Anelasticity of Phase Transitions and Magnetostriction in Fe-( 27-28 % ) Ga Alloys

Several aspects of anelastic behavior of Fe-(27-28%)Ga-Tb alloys are considered in this paper: (i) the phase transitions from a metastable to an equilibrium phase and phase transitions between equilibrium phases at higher temperatures in as cast alloys, (ii) the nature of corresponding three transient anelastic effects, (iii) the formation of an intrinsic composite microstructure with a different ratio between the bcc-derivative metastable and fcc-derivative equilibrium phases that have different magnetostriction and the effect of alloys doping by Tb to stabilize the metastable phase with high positive values of magnetostriction.


Introduction
Fe-Ga alloys exhibit attractive functional properties, such as magnetostriction that can be varied from giant positive values to negative values including zero magnetostriction 1,2 .This effect relates to a complex phase transformation sequences in the Fe-27%Ga alloys.Phase transitions in Fe 3 Ga-type alloys were first studied by X-ray diffraction [3][4][5] , and more recently by in situ neutron diffraction 2,6,7 to characterize the bulk alloy.Different ratios between metastable and equilibrium phases characterized by magnetostriction values of different signs vary the magnetostriction values of the alloy, λ S , from positive to negative, including the value of λ S = 0 2 .The phase transition from metastable D0 3 to stable L1 2 is accompanied with a well-pronounced transient anelas tic effect at about 450-550°C depending on the heating rate 8 .Anelastic effects for phase transitions at higher temperatures have as yet not been reported in the literature.Doping Fe-Ga with rare earth elements enhances magnetostriction in Fe- (17-19)Ga alloys by creating local inhomogeneities [9][10][11] and stabilizes the metastable bcc-derivative phase in as cast Fe-27Ga alloys by preventing L1 2 phase nucleation on the D0 3 grain boundaries 7,[12][13][14] .The structure of metastable Fe-27Ga type alloys before their transition to equilibrium L1 2 phase may show several metastable phases (A2, B2, D0 3 ), the size of which, as reported in the literature, can be from nano (e.g. 15 ) to macro domains.
In this paper, we studied the following phenomena: (i) transient internal friction peaks accompanied by three firstorder phase transitions, namely: D0 3 to L1 2 , L1 2 to D0 19 , and D0 19 to A2, (ii) influence of doping by Tb on these transitions, (ii) tailoring magnetostriction by isothermal phase transitions in Fe-27%Ga alloys.

Methods
Several Fe-27at.%Ga and Fe-27.4at.%Ga-Tballoys were produced by rapid solidification in a copper mold using pure Fe and Ga by arc melting under the protection of a high-purity inert argon gas using an Arc 200 mini vacuum furnace (Arcast Inc, USA).In this paper, all the compositions are given in atomic percent.We used energy dispersive spectroscopy to measure the chemical compositions of the cast buttons with ±0.2% accuracy as 27.4 to 27.8%Ga in binary alloys and Fe-27.4%Ga-0.3%Tb.
The phase transitions were characterized by in situ neutron diffraction measured with a high-resolution Fourier diffractometer (HRFD) 16 at the IBR-2 pulsed reactor in JINR (Dubna).The tests were performed on rectangular samples with a size of 4×8×40 mm.The acquisition time for each diffraction pattern was 1 min.Heating of the samples was carried out in a specialized furnace (ILL standard) with vanadium screens up to 850°C at a temperature rate of 1 or 2 K/min.Further details can be found in Refs. 2,6.
The internal friction (IF, or Q -1 ) was obtained by measuring the phase lag φ between the applied cyclic stress and the resulting strain: σ = σ 0 cos(ωt) and ε = ε 0 cos(ωt+φ), correspondingly.ω = 2πf and φ is the phase or the loss angle.It can be demonstrated that IF = tan(φ) if φ is small.The measurements were performed by two types of equipment: 1.An inversed torsion pendulum working in a forced mode.Most of the spectra were recorded at a frequency of 0.02 Hz to enhance the peak amplitude in case of phase transformations.The strain amplitude of the oscillations was 5×10 -5 .
Additionally, the spectra were also recorded at 0.3 and 1 Hz.The sample size was 1×4×30 mm between the clamps.The temperature was recorded by a thermocouple inserted in the lower clamp and touching the specimen.Therefore, a relatively good accuracy (± 2 K) can be expected.The heating and cooling rate was 1K/min.All the measurements were performed under a vacuum better than 5×10 -5 mbar.Each cycle presented here was confirmed at least once by another measurement in the same pendulum and sometimes even in another apparatus.2. A dynamical mechanical analyzer DMA Q800 TA Instruments.The measurements were conducted as a function of temperature between 0 and 600 °C using forced bending single cantilever vibrations in a range between 0.1 and 30 Hz with ε 0 = 7×10 -5 with a heating and cooling rate of 1 or 2 K/min.A DualScope C26 magnetic force microscope, MFM (DME Company, Copenhagen, Denmark) was used to obtain the magnetic force gradient image with a lift height of 250 nm and a high-moment Co-coated tip magnetized normal to the sample surface.The magnetostriction was measured using a hand-made experimental setup based on a strain gauge method up to a saturated magnetic field value of 350 kA/m.

