J. Phys. Chem. Solids, 1971, Vol. 32, pp. 2351-2356. Pergamon Press. Printed in Grea t Britain

E L A S T I C M O D U L I O F T H U L I U M A N D Y T T E R B I U M

F R O M 4.2 T O 300~

M. ROSEN

Nuclear Research Centre, Negev, P.O.B. 9001, Beer-Sheva, Israel

(Received 5 January 1971 )

Abstract--The longitudinal and transverse acoustic velocities of high-purity polycrystalline thulium and ytterbium have been measured by a pulse technique at a frequency of 10 MHz between 4.2 and 300~ The variation with temperature of the Young moduli E, shear moduli G, adiabatic compres- sibilities Ks, and Debye temperature 0o have been determined. The anomalies in the elastic properties observed in thulium were correlated with the magnetic changes known to occur in this metal at low temperatures. The shape of the compressibility curve suggests that the transition from the antiferro- magnetic to the ferromagnetic state in thulium is a gradual process, extending over a temperature interval. No anomalies were observed in the elastic properties of the nonmagnetic ytterbium. The limiting values of the Debye temperatures (at 0~ of thulium and ytterbium are 200 and 117.5~ respectively.

1. INTRODUCTION

THE LOW-TEMPERATURE behavior of the 'heavy' rare-earth metals, from gadolinium to lutetium, has extensively been investigated from both theoretical Ill and experimental[2] standpoints. Most of these metals display complex magnetic spin structures with strong dependence on temperature. The magnetic contribution to the total energy in the vicinity of the transition points generally affects the elastic moduli since they are the second derivatives of the thermodynamic potential of the lattice with respect to strain.

The present paper is concerned with the temperature dependence of the elastic moduli, adiabatic compressibility, and Debye tempera- ture of polycrystalline thulium and ytterbium. The elastic properties of the other 'heavy' rare-earth metals in polycrystalline form (gadolinium, terbium dysprosium, holmium and erbium) have previously been reported in detail [3]. The elasticity and anelasticity of the 'light' rare-earth metals, with the exception of lanthanum, have also been presented in a series of papers [4-6].

2. EXPERIMENTAL DETAILS

The high-purity (99.9%) polycrystalline specimens, supplied by Leytess Metal and Chemical Corp., New York, were in the form of fiat disks 6 mm in dia. by about 5 mm thick. The specimens were hand-lapped to a paral- lelism of faces of better than 2 parts in 104. The thickness of the disks was measured by means of a calibrated indicator stand to within 5 x 10 -4 ram. Room temperature densities (Table 1) were determined to +__0.003 gcm -3 by a fluid-displacement method using mono- bromobenzene. The temperature variation of the acoustical path lengths of the specimens was calculated by using the average coeffi- cients of thermal expansion given by Gschneidner[7] and listed in Table 1.

The elastic moduli, adiabatic compressi- bilities and Debye temperatures were deter- mined from the experimentally measured longitudinal and transverse sound-velocities. An ultrasonic pulse technique was employed at a frequency of 10MHz. Experimental details and method of data analysis were described elsewhere[3]. The estimated error

2351

2352

Table

M. ROSEN

1. Average room-temperature densities and thermal expansion coefficients of the specimens

Average linear coefficient of Densi ty thermal expansion

Element (g cm -a) (~ -~)

Tm 9.288 13.3 x I0 -n Yb 6.991 24.96 x 10 -6

in the absolute values of the elastic moduli is 0.4 per cent. The relative, point to point, precision is better by a factor of 4.

3. RESULTS AND DISCUSSION

3.1 Thulium Neutron diffraction measurements [8] indi-

cate that thulium is paramagnetic above 56~ and sinusoidally-modulated antiferromagnetic between 56 and 38~ Below 38~ the metal has a ferrimagnetic-type antiphase domain structure with a net moment parallel to the hexagonal c axis. In the ferromagnetic region, below 22~ the magnetic layers are parallel to the c axis. The structure is in the form of 4 layers with spins up followed by three layers with spins down. Magnetization data[9, 10] are in qualitative agreement with neutron diffraction measurements. The transition point from the paramagnetic to antiferromagnetic state, at about 56~ is clearly displayed in various transport properties[l 1-14], as well as in the temperature dependence of the specific heat[15]. However, no unusual behavior was observed at the ferromagnetic transition point, in the vicinity of 22~

The temperature variation of the Young (E) and shear (G) moduli of polycrystalline thul- ium is shown in Fig. 1. With decreasing temperature, from the ambient down to about 200~ both moduli increase in the normal manner. Below this temperature E and G exhibit a very small temperature dependence followed by a minimum at Tu (55~ With decreasing temperature, in the antiferro- magnetic region, the elastic moduli of poly- crystalline thulium increase abruptly by 5 per

cent of their values over a rather limited tem- perature interval of about 20~ The sharp increase of the moduli with decreasing tem- perature is halted at about 35~ where the sinusoidally modulated antiferromagnetic structure changes to the ferrimagnetic-type structure [8].

