Elastic properties 1. lntroduction

X- RAY SCATTERING STUDIES OF LANTHANIDE MAGNETISM 81 lanthanide series seems well accounted for. The same arguments apply to the variation of

7. Elastic properties 1. lntroduction

The magnetic structure has a strong influence on the elastic properties causing the appearance o f various anomalies at magnetic transition points, and a strong magnetic field dependence in the magnetically ordered states. A large amount o f the experimental data on the temperature dependence o f the elastic constants o f the rare earth metals can be found in the review o f Scott (1978). Here the main attention will be paid to the models describing the behavior o f the elastic constants related to the existence o f magnetic order and to results that were not reviewed earlier.

7.2. Anomalies near magnetic transitions 7.2.1. Thermodynamic consideration

In a single-domain ferromagnet the Young's modulus anomaly at the Curie temperature arises from an additional magnetostriction deformation o f the sample caused by the change o f spontaneous magnetization under the influence o f a mechanical stress acting on the sample during measurement. This is called the "mechanostriction o f the paraprocess".

The expression for such an anomaly was obtained by Von Döring (1938) on the base o f a thermodynamic approach

A E (ô~/OH) 2

- , (84)

EE ° Xhf

STATIC AND DYNAMIC STRESSES 141 where E is Young's modulus in magnetically ordered state, AE = E - E ° is the Young's modulus change due to the change of magnetic ordering, O)~/OH is the forced linear magnetostriction of the paraprocess, and Xhf is the high field susceptibility. The Young's modulus anomaly of this type is always negative.

In helical antiferromagnets an additional contribution to the Young's modulus anomaly is observed which is due to the influence of the stress on the spiral spin structure. It was considered by Lee (1964) and Landry (1967) in the framework of a model proposed by Herpin and Meriel (1961) and Enz (1960). It was assumed that the magnetoelastic interaction which caused the Young's modulus anomaly, was due to the variation of exchange integrals I1 and I2 under the variation of lattice constant e. The Young's modulus anomaly measured along the c-axis (AE«) was obtained to be:

AEc

Ec~

$23 ( O ~ C sin cp + 20~n~nc sin 2q0 ~ . 012 ) d q 0

2s~3 + $33

(85)

This anomaly is also negative in heavy lanthanide metals (according to experiment dcp/dP > 0, and OIi/Oln c ». 0).

The negative anomaly of elastic constant c33 and Young's modulus E« at the temperature of transition to a magnetically ordered state was observed experimentally in all heavy lanthanide metals (see for example Scott 1978, Spicbkin et al. 1996a,b, Bodriakov et al.

1992). However, at further cooling in some of these metals an increase of c33 and Ec was observed in spite of the fact that AE should have a negative value in the magnetically ordered state due the mechanostriction. Belov et al. (1960) and Kataev (1961) used the thermodynamic theory of second-order phase transitions of Landau and Lifshitz (1958) to describe the Young's modulus anomaly at the Curie temperature in a single-domain isotropic ferromagnet. In the simplest case the magnetoelastic energy can be written as a power series of the mechanical stress p

Eme = --(ypO "2 + 6/)20 "2 + ' " '), (86)

where y and e are the thermodynamic coefficients.

Belov et al. (1960) took into account in the free energy of the system the first term in eq. (86), which is linear in mechanical stress. As a result the equation for the Young's modulus anomaly, which was derived for the static case in the absence of an external magnetic field, has the form

AE 72

EE ° - 2fi' (87)

where fi is the thermodynamic coefficient (see eq. 75). Equation (87) is analogous to eq. (84) derived by Von Döring (1938).

The second term in expansion (86), squared in mechanical stress, was taken into account by Kataev (1961). For an isotropic ferromagnet the following formula was derived:

AE y2

E E ° - 2 f i + e ß . (88)

The second term in eq. (88) was related to the change of the bonding forces between the atoms in a crystalline lattice due to the magnetostriction deformations in the magnetically ordered state.

The Young's modulus anomaly arising at the transition from the paramagnetic to the helical phase in a single-domain spiral antiferromagnet, which belongs to the 6 m m (D4h) Laue class, was considered by Spichkin and Tishin (1997).

