Aluminum hyperfine interactions in ruby

J. Phys. Chem. Solids

Pergamon

ALUMINUM

Printed in Great Britain.

Press 1962. Vol. 23, pp. 515-531.

HYPERFINE N. LAURANCE,

INTERACTIONS

E. C. MCIRVINE

IN RUBY

and J. LAMBE

(Received 9 Octobw 1961)

Scientific Laboratory, Ford Motor Company, Dearborn, Michigan Abstract-ENDOR spectra, eon.sisting of some 200 spectral lines in the region S-IO MC/S,have been observed in dilute ruby. The observed lines may be ascribed to 5 inequivalent aluminum nuclear sites neighboring the chromium ion impurity. The spectra are classified and analyzed in terms of the magnetic hyperfine and quadrupole interaction parameters of these nuclei with the chromium electrons. A semi-quantitative model is presented for explaining the magnitude of these interactions. The salient features of the model are (1) the electrons are principally confined to the volume of the chromium ion so that the dipole interaction is approximately that due to a point source; (2) there is a moderate amount of local distortion about the chromium ion, of the order of 2 per cent of a lattice spacing, yielding local variations in the quadrupole coupling constant; (3) there is an appreciable degree of covalency in the bonding of the AlsOs lattice, of the order of 20 per cent, which permits the transmission of the Fermi contact interaction through the bonding states of the crystal.

1. INTRODUCTION

THE ENDOR technique is potentially a powerful tool in the spectroscopy of solids. Since it depends on the interaction of an electron with some nuclei, it can be viewed alternately as a method of nuclear spectroscopy or as a method of electron spectroscopy. In principle, it allows the determination of the electron spin density at the nuclear site, a quantity of fundamental interest in inferring details of the electron distribution. The ENDOR effect has been seen in a wide variety of materials but the use of the technique to its fullest advantage as a spectroscopic tool has been limited to a few substances. To date, ENDOR has been used in measurements of the donor states in silicon,(2) of the distribution of the F-center wave function in alkali halides,@-s) and of the unpaired electron in various organic free radicals(‘). The technique of performing ENDOR measurements may be summarized as follows: an electron spin resonance transition is observed by conventional microwave techniques. Sufficient microwave power is applied to the sample to partially saturate the absorption signal. An auxiliary radio frequency field is then applied to the sample through a small coil. When the frequency of the auxiliary field is in resonance with some nuclear

transition the level of the EPR signal is affected, indicating the resonance. The mechanism by which this effect occurs is incompletely understood,(lv 2~8) although progress has been made in this field, but from the spectroscopic point of view the mechanism is irrelevant. It is sufficient that the electrons act as a detector for resonances in the nuclear system, with sensitivity greatly increased over that obtainable from ordinary nuclear magnetic resonance. We can classify nuclei into three general types in terms of the ENDOR effect: (1) nuclei located at the center of the electron distribution, e.g. an impurity in an insulator or a donor in a semiconductor, (2) nuclei so far removed from the electron that the electron causes no perturbation of their resonance frequency, and (3) nuclei in the immediate vicinity of the electron distribution. ENDOR effects have been observed in each of these categories, but of the three the last is probably the most interesting. The principal information derived from the first type of ENDOR concerns the nuclear constants, magnetic moment, quadrupole moment, etc. It is similar to NMR in this respect, except for the greater sensitivity. The existence of the second type of ENDOR is of great physical interest since it appears to be related 515

516

N. LAURANCE,

E. C. MCIRVINE

to the process of dynamic nuclear polarization.(*) However, the resonance frequencies observed are identical to those seen in nuclear magnetic resonance and no new spectroscopic information is obtained. The third type of effect depends directly on the details of the electron distribution and represents a refinement of EPR spectroscopy. The present work will be concerned with this last type of ENDOR effect. We have observed ENDOR spectra in ruby which can be associated with the 5 aluminum sites nearest to a chromium impurity. From these lines, we have obtained a set of hyperfine interaction parameters characterizing the 5 sites which we will attempt to explain, at least qualitatively, on the basis of a simple model for the electron interaction and a moderate amount of distortion around the impurity. Most of the measurements were performed on crystals having Crs+ ion concentrations of 0.01 per cent. Some additional experiments necessary to check level assignments were performed on samples having Crs+ ion concentrations of 0.001 per cent. The Crs+ ion replaces an Al ion substitutionally in the lattice and is responsible for the paramagnetic absorption of the material as well as.for the pink color. In the following paragraphs, we will describe briefly the lattice structure and the information gained from optical measurements. The ruby lattice (AlsOs) has rhombohedral symmetry and belongs to the space group D& (R%).(9) The metal ion is located at a site whose local symmetry is C’s, surrounded by a distorted octahedron of 0 ions. The crystalline field may be described by a cubic term, representing the nearly cubic environment due to the 0 ions, a smaller trigonal component, arising from the distortion of the octahedron, and a third term (hemihedral) representing the lack of inversion syrnmetry.(rs) Since the cubic field is several orders of magnitude larger than the trigonal field, the two terms may be treated in successive orders of perturbation theory. The hemihedral component does not introduce any additional splittings in the energy levels since the only degeneracies left under the action of the trigonal field (symmetry CsV) are those associated with time reversal invariance. The hemihedral component mixes the parity of the states, permitting optical transitions which would be forbidden if the ion were at a center of inversion symmetry. The lattice has an

and J.

