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Physics 243A-Surface Physics: Spectroscopy
Problem Set 1
[1.1] If we wish to produce a thin-film magnetic storage device with 100 Gbits/sq.in., each bit is to be 5 times as big in one dimension as the other, the total amount of open area between bits is to be the same as the total area occupied by the bits, and the film thickness of the medium storing the information is to be 10 nm, how many atoms are involved in each bit? Assume for simplicity that the film is pure Co, with a density of 9.0 x 10 22 atoms/cm^3.
[1.2]
(a) Begin with the Maxwell-Boltzmann distribution for molecular velocities in an ideal gas as expressed in x,y,z coordinates, and derive by integration the formula for the rate at which molecules strike a flat surface of unit area perpendicular to one of the axes. From this, determine the time necessary to form a monolayer of gas on a surface, assuming a general sticking probability of P (^) S. (b) Now make the assumption that only the bare surface area remaining on the surface after a given exposure time is active for a particular gas adsorption, and that P (^) S = unity on the bare area, but zero elsewhere. Derive the general form of P (^) S as a function of time for this case. (c) If a surface is exposed to 10 -9^ torr of CO at ambient temperature, how long will it take to form the first monolayer: (i) If P (^) s = unity? (ii) If P (^) s follows the relationship of part (b)?
[1.3] (a) What would be the minimum energy required to take a cube of Pt metal 1.0 cm on a side at room temperature and disperse it into tiny cubic "nanoparticles" of 10 -6^ cm on a side? Assume that this is done in a perfect ultrahigh vacuum environment, with the surfaces in equilibrium with the very low vapor pressure of Pt. (b) Estimate the fraction of the atoms that are on the surface of the 1 cm cube. Of the 10 -6^ cm cube. Assume that the density of Pt atoms is 6.62 x 10 22 cm-3.
[1.4] (a) Use the tables of bulk cohesive energies and densities in a lecture slide to estimate the surface tensions of the elements from Ar to Kr using the method described in lecture) in which a certain fraction of bonds are broken in forming a surface (note a key factor of two relative to the equation presented in Zangwill). Begin by using the density of each element to estimate the average no. of atoms per unit area of surface (N (^) s) for it. (b) Compare the values so derived with those for liquid surface tensions as handed out in lecture or given in Zangwill by plotting them on the same figure. Discuss any systematic discrepancies.
[1.5] (a) Liquid water has a surface tension of 72 erg/cm 2 in air and a contact angle when placed on a clean glass surface of 150°. What does this tell you about the relative values of the surface tensions for the water-glass surface and the glass-air surface? (b) Now assume that a surfactant (e.g. a detergent) has been added to the water before the drop is placed on the surface, and that this lowers the water-air surface tension to 36 erg/cm 2. Also assume that the surfactant molecules are concentrated at the water-air interface, and so do not alter the water- glass surface tension. What can you now say about the contact angle between water+surfactant and glass?
[1.6] Consider the surface of an alloy of the noble metals Cu and Ag whose bulk composition is 90% Cu and 10% Ag in a perfect vacuum with no residual gas reactants. What would be your qualitative expectation for the surface composition of this alloy if it is allowed to reach equilibrium?
[1.7]
On a simple surface with a square lattice (e.g., as Ni(001)) show that there are a well-defined set of possible overlayer structures which can be written in Wood notation as:
(√2 x √2)R45º (√5 x √5)R(tan-1(1/2) = 26.6º) (√10 x √10)R(tan -1(1/3) = 18.4º) Etc. (a) On the following sheet, with represents the substrate square lattice, sketch the unit cells of these three overlayers, and add the fourth member of the series. (b) What will be the coverage in monolayers (ML) relative to the substrate if a single adsorbate atom goes into each corner of the unit cells above? (c) Consider now Ni(001)(√2 x √2)R45º-O, and use the simple “2D diffraction grating” method outlined in lecture to calculate the polar and azimuthal angles of the innermost thirteen LEED spots around the specular beam that would result if a beam of electrons at 150 eV is incident normal to the surface. Indicate which spots would be there for the clean surface, and which spots would be added when the O is adsorbed. Note that some spots arise from both the clean surface and the adsorbate overlayer.
[1.8]
The current in an STM experiment can be approximately calculated from the simple proportionality relation
I ∝ e -2κL^ ,
with L the distance between tip and surface and κ the usual quantity in barrier tunneling problems.
The potential barrier inside κ can be taken to be the work function of the surface, and we will take it to be 4.0 eV.
