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Zhiyong Dai, Chao Zhou, Qizhong Zhao, Kaiyan Cao, Zhengming Zhang, Dunhui Wang, Dezhen Xue, Adil Murtaza, Yin Zhang, Fanghua Tian, Wenliang Zuo, [Yoshitaka Matsushita](https://orcid.org/0000-0002-4968-8905), Sen Yang, Xiaoping Song

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This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Zhiyong Dai, Chao Zhou, Qizhong Zhao, Kaiyan Cao, Zhengming Zhang, Dunhui Wang, Dezhen Xue, Adil Murtaza, Yin Zhang, Fanghua Tian, Wenliang Zuo, Yoshitaka Matsushita, Sen Yang, Xiaoping Song; Effects of Ag doping on texture and magnetic properties of directionally solidified Fe-17%Ga alloys. Appl. Phys. Lett. 20 May 2024; 124 (21): 212406. and may be found at https://doi.org/10.1063/5.0200456.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Effects of Ag doping on texture and magnetic properties of directionally solidified Fe-17%Ga alloys](https://mdr.nims.go.jp/datasets/47dbd4bd-860a-416a-9a77-13a134c78fda)

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1  Effects of Ag Doping on Texture and Magnetic Properties of 1 Directionally Solidified Fe-17%Ga Alloys 2 Zhiyong Dai 1, Chao Zhou 1,*, Qizhong Zhao 1, Kaiyan Cao 1, Zhengming Zhang 2, Dunhui 3 Wang 2, Dezhen Xue 3, Adil Murtaza 1, Yin Zhang 1, Fanghua Tian 1, Wenliang Zuo 1, Yoshitaka 4 Matsushita 4, Sen Yang 1,3,*, Xiaoping Song 1 5 1 MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Physics, Xi’an 6 Jiaotong University, Xi’an 710049, China 7 2 Division of Microelectronic Materials and Devices, Hangzhou Dianzi University, Hangzhou 310018, China 8 3 State Key Laboratory for Mechanical Behavior of Materials, School of Materials Science and Engineering, Xi’an 9 Jiaotong University, Xi’an 710049, China 10 4 National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan 11 Abstract 12 FeGa-based alloys are potential candidates in magneto-mechanical transversion applications. It 13 has been demonstrated that doping with certain elements results in the increase of λ100 and 14 modification of the texture, both of which effectively enhance the magnetostriction of FeGa. 15 However, the contribution of each factor is difficult to distinguish because of lacking an effective 16 physical model. In this work, based on independence of saturation magnetostriction (𝜆𝑠 ) on 17 magnetic domain distribution, a paradigm that combines the orientation distribution function and 18 magnetostriction tensor of single crystals is employed to quantify effects of texture on 19 magnetostriction and predict λ100. Then this paradigm is applied to Ag-doped FeGa and the results 20 reveal that the enhancement of λ100 plays a more crucial role in the enhancement of 𝜆𝑠 after Ag 21 doping. Our work clarifies the contribution to magnetostriction enhancement from texture and λ100 22 in element-doped FeGa alloys, and may help develop more high-performance FeGa alloys. 23 _____________________________ 24 Authors to whom correspondence should be addressed: chao.zhou@xjtu.edu.cn; 25 yang.sen@xjtu.edu.cn.  26 mailto:yang.sen@xjtu.edu.cn 2  Magnetostrictive materials have wide applications in magnetic sensors, actuators, 27 transformers, and energy harvesters. Compared to ferromagnetic martensite alloys [1] and Laves 28 phase rare earth-transition metal compounds [2], FeGa-based alloys exhibit comparatively lower 29 magnetostriction. Many efforts have been devoted, e.g., doping with a trace amount of rare earth 30 elements proves to have a significant effect on magnetostriction of FeGa alloys. By doping with 31 0.2% Dy in Fe-17% Ga, magnetostriction increases from 70 ppm to 150 ppm at 1000 Oe [3]. 