Giant volume magnetostriction in the Y 2 Fe 17 single crystal at room temperature

at room temperature S. A. Nikitin, N. Yu. Pankratov, A. I. Smarzhevskaya, G. A. Politova, Yu. G. Pastushenkov, K. P. Skokov, and A. del Moral Physics Faculty, Lomonosov Moscow State University, Moscow 119992, Russia Physics Faculty, Tver State University, 170100 Tver, Russia Laboratorio de Magnetismo de Slidos, Departamento de Fisica de Materia Condensada and Instituto de Ciencia de Materiales de Aragn, Facultad de Ciencias, Universidad de Zaragoza-C.S.I.C, E-50009 Zaragoza, Spain


I. INTRODUCTION
The magnetostriction is the occurrence of the mechanical deformation variations of magnetic samples due to the changes in the magnetization degree or in the magnetization direction.The microscopic origin of the magnetostriction involves the interaction of the orbital atomic moment with the electric charges in the crystalline lattice (the crystal field).The magnetostriction is also observed when a magnetic field is applied to a magnetized sample; in this case, it is called forced magnetostriction. 1 The forced magnetization contributes to a number of effects such as the forced magnetostriction, thermal expansions, and magnetocaloric effect. 1 The forced magnetostriction of intermetallic compounds of 3d-transition and rare-earth metals (REM) was investigated in two temperature regions.3][4] It is known that the forced magnetostriction is insignificant at low temperatures (T ( T C ); however, it is large close to T C . 1,3he intermetallic compounds of 4f-and 3d-transition metals form an important class of materials with giant magnetostriction.It is possible to achieve giant values of magnetostriction in some compounds based on the REM, which is very important for practical application. 6,7The theory of magnetostriction of 3d-4f intermetallics has been reported in a number of scientific papers; 8,9 however, no research has been conducted on the forced magnetostriction contribution near the Curie temperature.The R 2 Fe 17 compounds are particularly useful for this kind of investigations.
The R 2 Fe 17 compounds are regarded as two-sublattice magnetics with the magnetic sublattices of the REM and iron.][15][16] It is known that extremely high values of the DM effect (the magnetization change due to the pressure change) 17,18 and the forced magnetostriction [19][20][21] are caused by the strong dependence of the exchange interaction and the width of the 3d-electron energy band on the unit cell volume and the interatomic distance.The high values of the DM effect and the magnetostriction are explained by the unit cell deformations due to the magnetization increase at the forced magnetization.It is known that the R 2 Fe 17 (R ¼ Dy-Lu, Y) compounds crystallize into a hexagonal Th 2 Ni 17 -type structure with P6 3 /mmc space group. 22The unit cell contains two formula units with 34 atoms of Fe that occupy four inequivalent positions.The mentioned data concerning the magnetoelastic properties of the R 2 Fe 17 compounds indicate that these compounds show some interesting physical properties and they are excellent model objects for analyzing physics of the magnetic phenomena.However, the important physical properties of the R 2 Fe 17 compounds, including magnetoelastic characteristics (the magnetostriction constants, volume magnetostriction, and dependence of the volume magnetostriction on the magnetization) in the temperature region close to the magnetic phase transition, have not been investigated thoroughly.At the same time, the effects connected with the forced magnetization in this temperature region can be clearly observed because the forced magnetization in the given case substantially contributes into the magnetization process.The peculiarities of the magnetoelastic effect in the R 2 Fe 17 compounds in comparison with classic ferromagnet occur at the forced magnetization due to the coexistence of both positive and negative exchange interactions. 1,4The concepts of dumbbell pairs of iron atoms and 3d-band model help explain different characteristics of these compounds. 4,23urthermore, it is necessary to obtain new data on magnetoelastic and magnetic characteristics of single crystals of these compounds.