LAMBE

appreciable field gradient at the Al site as evidenced by a quadrupole splitting in the Al NMR spectra of approximately 360 kc/s.(rl) The aluminum nuclei with which we shall be concerned in the ENDOR spectra are the 13 nearest neighbors to the Cr ion, located on 5 inequivalent sites. The co-ordinates of the first 20 neighbors are given in Table 1 in terms of the rhombohedral basis vectors. The sites we shall be concerned with are those labelled 1 through 4’. Sites 5, 6, 6’, and the 0 ion sites are included here for reference. In this notation, the “9 or “K” direction corresponds to the crystalline “c” axis. The notation follows that of WYCKOFF@) except that the origin of co-ordinates coincides with the Cr site. The number of equivalent sites of each type is in the last column. The co-ordinates for one site of each type is given in Table 1; the co-ordinates of the others may be obtained by permutation. Sites 1 and 5 are on the “c” axis which passes through the Cr ion; the remaining sites are not. In subsequent discussion, we shall refer to sites as being either “on axis” or “off axis” with reference to this fact. Sites 4 and 4’ are related to each other by an inversion through the Cr ion. The lack of inversion symmetry leaves them inequivalent. The groups of 0 ions listed are those which form the distorted octahedron about the Cr ion. These two groups are not related to each other by an inversion transformation, but would be if the parameters u and w were changed to u = l/12 = 0.083 and e, = l/3 = 0*333.(1s) Figure 1 shows the ions surrounding the Cr ion including Al site 1, the three equivalent Al ions in site 2, and the six 0 ions. The threefold symmetry is apparent from this figure. Figure 2 shows a part of the lattice indicating the relative positions of the remaining ions in Table 1. Only one site of each type is included here for reasons of clarity. The electronic configuration of the Crs+ ion is 3d3 and the ground state of the free ion is 4F.(ls* ls) The cubic field splits the ground state into three states which have the symmetries As, Ts and Tl (See Fig. 3). The As state lies lowest and the Ts and Tl states are removed from the ground state by 18,000 cm-l and 25,000 cm-r, respectively. The cubic field does not affect the spin character of the states, so each of the resulting states is also a quartet. Since the ground state is an orbital singlet, the trigonal field alone causes no further

ALUMINUM

Ion type

Cr Al 1 Al2 Al 3 Al4 Al 4 Al 5 AI 6 Al 6’ 0 ions 0 ions

HYPERFINE

INTERACTIONS

Location*?

no,

Distance to Cr ions

0)

(-2u, -2u, -2u) (f-2u,&2u, -Q-22() (l-2& -2u, -2u) (4, %, -4) (8, -4, -It) ii-2 &2;2u>f-W (1: o,‘-1) t-u, -u-w, -u-#-v) (a-u-o, B-u, &tJJrei)

IN

RUBY

Number

0

1

0.53197 0+%684 0.62017 0.68209 068209 O-73463 0.92789 O-92789 0.38705 0*36001

1 3 it 3 1 3 3 3 3

* Coordinates are in terms of the basis vectors a1 = 010(sin ef +cos e& as = CIO (-4 sin @+2/3/2 sin B$+cos &) K; 1 ;;Oo(t,sin 81- 2/3/2 sin Oj+cos 0&) ? Far Al&s, the para’meters are il = O-105 and &r= O-303. (See Ref. 9.) $ Distances are given in terms of the rhombohedral cell dimension as = 5 *137 A.

I ‘C” AXIS Fra. 1, A portion of the corunduxp lattice showing the three-fold symmetry about the “c” axis. The light balls are aluminum or chromium ions; the dark balls are oxygen ions. For notation see Table 1,

517

518

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LAURANCE,

“C”

E.

C.

MCIRVINE

and

J.

LAMBE

AXIS

FIG. 2. A portion of the corundum lattice showing the first 7 inequivalent aluminum neighbors to a Cr site. The dark balls are oxygen ions. There are 3 equivalent sites each for sites 2, 3, 4, 4’ and 6; only one of each type is shown for clarity. For other data see Table 1. The rhombohedral basic cell dimensions, (IO, is also shown.

splitting. However, through the combined action of the trigonal field and the spin orbit coupling. the spin degeneracy is partially removed, and the state splits into two Kramers doublets. The doublet corresponding to S, = + 3/2 lies lower,(rs) and the splitting is approximately 0.3 cm-r. The effect of a magnetic field is also shown in Fig. 3. The case illustrated is for the external field parallel to the “c” axis. In other orientations the spin sub-

levels mix and S, is not a good quantum number. All the experiments to be reported here were performed with the external field parallel to the “c” axis, and thus we shall not discuss the other possible cases. The vertical lines shown on the figure indicate the electron paramagnetic resonance transitions which may be observed at X-band frequencies. Experiments were performed on each of the transitions indicated. The EPR absorption

ALUMINUM

HYPERFINE

as a function of magnetic field is included at the bottom of the figure. Experiments were also performed on the low field line at Ku-band frequencies. For this case the transition energy is larger then D, the zero field splitting, and the transition observed is from S, = -312 to & ,= -l/2. 2. THEORY

The principal interactions with which we shall be dealing are described in the following spin Hamiltonian X

= g/3H*s+D[s,z-gs(s+l)]

The first two terms refer to the interaction of the unpaired electrons of the chromium ion with the crystalline and magnetic fields, and the remaining terms involve interactions of the nuclei with the field and with the electrons. In principle, the summation is to be taken over all the nuclear sites of the lattice. In practice, however, it is found that only the 13 aluminum ions nearest to the chromium impurity produce detectable ENDOR transitions, so we shall confine our attention to these nuclei. Since the first two terms do not involve nuclear co-ordinates, they do not enter into the expression for ENDOR transition frequencies and consequently will be ignored for the remainder of the discussion. In writing the Hamiltonian (l), we have assumed a co-ordinate system such that the zero field splitting, D, and the nuclear quadrupole interaction, Q’, are diagonal. Experimentally, it is observed that in order to detect the ENDOR spectrum, the external field must be parallel to the zero field axis of quantization to within 2”, and we will restrict our discussion to this case. In this case, the entire H~ltonian is diagonal with the exception of the last term, the dipole-dipole

INTERACTIONS

IN

RUBY

519

interaction. (Suitable treatment of this term will involve changes in the nuclear quadrupole interaction, a point to which we shall return shortly.) In discussing the Hamiltonian, it must be remembered that each term in the summation deals with the energy levels of a single nucleus. Since we actually messure the nuclear transitions (through the intermediary of the electron spin resonance), it is possible to discuss each term in the summation individually. The ENDOR spectrum is a combination of several spectra, one for each inequivalent nucleus. In order to take account of the interaction terms between the nuclei and the electron, we introduce the idea of an effective field acting on the nucleus. This is equivalent to discarding the offdiagonal elements of the electron spin operator in the dipole-dipole term. (The other terms are already diagonal in S.) This may be justified by a consideration of the energies involved; the nuclear transition energies are 5-10 MC, while the energy separation between electron spin states is about 10,000 MC. Such off-diagonal terms in general will contribute only 0.1 per cent of the diagonal terms. The electron spin, therefore, can be treated as a classical magnetic dipole aligned with the external magnetic field. In this approximation the nuclear interaction Hamiltonian becomes 3?=

-

3, &rt-

Q’[I; +(I+

l)],

(2)

Here we have dropped the index i from the nuclear variables and have replaced the relative position co-ordinate r-r( by r, denoting the distance from the electron to the nucleus under consideration. It can be seen that in general the effective field is not parallel to the external field. For convenience, we will talk about the longitudinal and transverse components of the effective field, referring to the two parts of equation (3). We introduce the following parameters, having the

520

N.

