Volume 118, number 2 PHYSICS LETTERS A 29 September 1986

NUCLEAR Q U A D R U P O L E C O U P L I N G T E N S O R S OF ~70 IN FORSTERITE, M g 2 S i O 4

R. FRITSCH a D. B R I N K M A N N b S.S. H A F N E R a S. HOSOYA a j. LORBERTH ~ and J. ROOS b

Institut for Mineralogie der Universitiit Marburg, 3550 Marburg, FRG h Physik-lnstitut der Unwersitiit Ziirich, 8001 Zurich, Switzerland ' Fachbereich Chemie der Universitiit Marburg, 3550 Marburg, FRG

Received 11 March 1986; revised manuscript received 11 July 1986; accepted for publication 31 July 1986

The nuclear quadrupole coupling tensors (eigenvalues and eigenvectors) of 170 were determined in a single crystal of Mg2SiO 4 using pulsed nuclear magnetic resonance. The tensors are of the same order of magnitude as in AI203 and TiO 2.

1. Introduction

Since the first studies of nuclear quadrupole Coupling tens'Ors in crystals more than three de- cades ago a large number of tensors were de- termined in various crystal structures, mainly using nuclear magnetic resonance, nuclear quadrupole resonance, and the MiSssbauer effect. In oxide crystals, primarily tensors of cationic nuclei were studied. There are almost no data on oxygen. Although the nuclear spin 1 = ~, the quadrupole moment Q = - 0 . 0 2 5 7 b, and the gyromagnetic ratio y/2v = 0.5772 M H z / k G of 170 allow the observation of resonance of this isotope under conventional conditions, the natural abundance is only 3.7 x 10 -4. It is for this reason that only few tensor data of this important atom were investi- gated in crystals.

Magnetic resonance of lvO in a solid was ob- served first in MgO [1] and MnO and CoO [2]. Quadrupole coupling tensors of 170 were de- termined in corundum A1203 [3] and rutile TiO 2 [4] using dynamic nuclear polarisation. We have studied the 170 coupling tensors at the three dis- tinct positions of oxygen in forsterite Mg2SiO 4 which possesses the orthorhombic space group Pbnm. In the crystal structure of forsterite oxygen O1 and 02 occur at two different positions 4c with point symmetry m and 03 is assigned to a general position 8d without symmetry [5]. Fig. l

(a) < b

[

(

I 0(2)o(3> oI,l oc3 0(2) 0(3) o(0 I

Ib)

Fig. 1. Structure of forsterite (figs. 11-15 of ref. [6]). ®: Mg(1); ~,: Mg(2); small black or open circles: Si; large or medium sized open circles: O1, 02, 03 . (a) projection on the

(a, b) plane; (b) projection on the (a, c) plane.

98 0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 118, number 2 PHYSICS LETTERS A 29 September 1986

shows the projection of the unit cell on the (a, b) and (a, c) plane.

2. Experimental

transition m ~ m - 1,

( ] )_ eQ [3m 2 I ( I + 1)] H - 4 1 ( 2 1 - 1 ) - ( 2 )

II. H, the electric field gradient along the z axis of

A single crystal with a diameter of about 20 mm in every direction was grown from the melt at 1800°C by the Czochralski method [7-9]. Oxygen in the crystal was enriched to about 15 percent in ~70 via MgO which was obtained from Mg(CHs) and H:O mixed with H2170. The crystal frag- ments used for the experiment had a size of about 3 × 8 x l l mm 3.

The experiments were performed at room tem- perature in a superconducting magnet of 5.17 T field strength with a pulsed nuclear magnetic reso- nance spectrometer at frequencies between 29 and 31 MHz. The ~70 free induction decay signals were digitized by a transient recorder and accu- mulated at least 128 times per frequency setting at a repetition rate of about 0.2 Hz. Subsequently, Fourier transformation was carried out by an on- line computer. In order to measure the depen- dence of the resonance frequencies on the orienta- tion of the crystal to the magnetic field, the crystal was rotated around its three crystallographic axes, a, b, c, the rotation axis being perpendicular to the magnetic field. The resonance frequencies were determined in steps of two degrees for each rota- tion.

3. Results

The basic formula for pure Zeeman and quadrupole interaction is given by the hamiltonian

eQ Vz z j~o= - yH . I =

4 1 ( 2 1 - 1)

× (1)

where I and Q are the nuclear spin and quadru- pole moment and V is the electric field gradient tensor in the principal axis system (see e.g. ref. [IOD. First-order perturbation theory yields an expression for the quadrupole energy shift of the

a H I I c . o ' ' . . . .

o . o ° * . ° • ° , ,

30 .~ . ,

, . . .