The total thermodynamic potential of the system in a magnetic field H was written in the form

F = ½acr 2 + lÆo*4 - 02(//1 c o s (/) 4-//2 c o s 2q~)

-- M i j , k l ~ i @ Tkl - Rij,k#,oOiCrj T«z Tno - gSij,kt To Tkl - H i a i . 1

(89)

The first and second terms here represent the exchange contribution to the free energy. The third term is the energy of the helical spin structure written down in the form proposed by Herpin and Meriel (1961) and Enz (1960), where nl and n2 are the exchange parameters related to the exchange integrals/1 and/2 by the expression

2 ( g j - 1) 2 , (90)

n i - - -2~-aT.,~ li"

g j l v pt B

The parameters nl and n2 were assumed to depend on the crystal lattice parameter c in the same way as proposed by Lee (1964) and Landry (1967):

ni = nio + 0~n-~nce33. (91)

The fourth and fifth terms are the magnetoelastic contribution written down in the tensor form (Mason 1951). Here the terms, which are linear and squared on the mechanical stress tensor components Tg, were taken into account. The Cartesian axes x, y and z coincide with crystallographic axes a, b and c, respectively. The nonzero components of the tensors M/j,kl and Rij,ktno for a crystal with a 6 m m symmetry class were derived by Fumi (1952). The sixth term is the elastic energy.

The spontaneous magnetization as, the magnetization a in a magnetic field and the equilibrium value of the helical turn angle can be derived by minimizing the free energy F, which is considered to be a function of a and q0. With the magnetization found one

STATIC AND DYNAMIC STRESSES Table 11

The values of coefficients B« and D« from eqs. (93) and (94) (Spichkin and Tishin 1997) 143

Measurement conditions Measured value B« D«

H or o5 aligned along SII; lB-Ea/EaE°a (Yll + s 1 3 . / ) 2 Elll crystallographic a-axis; s22; ~b/Eb EO (]/12 -}- S13~) 2 a«

O" 1 = 0- :~é 0; 0" 2 = 173 = 0 $33; ~kEr/EcLg°e (Y13 @ $33./) 2 6133

821 (Yl 1 q- si3 ~)(YI2 -}- si3 O) el 12

531 (]/11 q- ~g13 ~)(]/I3 -~ s33 ~) E113

H or o~ ali~õaed along sH; l~a/EaE°a (]/12 -~s13~) 2 E211 crystallographic b-axis; sa2; ~b/Eb EO (]/11 + S 1 3 ~ ) 2 e222

0" 2 = a v" 0; 0" I = 0-3 = 0 s33;

AEJEc E°

(Y13 + s33'/)2 e133

S21 (]/11 -~- S13 ~)(]/I2 4- S13 ~) a',

$3I (]/11 -k S13 */)(]/13 + $33 */) e123

H or 0-~ aligned along Sll; ~a/Ea EO ]/~1 C311

crystallographic c-axis; s22; t~b/EbE~ ]/21 ~311

0" 3 = 0- ~é 0; 0-1 = 0"2 = 0 $33 ; [~~,c/EcE°e ]/23 E333

s21 ]/32~ e3~2

$31 ]/31]/33 E313

can calculate the e n e r g y o f the m a g n e t i c a l l y o r d e r e d state and then b y m e a n s o f the t h e r m o d y n a m i c r e l a t i o n

02F

s , « - 0T/0Tj ' (92)

w h e r e the elastic c o m p l i a n c e s s/j and Young's m o d u l u s

Ei

are for the m a g n e t i c a l l y o r d e r e d state. H e r e the two i n t e r c h a n g e a b l e i n d i c e s are r e p l a c e d b y a single i n d e x a c c o r d i n g to the equations: 11 = 1; 2 2 = 2 ; 33 = 3 ; 1 2 = 2 1 = 6 ; 13 = 3 1 = 5 ; 23 = 3 2 = 4 . The results o b t a i n e d can be s u m m a r i z e d in the f o l l o w i n g g e n e r a l f o r m u l a s ( S p i c h k i n and T i s h i n 1997):

sij=s°.+(2[3o~~H/a De) O 2,

Be (93)

Z~-,i __

Be

EiEO i (21305+H/o De) o 2,

(94)

w h e r e the coefficients Be and De are g i v e n in table 11 for different cases o f m e a s u r e m e n t . In table 11 the f o l l o w i n g notations are m a d e

y,j = 2M~; eU~ = 2R~jk; (95)

a e = E l l 1 q- E21 1 - - E222; cl e = E l l 1 -4- e l l 2 - - E222.