LAURANCE,

E.

C.

CUBIC FIELD r4

+

MCIRVINE

AXIAL FIELO SPIN ORBIT

+I

-2

1

+3 -7

D

and 3. LAMBE

MA~~~~‘c

I I

D= 0.3 cm-’ = t I.4 kMc/s

FIG. 3. Splitting of the chromium free ion energy level under the successive application of (1) a cubic nystalline field, (2) an axial crystalline field and spin-orbit interaction and (3) an external magnetic field. The transition shown are for the X-band frequency, N 9000 MC/S.

dimensions of an energy, related to the components

of the effective field. A = + Gus

of the operator 1 in this plane by It and I,, parallel and perpendicular to the transverse field respectively, we can write our Hamiltonian (2) as iTi@= -+lo+AI&+B&

+&hSz+ &=

“[adY~ I

+3gBi

The transverse component of the effective field lies in the X-JJ plane, parallel to the component of r in the x-y plane. If we denote the components

Q'fr, +(I+

l)].

(5)

AS a result of the dipole field of the electron, the nuclear angular momentum will be quantized along the effective field and not along the external field. To take this into account, we transform the coordinate system at the nucleus so that the nuclear spin is lined up with the effective field, When this

ALUMINUM

INTERACTIONS

HYPERFINE

is done, however, the nuclear quadrupole interaction is no longer diagonal. The transformation is quite straightforward, with the result

ti(3cos3

i@ = -$Heff+~

[I;” -+(I+

l)]

+ sin7 cos~[(1;-_3)11+ +a sins#;2+r1_2]

(Ii+$)II]

, 1

(6)

where 77is the angle the effective field makes with the crystalline axis, and 1; is the component of I in the direction of the effective field. The diagonal values of this Hamiltonian are easily obtained by straightforward application of perturbation theory. We will not quote the results here but will give instead an expression for the difference in energy between adjacent nuclear states. A+2

+-m+

1) = f&-P”

+(m+$)P’+(m+$)sp”,

(7)

where wC = tHeff

I

ways: first, the details of the electron distribution on the chromium ion will change the fields in its vicinity; second, the disparity in size between the chromium ion and the aluminum ion implies some local distortion in the neighborhood of the impurity. Although no detailed construction of a ground state wave function for the chrcmium ion in ruby has been made, many of the features which are expected to occur in the ENDOR spectrum can be inferred from a simple free ion wave function conforming to the crystal symmetry. In this subsection, we shall construct such a wave function. In a cubic field the ground state of the chromium ion is of symmetry type As.(1sJs) The electronic configuration of the ion is 3ds. The cubic field splits the individual d-electron states into a doublet and triplet, symmetry types e and ts respectively, the triplet ts lying lower. For the ground state, then, one constructs a 3-electron wave function from three of the ts states, the resulting wave function to have symmetry As. Since the resulting state must have S =3/2, the spin part of the wave function must be symmetric in the interchange of electrons. This implies that the space part of the wave function must be antisymmetric under the interchange of electrons. The simplest function satisfying these requirements is a Slater determinant of three ts functions.

P = Q(3 cossq - 1) Y= p’

= ig

sir&$17 co&~-l]

(8)

WC

P”’ = 2E sin2q[9 cos2r] - 11. WC This energy difference should equal the ENDOR transition frequency for the nucleus under consideration. There are three parameters at our disposal to explain the ENDOR spectrum of a given nucleus, A + B,, iYt, and Q’. Experimentally, it is impossible to separate A from BZ, as inspection of equation (5) will show, because it is impossible to obtain the ENDOR spectrum as a function of angle. Notice that Q’ can vary from site to site. This variation is due to the fact that the impurity alters the crystal field gradient at the nearby nuclear sites. The alteration of Q’ can come about in two

521

IN RUBY

-&

&l(l)

M4

&l(3)

Ml)

k(2)

k(3)

&3(l)

h&)

hz3(3)

*

(9)

If for the functions 4 we use the one electron 3d functions, this determinantal wave function has the required cubic symmetry As. The function has nodes along the octahedral directions, i.e. it would avoid negative ions placed at the octahedral positions. The real symmetry of the lattice, however, is trigonal, with the oxygen ions displaced from the octahedral positions. The unpaired electron wave function is distorted and we can partially take this into account by requiring that the unpaired electron be orthogonal to the core states of the oxygen ion. This distortion has been referred to as the Pauli distortion of the unpaired electron wave function, and results in an admixture of the core states of the oxygen ion into the unpaired electron wave function. The above procedure is equivalent to mixing in higher

522

N. LAURANCE,

E. C. MCIRVINE

excited states of the chromium ion into the ground state, but the present approach is more advantageous for discussing hyperfine interactions. We therefore write for 6:

and J. LAMBE

Unfo~unately, no “Gauss’s” theorem holds for this integral. As a first appro~ma~on, however, we also use the values calculated from a point dipole model. The parameter A, the Fermi contact interaction, is a measure of the density of unpaired electron at the aluminum nucleus. Since all the where $ar is the unorthogonalized wave function aluminum nuclei are well shielded from the chrofor a chromium 3delectron, and (#Q~) are the core mium ion by the presence of the oxygen ions, one states of the oxygen ion (lf, 2~~2~). The normalizaexpects the amount of direct overlap of the chrotion factor, Ar, is mium eIectrons on the aluminum nuclei to be quite small. An order of magnitude estimate indicates that this direct interaction will account and the index rn runs over the oxygen core states for only about 10-s of the observed value. Thus for the six oxygen ions. In equation (10) the only one is led to search for an indirect method for oxygen core states which need be considered are transmission of the spin polarization from the those with their spin parallel to the spin of the chromium ion to the aluminum ion. We will anticipate some results of the following chromium electrons, since the states with spin opposite are already orthogonal to the chromium sections here to make the subsequent discussion more plausible. Experimentally, it is observed electron wave function. The lon~tudinal dipole interaction at an alu- that ENDOR spectra are obtained only for 5 inequivalent aluminum sites, These sites have one minum nucleus due to the electron ~stribution around the chromium ion is striking feature in common: they all have for a nearest neighbor at least one oxygen ion which is also a nearest neighbor of the chromium ion. This fact suggests that the appearance of the ENDOR spectra may be related to some covalent where $(r) is a properly normalized wave function of the electrons, R is the co-ordinate of the alu- bonding between the aluminum ion and the oxyminum nucleus and 6 is the angle between I and gen ion. BERSOHN(~~)concluded that the bonding R. Lacking a detailed chromium wave function, in Also3 is largely ionic, but this does not preclude we cannot properly evaluate this integral. However, a small degree of covalency in the bond. In fact, we can estimate the order of magnitude of this using Pauling’s formula relating the degree of covalency to the electronegativity difference,(ls) term by replacing the actual electron distribution by a spherical distribution within a reasonable one concludes that Al303 should be 20 per cent volume. It can easily be shown that an analogue covalent. To investi~te the effects of bonding, we have constructed the appropriate hybridized of Gauss’s theorem holds for this integral, i.e. bond orbitals for the AlsOs lattice. Referring to for a spherical distribution of dipoles, the value of Fig. 1, we see that each aluminum site is surrounthe field at any point R is equal to the value obtained if all the dipoles within a radius R were ded by six oxygen ions, arranged in three-fold about the “8 axis. Three ions are loconcentrated at the center of the sphere. The dipoles s~etry residing outside the sphere of radius R produce cated at an angle of 46”32’ with respect to the “c” no field at the field point. At reasonably large axis, and three are located at angle of 116”44’. values of R this is probably a good approximation We shall refer to bonds between the aluminum ion to the dipole interaction. We shall refer to this and the oxygen ions located at 46”32’ as acute bonds, and to bonds between the aluminum ion value as the point dipole approximation. The transverse component of the dipole interaction is and oxygen ions located at 116’44’ as obtuse bonds. Actually the distance from the aluminum ion to given by the two different types of oxygen ions is not the same; however, we shall ignore this difference in bond length in the discussion. A general

ALUMINUM

HYPERFINE

prescription for constructing orthogonal hybrid bond orbitals with maxima in arbitrary directions has been given by HULTGRJZN.(~‘) Using his method, we have constructed orbitals composed of aluminum 3s, 3p and 3d functions, with maxima in the direction of the neighboring oxygen ions. The results of this construction are given in Table 2. The electrons in the bond can interact

Table

2.

Composition of hybrid bond orbital appropraate to the corundum lattice

Type of

bond

Acute Obtuse

s character (%I

p character (%)

d character (%I

11.9 21.3

40.9 54.9

47.2 23.8

with the aluminum nucleus through the s character of the bond. We can then picture the contact interaction as follows. Since the orbitals occupied by the chromium unpaired electrons (xy, yz, XX) are to a good approximation orthogonal to the orbitals used for the chromium oxygen bond, the unpaired electrons will interact with the bonding orbitals through the exchange interaction. This tends to produce some negative polarization in the bond which results in a negative spin polarization at the oxygen nucleus. The negative polarized core states of the oxygen ion now interact via the exchange interaction with the bonding orbitals of the neighboring aluminum nuclei, producing a slight degree of unpairing of the bond. The degree of unpairing of the bond is given by the exchange interaction of the bond states with the oxygen core states divided by the excitation energy needed to raise the bond from the paired (singlet) state to the unpaired (triplet) state. The degree of unpairing has been referred to as the “spin attenuation factor”.(l** 1s) The bond then interacts with the aluminum nucleus because of its s character. The quadrupole interaction parameter, Q’, for the aluminum nuclei in the immediate vicinity of the chromium ion will depart markedly from the

INTERACTIONS

IN

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523

value of Q’ for nuclei far removed from the chromium ion. This change in Q’ arises from changes in the local value of the crystal field gradient due to the presence of the chromium ion. The change in crystal field gradient can arise in several ways. The gradient will be altered due to the presence of a non-spherical distribution of charge on the chromium ion. It can be shown that this effect has the same analytic form as the longitudinal dipole field term (12), and to the extent that we may replace (12) by an integral over a spherical distribution of charge, this contribution to the field gradient will vanish. The presence of an appreciable amount of covalent character in the bonding will contribute to the quadrupole interaction, but since we suppose that the degree of covalency is uniform throughout the lattice, the covalent bonds will contribute to the effective field gradient uniformly, and will not explain the change in Q’ for sites located near the chromium impurity. The other source of local variations in Q’ is the direct result of lattice distortion on the ionic crystal field gradient. In order to estimate the possible effect of distortion on the field gradient, a program was prepared for an IBM 704 computer which calculated the field gradient at several points on the “c” axis near the aluminum site. The method of computation used is due to NIJBOER and DE WETTJ+~), and the convergence is very rapid. Incidental to this work was a recalculation of the corundum field gradient. We arrive at a value for the gradient of 11.37 e/a”, as compared to 10.42 e/a”, computed by BERSOI~N(~~).It is thought that this difference in values is due to the improved convergence of the NIJBOER-DE WETTE method as compared to the direct summation employed by BERSOHN. To investigate the effects of distortion, the field gradient was calculated at a point slightly removed from the origin along the “c” axis. Put another way, the lattice was held fixed and the central aluminum ion was moved a small distance away from its nearest neighbor. It was found that if the aluminum ion was displaced 1 per cent of a lattice spacing from its normal position the field gradient rose to 16.58 e/a:, and for a displacement of 2 per cent field gradient was 22.82 e/a:. In other words, the field gradient had doubled with a displacement of O-1 b. Although this displacement of a single ion does not approximate the real

524

N.

LAURANCE,

E.

C.

