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Transcript
Chapter 2
STRUCTURE AND PROPERTIES OF ATOMS
The hydrogen atom
A set of elementary particles, protons, neutrons and
electrons can build an atom.
Our model in this chapter is the isolated atom, the atom
that does not treat into interactions with its environment.
A low pressure gas is its good realisation.
The hydrogen atom is the simplest atom
therefore this study begins with it.
The structure of the hydrogen atom
The description of the hydrogen atom by quantum
mechanics is not complicate. One have to solve the
Schrödinger equation:
ˆ   E
H
Ĥ is the Hamilton (energy) operator, is the eigenfunction

(a wave function), E is the energy eigenvalue.
electron
The H atom model:
r
proton
The calculations are simple using polar coordinates:
r is the radius,
 is the polar angle,
q is the azimuthal angle.
For solution of the Schrödinger equation is assumed
 r , ,q   Rr Y  ,θ 
Hamilton operator: sum of the kinetic (T) and
potential (V) energy operators: Ĥ  T̂  V̂
The potential energy operator:
Vˆ  V .
Potential energy of the hydrogen atom:
1 e2
V 
4o r
(e: absolute value of the electron charge)
2
2
The kinetic energy operator: T̂   2m T̂r  T̂ ,θ   2m T̂ p
e
p
Right side: first term refers to the electron, the
second term to the proton of the hydrogen atom.
Born-Oppenheimer approximation:
the movement of the electron is far faster than that of the
proton (cosidering the mass rate: mp/me1850.
the proton kinetic energy term is negligible,
practically unmoved proton.
Substituting the expressions on the kinetic and potential
energy operators in the Schrödinger equation:
 2

2
Y  ,θ 
T̂r  V̂  Rr  
Rr T̂ ,θ Y  ,θ   ERr Y  ,θ 
2me
 2me

Separating this differential equation according to the
angular and radial variables:

1  2
2
1
T̂r  V̂  Rr  
T̂ ,θ Y  ,θ   E

Rr   2me
2me Y  ,θ 

Born-Oppenheimer approximation:
the movement of the electron is far faster than that of the
proton (cosidering the mass rate: mp/me1850.
the proton kinetic energy term is negligible,
practically unmoved proton.
Substituting the expressions on the kinetic and potential
energy operators in the Schrödinger equation:
 2

2
Y  ,θ 
T̂r  V̂  Rr  
Rr T̂ ,θ Y  ,θ   ERr Y  ,θ 
2me
 2me

Separating this differential equation according to the
angular and radial variables:

1  2
2
1
T̂r  V̂  Rr  
T̂ ,θ Y  ,θ   E

Rr   2me
2me Y  ,θ 

4
e
me
RH
The solution 1: En  
 hc 2
2 2
32o n
n
That means, the electron energy is quantized, it depend on
the quantum number n, the principal quantum number. The
electron energy has only discrete values, its eigenvalues ,the
energy levels of the electron.
The solution 2:  n ,l , m  N n,l,m
1 l  2r
Rn 
r  na o
 m
Yl  ,q 

These are the eigenfunctions, called also orbitals, with
4π o 
ao 
 52.9 pm the Bohr radius, the radius for n=1.
2
me e
2
Nn,l,m is a normalization factor to fulfil the
request the normalization to 1:
*

