STATIC AND DYNAMIC STRESSES 151 a significant magnetic field dependence. On the base o f these c33(H ) measurements a simplified magnetic phase diagram o f D y was constructed by Isci and Palmer (1978) which is shown in fig. 26.
Jiles et al. (1984) studied the isotherms
c33(H )
and isofields c33(T) o f high purity terbium and constructed its magnetic phase diagram. Kataev et al. (1989b) investigated Young's modulus Eö in high purity Tb (R300/R4.2 ~ 100) in fields up to 2.5 kOe. The phase diagram obtained in this work is shown in fig. 27. The anomalous behavior o f Her observed near 223.5 K was associated with the commensurate effect by the authors.The field dependencies o f Ea and Eb were measured for a Tb0.»Dy0.5 single crystal by Kataev et al. (1989c). Like Tb and D y this alloy is a helical antiferromagnet with an easy basal plane (Bykhover et al. 1990). The field behavior o f E» and Ea in the H A F M phase is analogous to that observed for c33 in D y and Tb by Isci and Palmer (1978) and Jiles et al. (1984) (see fig. 28). The magnetic phase diagram Hcr(T) o f Tb0.»Dy0.» constructed
156 K / / / ~ 159
_ /~~//~ 162
45 ~ ~ , . . ~ ~ ~ f ~
181
~~i~ ~ ~~,~
£~
43181
4 5
44
43
I I I I I I I
4 8 12
Hcr (kOe) 5
FM
4
,/
3 //"
I ,it
2i
I
PM 10 140 160 180 200 T(K)
Fig. 29. Magnetic phase diagram of Tb0»Dy• » obtained from field measurements of Young's moduli Eo (dashed lines) and Eb (solid lines) (Kataev et al. 198%).
on the base of the data on Eh(H) and Ea(H) is shown in fig. 29. As for Tb and Dy three magnetically ordered phases were distinguished: HAFM, fan and FM. In spite of some anisotropy in the field behavior of Ea and E» (see fig. 28, the curve for 181 K), the phase diagrams obtained from Ea(H) and Eó(H) curves are quite similar. The magnetic phase diagrams of Tb, Dy and Tb0.sDy0.5, which were constructed on the base of the data on elastic constants and moduli, are consistent with those obtained by means of magnetization measurements.
7.3.2. Ferromagnetic phase
Southem and Goodings (1973) considered theoretieally the field dependence of the elastic constants c~j in the ferromagnetic state in the lanthanide metals with a hexagonal crystal structure. It was found that for H applied parallel to the c-axis c11, c33 and c66, and for H aligned along the a- or b-axis in the basal plane eil and c33 were not affected by the magnetic field. For the c44 and c66 field dependencies complex expressions were obtained.
In the case of a magnetic field aligned parallel to the c-axis the change of the C44 elastic constant has the form
o(h - ho)(b« ~r 3 ± B )
~C44 =
c44(H)-c44(~-I0)= ~1 ; ~ ; ~
(102)where
B = -3kl - 10k2o 4° - 21k6(721. (103)
Here b e is the magnetoelastic coupling constant cormected with the constant B c from eq. (68); the constants kl,k2 and k66 describe the tmiaxial and basal plane magnetocrystalline anisotropy, respectively; er, the reduced magnetization; H0, the magnetic field large enough to overcome the magnetocrystalline anisotropy and to ensure really complete alignment of domains; and h = NgjIABHJ, the reduced magnetic field. The sign in the last term of the numerator in eq. (102) depends on the direction of propagation
STATIC AND DYNAMIC STRESSES 153
4 0 I
Tb
3 0
1.0
20
.«'/" 0.7
10 ,/" 0.5 ... ~-
, " ' Y ' ... 0.3
0 2 4 6 8
Fig. 30. AcöJc66 and AC44/C44 a S a function of (H - Ho)/H o calculated for terbium by Southern and Goodings (1973) (magnetic field along the b-axis). The values near the curves are the reduced magnetization. Solid curves: Æc44/¢44 for sound wave propagation along the c-axis; dashed curves, Ac66/c66 for propagation along
the b-axis.
and polarization of the sound wave: the plus refers to the wave propagating along the c-axis and polarized along the a- or b-axis, and the minus is for the opposite situation.
