Spin-lattice interactions of ions with unfilled F-shells measured by electron spin resonance in uniaxially stressed crystal

Black, T. D.

Donoho, P. L.

English

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Spin-Lattice Interactions of Ions with Unfilled f-Shells 
Measured by ESR in iiniaxially Stressed Crystals 1- 
T. D. Black" and P. L. Donoho 
William Marsh Rice University 
Houston, Texas, U. S. A. 
Abstract 
The effect of uniaxial stress on the electron-spin- 
resonance spectra of U 3 +  and Nd3+ ions in CaF2 has been in- 
vestigated. The uniaxial stress wzs directed along the [OOl] 
and [ l o l l  axes, and its effects on the spectra of ions in 
sites of tetragonal symmetry were observed. Measurements 
were made at a temperature of 4 . 2 O K  and at a frequency of 
9.4 GHz. The effect of the applied stress was a shift in 
the position of a resonance line proportional to the magni- 
tude of the stress. The dependence of this shift on the 
magnitude and direction of the applied magnetic field was 
found to be in excellent agreement with that predicted by 
the inclusion in the spin Hamiltonian of a spin-lattice 
perturbation of the form, HSL = 6hijSiHj, where hij is a 
tensor linearly dependent on the strain in the crystal 
lattice: hij - Gijklekl. 
turbation added to the normal 9-tensor for the ion. For 
tetragonal symmetry, there are seven independent components 
of the G-tensor, and these have all been determined for the 
ions under consideration. 
In effect, hij is simply a per- 
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Spin-Lattice Interactions of Ions with Unfilled f-Shells 
Measured by ESR in Uniaxially Stressed Crystals 
T. D. Black and P. L. Donoho 
William Marsh Fice University 
Houston, Texas, U. S. A. 
The spin-lattice interaction for Nd3+ and U3+ ions at 
sites of tetragonal symmetry in CaF2 has been studied by in- 
vestigating the effect of uniaxial stress on the electron- 
spin-resonance spectra of these ions. Measurements were 
made with stress directed along 'che [OOl] and [loll axes, 
and were performed at a temperature of 4.2OK and a frequency 
of 9.4 GHz.  The application of stress resulted in a shift 
in the position of each resonance line proportional to the 
magnitude of the stress. The observed shifts were found to 
be typically 5 to 10 Oe for stresses which resulted in 
lattice strain of magnitude approximately 3 x 
The strain component along the stress axis was measured 
by means of a constantan foil strain gauge bonded directly to 
the stressed crystal. The stated gauge factor supplied by 
the manufacturer was reduced for operation at 4.2OK by approxi- 
mately 10 per cent, in accordance with the measurements of 
McClintock (1959). Although the indicated strain was dependent 
upon the value of the applied magnetic field, presumably because 
of magnetoresistive effects, the incremental strain due to 
applied stress was found to be, indepen3ent of the magnitude 
of the field, and a field-independent gauge factor could thus 
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Research Centers Only. 
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2 .  
be used. The estimated accuracy in the measured strain was 
The shift in the resonant magnetic field was measured 
by means of an NMR magnetometer to an estimated accuracy of 
0.1 Oe. 
The dependence of the observed line shifts on the magni- 
tude and direction of the applied magnetic field was found to 
be in excellent agreement with that predicted by the inclusion 
of a perturbation term in the s2in Hamiltonian of the form 
HSL = Bh..S.H 
In this expression 6 is the Bohr magneton, S is the effective- 
spin operator, H is the applied nagnetic field, and hij is 
a tensor linearly dependent on the lattice strain: 
In this experiment, the components of the strain tensor could 
(summation over repeated indices is implied). 
11 1 j 
- 
hij - Gijkl ekl* 
be determined from measurement of the component along the stress 
