Download 1. dia

Atomos = indivisible
The structure of the atoms
‘All that exists are atoms and empty space;
everything else is merely thought to exist.’
Democritus, 415 B.C.
University of Pécs, Medical School,
Dept. Biophysics
November 2013
Joseph John Thomson
Thomson model (1902)
The atom models
Atoms are stable
Their chemical properties show periodicity (Mendeleev 1869)
After excitation they emit light, and their emission spectra is linear
Johann Jacob Balmer’s
empirical formula (1885):
1

1 1 
 R  2 
4 n 
n: 3,4,5……
R: Rydberg constant
1897 - electron
„Plum pudding”
Ernest Rutherford
Rutherford model (1911)
(R = 10 973 731.6 m -1)
Rutherford’s conclusions
1. The majority of matter is „empty space”!
2. The positive charge is concentrated into a tiny space (nucleus ~10 -15 m).
3. Electrons are revolving around the nucleus, like planets around the Sun.
1
The conclusions of the Bohr’s model
Niels Bohr
Bohr’s model
1. Radius of the 1st orbit: r1 = 5.3 ·10-11 m (Bohr-radius)
Bohr’s postulates:
1. Electrons in an atom can only have defined orbits. The formula defining
the radius of the allowed orbits is:
r2 = 4r1, r3 = 9r1…..
2. Energy of the first orbit: E1 = -13.6 eV (because it is bound)
E2 
h
L  mrv  n
2
rn = n2r1
E1
4
E3 
E1
9
En 
E1
n2
Stationary wave!
2r  n  n
2.
h
mv
When the electron jumps from one allowed orbit to another, the energy
difference of the two states is emitted as a photon with the energy of hν:
hv  E2  E1
The proof of the Bohr’s model
The Frank-Hertz experiment
Quantum mechanical atom model
Matter wave – wave
function ()
Described by the
Schrödinger’s equation
Probability of
occurrence of an
electron:  2
Atoms can absorb only precisely given amounts of energy. The Hg
atoms e.g. 4,9 eV. The 4,9 eV is equals to the energy difference
between the ground state and the first excited state of a Hg atom.
Heisenberg’s uncertainty
principle (1927)
It is impossible to precisely determine the position and the momentum of the
particle at the same time. The multiplication of the uncertainty (error) of two
measurements at the same time is always higher than h / 2 :
The position of the ground state electron of a hydrogen
atom, around the nucleus.
The density of the spots is proportional to the finding
probability of the electron.
The graph shows Ψ2 in the function of the distance
measured from the nucleus.
An example to the Heisenberg’s
uncertainty principle
The Large Hadron Collider ( LHC ) at CERN will be accelerating protons close to the speed of light,
C, whose rest mass is
Before achieving smashing protons at close to C,
let's suppose that the protons are speeding at
with a 1% measurement precision or
x  p x 
h
2
Therefore, the uncertainty in measurement of proton velocity is
and by the Heisenberg Uncertainty Principle, the uncertainty in simultaneously determining proton velocity and position is given as follows:
The relation gives a limit of principle: the multiplication of
the measured uncertainty of the two quantities can not be
smaller than h / 2.
http://www.relativitycalculator.com/Heisenberg_Un
certainty_Principle.shtml
2
Quantum numbers
Quantum numbers
Quantum numbers describe values of conserved quantities in the dynamics of
the quantum system. They often describe specifically the energies of
electrons in atoms, but other possibilities include angular momentum, spin
etc.
It is already known from the Bohr’s atom model that the energy of the
electrons is quantized so they can have only one value. The energy values
are determined by the n principal quantum number.
The quantum mechanics is proved that there are sublevels of the given
energy levels that is why the n principal quantum number is not enough and
more other quantum numbers are needed.
The principal quantum number (n)
It is known that the principal quantum number defines the energy, and an energy
value belongs to every n value ( n
En ). The electrons with given n values are
forming shells which are named with K, L, M, etc. letters. There can be more other
states inside a shell which states are determined by the orbital quantum number.
Bohr had predicted the positions of orbits with amazing accuracy but did not take
count that this is not the only position of electron, this is the place where the
electron can be found with the highest probability.
Quantum numbers
Quantum numbers
The orbital quantum number (l)
The orbital quantum number (l)
It defines the magnitude of the angular momentum of an electron.
It defines the magnitude of the angular momentum of an electron.
Angular momentum:
The angular momentum of a body which is revolving around an r radius orbital
with v speed is a vectored quantity.
Its value is L = mvr.
Its direction is perpendicular to the
plane of the velocity.
Its value is L = mvr. Its direction is perpendicular to the plane of the
velocity.
