lecture viii hydrogen atom and many electron atoms dr hab. ewa popko
TRANSCRIPT
![Page 1: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/1.jpg)
Lecture VIII
Hydrogen Atom and
Many Electron Atoms
dr hab. Ewa Popko
![Page 2: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/2.jpg)
Niels Bohr1885 - 1962
![Page 3: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/3.jpg)
Bohr Model of the Atom• Bohr made three assumptions (postulates)• 1. The electrons move only in certain circular orbits, called
STATIONARY STATES. This motion can be described classically
• 2. Radiation only occurs when an electron goes from one allowed state to another of lower energy.
• The radiated frequency is given by
hf = Em - En
where Em and En are the energies of the two states• 3. The angular momentum of the electron is restricted to
integer multiples of h/ (2)
mevr = n
![Page 4: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/4.jpg)
zyxEzyxzyxVzyxm
,,,,,,2 2
2
2
2
2
22
zyxEzyxH ,,,,ˆ
The Schrödinger equationThe hydrogen atom
The potential energy in
spherical coordinates
(The potential energy function is spherically symmetric.)
Partial differential equation with three independent variables
r
erV
2
04
1)(
![Page 5: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/5.jpg)
The spherical coordinates
(alternative to rectangular coordinates)
![Page 6: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/6.jpg)
For all spherically symmetric potential-energy functions:( the solutions are obtained by a method called separation of variables)
)()(,,,),,( , ll mmlnllmnl rRYrRrzyx
Radial function Angular function of and
The hydrogen atom
The functions and are the same for every spherically symmetric potential-energy function.
Thus the partial differential equation with three independent variables
three separate ordinary differential equations
![Page 7: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/7.jpg)
The solutionThe solution is determined by boundary conditions:
- R(r) must approach zero at large r (bound state - electron localized near the nucleus);
and must be periodic:
(r,and(r,describe the same point, so
and must be finite.
Quantum numbers:
n - principal l – orbital ml - magnetic
![Page 8: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/8.jpg)
2220
2
4 1
32
n
eEn
Principal quantum number: n
The energy En is determined by n = 1,2,3,4,5,…;
2
16.13
neVEn
E = - 13.6 eV
- 3.4 eV
Ionized atom
n = 1
n = 2
n = 3
reduced mass
Ne
Ne
mm
mm
![Page 9: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/9.jpg)
)1( llL ...,2,1,0l
Quantization of the orbital angular momentum.
The possible values of the magnitude L of the orbital angular momentum L are determined by the requirement, that the function must be finite at and
There are n different possible values of L for the n th energy level!
Orbital quantum number
![Page 10: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/10.jpg)
)1( llLLz
lmllm ll )1( lml ...,2,1,0
Quantization of the component of the orbital angular momentum
lz mL
![Page 11: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/11.jpg)
Quantum numbers: n, l, m
l – orbital quantum numberl - determines permitted values of the orbital angular momentum
n – principal quantum numbern – determines permitted values of the energy
l = 0,1,2,…n-1;ml - magnetic quantum numberml – determines permitted values of the z-component of the orbital angular momentum
lml ...,2,1,0
n = 1,2,3,4...
![Page 12: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/12.jpg)
Wave functions
n,l,m
l = 1 m = ±1
l = 0n = 1
n = 2
n = 3
l = 0,1
l = 0,1,2
rerR ~)(
)()(,, , ll mmlnl rRr
polynomial
~ ie
![Page 13: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/13.jpg)
Quantum number notation
Degeneracy : one energy level En has different quantum numbers l and ml
l = 0 : s states n=1 K shell
l = 1 : p states n=2 L shell
l = 2 : d states n=3 M shell
l = 3 : f states n=4 N shell
l = 4 : g states n=5 O shell
. .
. .
![Page 14: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/14.jpg)
1&1
0&1
1&1
0&0
2
l
l
l
l
ml
ml
ml
ml
n
0&0
1
lml
n
2&2
1&2
0&2
1&2
2&2
1&1
0&1
1&1
0&0
3
l
l
l
l
l
l
l
l
l
ml
ml
ml
ml
ml
ml
ml
ml
ml
n
Electron states
1s
2s
2p
3s
3p
3d
M
L
K
![Page 15: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/15.jpg)
S-states probability
![Page 16: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/16.jpg)
P-states probability
![Page 17: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/17.jpg)
Spin angular momentum and magnetic moment
Electron posseses spin angular momentum Ls. With this momentum magnetic momentum is connected:
se
es Lme
g
2
where ge is the gyromagnetic ratio
For free electron ge=2
se
s Lme
![Page 18: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/18.jpg)
Allowed values of the spin angular momentum are quantized :
)1( ssLs
spin quantum number s = ½ 2
3sL
Własny moment pędu - spin
The z – component of the spin angular momentum:
ssz mL
2
12
1
sm
Spin angular momentum and magnetic moment
![Page 19: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/19.jpg)
To label completely the state of the electron in a hydrogen atom, 4 quantum numbers are need:
name label magnitude
Principal quantum number
n 1, 2, 3, ...
Orbital quantum number
l 0, 1, 2, ... n-1
magnetic quantum number
ml od –l do +l
Spin quantum number
ms ± 1/2
![Page 20: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/20.jpg)
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 )
![Page 21: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/21.jpg)
Shells and orbitals
Nmax - maximum number of electrons occupying given orbital
n shell orbital
1 K 0 s
2 L 0 s
L 1 p
3 M 0 s
M 1 p
M 2 d
4 N NNN
01
23
sp
df
Nmax
2
2
2
6
6
6210
1014
![Page 22: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/22.jpg)
Shells K, L, M
n 1 2 3
0 0 1 0 1 2
m 0 0 -1 0 1 0 -1 0 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
1s22s22p2
1s22s22p4
carbon
oxygen
Hund’s rule - electrons occupying given shell initially set up their spins paralelly
![Page 23: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/23.jpg)
The periodic table of elements
![Page 24: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/24.jpg)
Atoms of helium, lithium and sodium
n =1, = 0 n =1, = 0 n =1, = 0
n =2, = 0 n =2, = 0n =2, = 0
n =2, = 1 n =2, = 1
n =3, = 0
Helium (Z = 2) Lithium(Z = 3) Sodium (Z= 11)
1s
2s
2p
3s
![Page 25: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/25.jpg)
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10
6p6 7s2 6d10 5f14
110
25
15
23
22
21
26
162
43:
43:
43:
43:
43:
43:
43:
431:
sdCu
sdMn
sdCr
sdV
sdTi
sdSc
spCa
spsK
Electron configuration – the occupying of orbitals
![Page 26: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/26.jpg)
Example: l = 1, s = ½
1 jjJ
21
21
23
21
21
23
21
21
23
21
, lub,,,
1lub1
jj mm
jj
j = 3/2 j = 1/2
SLLJ
Possible two magnitudes of j : l-sjslj or
jjjjmmJ jjz ,1,,1,,
Total angular momentum - J
![Page 27: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/27.jpg)
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.
![Page 28: Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649ee65503460f94bf6502/html5/thumbnails/28.jpg)