Pauli Exclusion Principle

- Consider many electron atoms. What determines the quantum state (characterized by its quantum numbers) that the electrons in such an atom occupy?

- Are all the electrons in the same quantum state? This is unlikely as atoms with different atomic number Z have very different chemical properties hinting at different electron configurations in the atom, e.g. the chemically active halogen fluorine Fz=9, the inert noble gas neon NeZ=10, the metal sodium NaZ=11

- exclusion principle: No two electrons in an atom can exist in the same quantum state. Each electron must have a different set of quantum numbers n, l, ml, ms.

- fundamental principle governing the electronic configuration in atoms discovered by Wolfgang Pauli in 1925

- principle found through the study of spectra and their relation to quantum numbers

- e.g. in Helium no transitions are observed to and from the ground state with both electron spins pointing in the same direction; transitions are only observed to and from ground states with electron spins pointing in different directions.

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Nobel Prize in Physics 1945

"for the discovery of the Exclusion Principle, also called the Pauli Principle"

Wolfgang Pauli AustriaPrinceton University Princeton, NJ, USAb. 1900d. 1958

Pauli's observation: every unobserved state or unobserved transition involves two or more electrons with identical quantum numbers.

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Symmetries of wave functions: Fermions and Bosons

- the wave function ψ of a quantum system that is composed of n non-interacting indistinguishable particles can be written as a (tensor) product of its component wave functions ψI

example: for two indistinguishable particles in two quantum states a and b

the labeling of the particle should make no difference whatsoever for the probability density |ψ|2 of the combined system.

the probability density is independent of the symmetry of the wave function

symmetric 2-particle wave function

anti symmetric 2-particle w.f.

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possible wave functions for two particles

symmetric wave function:

anti-symmetric wave function:

- for indistinguishable particles the cases I and II can not be distinguished. Thus they are equivalent and can appear with equal probability.

- the pre factor reflects the equal probability with which the two possibilities occur

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Consequences:

- two particles in the same state in a symmetric wave function are allowed

- two particles in the same state in an antisymmetric wave function are not allowed

We find that such an anti symmetric wave function would be consistent with the Pauli exclusion principle for electrons where no two particles can be in the same quantum state, i.e. the probability of finding such a state would vanish. This is realized by an anti symmetric wave function.

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Particles with a half integer (1/2, 3/2, ...) spin (electrons, protons, neutrons) are described by an antisymmetric wave function and are called Fermions. They obey the exclusion principle.

Particles with an integer (0, 1, 2, 3, ...) spin (photons, alpha particles, Helium atoms, ...) are described by symmetric wave functions and are called Bosons. They do not obey the exclusion principle, i.e. an arbitrary number of particles can occupy a quantum state with the same quantum number.

5 fermions at T = 0

5 Bosons at T = 0

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Periodic Table of Elements:

Dimitri Mendeleev (1869): When the elements are listed in order of atomic number Z, elements with similar chemical and physical properties recur at regular intervals.

- this observation was made long before quantum theory was developed and even before the concept of atomic number Z and atomic masses was known

- nevertheless he managed to set up a table of the then known 63 elements sorted by their chemical properties

- gaps in his table were indicating at the existence of elements that were not yet discovered at that time

In the periodic table elements are arranged in sequence of their atomic number Z forming the rows of the table. Elements with similar chemical properties are arranged in the same column.

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groups: elements with similar chemical and structural properties are arranged in groups which form the columns of the periodic table

periods: rows of elements in the periodic table. Along the period transition from chemically active metals, to less active metals, to active non-metals and inert gases is observed.

The majority of the elements are metals. There are a number of nonmetals and few the inert gases

variation of chemical activity across the periodic table

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some important groups

group I: hydrogen (H) and the alkali metals lithium (Li), sodium (Na), potassium (K) etc.- properties: soft metals, low melting points, chemically very active

group VII: fluorine (F), chlorine (Cl), bromine (Br), iodine (I) etc. are the halogens- properties: chemically active, oxidants, form diatomic molecules

group VIII: the nobel gases helium (He), neon (Ne), argon (Ar) etc.-properties: chemically inactive, form almost no compounds, do not form molecules

transition elements: elements between group 2 and three in periods 4 and higher- properties: metals, hard, brittle, high melting points, similar chemical behavior

lanthanides (rare earths): transition elements in period 6

actinides: transition elements in period 7

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Atomic Structure

principles determining the structure of atoms:- only one electron can exist in a particular quantum state in an atom (Pauli principle)- a system of particles is stable when its total energy is at minimum

note:- these principles determine the distribution of all electrons in an atom to the different states

- electrons interact with each other in a many electron atom

- it is useful to consider an electron in the electrostatic potential of the nucleus with charge Ze reduced by the other electrons screening this charge (see example)

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Atomic Shell:

Electrons with the same principal quantum number n have a similar average distance from the nucleus interacting with a similar electric field and thus also have a similar energy En. Such electrons are said to occupy the same atomic shell.

labeling convention for atomic shells:

principal quantum number

shell label

- In many electron atoms the energy of a particular electron strongly depends on its principal quantum number n but also depends on its angular momentum quantum number l.

- The probability distribution |ψ|2 of finding the electron at a certain distance from the nucleus depends on n and l.

- For small l the electron is closer to the nucleus (where the screeing by the other electrons is less effective) and thus is more strongly bound, i.e. it has a more negative energy.

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Electron binding energy versus atomic number Z

- increasing binding energy with Z

- decreasing binding energy with increasing n

- decreasing binding energy with increasing l

Subshells:

- Electrons with the same angular momentum quantum number l are said to be part of the same subshell.

- The dependence of their energy on magnetic quantum number ml and spin quantum number msis small in comparison to the n and l dependence.

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Occupancy Electronic States in Many Electron Atoms

example: sodium (Na) 1s2 2s2 2p6 3s1

- principal quantum number n- followed by angular quantum number l represented by the associated letter- superscript indicating the occupation of the subshell with electrons

Capacity of shells and subshells:

total number of electrons

M shell (n = 3) capacity:

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The Periodic Table

closed shell: A shell in which all possible electron states are filled is called closed.

- closed s shell: 2 electrons- closed p shell: 6 electrons- closed d shell: 10 electrons- closed f shell: 14 electrons

properties:- closed subshells have zero total angular momentum- closed subshells have zero spin momentum- electrons in a closed shell are strongly bound because of the large only weakly screened charge of the nucleus- an atom with closed shell has no dipole moment, it interacts only weakly with other atoms- closed shell atoms such as the noble gases are chemically relatively inert

closed l = 1 subshell closed l = 2 subshell

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Ionization Energy

variation of the ionization energy with atomic number Z

notes:- inert gases have the highs ionization energy- alkali metals have the lowest ionization energy, i.e. they are easy to ionize and form positively charged ions- decrease of ionization energy with in group for increasing Z- increase in ionization energy with increasing Z within period due to increasing nuclear charge- halogens can lower their energy by adding an electron to their incomplete shell forming negatively charges ions

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Size of Many Electron Atoms

variation of atomic radii with atomic number Z

extracted from atom-atom spacing in crystal lattices of element in a solid configuration

note:- alkali metals have larger atomic radii due to weakly bound outer electron- with increasing charge Z in each period the effective radius is reduced due to imperfect screening- relatively small variation in atomic radius with Z - atoms with Z ~ 90 have only a radius of 3 times that of the hydrogen atom- the largest atom is cesium (Cs) with a radius of 4.4 times that of the hydrogen (H) atom

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