UNIT-III ATOM WITH ONE ELECTRON

Bohr’s Model Niels Bohr suggested that the problem about hydrogen spectrum can be solved if we can make some assumptions. According to classical theory, the frequency of the electromagnetic waves emitted by a revolving electron is equal to the frequency of revolution. As the electrons radiate energy, their angular velocities would change continuously and they would emit a continuous spectrum against line spectrum actually observed. So, Bohr concluded that even if electromagnetic theory successfully explained the macroscopic phenomenon, it could not be applied to explain microscopic phenomenon, that in atomic scale.

An unsatisfactory model for the hydrogen atom According to classical physics, light should be emitted as the electron circles the nucleus. A loss of energy would cause the electron to be drawn closer to the nucleus and eventually spiral into it.

Bohr's Postulates Every atom consists of nucleus and suitable number of electrons revolved around the nucleus in circular orbits. The force of attraction between the electron and the nucleus provide necessary centripetal force for the circular motion. Electrons revolved only in certain non-radiating orbits called stationery orbits for which the total angular momentum is an integral multiple of h/2Л, where h is planck's constant and L is the Angular momentum of the revolving electrons L=mvr=nh/2Л Radiation occurs when an electron jumps from higher orbit to a lower orbit i.e., E2 - E1 = hf, where f is frequency of radiation

Bohr’s Model of the Hydrogen Atom

Hydrogen Atom Spectrum Line spectra in other spectral regions also were observed: Lyman series ultraviolet Balmer visible Paschen, Brackett, Pfund infrared wavelengths of emission is given by where m = 1, 2, 3,… and n = 2, 3, …(always at least m + 1) Longest wavelength observed when n = m + 1. Shortest wavelength observed when n = .

Correction for finite nuclear mass

Bohr’s correspondence principle The predictions of the quantum theory for the behaviour of any physical system must correspond to the prediction of classical physics in the limit in which the quantum number specifying the state of the system becomes very large: quantum theory = classical theory At large n limit, the Bohr model must reduce to a “classical atom” which obeys classical theory

Bohr’s Quantum Condition

Modification of the Bohr Model Modifications to the Bohr Model Elliptical orbits Orbital quantum number Orbital magnetic quantum number Spin magnetic quantum number Successes of the Bohr Theory Model of the Atom Explained Atomic Spectra Predicted Rydberg Constant Gave expression for the radius of an atom Predicted the energy levels of the hydrogen atom

Sommerfeld’s model Fine structure: a splitting of spectral lines due to spin-orbit interaction Sommerfeld’s explanation for an elliptical orbit:

Frank-Hertz experiment Shows the excitation of atoms to discrete energy levels Mercury vapour is bombarded with electron accelerated under the potential V (between the grid and the filament) A small potential V0 between the grid and collecting plate prevents electrons having energies less than a certain minimum from contributing to the current measured by ammeter

The electrons that arrive at the anode peaks at equal voltage intervals of 4.9 V As V increases, the current measured also increases The measured current drops at multiples of a critical potential V = 4.9 V, 9.8V, 14.7V

Interpretation As a result of inelastic collisions between the accelerated electrons of KE 4.9 eV with the the Hg atom, the Hg atoms are excited to an energy level above its ground state At this critical point, the energy of the accelerating electron equals to that of the energy gap between the ground state and the excited state This is a resonance phenomena, hence current increases abruptly After inelastically exciting the atom, the original (the bombarding) electron move off with too little energy to overcome the small retarding potential and reach the plate As the accelerating potential is raised further, the plate current again increases, since the electrons now have enough energy to reach the plate Eventually another sharp drop (at 9.8 V) in the current occurs because, again, the electron has collected just the same energy to excite the same energy level in the other atoms