The ferromagnetic transition point is not clearly displayed in the temperature variation of the elastic moduli, Fig. 1. However, a shallow minimum, followed by a well-defined dip were observed at 25 and 13~ respec- tively. But neither of these two temperatures coincides with the Curie point of 22~ as determined from neutron diffraction data [8].

The temperature dependence of the adia- batic compressibility Ks, of thulium is shown in Fig. 2. The normal decrease of the com- pressibility with decreasing temperature, in the range between 300 and 200~ is followed by a shallow change, and subsequently by a sharp maximum at the Nrel temperature (55~ At lower temperatures the behavior of the adiabatic compressibility is consistent with that of the elastic moduli, displaying two maxima at 25 and 13~ The shape of the broad maximum in the compressibility, peaked at 25~ would suggest that the transi- tion from the antiferromagnetic to the ferro- magnetic state in thulium is not a usual second-order type cooperative phenomenon. It appears that the low-temperature transition in thulium has a gradual character and extends over a wide temperature range. This is appar- ently the reason why the Curie point of thul- ium is indiscernible in various transport properties of this metal [10-14].

ELASTIC MODULI OF THULIUM AND YTTERBIUM 2353

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2354 M. ROSEN

The temperature dependence of the Debye temperature, 0o, is shown in Fig. 2. It is very similar to the variation of the elastic moduli, Fig. 1. The limiting value of 0o, extrapolated to 0~ is 200~ This value is significantly higher than 167~ obtained from specific heat measurements [ 15].

3.2 Ytterbium Ytterbium metal is different from the other

rare-earths in several respects. Contrary to the other metals of the series which have an h.c.p, structure at room temperature, ytter- bium (and also europium) is f.c.c. Further- more, ytterbium is divalent in oontrast to most rare-earth metals whose properties in the metallic state can be explained in terms of a trivalent model for the ion cores. The 4fshell in ytterbium is full, leaving only two electrons for conduction processes. Therefore, the physical properties of this metal are markedly

different from the other rare-earths. Lock [ 16] found that ytterbium metal is paramagnetic with a Curie constant of 56 x 10 -6 e.m.u, g-1. But recent magnetization data[17] suggest that very pure ytterbium may be diamagnetic. Magnetoresistance measurements[18] show that ytterbium is a compensated metal with a closed Fermi surface. No low-temperature anomalies were observed in the electrical resistivity [ 19,20], thermoelectric power [ 12], and specific heat measurements. The density of ytterbium is much lower than that of its neighbor thulium, Table 1.

The Young (E) and shear (G) moduli of polycrystalline ytterbium are shown in Fig. 3. The absolute values of the moduli of this metal are significantly lower compared with those of thulium. The temperature variation of the elastic moduli of ytterbium, Fig. 3, is smooth and well-behaved as would be ex- pected from a metal that does not possess any

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Fig. 3. Temperature variation of the Young (E) and shear (G) moduli of polycrystalline ytterbium.

ELASTIC MODULI OF THULIUM AND YTTERBIUM 2355

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magnetic transitions at low temperatures. The compressibility of ytterbium is rather high, therefore it is not surprising that the elastic moduli E and G increase by about 8 per cent with decreasing temperature from the ambient to 4.2~

The temperature dependence of the adia- batic compressibility, Ks, of ytterbium is shown in Fig. 4. From the ambient down to about 100~ Ks decreases smoothly, then levels off with decreasing temperature. According to the third law of thermodynamics, the compressibility should meet with zero temperature. However, the reason for the slight increase in Ks (Fig. 4) at temperature below 80~ is not clear.

The limiting Debye temperature, extra- polated to 0~ is 117.5~ Fig. 4. This value is in good agreement with l18~ as deter- mined from low-temperature specific heat measurements [21]. It should be noted that the

temperature dependence of 0o, as displayed in Fig. 4, does not exhibit the reported[21] minimum at 14~ as was calculated from heat capacity data.

Acknowledgements-The author wishes to express his gratitude to A. Ha lwany , D. Kalir and B. C o h e n for their ass i s tance in the various phases o f this work.

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2356 M. ROSEN

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