If the tip is brought to within 5 Å of the surface, with what precision in % would the tunneling current have to be measured to reliably detect a change in surface height of 0.1 Å? Note that the precision will have to be about 10x better than the effect you want to see to be able to measure it accurately.
[2.3]
The adsorption of molecular nitrogen on the (110) surface of Ni has been studied with various techniques as a function of temperature and pressure. A stable overlayer structure is found to form at a coverage of 1/2 monolayer. In one such experiment, the N 1s core photoelectron intensity was monitored as a function of temperature and at various constant pressures for this system, with the results of these experiments attached. The N 1s intensity can here be assumed to be proportional to the no. of N atoms present on the surface.
(a) Determine the dependence of the isostearic heat of adsorption on coverage. Is this a chemisorption or a physisorption process; explain your answer.
(b) Do you see any evidence in your results of the point at which the stable overlayer is formed?
[2.4]
On p. 207, Zangwill asserts that "the isotherms measured for carbon monoxide adsorbed at low coverage onto a single crystal Pd(111) surface follow the Langmuir isotherm at high temperature (Fig. 9.2)". Show this quantitatively for several coverages and temperatures, and thus indicate the region of Fig. 9.2 over which the adsorption kinetics is in fact Langmuir in character. A hint here is to rearrange the fundamental Langmuir equation so that it yields a linear relationship between 1/(coverage) and 1/(pressure), and then see which portions of the data fit this variation.
[2.5]
Consider a sample of arsenic (As), which has a free-atom electronic configuration of 1s 2 …4s 2 4p^3 , that is exposed to incident soft x-rays of 2,000 eV energy. Use the X-Ray Data Booklet handed out in class, in particular Table 1-1: electron binding energies, Fig. 1-1: x-ray emission line nomenclature, Table 1-2: x-ray emission energies, Fig. 1-3: energies of most intense Auger peaks, and Table 5.2, which includes electron configurations and other information, to answer the following.
(a) Are all of the electron binding energies of As present in the table of experimental values in Table 1-1? If not, explain why some might be routinely left off such tabulations.
(b) Predict the kinetic energies of all photoelectron peaks that would be observed, noting that the relativistic spin-orbit interaction splits any non- s (l ≠ 0) subshell binding energy into two components: j = l ± 1/2 and that an alternate nomenclature for the one-electron energy levels is 1s = K, 2s = L 1 , 2pj=1/2 = L 2 , 2pj=3/2 = L 3 ,…etc.
(c) Use appropriate binding energies to calculate the Kα 1 , Kα 2 , and Lβ 1 x-ray emission energies for As, and compare your answer to those given in the handbook.
(d) Predict all of the Auger transitions of As having an L-shell (n = 2) electron as the lowest level, and two M-shell (n = 3) electrons as the highest levels (i.e. the L (^) jMkMl Auger series, which has 45 possible members), and compare your answers to those of the most intense lines in Fig. 1-3 of the handbook. Use the most accurate formula for predicting these energies from binding energies, as given in Eq. 3.25 of Woodruff and Delchar, which for present purposes can be written as:
Ekin (LjMkMl) = EbZ^ (Lj)-1/2[EbZ^ (Mk)+ E (^) bZ+1(MI)+EbZ+1(Mk)+EbZ^ (MI)].
Assume also that the relative intensity of a given peak can be roughly estimated by just multiplying the degeneracies of the levels involved, which are always 2j+1, so that
I(LjMkMl) ≈ (2jj+1)(2jk+1)(2jl+1),
and finally plot your results as a bar graph versus kinetic energy. Do your calculations, with qualitative allowance for some kind of broadening due to experimental resolution) qualitatively or quantitatively agree with the positions expected for the strongest peaks?
Data for Problem [2.3]: 1 mbar = 10 -3^ atm = 10 -3^ (760 torr) = 0.76 torr
values of m l and m l ' in the subshells involved, and Kn l ,n' l ' = an analogous average for an exchange
integral that is relevant when the two electron spins in n l and n' l ' are parallel.
(a) Make a bar plot of the average radii along r and discuss whether there is there evidence for a principal quantum no. "shell structure" in these nos.
(b) Which is larger, J1s ,2 s or J1s ,3 s and why?
(c) Which is smaller, J1s ,4 f or K1s ,4 f and why?
(d) Which is larger, J (^) 4d ,4d or J (^) 4d ,4 f and why?
(e) Which is larger, J2 p (^) x ,2 px or J (^) 2 p (^) x ,2 py and why?