For the case of a ferromagnetic phase two-ion magnetoelastic interactions were not taken into account. The calculations made by Southern and Goodings (1973) for Gd, Tb, Dy and Er, which were based on experimental data of the anisotropy and magnetoelastic coupling constants, showed a nonlinear increase of c44 and c66 with the field aligned in the basal plane. Field saturation of Ac66 was observed for H ~ 9H0 (H0 = 10kOe) for Gd, Dy and Ho. The dependencies c44(H) do not reveal saturation in Tb and Dy. The maximum saturated value of Ac66/c66 was calculated to be about 10 2-10-1 for H = 9Ho for Th, Dy and Ho (for a = 0.5) and the minimum (~ 10 4) for Gd. Figure 30 shows the calculated ÆC44/C44 and Ac66/c'66 as functions of the field directed along the b-axis for Tb. Experimental measurements of c33 in Dy and the Ho0.»Tb0.5 alloy made by Isci and Palmer (1977, 1978) reveal a significant field dependence in high fields which is not consistent with results of Southern and Goodings (1973). Isci and Palmer attributed this effect to the incomplete alignment of spins or to the existence of ferromagnetic and fan domains even in such high fields above H f .
Jensen and Palmer (1979) studied the behavior of c66 in the ferromagnetic phase of Tb under magnetization along the hard a-axis in the basal plane. Application of a
154 A.M. TISHIN et al.
magnetic field of sufficient magnitude (~ 32 kOe) leads to the alignment of moments in this direction. Below this critical field value the moments make some angle with the a-axis. The transition between the two phases is of the second order and the transverse differential magnetic susceptibility diverges. Under these conditions the action of elastic stresses will give rise to a strong variation of the magnetic structure leading to a large softening of the elastic constants. It was shown with the help of the random-phase theory of a coupled spin-lattice system (Jensen 1971, Liu 1972, Chow and Keffer 1973) that at the field corresponding to the above described transition, c66 should go to zero if the shear wave propagates along the hard a-axis and it should display a deep minimum if it propagates along the easy axis. Experimental results obtained by Jensen and Palmer (1979) confirm the theoretical predictions.
Kataev et al. (1985) measured the field dependencies of Young's modulus along the easy b- and hard a-axis in the ferromagnetic alloy Tb0.4Gd0.6 for H directed along the b-axis. In the low field region (~ 2 kOe) a negative AE effect (AE = E(H) - E ( H = 0)) and hysteresis of the AE/E(O) curves were found. For an explanation of this behavior the domain model was used. Two utmost cases were considered: (a) the domain walls are displaced by the field and applied mechanical stresses and spin rotations are absent;
(b) the same factors alter the equilibrium spin direction in the domains and the domain walls are fixed. It was shown that both mechanisms can cause the negative AE effect and the minimum on Eh(H), but the main role is played by the rotation of the spins.
It happens in fields where the domain structure is in a nonequilibrium stare, and the abrupt reorientation of the non-180 ° domains to the easy basal plane axis closest to the field direction takes place. This can lead to a considerable softening of E, and its field hysteresis. A further increase of Young's modulus in high fields can be related to the process of domain moments aligument.