The p a r a m e t e r t / c h a r a c t e r i z e s the effect o f the elastic stresses on the h e l i c a l spin structure and has the f o r m

2nlon20 0@n~c + 8(n20 _ .2 x

nlO) O~c On2

t / = 4n~0 (96)

For m e a s u r e m e n t s in a m a g n e t i c field h i g h e n o u g h to destroy the h e l i c a l strucmre, tl = 0.

IA210

°I~ó °

ù. ! . ~ - 2 - - 73

o

O 0 O

220 23J~._ T(K)

~~. °

e°o_ °

%.. o ° • OOoo. o

'-u 0 0 0 ' ° • ~dO0

Q 2

Fig. 21. Temperatur° dependence of AE, (curve 1) and AE b (curve 2) for terbium in the range of the magnetic phase transitions (Spichkin and Tishin 1997).

Near the Curie point according to the theory of second-order phase transitions the following field dependence of magnetization takes place:

(it = [3 1/3H1/3, (97)

which leads to a H 2/3 field dependence of

Ei AEi _ (Be DeH 2/3

EiEOi

- 3 B /32/3 ] " (98)

Since the experimental measurements of the elastic properties are made in a dynamic regime, when the elastic waves are excited in the sample, it is necessary to take into account the relaxation effects. According to Belov et al. (1960) and Kataev (1961) this gives the following relaxation relations:

zXEi _ ( AE~ EiDea2 )

(99)

E°i 1 + o02 r 2 '

B e 0 2 - D e O 2, (100)

SO" = sO + (2/30"2 + H / a ) ( 1 + o 2 T 2)

where A& is the Young's modulus relaxation degree,

E° (101)

A& - 2/302 +

H/a'

~o is the frequency of sample oscillations, and z" is the relaxation time (see sect. 6.2.2).

Spichkin and Tishin (1997) and Spichkin et al. (1996a) studied the temperature and field dependencies of Young's modulus Ea and

Eh

in Tb at acoustic frequencies.

Figure 21 shows AE«(T) and AEb(T) - the parts of Young's modulus due to the existence of a magnetic structure in the region near TN and To The pure lattice Young's

STATIC AND DYNAMIC STRESSES 145

>I

-o

O

~7

o i-4

~a

41 /

^{7 8}

5 4

2 0 0 4 0 0 6 0 0 H 2/3 ( O e 2n )

Fig. 22. AEJEbE ° as a function of H 2/3 for terbittm near T N. (1) T - 246K; (2) 240K;

(3) 235K; (4) 232K; (5) 229K; (6) 226K;

(7) 220K; (8) 210K. The magnetic field was applied along the b-axis (Spichkin and Tishin 1997).

modulus E°(T) was calculated by means of the model proposed by Lakkad (1971) and experimentally tested by Tishin et al. (1995) on yttrium. The observed temperature dependencies of Ec and E» are analogous to that of c33 and Cll obtained earlier by ultrasound methods (see Jiles et al. 1984, Scott 1978). Field measurements of Eb (H along the b-axis) revealed near TN the linear dependence of AE»/EóE ° on H 2/3 in accordance with eq. (98) - see fig. 22. Far from TN in the paramagnetic phase a deviation from this dependence was observed. The experimental values of Ytt (obtained from AEb(H) curves),/3 (obtained from magnetization measurements), s/j from the work of Salama et al. (1972) and 0Ii/Õlnc data of Bartholin and Bloch (1968a) allowed one to calculate A E J E » E ° = - 1 . 5 x 1 0 - 1 ° c m 2 / d y n at TN using eq. (94); this is consistent with the experimental value o f - 0 . 7 x l 0 - m c m 2 / d y n . I f the coefficients E133 and E222 (E222 ~ 4 . 5 × 1 0 - m G 4 as obtained from experiment) are positive then the magnetic anomalies AE« and AEb below TN should decrease in absolute value upon cooling because of the spontaneous magnetization. Such a behavior was observed experimentally near TN (see fig. 21).

7.2.2. Microscopic models

Southern and Goodings (1973) and Goodings (1977) used the theory developed by Brown (1965) and Melcher (1972) for the description of elastic constant anomalies and their field dependence in the ferromagnetic state and near the transition from the paramagnetic

to the magnetically ordered state in the lanthanide metals with a hexagonal structure.