Anelastic effects due to phase transitions
Several internal friction (IF) peaks of different nature can be observed at Temperature Dependent IF (TDIF) curves for the Fe-27Ga-type alloys.At least three of them are Debye-type thermally activated peaks (P1, P2, P3, P4) and three transient effects (Tr1, Tr2, Tr3) (Fig. 1).
The activation parameters for P1 and P2 peaks were collected in our previous papers, e.g. 8,17, and assigned to Snoek-type and Zener relaxations.
In this paper, we focus on transient effects: in situ neutron diffractions were run with practically the same heating rate as mechanical spectroscopy to define phase transitions in the alloys (Fig. 2). Figure 2 shows intervals for phase transitions, phase transition reaction rates, and atomic volume of the phases in Fe-27Ga.The D0 3 → L1 2 transition is accompanied by a pronounced jump in the atomic volume, the D0 19 → А2 transition is also accompanied with a jump in the atomic volume, whereas this effect is negligibly small in the case of the L1 2 → D0 19 transition.The change in the atomic volume in the range of the co-existing phases (D0 3 and L1 2 or D0 19 and А2) leads to an increase in micro stresses in the alloy.
It is important to note that each time the transient IF peak is associated with a modulus dip.The drop in the elastic shear modulus is due to a strong coupling between the stress and the anelastic strain: similarly to martensitic transformation, phase interfaces in Fe-Ga alloys propagate during the transformation.Misfit dislocations migration may also contribute.From the temperature of the modulus dip and considering the precision of the measurements, we cannot state that some precursor soft mode activates the transformation.Below we consider anelastic effects accompanied by these phase transitions.