The aim of this paper is to thoroughly investigate the magnetization, magnetostriction, and thermal expansion of the Y 2 Fe 17 single crystal in the Curie temperature region and also to determine the dependences of the magnetostriction on the magnetization, the magnetostriction constants on the temperature, and the value of exchange interaction on the unit cell volume.Special attention is given to the study of the field induced volume magnetostriction, which is connected with the unit cell deformation dependences on the width of 3d-electron energy band and the exchange interactions. 4,20This kind of magnetostriction is clearly noticeable at the Curie temperature (T C ).The magnetostriction characteristics in the Curie temperature region are also discussed in accordance with the Landau theory 1,24 and the Standard Theory of Magnetostriction (STM). 4n this work, the rare-earth ions were replaced by yttrium.This metal has a very small atomic magnetic moment ($0.4 l B ), whereas the local magnetic moments of the Fe atoms in different crystallographic positions have the values of 1.9-2.4l B . 25 Yttrium is a Pauli paramagnet and its ions are located in the crystal lattice as three-valence ions, which is similar to rare-earth ions of the yttrium sub-group.At the same time, the ion radius of yttrium is close to the radius of rare-earth ions. 26Therefore, the studying of the Y 2 Fe 17 single crystal magnetism provides important information on the Fe sublattice for the whole class of the R 2 Fe 17 compounds.

II. EXPERIMENTAL TECHNIQUES AND SAMPLE PREPARATION
The initial Y 2 Fe 17 alloy was produced by the induction melting method from 99.9% pure yttrium and 99.998% pure iron in the atmosphere of pure argon in the alumina crucible.The ingot was cooled down (10 K/min) to provide conditions appropriate to grow large single crystal grains.It was found out that at the annealing at 1800 K with the subsequent cooling to 1500 K followed by slow cooling to 1400 K for 3-8 h, it is possible to obtain single crystals of the main phase.The large single crystal grains (up to 200 mg) were separated from the ingot.
The phase composition investigation was carried out by the methods of optical metallography, X-ray diffraction phase analysis, and scattering X-ray spectroscopy.The quantitative phase analysis was conducted by the Rietveld method with the PowderCell program using the data of X-ray phase analysis.The X-ray data were obtained by the diffractometer DRON-8 (Fe K a radiation).It was shown that the crystal structure of the Y 2 Fe 17 alloy is single phased with hexagonal crystal lattice of the Th 2 Ni 17 -type (space group D 4 6h or P6 3 /mmc) and the following lattice parameters: a ¼ 0.8467 nm and c ¼ 0.8318 nm.
The magnetization was carried out by the SQUID magnetometer (Quantum Design Magnetic Property Measurement System (MPMS) 5-S) in magnetic fields up to 3 T in the temperature range of 5-350 K.
For the magnetostriction measurements, single crystals were oriented by the Laue back-reflection method and were cut in the form of a disk with its plane containing two out of three main crystallographic directions (a, b, and c).To remove the strain from the surface of the samples, electrochemical polishing was carried out.
The magnetostriction and thermal expansion were obtained by the strain gauge technique with the compensation of strain gauge magnetic resistance. 4,5The linear size of the strain gauge was $5 mm, and the resistance was 120 X.The first gauge was glued to the surface of the sample and the second one was glued to the quartz plate.Both gauges were connected to the opposite arms of the Wheatstone bridge to compensate magnetic resistance.The accuracy of measurements of magnetostriction and thermal expansion was $3%.The temperature was detected by the differential copperconstantan thermocouple with its signal measured by a microvoltmeter.The accuracy of temperature determination was 0.1 K.The temperature and field dependencies of both longitudinal and transversal magnetostriction were studied in the temperature range of 100-350 K in fields up to 1.2 T.

A. Magnetization
The magnetization (M) along the main crystallographic directions of the Y 2 Fe 17 single crystal was studied in the temperature region of 5-350 K in magnetic fields up to 3 T. Fig. 1(a) shows the temperature dependencies of the magnetization M(T) along the easy magnetization axis (crystallographic direction a in the basal plane).It is seen that all curves at H k a are characterized by the monotonous decrease in magnetization with the increase in temperature.