(f) Which is larger, J2 p (^) x ,2 px or J (^) 3 px ,3 px and why?
(g) Is the effective nuclear charge seen by the 4s electrons greater than or less than that seen by the 4d electrons? Explain your answer.
(h) Estimate the values of J1s ,4 s , J1s ,5 s , and J1s ,6 s by means of a classical approximation in which the actual charge distributions are replaced by thin spherical shells of charge at rn l. How do your values compare with the exact calculated nos. of 2.726, 1.080, and 0.326 a.u. (1 a.u. = 1 Hartree = 27.21 eV).
(i) Is there any way to use a classical approximation to estimate K1s ,4 s , K1s ,5 s , and K1s ,6 s? Explain why or why not?
[3.3]
As a very simple example of a many-electron atom, let us consider some energy shifts and splittings in the 3-electron atom Li with initial configuration 1s 2 2s 1 , and ground-state L-S coupling as 2 S (total S = 1/2, total L = 0).
(a)-(c) First consider the change in the 1s binding energy associated with the removel of a 2s electron to form Li +1^ 1s 2.
(a) Use a Koopmans' Theorem argument to estimate this "chemical shift" in binding energy 1 0 1 0 ΔE (1s,Lib Li ) ε1s ,Li (^) + ( ε1s ,Li )
- (^) − = − − − in terms of coulomb and exchange integrals, assuming
that the 1s and 2s orbitals are the same in both species. You need not evaluate any integrals specifically. Note that the two 1s electrons might have a slightly different shift in this simple picture. (b) Now use a simple classical argument to estimate this shift in binding energy, noting that accurate Hartree-Fock calculations yield values of r1s = 0.573 Bohr and r2 s = 3.87 Bohr. (c) The experimental 2s binding energy in Li 0 is^ ∼0.4 Rydberg = 5.4 eV (Phys. Rev A 13, 1466 (1976)). Estimate this binding energy from a purely hydrogenic model via the equivalent core approach, and see how close you come. Explain qualitatively the origin of any error you find. By what factor would you have to multiply the 1s charge of 2e in the equivalent core analysis to give the correct binding energy? Such thinking leads to what is referred to as the "fractional screening" of a given subshell.
(d)-(f) Now consider the energy splitting associated with the two final states of 1s photoemission from atomic Li with final configuration 1s 1 2s 1 and two different total spins S = 0 (singlet) and S = 1 (triplet). We desire to prove that the so-called Van Vleck Theorem as stated in Eq. (148) in "Basic Concepts of XPS" Δ[ E ( ns )]b = ( 2S + 1 )Kns ,n l , is valid for this case, where
S is the total initial spin of the system , ns is a core s-subshell, and n l is a partially filled outer
subshell.
(d) Write down the full Hamiltonian H
for the final Li+1^ state involved, labelling each term to indicate its significance.
(e) Write down also the correct two-electron wave functions for two of the multiplet-split final L-S states of the ion corresponding to the singlet and triplet spin states possible, specifying the space and spin parts fully. Here, you can use the model of the excited states of the two- electron He atom that is described in any quantum mechanics/quantum chemistry textbook (see appended pages from Levine, Quantum Chemistry, for example), and as quantum numbers for the final states, you can use specifically: Ψ 0 ( L = 0, ML = 0; S = 0, MS = 0 ) and Ψ 1 ( L = 0,ML = 0; S = 1,MS = 0 ). Assume again that the 1s and 2s orbitals are identical between these two states for simplicity. Now calculate Ψ 0 | H |ˆ Ψ 0 and Ψ 1 | H |ˆ Ψ 1 with the Hamiltonian of part (d), and thus show finally that the energy difference E 1 − E 0 so obtained is in complete agreement with the Van Vleck Theorem. Again, you need not evaluate specifically any angular or radial integrals here, as many terms will just cancel out in the energy difference.
(f) The integral K1s ,2 s has been calculated for Li to be 0.0285 Hartree (1 Hartree = 27.21 eV).
Calculate the multiplet splitting in photoelectron emission for this case, and indicate whether it should be observable in a spectrum with an overall experimental resolution of 0.1 eV.
other parameters involved, including electron inelastic attenuation length from the so-called TPP-2M formula of Tanuma, Powell, and Penn (Eq. 38 in handout by Jablonski and Powell), and indicating the sources of all your data and any simplifying assumptions that you have made. Plot this intensity finally as a function of the electron emission direction θ.