The D0 3 to L1 2 transition (Tr1) and tailoring magnetostriction of the alloy
The relatively low-temperature D0 3 to L1 2 transition (Tr1) was mainly studied using DMA equipment with varying frequencies from 0.1 to 30Hz.An example of these tests is given in Fig. 1a.The dependence of the Tr1-peak height on frequency and the heating rate is plotted in Fig. 3a.The dependence: (1)   where ω = 2πf is the frequency, (dn/dT) is the volume fraction of the transformed phase during the transition with an increase in T -temperature, J is compliance, k and m are coefficients where m ≈ 0.4 5 is typical for a shear (martensitelike) transition.In contrast with 18 , no evidence of martensite or any other phases were detected by our in situ neutron diffraction tests.Moreover, careful study of the D0 3 to L1 2 transition shows that this transition goes through diffusion controlled disordering D0 3 to A2 followed by A2 to A1 transition and L1 2 ordering of the A1 phase 2,7 .Significant local stresses that appeared at D0 3 to L1 2 transition are strongly anisotropic and the average lattice deformation depends on D0 3 and L1 2 phase fraction 2 : an increase of the volume fraction of the L1 2 phase leads to an increase of the stress mainly in the D0 3 phase, whereas the stress in the L1 2 phase remains at the same level, which is relatively high from the beginning of the appearance of this phase.
If during the TDIF tests the heating is interrupted at the temperature of the Tr1-peak maximum, i.e., at about 470-475°C (Fig. 3b), the IF level decreases with holding time.A similar effect, which is slower, occurs in NiTi-or CuAl-based alloys with a thermoelastic martensite transition 19 .The isothermal in situ neutron diffraction tests carried out at the same temperature demonstrate practically the same rate for the D0 3 to L1 2 transition rate (see inset in Fig. 3b).Thus, the Tr1-peak height is, indeed, proportional to the transition rate for the D0 3 to L1 2 reaction, as predicted by Eq. ( 1), both at instant heating and isothermal annealing.
Isothermal annealing in the range of the D0 3 to L1 2 transition leads to another important consequence.Isothermal heat treatment offers an excellent opportunity to vary the magnetostriction values in the alloys by interrupting the transition to obtain a different ratio between the phases.The metastable D0 3 and the equilibrium L1 2 phase are both ferromagnetic phases, but with a different sign of magnetostriction.The difference in the ratio between these phases in the same sample, as a result of incomplete D0 3 to L1 2 transition, may change magnetostriction of the sample from positive (in D0 3 ) to negative (in L1 2 ) values.This effect was recently reported by Chinese researchers 1,20 and by the present authors 2,7 .
The domain wall motion of the D0 3 phase with positive magnetostriction results in elongation up to the maximum value at lower magnetic fields.With an increase in the magnetic field, the L1 2 phase with stronger magneto-crystalline anisotropy reduces magnetostriction and contracts the sample (Fig. 4).Annealing at 400 and 435°C leads to different dispersion of the L1 2 phase, which nuclei on the grain boundaries of the D0 3 phase.One can observe a different sensitivity of magnetostriction in the samples with differently dispersed L1 2 phase to the magnetic field.Both the maximum positive magnetostriction (red arrows) due to contribution of the D0 3 phase in low fields and the leveling-off magnetostriction values at higher magnetic field (black arrows) due to the compensating contribution of the L1 2 phase with negative magnetostriction occur at lower values of the applied field in the sample annealed at 400°C with a more dispersed L1 2 phase as compared with the samples annealed at 500°C.The magnetic domain structure in the Fe-27.8Gasample annealed at 400 °C for 600 min was studied by MFM.The magnetic domain structure of the D0 3 and L1 2 phases is rather different: the L1 2 phase contains random and irregular magnetic domains, whereas the D0 3 phase has plate-like magnetic domains with a distinct magnetic substructure.Fig. 5a and b display the topography and the MFM images in a different region of the sample.Fig. 5c shows the enlarged region of the marked area (white rectangle), in Fig. 5b with irregular magnetic domain patterns in the L1 2 phase and a large (width: 3-5 µm, length: about 40 µm) well-aligned stripe domains are observed in the D0 3 phase.The domain walls in the well-aligned stripe domains are more easily movable as compared to the irregular magnetic domains.