axis by using the known elastic constants and the fact that the 
stress was uniaxial. 
This form for HSL 
as that first employed 
lattice relaxation for 
is derived essentially in the same manner 
by Van Vleck (1940) to explain spin- 
certain iron-group ions. That is, the 
stress-induced distortion of the lattice produces a perturba- 
tion of the crystalline electric field acting on the ion. To 
first order, this perturbation to the total electronic Hamil- 
As observed by tonian is linear in the strain: HI - Vij eij. 
Orbach (1961), the Kramers-doublet ground state of the ions 
- 
considered here cannot be split by a perturbation of the 
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crystalline electric field alone. As a result, the effect 
of HI can be observed only in the presence of a magnetic 
field, just as in the case of a naturally occurring crystal- 
field component, which would merely lead to a non-iostropic 
g-tensor for the ion, in the spin-Hamiltonian formulation. 
Thus, the effect of the stress-icduced perturbation is simply 
a variation of the normal g-tensor. This variation of the 
g-tensor is just the h-tensor introduced above. Of course, 
a more complicated form for the spin-lattice perturbation to 
the spin Hamiltonian would occur if non-linear strain effects 
or higher-order perturbation theory were employed. 
the experimental results reported here are in excellent 
agreement with the simple form given above for HSL, and no 
more complicated form is considered here. 
However, 
In order to calculate the effect of applied stress on the 
position of a resonance line, it is necessary to determine the 
eigenvalues of the total effective-spin Hamiltonina, which, 
neglecting hyperfine terms, takes the form Hs = B(gij + hij)HiSj. 
Since the ion has a Kraners-doublet ground state, for which 
S = %, a particularly simple form for the rate of change of 
the resonant field with respect t o  the component of the strain 
along the stress axis is found: 
aH/ae = (-hv/ag3) cjii b.b .G ijkl (dekl/de). 
In the above expression, hv = gCH is the unstressed resonance 0 
condition, b. are the direction cosines of the magnetic field 
with respect to the coordinate system in which the g-tensor 
1 
is defined, and dekl/de is a tensor whose components are 
simple functions of the elastic compliance tensor. 
XCRO 
cony 
-. 
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4. 
The electronic configurations of the ions U 3 +  and Nd3+ 
are, respectively, 5f3 and 4 f 3 .  In CaF these ions are 
found primarily in sites of tetragonal symmetry, in which 
ground state splits into five Kramers doublets. the 41 
According to the model proposed by Bleaney, Llewellyn and 
Jones ( 1 9 5 6 ) ,  the tetragonal syrmetry is a consequence of an 
interstitial charge-compensating F- ion which may be located 
along any one of the three principal cube axes, resulting in 
symmetry at the site of the ion under consideration. 
Although a small percentage of ions is found in sites of dif- 
ferent symmetry, such ions were not considered in the work 
reported here. The tetragonal symmetry, in a manner similar 
to that encountered in a considera-tion of the elastic con- 
stants, simplifies the interpretation of the experimental 
results by greatly reducing the number of independent com- 
ponents of the G-tensor. These components can, therefore, 
be evaluated with a relatively small nunber of measurements. 
9 / 2  
c4v 
Because of the tetragonal symmetry, the h-tensor is 
effectively symmetric to an i2terchange of its indices, and 
the six-dimensional Voigt notation can be employed wherein 
each second-rank tensor becomes a column matrix and each 
fourth-rank tensor becomes a square natrix in six dimensions. 
Thus, using the same notation for the quantities of interest 
here, it is possible to write hm = Gmn en. Again because of 
the C4v symmetry, many of t h e  components of the G-tensor are 
zero and many others arc equal to cach other, and this tensor 
