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EFFECT OF UNIAXIAL STRESS ON THE HELIMAGNETIC-FERROMAGNETIC PHASE TRANSITION IN SINGLE-CRYSTAL DYSPROSIUM
L. Benningfield, Jr., P. Donoho
To cite this version:
L. Benningfield, Jr., P. Donoho. EFFECT OF UNIAXIAL STRESS ON THE HELIMAGNETIC- FERROMAGNETIC PHASE TRANSITION IN SINGLE-CRYSTAL DYSPROSIUM. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-233-C1-234. �10.1051/jphyscol:1971175�. �jpa-00214502�
PROPRIETES MAGNETOELASTIQUES ET DE TRANSPORT DANS LES TERRES RARES
EFFECT OF UNIAXIAL STRESS
ON THE HELIMAGNETIC-FERROMAGNETIC PHASE TRANSITION IN SINGLE-CRY STAL DYSPROSIUM
(")L. V. BENNINGFIELD, Jr. and P. L. DONOHO Rice University, Houston, Texas 77001
Rksumk. - L'effet d'une tension uniaxiale sur le champ critique de la transition de phase htlimagnetique-ferromagne- tique a Cte etudie pour le dysprosium monocristallin. La tension uniaxiale de 600 bars a kt6 appliquee sur l'axe hexagonal a des temperatures entre 85 OK et 179 OK. L'augmentation du champ critique lors de l'application de la tension s'accorde bien aux predictions d'une theorie phenomenologique, et cette augmentation a lieu a cause du couplage magnetodlastique a deux ions.
Abstract. - The effect of applied uniaxial stress on the critical field for the helimagnetic-ferromagnetic phase transi- tion has been studied in single-crystal dysprosium. Uniaxial stress as large as 600 bars was applied along the hexagonal axis, at temperatures between 85 OK and 179 OK. The observed increase in the value of the critical field when stress was
applied is in good agreement with the predictions of a theory based on a phenomenological Hamiltonian, and this increase is primarily due to the two-ion magnetoelastic coupling.
The effect of applied uniaxial stress on the critical field for the helimagnetic-ferromagnetic transition has been studied experimentally in single-crystal dyspro- sium. Stress was applied along the crystallographic c-axis, and the resulting shift in the susceptibility maximum, which occurs at H,, was measured. A maxi- mum stress of 600 bars was employed, giving rise to strains comparable to the normal magnetostrictive strain which occurs a t the transition. At all tempera- tures in the helimagnetic range (85 OK t o 179 OK) H, increased linearly with stress, and increases as large as 1.5 kOe were observed. The increase in H, with applied stress can be attributed qualitatively to the fact that the application of compressive stress opposes the normal magnetostrictive expansion of the c-axis which accompanies the phase transition. As a result, a larger magnetic field is required to overcome the tendency toward helical ordering attributable to the indirect exchange energy. Theoretical predictions based on a phenomenological Hamiltonian show that the increase in the critical field is due primarily to the two-ion exchange magnetoelastic coupling.
The phenomenological spin Hamiltonian employed to explain the experimental results utilizes a set of approximately temperature-independent constants t o explain many of the observed magnetic properties of the rare earths. In the calculations discussed here, the molecular-field approximation has been employed, and for further simplification it has been assumed that the transition proceeds directly from an undistorted helix to a perfect ferromagnet, thereby ignoring the Zeeman energy in the helical state and the possibility of an intermediate fan-like state.
The magnetic properties of dysprosium can be considered the consequences of a Hamiltonian of the form : H = Ha
+
He,+
H,, f H,+
Hz. The first term, Ha, represents the crystal-field anisotropy, but it does not enter into the work discussed here, since measurements [I] indicate that the axial anisotropy is so large that all moments lie in the basal plane and that basal-plane anisotropy is very small for tempera- tures above 110 OK. The second term, He,, representsthe exchange energy due to the RKKY indirect mecha- nism. The contribution of this term to the free energy can be expressed, using the molecular-field approxi- mation, in terms of the reduced magnetization, o, the helical turn angle, 9, and three exchange constants, I,, I,, and I,, which are related to the first-through third-nearest-neighbor exchange integrals. Both these terms refer to the unstrained hexagonal lattice. The fifth term in the Hamiltonian represents the normal Zeeman energy of the system in an applied field.