 d  1
The quantum numbers n, l and m of the hydrogen atom are:
n: principle quantum number , n=1,2,3,... ,positive integers
l: angular moment quantum number, l=0,1,2,...,n-1
m: magnetic quantum number, m=-l,-l+1,...,0,...,l-1,l
The electron states of the hydrogen atom are degenerated:
the same energy belongs to more eigenfunctions (orbitals).
Electron orbitals with the same n build an electron shell,
orbitals with the same n and l build a subshell. Shells are
labelled with capitals, subshells with low case letters:
n= 1
label K
2
L
3
M
4
N
5
O
6
P
etc.
etc.
l=
0
label s
1
p
2
d
3
f
4
g
5
h
etc.
etc.
There exist a special angular moment of the electron
from its nature, the spin. The eigenvalue equation for the
spin component in the direction of the external field (z):
ˆs z    ms   
The quantum number ms is the magnetic spin quantum
number, and has only two possible values: +1/2 and -1/2,
labelled  and , respectively.  is the so-called "spin
coordinate". The spin quantum number is labelled with s
and has similarly the same values.
Considering the restrictions for the l, m and s, if the
principal quantum number is n, the possible maximal number
of electrons with n is 2n2.
Hydrogen
eigenfunctions
The surfaces where the orbitals have zero values
are called nodal surfaces, they depend on functions of r,
 and q :
1 3/2
1 3/2
R1 
a o exp  ρ , R 2 
a o exp  ρ/2  and
π
8 π
1
R3 
a o3/2exp  ρ/3 have one nodal surface in the infinity.
81 π
The r polynoms have n-l-1 solutions zero for r. The
geometric place of points with a given r value build a
sphere. An orbital has n-l spheric nodal surfaces, one of
them is in the infinity.
The points with common q build a cone; at 90o cosq=0,
cone deforms to a plane; their number is: l - m
The points with common polar angle build planes (sin=0 or
cos=0); the number of planar nodal surfaces is m .
Altogether, n nodal surphaces belong to the
principal quantum number n.
The form of the
hydrogen
eigenfunctions,
indices according to
the n, l, and m
quantum numbers.
Angular and magnetic moments of the hydrogen atom
The energy levels of the hydrogen atom are degenerated.
External E and B fields split them. They are defined with
their z components and absolute values.
The eigenfunction of the square of the orbital angular
moment operator (l) is the radial part of the energy
eigenfunction:
ˆl 2Yl m  l l  1 2Yl m
So its absolute value:
The z component:
Therefore
l  l l  1  l * 
l*  l l  1
l̂ z Yl m  mYl m
l z  m
Both l and lz are quantized by l and m, respectively.
If l=3, l*= 12 , possible m values: 3, 2, 1, 0, -1, -2, -3.
m
The angle of angularmomentum to z axis:   arccos * 
l 
The orbital magnetic moment:
absolute value:
z component:
m  Bl
m z   B m
*
Spin angular moment:
Eigenvalue equation:
e
l  2m
me
e
B 
2me
ˆs 2    ss  1  
: spin coordinate, s: eigenvalue, : eigenfunction.
absolute value:
z component:
 s  ms   s
s  ss  1  s* 
s z  ms 
s=1/2 only!
ms= -1/2, +1/2
*
m


2

s
Spin magnetic moment: absolute value
s
B
z component:
m s ,z  2 B m s
Total angular moment of the electron
jls
From solution of the eigenvalue equation for ˆj 2
The absolute value j 
j  j  1  j * 
 j  l  s
j: total angular moment quantum number
 j  m
  j
j
z component: j z  m j 
mj is the corresponding magnetic quantum number.
Corresponding magnetic moment: like in case of angular
magnetic moment:
m j    B j*
Nuclear spin
Iˆ 2  I I + 1 2
I: the nuclear spin quantum number.
Absolute value: I  I I  1  I * 
1
I

for hydrogen
2
z component, eigenvalues: I z  M I 
1 1
For hydrogen M I   , .
2 2
*
M

g
μ
I
Corresponding nuclear magnetic moment:
I
p N
gp: Landé factor for proton.
z component: M I ,z  g p  N M I
Selection rules for the hydrogen spectrum
The necessary conditions for its generation
- energy difference between the ground and excited states;
it is equal to the energy of the photon: ΔE  Ei  E j  hν
*
- the transition moment vector is not zero P   i p jd  0
Change in dipole moment: p  er
Cartesian components 1s 2s transition:
p x  er sin q cos  p y  er sin q sin  pz  er cos q
The space element: d  r 2 sin qddqdr
  2
Px  e   i* r , ,q  j r , ,q r 2 r sin 2 q cos ddqdr
0 0 0
  2
Py  e   i* r , ,q  j r , ,q r 2 r sin 2 q sin ddqdr
0 0 0
  2
Pz  e   i* r , ,q  j r , ,q r 2 r sin q cos qddqdr
0 0 0
Integrating with respect to q from 0 to 
all integer powers of sinq: the result is non-zero,
even powers of cosq: the result is non-zero,
odd powers of cosq: the result is zero.
Integrating with respect to  from 0 to 2
even powers of sin or cos : the result is non-zero,
odd powers of sin and cos: the result is zero.
Angle functions for excitation from 1s (100) state to 2s
(200) state the
integrals are zero (no transition), since: 1s 2s transition
x component contains sin2q cos
is not possible.
2
y component contains sin q sin
z component contains sinq cos q
Excitation from the 1s state (100) to 2px state (211) the angle
function parts of the integrals are
for the x component: sin3qcos2
the integral is non-zero
for the y component: sin3qsincos the integral is zero
for the z component: sin2qcosqcos the integral is zero
Transition in direction x.
n  0
Similarly 1s 2py is allowed in direction y,
1s 2pz in direction z.
Selection rules:
necessary:
sufficient:
n  0
l  1
The electronic spectrum of the hydrogen atom
The hydrogen atom is excited in electric arc or with
discharges. The wavenumber of the spectral line is
 1

