −e. In this calculation it is assumed that the mass
of the electron is negligible (in comparison to that of the nucleus) and that the
velocity of the electron is small compared with the speed of light4 .
For the electron to stay in its orbit, the electrical force should be equal to
the centripetal force.
mv 2
e2
=
= mr(2πf )2
(3)
2
r
r
The total energy of the electron is the sum of its kinetic and potential energy.
1
e2
e2
e2
e2
+ mv 2 = − +
=−
(4)
r
2
r
2r
2r
Combining these two formulas results in an expression for both the frequency
of revolution and radius of the system, depending on the total energy.
√
3
2e2
2 |W | 2
√
,
r
=
(5)
f=
π e2 m
|W |
W = Epot + Ekin + = −
If the electron had radiated energy, W would not be constant. The frequency
could not be constant either, neither could the radius. It is now clear to see
that if this were the case, it would be impossible for the electron to describe
a stationary orbit. Atoms, howerever, seemed to have strict dimensions and
frequencies, what does not correspond to the model described above. Bohr
made a very daring and surprising assumption:
Let us at first assume that there is no energy radiation. In this case
the electron will describe stationary elliptical orbits’ 5
He did not reject the saturnian model, but he rejected the mechanical world
in which it was living! The new situation was a nucleus with an electron,
now orbiting in certain possible stationary states with certain corresponding
energies (although he did not know which states existed). The very important
next step he made, was using the quantum hypothesis Planck postulated in
1900. This hypothesis said that the energy of charged oscillators is quantized.
The oscillator can only emit or receive amounts of energy equal to an integer
multiple of a constant times the oscillation frequency of the oscillator: E = nhν.
4 Bohr(1913)
5 Bohr
p.3
(1913), p.3
7
The constant h is now called the Planck constant. At the moment he postulated
his hypothesis nobody (not even Planck himself) saw the huge implications of
Plancks discovery. It was Bohr who made a very important next step. He linked
the differences in energy between the different stationary orbits to differences in
frequency. The way he did this is once again remarkable, his second assumption
was:
Let us now assume that: during the binding of the electron, a homogeneous radiation is emitted of a frequency ν, equal to half the
frequency of revolution of the electron in its final orbit (my italics)6
He did this without any further justification, just because it seemed to work.
Although Bohr came up with some possible explanations, it is important to
see that these explanations were not the reason Bohr came up with the factor
1
2 in the first place. To implement his postulated relation between radiation
frequency and frequency of revolution Bohr makes another, not implausible
assumption: following Planck he takes the energy of the radiation that is emitted
when an electron goes from a free state to a bound state to be equal to nhν,
with n an integer and ν the frequency of the radiation. Thus:
W = nhν
Using his assumption for the relation between f and ν Bohr then finds:
f
(6)
2
(I stuck with the symbol f for frequency of revolution, like I did before). With
the expressions for frequency and energy he found before, he could now get a
series of values for W , f and r, corresponding to a series of configurations of
the system.
W = nh
W =
4.1
2π 2 me4
n2 h2
f=
4π 2 me4
n3 h3
2r =
n2 h2
2π 2 me2
(7)
Line spectra
This system can emit energy (because of the spectral lines), but it does not do so
in a stationary state. It radiates when it passes between the different stationary
states. Therefore, the frequencies (or energies) emitted by this system, are
the energy (frequency) differences between the different stationary states! An
expression for these differences follows from formula 7.
1
2π 2 me4 1
− 2
(8)
W n2 − W n1 =
h2
n22
n1
The now found energy differences should, according to Planck, be equal to hν.
The emitted frequencies then are:
6 Bohr
(1913 p.4
8
ν=
2π 2 me4
h3
1
1
− 2
2
n2
n1
(9)
Putting n2 = 2 and varying n1 , he got the well-known experimental Balmer
formula 2! He obtained a value for the Rydberg constant in terms of atomic
2
4
constants: R = 2π hme
. The agreement between the value he found and the
3
experimental value was within the experimental marges of uncertainty. Bohr
could now also explain why only some of the lines of the Balmer series could
be observed. According to 7, the radius of the atom is proportional with n2 ,
therefore it is possible to find an n, for which the diameter of the orbit is of the
dimension of the distance between the molecules in a gas. To observe spectral
lines with a high n, the gas has to be very dilute in order to keep the distance
between the molecules large enough. This explains why spectral lines that can
not be observed on earth, sometimes can be observed in stars: the pressure
there is much lower, as is the density. These facts indicated strongly that Bohr
was on track. But he still needed a justification of equation 6. He gave three.
