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AH (CHCO) as 152+2 kcal/mol (635 kJ/mol) and AH (CH CO) as 11.4 kcal/mol (47.7 kJ/mol) [24], it is easily shown that P.A. (acetone) is 203±2 kcal/mol 849±8 kJ/mol). An upper limit for the P.A. (acetone) may be estimated either by determining (i) whether or not (CHCOCH)H will transfer a proton to a molecule (M) which has a known (and presumably higher) P.A. than acetone or (ii) by determining that the reaction

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does not occur, and is therefore presumably endothermic. This combined method has been used successfully by Munson [16] to establish the relative vapor phase basicities for a variety of polar molecules. We have used the second alternative mentioned above to estimate an upper limit for the P.A. of acetone since we found it difficult to generate large yields of (CHCOCH)H by any suitable combinations of reactants and additives. Our approach was to establish whether or not NH; would participate in a proton transfer reaction with acetone. The most recent estimate of the P.A. of NH is 207 kcal/mol (865 kJ/mol) [12b], which is very close to the lower limit for acetone derived above. The results of a typical experiment in which NH3-CH COCH mixtures were photolyzed at 106.7-104.8 nm are shown in figure 3. It is apparent that the major initial reaction involving the primary ions CH COCH and NH is proton transfer to NH to yield NH;. As the total pressure is increased, NH; is found to react with CH COCH only via condensation:

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The small yield of (CH3COCH)H which is observed. is consistent with that expected from the interaction of CH COCH; and CHCO with the acetone component in the mixture. The solvation of NH; by NH to form (NH)H is also observed, as well as the further solvation of (CHCOCH)NH; and (NH)2H* by acetone:

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PERCENT OF TOTAL IONIZATION

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(AC)2NH

NH3

(AC) NH4+

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6 PRESSURE (millitorr)

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in any of the mixtures investigated. Based on the assumption that Reaction 15 would have occurred if exothermic, which appears to be valid for this class of reactions according to Munson's results, we conclude that P.A. (NH3) > P.A. (acetone), or that 203±2 kcal/mol < P. A. (acetone) < 207 kcal/mol.

3.3. Absolute Rates for Process 1

As stated in the Introduction, one of the objectives of this study was to provide a basis for converting the relative rate constants for Process 1, which were previously measured in this laboratory [1], to absolute values. Our experimental value for Reaction 2 is lower than the value assumed in that study by a factor of approximately 4.4, and the reported values should be adjusted accordingly. The fact that the rate coefficients for Process 1 are significantly lower than even originally thought adds further credence to the conclusions of Ausloos and Lias [1] that the rates of the hydride (14) transfer process involving t-CH, and higher hydro

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carbons are influenced by the combined effects of [12] Average of three most recent values as given in: activation energy and steric hinderance.

We are indebted to P. Ausloos for his encouragement and counsel during the course of this study.

4. References

[1] Ausloos, P., and Lias, S. G., J. Am. Chem. Soc. 92, 5037 (1970).

[2] Munson, M. S. B., J. Am. Chem. Soc. 90, 83 (1968). [3] Field, F. H., J. Am. Chem. Soc. 91, 2827 (1969).

[4] Vrendenberg, S., Wojcik, L., and Futrell, J. H., J. Phys. Chem. 75,590 (1971).

[5] Sieck, L. W., Searles, S. K., and Ausloos, P., J. Res. Nat. Bur. Stand. (U.S.), 75A (Phys. and Chem.), No. 3, 147153 (May-June 1971).

[6] Lias, S. G., and Ausloos, P., J. Chem. Phys. 43, 2748 (1965). [7] Field, F. H., and Lampe, F. W., J. Am. Chem. Soc. 80, 5587 (1958).

[8] Sieck, L. W., Searles, S. K., and Ausloos, P., J. Am. Chem. Soc. 91, 7627 (1969).

[9] Sieck, L. W., and Searles, S. K., J. Am. Chem. Soc. 92, 2937 (1970).

[10] Steiner, B., Giese, C. F., and Inghram, M. G., J. Chem. Phys. 34, 189 (1961).

[11] Beckey, H. D., Z. Naturforsch. 16a, 505 (1961).

a. Beauchamp, J. L., and Butrill, S. E., J. Chem. Phys. 48, 1783 (1968).

b. Haney, M. A., and Franklin, J. L., J. Chem. Phys. 50, 2028 (1969).

c. Sieck, L. W., and Searles, S. K., J. Chem. Phys. 53, 2601 (1970).

[13] Munson, M. S. B., 159th A. C. S. Meeting, Feb. 22-27 Houston, Texas (1970).

[14] Beauchamp, J. L., and Dunbar, R. C., J. Am. Chem. Soc. 92, 1477 (1970).

[15] Freeman, G. R., Rad. Res. Rev. 1, 1 (1968).

[16] Munson, M. S. B., J. Am. Chem. Soc. 87, 2332 (1965). [17] Unless otherwise noted, Heats of formation and ionization potentials are taken from Franklin, J. L., Dillard, J. G., Rosenstock, H. M., Herron, J. T., Draxl, K., and Field, F. H., Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. (U.S.), 26, 289 pages (June 1969).

