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1AMQ P.H. Regan & W.N.Catford 1 1AMQ, Part III The Hydrogen Atom 5 Lectures n Spectral Series for the Hydrogen Atom. n Bohr’s theory of Hydrogen. n The Hydrogen atom in quantum mechanics. n Spatial quantization and electron spin. n Fine Structure and Zeeman splitting. K.Krane, Modern Physics, Chapters 6 and 7 Eisberg and Resnick, Quantum Physics, Chapters 4, 7 & 8
1AMQ P.H. Regan & W.N.Catford 2 Atomic Line Spectra Light emitted by free atoms has fixed or discrete wavelengths. Only certain energies of photons can occur (unlike the continuous spectrum observed from a Black Body). Atoms can absorb energy (become excited) by collisions, fluorescence (absorption and re- emission of light) etc. The emitted light can be analysed with a prism or diffraction grating with a narrow collimating slit (see figure below). The dispersed image shows a series of lines corresponding to different wavelengths, called a line spectrum. (note you can have both emission and absorption line spectra)
1AMQ P.H. Regan & W.N.Catford 3 Hydrogen-The Simplest Atom Atomic hydrogen (H) can be studied in gas discharge tubes. The strong lines are found in the emission spectrum at visible wavelengths, called H and . More lines are found in the UV region, more which get closer and closer until a limit is reached.
1AMQ P.H. Regan & W.N.Catford 4 The visible spectrum lines are the Balmer Series. Balmer discovered that the wavelengths of these lines could be calculated using the expression, where, R H = the Rydberg constant for hydrogen = 1.097x10 7 m -1 = 1/911.76 angstroms Balmer proposed more series in H with wavelengths given by the more general expression, n 1 >n 2 for positive integers. These series are observed experimentally and have different names n 2 = 1 Lyman Series (UV) = 2 Balmer Series (VIS) = 3 Paschen Series (IR) = 4 Bracket Series (IR) = 5 Pfund Series (IR)
6 These lines are also seen in stellar spectra from absorption in the outer layers of the stellar gas.
1AMQ P.H. Regan & W.N.Catford 7 The Rydberg-Ritz Combination Principle is an empirical relationship which states that if and 2 are any 2 lines in one series, then | is a line in another series. Electron Levels in Atoms. Discrete wavelengths for emis/abs. lines suggests discrete energy levels. Balmer’s formula suggests that the allowed energies are given by (cR H / n 2 ) (for hydrogen). A more detailed study is possible using controlled energy collisions between electrons and atoms such as the Franck-Hertz experiment. Excitation Energy ground state, (ie. lowest one) 1 st excited state e - s with enough energy can cause this transition.
9 Bohr Theory: Bohr’s postulates defined a simple ordered system for the atom.
1AMQ P.H. Regan & W.N.Catford 10 a 0 is the Bohr radius (ie. smallest allowed) For hydrogen, Z=1 and a 0 =0.529Angstroms, ie model predicts ~10 -10 m for atomic diameter. For these allowed radii, we can calculate the allowed energies of the levels in the Bohr atom
1AMQ P.H. Regan & W.N.Catford 11 Thus by substitution, we obtain ie. quantization of the angular momentum leads in the Bohr model to a quantization of the allowed energy states of the atom.
1AMQ P.H. Regan & W.N.Catford 12 For H, the ground state energy is given by
1AMQ P.H. Regan & W.N.Catford 13 The allowed energies let us calculate the allowed frequencies for photons emitted in transitions between different atomic levels ie. h E initial -E final If n i and n f are the quantum numbers of the initial and final states, then, for H, if n f < n i This prediction of the Bohr model compares with an expt. value of 1/911.76Angs, ie. accurate to with 0.05% in H. (Exact agreement if motion of nucleus is included. ie nucleus and electron move around the atoms centre of mass.) Balmer series correspond to when n f =2 Lyman series correspond to when n f =1
1AMQ P.H. Regan & W.N.Catford 14 The Lyman series correspond to the highest energy (shortest wavelength) transitions which H can emit.