32 Besides, doping with Tb [4], Y [5], and Ce [6] shows similar effects.  33 The magnetostriction enhancement after doping with certain elements is attributed to the 34 increase of tetragonal magnetostrictive constant (λ100), e.g., Tb [7], or modification of the preferred 35 orientation of crystals in polycrystalline aggregates (i.e., texture) [8], e.g., Tb [4], Er [9], Pt [10], Sm 36 [6]. Because of lacking an effective physical model, the contribution of each factor is difficult to 37 distinguish, making it impossible to systematize theoretical research and engineering practice.  38 Here, we propose a paradigm to distinguish the contribution of each factor. For single crystals, 39 𝜆∥ and 𝜆⊥ (measured in a direction when the applied field is parallel and vertical to measurement 40 direction, respectively) are dependent on initial magnetic domain distribution, but saturation 41 magnetostriction 𝜆𝑠  (= 𝜆∥ − 𝜆⊥ ) is not and remains constant [11]. For polycrystals, 𝜆𝑠  is also 42 independent of initial domain distribution [12]. Thus, 𝜆𝑠 of a particular distribution of magnetic 43 domains is equal to that of ideal demagnetization, which only depends on texture. Orientation 44 distribution function (ODF) is combined with the magnetostriction tensor of single crystals, by 45 which effects of texture on magnetostriction can be quantified. Rare earth element-rich phases are 46 easy to precipitate from rare earth element-doped FeGa alloy [3-6, 9]. Approximately, 47 magnetostriction of FeGa alloy which is composed of different phases, obeys a rule of mixture 48 relationship [13]. So, apart from A2 phase (alpha iron-based disorder phase), characterization of 49 rare earth element-rich phases (ODFs of textures and magnetostriction tensors of single crystals) 50 needs to be taken into consideration, which makes computation complicated. For simplicity, we 51 choose a non-magnetic element as an example, which will only include A2 phase for consideration 52 and omit possible element-rich precipitates for their negligible volume and magnetostriction. By 53 DFT calculation, Ag and Cu are potential elements that can effectively enhance λ100 [14]. So, in the 54 present study, Ag is chosen to be the doping element as an example, to quantify the effects of 55 doping on λ100 and texture in FeGa alloy. Besides, directional solidification with a high withdrawal 56  3  speed is used to decrease precipitation of Ag and obtain good textures [15]. Microstructure and 57 magnetic properties of Fe83Ga17 and (Fe0.83Ga0.17)99.8Ag0.2 were investigated, while distribution of 58 magnetic domains was also studied by magnetic domain patterns.  59 The (Fe0.83Ga0.17)100−xAgx polycrystalline samples (x represents the atomic percentages; x = 60 0, 0.2) were prepared by arc-melting techniques under argon atmosphere, using high purity metals 61 of Fe (99.95%), Ga (99.99%) and Pt (99.99%). To ensure compositional homogeneity, each sample 62 was melted four times [16]. The directionally solidified samples were prepared at 1650°C with the 63 withdrawal rate of 1800 mm/h. The samples were cut by the electric-spark method along the 64 directional solidification direction and the fin al size of samples for tests is 8 mm × 8 mm × 1.5 65 mm. The macrostructure was investigated by optical microscopy and columnar grains were 66 confirmed to be arranged parallelly to growth direction in Fig. S1. The X-ray diffraction (XRD) 67 patterns were measured by using a Bruker D8 ADVANCE Diffractometer (Cu–Kα, λ = 1.5406 Å) 68 at room temperature. For texture analysis, pole figures from the (110), (200), and (211) reflections 69 were collected at room temperature. To cover the pole sphere, 2θ = 30°~90°, Ψ = 0°~75° and φ = 70 0 to 360˚ in 5˚ increments. Iron alpha (A2 phase) is expected to be contained. The raw XRD data 71 were processed using Bruker DIFFRAC.TEXTURE software (the harmonic series expansion 72 method, thirty-four series rank L = 34, orthotropic sample symmetry). Then the data were analyzed 73 by the MATLAB toolbox MTEX to obtain the orientation distribution function (ODF) plots [17]. 