The temperature and field dependencies of the elastic constants of Er were studied by du Plessis (1976), Jiles and Palmer (1981) and Eccleston and Palmer (1992). The transitions at Ty, Tcy and Tc manifest themselves in anomalies of the temperature dependencies of c~1, c33 and e44 (see fig. 31). A magnetic field applied along the c-axis softens c33 and c44 in the longitudinal spin wave (LSW) and eycloidal phases, and restiffens cll and c66 from the values in zero magnetic field. Jiles and Palmer (1981) studied the field dependencies of clt and c33. For the c33(H) curves measured below Tc for H aligned along the b-axis a lowering of c33 at ~ 17kOe was observed, and in the temperature range 3 0 - 5 0 K for a c-axis field a rapid increase in cll occurred between 5 and 12kOe depending on temperature. The authors thought this behavior was related to the changes in spin structure: the destruction of the basal plane spiral in the first case and with a collapse of cycloidal structure into a cone in the second case. Above Tcy and in the paramagnetic phase the field had little influence on c33. The zero field measurements of c33, c11, c44, and C66 made by Jiles and Palmer (1981) and Eccleston and Palmer (1992) confirmed the general form of the cij(T) curves obtained by du Plessis. In the cycloidal phase these authors observed various inflection points and changes in the slope of the eij(T) dependencies at 27, 33, 41 and 48 K. These peculiarities were related to commensurate effects in this temperature interval, and these are discussed below.
STATIC AND DYNAMIC STRESSES 155
O
,Y 8 . 8
8 . 4
I I r I L I I I
I
8 0
7 . 6 ! 0
o o [ ] o [ 3
oO.O~°. ~o2 oo o
o e •
o D e D
I
O o o ° o % 0 o ° ~ o o ~ ~ ~ ~ o o o •[]
~o I - o . .
o o z~ •
• ~ • [] •
A
oÆ z~ •
U []. g • •
~.'«[]
• ( a )• o •
I I I I I I ~ I
4 0 8 0
I
o - -
r
I
1 2 0
9.0 L I i i i I I I I I i i I I I i I |
q
I •
o_~C b
8 . 6 ' -
8 . 2 -
7 . 8 -
I I I I I I I I I I I I I I I I
0 2 0 4 0 6 0 80
T e )
Fig. 31. Temperature dependencies of elastic constants c33 (a) and Cll (b) o f erbium for various values of the magnetic field applied along the c-axis. Open circles, zero field; solid squares, 10 kOe; open triangles, 15 kOe;
solid circles, 20kOe; open squares, 24kOe; solid triangles, 39kOe (du Plessis 1976).
x _ , ,
~9
Since the gadolinium was investigated the most thoroughly of all o f the lanthanide metals it will be considered separately. All investigators observed anomalies in the elastic constants of gadolinium in the vicinity of the Curie temperature (see Scott 1978).
The temperature dependence of the elastic constant c33 reveals two dips: at the Curie temperamre Tc and at the spin-reorientation transition temperature Tsr (see fig. 32).
8.0
7.8
7.6
7.4
7.2
7.0 I i I I
I
I 180
I I I I l I
7.4. Gadolinium
~
Ff ~
L
5.75 K b a r 4.15 Kbar
~
1 5 , KUar_0 Kbar
I I
220 260 300 340
T(K)
Fig. 32. Temperature dependencies of the elastic constant c33 of gadolinium at various hydrostatic pressures (Klimker and Rosen 1973).