Fig. 2 shows an abrupt decrease of magnetization in a low field of l 0 H ¼ 0.4 mT along easy axis in the temperature range of 315-330 K, which arises during the transition from the ferromagnetic to the paramagnetic state.The Curie temperature (T C ¼ 324 K) was determined as the inflection point on the M(T) curve in the field l 0 H ¼ 0.4 mT.
Fig. 1(b) displays the temperature dependencies of the magnetization along the hard magnetization axis (crystallographic direction c).It is seen that the curves at H k c are characterized by the maximums at T < T C .It is important to notice that the maximum values on M(T) curves are observed in relatively strong fields l 0 H ' 1 T not only close to T C but also at lower temperatures-for example, at 20 K lower than T C .Therefore, these values cannot be explained by the Hopkinson effect, which is observed in ferromagnetic materials close to T C only in the low field region.
The temperatures of the maximums are shifting towards lower values with the increase of the magnetic field.Similar behaviour of the magnetization along the hard magnetization axis has been observed in other rare-earth magnets. 27This phenomenon has been explained by low-symmetry and highsymmetry magnetic phases, which existence depends on the value of the magnetic field H k c. 27 The low-symmetry phase is observed in a magnetic field with a value less than critical (H cr ).This phase is characterized by the coexistence of both components of the magnetization vector (along and perpendicularly to the c axis).Therefore, in this phase, the magnetization vector is not parallel to the field direction and the c axis.The high-symmetry phase with M k c (the magnetization vector is aligned to the field H k c) exists at H > H cr .The maximum on the M(T) curve occurs at the temperature of transition from the low-symmetry phase to the highsymmetry one.The dotted curve in Fig. 1(b) shows the border that separates the high-symmetry and low-symmetry phases at H k c.The magnetization significantly decreases in a low field at T > T C (Fig. 2), which indicates that there are no extraneous phases in the sample.However, in strong magnetic fields higher than 0.3 T, magnetization decreases relatively slowly at T > T C , which can be explained by intensive paraprocess both close to T C and at T > T C .

B. Magnetostriction
The magnetostriction value was determined with regard to the direction of the magnetic field and the direction of deformation.Since the strain gauge and the magnetic field were oriented along main crystallographic directions, two indexes were used to designate the magnetostriction.The first index is the direction of the applied field and the second one is the direction of the magnetostriction measurement.
The temperature dependencies of longitudinal and perpendicular magnetostrictions k cc ; k ca ; k aa , and k ac are shown in Fig. 3. Here, both k ca and k aa are the strains along the a-axis in a field along and perpendicular to the c-axis, respectively.And both k cc and k ac are the strains along hexagonal axis c in a magnetic field along and perpendicular to the c-axis, respectively.Experimental points were obtained from magnetostriction field dependencies considering magnetostriction measurements that were carried out starting with point k ¼ 0. It is seen that the maximum of the magnetostriction is observed near T C .These diffused maximums on the temperature dependencies of the magnetostriction in the T C region were explained by the strong dispersion of the exchange integrals due to the coexistence of both positive and negative exchange interactions.Therefore, both the magnetostriction and magnetization data indicate the temperature of magnetic phase transition "paramagnetism-ferromagnetism" at T C ¼ 324 K.
In the region of magnetic phase transition (T > 250 K), the values of magnetostrictions k ca ; k cc ; k aa , and k ac are positive.The magnetostrictions reach their maximums at T C (see Fig. 3).The maximum values of the magnetostrictions are different.For example, k ca ¼ 85 Â 10 À6 , k ac ¼ 155 Â 10 À6 , k aa ¼ 105 Â 10 À6 , and k cc ¼ 120 Â 10 À6 in the field of 1.2 T. The magnetostriction close to T C is forced 4 for H k a (easy magnetization axis).The values of k aa ; k ca ; k ac , and k cc maximums increase together with the magnetic field growth as the result of the forced magnetization in the magnetic phase transition region.
