Complex atoms
Structure of nucleus
PHY123
7/24/2016
Lecture XVIII
1
Concepts
•
•
•
•
•
•
Quantum numbers, quantum state
Pauli principle
Periodic table
Isotopes
Binding energy
Fission and fusion
7/24/2016
Lecture XVIII
2
Pauli principle
• Electron has spin 1/2
• Electron is a fermion:
– Not more than one electron can be in each quantum state (Pauli
principle)
• Pauli principle is responsible for periodic table
(Mendeleev)
• NB. If particle has an integer spin (0 or 1) it is a boson all particles tend to fall in the same state.
• Example – photons (lasers use this principle)
7/24/2016
Lecture XVIII
3
Electron quantum state
• Principle quantum number n=1,2,3,4,…
– determines energy level, higher E for higher n
• Orbital quantum number l
– For each n l can be 0,1,2,3, …(n-1)
– l states are leveled by letters
– s: l=0;
p: l=1;
d: l=2;
f: l=3;
g:l=4
• E.g. n=5, then l can be 0, 1, 2, 3, 4
– Possible l states are s,p,d,f,g
• n=1, only l=0 s-state is possible
7/24/2016
Lecture XVIII
4
Electron quantum state
• Orbital quantum number is a
vector length l
• Its projection on z axis is
another q.n. – magnetic
quantum number ml
• ml can be only integer
ml  l;(l  1);(l  2),...  2,1,0
7/24/2016
Lecture XVIII
z
l  2; ml  2
l  2; ml  1
l  2; ml  0
l  2; ml  1
l  2; ml  2
5
Electron quantum state
• All electrons have spin=1/2
• It is a vector
• Its projection on z axis is
another q.n. – spin ms
• ms can be only
1
ms  
2
7/24/2016
Lecture XVIII
z
1
1
s  ; ms 
2
2
1
1
s  ; ms  
2
2
6
Possible number of states
• For each given n,l,ml  2 possible states:
– ms=+1/2; ms=-1/2
• For each given n,l  2(2l+1) states:
– ml=+l, +(l-1), …+2,+1, 0, -1, -2, …-(l-1), -l  (2l+1)
– For each ml: ms=+1/2; ms=-1/2  2
• For each given n  2n2 possible states
– l=0,1,2,3,4,….(n-1)
– For each l: ml=+l, +(l-1), …+2,+1, 0, -1, -2, …-(l-1), -l  (2l+1)
– For each (l,ml) ms=+1/2; ms=-1/2
7/24/2016
Lecture XVIII
7
State labeling
• 2p3 means “3 electrons in p state (l=1) at n=2
level”
• 4f5 means “5 electrons in f state (l=3) at n=4 level”
• States filled up from lower to higher n
• For each n from lower to higher l
7/24/2016
Lecture XVIII
8
Periodic table
• Elements in increasing mass • H Z=1 1 electron
– n=1, l=0 (s), ml=0
– 1s1
– MA=NpMp+NnMn
• But mass does not
determine chemical
properties
• Atomic number Z does.
Z=Np- charge of nucleus =
number of electrons
7/24/2016
• He Z=2 2 electrons
– n=1, l=0 (s), ml=0
– 1s2
– 2 e: spin up, spin down
• Li Z=3 3 electrons:
Lecture XVIII
– 2 e on 1s shell
– 1 e on n=2, l=0 (s), ml=0
– 1s22s1
9
Electron structure
•
•
•
•
n=1,2,3,4….