The origin and field dependence of the anomaly at Tsr were considered by Levinson and Shtrikman (1971), Freyne (1972) and Klimker and Rosen (1973). The mechanostriction mechanism was assumed to cause the anomaly. The application of strain along the c-axis induces a spin rotation back towards the c-axis which changes the magnetization and causes additional magnetostrictive deformations leading to the drop in c33. Using a system Hamiltonian consisting of the isotropic energy term, the elastic energy associated with homogeneous strains, the energy of the magnetocrystalline anisotropy, and the energy of single-ion and two-ion magnetoelastic coupling squared in spin components and linear in
STATIC AND DYNAMIC STRESSES 157
strain, Klimker and Rosen (1973) derived the following expressions for the c/j change at Tsr:
( b l l ( ¢ ) ) 2 ( b l l ( ~ ) ) 2 Cll = C l l ( 0 ) - - , C12 = C12(0)
2K2 2/£2
(b33(~)) 2 b 3 3 ( O ) b l l ( 0 ) 833 = c33(0) - - , c13 = c13(0)
2K2 2K2
C44 = C44(0), Coo = C66(0),
(104)
where co(0) are the elastic constants for ~b = 0; b/j(~) are the constants of magnetoelastic coupling which depend on the angle q~ and are connected with the constants B~I and B~2 from Callen and Callen (1965). The constants
bij(~)
were calculated with the help of the available magnetostriction and elastic constant data and it was shown that near TsrI b11(0)
[«1 b33(~) I. This means that the noticeable c~ anomaly at Tsr in gadolinium should take place only for c33. The experimental measurements of Cll and c44 made by Palmer et al. (1974) and Savage and Palmer (1977) confirm the validity of this conclusion.The hydrostatic pressure dependence of C33 was also studied by Klimker and Rosen (1973) (see fig. 32). The increase ofc33 under pressure is ascribed to the variation ofc33(0) due to the higher-order elastic constants. The observed increase of the spin-reorientation anomaly dip under pressure was explained by the pressure dependence of magnetoelastic coupling parameters and changes in magnetic ordering due to the shifts of Tc and Tsr.
Freyne (1972) used the same method and took into account the energy of the external magnetic field,
Hiß,
in the total Hamiltonian of the system. According to Freyne's result the anomaly at Tsr should become shallower with increasing field and the position of the dip should shift to lower temperatures for a field applied in the basal plane.Experimental measurements of the field behavior of c33 in the region of spin-reorientation were carried out by Long et al. (1969) (see fig. 33), and are in qualitative agreement with Freyne's theory. However, a quantitative estimate of the field where the anomaly in c33 should vanish in the framework of Freyne's theory gives a value of about 100Oe (Levinson and Shtrikman 1971), which is inconsistent with the experimental results of Long et al. (1969). To overcome this discrepancy Levinson and Shtrikman included in the total Hamiltonian of the system the self-magnetostatic energy contribution and took into account the corresponding domain strucmre. According to their model in a weak magnetic field the sample consists of the domains whose magnetization lies closest to the applied field direction, i.e. "plus domains", and the domains with magnetization largely opposed to the field (the "minus domains"). In such a domain structure the spin rotation induced by the strains will not alter the magnetostatic energy since the magnetization component in the field direction will remain tmchanged. However, for sufficiently large magnetic fields the sample consists only of "plus" domains and the strain induced rotation will strongly change the magnetostatic energy. This will effectively restrict the ability of the magnetic moments to rotate under application of strain and consequently, to lower c33.
Since Long et al. (1969) used a cylindrical specimen with a large demagnetizing factor,
158 78
A.M. TISHIN et al.
78 I-- I
77 ?
L
76
d
7, i
741 ~a,,
100 140
77 r
~76 ~ ~
75
,q 74 -- (b) .
I r I I I_ I I I I I
180 220 1 O0 140 180 220
T(K) T(K)
Fig. 33. Temperature dependencies of the elastic constant c33 of gadoiinium in various magnetic fields. (a) Field in the basal plane; (b) field along the c-axis. Solid circles, zero field; solid triangles, 1 kOe; open circles, 2 kOe;
open sqnares, 3 kOe; open triangles, 4 kOe; solid sqnares, 5 kOe; open inverted triangles, O kOe; solid inverted triangles, 8 kOe (Long et al. 1969).
then the spin-reorientation anomaly vanished in strong enough fields (see fig. 33). The results o f the modified calculations made by Levinson and Shtrikman agree well with the experimental field measurements o f c33 in the spin-reorientation temperature range.