74 The magnetic characterization was carried out on the superconducting quantum interference device 75 - vibrating sample magnetometer (MPMS-SQUID VSM-094) at 300 K. The magnetostriction was 76 tested at 300 K, with standard strain gauges which were pasted on samples and parallel to rod 77 growth direction. Longitudinal and transverse measurements were taken by applying the magnetic 78 field parallel and transverse to the strain gauges respectively, while keeping the magnetic field in 79 the slab plane. Strain gauges are BX120-1AA (120 Ω resistance, sensitivity coefficient 2.08 ± 80 0.01), produced by Ningbo YaoNan Electromechanical Equipment Co. (Ningbo, China). The 81 magnetic domain was observed by magnetic force microscopy (MFM, Bruker Innova) at room 82 temperature and pictures were exported by NanoScope Analysis. The MFM samples were firstly 83 polished by silicon carbide sandpapers and at final treatment by electrolytic polishing. 84 Fig.1(a) presents XRD patterns of directionally solidified (Fe0.83Ga0.17)100−xAgx (x = 0, 0.2) 85 polycrystalline samples. Only A2 phase is detected for both samples, and peaks from phases 86  4  related to Ag are not manifested, like Fe-based ribbons doped with 1 at.% Ag [18]. The scheme of 87 the relationship between specimen coordinate system (the rolling direction (RD), the transverse 88 direction (TD), and the rolling plane normal direction (ND)) and crystal growth direction is 89 illustrated in Fig.1 (b). Fig. 1(c) and 1(d) show ODF plots of (Fe0.83Ga0.17)100−xAgx (x=0, 0.2) 90 samples from XRD texture analysis. The orthotropic sample symmetry was imposed on the 91 calculation, mainly due to the symmetry of the magnetostriction tests along three directions, i.e. 92 RD, TD, and ND. The ODF plots demonstrate that, by doping with Ag, sample texture changes 93 from near Goss texture {011}<100> to near Cube texture {001}<100>. Since the “cube texture”, 94 {001}<100>, is better than Goss texture {011}<100> for enhancing magnetostriction [19], it is 95 expected that the formation of a preferred texture presented by ODF plots enhances 96 magnetostrictive properties [20]. In consideration of the increase of intensity in ODF plots far from 97 Goss texture and Cube texture, the total effect of texture change by Ag doping still needs to be 98 studied. 99  5   100 Fig. 1 (a) X-ray diffraction patterns of directionally solidified (Fe0.83Ga0.17)100−xAgx (x = 0, 0.2) samples at room 101 temperature. (b) the scheme of the relationship between specimen coordinate system (RD, TD, and ND) and crystal 102 growth direction. (c) ODF plot (φ2 = 45°) of directionally solidified Fe0.83Ga0.17 sample at room temperature. (d) ODF 103 plot (φ2 = 45°) of directionally solidified (Fe0.83Ga0.17)99.8Ag0.2 sample at room temperature. 104 Magnetostriction curves of two samples are shown in Fig. 2(a) and 2(b). 𝜆∥ is measured along 105 RD with the magnetic field along RD, while 𝜆⊥ is measured along RD with the magnetic field 106 along TD. For Fe0.83Ga0.17, magnetostrictions at H = 3000 Oe (magnetization is saturated and the 107 direction is parallel to the applied magnetic field, which will be shown in the following Fig. 3) are 108 𝜆∥ = 120 ppm, 𝜆⊥ = -52 ppm and 𝜆𝑠 = 172 ppm. For (Fe0.83Ga0.17)99.8Ag0.2, magnetostrictions at H 109 = 3000 Oe are 𝜆∥ = 100 ppm, 𝜆⊥ = -130 ppm and 𝜆𝑠 = 230 ppm. By doping with Ag, saturation 110 magnetostriction 𝜆𝑠 increases.  