This theory is based on the finite deformations approach and imposes the requirement that the Hamiltonian of the system should be invariant under rigid rotation of magnetic and lattice subsystems. The Hamiltonian included the Heisenberg exchange term (in molecular field approximation), crystal field, magnetoelastic, elastic energy terms and the magnetic energy due to the external magnetic field. For the paramagnetic state the magnetoelastic energy in terms of the first and second order in small strains originating in either the single-ion or two-ion interactions was taken into account, and complex formulas of the field changes for the elastic constants c1~ and c33 were obtained (the case of the ferromagnetic state will be discussed later). The main consequence from these formulas is

t h a t Æcii (here Acii is the change o f c i i in magnetic field, ÆCii = c ü ( H ) - ¢ii(H = 0 ) ) should be directly proportional to the second order of the magnetization. Since magnetization is supposed to be a = x(T)H (where susceptibility Z is not dependent on H ) this leads to a H 2 dependence of ACii. Such a dependence in the paramagnetic region was observed experimentally in Gd, Tb, Dy and Ho by Moran and Lüthi (1970) and Jiles and Palmer (1980), in Dy by Isci and Palmer (1978), in Tb by Salama et al. (1974) for temperatures far from TN. Near TN deviation from the H 2 dependence of Acü was observed. This may be related to the field dependence of the magnetic susceptibility. The nonlinear dependence of ~ on H following from the theory of second-order magnetic phase transitions leads to a H 2/3 dependence of AE (see eq. 98) and was observed experimentally in Tb for AEb and H applied along the b-axis (see fig. 22).

Vasconcelos (1982) by means of the model of Southern and Goodings (1973) considered the c33(T) behavior near TN in erbium. Good agreement of the calculated

C33(T )

dependence with the experimentally observed drop in c33 near TN was found.

7.3. Magnetically ordered state of the heavy lanthanide metals and their alloys 7.3.1. Helical phase

The behavior of the elastic constants cll and c33 , and Young's moduli Ea and Eb of the lanthanide metals in the helical antiferromagnetic phase has been studied most. The temperature dependencies of these values reveal peculiarities not only at the transition to the magnetically ordered state, TN, but also at the temperature of the transition to the phase with the ferromagnetic component, Te (see Scott 1978).

The temperature hysteresis of c33 in the helical state was observed in Dy, Tb0 »Ho0.5 and Gd06Y0.4 alloys by Palmer (1975) and Blackie and Palmer (1980). Figure 23 shows the result of thermal cycling near Tc in Dy. First the sample was cooled below Tc into the ferromagnetic state and then warmed into the helical phase. Under this condition a drop in c33 (compare with the value obtained under cooling conditions) was observed.

This effect was explained by Palmer (1975) to be due to the existence of spiral domains and domain walls between them as discussed in sect. 6. Compressive and expansive stresses parallel to the c-axis produced by the sound wave causes the rotation of the magnetization in the walls, changing their thickness and the dimension of the domains.

This leads to the development of additional magnetostrictive strains reducing the elastic

STATIC AND DYNAMIC STRESSES 147

0

O i--4 v

O

I

8 . 2 -

8 . 0 -

7 . 8 -

I I ~ I r I I\ I

90 i00 120 140 160 180

T (K)

Fig. 23. Temperature dependence of the elastic constant c33 in Dy near T c (Blackie and Palmer 1980).

constants (i.e., this is the mechanostriction mechanism discussed above). The difference in the value of c33 observed on cooling from the paramagnetic state and warming from the ferromagnetic state is connected with the difference in the number of spiral domains in the two cases. The maximum number of spiral domains exists near Tc and it decreases as TN is approached. Blackie and Palmer (1980) measured

c33(T )

when the sample was cooled from various temperatures in the helical phase (see fig. 23). In this case the number of spiral domains remains approximately constant and all c33 ( T ) c u r v e s were nearly parallel down to Tc.

Hysteresis was observed also in other rare earth alloys with the helical phase - Gd0.6Y0.4 (see fig. 24), Gd0.654Y0.346 and Gd0.o89Y0.311 by Palmer et al. (1977), Sousa et al. (1982) and Blackie and Palmer (1982), respectively. At the same time the elastic constant in the basal plane c]1 and shear elastic constant c44 do not reveal any hysteresis of this type. According to the model proposed by Palmer (1975) the number of spiral domains interacting with the longitudinal sound wave propagating in the basal plane is the same on cooling from the paramagnetic phase and warming from the ferromagnetic state as was discussed in sect. 6. The shear sound wave is coupled with spins only in second order and its interaction with spiral domain walls is small. In the ferromagnetic alloy Gd0.703Y0.297 no hysteresis was observed (Blackie and Palmer 1982).