The angular momentum resulting from
the movements of the electrons on
their orbital can only be:
L  l (l  1)
h
2
where h is the Planck constant and l is the orbital quantum number, which can be
an integer between 0 and n-1.
Sample calculation:
The Moon:
Mass = 7.344·1022 kg
Average orbital speed = 1.025 km/s
Distance from Earth (average) = 384 400 km
Angular momentum = ?
kg·m 2/s, vagy N·m·s
Example: n = 2; l = 0 (2s state): L = 0
h
l = 1 (2p state): L  2 2
Quantum numbers
The magnetic quantum number (m)
It defines the direction of the angular momentum of an electron. That is why the
angular momentum can be set only in given directions. The projection of the
angular momentum on the direction of the outer magnetic field can only be:
Lz  m
h
2
where m is the magnetic quantum number which values are whole numbers
between -l and +l. This determines the direction of the angular momentum
definitely.
How can it define the angular momentum:
Example: if n = 2; l = 0, 1; m = -1, 0, +1
Zeeman effect I
When an atom turns from an
initial higher energy level to
a stationary level with lower
energy then the energy
difference can be emitted as
a photon. This may gives a
line in the visible spectrum.
In the presence of an external magnetic field, these different states will
have different energies due to having different orientations of the magnetic
dipoles in the external field, so the atomic energy levels are split into a
larger number of levels and the spectral lines are also split. The rate of split
is proportional to the applied magnetic field. The new lines appear
symmetrically on the right and on the left side of the original line. This is
the so-called Zeeman effect (normal Zeeman effect).
3
Quantum numbers
Quantum numbers
The magnetic spin quantum number (ms)
The spin quantum number (s)
It defines the value of the spin angular momentum of the electron. It is
imagined as the electron (like the Earth) not just revolving around its orbit
but it is spinning around its own axis. The electron’s own angular
momentums can only be:
S  s( s  1)
h
2
It defines the direction of spin angular momentum of an electron.
The projection of the angular momentum on the direction of the outer
magnetic field (z) can only be:
S z  ms
where s is the spin quantum number. The spin quantum number can only be
½. It does not defines other sublevels.
Zeeman effect II
h
2
where ms is the magnetic spin quantum number which is ½ or -½, so the
spin (owned angular momentum) can be set only in two directions.
The Stern-Gerlach experiment
As an atom is ”placed” into a magnetic field each of
its fine structure lines further splits into a series of
equidistant lines with a spacing proportional to the
magnetic field strength.
For some atoms, the spectrum displays a more complex pattern of splittings. That is the
so-called anomalous Zeeman effect.
In these cases, it is found that the number of Zeeman
sub-levels is actually even rather than odd. This cannot be
explained within the normal Zeeman theory. However, it
suggests the possible existence of an angular momentum like
quantity. The proximate proof for the evidence of spin angular
momentum was found by the Stern-Gerlach experiment.
The Stern-Gerlach experiment
Conclusions:
1922
1st The experiment proves that the angular momentum is quantized.
2nd Why is the beam deflected into two beams?
if l=0 => m=0 => no deflection
if l=1 => m=0,  1 => deflects into three beams
(that is why a two-beam deflection can not caused by direction of the angular
momentum)
Phipps and Taylor reproduced the effect using hydrogen atoms in their ground
state in 1927.
1925 – Uhlenbeck and Goudsmit formulated their hypothesis of the existence of
the electron spin.
The Stern-Gerlach experiment involves sending a beam of particles
through an inhomogeneous magnetic field and observing their deflection.
The particles passing through the Stern-Gerlach apparatus are deflected
either up or down by a specific amount. This result indicates that spin
angular momentum is quantized (it can only take on discrete values), so
that there is not a continuous distribution of possible angular momenta.
The Einstein-de Haas effect
A freely suspended body consisting of a ferromagnetic material
acquires a rotation when its magnetization changes.
Because of the change of the
external
magnetic
field
mechanical rotation of the
ferromagnetic
material
is
happened associated with the
mechanical
angular
momentum, which, by the law
of conservation of angular
momentum,
must
be
compensated by an equally
large and oppositely directed
angular momentum inside the
ferromagnetic material.
4
Quantum numbers
Quantum number
Quantum numbers
Symbol
Quantized value
Values
Principle
n
Energy
1,2,3…
Orbital
l
Value of angular momentum
0,1……n-1
Magnetic
m
Direction of angular
momentum
-l, -l+1…0…l-1, l
Spin
s
Value of own angular
momentum
½
ms
Direction of own angular
momentum
http://dilc.upd.edu.ph/images/lo/chem/Quantum/quantum.swf
Magnetic spin
–½, +½
5