111  6   112 Fig. 2 Magnetostrictions of directionally solidified (Fe0.83Ga0.17)100−xAgx samples at 300 K: (a) x = 0, (b) x = 0.2. 113 Anisotropy of many properties can be evaluated from the knowledge of the single-crystal 114 tensors and ODF of crystals in polycrystals [21]. Magnetostriction in polycrystalline textured 115 materials can be described by a 4-rank tensor λ𝑖𝑗𝑘𝑙(g) defined in the specimen reference frame [22]. 116 Here, g = (φ1, Φ, φ2), describing the crystallographic orientation. ODF, which will be denoted by 117 f(g), represents the normalized probability density that quantifies the occurrence of the 118 crystallographic orientation, g. Then the Voigt average of a specimen is [22],  119  λijkl = ∮ λijkl(g)f(g)dg  (1) 120 Furthermore, λijkl(g) can be expressed by the single crystal constants 𝜆𝑚𝑛𝑜𝑝0  [23], 121  λijkl(g) = λmnop0 ⋅ gim ⋅ gjn ⋅ gko ⋅ glp (2) 122 and gij is rotation. Hence, Eq. (1) can be written as [23], 123  λijkl̅̅ ̅̅ ̅ = λmnop0 ⋅   Tijklmnop̅̅ ̅̅ ̅̅ ̅̅  (3) 124 where the quantities 𝑇̅ depend only on the texture [23], 125  Tijklmnop̅̅ ̅̅ ̅̅ ̅̅ = ∮ gim ⋅ gjn ⋅ gko ⋅ glp ⋅ f(g) ⋅ dg (4) 126 For a single cubic crystal at the ideal demagnetized state, the magnetostriction tensor matrix is [24],  127  7   𝜆𝑚𝑛𝑜𝑝0 =|||λ100 −12λ100 −12λ100 0 0 0−12λ100 λ100 −12λ100 0 0 0−12λ100 −12λ100 λ100 0 0 00 0 034λ111 0 00 0 0 034λ111 00 0 0 0 034λ111||| (5) 128 So, we can predict magnetostriction of polycrystals grown by directional solidification at the ideal 129 demagnetized state. For Fe0.83Ga0.17 alloy, λ100 = 200 ppm, λ111 = -11 ppm [25]. Calculated by the 130 function calcTensor in MTEX [17, 21] (a Matlab toolbox) with ODF of Fe0.83Ga0.17 sample, λ|| = 127 131 ppm, and λ⊥ = -51 ppm, which is consistent with the experimental results. The calculated saturation 132 magnetostriction 𝜆𝑠  (= 178 ppm) is close to the experimental result of 172 ppm with a 3.5% 133 difference. This means that the calculation is fairly successful for saturation magnetostriction 𝜆𝑠, 134 and the sample is close to the ideal demagnetized state. Thus, magnetostriction of samples behaves 135 like the Voigt average of polycrystalline aggregates. It also has been reported that Fe-Ga films 136 grown on glass display a strong Voigt-type elastic behavior [26]. The magnetostriction enhancement 137 may originate from not only texture change induced by Ag doping, but also changes of λ100 and 138 λ111. To distinguish the effect of texture change on magnetostrictive properties, induced by Ag 139 doping, λ100 and λ111 of Fe0.83Ga0.17 alloy are put into calculation with ODF of (Fe0.83Ga0.17)99.8Ag0.2 140 sample. We obtain that λ|| = 100 ppm, λ⊥ = -42 ppm, and 𝜆𝑠 = 142 ppm. Comparing two calculation 141 results of 𝜆𝑠 (142 ppm is 79.8% of 178 ppm), it is concluded that the total effect of texture change 142 induced by doping with 0.2% Ag actually deteriorates the magnetostrictive properties of FeGa 143 alloy.  144 We can also try to get a prediction of λ100 from our experimental results. Actually, predictions 145 of λ|| and λ⊥ are 𝜆1111̅̅ ̅̅ ̅̅ ̅ and 𝜆1122̅̅ ̅̅ ̅̅ ̅. So, 146  𝜆1111̅̅ ̅̅ ̅̅ ̅  −  𝜆1122̅̅ ̅̅ ̅̅ ̅  =  𝜆𝑠 (6) 147 If components of 0 in magnetostriction tensor 𝜆𝑚𝑛𝑜𝑝0  are changed into a negligible value, e.g., 0.1, 148 magnetostriction tensor matrix has an inverse matrix now (using function inv in MTEX). We mark 149 a guess of magnetostriction tensor (0 changed into 0.1) as 𝜆𝑚𝑛𝑜𝑝0 ′, and its inverse matrix as 150 (𝜆𝑚𝑛𝑜𝑝0 ′)−1. We calculate magnetostriction tensor