4.1.1
Justification 1
The first justification he gave, was the following: the radiation emitted during
the passing of the system must be equal to an integer multiple of hf . Instead
of 6 this gives W = x(n)hf , and the expressions for W and f , instead of 7,
become:
W =
π 2 me4
2h2 x2 (n)
f=
π 2 me4
2h3 x3 (n)
And instead of 9, the formula for the allowed frequencies becomes:
π 2 me4
1
1
ν=
−
2h3
x2 (n1 ) x2 (n2 )
(10)
(11)
To get an expression of the form of the Balmer series, x(n) has to be of the form
x(n) = cn. It is now possible to determine c by considering the passing of the
system between 2 stationary states n1 and n2 .
n1 = N and n2 = N − 1 gives:
π 2 me4
1
1
π 2 me4
2N − 1
ν=
−
=
(12)
2h3 c2 N 2
(N − 1)2
2h3 c2 N 2 (N − 1)2
For large N, this is approximately equal to
ν=
π 2 me4
π 2 me4 2
=
2h3 c2 N 3
h3 c2 N 3
Using formula 10, he could find the ratio between the frequency before and after
the emission.
9
fN
fN −1
=
(N − 1)3
N3
(13)
For large N , this will be very close to 1.
The ratio between the frequency of radiation emitted and the frequency of
revolution, should then according to ordinary electrodynamics also be very close
to 1, that is ν should be close to equal to f . Since (still in the limit for large
2
4
π 2 me4
1
N ) f = 2cπ3 hme
3 N 3 , and ν = c2 h3 N 3 , this condition will only be satisfied if c = 2 .
4.1.2
Justification 2
The second solution Bohr supplied was that the mean frequency with respect to
the nucleus should be used. Because the initial frequency is zero, the average of
both frequencies, like in the formula, is half the frequency at which the electron
is orbiting.
4.1.3
Justification 3
Bohrs hypothese put a condition on the possible stationary states, this condition
can be translated in a quantum number that gives the allowed angular momenta
of the electrons. The angular momentum of an electron (L), expressed in kinetic
kin
. In a circular orbit,
energy (Ekin ) and frequency of revolution (f ), is L = Eπω
Ekin = W . Using condition 6, the expression for the angular momentum is
nh
(14)
2π
Denoting it this way, stationary states of an electron can be described by a
quantumnumber n.
L=
4.2
The Rydberg constant
Although his justifications were really ad hoc, Bohrs theory gained ground by
giving a right value for the Rydberg constant. As said before, with his theory
he could express this constant in terms of atomic constants. Besides that, he
also proved that this constant is the same for all elements.7 For that he used
the most general formula for spectral lines (i.e. this formula isn’t restricted to
hydrogen lines):
ν = Fr (n1 ) − Fs (u2 )
With n1 and n2 entire numbers, K a universal constant, equal to the factor outside the bracket in formula 9, and F1 , F2 , F3 ... functions of n equal to
K
K
(n+a1 )2 , (n+a2 )2 , .... First he remarks that this formula is again a difference between two entire functions. This suggests that not only the sprectral lines of
hydrogen, but all spectral lines are caused by elements passing between their
7 Bohr
p.11
10
∼ k2 . Formula 7 gives another exdifferent stationary states. Now for large n, F =
n
2π 2 me4
2
pression for F , namely F = W
,
for
K
this
yields:
K = F ·n2 ∼
.
=W
h
h ·n =
h3
The limit for n is large gives thus always the same value for the Rydberg constant.
4.3
General acceptation of Bohr theory
There was one series of spectral lines that caused Bohr trouble, the Picker1
ingseries: νn = R( 14 − (n/2)
2 ). In Bohrs formula there was no place for half
integers, because no half quanta could be emitted. Bohr fitted them in his
theory, by rewriting the series as ν = 4R( 412 − n12 ) and attributing it to ionized helium. With a nuclear charge of 2, the Rydberg constant should be four
times that of hydrogen. The Balmer formula did agree very well with the experiment, the Pickering formula not so much. Bohr was very inventive again
and found a way to correct Pickerings formula. He said to have neglected the
small motion of the nucleus, the mass m should be replaced by the reduced
m
. He required: RHe \RH = 4.00163. The experimental ramass m0 = mmZZ+m
tio found was 4.0016. Although Thomson still wasn’t convinced, Bohrs theory
fastly gained ground. Einstein said to be astonished by this fact, and he called
it ’an enormous achievement’ (p47 Heilbron 1).