E
1
~
 
 RH  2  2 
ni  n j


hc
 n j ni 
RH is the Rydberg constant The distance between
RH
~
the neighbouring lines decreases, the limit is   Ti  2
ni
The Ti like expressions are called terms.
The spectral series of the hydrogen atom
Many-electron atoms
More than one electrons move around the nucleus. Electric
and magnetic interactions between the electrons of the
electron cloud.
Hydrogenic atoms (or ions) have only one electron on their
outside (valence) shell. This is a good approximation for these
type atoms. All other atoms are far from this model, their
description is very complicate.
Hydrogenic atoms
z 2 hc RH
Rz
En  
 hc 2
2
n
n
where Rz is the Rydberg constant for the atom with atomic
number z. Comparing this equation with the spectra


1
1
~  Rz 


2
2
 n j  a j  ni  a i  
The corrections express the effect of the inner shell
electrons. The correction terms with aj and ai split in some
cases, i.e. the spectral lines may split besides their shifts.
Spectral series of alkali metal atoms
Other many-electron atoms
Strong electron interactions exist in open (not absolutely
closed) electron shells and clouds. The Hamilton operator:
2
2
2
ze
e
2
Ĥ  


 

2 me i
ri
i
i j i rij
Sums are extended to all electrons; ri: electron-nucleus
distance; rij is the electron-electron one. First term: kinetic
energy operator, second term: electron-nucleus, third
term: electron-electron potential energy.
The atomic energy depend at these atoms also on the
angular moment quantum number l.
Quantum mechanical calculations:
energy level series for the subshells.
E1s  E 2s  E 2p  E3s  E3p  E4s  E3d  E4p  E5s ...
Hund's rule: the atom in its ground state adopts the greatest
possible number of unpaired electrons.
The electron configuration of the atoms is labelled by giving
- the principal quantum number,
- the letter for the angular quantum number,
- as superscript the number of electrons in the subshell
E.g. for sulfur: 1s22s2p63s2p4. This configuration is labelled as
Moments of the many- electron atoms
angular moment of all the electrons
vector: L   l i
i
absolute value: L  LL  1  L* 
0  L  l1  l 2
quantum number: L
angular magnetic moment:
M    B L*
States with different values of L are labelled
L=
0
label S
1
P
2
D
3
F
4
G
5
H
etc.
etc.
Spin vector for all the electrons
S   si
i
absolute value:
S  S S  1  S * 
S is spin quantum number of all electrons.
For two electrons S = 1 or 0.
The total angular momentum for all electrons J
depends on the type of interactions:
A. electrostatic interactions
B. magnetic (spin-orbital) interactions
If the atomic number z < 40 then A>B, case 1
if the atomic number z  40 then B>A, case 2
1. Russel-Sounders or ls coupling
ˆ  Sˆ
Jˆ  L
J  J J  1  J * 
Jz  M J
J = L + S, L + S - 1,..., L - S
 J  MJ  J
Multiplicity of the state:
M J  M L  MS
  2S  1
  2L  1
L  S, more often 
L  S, rare 
Names of the states are:
if  =1 this is a singlet state,
if  =2 this is a doublet one,
if =3 this is a triplet one, etc.
2. jj coupling
Jˆ   ˆji
i
for two electrons : J  j1  j 2 , j1  j 2  1,..., j1  j 2
Example: the electron states of two p electrons, ls coupling.
The individual quantum numbers:
first electron
l1 = 1
l1*
 2 s1 
1 *
3
s1 
2
2
second electron
3
1 *
s

s

l2 = 1 l  2 2
2 2
2
*
2
Quantum numbers for all electrons
L = 0, 1, 2
L*  0, 2 , 6
S = 0, 1
S *  0, 2
Possible vector sums of angular moments and spins (in  units):
 2
L*=0
L*
S*=0
S*  2
L*  6
l