• ~' . ! ~ . . . . .:! • . • ' . ,

30..~ I : , '

• •; . t - t ' , . . ~

30,.j | " ' . . . . . . . , ; ' " • . *•. ". . " ° . . . . ,~ • ~ ' ; ~ , . . ~ i , ~

, : " ; ~ . . , ° " ' . : " . , i . i

30.~ ~ " " ~ , s~s

. . , ,~ / : ~ / . , . •

e ' . o . °

5 " o ; I *~ t~ . ~ - , . . ° , .

2 I ' j . . . . . . . . > . ~ ~b,,~i,| o , • ' . ' " ' .

Z *~ , i , s " .~ . . ° ~.J • • ' . , " , , ° b ~ ' • ."

I L l I , ' ' ' , ° ' " ~ "

. . . : . , " , : ] : . , , ; t , : ; : ~ I ~ . . . : ; ~ ~ ~ : . . .

. ~ ." . . . . . . : i ~ . . . . " ' " . .

. . ".~:'.o.. ",~;.~" 29.ai ' ' . .

' • . . "

29.,.3 " " ' " " . . . . " ' " . " ' " ' . ' . , . . o , °

R O T A T I O N A N G L E

Fig. 2. Resonance frequencies of 170 in forsterite as a function of the angle of rotation around the crystallographic b axis. The constant magnetic field B of 5.17 T is perpendicular to the rotation axis. The three distinct positions are: 4c(1) (crosses),

4c(2) (triangles) and 8d (dots).

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Volume 118, number 2 PHYSICS LETTERS A 29 September 1986

the laboratory frame can be expressed in terms of the rotation angle 0, around the crystal i axis ( i = a, b, c):

v : . = o . s ( v # + v;;) +0.5(V( ' {' V ( sin 20,, (3) ,:/ - Vxk) cos 2 0 , - ix

i, j , k cyclic. V C is the electric field gradient in the crystal frame [11].

The pattern of the resonance frequencies in MHz obtained from the rotation of the crystal around the b axis is shown in fig. 2. The multitude of points results from transitions due to spin I = of lVO and the multiplicities of the distinct anion sites.

It was possible to assign the resonance signals to oxygen positions at 4c and 8d of Pbnm. All observed signals fitted into this scheme without exception. According to the point symmetries of

the positions 4c and 8d we expect the multiplici- ties for the N M R transitions given in table 1.

Inspection of fig. 2 yields two sets of frequen- cies which correspond to point symmetry m of 4c(1) and 4c(2), and one set corresponding to 1 of 8d. Fig. 2 also shows that perfect orientation was not achieved for the rotations. The small misalign- ment illustrates the intrinsic multiplicities of the positions according to table 1, and was quite helpful for correct assignment of the sets.

Fig. 3 shows the angular dependence of the frequency separation of the inner satellites (i.e. the

---, ½ or - ½ ~ - 2 transition) for the three rota- tions.

The ]70 quadrupole coupling tensors were de- termined using second-order perturbation theory [11] for the frequency separation

3eQ (4) Av = ( m - - ½)v:H: I ( 2 I - - 1)h

400

200

I 2~

z l ' o

Q.

~ 200 z

©

-401

/ / / /

/ 1 / / c . LH

/ / / /

i i i / ./_2-"

allH I I I i 0 310

. / / "~ \

/ x

/ "\

/ " ~ I ~ ~

, , ' i " I " \ , / / / i \ \ x \

, , ' / i // / i \ ,, .+..- ' \ I , I I I t~/ ~ \ \ t / ~ /

I 11 X / / ,1~ ; ~ XX \ / / /

/ . \ / / i I ~ / / / ' 1 \~

a i M b .LH

_ i _ Fig. 3. Angular dependence of the frequency separation of the inner satellites (i.e. the ~ ---, ~ or ,

\ \ I \ \ , I

bllH c l IH

' ; o ' ' ' . . . . . , ; o ' ' I ' ' ~'o ' , ~> , , l 9 0 30 90 6 9 0

ANGLE OF R O T A T I O N ( D E G R E E )

transition) of the three rotations. The three distinct oxygen positions are: 4c(1) (dashed lined), 4c(2) (dash-dotted) , and 8d (solid).