[18]

Giovmousis, G. G., and Stevenson, D. P., J. Chem. Phys. 29, 294 (1958).

[19] Moran, T. F., and Hamill, W. H., J. Chem. Phys. 39, 1413 (1963).

[20] Lossing, F. P., and Semeluk, G. P., Can. J. Chem. 48, 955 (1970).

[21] Kebarle, P., Searles, S. K., Zolla, A., Scarborough, J., and Arshadi, M., J. Am. Chem. Soc. 89, 6393 (1967).

[22] Terry, J. O., and Tiernan, T. O., presented at the Sixteenth Annual Conf. on Mass Spectrometry and Allied Topics, Pittsburgh, Pa. May, 1968.

[23] Munson, M. S. B., J. Am. Chem. Soc. 87,5313 (1965). [24] Nuttal, R. L., Laufer, A. H., and Kilday, M. V., J. Chem. Thermodyn, (in press).

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JOURNAL OF RESEARCH of the National Bureau of Standards - A. Physics and Chemistry
Vol. 75A, No. 5, September-October 1971

Energy Levels, Wave Functions, Dipole and Quadrupole
Transitions of Trivalent Gadolinium lons in Sapphire

Paul H. E. Meijer

Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234

and

Physics Department, Catholic University of America, Washington, D.C. 20017

Jacques Lewiner

Ecole Superieure de Physiqué et de Chimie, Laboratoire d Electricité Générale, Paris 5o

(March 22, 1971)

A computation is made of energy levels, wave functions and transition matrix elements of the Gd3+ ion in Al2O3. The crystal field parameters are taken from Geschwind and Remeika's paramagnetic resonance experiments. The transition probabilities are given for dipole radiation in three polarization directions. For ultrasonic work we give the real and imaginary parts of the five matrix elements of the quadrupole transition. From these one can easily deduce the transition probabilities in any given direction.

The magnetic field directions are described by the angles 0 and 6, the polar and azimuthal angles with respect to the crystalline c axis. The values of 0 go from 0 to π/2 in six steps and two values of π are chosen, 0 and 2 π/3 for which the variation is largest. The magnetic field strengths are from 0 to 0.6 tesla (6000 gauss); beyond this value the spin can be considered as "free." Some consideration is given to the analytical behavior of the energy versus field diagram for the direction 0 == 0.

Key words: Corundum; spin Hamiltonian; energy levels of Gd+++ in Al2O3; quadrupole transitions;
transition probabilities; ultrasonic (paramagnetic) resonance; ultrasonic transition probabilities;
wave functions of Gd+++

1. Introduction

After the successful use of tables [1]1,2 for Fe3+ doped Al2O3 to predict possible field directions and strengths for double resonance we decided to investigate the Gd3+ doped material. It appeared at first that there were some uncertainties with regard to the selection rules for ultrasonic transitions which, it turned out were unfounded. One can with good certainty predict that the "points in H-space" are double, i.e., electron and acoustic paramagnetic resonance is possible with the usual quadrupolar selection rules for ultrasonic resonance.

Figures in brackets indicate the literature references at the end of this paper.

In this reference the following corrections should be made: The reference under figure 1 should be [11]: The <-> element in the matrix on page 244 should start with - 8D/3; The coefficients in the 1.6 and 2.5 wave functions should be a/C and a/D: The expression for A. should end with a2/9; The last two coefficients should be C and D2, rather than C and D. on the left hand side; On the right half top line of p. 245, 8g and d should be replaced by dg and d: In the first table, second line in the (-1/2) column should be -.999; In the second table, first line in the (4) column should be - 1.000.

The acoustic transition probabilities, which are of quadrupolar nature, are tabulated in a new way. The previous method was found too restrictive if the sound wave were different from a simple plane wave in x, y, or z direction.

The spin Hamiltonian is described in section 3. The angle is varied for 0 to 7/2 in steps π/12 and the angle =0 and 27/3. Only the positions of in which the energy levels change markedly with the change in are listed for both values of .

As before the number of values of the magnetic fields [the range and the steps] was a matter of compromise. The largest value is chosen such that one is in the free spin region. This value was divided into about 10 equal steps for preparation of the tables. In reality, finer steps were used in order to obtain a smooth curve near the noncrossing points.

2. The lon

The Gd3+ ion is in a (f)8S7/2 ground state. It has in common with the Fe3+ ion that the shell is half filled and that the orbital momentum is zero (according to Hund's rule). In such cases the crystal field splittings are relatively small, since the electric field acts on the ion through higher states with J=7/2 and L 0, rather than on the ground state directly, according to the general ideas of angular momentum in atomic physics. The actual interaction is very complex; there are at least eight different mechanisms proposed. Wybourne [3] calculated all of these for the lanthanum ethylsulphate lattice and still could not obtain agreement with the experimental data. There is no explicit calcu

lation of the Gd3+ wave functions in Al2O3 available in the literature as far as we are aware.