(c) The analyzer in (b) has a resolution of δE/Eo =0.01 (i.e. 1%). How much would the Fe 2p photoelectrons have to be retarded from their initial value Ekin to measure their energies with a spectrometer resolution of 1 eV?
(d) Would the spin-orbit splitting of the Fe 2p binding energy be resolvable in the above XPS experiment, and if so, what would you expect for the approximate intensity ratio I(Fe 2p3/2)/I(Fe 2p1/2)?
(e) Does Fe exhibit any Cooper minima in its cross sections, and if so, where? You can reason by analogy with the curves for its next-door neighbor Mn that were discussed in lecture, or go to the website http://ulisse.elettra.trieste.it/services/elements/WebElements.html where you can obtain the exact curves for Fe.
(f) If Fe 3d photoemission were to show resonant effects, at which energies might you expect this to occur?
(g) Finally, note that the photoelectric effect is the dominant absorption mechanism for x-rays of this energy as they travel through a solid. (See Figs. 3.1 and 3.2 in the X-Ray Data Booklet.) Consider x-rays incident normal to a surface of pure Fe. Use the total photoelectric cross section over all subshells per atom of Fe for AlKα radiation and the Fe atomic density to calculate the relative rate at which the photon flux Iz at depth z would be lost through absorption (dI/Iz)/dz. Show that this permits determining an exponential attenuation length for x-rays, and calculate its number for Fe. (See p. 1-38 and Section 1-5 in the X-Ray Data Booklet.) Check your answer by going into the website operated by the Center for X-ray Optics at LBNL (http://www-cxro.lbl.gov/optical_constants/) and calculating the attenuation length there. Comment on any discrepancies noted. Also comment on these values relative to typical electron attenuation lengths.
[4.4] The surface of a metal "M" has been studied by XPS (using non-monochromatized Mg Kα
radiation for excitation) and by AES (using electron beam excitation and expressing the results as derivatives dN/dE of normal spectra). Spectra have been obtained from both the clean surface and the surface after bombardment with a beam of reactive ions "X+". In the XPS experiment, the electrons were emitted normal to the surface, and the angle between photon incidence and electron exit was 45o. The bombardment was continued until a thin layer of a new compound with the simple formula MX was formed. Some spectra obtained in this study are shown on the next page.
(a) Identify all of the peaks labelled a, b, c,...m as to atom and/or process of origin. What atom(s) are present in the ion beam? What is the origin of the doublet in Fig. 3? What then is the compound MX?
(b) Use the intensities of peaks l and m in Fig. 3 to estimate the thickness of the layer formed, assuming for simplicity that the inelastic attenuation length in the layer is equal to that in the substrate. Assume also that the density of M atoms in the substrate is 6.0x10^22 atoms/cm^3 , and that the density of M atoms in the overlayer is 1/2 of that. Be as quantitative as you can here, indicating the sources of your inputs and any other simplifying assumptions that are needed to reach a final number.
determine the photon energy to within 0.1% at a photon energy of 500 eV? Be careful here to determine the distance of the detector from the grating, as it must lie on the Rowland Circle. How does the required detector resolution change if it is tilted so as to have a grazing angle of photon incidence on the detector of 5º?
Physics 243A-Surface Physics of Materials
Problem Set 5
[5.1] Use the LBNL Center for X-ray Optics Website (http://www-cxro.lbl.gov/optical_constants/) to
answer the following questions, inserting figures from it to illustrate your answer.
(a) What evidence do you find in the index of refraction for Ag in the energy range 2500-5000 eV of specific absorption edges and what are they? Has spin-orbit splitting been taken into account in this data?
(b) From this index of refraction data, estimate the critical incidence angle at which significant reflectivity will begin for 2500 eV photons? Does an actual calculation of reflectivity as a function of incidence (“grazing”) angle agree with your number?
(c) Calculate the x-ray attenuation length in Ag at 2500 eV as a function of incidence angle θ from
normal (90°) to parallel (0°) to the surface, including a blowup of the region below the critical
angle. Now replot this data as a function of sin θ, and determine over what range the x-ray
attenuation depth as measured from the surface follows a relationship like Λ h ν ( θ =0)sin θ that is
the same in form which we have discussed for electron escape depths from a surface.
[5.2] Core electron binding energies have been measured for various gas-phase chlorinated
benzenes, with structures and spectra shown below.
(a) Assign the peaks observed to various carbon atoms in the molecule, and predict the relative intensities of the two components for each case. Are your predictions consistent with experiment?