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Volume 118, n u m b e r 2 P H Y S I C S L E T T E R S A 29 Sep tember 1986

Tab le 1 N M R mult ipl ic i ty , i.e. the number of d is t inct quadrupo la r spl i t t ings, expected for a par t icu lar pos i t ion 4c or 8d due to the

o r i en ta t ion of the crystal with respect to the external magne t ic

field B.

Posi t ion Or ien ta t ion wi th respect to B N M R mul t ip l ic i ty

4c general 2 B in (a, b) p lane 2 B in (b, c) or (c, a) p lane 1 B paral le l to a, b or c axis 1

8d general 4 B in (a, c), (b, c), (a, b) p lane 2 B para l le l to a, b, or c axis 1

and the direction cosines between the crystal and principal axis system, which are presented in ta- bles 2 and 3.

The three rotation patterns left a two-fold am- biguity in determining the 8d tensor because of two possible combinations of each NMR transi- tions when the magnetic field is parallel to either of the crystallographic axis. The ambiguity could be removed by an additional rotation around the direction of the principal axis Y.

4. Discussion

Table 2 Quadrupo le coup l ing cons tan ts e2qQ/h and a symmet ry pa ramete r s 71 of 170 located at 4c(1), 4c(2), and 8d in forsteri te

Mg2SiO4.

Site eZqQ/h (MHz) r/

4c( l ) 2.77(2) 0.28(2)

4c(2) 2.53(2) 0.39(1) 8d 2.42(2) 0.18(1)

of a satellite pair. Transforming the tensors from the crystal sys-

tem to the principal axis system yields the quadrupole coupling constants e2qQ/h = eQVzz / h, asymmetry parameters ~ - (Vxx - Vvr ) / Vz z ,

The coupling constants of 170 at the three distinct positions are quite similar to each other. They are also similar to the coupling constants of 170 in corundum (2.167 MHz [3]) and rutile (1.497 MHz [4]). In contrast, the quadrupole constants of cation nuclei in oxides, e.g. of 27A1, vary by orders of magnitudes [12-23].

Somewhat different results were obtained from a study of forsterite powder [24]: e2qQ/h = 2.70, 2.35, and 2.35 MHz, and r /= 0.3, 1.0, and 0.2, respectively. Since this work was based on a line shape analysis it cannot be compared with direct information obtained from the rotation patterns of a single crystal. The present work yields more accurate data on the coupling tensors and gives the complete orientation of the principal axes with respect to the crystallographic axes.

Table 3 Direc t ion cosines be tween the pr inc ipal axes X, Y, Z of the quadrupo le coup l ing tensors of 4c(1), 4c(2), and 8d and the crys ta l lographic axes a, b, c. The signs in front of the cosines indicate the ambigu i ty cor responding to the four c rys ta l lographic equiva len t sites of 4c, and the eight equivalent sites of 8d. Note that oxygens at each site are re la ted in pai rs by an inversion center

and are magnet ica l ly equivalent . The symmet ry opera t ions are (from right to left): 1, b, m, n.

Site Pr incipal axis Crys ta l lographic axis

a b c

4c(1) X - +0 .0653 + +0 .9978

Y 0 0 Z + +0 .9979 + - 0 . 0 6 5 3

4c(2) X - + 0.8176 - - 0.5756 Y 0 0 Z + +0.5757 - +0 .8177

8d X + + - +0 .7984 - + + +0.2136 Y - - + - 0 . 3 3 8 5 - + + +0 .9324 Z - - + +0 .4979 + - - +0.2915

0 1

0

0 1

0 - + - - 0 . 5 6 2 9 - + - - 0 . 1 2 6 4

- + - +0 .8168

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Volume 118, number 2 PHYSICS LETTERS A 29 September 1986

W e cou ld u n i q u e l y ass ign the c o u p l i n g t enso r

at pos i t i on 8d to 0 3 . It was no t poss ib le to ass ign

the two tensors at 4c to e i ther O1 or 0 2 by m e a n s

o f e x p e r i m e n t . N e v e r t h e l e s s there are s o m e hints

for ass ign ing 4c(1) to O1. T h i s a s s u m p t i o n is

s u p p o r t e d by the 29Si N M R shif t [25] and the

Fe 3÷ E P R m e a s u r e m e n t s in a syn the t i c c rys ta l o f

fo r s te r i t e [26].

Acknowledgement

This w o r k was s u p p o r t e d by S F B 127 o f the

D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t . W e thank

D i e t r i c h Babel , D i e t e r M a t e i k a and H e l m u t R a g e r

for thei r suppor t .

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