The third term of the spin Hamiltonian represents the magnetoelastic coupling. The procedure employed here follows the very general phenomenological theory of Callen and Callen [2]. As a first approximation, the magnetoelastic coupling is assumed to be a linear function of lattice strain and a quadratic function of angular-momentum operators. Two sources of the magnetoelastic coupling have been considered : the variation in crystal-field energy of each ion with lattice deformation, and the variation of exchange energy with lattice deformation. If lattice symmetry is taken into account, the one-ion magnetoelastic coupling can be expressed in terms of five constants, B,, B,, B,, B,, and B, 121. If it is assumed that the dominant source of the two-ion coupling is the variation of the isotro- pic exchange with c-axis strain, then the two-ion magnetoelastic coupling may be expressed in terms of four constants, D,, D,, D,, and D,.
The fourth term in the Hamiltonian, He, represents the normal classical elastic energy, and it has been expressed in this work in terms of the usual second- order elastic constants.
With the above descriptiolz of the phenomenological Hamiltonian, the internal energy per ion in the helima- gnetic phase may be written :
Ul,, = - [I,
+
(I, -I- Dl e,) cos cp+
(I,+
D, e 3 )+ +
cos 2 (p] 02 - f ( ~ ) (el+
e,)-
g(o)e,+
U,.
In the ferromagnetic phase, the internal energy per ion may be written :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971175
C 1
-
234 L. V. BENNINGFIELD, JR. AND P. L. DONOHOIn ,these expressions, the ei are the equilibrium magne- tostrictive strains, /3 is the Bohr magneton, and g is the Lande factor for dysprosium. The functions f(c), g(o), and h(o) are defined as follows :
The symbol T5/2 represents the normalized modified Bessel function whose argument is the inverse Langevin function of o, following Callen and Callen [2].
In a manner similar to that of Landry [3], the Gibbs free energy per ion in either magnetic phase may be written :
In this expression, T, is the applied stress, a, is the coefficient of thermal expansion along the c-axis, E is the effective Young's modulus, and S is the entropy as defined by Landry. In order to obtain expressions for the helimagnetic turn angle or the equilibrium strains, the free energy in each phase is minimized with respect to variations in p and the strain.The phase transition occurs when the free energies of the two phases are equal, and this condition of equality yields an expression for the critical field, H,, which is, within the approximations employed here, a linear function of the applied stress :
AH, = Hc(T3) - Hc(0) =
= (oT,/gPJE) [D1(l
-
cos cp)+
2 D,(1 - cos2 p)].
I t may be seen in this expression that the stress- dependent change in the critical field depends upon the two-ion magnetostrictive constant Dl and D, and that this two-ion magnetoelastic effect is the dominant source of the variation in internal energy with c-axis strain. The temperature dependence of the slope, AHJT,, is due to the temperature dependence of o and cp. Thus, experimental measurements of this slope, together with known values of a, 9, and E will yield an accurate determination of the two-ion magneto- elastic coefficients Dl and D,.
Measurements were made with a single-crystal disk of Dy, 0.18 mm thick and 6.0 mm in diameter, with the disk axis parallel to the c-axis. The susceptibility was measured by means of a straightforward Iow- frequency method as a function of applied magnetic field. The magnetic field was applied in the a-direction,
and the abrupt increase in magnetization at the cri- tical field gave rise to a well-defined susceptibility peak. Stress was applied by means of a simple screw, lever, and piston mechanism, and the stress was cal- culated from the observed strain in the single-crystal sapphire piston.
The variation of H, with stress was measured in the temperature range 120- 170 OK, and at all tempe- ratures in this range the critical field increased linearly with increasing stress. The variation of H, with stress was completely reversible. The best fit of the experi- mental data to the expression obtained above for the stress-dependent shift in Hc gave the following values for the two-ion magnetoelastic constants :
Dl = 2.0 x lo-'' erg = 1.40 x lo5 OK ; D, =
-
0.51 x lo-" erg = - 0.37 x lo5 OK.The theoretical temperature dependence of the slope AH,/T3 utilizing these values for Dl and D, is compa- red with the experimental data in (Fig. 1).
FIG. 1. - Variation of AHc/T3 with temperature. Solid curve represents theoretical prediction.
These values for Dl and D, are in considerable disagreement with those of Landry [3], who obtained his values by fitting the variation of the turn angle to the theory. The reason for this discrepancy ,is not understood at present. Further work is in progress, including new measurements with stress along different axes and a new theoretical treatment containing fewer approximations.
References
[I] BEHRENDT @.), LEGVOLD @.) and SPEDDING (F.), Phys. Rev., 1958, 109, 1544.
[2] CALLEN (E.) and CALLEN (H.), Phys. Rev., 1963, 129, 578.
[3] LAND^.^.), Phys. Rev., 1967, 156, 578.