The field dependencies of k aa ; k ca ; k ac , and k cc magnetostrictions at the Curie temperature (T C ¼ 324 K) are shown in Fig. 4. Figs. 3 and 4 indicate that the magnetostrictions are positive at the "ferromagnetism-paramagnetism" phase transition and increase together with the growth of the applied magnetic field.The speed of the increase is not the same for different crystallographic magnetic field directions.Therefore, the anisotropic change of the sample volume occurs due to the forced magnetization.
The magnetostriction saturation at Curie temperature T C ¼ 324 K is not observed.It is known that the saturation of magnetostriction is observed for the processes of rotation of magnetization vectors. 1,4Therefore, the forced magnetostriction is responsible for the volume magnetostriction near T C in the Y 2 Fe 17 compound.
We should emphasize that the forced magnetostriction in the Y 2 Fe 17 compound is quite large in moderate magnetic fields close to the Curie temperature.Both longitudinal and transversal magnetostrictions close to T C are positive and demonstrate high values.
These data clearly indicate that the volume expansion occurs under the influence of the magnetic field.The abovementioned data indicate that the volume change makes the largest contribution to the magnetostriction close to T C , which means that large field induced volume magnetostriction occurs in this region.
It is noticeable that the k aa ðTÞ and k ac ðTÞ curves (see Fig. 3) are almost symmetrical close to T C , while the k ca ðTÞ and k cc ðTÞ curves are asymmetric (see Fig. 3) and have minima at T below T C .The asymmetry of these curves is explained by the complicated nature of the magnetization process along the hard magnetization axis in the anisotropic ferromagnet in the field H k c at H < H cr and T < T C . 27In this case, the increase of magnetization occurs due to both the forced magnetization and magnetization vector rotation from the basal plane (easy magnetization plane) to the direction of the magnetic field (H k c).The negative contribution to the magnetostriction k ca and k cc occurs due to the magnetization vector rotation to the direction of hard magnetization axis under the magnetic field H k c that leads to the presence of the minimums on the k ca and k cc temperature dependences at T < T C .

C. Thermal expansion
Our data of thermal expansion along the hexagonal axis c ðD'='Þ c and the a-axis in the basal plane ðD'='Þ a obtained by the direct measurements in the temperature range of 100-350 K are shown in Fig. 5 as triangles.For comparison purposes, the thermal expansion at the temperatures up to 750 K calculated according to the X-ray analysis data 15,16 is presented on the same diagrams as filled circles.
It is seen that the thermal expansion detected by the direct measurements and calculated from the indirect measurements coincide precisely in the temperature range of 100-350 K.At the temperatures higher than the Debye temperature ðT D Þ, the thermal expansion has a linear behaviour that allows to extract the phonon contribution ðD'='Þ ph using the Debye theory with two parameters (inclination of the line and T D ) taking into account the value of the Debye temperature T D ¼ 440 K. 15 The phonon contributions are shown in Fig. 5
were found by subtracting ðD'='Þ ph from the experimental values of the thermal expansion ðD'='Þ.Fig. 5 shows the contributions K c ðTÞ and K a ðTÞ along ðD'='Þ ph considerably change the temperature dependences of the thermal expansion due to their higher values.It should be noticed that the linear size along the axis c abnormally increases at cooling below T C and does not decrease.
