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Lecture VI
Many - electron atoms
dr hab. Ewa Popko
S-states probability
P-states probability
The Zeeman effect
The Zeeman effect is the splitting of atomic energy and the
associated spectrum lines when the atoms are placed in a
magnetic field. This effect confirms experimentally the
quantization of angular momentum.
potential energy
The potential energy of an object in a magnetic field
depends on the magnetic moment of the object and the
magnetic field at its location
 
U    B


S
N
S
N
magnetic moment of a current loop
The magnetic moment of a wire loop carrying current depends
 
 loop.
a and the area
a A of the
current
 Iin the
 loop
on the
 1  2  3  4  Ib  nˆ  B  Ib  nˆ  B  0  0  IA  B
2 
2
  IA
The Zeeman effect
The orbiting electron is equivalent to a current loop
2
with radius r and area r .
e

The average current I is the average charge per unit
time T for one revolution, given by T=2r/v.
e
 ev 2  e
e
  iA 
A
r 
 me vr 
L
T
2r
2me
2m e
The Zeeman effect
Suppose B is directed towards z-axis. The interaction energy of
the atom magnetic moment with the field is:
U   z B
where z is the z-component of the vector . On the
other hand:
e
z  
Lz
2m
and Lz=ml 
with
m l  0,1,2,3...... Thus
e
U    z B  ml
B  ml  B B
2m
B  Bohr
magneton
Zeeman effect
The values of ml range from –l to +l in steps of one, an energy level
with a particlular value of the orbital quantum number l contains
(2l+1) diffrent orbital states. Without a magnetic field these states all
have the same energy; that is they are degenerate. The magnetic field
removes this degeneracy. In the presence of a magnetic field thy are
split into (2l+1) distinct energy levels:
U  ml  B B
with
ml  0,1,2,...
Adjacent levels differ in energy by
U  (e / 2m) B   B B
μB 
e
eV
 5.79  10 5
2me
T
The Zeeman effect
Energy diagram for
hydrogen, showing the
splitting of energy levels
resulting
from
the
interaction
of
the
magnetic moment of the
electron’s orbital motion
with
an
external
magnetic field.
The Zeeman effect
Splitting of the energy levels
of a d state caused by an
applied
magnetic
field,
assuming only an orbital
magnetic moment.
Selection rules
The photon carries one unit (  ) of angular momentum. Therefore the
allowed transitions: l must change by 1 and ml must change by 0 or  1
Solid lines
– allowed
transitions;
dashed
forbidden
Nine solid
lines give
only three
energies:
Ei-Ef ;
Ei-Ef +BB;
Ei-Ef -BB
The Zeeman effect
Conclusions: spectrum lines corresponding to transitions
from one set of levels to another set are correspondingly
split and appear as a series of three closely spaced spectrum
lines replacing a single line.
Anomalous Zeeman effect
Spin angular momentum and
magnetic moment
Electron posseses spin angular momentum Ls. With this momentum
magnetic momentum is connected:
e 
 s   Ls
me

e 
s   ge
Ls
2me

where ge is the gyromagnetic ratio
For free electron ge=2
Spin
angular
Własny
momentmomentum
pędu - spin and
magnetic moment
Allowed values of the spin angular momentum are quantized :
Ls   s( s  1)
spin quantum number s = ½
Ls  
3
2
The z – component of the spin angular momentum:
Lsz  ms 
 1
 2
ms  
 1
 2
The z- component of the spin
magnetic moment
 sz  
e
e
Lsz  
me
me
 1
 
 2
e
 sz  
 B
2me
Ls
ms  
1
2
ms  
1
2
 sz
Electron in a magnetic field
E  E0   sz B
ms  
1
2
ms  
1
2
To label completely the state of the electron in a
hydrogen atom, 4 quantum numbers are need:
name
label
magnitude
Principal
quantum
number
Orbital
quantum
number
magnetic
quantum
number
Spin
quantum
number
n
1, 2, 3, ...
l
0, 1, 2, ... n-1
ml
ms
od –l do +l
± 1/2
Many – electron atoms and the exclusion
principle

Central field approximation:
- Electron is moving in the total electric field due to the nucleus and
averaged – out cloud of all the other electrons.
- There is a corresponding spherically symmetric potential – energy
function U( r).
Solving the Schrodinger equation the same 4 quantum numbers are
obtained. However wave functions are different. Energy levels depend
on both n and l.
• In the ground state of a complex atom the electrons cannot all be in
the lowest energy state.
Pauli’s exclusion principle states that no two electrons can
occupy the same quantum – mechanical state. That is, no
two electrons in an atom can have the same values of all four
quantum numbers (n, l, ml and ms )
Shells and orbitals
n
1
shell
K
l
orbital
0
s
2
L
0
s
L
1
p
M
M
M
N
N
N
N
0
1
2
0
1
2
3
s
p
d
s
p
3
4
d
f
Nmax - maximum number of electrons occupying given orbital
Nmax
2
2
6
2
6
10
2
6
10
14
Shells K, L, M
n
1
2
l
0
0
ml
0
0
3
1
-1
0
0
1
0
1
-1
0
2
1
-2
-1
0
1
2
ms
N
2
8
18
N : number of allowed states
state with ms = +1/2
state with ms = -1/2
Hund’s rule - electrons occupying given shell initially set up
their spins paralelly
carbon