can be written in Voigt notation in the following way: 
1 ,  I 
5. 
G 1 2  G1 3 0 0 0 
G 1 1  G1 3 0 0 0 
G 3  1 G3 3 0 0 0 
C 0 G4 4 0 0 
0 ! 0 0 0 G44 
K O  0 0 0 G66_1 
It should be observed that Gpn is not symmetric to an inter- 
change of indices, so tyat G 1 3  and G 3 1  are not equal. Thus, 
there are seven independent compoDents of the G-tensor, all 
of which have been determined by the method reported here 
for both of the ions under consideration. 
In general, each measured line shift was due to a com- 
bination of several of the components of the G-tensor, although 
a few special cases occurred in which the shift was due to a 
single component. Because of the presence of ions with three 
different symmetry axes in each experimental sample, however, 
it was possible to obtain sufficient data for the calculation 
of all components of the G-tensor with only two samples. In 
one of these, the stress was applied along the [ O O l ]  axis, 
and in the other it was applied along the [lo13 axis. From 
measurements on these two samplzs, it was possible to obtain 
a least-squares fit of the predicted line shifts to the ex- 
perimental data, resulting in the following values for the 
components of the G-tensor for the two ions: 
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6. 
G 1 1  G 1 2  G1 3 G 3  1 G 3  3 G 4 4  G 6  6 
Nd3+: 4.6'1.0 3.1'1.0 4.6'1.5 1.8'1.0 1.7'1.0 -8 .9 '9 .1  G.l*l.c 
u 3 + :  1.520.5 1.3'0.5 1.050.5 2.P'l.O 3.2'1.0 -8.9'1.0 -5.9'0.1 
I 
The indicated error incluc?es the estimated errors in the 
measurement of the strain and field values, probable errors 
in the elastic constants, and other errors which arise from 
in accuracy of crystal orientation, nonuniformity of stress, 
etc. 
Examples of the comparison of the experimental data with 
the predicted line shifts using the above values for the G- 
tensor are shown in Figures 1 ;?nd 2. In both cases, the 
stress axis was in the plane of rotation of the magnetic field, 
and e = 90° corresponds to a situation where the field and 
stress were parallel. The error bars on the experimental 
points represent the sprea? in reFezted experimental measure- 
ments. 
The direct one-phonon s?in-lattice relaxation times for 
these ions have been computed, using the spin-lattice Hzmil- 
tonian introduced here, by Doncho acd Black (1966). Although 
some agreement with experimental measurements was found in 
the case of U3' ions, an enormous disagreement wa.s found in 
the case of Nd3+ ions. This large 6isagreement is surprising 
in view of the good agreement between the results of the wor:; 
reported here and the theory oxtlined In this paper. 
As a verification of this work, an attempt was made by 
'Vetsel and Donoho (1965) to measure the spin-lattice inter- 
action for U3+ ions by the nethod of ultrasonic resonance. 
7 .  
No resonance absorption was observed, however, since the 
maximum absorption pre2..i,-ed from the measured G-tensor 
was too small to be detected. 
X F F O  
‘01 Y 
.- 
8 .  
B. Bleaney, P .  M. L lewel lyn ,  and D. A .  Zones ,  Proc.  Phys. 
SOC. - B69, 858 (1956). 
P. L .  Donoho and T .  9. Black, accoi?.panying paper .  
R. ?I. NcClintock, Rev. S c i .  I2str. - 3 0 ,  715 ( 1 9 5 9 ) .  
R .  L. Orbach, Proc.  Zoy. SOC. A.26Ar 458 (1961). 
J. H. Van Vleck, Phys. Xev. - 57, 426 (1940). 
G. C. Wetsel, Jr. and P. I,. DoncL-, Phys. R e v .  - 1 3 9 ,  A 3 3 4  
(1965). 
I 
9 .  
Figure Capticns 
Figure 1: aH/ae for E 3 +  icns w i t h  synT.ztry about [ O l O I ,  
and with s t r e s s  apyIFec3 along [ l o l l .  The angular 
dependence is simply cos (29). Smooth curve rep- 
resents lez-t-squares fit. 
3; . Figure 2 :  aH/ae for Ne.- ..oris with symmetry about [lOO], 
and with stress applied along [ l o l l .  Smooth curve 
represents least squares fit. 
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Spin-lattice interactions of ions with unfilled F-shells measured by ESR in uniaxially stressed crystals

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