J. Phys. Chem. Solids, 1974, Vol. 35, pp. 1333-1338. Pergamon Press. Printed in Great Britain

SINGLE CRYSTAL ELASTIC CONSTANTS AND MAGNETOELASTICITY OF HOLMIUM

FROM 4-2 TO 300 K

M. ROSEN, D. KALIR and H. KLIMKER Nuclear Research Center-Negev, B.P. Box 9001, Beer Sheva, Israel

(Received 7 Nooember 1973)

Abstract--The five independent elastic coefficients of holmium single crystals have been determined by means of an ultrasonic pulse technique at a frequency of 10 MHz, between 4-2 and 300 K. From the elastic constants the temperature variation of the directional adiabatic compressibilities, the limit- ing Debye temperature and the elastic anisotropy ratio were calculated. The elastic coefficients exhibit anomalies at the magnetic ordering transitions known to occur in holmium. Anomalous behavior in the elastic constants was also observed at about 80 K. The limiting value of the Debye temperature was found to be 191.5 K. The present measurements of the elastic constants, and the reported magnetostriction and thermal expansion data, enabled the calculation of the magnetoelastic contribution to the total Hamiltonian of holmium in the magnetically ordered states. A very small discontinuity in the temperature dependence of the magnetoelastic energy was observed at the Curie point of holmium. Below the Neel temperature, the magnetoelastic energy varies smoothly with decreasing temperature, attaining a value of--2.13 J cm -a at liquid helium temperature. The tempera- ture dependence of the magnetoelastic energy in the vicinity of the Curie point in holmium suggests that the magnetic transition from the antiferromagnetic arrangement into the ferromagnetic state is of second order.

1. INTRODUCTION

The temperature variation of the physical pro- perties of holmium indicates the existence of two magnetic transitions, at 20 and 132 K. Below 20 K holmium is a conical ferromagnet[1, 2] of a mean cone angle of 80 ° from the c-axis, and a turn angle of 30 ° . In each hexagonal basal layer the projection of the magnetic moments are parallel. This struc- ture is similar to that of ferromagnetic erbium [3]. Between 20 and 132 K holmium has an antiferro- magnetic helical spin arrangement. The magnetic moments are parallel within each hexagonal layer, and rotate by a certain angle per layer in the successive planes along the c-axis. Similar to dy- sprosium [4], the turn-angle in holmium is tempera- ture-dependent. It decreases linearly with decreas- ing temperature[2] from 51 ° at 132 K to 30 ° at 20 K. Magnetization data [5] confirm that the heli- cal-to-ferromagnetic transition in holmium is less drastic than in dysprosium, and takes place via a ferromagnetic helix in which the magnetic moment has components both along the hexagonal axis and in the basal plane. Several other physical pro- perties, e.g. specific heat [6], lattice parameters [7], electrical resistivity [8, 9], and thermal conductivity [9], indicate that the paramagnetic-to-helical transition at 132 K is of a more drastic character than the ferromagnetic transition at 20 K. How-

JPCS Vol. 35, No. 9---T

ever, holmium exhibits very large anisotropic magnetostrictive strains [I0, 11] of the order of 2 × 10 -a, and field-induced exchange magneto- strictions of about 3.5 × 10 -3.

The temperature variation of the elastic moduli and ultrasonic attenuation of polycrystalline holmium[12] reveals prominent anomalies at the ordering points. Both transitions, at 20 and 132 K appear to be of the second order. The longitudinal and transverse ultrasonic attenuation in poly- crystalline holmium exhibited pronounced ano- malies at 72 K, the nature of which was not clear. N o anomaly, in this temperature region, was observed in the temperature dependence of neutron diffraction [ 1, 2]. However , magnetization measure- ments[13] show that there are two successive jumps above 50 K when high magnetic fields are applied along the [0001] direction. The single- crystal elastic constants of holmium have pre- viously been measured [ 14, 15]. The latter set [ 15] was limited to the determination of the elastic constants at temperatures above 77 K, i.e. far above the ferromagnetic transition point of holmium. In the overlapping range, significant discrepancies between the two sets of data were found in both the absolute values o f the elastic constants, and in their temperature dependence. Such discrepancies can affect the results of the