oxygen


1s22s22p2

1s22s22p4
The periodic table of elements
Atoms of helium, lithium and sodium
n =3, l = 0
3s
n =2, l = 1
n =2, l = 1
2p
n =2, l = 0
n =2, l = 0
n =2, l = 0
2s
n =1, l = 0
n =1, l = 0
n =1, l = 0
1s
Helium (Z = 2)
Lithium(Z = 3)
Sodium (Z= 11)
Electron configuration – the occupying of orbitals
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14
5d10 6p6 7s2 6d10 5f14
K : 1s 2  3 p 6 4 s1
Ca :
3 p 6 4s 2
Sc :
3d 1 4 s 2
Ti :
3d 2 4 s 2
V:
3d 3 4 s 2
Cr :
3d 5 4 s1
Mn:
3d 5 4 s 2
Cu :
3d 10 4 s1
Total angular momentum - J
  
J  L  LS

J 
j  j  1
Possible two magnitudes of j :
J z  m j ,
j  l  s or j  l-s
m j   j, j  1,, j  1, j
Example: l = 1, s = ½
j  1  12 
3
2
lub
j  1  12 
1
2
m j   23 , 12 , 12 , 23 lub m j   12 , 12
j = 3/2
j = 1/2
TheStern-Gerlach experiment
Diamagnetics
.Diamagnetics
Shells totally filled with electrons. Total magnetic moment equals
zero. (In a filled orbital, the vectors for both the orbital angular
momentum and the spin angular momentun point in all posible
directions and thus cancel).
• Noble gas
- He, Ne, Ar…..
• diatomic molecule gas
- H2, N2…..
• solid states of ionic bonds
- NaCl(Na+, Cl-)…
• solid states of covalent bonds
- C(diamond), Si, Ge…..
• most organic materials
Paramagnetics
. Paramagnetics
Shells partially filled with electrons
Total magnetic moment different from zero.
ef  g J ( J  1)  B
The component of the magnetic moment directed
towards external magnetic field
 ef ,H  g M J  B
Fine and hyperfine structure
 Line splittings resulting from magnetic inetractions are
called fine structure.
 The nucleus of the atom has also magnetic dipole
moment that interacts with total magnetic moment of
electrons. These effects are called hyperfine structure.
NMR ( nuclear magnetic resonance)
Like electrons, protons also posses magnetic moment due to
orbital angular momentum and spin ( they are also spin-1/2
particles) angular momentum.
Spin flip experiment:
Protons, the nuclei of hydrogen atoms in the tissue under study,
normally have random spin orientations. In the presence of a
strong magnetic field, they become aligned with a component
paralell to the field. A brief radio signal flips the spins; as their
components reorient paralell to the field, they emit signals that
are picked up by sensitive detectors. The differing magnetic
environment in various regions permits reconstruction of an
image showing the types of tissue present.
An electromagnet used for MRI imaging
Wilhelm Roentgen 1895
Roentgen lamp
Roentgen 1895; X -ray: 10-12m – 10-9m
me v 2
hc
 eVAC  h max 
2
 min
X-ray continuum spectra
me v 2
hc
 eVAC  h max 
2
 min
min
hc

eV AC
X-ray spectra and Moseley law
min
hc

eV AC
The continous –spectrum radiation is
nearly independent of the target
material.
Sharp peaks (characteristic spectra)
depend on the accelerating voltage
and the target element. Frequencies
of the peaks as a function of the
element’s atomic number Z:
f  (2.48  10 Hz)(Z  1)
15
Moseley law
2
X-ray spectra and Moseley law - explanation
Characteristic x-ray radiation is emitted in transitions involving the
inner shells of a complex atom.
Let us assume, that due to electric field one of the two K – electrons
is knocked out of the K shell. The vacancy can be filled by another
electron falling in from the outer shells. Ka is the transition from
n=2 to n=1. As the electron drops down it is attracted by Z protons
in the nucleus screened by the one remaining electron in the K shell.
The energy before (Ei) an after (Ef) transition:
Ei  (Z  1) (13.6eV ) / 2
2
E f  ( Z  1) 2 (13.6eV )
2
E K  ( Z  1) (10.2eV )
2
a
E
15
2
f   (2.47  10 Hz)( Z  1)
h
X-ray diffraction pattern
X-ray diffraction pattern
X
Diffraction maxima:
2d sin   m