11
5
Sommerfeld and the old quantum theory
Bohr laid the foundation for what is now called the old quantum theory: the
principle that motion in an atomic system is quantised. His theory was, however, far from perfect. Not only did his theory only work for hydrogen(like)
atoms, with one electron rotating in a circular orbit, but his theory was also
unable to explain experimental results like the Stark effect or Zeeman effect.
It became clear, that more degrees of freedom were needed. Sommerfeld used
Bohr’s theory and extended it, resulting in new quantum states, now called the
Bohr-Sommerfeld quantisation rules.
I
I
I
pφ dφ = nh
pr dr = n1 h
pθ dθ = n2 h
(15)
With two new quantum numbers, more states were possible. Selection rules
(which described which combinations of quantum numbers were allowed), stated
which transistions were allowed. Experimental measurements showed that each
of the known spectral lines were, in fact, a few adjacent lines. Sommerfeld’s major succes was that he was able to explain this so-called fine structure of spectral
lines. The new transitions that could be found with his selection rules were in
agreement with the measurements on fine structures. Scientists kept extending
Bohr’s (and Sommerfeld’s) theory, by introducing more quantum numbers and
methods to generalize quantum conditions. An important example of such a
method is the adiabatic theorema of Ehrenfest, that states that an adiabatically transformed quantized system remains in its quantized state. With this
method, it possible to determine the allowed motions of any periodic system of
one degree of freedom if it is adiabatically related to the harmonic oscillator8 ,
by finding an adiabatic invariant I. The quantum condition for the new system
then reads I = nh. A lot of new quantum conditions could be found for many
new systems, but they were still conditions imposed on classical mechanics systems. The conditions were ad hoc and there was no physical explanation for
any of them. That’s where the older quantum theory stuck. In fact, a new
kind of mechanics with new formalisms was needed: quantum mechanics. This
new mechanics started to develop in the early 1920s, with famous names like
Schrödinger and Heisenberg.
8 Heilbron(1977)
p.99
12
6
Afterword
This paper is not meant to be a mere description of the development of the
atomic model, but above all it is meant to give an insight in how science moves
forward. Studying physics in high school or at a university sometimes gives
the wrong impression of how a new theory develops. I saw that very clearly
when I was writing this paper. In the textbooks I read and learned from,
scientific processes seem to be very logical, straightforward and smooth. I did
not learn about the dead ends that at first were so promising. What Bohr did
was very unlogical then, there weren’t more arguments than the right value of
the Rydberg constant he found. In order to progress, it is important that there
are people who dare to make assumptions no one believes in, even if they prove
to be wrong. Science isn’t a straight line, there are many dead ends along the
way. Maybe we are heading for a dead end now. But it is also possible that we
develop theories that people will marvel at in a hundred years.
13
7
References
1. Bohr N. On the constitution of atoms and molecules. In: Philosophical
Magazine; Series 6, Volume 26; 1913. p. 1-25.
2. Brown LM, Pippard B, Pais A, editors. Twentieth Century Physics. Taylor & Francis; 1995. p. 43-141.
3. Heilbron J. Bohr’s first theories of the atom. In: Physics Today; Volume
38, Issue 10;1985. p. 28-36.
4. Heilbron J. Lectures on the history of atomic physics 1900-1922. In: C.
Weiner (Ed); History of twentieth century physics (= Rendiconti della
scuola internazionale di fisica Enrico Fermi 57); New York & London:
Academic Press, 1977. p40-108.
5. Jammer M. The conceptual development of quantum mechanics. New
York: McGraw-Hill; 1966. p.96.
6. Krach H. Quantum Generations, a history of quantum mechanics. Oxford:
Clarendon; 1981.
7. Pais A. Niels Bohr’s times in physics philosophy and polity. Oxford: Oxford university press; 1991.
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