matrix of polycrystal by the function calcTensor 151 in MTEX, 152  8   𝜆𝑖𝑗𝑘𝑙̅̅ ̅̅ ̅̅ ′ = 𝜆𝑚𝑛𝑜𝑝0 ′ ⋅   𝑇𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝̅̅ ̅̅ ̅̅ ̅̅  (7) 153 Then we calculate in Matlab with the equation, 154  (𝜆𝑚𝑛𝑜𝑝0 ′)−1: 𝜆𝑖𝑗𝑘𝑙̅̅ ̅̅ ̅̅ ′ = (𝜆𝑚𝑛𝑜𝑝0 ′)−1: (𝜆𝑚𝑛𝑜𝑝0 ′ ⋅   𝑇𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝̅̅ ̅̅ ̅̅ ̅̅ ) (8) 155 where double dot is inner product between tensors defined by MTEX. 156 We evaluate the equation symbolically, 157  𝜆𝑚𝑛𝑜𝑝0 : ((𝜆𝑚𝑛𝑜𝑝0 ′)−1: 𝜆𝑖𝑗𝑘𝑙̅̅ ̅̅ ̅̅ ′) = 𝜆𝑚𝑛𝑜𝑝0 : ((𝜆𝑚𝑛𝑜𝑝0 ′)−1: (𝜆𝑚𝑛𝑜𝑝0 ′ ⋅   𝑇𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝̅̅ ̅̅ ̅̅ ̅̅ )) (9) 158 Then we get two equations which include three unknown parameters, i.e., 𝜆1111̅̅ ̅̅ ̅̅ ̅, 𝜆1122̅̅ ̅̅ ̅̅ ̅, and λ100 159 (prediction). Combining these two equations with Eq. (6), we can get values of 𝜆1111̅̅ ̅̅ ̅̅ ̅, 𝜆1122̅̅ ̅̅ ̅̅ ̅ and 160 λ100 (prediction). Using the above steps, we get values of 𝜆1111̅̅ ̅̅ ̅̅ ̅, 𝜆1122̅̅ ̅̅ ̅̅ ̅ and λ100 (prediction) of 161 Fe0.83Ga0.17 and (Fe0.83Ga0.17)99.8Ag0.2 alloys, which are summarized in Table 1. From Table 1, the 162 prediction of λ100 of Fe0.83Ga0.17 is close to the experimental value with a 3.5% difference. The 163 predictions of λ|| and λ⊥ of (Fe0.83Ga0.17)99.8Ag0.2 are quite different from experimental values. 164 This demonstrates that the initial demagnetized state is far from ideal. The prediction of λ100 of 165 (Fe0.83Ga0.17)99.8Ag0.2 is 325 ppm, which means a huge improvement (62.5%) from 200 ppm of 166 Fe0.83Ga0.17 alloy. From the analysis of the total effect of texture change and λ100, it can be 167 concluded that the main contribution to enhancement of saturation magnetostriction 𝜆𝑠  comes 168 from the enhancement of λ100 by doping with Ag. In Fig.1(a), in comparison with Fe0.83Ga0.17 169 sample, a clear asymmetry or splitting is shown in (200) peak of (Fe0.83Ga0.17)99.8Ag0.2 sample. 170 This peak splitting originates from A2 tetragonal distortion, and was also reported by Yangkun He 171 et al. in Fe0.83Ga0.17 ribbons [27] and by Yijun Chen et al. in directionally solidified 172 (Fe0.81Ga0.19)99.9Tb0.1 crystals [28]. Symmetry lowering upon ferromagnetic transition is a general 173 effect for all cubic ferromagnets [29], and symmetry in the calculations should be set as that before 174 lowering. D03 phase is excluded, because no evidence of any D03 superlattice reflections is 175 observed [30]. The modified-D03 nanoinclusions [27] are omitted because of their extremely low 176 volume fraction and complexity of distinguishing (due to similar atom scattering factors of Fe and 177 Ga atoms). So, the percentage of the tetragonal A2 phase in (Fe0.83Ga0.17)99.8Ag0.2 sample is nearly 178 100%, as shown by the XRD data. The XRD data of (Fe0.83Ga0.17)99.8Ag0.2 sample were analyzed 179 by Rietveld refinement in Supplementary Fig. S2. A2 phase is transformed into a body-centered 180 tetragonal lattice Aa phase (space group: I 4/m m m, 139) to obtain a and c values. For 181 (Fe0.83Ga0.17)99.8Ag0.2 sample, they are 2.89711 Å and 2.90719 Å, respectively. Thus, the 182  9  tetragonality, (c-a)/c = 0.003479. By doping with Ag, the tetragonality increases and is manifested 183 in the XRD data. 184 Table 1 Comparison between experimental values and predictions for λ||, λ⊥  and λ100 in Fe0.83Ga0.17 and 185 (Fe0.83Ga0.17)99.8Ag0.2 alloys 186  λ|| (experimental) λ|| (prediction) λ⊥ (experimental) λ⊥ (prediction) λ100 (experimental) λ100 (prediction) Fe0.83Ga0.17 120 122 -52 -50 200 [25, 31] 193 (Fe0.83Ga0.17)99.8Ag0.2 100 162 -130 -68 No data 325  187 Fig. 3 presents magnetization hysteresis loops of directionally solidified (Fe0.83Ga0.17)100−xAgx 