1333

1334 M. Rosen et al.

calculation of the magnetoelastic contributions to the total energy of holmium, which was the main purpose of the present study. Preliminary measure- ments of the elastic constants of holmium at temperatures above 77 K in this work, employing accurate ultrasonic techniques revealed satis- factory agreement with the data reported in Ref. [15]. However, for the calculation of the magneto- elastic contribution it was necessary to have available an accurate set of the elastic constants of holmium from liquid helium to the ambient.

The objective of the present study was to deter- mine the temperature dependence of the magneto- elastic contribution to the total Hamiltonian in the helical-antiferromagnetic and conical-ferromag- netic phases of holmium. For this purpose, the five independent elastic coefficients of hexagonal holmium single crystals were remeasured for liquid helium to room temperature. Calculations of the magnetoelastic energy were carried out using the reported values of the anomalous thermal expansion of holmium [ 11 ].

2. EXPERIMENTAL

The holmium single crystals, supplied by Metals Research Ltd. Cambridge, England, were prepared by a zone melting technique. Their purity, as quoted by the supplier, was better than 99.9 per cent. The major impurities being other rare earth elements. The three single crystals were in the form of flat disks, 6 mm in diameter by about 4 mm thick. The crystals were oriented, respec- tively, parallel, perpendicular, and at 45 °, to the hexagonal c-axis. X-ray back reflection Laue photographs indicated negligible deviations from the nominal orientations.

The acoustic path length of each crystal disk was measured by a calibrated indicator to within -----5 × 10 -4 mm. The room-temperature specimen density was determined by means of a fluid- displacement technique. The average density of the three single crystals was found to be 8-804g cm -a. The temperature dependence of the acousti- cal path lengths was calculated using the average coefficients of thermal expansion of holmium [ 16].

The sound-wave velocities were measured by means of an ultrasonic pulse technique at a fre- quency of 10MHz. Experimental details, con- cerning ultrasonic couplant preparation, cryo- genics, and method of data analysis were described elsewhere[12,17]. From the temperature de- pendence of the elastic constants the variation of the linear adiabatic compressibilities, the elastic

anisotropy, and the Debye temperature were calculated.

The total estimated error in the absolute values of the elastic coefficients cH, c~, c44 and c,6 is about 0.5 per cent. The estimated error in the ab- solute value of c,2 is about 1 per cent, whereas that of c~3 is 2"5 per cent. However, the relative point- to-point precision of all the elastic constants as a function of temperature is better than the absolute one by a factor of 4.

3. RESULTS AND DISCUSSION

3.1 E l a s t i c c o n s t a n t s

Figure 1 shows the temperature dependence of the dilatational elastic constants of holmium. The compressional sound wave along the hexagonal c-axis yields the ca:3 elastic constant, while the compressional wave normal to c gives Cl,. With decreasing temperature, from the ambient, both c,, and c3.~ increase in a normal manner, although the temperature dependence of cH is sharper than that of c33. A similar behavior was observed in the temperature dependence of these elastic constants in dysprosium[17] and erbium[18]. Figure 1 displays a drastic anomaly in c33 at the Neel point (TA, = 132 K) of holmium. In contrast, cN behaves almost in a normal manner at TA,, except for a minor change in slope. The absolute values, as well as the temperature variation of c3~ and c,,, as shown in Fig. 1 are in satisfactory agreement with the measurements of Salama et al. [15]. With decreasing temperature, below TN, cH exhibits a change in its temperature dependence at about 80K. Polycrystalline holmium[12] was found to show sharp anomalies in the vicinity of this temperature. The nature of these anomalies was not elucidated. The Curie point (To = 20K) is marked by minima in both ca3 and c,,.

B.40 ~" 8.30 E ~ 8 .20

o 8.,0 8.00

"o 7.s0

7.8o

7.70

~ x ~ . " , - . . . . C e _ _ . .