The L1 2 to D0 19 transition
According to the in situ neutron diffraction, the phase transition between two closed packed phases is not accompanied by a change in the atomic volume and a pronounced increase in internal stresses.Moreover, heating of the sample above the D0 3 to L1 2 transition leads to a smooth decrease in microstresses in the sample 21 .Thus, most probably, the IF peak (Tr2) in the case of the L1 2 → D0 19 (or fcc → hcp) phase transition is due to a long-range motion of the Shockley dislocation, which assists transition between these two closed packed phases.In accordance with our neutron diffraction data, the L1 2 → D0 19 transition also proceeds via disordering: L1 2 → A1 → A3 → D0 19 (these data are not presented in this paper).We carried out several tests with five frequencies (Fig. 6a) and four heating rates (Fig. 6b) in several subsequent runs, with the peak height changing roughly in agreement with Eq. ( 1).The effect of the frequency on the transient component of the martensitic transition peak for fcc -hcp transition in pure Co was studied by Bidaux et al. 22 .They demonstrated that the main (transient) component of internal friction must depend on Ṫ/ωσ 23 .An increase in the heating rate expectably leads to a shift of these diffusion controlled phase transitions to a higher temperature: the red dash lines for 0.5 K/min and the blue dot lines for 3 K/min show the range of existence of D0 19 phase in Fig. 6b.

The D0 19 to A2 transition
The effect of this first-order phase transition is similar to that of discussed in section 3.1.2.Here, we can only specify that it is the D0 19 → B2 → A2 transition.According to the in situ neutron diffraction study, if the heating rate is 2 K/ min, there is not enough time for the B2 → A2 reaction, whereas in the case of heating with a rate of 1 K/min, the neutron diffraction detects weak B2 ordering of the A2 phase, as discussed in 6 .The Tr3 peak splitting with an increase in the heating rate (Fig. 6b) can be related to this effect, i.e., to the formation of the A2 phase from D0 19 followed by a B2 → A2 reaction.This effect needs a more careful study in future.A little IF peak at about 800°C, which probably corresponds to the B2 → A2 reaction that is in agreement with the equilibrium phase diagram, is not included in Fig. 6.
It is important to note that each time the transient IF peak is associated with a modulus dip.The presence of a drop in the elastic shear modulus witnesses a strong coupling between the stress and the anelastic strain, which means that, similarly to the martensitic transformation, the phase interfaces propagate during the transformation.Misfit dislocations migration may also contribute.From the temperature of the modulus dip and considering the precision of the measurements, we cannot state that some precursor soft mode is activating the transformation.

Doping Fe-27Ga by Tb
Doping the Fe-27Ga alloys by Tb significantly increases saturation magnetostriction in the as cast alloy with the D0 3 structure and slows down the D0 3 to L1 2 transition both upon isothermal annealing (Fig. 7a) and instant heating (Fig. 7b).The effect of Tb on magnetostriction is supposed to be caused by enhancing a tetragonal distortion in the nano-sized ordered domains in the bcc derivative lattice.The effect of Tb on the kinetics of the phase transition from bcc to fcc derivative lattice in the Fe-Ga alloys was also confirmed by neutron scattering 13,24 .Consequently, Tb smears the D0 3 to L1 2 transition, shifts it to higher temperatures and decreases a volume fraction of the L1 2 phase at constant heating 13 .As a result, the Tr1-peak is not well recorded in most of our tests of the Tb-doped Fe-27Ga alloys.In contrast, the Tr2-and Tr3-peaks at higher temperatures are well recorded (Fig. 7).Their temperature positions are in excellent agreement with the L1 2 to D0 19 and D0 19 to B2 transitions similarly to this effect in the binary alloys.In contrast with the binary alloys, in the Tb-doped samples, the L1 2 and D0 19 phases coexist with the B2 phase (the volume fraction of the B2 phase is at least twice larger as compared with these closed packed structures), as the D0 3 to L1 2 transition is not complete in the alloys doped by Tb.
The apparent activation parameters of the P3 peak are: H = 2.5 eV, τ 0 = 2.5×10 -14 , which is typical for a point defect peak with a broadening parameter ≈1.Our estimation of the activation energy for the P4 peak gives a value of 2.84 eV, which is close to those for Fe-Al-Si (H = 2.7 eV 25 ) and Fe-Al-Cr (H = 2.3-2.5 eV 26 , 2.7-2.9 eV 27 ) alloys early reported in [25][26][27] and assigned to dislocation peaks at elevated temperatures.The peak height in all these systems significantly increases when it is measured at cooling (not shown in the figure) as compared with the measurements at heating.In the ordered alloys, independently of the type of order, the mobility of the dislocations and the grain boundaries is significantly reduced, which results in low values of damping.In contrast, in the disordered state after heating to high temperatures and  subsequent cooling, the dislocation and the grain boundary mobility increase, which leads to an increase in damping as compared with the ordered state 28 .Thus, we can preliminary attribute the P4 peak to dislocation effects.