The electric field is produced by the O--ions surrounding the Al sites (the Al sites are the places where

the Gd ions go), and is rather strong. Geschwind and Remeika [4], whose parameters we use, report that the resulting splitting is the largest observed for this ion (1.24 cm-1).

The two Al sites have C3 point symmetry, one being rotated by 27/3 with respect to the other. Both share the same z axis, which we let coincide with the c axis of the crystal. Contrary to expectations, the two sites were not equally occupied. Since this was assumed by Geschwind and Remeika to be due to a "fluctuation" in the crystal during the growth, we omit consideration of this effect, but one should be aware of this possibility if one compares the transition probabilities with the experiment.

m=0 with n = 2, 4, 6

m=3 with n = 4, 6 and m=n=6.

One can try to obtain some a priori ideas about the value of these coefficients either by making a point charge calculation, which is usually not very reliable. or by simply trying to establish which coefficients will roundings. The latter method was used for Fe3+ and be dominant on the basis of the description of the sursmall cubic field added to it. The field of cubic symwe assumed a predominantly cylindrical field with a metry was oriented in such a way that its body diagonal

lay along the c axis. There is no compelling reason to consider the field from this point of view, although it was done in the Fe3+ case, since the large diameter of the Gd3+ ion may distort the lattice and hence dis

place the O- ions. Since these ions are held responsible for creating the electric field, this will have influence on the crystal field parameters. Moreover, the assymmetric occupation mentioned above seems to suggest that once one site is occupied the other is avoided. which seems to confirm the distortion.

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Also a look at the coefficients determined by Geschwind and Remeika shows no hint of a cubic field. According to the transforms in Hutchings' paper (ref. [2] eq 6.15) the ratio of the coefficients of Og to O should be 1:20 V2, and the ratios of the coefficients | of Og to 0 to Og should be 1: 35 V2/4: 7718. None of the values is in this neighborhood. (Compare table 1.)

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n = 1 m = }}

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The factors a, ẞ, y in Stevens' paper are effectively 1, since we deal with an indirect interaction. General symmetry considerations as described in section 2 of reference [1] lead to the absence of certain coefficients B; those remaining in the case of f-electrons are:

The most general way to study the wave functions is to use group theory [6, 7]. For trigonal symmetry and J=7/2 we find, using the double group, the representations I and I, three times each and I once. (Compare with appendix I.) Luttinger and Kittel [8] following Bethe [6], obtained this decomposition for cubic symmetry and found 6, г7, and Is, two two-fold and one four-fold degenerate level. Since the two-fold levels in zero field are at comparable distance in our case, this shows again that there is no trace of "cubicity" in the crystal field parameters.

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E6

E7

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=

=

H77-(5/2)b
H33+ (3/2)b
H66-(3/2)b

(2)

If the magnetic field is along the z direction (0=0) the matrix has only diagonal Zeeman terms and it decomposes in two 1 by 1 and two 3 by 3 matrices. The singlet states are associated with | ±3/2 >. The bases for the other two are formed by the 3 linear combina- for the approximate eigenvalues, and tions of | 7/2>, |> and | -5/2> and the 3 linear combinations of - 7/2 >, | − 1⁄2 > and | 5/2 >.

From the diagonal matrix elements of the crystal field it is easy to see that the | > levels lie lowest, the 3/2 are next, followed by the | ±5/2 > and the

7/2 are highest. The result is that the │7/2>, which goes up in energy if the magnetic field along the (positive) z axis is introduced, can be considered to be isolated for all practical purposes. Consequently we also can consider the >, [-5/2> linear combinations as independent of 7/2 >. If we look at the 3 combinations of -7/2>, |-> and 5/2> it is clear, from the small magnitude of the H28 matrix element that we can consider these as two linear combinations of -7/2 and -> with a small admixture of 5/2> and a 5/2> with a small admixture of 7/2 > and ->. In this way we reduced the diagonalization calculation, in first approximation, to a problem that requires not more than a 2 by 2 diagonalization.

Using arbitrary labels (b stands for guB), we have:
E1 =H1+ (7/2)b

11

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│1 >= 7/2 > +

|

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2>=cos a1 | -7/2 > + sin a1 | −>+ €26 | 6> 0 |3>= sin a1 | -7/2 > + cos a1 |- > + €36 | 6 > 0 | 4>=cos α2-5/2>+ sin a2+ 15 >= sin a2 | -5/2>+cos a2 | 16>=15/2>+€62 | 2>0+ €633> 0 3/2 >

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3/2 >

+

tg(2α1) = 2H58/ (H55-H88 +3b)
tg (202)+2H47/ (H44-H 77+3b)

for the eigenfunctions. We inserted prematurely the admixtures occurring when the Og matrix elements are not ignored (see below).

The conclusions with respect to the energy level diagram for 0=0 are the following. If the spin were free the arrangement of the zero magnetic field splitting is such that there are 12 potential crossing points (compare fig. 1). The functions 7 and 8 >, associated with m = ±3/2, will cross all other levels. Since they behave like free spins, the energy will be exactly a linear function of the field. The members of the [1>.

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