- \ \ ~ C -- x\\ \ I

,J 11

i, I i ~ t i f ~ " ~ 0 50 100 150 200 250 300

TEMPERATURE ("IR) Fig. 1. Temperature dependence of the dilatational elastic coefficients c,, and c33 of holmium single crystals.

Elastic constants and magnetoelasticity of holmium 1335

2.90

2.85 u

2.80

% ~.TS "3.- u 2.70

o 2 . 6 5 =

u 2 . 6 0

2.55

4 4

- " c.~--_,

I I I I 1 "1

0 50 100 150 200 250 300

TEMPERATURE (*K)

Fig. 2. Temperature dependence of the shear elastic coefficients c,, and c~ of holmium single crystals.

~u

_J

03 tu

g u

0.79

0.77

0.75

0,73

1 ~ I K S j ~ I

I I

I l I I , 1 I

SO 1OO 150 200 250 300

TEMPERATURE ('K)

0:87

E °

0.85

0.83 _~

o.8,

I J

Fig. 4. Temperature dependence of the adiabatic direc- tional Ks~ (parallel) and Ks~ (perpendicular) compres-

sibilities of holmium single crystals.

The temperature variation of the shear con- stants c4~ and c~ is shown in Fig. 2. Similar to the behavior of these elastic constants in dysprosium [17] and erbium [18], both c4a and c,, are very little affected at TN. However , the temperature depen- dence of c4a changes its slope, rather drastically at 132 K. The smooth behavior of c~ is influenced by the variation of cN with temperature, since c~=l /2 (cH-c~o . ) . The cross coupling elastic coefficients c~_~ and cr~ are shown in Fig. 3. The salient feature of their behavior is that cv., is almost the mirror image of the temperature dependence of qa. Below T~,, c~_~ displays a "stiffening" whereas the elastic coefficient c~ shows a "softening" behavior. The inflection point of these opposite temperature dependences is at 80 K, where the ultrasonic attenuation in polycrystalline holmium showed prominent maxima[12]. The absolute values of c~z, are substantially higher than those of q3, over the whole temperature range investi- gated. The general behavior of cr, and c~a is in agreement with previous measurements of the elastic constants in holmium [14].

The adiabatic directional compressibilities of holmium, parallel (Ks~O and perpendicular (Ks±) to the hexagonal c-axis are shown in Fig. 4. The directional compressibilities were derived from the calculated elastic compliances[19] s~, s33, s44, s~ and s~.~. The compressibilities are defined as follows:

Ks,, = ( c,, + c,~-- 2q3) /[ ( c,, + cry) c,s--2c13)] z

K sx = (ca3 -- c,a)/[(c,1 + c,2) c,a -- 2qa] 2

and the adiabatic volume compressibility, K s ( V ) , shown in Fig. 5,

K s ( V ) = Ksll-l- 2Ks.L.

The directional compressibilities Fig. 4, show anomalies at the magnetic transition points of holmium. The parallel compressibility is almost temperature-independent between room tempera- ture and about 160 K, whereas the perpendicular compressibility decreases linearly in this tempera-

I I [ I I I 2~70 / ~

( ~ / ~" . . . . . . 2.1 S / C,,

g; 1 .85 2 .50 i i I I I

0 50 too 150 200 250 300

TEMPERATURE ('K)

Fig. 3. Temperature dependence of the cross-coupling elastic coefficients ct= and c,~ of holmium single crystals.

2.66

"2 2.62 g-

% 2 .s8

G 2,54

E 2.05 g

"o

1.95 G

"2

2.43 ~u 'o

2.41

2.39

o- 2 . 3 7

( J

0

/~ I /-~ I I I I I

! ~ I / / A \ \ / /

f /

Ks(v) \ i// /

I I 1 I I I

50 I00 150 200 250 300

TEMPERATURE ('K)

A

o 1.01

1 . 0 0 ~

0.99 ,,~

Fig. 5. Temperature dependence of the adiabatic volume compressibility Ks (V), and of the elastic anisotropy ratio

A = (c,,-c,J2c,,) of holmium single crystals.