The coefficients of thermal expansion (c a and c c ) were found by differentiating the curves of thermal expansion ðD'='Þ a ðTÞ and ðD'='Þ c ðTÞ with the temperature The temperature dependencies of thermal expansion coefficients along the hexagonal axis c (c c ) and in the basal plane (c a ) are shown on the insertions in Fig. 5.The coefficients of thermal expansion have the k-type temperature dependences close to T C .Some peculiarities in their behaviour are also seen in the temperature region above T C .They are the sum of two contributions: an expansion due to the stimulation of the phonons (c ph ) and the magnetostrictive contribution.To explain the abnormal behaviour of (c c ) and (c a ), we should consider that c ph > 0 according to the Debye theory and the experimental data for the without a spontaneous magnetic moment.At some temperatures close to T C , the positive contribution due to the stimulation of the phonons and the negative magnetostrictive contribution compensate each other, which causes zero values of the thermal expansion coefficient (it is also typical for the invar Fe-Ni alloys). 4At the temperatures below T C , the negative magnetostrictive contribution is dominant, which causes the negative values of the c c and c a coefficients.It is important to notice that the thermal expansion anisotropy is observed: the (c a ) and (c c ) coefficients (along a and c axes) are significantly different.

IV. DISCUSSIONS A. Calculation of volume magnetostriction and magnetostriction constants within the framework of phenomenological approach
It is well known that in the region of the second order phase transition from the ferromagnetic to paramagnetic state, the orthorhombic distortion and the shear deformations may be disregarded. 4This conclusion is confirmed by the received experimental results, according to which the magnetostrictions along both a and b axes in the basal plane (within the limits of possible experimental error $3%) practically coincide in the temperature range of 200-350 K. 20 Without considering the orthorhombic distortion and the shear deformations as both of them are not observed in the investigated temperature range, the magnetostrictive deformation can be written according to the STM 4,28 ; 9][30] On the one hand, the magnetostriction constants k a 11 and k a 21 are determined by isotropic pair exchange and pseudodipole interactions of two magnetic ions.On the other hand, k a 12 and k a 22 are determined by anisotropic interaction of the magnetic ion with the crystal field, which is created by the surrounding ions of the crystal lattice.Exchange interactions make an isotropic contribution to the magnetostrictive deformation, which does not depend on the directing cosines of the magnetization vector M. At the same time, the single-ion interactions make anisotropic contributions depending on the direction of the magnetization vector.
The volume magnetostriction as a function of variables (T, H) is presented as 28 From (3), it is possible to calculate the volume magnetostrictions x a ðH k aÞ and x c ðH k cÞ. 28In case when the external magnetic field H exceeds the anisotropy field (thereby, vector M is parallel to the direction of H), from ( 2) and ( 3) we obtain Thereby, the field induced volume magnetostriction is anisotropic relatively to the direction of the applied field H, as a result of the single-ion k a 12 contribution.It is worth noting that the axis ratio c/a of the lattice parameters also changes simultaneously with the volume change.For ðH k aÞ from (2) it is as follows: For ðH k cÞ, it is as follows: Fig. 6 shows the temperature dependencies of the field induced volume magnetostrictions x a and x c calculated by Eqs. ( 4) and ( 5) using the experimental values of the magnetostrictions k ca ; k aa ; k ac , and k cc .It is seen that x a and x c have the giant magnitude close to the Curie temperature: x a ¼ 360 Â 10 À6 and x c ¼ 250 Â 10 À6 in the field l 0 H ¼ 1.2 T. As expected, the x increased together with the field as in this case the forced magnetostriction was dominant (the forced magnetostriction increases linearly with the field).
Using the experimental data on magnetostrictions, the contributions of a crystal field to the axis ratio c/a described by the magnetostriction constants k a 21 and k a 22 were calculated by ( 6) and (7).The temperature dependencies of the magnetostriction constants k a i;j ðT; HÞ ði; j ¼ 1; 2Þ in the field l 0 H ¼ 1.2 T are shown in Fig. 7 12 and k a 22 , the maximums are shifted to lower temperatures.This behaviour is caused by the negative contribution into the magnetostriction from the processes of magnetization vector rotation in the low-symmetry phase at H < H cr (see Fig. 1(b)).In the field along the hard axis c, the minimum on k a 12 ðTÞ is correlated with the transition from the low-symmetry phase (M is not parallel to H) into the high-symmetry phase (M k H) in the field l 0 H ¼ 1.2 T (see Fig. 1(b)). 27The border that separates the high-symmetry and low-symmetry phases is shown in Fig. 1(b) by the dotted curve.Thus, the magnetostriction constants k a 12 and k a 22 are determined by the contributions of the magnetization vector orientation relative to the axis c.