l=0,1,2,…(n-1)
ml=-l,…-1,0,1,…+l
ms=+1/2; -1/2
• 3p4: 4 electrons on
n=3, l=1 subshell
7/24/2016
• 2n2 states for each n
• 2(2l+1) states for each n,l
• 2 states for each n,l,ml
•
•
•
•
•
s(l=0), p(l=1), d(l=2), f(l=3)
2 states in each s subshell
6 states in each p subshell
10 states in each d subshell
14 states in each f subshell
Lecture XVIII
10
Test problem #1
• The following configuration is allowed
1s22s22p63s3
– True
– False
7/24/2016
Lecture XVIII
11
Test problem #2
• The following configuration is allowed
1s22s22p63s23p54s2
– True
– False
7/24/2016
Lecture XVIII
12
Test problem #3
• The following configuration is allowed
1s22s22p62d1
– True
– False
7/24/2016
Lecture XVIII
13
Atomic spectra
• Forbidden (not completely though), allowed
transitions (Dl=±1), because photon has spin 1
and angular momentum is conserved
• Inner energy shells (called K-shells) “see” almost
full potential of the nucleus proportional to (Z-1)2
• Observe atomic spectra in X-ray scattering
7/24/2016
Lecture XVIII
14
Nucleons and nuclei
• Proton and neutron are called nucleons
• 1u=1.66054x10-27kg=931.5MeV/c2
• Protons have positive charge , neutrons have zero charge
• A new kind of force – “strong” force – holds nucleons
inside nucleus. Unlike EM force strong force is “short
distance”  nuclei are very tightly packed
7/24/2016
Lecture XVIII
15
Nucleus radius
• Volume proportional to the number of nucleons:
4 3
V
r A
3
1/ 3
15
1/ 3
r  A ; r  1.2 10 mA
7/24/2016
Lecture XVIII
16
Isotopes
• Mass of the nucleus does not
determine chemical properties
Atomic number Z does.
• Same chemical element
(Z=Np) can have different
number of neutrons –isotopes:
• Carbon Z=66 p, but it can
have
• 5n: 11C, 6n: 12C, 7n: 13C,
8n: 14C, 9n: 15C, 10n: 16C
• Only 12C is stable (98.9% of C
on Earth)
7/24/2016
Lecture XVIII
17
Masses of nuclei and nucleons
• Let’s compare nucleus mass with the sum of nucleon masses (from
appendix D in Giancoli):
• Al: Z=13  13 p, A=27 27-13=14 n
• M(13p+14n)=13x1.007276u+14x1.008665=27.215898u
• M(Al)=26.981538u
• Where did (27.215898-26.981538=0.23436u) go?
• Recall that mass is a form of energy
• To extract a nucleon from the nucleus you must supply energy W:
• M(Al)+W=M(13p)+M(14n)
• Hence M(Al)<M(13p)+M(14n)
• W is called binding energy
7/24/2016
Lecture XVIII
18
Binding energy per nucleon
Fusion
Fission
Too many protons
Electrostatic repulsion
Most tightly woven nuclei
in the middle of the Mendeleev’s table
Energetically most favorable place
Not enough nucleons to
build up strong force
7/24/2016
Lecture XVIII
19
Nuclear fission
• Discovered by German scientists Otto
Hahn and Fritz Strassmann in 1938
• 235U splits into two nuclei when
bombarded by neutrons
• More neutrons are produced in the
reaction:
• n+ 235U141Ba+92Kr+3n
• This means a chain reaction is possible
when the minimum mass of uranium
reaches critical mass
• Tremendous amount of energy is
released
7/24/2016
Lecture XVIII
20
Chain reaction
Uncontrollable chain reaction – fission bomb
7/24/2016
Controllable
Lecturechain
XVIII reaction – nuclear reactors
21
Nuclear reactor
7/24/2016
Lecture XVIII
22
Fusion reactions in the Sun
• The key is to bring two protons together close enough
for the strong force to overcome electrostatic repulsion –
need high T:
• 1H+1H2H+e++n
• 1H+2H3He+g
• 3He+3He4He+1H+1H
• This is how heavier (Z>1) elements were produced in the
first place (only upto iron or nickel)
• Heavier elements are produced in the middle of the
starts or supernova explosions by absorbing extra
energy
• Fission bomb is used to ignite fusion bomb – hydrogen
bomb
7/24/2016
Lecture XVIII
23