CHAPTER 8 Atomic Physics n Schrdinger equation for

CHAPTER 8 Atomic Physics n Schrödinger equation for more than two particles n Bosons and fermions, Pauli’s exclusion principle n 8. 1 Atomic Structure and the Periodic Table What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior. - J. Bronowski Suffices for this chapter, derived results are numerically nearly correct, also we do allow for an inclusion of effects of the forth dimension (by multiplying what goes on in 3 D with the spin wave function) 1

There is no path for a quantum mechanical object to follow, uncertainty principle forbids this 2

If that made sense, the particle that we find at x = L/2 needs to be always the one in state n = 1, if we were to change labels the same condition would apply so we would violate the condition that quantum mechanical particles are indistinguishable which results from the uncertainly principle, so it cannot make sense 3

Two basis types of particles, bosons (integer spin) and fermions, (half integer spin) Matter is composed of fermions, half integer spin, Paraphrasing Winston Churchill: not everybody at the horse races is a crook, but all the crooks are at the 4 horse races: Not all bosons are force particles, but all force particles are bosons

Pauli Exclusion Principle n To make sense of atomic spectroscopic data of the anomalous Zeeman effect, Pauli proposed his famous exclusion principle: No two electrons in an atom may have the same set of quantum numbers (n, ℓ, ms). n It applies to all particles of half-integer spin, which are called fermions, electrons and composite particles (protons and neutrons) in the nucleus are fermions. Each of the latter (composite) particles is composed of three quarks, spins add up, so no chance for them to become a boson) The whole periodic table (chemical properties) can be understood by two rules on the basis of the hydrogen atom: 1) The electrons in an atom tend to occupy the lowest energy levels available to them. 2) Pauli exclusion principle. 5

Hydrogen atom model, Schrödinger plus spin The principle quantum number also has letter codes. n= 1 2 3 4. . Letter = K L M N … n = shells (e. g. : K shell, L shell, etc. ) nℓ = subshells (e. g. : 1 s, 2 p, 3 d – where leading number refers to principal quantum number in each hydrogen orbital (3 D spatial wavefunctionsquared) up to two electrons with opposite spin 6

Since n = 3, three subshell types, first is called 3 s (l = 0), second 3 p (l = 1), and third 3 d (l = 2), 18 electrons max when all 9 sub-shells are filled 3 5 M shell L shell K shell 1 1 1 Since l = 0, just one sub shell called s 3 Since n = 2, two subshell types, one is called s (l = 0) the other p (l = 1), 8 electrons max in this shell when all 4 sub -shells are filled Filled and half-filled shells and sub-shells result in spherical symmetric electron density distributions for the corresponding atoms, (Unsoeld’s theorem) 7

Atomic Structure Hydrogen: (n, ℓ, ms) = (1, 0, 0, ±½) in ground state. Both spin states with same probability n In the absence of a magnetic field (and ignoring the hyper-fine structure), the state ms = ½ is degenerate with the ms = −½ state. Helium: (1, 0, 0, ½) for the first electron, (1, 0, 0, −½) for the second electron. n Electrons have anti-aligned (ms = +½ and ms = −½) spins, they are being paired and cancel, total spin becomes an integer (0), i. e. the whole particle becomes a boson, composed of fermions (which are subject to the Pauli exclusion principle, nuclear spin cancel also, happens at there are two protons and two neutrons). Electrons for H and He atoms are in the K shell. H: 1 s He+: 1 s 1 He: 1 s 2 Li++: 1 s 1 just like H There is no sub-shells at all for n = 1, because l = 0, meaning ml also = 0, so just one set with spatial (3 D) quantum numbers (1, 0, 0) Number of sub-shells is number of sets with unique spatial (3 D) quantum numbers 8

Hund’s rule, rather than joining an orbital that is already occupied by one electron, the next electron goes into an orbital all by itself to minimize total energy Ne 9

Atomic Structure How many electrons may be in each shell and subshell? Total For each mℓ: two values of ms 2 For each ℓ: (2ℓ + 1) values of mℓ 2(2ℓ + 1) apparent irregularities Recall: ℓ = 0 1 2 3 4 5 … letter = s p d f g h … ℓ = 0, (s state) can have two electrons. ℓ = 1, (p state) can have six electrons, and so on. The lower ℓ values have more elliptical orbits than the higher ℓ values. Electrons with higher ℓ values are more shielded from the nuclear charge and have, therefore, higher energy levels than those with lower ℓ values. 4 s fills before 3 d – it’s an effects of interactions between electrons 10

La Ac Lu Lr There are 14 boxes, but both Ce and Th just start with two electrons in these boxes, so it is not obvious if La should be in the same column as Se and Y, or if Lu should be in the same column as these two. 11

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note that the f-block is just 14 boxes long, in it the seven f subshells get filled up, this is achieved when we come to Yb and No, then this block ends 13

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Standard long periodic table 15

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Note the closed subshells for any n at the nobel gasses 17

Note the closed subshells for any n at the nobel gasses Lr 18

Groups and Periods in Periodic Table Groups: q Vertical columns. q Same number of electrons in the ℓ orbits. q Can form similar chemical bonds as these are determined by the outermost (most loosely bounded) electrons all atoms have about Periods: the same size q Horizontal rows. q Correspond to filling of the sub-shells. n Beginning of each period shows in atomic radii plot, end of each period shows more or less in ionization energy. 19

The Periodic Table Inert Gases: n Last group of the periodic table n Closed p sub-shell except helium (which has closed s sub-shell) n Zero net electronic spin and large ionization energy n Their atoms interact only very weakly with each other Alkalis: n Single s electron outside an inner core, largest atomic radii n Easily form positive ions with a charge +1 e, highly reactive n Lowest ionization energies n In chemical compounds with valence number 1, e. g. Li 2 O (lithia, 8 Li cations and 4 O anions per unit cell of a crystal), for molecules: H 2 O n Electrical conductivity in solids is relatively good as the electron joins the free electron cloud easily Alkaline Earths: n Two s electrons in outer sub-shell n In chemical compounds with valence number 2, e. g. Mg. O 20 (magnesia), 4 Mg + 4 O per unit cell of a crystal

The Periodic Table Halogens: n Need just one more electron to fill outermost subshell n Form strong ionic bonds with the alkalis, e. g. Na. Cl n More stable configurations would occur when p subshell is completely filled, therefore highly reactive Transition Metals: n Three rows of elements in which the 3 d, 4 d, and 5 d are being filled n Properties primarily determined by the s and p electrons, rather than by the d subshell being filled n Most have d-shell electrons with unpaired spins n As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced 21

The Periodic Table Lanthanides (rare earths): n Have the outside 6 s 2 sub-shell completed n As occurs in the 3 d sub-shell, the electrons in the 4 f sub-shell have unpaired electrons that align themselves n The large orbital angular momentum contributes to ferromagnetic effects Actinides: (all radioactive): Inner sub-shells are being filled while the 7 s 2 sub-shell is completed n Difficult to obtain chemical data because they are all radioactive (last stable atom is Bi, # 83) n Commercial usage of U, Pu, Am 22

Summary Physical foundations are electronic structures their consequence s are all of chemistry !!! All atoms in crystals are of about the same size, 0. 1 – 0. 5 nm diameter, in fact, their size is inferred on how much space they take up in crystals missing in nature 23

Periodic physical and chemical properties of atoms are due to periodic electronic structure, chemical properties depend strongly on the outermost electrons 13. 6 Th 24