If bombared by electron with Ke = 4 If bombared by electron with Ke = 4.9 eV excitation of the Hg atom will occur. This is a resonance phenomena Hg First excitation energy of Hg atom DE1 = 4.9eV Ke reaches 4.9 eV again here Third resonance initiated Ke reaches 4.9 eV again Ke= 4.9eV second resonance initiated Ke= 0 after first resonance Ke= 0 Ke= 0 after second resonance Hg Hg Hg Electron continue to be accelerated by the external potential until the second resonance occurs Electron continue to be accelerated by the external potential until the next (third) resonance occurs First resonance at 4.9 eV Plate C Plate P electron is accelerated under the external potential V = 14.7V

The higher critical potentials result from two or more inelastic collisions and are multiple of the lowest (4.9 V) The excited mercury atom will then de- excite by radiating out a photon of exactly the energy (4.9 eV) which is also detected in the Frank-Hertz experiment The critical potential verifies the existence of atomic levels

Quantum Numbers Schrödinger’s equation requires 3 quantum numbers: Principal Quantum Number, n. This is the same as Bohr’s n. As n becomes larger, the atom becomes larger and the electron is further from the nucleus. ( n = 1 , 2 , 3 , 4 , …. ) Angular Momentum Quantum Number, l. This quantum number depends on the value of n. The values of l begin at 0 and increase to (n - 1). We usually use letters for l (s, p, d and f for l = 0, 1, 2, and 3). Usually we refer to the s, p, d and f-orbitals. Magnetic Quantum Number, ml. This quantum number depends on l . The magnetic quantum number has integral values between -l and +l . Magnetic quantum numbers give the 3D orientation of each orbital.

Quantum Number Symbol Values Description Principal n 1,2,3,4,… Size & Energy of orbital Angular Momentum l 0,1,2,…(n-1) for each n Shape of orbital Magnetic ml l…,0,…+ l for each l Relative orientation of orbitals within same l Spin ms +1/2 or –1/2 Spin up or Spin down Angular Momentum Quantum Number Name of Orbital s (sharp) 1 p (principal) 2 d (diffuse) 3 f (fundamental) 4 g

Orbital Magnetic Moment

Electron Spin 1925: S.A Goutsmit and G.E. Uhlenbeck suggested that an electron has an intrinsic angular momentum (i.e. magnetic moment) called its spin. The extra magnetic moment μs associated with angular momentum S accounts for the deflection in SGE. Two equally spaced lined observed in SGE shows that electron has two orientations with respect to magnetic field.

The Stern-Gerlach Experiment The Stern-Gerlach Experiment (SGE) is performed in 1921, to see if electron has an intrinsic magnetic moment.

A beam of hot (neutral) Silver (47Ag) atoms was used. The beam is passed through an inhomogeneous magnetic field along z axis. This field would interact with the magnetic dipole moment of the atom, if any, and deflect it. Finally, the beam strikes a photographic plate to measure,if any, deflection.

Why Neutral Silver atom No Lorentz force (F = qv x B) acts on a neutral atom, since the total charge (q) of the atom is zero. Only the magnetic moment of the atom interacts with the external magnetic field. Electronic configuration: 1s2 2s2 2p6 3s2 3p6 3d10 4s1 4p6 4d10 5s1 So, a neutral Ag atom has zero total orbital momentum. Therefore, if the electron at 5s orbital has a magnetic moment, one can measure it. Why inhomogenous magnetic Field In a homogeneous field, each magnetic moment experience only a torque and no deflecting force. An inhomogeneous field produces a deflecting force on any magnetic moments that are present in the beam.

In the experiment, they saw a deflection on the photographic plate In the experiment, they saw a deflection on the photographic plate. Since atom has zero total magnetic moment, the magnetic interaction producing the deflection should come from another type of magnetic field. So electron’s (at 5s orbital) acted like a bar magnet. If the electrons were like ordinary magnets with random orientations, they would show a continues distribution of pats. The photographic plate in the SGE would have shown a continues distribution of impact positions. However, in the experiment, it was found that the beam pattern on the photographic plate had split into two distinct parts. Atoms were deflected either up or down by a constant amount, in roughly equal numbers. z component of the electron’s spin is quantized.