1AMQ P.H. Regan & W.N.Catford 15 Deficiencies of the Bohr Model No proper account of quantum mechanics (de Boglie waves etc.) It is planar and the `real world’ is three dimensional. It is for single electron atoms only. It gets all the angular momenta wrong by one unit of h/2
1AMQ P.H. Regan & W.N.Catford 16 The Hydrogen Atom in Quantum Mechanics The e- is bound to the nucleus (p) by the Coulomb pot. This constraint leads to energy quantization. The Time Independent Schrodinger Equation can be used. For hydrogen, (Z=1) It is easier to solve this in spherical polar rather than cartesian coordinates, this we have This equation is said to be separable ie, z x y r
1AMQ P.H. Regan & W.N.Catford 17 The three spatial dimensions (r, lead to 3 quantum numbers, which relate to How far the orbital is from the nucleus (n) How fast the orbit is (ie. angular momentum) (l) Then angle of the orbit in space (m l ). The quantum numbers and their allowed values are n principle quantum number, 1,2,3,4,5… l angular momentum q.n. 0,1,2,3,4..(n-1) m l magnetic q.n. -l-l+1,...-1,0,1,….l-1,l
1AMQ P.H. Regan & W.N.Catford 19 Energies and Degeneracies: Each solution, n,l,ml has an energy that depends only on n (E n =Bohr value) and there are n 2 solutions (ie. all the possible values of l and m l ) for each energy E n. Radial Wavefunctions. Determined by n (main factor in determining the radius) and l measures the electrons angular momentum. If L is the angular momentum vector, then
1AMQ P.H. Regan & W.N.Catford 22 Spatial Quantization and Electron Spin The angular wavefunctions for the H atom are determined by the values of l and m l. Analysis of the w.functions shows that they all have Ang. Mom. given by l. and projections onto the z-axis of L of L z = m l h/2 L Z Quantum mechanics says that only certain orientations of the angular momentum are allowed, this is known as spatial quantization. For l=1, m l =0 implies an axis of rotation in the x-y plane. (ie. e - is out of x-y plane), m l = +1 or -1 implies rotation around Z (e - is in or near x-y plane)
1AMQ P.H. Regan & W.N.Catford 23 Krane p216 The picture of a precessing vector for L helps to visualise the results
1AMQ P.H. Regan & W.N.Catford 24 Krane p219 Product of Radial and Angular Wavefunctions n=1 spherical n=1, l=0 spherical, extra radial bump l=1, m l =+-1 equatorial l=1, m l =0 polar n=3 spherical for l=0 l=1,2 equatorial or polar depending on m l.
1AMQ P.H. Regan & W.N.Catford 25 Moving charges are currents and hence create magn. fields. Thus, there are internal B- fields in atoms. Electrons in atoms can have two spin orientations in such a field, namely m s =+-1/2….and hence two different energies. (note this energy splitting is small ~10 -5 eV in H). We can estimate the splitting using the Bohr model to estimate the internal magnetic field, since In the Bohr model the magnetic moment, , is Magnetic Fields Inside Atoms Electron Spin Electrons have an intrinsic spin which is also spatially quantized. Spinning charges behave like dipole magnets. The Stern-Gerlach experiment uses a magnetic field to show that only two projections of the electron spin are allowed. By analogy with the l and m l quantum numbers, we see that S=1/2 and m s =+-1/2 for electrons.
1AMQ P.H. Regan & W.N.Catford 27 This energy shift is determined by the relative directions of the L and S vectors. The -ve sign indicates that the vectors L and L point in opposite directions. The z-component of L is given in units of the Bohr magneton, B, where r L e - i
1AMQ P.H. Regan & W.N.Catford 28 Fine Structure and Zeeman Splitting For atomic electrons, the relative orbital motion of the nucleus creates a magnetic field (for l=0). The electron spin can have m s =+-1/2 relative to the direction of the internal field, B int. The state with s aligned with B int has a lower energy than when anti- aligned.
1AMQ P.H. Regan & W.N.Catford 29 Spin-Orbit Coupling For a given e -, L and S add together such that J=L+S
1AMQ P.H. Regan & W.N.Catford 30 Atomic Doublets Levels with l=0 are split into energy doublets, called fine structure, due to spin-orbit coupling. The fine structure is approx. 10 -5 eV in Hydrogen and increases as Z 4 for heavier elements. We can use the Bohr model get an estimate of the spin-orbit splitting, by assuming an electron orbit of radius r, carrying current i establishing a magnetic field, B at the centre of loop. Thus,
1AMQ P.H. Regan & W.N.Catford 31 Zeeman Effect If the atom is placed in an external magnetic field, the e - orbital angular momentum (l) can align with the field direction.While this magnetic field is `switched on’ there will be an extra splitting of the energy levels (for l=0).
1AMQ P.H. Regan & W.N.Catford 32 Splitting is further affected by m s =+-1/2 n=1, l=1, j=1/2 splits into m j =-1/2, +1/2 n=1, l=1, j=3/2 splits into m j =-3/2,-1/2,+1/2,+3/2