Conclusions
Three transient IF effects or peaks are recorded in the studied Fe-Ga alloys.These peaks correspond to the D0 3 → L1 2 → D0 19 → A2 (or B2) phase transitions.The transient peaks (the best studied is the Tr1 peak of the D0 3 → L1 2 transition in the binary alloys) have similar features with respect to transient peaks in the thermoelastic martensitic transition.Nevertheless, the in situ neutron diffraction does not support the idea about the appearance of the tetragonal phases by shear deformation.Moreover, the diffusion controlled disordering-ordering processes are confirmed.The D0 3 → L1 2 and the D0 19 → A2 (or B2) phase transitions occur with a change in the atomic volume and a rise of the local stresses and strains, which lead to these anelastic peaks.The presence of the dips, i.e., shear modulus minima at the transition means that a softening of the elastic constants in some directions triggers the transition or maybe just a dislocation motion.One should note the pre-transition hardening that is better observed in the pendulum before the D0 3 → L1 2 transition.This behaviour is probably due to the metastable nature of the D0 3 phase.
The L1 2 → D0 19 phase transitions are accompanied neither by a change in the atomic volume nor by a significant rise of the local strains.Most probably, the IF peak in this case is due to a long-range motion of the Shockley dislocation that assists this transition.
The D0 3 → L1 2 phase transition is a unique tool to vary magnetostriction in the Fe-27Ga type alloys.The parameters of magnetization vs. field curves depend on the dispersion of the L1 2 phase in the two-phase (D0 3 + L1 2 ) samples.

Figure 1 .
Figure 1.TDIF curves measured at heating with 1 K/min rate using bending (up to 600°C, DMA Q800; left figure) and torsion (up to 800°C, inverted pendulum; right figure) vibrations with different frequencies from 0.02 to 30 Hz (see in figures).

Figure 2 .
Figure 2. Phase transitions in the Fe-27.0Ga(a) samples heated with a rate of 2 K/min obtained by normalizing the intensities of the characteristic diffraction peaks at their maximum values (reduction of the A2 phase above 700°C is associated with a gradual decrease in the intensity of the magnetic contribution to the peaks); (b) First derivatives of the above curves-the rate of the phase transitions-illustrate the rate of growth and the dissolution of different phases.(c) Atomic volumes of the phases in the same temperature range.

Figure 3 .
Figure 3. a) The Tr1-peak height as a function of inverse frequency for three different heating rates; b) TDIF curve with interrupted heating at 470°C.Inset: The decrease of the Tr1 peak as a function of time of isothermal annealing (green circles) and intensities of neutron diffraction lines from decaying D0 3 phase (blue down triangles) and growing L1 2 phase (red up triangles) under the same conditions.

Figure 6 .
Figure 6.Second and third transition peaks in the Fe-27.8Gasample: a) influence of the frequency of vibrations (from 0.05 to 1 Hz, heating rate 1 K/min), b) influence of the heating rate (from 0.5 to 3 K/min, 1 Hz).

Figure 7 .
Figure 7. (a) Amount of L1 2 phase determined by X-ray analysis after isothermal annealing in the Fe-27.4Ga(yellow circles) and in the Fe-27.4Ga-0.5Tb(red circles) samples; (b) Phase transitions in the Fe-27.4Ga-0.3Tbsample according to the in situ neutron diffraction (Intersity, left Y-axe) and TDIF (black curve corresponds to right Y-axe).