The above-mentioned data indicate the coexistence of both exchange and magnetocrystalline contributions to the forced magnetostriction.It is worth noting that the field induced volume magnetostriction in the Y 2 Fe 17 compound is higher than in other intermetallic compounds at the room temperature. 4

B. Magnetization dependence of the forced magnetostriction according to the thermodynamic theory
The experimental data of both temperature and field dependencies of the volume magnetostriction in the Curie temperature region were explained on the basis of the Landau thermodynamic theory for the second order phase transitions. 24For a ferromagnetic with the hard [This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: magnetization axis along the hexagonal axis c and the easy magnetization axis in the basal plane (a, b), using two components of the order parameter is sufficient: the projection of the magnetization vector (M k ) on the axis c and on the basal plane (M ?).It has been shown by Nikitin et al. 27,31 for the ferromagnetic with uniaxial anisotropy in the magnetic field H and under the applied pressure P, the thermodynamic potential G can be written as where m ¼ M=M 0 is relative magnetization, M is spontaneous magnetization at temperature T, M 0 is spontaneous magnetization at 0 K, and a 1 , a 2 , b 1 , b 2 , and e are thermodynamic coefficients.According to the theory, 27,31 we receive where K 1 ; K 2 , n 1 , n 2 , and n 3 are magnetoelastic coefficients and a 1H , a 2H , b 1H , b 2H , and e H are thermodynamic coefficients at P ¼ 0. The stable phases and the limits of their existence can be achieved from the minimization of the thermodynamic potential. 27n the presence of external pressure, the thermodynamic coefficients a 1 and a 2 have zero values at the corresponding temperatures H ð1Þ ðPÞ and H ð2Þ ðPÞ, where H ð1Þ ðPÞ ¼ H ð1Þ ð0Þ À K 1 P a 1H and H ð2Þ ðPÞ ¼ H ð2Þ ð0Þ À K 2 P a 2H .The volume change x can be found directly from the thermodynamic potential (8) When the applied magnetic field exceeds the magnetic anisotropy field, the magnetization vector is aligned to the magnetic field and the high-symmetry phase appears. 27ow let us consider three different states of a ferromagnetic in the high-symmetry phase: (1) H ¼ 0, m k ¼ 0, and m ?¼ m S ; in this case, the spontaneous magnetostrictive deformation is (2) In field along easy magnetization axis (H?c), the magnetization vector is aligned to field direction (m k ¼ 0) and the volume magnetostriction x a in the highsymmetry phase equals where x ?ðHÞ is magnetostrictive deformation at H?c. (3) For the field H k c (along the hard axis), the field induced volume magnetostriction x c in the high symmetry magnetic phase (m ?¼ 0) is equals to where x k ðHÞ is magnetostrictive deformation for H k c.It should be noticed that change of both x ?ðHÞ À x S ð0Þ and x k ðHÞ À x S ð0Þ are the effects induced by the external magnetic field.The case when H?c (the field applied along the easy axis) corresponds to the forced magnetization as spin fluctuations close to T C is suppressed by the magnetic field.In case when the field is applied along the hard axis (H k c) in the low symmetry phase where m ?6 ¼ 0; m k 6 ¼ 0, the process of magnetization at the phase transition is more complicated: with addition to the forced magnetization, the magnetization vector sequentially turns towards the direction of the hard axis, and at the same time the stimulation of the transverse spin waves is also suppressed. 4,32The dependence of the field induced volume magnetostriction x a on the square of magnetization m 2 at the temperatures near T C is shown in Fig. 8.It should be noticed that the thermodynamic theory (12) is well confirmed subject to the terms proportional to the squared magnetization m 2 ? .Thus, the contributions to the field induced volume magnetostriction proportional to the fourth power of the relative magnetization are insignificant.According to the relation (12), the segments cut off by the straight lines x a ¼ f ðm 2 ?