MCIRVINE

distortion around an impurity, it shows that the field gradient is sufficiently sensitive to local coordinates to explain any variations in the quadrupole interaction in the neighborhood of the impurity. 3. EXPRRIMENTAL

PROCEDURE

The measurements reported here were made on single crystals of ruby cut from disc bottles obtained from the Linde Company. Most of the measurements were made on crystals having a concentration of chromium of O-01 per cent. Some additional measurements were performed on samples with a chromium concentration of 0.001 per cent. The samples were approximately 3 mm x 3 mm X 15 mm with the “c” axis parallel to one of the short dimensions of the specimen. The samples were mounted in the center of an X-band microwave cavity on a sample mount which also held the small r.f. coil for exciting the nuclear transitions. The coil was driven by a Hewlett-Packard 606A oscillator and an IF1 broad band amplifier capable of delivering 3 W into 185 Q. Details of the cavity, the spectrometer and the general measuring technique have been reported previously.(*) All spectra were obtained with the samples immersed in liquid helium. Typically, spectra were run with a microwave power level of 35 dR down from a 100 mW source. A 5 kc/s magnetic field modulation technique was used to observe the electron spin resonance. The signal was applied to one channel of a Texas Instrument dual trace recording potentiometer. The level of the EPR signal was observed as a radio frequency field was applied to the coil surrounding the sample. When the frequency coincided with a transition frequency of one of the nuclei, the EPR signal level changed abruptly. The frequency of the oscillator was measured continuously with a HewlettPackard counter and was recorded on the other channel of the dual trace recorder. In this manner, the recorder chart contained both the signal and the frequency measurement on the same record. The frequency determinations are thought to be accurate to 1 kc/s. The EPR lines of interest are inhomogeneously broadened with a line width of approximately 12 G. The appearance or non-appearance of the ENDOR effect was found to be dependent on the position of observation within the EPR line. The following prescription describes the setting used to obtain the largest effect. First, the line was observed at very low power, low enough so that the line was not saturated. The magnetic field was adjusted to coincide with the maximum of the derivative signal, i.e. the inflection point of the absorption. At this setting, the power was increased to partially saturate the resonance. Finally, small adjustments in the magnetic field were made to maximize the ENDOR signal. In addition, the appearance of the ENDOR effect is critically dependent upon the alignment of the crystal in the magnetic field. It was found that if the “c” axis of the crystal was more

and

J.

LAMBE

than 5” away from the direction of the magnetic field, the ENDOR effect vanished. In most measurements, therefore, the sample was aligned to within & Some measurements were made modulating the radio frequency field at 5 c/s and detecting the resulting signal synchronously. In many cases, this improved the signal to noise ratio and had the added advantage of eliminating drift instability. In addition, some measurements were made while pumping on the liquid helium to lower the temperature below the X point. This procedure afforded no great experimental advantage, however, and was discarded. Some additional experiments were performed at Ku-band frequency to accurately ascertain level assignments. The apparatus used for these experiments was similar to that used at X-band, except that the spectrometer was operated at 14 kMc/s. The transition of interest in this case is between the levels S, = -3/2 to SE = -l/2 (cf. Fig. 3). At 14 kMc/s, this transition occurs at about 900 G. This is very nearly equal to the field necessary for the SL = l/2 to SC = 3/2 transition at X-band frequency. Since the ENDOR transition frequencies depend only on the magnetic field and not on the spectrometer frequency, the spectra obtained in these two cases are almost identical. The only difference is in the relative intensities of the lines, a fact which aids us in assigning levels to the transitions.

4. EXPERIMENTAL

RESULTS

A portion of a typical ENDOR spectrum is shown in Fig. 4. This spectrum was run at X-band frequency on a crystal containing 0.01 per cent chromium on the high field magnetic transition (Cf. Fig. 3). The frequency of the r.f. oscillator is given on the figure. Such spectra were obtained for each of the three EPR transitions. These spectra are composites of the transitions of several different nuclei. The process of identification of the transitions in the spectra is largely one of trial and error, but we have several physical principles to guide us in the classification. In the first place, we expect the aluminum nuclear transition frequencies to occur in sets of five, almost equally spaced, with a spacing in the neighborhood of 360 kc/s. In addition, we know that some lines arise from the transitions of three equivalent nuclei located off axis. When the magnetic field is rotated slightly away from the “c” axis (about 27, the three nuclei are no longer equivalent and the lines corresponding to their transitions split into three components. Transitions associated with nuclei on axis, however, do not split. In this way, we are able to determine whether a given line arises from a nucleus on axis or off axis. The

ALUMINUM

HYPERFINE

INTERACTIONS

IN

RUBY

525

FIG. 4. A portion of a typical ENDOR spectnmx. The lines along the abscissa indicate the predicted lines from the three parameter theory, and the letters indicate to which ENDOR set the lines belong. This data was obtained on 0.01 per cent Cr-doped ruby at a magnetic field of 7.517G.

of the lines also aids in grouping them into sets. Thus the spectrum is classified into 5 different sets, corresponding to 5 inequivalent nuclear sites. Of these sites only one is located on axis, the other four being off axis. Each of the sets of ENDOR lines is then analyzed to yield the three theoretical parameters, A+B,, Bt and 8’. An example of a fit to a set of experimental lines is given in Table 3. These lines are associated with a nuclear site which we denote 1, and were obtained on the high field EPR transition. It can be seen that the theoretical three parameter fit to the experimenta lines is excellent. Since each nucleus has 5 transitions, in the local hyperfine field of each of 4 possible electron spin states, which could be observed in each of 3 applied magnetic fields, each nucleus gives rise to 60 observable ENDOR lines. Some of these lines fall in inaccessible portions of the spectrum, however, so that about 40 lines for each nuclear site are actually seen. Although the EPR transition being saturated intensity

in the experiment involved only 2 of the 4 electron spin states, ENDOR transitions were observed which could be associated with nuclei near electrons in each of the 4 electron spin states, The question of whether a given set of lines should be associated with S, or with -S, was uncertain at this point. This uncertainty was resolved by observing the spectrum in a more dilute specimen of ruby (0.~1 per cent Cr.) In this case, the ENDOR transitions associated with the electronic spin states actually being saturated remained large, while those associated with the other spin states became very small. In this way, a positive identification of the proper electron spin state was made. Experiments performed on this sample at X-band frequency enhanced the ENDOR transitions corresponding to S, = t-312 and +1/2, while the same experiments performed at &-band frequency enhanced the lines corresponding to S, = -312 and - l/2. In addition, comparison of the ENDOR frequencies at X-band and at &-band indicate that A+B, is positive for all the nuclear

526

N.

LAURANCE,

and J.