The details o f the domain structure and domain rearrangement process strongly depends on the sample purity, internal stresses in the sample, and on the orientation o f the applied field with respect to the crystallographic axes. All these factors will affect the magnitude o f the spin-reorientation anomaly on c33 in gadolinium. Savage and Palmer (1977) did c33(T) measurements on a high purity Gd single crystal and found a much shallower spin-reorientation anomaly than other investigators. Evidently the difference in the spin- reorientation anomaly value obtained by various authors (see Jiles and Palmer 1980, Savage and Palmer 1977, Klimker and Rosen 1973) can be explained by different sample purities and the existence o f internal stresses in the samples.
The position o f the dip in c33 at the Curie temperature was found by Jiles and Palmer (1980) to be weakly dependent on magnetic field and became broad and indistinct in strong magnetic fields. In the paramagnetic phase the decrease o f c33 was found to be proportional to H 2 which is consistent with the theory o f Southern and Goodings (1973).
7.5. The effect of commensurate magnetic structures on the elastic properties
As was noted in sect. 6 the maxima in the temperature dependence o f the shear attenuation coefficient a44 observed in Ho and D y by Tachiki et al. (1974), Lee et al. (1975) and
STATIC AND DYNAMIC STRESSES
(b)
159
/ ~ ~ (d)
Fig. 34. Spin-slip structures in Ho: (a) s = 2; (b) s = 5; (c) s = 8;
(d) s = 11. The arrows show the singlets and the thin lines the doublets.
Greenough and Isci (1979) was interpreted by means o f incommensurate-commensurate domain effects. Greenough et al. (1981) explained the changes o f 0c44/0T observed in D y in the temperature range Tc to Ty by the effects o f the commensurability o f magnetic and crystal structures. Kataev et al. (1989b) constructed the magnetic phase diagram o f Tb (see fig. 27) on the basis o f Eh measurements in a magnetic field. The anomaly at 223.5 K in this phase diagram was coimected with the commensurate value o f the helical turn angle at this temperature.
Later the influence o f commensurate effects on the elastic properties o f Ho was intensively studied by Bates et al. (1988), Venter et al. (1992a,b) and Venter and du Plessis (1995). For an explanation o f the observed results they used the spin-slip structure model proposed by Gibbs et al. (1985) which was based on X-ray scattering results on holmium. According to this model the periodic magnetic structure, such as observed in the helical phase o f Ho, can be regarded as made up o f sequential blocks o f m or m - 1 atomic layers; the spins o f the atomic layer in each block being associated with the same easy direction (Bohr 1991, Bohr et al. 1986). A block o f length m - 1 introduces the discommensuration in the structure and is called a spin-slip. The shorthand notation o f the magnetic structure consists o f integers and dots (*d), where the integer d gives the number o f blocks o f length m and a dot represents a block o f length m in the sequence. For holmium m = 2, so the helical spiral o f holmium consists o f doublets and singlets. The sequence o f doublets and singlets is a unique property o f magnetic wave vector Q characterizing the magnetic structure. Thus, in such structures the regions o f commensurate spin structure are separated by spin-slips. I f spin-slips occur at regular intervals then we get a supercommensurate structure which is characterized by a unique nurnber (s) o f interplanar distances between two successive spin-slips and by the wave vector Q» Some o f the supercommensurate spin-slip structures which may arise in holmium are shown schematically in fig. 34. At the spin-slip the helical turn angle experiences an additional change as if there is an extra atomic layer in the structure. This causes a magnetoelastically induced lattice modulation and a change o f the hexagonal symmetry. There are three possible mechanisms o f such a modulation: (a) distance
160 A.M. TISHIN et al.
r~ I I I I I
2.85 f ~ ~ ~ 2.75
i
L I I I I I
ù= 60
2.88
2.87,
f
I
286L (b)
I I I I I
13
I I I I I I l l l l l l I I - - q
I ) I I I _1 f i I I ; I
120 180 240
I I I I I 2.83
LOo,o , , , ,
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o
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-- (C)
I I I I I ~ ~ ~ ~
25 37 91 97 103