1336 M. ROSEN et al.

ture range. A similar behavior was observed in erbium[18]. The absolute values of the parallel compressibility are higher than those of the perpendicular compressibility over the whole temperature range investigated, between liquid helium temperature and the ambient.

The volume compressibility Ks (V) and the elastic anisotropy ratio (A = (cn--c~2)/2c44) are shown in Fig. 5. The temperature variation of Ks(V) displays the expected maxima at TN (132 K) and at T~ (20K). In addition a rather broad peak is formed at about 80 K, indicating the possibility of a change in the antiferromagnetic helical spin arrangement, possibly by a change in the rate of decrease of the turn angle in holmium. A major change in the antiferromagnetic arrange- ment, similar to that in erbium [18] at 54 K, seems unlikely, since it was observed neither by neutron diffraction measurements [!, 2], nor by the present elastic measurements, except for some slope changes, Figs. 1-4 and the broad maximum in Ks(V), Fig. 5. However, the major peaks in the ultrasonic attenuation of polycrystalline holmium [12], at 72 K still remain unaccounted for. I t is suggested that careful magnetization and neutron diffraction measurements should be performed in order to elucidate this anomalous behavior of holmium at about 80 K.

The elastic anisotropy ratio A = (c,-c~2)/2c44 is equal to unity for an elastically isotropic crystal [20]. Figure 5 Shows the temperature dependence of A in holmium. The room temperature value is 1-02, and decreases with decreasing temperature to 0.99 at TN. It is noteworthy that the temperature dependence of A is compatible with that of Ks (V), Fig. 5. Assuming that the low temperature ano- malies in the adiabatic compressibility in holmium are of a magnetoelastic nature, such a behavior of A is expected. The elastic anisotropy ratio in erbium[18] displayed a maximum at 54 K, where a change in the antiferromagnetic arrangement occurs, accompanied by a possible change in the crystal symmetry of erbium. The temperature variation of A in holmium between 132 and 20 K, is not indicative of a linear variation with tem- perature of the turn angle in holmium, but rather of the occurrence of a maximum.

The low temperature elastic constants enable the computation of the limiting value of the Debye temperature, 0o 0, at 0 K. The elastic 0o ° is of particular importance for the analysis of the transport properties of magnetically ordered substances. 0~ 0 is related to the velocities of sound by the equation [20]

00 = (h/kn) (3Np/4zrM)l/3v',,

where h and kB are Planck's and Bottzmann's constants, respectively. N is Avogadro's number, p is the metal density, M is the atomic weight, and v,, is the mean sound velocity. In the present study, Anderson's velocity-averaging procedure for hexagonal crystals[21] has been employed in order to calculate v,,, and subsequently 0o ° for holmium. The limiting Debye temperature was calculated to be 191.5 K. This value is in good agreement with the OD ° = 194 K obtained from low temperature elastic moduli of polycrystalline holmium[12], but much higher than the value of 161 K obtained from heat capacity measurements [16].

3.2 Magnetoelasticity Large magnetostrictive effects were observed in

holmium[l 1] at temperatures below TN(132K). The magnetostrictive strains contribute to the energy of the ordered states via a magnetoelastic interaction. Using the experimentally determined values of the elastic constants of holmium, shown in Figs. 1-3, and magnetostriction da ta[ l l ] , in conjunction with the local strain formalism[22], one can evaluate the temperature dependence of the magnetoelastic contribution to the total energy of holmium.

In the expression for the total Hamiltonian of the crystal [23]:

H = H, , ,+He+Ht +H2. (1)

The strain-dependent terms Hs will be con- sidered, where

Hs = He+H1 + H2 (2)

and He is the elastic energy. In the notation of Calten and Callen[23]. Using the standard sym- metric representation

c r, are the elastic constants, and et r'~' are the uni- form strains.