E. C. MCIRVINE

Table 3. Typical expimental

LAMBE

results and thmretakal fit

Parameters for beat fit A+&

= 2.418 M& Bt = 0444 Mcfs

ENDOR Set I Ho = 7517 G

2Q’ = 0.275 MC/S

-312

-112

Line 1 Theory Experiment

12.534 12.53

10.099 10.10

7.681 7.68

5.296 -

Line 2 Theory Experiment

12.260 12.26

9.825 9.82

7.407 7.405

5.024 -

Line 3 Theory Experiment

11.986 11.99

9.551 9.55

7.133 7.12

4,756 -

Line 4 Theory Experiment

11.712 11.71

9.277 9.27

6.859 6.845

4.490 -

Line 5 Theory Experiment

11.438 1144

9.003 8.99

6.585 6.57

4.228 -

Electron Spin State

sites observed. This fact was inferred from the observation that the ENDOR transition frequencies

4.312

+1/2

reference. The accuracy in the determination of the quantities A+ Bz and Q’ is about 0.02 MC/S. The ENDOR transition frequencies are much less sensitive to 38, and the accuracy in this value is only about O-10 MC/S. Of the 5 sets observed, only the set labelled G does not show splitting as

increased with an increase in magnetic field.

The three parameters obtained for each of the 5 sets of ENDOR lines are listed in Table 4. We have labelled the 5 sets G, I, J, K and L for

for the obsmed sets of EN~R

Table 4. ~x~~i~ta~~a~t~~

wipnents

&es and t~~tiQe

to n&ear s&es

A f Ss (MeIs)

2*:9

I 2.42

K 2.06

2.45

1.39

:;I

0.53 0.00

8::

0.29 0.84

0.34 0.89

0.35 0.88

Associated with nuclear site

1

2

3

4’

4

2.01 0.00

-0.82 0.55

-O*lS 0.83

0.07 0.70

0.07 o-70

0.68

3.24

2.21

2.38

1.32

ENDOR set

Point dipole energies BS Bt Contact interaction (to fit)

A

J

L

ALUMINUM

HYPERFINE

the magnetic field is rotated off axis. This implies

that G corresponds to a nuclear site which is located on axis, and the other 4 sets correspond to sites located off axis. On this basis it is reasonable to associate set G with the nuclear site 1. (Cf. TabIe 1 and Fig. 2.) The remaining 4 sets of lines must be associated with 4 nuclear sites which are off axis. Sites 2, 3, 4 and 4’ are logical candidates for this association. Site 5, the next nearest aluminum site, is on axis and need not be considered. Site 6, another possibility, is off axis, but it is located in the x-y plane containing the chromium ion, and it should have zero transverse dipole term. (Cf. equation 13.) The remaining nuclear sites appear to be too far from the chromium ion to merit consideration. We are thus led to conclude that the four off axis sets must correspond to the four sites, 2, 3, 4 and 4, The correct association among these sites is uncertain; however, on the basis of the dipole terms calculated from the point dipole model we have made the tentative assignment given in Table 4. As has been noted above, sites 4 and 4’ are related to each other through an inversion through the chromium site. If the chromium ion were at a center of inversion symmetry, the two sites would be equivalent and we should obtain only one set of ENDOR lines for sites 4 and 4’. The difference between the spectra due to 4 and 4’ is a measure of the lack of inversion symmetry of the chromium site. Put another way. since the ground state of the chromium ion is principally of even parity, the difference in the spectra of sites 4 and 4’ measures the admixture of odd parity

INTERACTIONS

IN RUBY

527

components into the ground state. We have also included in Table 4 the dipole terms calculated on the point dipole model for the various sites and the amount of contact interaction, A, necessary to fit the observed results. 5. DISCUSSION An adequate explanation of the observed spectra

must include values of the contact interaction, the dipole field and the variation of the quadrupole interaction. In this respect, the limitation of obtaining spectra at only one angle is a serious one, since we cannot separate the contact term from the longitudin~ dipole term. Hence our discussion will be a strictly qualitative one, based on the dipole fields calculated from the point dipole model. In the tentative assignments made in Table 4, the values of the transverse dipole field guided us in distinguis~ng between site 2 and the other ofI axis sites. Sites 4 and 4’ were assigned to the sets of lines L and J because of the similarity of their quadrupole interaction constants, Q’. We thus arrive at the values of the contact interaction given in the last row of Table 4. We have indicated an explanation for the contact interaction above. In Table 5 are listed the types of Al-O bonds which join each of the various sites to an oxygen ion neighboring a chromium nucleus. We know that the s character of an acute bond is about 12 per cent, while that of an obtuse bond is about 20 per cent. (Cf. Table 2.) The contact interaction carried by these two types of bonds, therefore, should be in the ratio of 5 to 3, Inspection of Table 5 shows that for the sites 2, 3,

Table 5. iPredicted and observed ratios of eontad interactions on the cowlent bond wwdel Site

Associated ENDOR set

Types of bondsto oxygenions neighboring the chromium ion Acute

: 3 4 4’

IG K J”

31

Obtuse

1 1

1 1

* Based on contact interaction

Ratio of contact* interaction Predicted

Observed

3.00 2.79 1.79 l-00 1.79

0.506 246 1.67 1.00 l-80

= l-00 for site A

528

N. LAURANCE,

E. C. MCIRVINE

4 and 4’ there is good qualitative agreement between this simple prediction and the experimentally derived values of the contact interaction. Site 1 does not agree with this prediction at all and, in fact, the value of A appears anomalously low considering the proximity of this site to the chromium ion. It is not difficult to account for the size of the contact interaction on the covalent bond model. Using the IIartree-Fock wave function for Al*+ calculated by Klr’rr~t~~ctr(*r), one arrives at a value of A = 4200 MC/S for the contact interaction of a 3s electron with the aluminum nucleus. The obtuse hond has only 20 per cent s character. This reduces the available interaction to 840 MC/S. The degree of covalency, which mcasurcs the fraction of an electron actually present in the hond, is hard to estimate. WC will use 20 per cent, the value obtained from P~t_rl.rsc’s semi-empirical formula.‘ts) This reduces the available interaction to 168 >Ic;s. The dcgrce of spin unpairing present at the oxygen nucleus is yet another unknown; one can only make reasonable estimates of its size. Guided by the values obtained for KSInFs, as quoted hy FMExthr and \~AIxIs(“), we will assume IO per cent for this unpairing. This reducts the available interaction to 16-g hlc,5. Finally, the last term to he considered is the degree of unpairing in the aluminum oxygen bond. hlcCossn.r.(ls) has calculated the spin attenuation factor for a carbon-hydrogen bond in organic materials and obtained a value of 20. In the present case, we need a factor of 8 to reduce the interaction to the experimental value, 2 hlc,‘s. It should hc obvious that there is sufficient latitude in the choice of these paramctcrs, lacking more detailed knowlcdgc of the interactions, to fit experiment over a wide range of possible experimental values. However, the size of the contact interaction is at least in order of magnitude agreement with the model, while the ratio of contact Interaction constants for the various sites is in good agreement. The contact interaction of site 1 is not in agrecment with the ahovc model. On our simple prcdiction, we would expect an interaction constant of about 4 MC/S, roughly 3 times that of site 4, since it is linked to the chromium ion by 3 acute bonds. There are several posgihle explanations for this discrepancy which we till diecuss briefly.