Þ on the m ?axis provide the value of the squared spontaneous relative magnetization m 2 S , and the segment cut off on the ordinate axis provide the value of the spontaneous volume magnetostriction with the minus.The experimental data for x a and m 2 ?allow to determine the thermodynamic coefficient K 1 ¼ 4.8 Â 10 À3 in Eq. (12).The giant forced magnetostriction and the sharp anomalies of thermal expansion below T C in Y 2 Fe 17 indicate the strong dependence of the exchange integral on the unit cell volume V and the interatomic distances in the Y 2 Fe 17 compound.To determine the dependence of the exchange integral (A) on the volume of the unit cell in terms of quantity, this dependence was calculated on the basis of thermodynamic correlations.These correlations suggest that the forced magnetization is the function of the T=T C ratio and the magnetic field H.The widely known thermodynamic relation 1,4 was used for this purpose: As M is the function of T=T C and H, the following equation was obtained: It is known that compressibility k equals Therefore, So, at T ¼ T C follows from Eqs. ( 14)-( 17), According to the experimental data in the field higher than the field of technical magnetization where the forced magnetization occurs, we obtain: dx=dH ¼ 2.5 Â 10 À3 m/kA and dM/dT ¼ À10 À3 T/K for the field of 1 T. The compressibility of Y 2 Fe 17 is k ¼ 1.05 Â 10 À11 Pa À1 . 33The exchange integral variation with the change of the unit cell volume was found by (18).According to the fact that T C is proportional to the exchange integral A between the Fe atoms, it follows: It was found that for Y 2 Fe 17 C ¼ 9.1, which correlates with the values of C for R 2 Fe 17 that were published by different authors in the past. 35For example, Radwanski in his work 23  The analysis of previously published works on the value of C revealed a spread in the values of C for R 2 Fe 17 .This may be related to two factors: (1) most of the works used polycrystalline samples and (2) different methods were used to estimate the parameter.We used single crystal samples, which allowed us to obtain a more precise value of C. Nevertheless, we can conclude that the exchange integral change at the unit cell volume change is extremely high in R 2 Fe 17 .
Similarly to this, the field induced volume magnetostriction in Y 2 Fe 17 reaches high values.Therefore, our investigation shows that Y 2 Fe 17 is a material with a giant field induced volume magnetostriction in moderate magnetic fields (x a % 360 Â 10 À6 in the field of 1.2 T) at the Curie temperature T C ¼ 324 K (in the room temperature region).The mentioned magnetostriction is caused by the forced magnetization near the Curie temperature.This property was explained 3,19,23 by existence of abrupt change of the exchange integral between Fe atoms in 4f positions that form dumbbell-like pairs perpendicular to the hexagon composed of the nearest Fe atoms in the positions 12j.These interactions turn to be negative, 3,19,23 when the distance between two Fe atoms in the positions 4f less than critical value.According to the theory of the forced magnetostriction, 4,20 the anisotropic single-electron jumps occur in Y 2 Fe 17 in "dumbbells" of the Fe atoms in the positions 4f.These "jumps" are very sensitive to the both distances between the Fe atoms and unit cell volume.As a result of this process, 4,20 one may observe a strong dependence of the Hubbard matrix of the "jumps" t FeÀFe on deformation for 3d-electrons together with the strong decrease of the Coulomb electron repulsion (U) which leads to the giant forced magnetostriction in Y 2 Fe 17 .