L*

l1*

l*
2

s


S*


l

s*
1

s


2
s*
2




The possible combinations of the two moments are
L=0 & S=0, L=1 & S=1, L=2 & S=0.
The effect of the interactions: Superscipt: multiplicity,
subscript: total quantum number (J)
the angular moment
electrostatic interaction
magnetic interaction
(spin-orbit interaction)
external magnetic field
(possible values of MJ)
S
1S
1S
o
0
3P
o
0
P
3P
3P
1
3P
2
-1,0,1 -2,-1,0,1,2
For 3P: L=1 & S =1,there are 3 possibilities:
3P : L* and S* have opposite direction
o
3P : L* to S* angle is between 90o and 0o
1
3P : L* and S* have the same direction
2
Look at the splitting of the energy levels!
D
1D
1D
2
-2,-1,0,1,2
1S
1S
o
MJ
1D
1D
2
2
0
-2
3P
2
2
3P
p2
H atom level
-2
3P
1
3P
o
electrostatic
interaction
magnetic
interaction
1
-1
0
external
magnetic field
Splitting of the energy levels
Interaction with external magnetic field
The energy of the electronic states (Eo, without interactions)
depends on the electrostatic interactions (on J*) and in
presence of weak or modest magnetic fields B on MJ:
 
1
ls splitting): E  E o  A J *
2
A is a coupling constant,
g is the Landé factor:
2
 g B M J B

J   S   L 
g  1
2J 
Landé factor of a free electron g=2
* 2
* 2
* 2
* 2
(l *
*  3 j*  3 )
s
0
2
2
Zeeman effect: electronic levels split according to MJ.
Paschen-Back effect: very strong splitting,
strong magnetic field the magnetic interactions dominate
(jj coupling)
E  E o  AM L M S  gM L  2M S B
The energy levels of the nucleus splits also in magnetic field:
EE g  M B
o
N N I
The splitting of the energy levels in external magnetic field
is important in the methods of nuclear magnetic resonance
and electron spin (paramagnetic) resonance.
Interaction with external electric field
The electronic energy levels split also in
external electric field (E)
1 2
 2 1 * 2 
E  E o  E a  2b M J  J  
2 
3


a and b are constants.
The levels split in this Stark effect only according
the absolute values of MJ since the MJ quantum
number is here squared.
Therefore the levels split to MJ +1 ones.
Interpretation of the electronic spectra
The emission of the photon is either spontaneous or induced.
The excitation with photons is called absorption.
E* excited state
i
absorption
emission
E
j
ground state
Beside the photon absorption (optical excitation) the
excitation is possible with electric energy, with collisions and
with thermal energy.
Absorption and the emission is possible if the
transition moment is not zero.
Selection rules for one- and many-electron atoms
= 2,
.
Therefore the characteristic quantum numbers
are: L = l = 0, S  s  1 , = 2, J  j  1
2
2
The possible spectral series are
Lines of sharp and the principal series split into 2 lines
(doublets), those of the diffuse and the fundamental series
split into three lines (triplets).
Next slide: term diagram (Grotrian diagram) of the
potassium spectrum, transitions of these series, the lines of
these series, and their sum, the electronic emission
spectrum.
The measurement of the atomic spectra
Simple emission spectrometer. The computer controls the
system and stores the measured data. The excited sample
emits the spectrum. The monochromator resolves the
electric
spectrum. The detector detects the optical signal
signal. The electronics under the control of the computer
forms the signal and transfers it to the computer (data
acquisition).
sample
monochromator
detector
stepping
excitation
electronics
control
control
computer
Emission spentrometer
data acquisition
Absorption spectrometer.
The sample is here passive, a light source is necessary
for the working. A part of the light will be absorbed by the
sample. The monochromator resolve the light, the
detector transforms the optical signal to electric one.
The speciality of the absorption spectrometer is the
necessity of a reference. The reference will be measured
either by dividing the source light into a sample and a
reference beam; the measurement of them will detected in
short time spans during the same mesurement.
Otherwise, the sample and the reference will be
measured in different measurements. The resulted
spectrum will be produced by the computer.
sample
monochromator
detector
stepping
light source
electronics
control
control
computer
data acquisition
Simple dispersion absorption spectrometer
monochromator: resolves the spectrum using a (reflecting)
grating or a prism, transparent in the investigated region
(or a Fourier transform spectrometer is used).
detector: transforms the optical signal to electric one.
- photocell, photodiode, photodiode array,
photomultiplier (PMT: photomultiplier tube),
photoelement in the UV and VIS regions,
CCD (coupled charge detector);
- thermoelement in the VIS and IR regions;
- piezoelectric crystal, Golay cell (IR, far IR);
- chilled semiconductor detectors (IR, near IR).
electronics: signal elaboration (signal difference for doublebeam spectrometers, amplitude or phase modulation,
amplification), digital computer, printer and plotter.
The spectral information:
- frequency, i.e. position of the line;
- intensity, i.e. area of the line;
- profile or shape of the line.
Applications of the atomic spectra:
mostly quantitative and qualitative chemical
analysis, rarely physics (theory of atomic structure).
Ions
Atomic ions and molecular ions
Ionization
Ionization is generation of positive ions.
Generation of a positive ion from a neutral atom: A  e   A 
The external (outside) ionization is the ablation of an
electron from the outmost (valence) shell of the atom.
The internal (inner) ionization is called the ablation of an
electron from the core (inner electron shells) of the atom.
The electron gain is the building of a negative (or more
negative) ion by receiving an electron:

Ae  A



A e  A
2
The ionization energy of an atom (I) is the necessary energy
to ablate an electron from an atomic (molecular) orbital.
First and second ionization energies of some atoms (kJ mol-1)
First column of the periodic table: both first and second
ionization energies decrease with increasing atomic number:
the valence shell is more and more far away from the nucleus.
The second ionization energy is always higher than the first.
Boron: unpaired p electron, oxygen: triplet state.
Electron affinity (A) is the energy gain that we get
if an electron is absorbed by an atom or a molecule.
Relation to ionization energy, e.g. oxygen: IO    AO
Electron affinities of some atoms (kJ mol-1)
The hydrogen has very low electron affinity, fluorine and chlorine
have a higher onces, here is the maximum, the heavier halogens
have lower and lower electron affinity.
Methods for ionization of atoms (A) and molecules (M):
1. Collision in vacuum with high energy electrons,
partly inelastic scattering:


A  e  A  2e

M  e  M  2e
Mass spectroscopy, electron impact (EI) ionization,
generation of negative ions, like ABC  e  AB  C
2. Ionization with photons (inelastic scattering with particle
change). The photon energy must be higher than the


ionization energy of A or M: A  h  A  e
There is a break in the spectrum at the frequency of this photon:
the spectral line density increases and approaches a limit, at
higher frequencies a diffuse region appears.
This method is applied in photoelectron spectroscopy.
3. Penning ionization. The atom A is excited with photon, the
absorption transfers its energy to atom B with collision (partly
inelastic scattering). If the transferred energy is high enough:
A*  B  A  B  e
4. Dissociative ionization. The atom A excited with photon
absorption, collides with the molecule BC.
A*  BC  AC  B  e
5. Ionization through collision. The excited atom A collides
with atom B. The result is
A*  B  A  B  e
or
A*  B  A  B
6. Associative ionization. The excited atom A associates
with atom B to a molecular ion:
A*  B  AB  e
7. Auger effect. After an inner ionization an electron gap
appears on an inner electron shell. This gap will be filled by
an electron jumps from a higher level. The energy difference
between the two states can be assign to the emission of an
X-ray photon or to a second ionization: to the emission of a
second photon. The latter is the Auger effect.
8. Chemical ionization (CI). This is a special method for the
ionization of molecules. A high velocity molecular ion (let us
denote it with X+) collides with M (mass spectroscopy):
X  M  X  M or
X   M  XM 
If X contains hydrogen atoms (XH):
XH  M  X  MH
9. Fast atomic bombardment (FAB). The molecules are
bombarded with high velocity atom or ion radiation (Ar, Xe,
Cs+), applied in mass spectroscopy and secondary ion
emission spectroscopy.
Interactions of ions
The interaction of ions with electromagnetic waves is
similar to those of the neutral atoms: spectral line series
appear in the emission and absorption spectra. Electric and
magnetic fields act on ions having Q charge and v velocity:
Coulomb force: F  QE Lorentz force: F  Qv  B
Applying these equations ions can be direct detected.
Constant B, ions can be detected by velocities.
Electrons of different velocities are detectable with the
same (in the space fixed) detector.
Changing B, particles with different velocity can be
directed to the same in space fixed detector.
These methods are applied in mass spectroscopy
and in different electron spectroscopic methods.