and

J. LAMBE

First, we have ignored the fact that the acute bond lengths ate longer than the obtuse by about 7 per cent, and this fact will certainly influence the relative effectiveness of tranfmission of spin polarization. Close to the chromium ion there may be enough distortion to change the bond length even more, making the acute bonds very ineffective. Site 4, with which we have been comparing site 1, is linked to the chromium ion through an acute bond followed by an obtuse hond, while site 1 has threefold linkage through an acute bond followed by an acute bond. Secondly the fact that there is a degree of unpairing of the core states on the oxygen ions means that in addition to the chromium ion these ions act as sources of dipole fields. Since their spin polarixation is opposite to that of the chromium ion they will tend to decrease the cffcctive dipole fictd present at an aluminum neighbor. (At some angular positions their effects may add, hut at site I they arc opposed.) This would imply that the point dipole value given in ‘I’ahlc 4 is too large and hcncc that the real contact interaction is proportionally larger. However, this cffcct dccs not appear to he large enough to explain the observed discrepancy. Since site 1 is closer to the chromium ion than are the other aluminum sites, another possibility exists. h.xuee of the presence of the chromium ion electrons, the exchange potential for the core state aluminum electrons is different depending on whether the aluminum electrons have spin parallel or antiparallel to the chromium clcctrors. This should have the effect of distorting the orhitals with spin parallel towards the chromium ion and therefore leaving an excess of antiparallel spin at the aluminum nucleus. This effect would give a ncgativc contribution to the contact interaction, hut it is difficult to see how this term could he large enough to product the observed interaction. A fourth possibility exists which we mention for the sake of complctcncss. Site 5, on the hasis of its distance from the chromium ion site alone, might fit the parameters for the set of ENDOH linrs wc have ascribed to site 1. While attractive in some ways, this asuignmcnt is dubious. First, a site 5 aluminum ion is not hondcd to my oxygen ions in commo11 with the chromium ion, which bonding appears to be necessary for the appearance of ENDOH. Secondly if these parameters belonged to site 5, there should

ALUMINUM

HYPERFINE

INTERACTIONS

IN RUBY

529

effect. On the basis of measurements of LEW and be another set of ENDOR transitions at higher WESSEL,(~~)DAS and HAHN@~)give the value for the frequencies corresponding to site 1. This area of field gradient at a nucleus due to a single 3p the spectrum was searched many times in several electron as - 7.23 x 1024 e/ems. The fact that the different runs, always with negative results. For bonds are almost in octahedral co-ordination means these reasons it is felt that ENDOR set G should that the combined effect of the six bonds will be associated with site 1. almost vanish. Indeed, considering a p electron The origin of the variation in the values of Q’ directed at each of the neighboring oxygen ions, almost certainly is due to physical distortion the combined effect of the six electrons is around the chromium impurity. If the wave -29*0x 1022 e/ems. We know from our confunction of the electron were diffuse and spread out appreciably over several nuclear sites, this struction of the proper hybridized bond orbitals that the acute bonds have 40.9 per cent p character would also affect the quadrupole interaction. But the fact that the values of the dipole fields while the obtuse have 54.9 per cent p character. Using these weighting factors, we arrive at an calculated from the point dipole model are in effective field gradient at the aluminum nucleus reasonable agreement with the experiments inof + 47.8 x 1022 e/cma. Notice that the use of the dicates that the electron distribution is principally proper weighting factors has changed the sign of confined to the chromium ion volume. In addition, the contribution so that it now agrees with the any appreciable amount of overlap of the positive sign we demanded above. The Sternheimer chromium electrons with the aluminum nuclei anti-shielding factor for such an electron is nearly would produce a much larger value of A than that l*O;(as) we will assume it to be 1.0 for the present which is observed. For this reason, it seems discussion. The above is then the field gradient reasonable to ascribe the variations in Q’ to local produced in a 100 per cent covalent crystal. We distortion. The calculation described in Section 2 shows that a very small amount of distortion is have ignored the contributions of the d character of the bonds; they will change the numbers adequate to explain the observed effects, although that calculation makes no pretense of representing slightly but not enough to change the conclusions. the distortion as it actually occurs. Indeed, since The effective field gradient required to satisfy there are many ions affected, it is difficult to predict the measurements of the quadrupole moment Q and of the quadrupole coupling constant Q’ is how the ions will move to make room for the 46.0 x 1022 e/u-r@. This is equal to the field gradient oversized chromium ion. times the anti-shielding factor. For the ionic The postulate of 20 per cent covalent bonding raises some questions about the quadrupole model BERSOHNcalculated q to be 7.72 x 1022 e/cm3 and concluded that 1 -R, the anti-shielding factor, interaction in the host lattice far removed from impurities. Based on the calculation of the field is 5.94. This is to be compared with a value gradient computed on the ionic model, BWSOHN(~@ calculated for the ionic case of 3.59.@7) The agreement improves if one uses the value we have has concluded that the ionic model sufficiently calculated for the ionic lattice, 8.42 x 1022 e/ems, explains the observed interaction. The question but not enough to make an appreciable difference. remains, is the postulate of 20 per cent covalent It can readily be seen that if one supposes the bondcharacter compatible with his conclusions? Since SILVER et aZ.(aa) have shown that Q’ is positive for ing to be 20 per cent covalent and 80 per cent ionic the values calculated here are in equally good sapphire, and the nuclear quadrupole moment, Q, is known to be positive from other work,(M) we accord with experiment. In fact, on the basis of conclude that the effective field gradient, q, must these values one may assume any percentage of also be positive. BERSOHN’Scalculation shows that covalency and still maintain a reasonable agreeq is positive for an ionic lattice, consistent with ment with the quadrupole interaction data. this conclusion. For the case of covalent bonds, An investigation of the Ala7 nuclear magnetic we need only consider the bonds on a given resonance(m) in Alp03 has shown partially resolved aluminum ion as contributing to the field gradient structure. If this resolution is ignored, the total at the nucleus; the bonds on ions other than the second moment is in good agreement with that one under consideration will give a much smaller calculated by the Van Vleck method. The Van 2u

S.