H~ and /-/2 are the one-ion and two-ion mag- netoelastic contributions [22, 23], respectively:

H, = - Z E Z ,~ E g,'"J (f)~,"J' ( f ) (4) 2/' f F j j ' i

H2 = - - E E E b~, E fi r" (fg) g r,j, ( fg) (5) fg F ~ ' i

Elastic constants and magnetoelasticity of holmium 1337

where, g~".J(f) and ~r,j(fg) are the one-ion and two-ion local strains, respectively..gr,j(fg) are the standard one-ion and two-ion spin functions, respectively,

The conical ferromagnetic structure of holmium, with a cone half-angle O of about 80 ° and an inter- layer turn angle @ of 30 °, is similar to the ferro- magnetic structure of erbium[24]. Above Tc (20K), the magnetic structure of holmium is similar to that of dysprosium in the antiferro- magnetic range [25]. The moments are constrained in the basal plane. In any basal plane all moments are parallel to each other, but in adjacent basal planes there is a turn angle @ that varies linearly from 50 ° at TA, (132 K) to 30°at Tc (20 K) [1,2].

For the determination of the magnetoelastic energy contribution in holmium it can be assumed that the difference between the conical ferro- magnetic structure and the antiferromagnetic helical spin arrangement is only in the conical half- angle O. In the antiferromagnetic state 0 = rr/2, whereas in the ferromagnetic state 0 =I= rr/2. This consideration greatly simplifies the solution of H~ and H2, equations (4) and (5).

Substitution for the appropriate local strains and spin functions, and applying the calculation pro- cedure used for erbium [ 18] the expressions ( H~ ) and (H2) in the two magnetic states of holmium were obtained. For the two-ion interaction two types of neighbors were considered, namely, those in the same basal plane, and those in dif- ferent layers. Such a procedure was also used in the case of dysprosium[22]. The expression for (H,) contain trigonometric terms in O and @. Terms of the form £ sin n @ and £ cos n @, where n is the

index number of the basal layer, do not contribute to volume strain, and therefore cancel out [22].

The final, non-zero, expressions for (H~) and (H2) of holmium are identical to those for erbium [18]. For the antiferromagnetic state of holmium

Table 1. Temperature dependence of the equilibrium strains below

N~el point (132 K)

T (K) e~,;/× 10 -a e~';, ~ X 10 -a

4-2 1.16 5-64 20 1.11 5-28 40 0.96 4.78 60 0.77 4.23 80 0.38 3.37

100 0.54 1-96 120 -0.01 0.83 132 -0.01 0.17

one has to substitute for 0 the value of 7r/2 whereas for the ferromagnetic state 0 = 80 °. The magneto- elastic energy in the magnetic state of holmium depends also on the magnitude of the equilibrium

,~a and .,2 strains Eeq Eet I as determined by minimization of the strain-dependent Hamiltonian H~ [18.22, 23]. These equilibrium strains are expressed in terms of the magnetoelastic coupling coefficients, and elastic constants in the form of the symmetric representation [23]:

1 c~'1 = ~ (2c, + 2c,2 +4c,a + Caa)

3 c ~ = - - - - 7 ( c , , - c,~ + c,:, + c = )

3V2 (6)

2 cz~ = ~ ( cn + c, z -- 4c, a + 2Caa).

From the experimental results in the present work it was found that the value of c~' 2 for holmium is close to zero over the whole temperature range investigated. Therefore, the cross terms in the equilibrium strains can be assumed to be negligible. Determination of e~61 and egd 2 from the anomalous thermal expansion data[l l] of holmium at tem- peratures below TN (132 K) as presented in Table 1, permits Calculation of the magnetoelastic contribution in the magnetically ordered states of holmium. The final expression for H~ is

Hs = --I/2 c~1 (%q) 1/2c2z, e~ - (7)

Using the elastic constants determined in the present investigation, and applying the reported values for the anomalous thermal expansion of holmium[l 1], the temperature dependence of the magnetoelastic contribution to the total energy was calculated. Its temperature dependence is shown in Fig. 6.