Later, Palmer et al. (1986) investigated the influence of the sample purity on domain patterns and the hysteresis in the helical phase in Tb and Ho. The domain patterns were observed by polarized neutron diffraction topography. It was found that the size of the spiral domains in the helical state was very sensitive to the sample purity. For the low purity material the size of the domains was below the resolution limit of the method, and

148 A.M. TISHIN et al.

7.70 7.50 7.30 7.10 6.90 6.70 2.55 2.50 2.45 2.40 2.35

ro

C33

2.30 2.25 2- 1.65 ~-

1.60 f 1.55

1.50 r

C13

0 100 200 300

7.6 7.4 7.2 7.0 6.8 2.65 2.60 2.55 2.50 2.45 2.40 2.35

T(K)

Fig. 24. Temperature dependence of the elastic constant ci2 » c13 , c33 , Cll and c44 for the Gd0.6Yo. 4 alloy for increasing and decreasing tempera~tre, c,j in units of 10 I1 dyrdcm 2 (Palmer et al. 1977).

for high purity Tb (resistivity ratio R3oo/R4.2 > 150) domains ranging from 0.15 m m to several m m were observed. The c33 temperamre dependence measurements in high purity Tb did not reveal the hysteresis on warming from the ferromagnetic state. Palmer et al.

(1986) explained this by the small number o f d o m a i n walls which exist in the high purity sample. They noted that in Ho the situation is quite different since even below Tc the spin

STATIC AND DYNAMIC STRESSES 149

7.7

7.5

1

^{7.3 7.1}

6.9

L L L L L I I

6.7 I I

0 20 40 60 80

Applied magnet-ic field (kOe)

Fig. 25. Magnetic field dependence of the elastic constant c33 at various temperatures in Dy (field along the a-axis). Open circles, 135 K; solid circles, 150 K; open triangles, 160 K; open squares, 168 K; solid squares,

175 K (Isci and Palmer 1978).

structa~re has a spiral character. So the domain pattern should not change when warming from the phase below Tc. This can explain the absence of hysteresis in Ho observed experimentally.

The field dependencies of the elastic constant c33 in the helical phase were measured in Gd06Y0.4 and Ho0.»Tb0.5 alloys, and in Tb and Dy by Palmer et al. (1977), Isci and Palmer (1977, 1978), Jiles et al. (1984). We will consider the results obtained for Dy by Isci and Palmer (1978) because the features of the c33(H) curves in the helical state are characteristic for all lanthanide metals with a spiral antiferromagnetic structure (see fig. 25). Just above Tc in the helical state a shallow minimum was observed which was attributed to the helical antiferromagnetic to the ferromagnetic transition. Above 125 K there is a sudden drop in c33 at what was assumed to be the H A F M - f a n strucmre transition followed by a rapid increase at the fan-FM state transition. The interval of the fan structure existence is governed by the basal plane anisotropy which brings

Hf

(the field corresponding to the transition from fan to FM struc~Jre) down to the same value as Hcr at low temperamres leading to one minimum at c33(H). With increasing temperature the plane anisõtropy decreases and two separate transitions are observed at the c33(H) curves.

The results of sound attenuation measurements show that the transition H A F M - F M phases are realized through intermediate phases. Above

Hf

the constant c33 still has

60

5O O

" o

-~ 40

o I13 c

3o

c 0) c

- 20

10

I 50

F~

HAFM

100 140 180

T(K)

PM

I 220

Fig. 26. Magnetic phase diagram of dys- prosium obtained from elastic constant c33 measurements in a magnetie field (Isci and Palmer 1978). Ferrornagnetic (FM), paramag- netic (PM), helical antiferromagnetic (HAFM) and fan phases are shown.

900 = , ò

© 600

300 t

ot

216

] i [ i

J t

)

222 226 230

T(IO

FM

220 224 228 232 236

TO()

Fig. 27. Magnetic phase diagram of terbium obtained from the field measurements of Young's modulus Eb (Kataev et al. 1989b).

The insert shows the behavior of H«~ near 223.5 K.

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~ ~ ~~,~

£~

⁴³

181

4 5

44

43

I I I I I I I

4 8 12