Huge magnetostrictive deformations that exist in a number of materials in the low temperature region abruptly decrease together with increasing temperature.These deformations are caused by magnetocrystalline interactions.The large field induced volume magnetostriction at the Curie temperature (near room temperatures) is observed in the following materials Gd (x % 300 Â 10 À6 ) 4,34 and Lu 2 Fe 17 (x % 150 Â 10 À6 ) 4,21 in the field of 1.2 T. Therefore, the investigated Y 2 Fe 17 compound has a giant field induced volume magnetostriction that exceeds the magnetostriction in the above-mentioned materials in the room temperature region, allowing technical application of this compound.This volume magnetostriction arises from the sharp dependence of the exchange interaction.The advantage of the Y 2 Fe 17 compound is its lower cost compared with rare-earth intermetallic alloys (such as Terfenol-D) due to the high content of Fe, which is important for the technical application in magnetostrictors.

V. CONCLUSIONS
The giant field induced volume magnetostriction exists in the Y 2 Fe 17 compound in the temperature range of ferromagnetic-paramagnetic phase transition.The values of the magnetostriction constants and their temperature dependencies are explained by the Standard Magnetostriction Theory 4 and the thermodynamic theory, where the strong dependence of the exchange interactions and magnetocrystalline on the deformations should be taken into account.It was shown that the volume change at the forced magnetization in a field had anisotropic character close to the Curie temperature.
It was found that the peculiarities of the magnetostriction temperature dependence in a field applied along the hard magnetization axis were caused by the fact that in addition to the positive contribution due to the forced magnetization, the contribution from the magnetization vector rotation also existed in the fields lower than the effective field of the magnetic anisotropy.The calculated values of the magnetostriction constants k a 11 and k a 12 show that the anisotropic single-ion contribution and isotropic pair exchange interactions contribution are comparable in the magnitude, but the exchange contribution to the magnetostriction dominates at the Curie temperature.
The origin of the giant values of the field induced volume magnetostriction is explained by the strong dependence of the electron "jumps" parameter and the width of the 3delectron energy band on the deformation.Special attention should be paid to the fact that the Y 2 Fe 17 compound shows the giant values of the field induced volume magnetostriction in moderate magnetic fields at the room temperature allowing using this effect in new hydraulic devices. 37

FIG. 1 .
FIG. 1. Temperature dependencies of magnetization of Y 2 Fe 17 single crystal along both a (a) and c (b) crystallographic directions in different magnetic fields.
FIG.3.Temperature dependencies of k aa , k ca ; k ac , and k cc magnetostrictions in magnetic fields of 0.5, 0.9, and 1.2 T along the hexagonal c axis (a) and a axis (b).

FIG. 5 .
FIG. 5. Temperature dependencies of thermal expansion along the crystallographic directions a and c.Inset: The thermal expansion coefficients vs temperature along crystallographic directions a and c.

FIG. 6 .
FIG.6.Temperature dependencies of the field induced volume magnetostriction x a and x c in different magnetic fields: 1-1.2 T, 2-0.9 T, and 3-0.5 T.

FIG.
FIG.The field induced volume magnetostriction x a versus the square of relative magnetization m ? .
. It was found that k a 11 ¼ 326 Â 10 À6 , k a 12 ¼ À238 Â 10 À6 , k a 21 ¼ 16 Â 10 À6 , and k a 22 ¼ À26 Â 10 À6 at T C in the magnetic field of 1.2 T. It is seen that near T C , the magnetostriction constants k a 11 and k a estimated this parameter as C ¼ 15 using relative differences in the volumes dV of crystal lattices and the difference in the Curie temperatures dT C in Y 2 Fe 17 and Lu 2 Fe 17 compounds.Buschow 36 estimated the value of C ¼ 12 for the polycrystalline sample Y 2 Fe 17 investigating the Curie temperature dependency from lattice parameters under pressure.Givord 10 found the value of dT C =dlnV ¼ 2030 K for the polycrystalline sample Y 2 Fe 17 , which corresponds to C ¼ 6.3, from thermal expansion data (neutron diffraction).