In contrast to the behavior of dysprosium[17] and erbium[18], the temperature dependence of the magnetoelastic energy exhibits only a minute discontinuity at Tc (20 K). Such a behavior can be understood if one considers the magnetic structures of holmium from both sides of Tc. The turn angle q~ remains constant, whereas 0 varies only slightly, from 90 ° in the antiferromagnetic phase to about 80 ° in the ferromagnetic state. This difference does not contribute significantly to the change in the magnetoelastic energy at Tc. The magnitude of the magnetoelastic energy of holmium at 4.2 K is -2.13 J cm -3, which is comparable with

1338 M. ROSEN et al.

0 .0

E 0.5

o

1.0 t,d

u

o Ig

2.0

I I

0 50 100 150

TEMPERATURE ('K)

Fig. 6. Temperature dependence of the magnetoelastic energy in holmium.

the value of--2-75 J cm -~ for dysprosium at liquid helium temperature [17]. However , whereas the magnetoelast ic energy in dyspros ium varies very slowly in the tempera ture range between TN and Tc, and displays a jump at Tc, the energy in holmium Fig. 6, varies ra ther smoothly over the whole temperature range. This behavior would suggest that the transit ion from the antiferro- magnetic arrangement in holmium to the ferro- magnetic one is o f the second order, and is not accompanied by a lat t ice distortion, as was the case in dysprosium.

R E F E R E N C E S

1. Koehler W. C., Cable J. W., Wilkinson M. K., and WoIlan E. O., Phys. Rev. 151, 414 (1966).

2. Koehler W. C., Cable J. W., Child H. R,, Wilkinson M. K. and WoUan, E. O., Phys. Rev. 158, 450 (1967).

3. Koehler W. C.,J. appl. Phys. 36, 1078 (1965). 4. Landry P. C., Phys. Reu. 156, 578 (1967). 5. Strandburg D. L., Legvold S., and Spedding F. H.,

Phys. Rev. 127, 2064 (1962). 6. Gerstein B. C., Griffel M., Jennings L. D., Miller

R. E., Skochdopole R. E. and Spedding F. H., J. chem. Phys. 27, 3094 (1957).

7. Darnell F. J., Phys. Reu. 130, 1825 (1963). 8. Colvin R. V., Legvold S. and Spedding F. H., Phys.

Reu. 120, 741 (1960). 9. Nellis W. J., and Legvold S., Phys. Reu. 180, 581

(1969). 10. Legvold S., AIstad J., and Rhyne J., Phys. Rev.

Lett. 10, 509 (1963). 11. Rhyne J. J., Legvold S. and Rodine-E. T., Phys. Reo.

154, 266 (1967). 12. Rosen M. Phys. Rev. 174, 504 (1968). 13. Flippen R. B.,J. appl. Phys. 35, 1047 (1964). 14. Palmer S. B. and Lee E. W., Proc. R. Soc. Lond.

A327, 519 (1972). 15. Salama K., Brotzen F. R. and Donoho P. L., J.

appl. Phys. 44, t80 (1973). 16. Gschneidner K., In: Solid State Physics (Edited by

F. Seitz and D. Turnbull), Vol. 16, p. 275. Academic Press, New York (1964).

17. Rosen M. and Klimker H., Phys. Rev. B1, 3748 (1970).

18. Rosen M., Kalir D. and Klimker H., Phys. Rev. B8, 4399 (1973).

19. McSkimin H. J.,J. appl. Phys. 26, 406 (1955). 20. de Launay J., In: Solid State Physics (Edited by F.

Seitz and D. Turnbull), Vol. 2, p. 219. Academic Press, New York (1956).

21. Anderson O. L., J. Phys. Chem. Solids 24, 909 (1963).

22. Evenson W. E. and Liu S. H., Phys. Rev. 178, 783 (1969).

23. CaUen E. and Callen H. B., Phys. Rev. 139, A 455 (1965).

24. Cable J. W., Wollan E. O., Koehler W. C. and Wilkin- son M. K., J. appl. Phys. 32, 49 S (1961).

25. Wilkinson M. K., Koehler W. C., WoUan E. O. and Cable J. W., J. appl. Phys. 32, 48 S (1961).