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Ch 41 Atomic Structure © 2005 Pearson Education

41.1 The Hydrogen Atom energy levels of hydrogen magnitude of orbital angular momentum components of orbital angular momentum © 2005 Pearson Education

Example 41.1 How many distinct (n, l, m l ) state of the hydrogen atom with =3 are there? Find the energy of these states. How many distinct (n, l, m l ) state of the hydrogen atom with =3 are there? Find the energy of these states.ANS: © 2005 Pearson Education

probability that the electron is between r and r+dr smallest r, Bohr model © 2005 Pearson Education Electron Probability Distributions

© 2005 Pearson Education 3-D probability distribution of hydrogen atom

© 2005 Pearson Education Cross sections of 3-D probability distributions

41.2 The Zeeman Effect © 2005 Pearson Education Zeeman effect is the splitting of atomic energy levels and the associated spectrum lines when the atoms are placed in magnetic field

Bohr magneton Magnetic moment: Vector Area For I=ev/2πr © 2005 Pearson Education In Bohr model, L=nh/2. For n=1, µ  µ B orbital magnetic interaction energy

© 2005 Pearson Education Selection Rules

41.3 Electron Spin © 2005 Pearson Education

components of spin angular momentum magnitude of spin angular momentum © 2005 Pearson Education Spin Quantum Number

41.4 Many-Electron Atoms and the Exclusion Principle allowed values of quantum numbers © 2005 Pearson Education

The Exclusion Principle No two electrons can occupy the same quantum-mechanical state in a given system. That is no two electrons in an atom can have the same values of all four quantum numbers (n, l, m l, m s ).

© 2005 Pearson Education

41.5 X-Ray Spectra © 2005 Pearson Education

Moseley’s law © 2005 Pearson Education

The Schrodinger equation for the hydrogen atom gives the same energy levels as the Bohrmodel. If the nucleus has charge Ze, there is an additional factor of Z 2 in the numerator of Eq. (41.3). The Schrodinger equation also shows that the possible magnitudes L of orbital angular momentum are given by Eq. (41.4), and that the possible values of the z- component of orbital angular momentum are given by Eq. (41.5). (See Example 41.1 and 41.2) © 2005 Pearson Education

The probability that an atomic electron is between r and r + dr from the nucleus is P(r) dr, given by Eq. (41.7). Atomic distances are often measured in units of a, the smallest distance between the electron and the nucleus in the Bohr model. (See Example 41.3) © 2005 Pearson Education

The interaction energy of an electron (mass m) with magnetic quantum number m l in a magnetic field along the +z-direction is given by Eq. (43.17) or (43.18), where is called the Bohrmagneton. (See Example 41.4)

© 2005 Pearson Education An electron has an intrinsic spin angular momentum of magnitude S, given by Eq. (41.20). The possible values of the z-component of the spin angular momentum are, where m s =± 1/2. (See Examples 41.5 and 41.6)

In a hydrogen atom, the quantum numbers n, l, m l, and m s of the electron have certain allowed values given by Eq. (41.26). In a many- electron atom, the quantum numbers for each electron are the same, but the energy levels depend on both n and l because of screening, the partial cancellation of the field of the nucleus by the inner electrons. The idea of screening is related to the central-field approximation, in which each electron moves in the electric field of the nucleus and of the averaged-out, spherically symmetric charge distribution of all the remaining electrons. If the effective (screened) charge attracting an electrons is Z eff e, the energies of the levels are given approximately by Eq. (41.27). (See Examples 41.7 and 41.8) © 2005 Pearson Education

Moseley’s law states that the frequency of a K α x ray from a target with atomic number Z is given by Eq. (41.29). Characteristic x-ray spectra result from transitions to a hole in an inner energy level of an atom. (See Example 41.9)

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