530

LAURXSCE,

Vlcck exprcssion126b as modified 0LLo~f(2iI is (for like spins) (A$)

E.

C.

MCIRVINE

by I
= h-2 4

x [. ;l(I+

--~w+1) -.- 1

2P(I+ l)+

1)2+3Z(Z+

l)+

+?

(14)

For Al*7 in &03, this sum was evaluated using values of 11 and BI calculated by an IHM 650 computer from the known crystal structure.Is)The solution is (AT?) = 7+0 (\ic/s)‘. Since hyperfine broadening explains the SXIR lint-width in Al?Oa, it is of interest to calculate the expected EPR line-width for the ClrS’ ion on the same basis, using the cxperimcntally detcrmined hype&e paramctcrs. For unlike spins, the Van Ylcck expression is different :

The this

addition of contact expression to

,‘lS>

interaction

will

= h-‘[t~zz(z+ I)]~(.4 + Hz);.

and J. LAMRE

“c” axis which passes through the Cr impurity and the other four are not. Observations made at differing levels of Cr concentration and at different values of the magnetic field allow us to positively assign quantum numbers to each of the transitions ohserved and to conclude that the quantity R + 8, is positive for each set. The five sets of lines arc associated with the five nearest inequivalent nuclear sites to a given chromium site. An a.ssociation of each site with a given set is made on the basis of calculations of the dipole field, assuming a point charge model for the clcctron distribution. ‘1’0 explain the magnitude of the contact intcraction ohservcd, it is necessary to postulate that the corundum lattice is about 20 per cent covalently bonded and that the interaction is transmitted to the aluminum nuclei through the aluminumoxygen hond. ‘I’hc quadrupolc interaction observed may hc explained by assuming that the chromium ion is displaced from the regular lattice site about 2 per cent of a lattice spacing away from the nearest aluminum neighbor. ‘I’hc point dip& calculations agree quite well with cxpcriment. indicatir:g that the clcctron is principally confined within the (distorted) octahedron of oxygen ions.

change

(16)

1

Using the values of (A +BL) determined experimentally for the first thirteen neighbors of the paramagnetic center and the calculated dipole values for the remainder of the lattice, we calculate a second moment of 29.3 (>Ic/s)* and predict a line-width of 9.7 G. The experimental line-width is 12 G, corresponding to a second moment of 44 (MC/S)*. Apparently the hype&e interaction with nearby Al*7 nuclei is the principal but not the sole source of the line width.

6. S-Y

ENDOR spectra comprising some 200 lines have been observed for ruby. The observed lines may be ascribed to 5 distinct nuclear sites, with a set of 3 hyperfine interaction parameters for each site. Analysis shows that one of the sites is vn the

.~rknotAfgrmen~r .-\Vc wish to thank R~c-~rawr)Acw for hlri assistance in performing these cxpcrimcnts. IVc wirh also to acknowlcdye several fruitful discurrsions concerning the intcrpwtation of the rcsuhs wth Drs. ‘I‘EwIY (:01X. A. w. O\:RHAI’SM. 11. \v. ‘I’?XlN~SI?

and ‘I‘. P. I).As.

RE?zRENcEs 1. F~tiaii G.. 1’h.v. Her. 103, 834 (1956): 105, 1122 (19573. 2. F&WX G.. I’hpr. Rm. 114. 1219 (1959). 3. Ihm <;.. Phvs. Rm. 105. 1122 (1957). 4. ImRD 5. w.; Phys. ReL~:IRffer* 1, 170 (1958). 5. HI.t’MRFJtG w. 1:. and FarlICit G., nua Amer. PbI. sot. (2) 5, 183 (1960). 6. IIWTOS w. <:.. L)I.tX 11. and SI.ICHlUl <‘. I’.. Phy. He. I_.trrm 5, 197 (1960). 7. COLE ‘I’., I~EI.I.ER C. and I.AMRP.J., /. Chm. I’hyr. 34, 1447 (1961). 8. LAWIE J., lnc~uscs S., \ICIHYISE E. C. and ‘I+.IICSP. R. \V.. Pbs. Rn-. 122, 1161 (1961).

9. ~V’YCKOWR. \V. (;., Cvrfa/ .Sfruclurrr, \‘ol. I. Intencicncc. Sew York (1948). IO. S~:C.ASO S. and ‘I’ASAIIE \‘.. J. Phyr. .%c. Japan 13, HXO(1958).

ALUMINUM

HYPERFINE

11. POUND R. V., Phys. Rev. 79, 685 (1950). 12. Gm J., J. Phys. Chem. Solidr 17, 18 (1960). 13. MCCLURE D. S., Solid State Physics (Edited by SEITZ F. and TIJRNBULL D.) Vol. 9. Academic Press, New York (1959). 14. MUKHERJI A. and DAS T. P., Phys. Rew. 111, 1479 (1958). 15. BER~~HN R., J. Chem. Phys. 29, 326 (1958). 16. PAULING L., The Nature of the Chemical Bond, Cornell University Press, Ithaca (1960). 17. HULTGIUXNR., Phys. Rew. 40, 891 (1932). 18. MCCONNELL H. M., J. Chem. Phys. 24,764 (1956). 19. MCCONNELL H. M. and CHE~NUT D. B., J. Chem.

Phys. 28, 107 (1958).

INTERACTIONS

IN

RUBY

531

20. NIJBOERB. R. A. and DE WETTEF. W., Physica 23, 309 ($957). 21. KATTERBACH K., Z. Astrophys. 32, 165 (1953). 22. FREEMAN A. J. and WATSON R. E., Phys. Rev. Letters 6, 343 (1961). 23. SILVER A. H., KUSHIDA T. and LAMBE J., to be published. 24. LEW H. and WESSEL G., Phys. Rew. 90, 1 (1953). 25. DAS T. P. and HAHN E.; So&f State Physics (E&ted by SEITZF. and TURNBULL D.) Suu~l. 2. Academic Press, New York (1958). ’ _26. VAN VL.ECKJ. H., Phys. Rew. 74, 1168 (1948). 27. KAMBE K. and OLLOM J. K., J. Phys. Sot. Japan 11, 50 (1956).