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Chapter 1
The size of atoms etc.
Introduction to atomic physics where a few important numbers are
recapitulated. The size of the hydrogen atom is calculated with dimensional analysis.
The masses of atoms are close to integer multiples of the unit mass 1.66·10−27
kg. This mass unit corresponds to an energy of
1 uc2 ≈ 109 eV.
(1.1)
The energy unit eV is the kinetic energy of one elementary charge going through
the potential drop of 1 Volt,
1 eV = 1.6201 · 10−19 J.
(1.2)
Atomic nuclei are composed of neutral particles called neutrons that have a
mass which is close to that of the proton, which has a positive charge. The nuclei
are held together by the nuclear (or ’strong’) forces. The charge of a proton is
identical in magnitude to that of the electron to the best of our (humankind’s)
knowledge, and a neutral atom, where the number of electrons matches the
number of protons is therefore completely electrically neutral to the outside
world. One may in fact define the atom as the smallest neutral unit of matter
one can have (ignoring neutrinos).
The mass of atoms are related to human scale masses via Avogadro’s number,
NA = 6.02 · 1023 , which gives the number of atoms in a number of grams equal
to the atomic mass. Thus 12 g of carbon contains NA atoms. The amount NA
is also called a mole. The molar masses found in the periodic table are not
always close to integers. This is because the values are averages over atoms
with different numbers of neutrons but the same number of protons. These are
known as isotopes. The number of protons in a nucleus defines what element
the atom is.
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2
CHAPTER 1. THE SIZE OF ATOMS ETC.
The size of the nucleus is measured in fm = 10−15 m. The size of the atom
is much larger, although still very small on human scale. We can estimate the
effective size of the atoms from properties of macroscopic matter. Densities of
ordinary condensed matter, water for example, is on the order of g/cm3 , and
molar masses are similarly on the order of 10 g, for not too heavy elements.
This gives the atomic size
a≃
1
3 N
1g/cm A
10g
!1/3
= 2 − 3 · 10−10 m.
(1.3)
This is the size of the atom in equlibrium positions in the presence of other
atoms.
We can estimate the size of an isolated hydrogen atom with a simple argument based on the Heisenberg uncertainty relations. The classical Hamiltonian
function is
p2
e2
−
.
(1.4)
H=
2m 4πǫ0 r
Combine this with Heisenberg’s uncertainty relation between position and momentum
~
(1.5)
∆x∆p ≥ .
2
We will now apply this to the hydrogen atom. First we identify ∆x and ∆p with
the quantities themself. Secondly, we replace the ” ≥ ” with an equality sign
in the Heisenberg uncertainty relations. These are reasonable approximations
when we consider properties of the quantum ground state. We then ignore the
fact that the problem really is three dimensional, and finally we idetify the
Hamiltonian with the energy. This all gives
1
2p
=
,
r
~
(1.6)
and
E=
p2
e2 2p
−
.
2m 4πǫ0 ~
(1.7)
e2 2m
.
4πǫ0 ~
(1.8)
2
(1.9)
This energy has a minimum for
p=
Inserted into the energy this gives
E = −2m
e2
4πǫ0 ~
This is four times the authorized value.
= −54.4 eV.
3
In summary:
me
mp
qe
qp
rp
=
1
1800
(1.10)
= 1
(1.11)
≃ 10−5
(1.12)
ratom
We can determine the size of an atom with didmensional analysis. The
relevant parameters are the constants of the Coulomb interaction between two
charges of magnitude e:
e2
,
4πε0
dimension Jm =
kgm3
,
s2
(1.13)
Planck’s constant:
kgm2
,
(1.14)
s
and the mass of the electron (really the reduced mass of the electron and proton):
dimension
~,
me ,
dimension kg.
(1.15)
The job is to form a length scale out of these numbers. With powers α, β,
and γ which are to be determined, we can write
2 α
e
[~]β [me ]γ = m
(1.16)
4πε0
where the square brackets mean ’the dimensions of’. Inserting the dimensions
we get
β
α kgm2
kgm3
kgγ = m, ⇒ kgα+β m3α+2β s−2α−β = m
(1.17)
s2
s
which is a set of three equations:
α+β+γ =0
(1.18)
3α + 2β = 1
−2α − β = 0
(1.19)
(1.20)
Eqs.1.19 and 1.20 give α = −1 and β = 2. Combined with Eq.1.18 we then
have γ = −1. This solution is unique, and gives the length scale of an atom. In
numbers it is
~2
= 0.529...Å.
(1.21)
a0 = e2
4πε0 me
This is the Bohr radius and corresponds to the length scale in the hydrogen
atom ground state. Quantum mechanically a0 is the characteristic distance in
the ground state wavefunction.
The salient point here is that if there were no parameter like Planck’s constant we would have no solutions to the set of equations, and the size of the
atom would be undetermined. This fact was already noted by N.Bohr in 1913.