Electron Spin Orbital motion of electrons, is specified by quantum number l. Along the magnetic field, l can have 2l+1 discrete values.

Similar to orbital angular momentum L, the spin vector S is quantized both in magnitude and direction, and can be specified by spin quantum number s. We have two orientations: 2 = 2s+1  s = 1/2 The component Sz along z axis:

It is found that intrinsic magnetic moment (μs) and angular momentum (S) vectors are proportional to each other: where gs is called gyromagnetic ratio. For the electron, gs = 2.0023

Spin Orbit Coupling Interaction of the spin of an electron with the magnetic field generated by its own orbital motion

Larmor Precession When a magnetic moment m is placed in a M.F.(B), it experiences a torque. For a static magnetic moment or a classical current loop, this torque tends to line up the magnetic moment with the magnetic field B, so this represents its lowest energy configuration. But if the magnetic moment arises from the motion of an electron in orbit around a nucleus, the magnetic moment is proportional to the angular momentum of the electron. The torque exerted then produces a change in angular momentum which is perpendicular to that angular momentum, causing the magnetic moment to precess around the direction of the magnetic field rather than settle down in the direction of the magnetic field. This is called Larmor precession.

The precession angular velocity called Larmor frequency is

ZEEMAN EFFECT

What is the Zeeman Effect In absence of magnetic fields, electrons emit photons of certain energy in atomic transitions In presence of magnetic fields, atom forms a dipole, wants to align itself with B-Field. So it has a potential energy of orientation This energy adds to the energy of atomic transitions, changes the energy of the photon emitted. Different energy = Different Wavelength. We can observe and measure the shifts in wavelength

Two Different Types Normal and Anomalous Zeeman effect Magnetic dipole is sum of orbital and spin If spins cancel, normal Zeeman effect If not, anomalous Zeeman effect

Normal Zeeman effect Observed in atoms with no spin. Total spin of an N-electron atom is Filled shells have no net spin, so only consider valence electrons. Since electrons have spin 1/2, not possible to obtain S = 0 from atoms with odd number of valence electrons. Even number of electrons can produce S = 0 state (e.g., for two valence electrons, S = 0 or 1). All ground states of Group II (divalent atoms) have ns2 configurations => always have S = 0 as two electrons align with their spins antiparallel. Magnetic moment of an atom with no spin will be due entirely to orbital motion.

Interaction energy between magnetic moment and a uniform magnetic field is Assume B is only in the z- direction. Hence, interaction energy of the atom is where ml is the orbital magnetic quantum number. This equation implies that B splits the degeneracy of the ml states evenly.

Normal Zeeman effect transitions Selections rules for ml: ml = 0, ±1. Consider transitions between l=0 and l=1 atomic levels. Allowed transition frequencies are therefore, Emitted photons also have a polarization, depending on which transition they result from.

Anomalous Zeeman effect Occurs in atoms with non-zero spin => atoms with odd number of electrons. In LS-coupling, the spin-orbit interaction couples the spin and orbital angular momenta to give a total angular momentum according to In an applied B-field, J precesses about B at the Larmor frequency. L and S precess more rapidly about J to due to spin-orbit interaction. Spin-orbit effect therefore stronger.

Interaction energy of atom is equal to sum of interactions of spin and orbital magnetic moments with B-field where gs= 2, and the < … > is the expectation value. The normal Zeeman effect is obtained by setting

In the case of precessing atomic magnetic system neither Sz nor Lz are constant. Only is well defined. Must therefore project L and S onto J and project onto z-axis

The angles 1 and 2 can be calculated from the scalar products of the respective vectors which implies that Now, using

so that Similarly, and

We can therefore write This can be written in the form where gJ is the Lande g-factor given by This implies that and interaction energy with the B-field is

Mathematical Relation Energy Shift Landé g-factor Bohr Magneton, Quantum number, due to combination of orbital and spin magnetic dipoles Magnetic Field

Anomalous Zeeman effect spectra Selection rules for J and mj

Na d-lines produced by 3p  3s transition