188 (x = 0, 0.2) samples. From these loops, we can get saturation magnetization of Fe0.83Ga0.17, 182.4 189 emu/g, and (Fe0.83Ga0.17)99.8Ag0.2, 190.4 emu/g. The saturation magnetization of Fe0.83Ga0.17 is in 190 good agreement with that result, 184.76 emu/g reported by Lijuan Zhao et.al [32]. By doping with 191 Ag, saturation magnetization increases. The increase of saturation magnetization may be due to 192 lattice expansion, like doping with Pt [10], and Tb [33]. For Fe0.83Ga0.17 alloy, first order 193 magnetocrystalline anisotropy constant, K1 = 35 kJ/m3 [34], and as-grown stress-induced anisotropy 194 𝐾𝑢 of 1~10 kJ/m3 [35]. So, quality factor 𝑄 = 𝐾/𝐾𝑑 = 𝐾/(0.5𝜇𝑀𝑠2) ≪ 1, where 𝐾 is anisotropy 195 energy and 𝐾𝑑 is stray field energy [36, 37]. This results in in-plane domain patterns that minimize 196 stray fields at the expense of anisotropy energy [38]. In this case, denser domain walls form in the 197 surface zone and increase coercivity which is dominated by domain wall pinning in textured soft 198 magnetic materials [37]. However, since volume of surface zone is small compared to bulk material, 199 the increase in the coercive field is trivial. The inset in Fig. 3 plots the coercive fields and remnant 200 magnetizations, which increase after doping with 0.2% Ag. 201  10   202 Fig. 3 Magnetization hysteresis loops of directionally solidified (Fe0.83Ga0.17)100−xAgx (x = 0, 0.2) samples at 203 300 K. The inset plots the coercive fields and remnant magnetizations. 204 Fig. 4 (a) and 4(b) demonstrate magnetic domain patterns of directionally solidified 205 Fe0.83Ga0.17 and (Fe0.83Ga0.17)99.8Ag0.2 samples in the slab plane by MFM, respectively. In Fig. 4(a) 206 and 4(b), the horizontal direction is parallel to RD and vertical direction is parallel to TD. These 207 typical in-plain domain patterns are usually called lancet domains and observed also by magnetic-208 optic Kerr microscopy in the previous study [39]. It demonstrates that Q<<1 also applies to 209 (Fe0.83Ga0.17)99.8Ag0.2 alloy. The domain arrangement is strongly dependent on the surface 210 orientation relative to the easy magnetization directions [40]. The (110) surface only contains one 211 easy magnetization axis (i.e., [001]), and lancet domains are often observed on slightly misoriented 212 (110) surfaces. So, crystallographic orientation of observed areas in Fig. 4(a) and 4(b) are both 213 slightly misoriented (110) surfaces, which are illustrated in Fig. 4(c). The flux is transported away 214 from the surface, either to the opposite surface or towards the neighboring domains [37], which is 215 shown in Fig. 4(d). The isolated lancets as well as the short kinks in the main walls in Fig. 4 (a) 216 and 4(b) are connected with internal transverse domains [37]. The density of supplementary 217 domains is related to the tilt misorientation θ of the grains. Domain width W and length L decrease 218 inversely with increasing misorientation angle θ [39]. Thus, misorientation in Fe0.83Ga0.17 is larger 219  11  than that in (Fe0.83Ga0.17)99.8Ag0.2, so transverse domain volume increases [41, 42]. The main 220 contribution to dimension change comes from 90° domain rotation, and 180° domain wall motion 221 has no effect on dimension change [11]. Thus, a larger volume of internal transverse domain induces 222 larger magnetostriction when applied field is parallel to RD, i.e., 𝜆∥. This partly confirms that 223 initial demagnetized state of Fe0.83Ga0.17 alloy is much closer than (Fe0.83Ga0.17)99.8Ag0.2 alloy to 224 ideal demagnetization from previous discussion. Since comparison of volume of internal 225 transverse domains cannot be inferred from near (100) or (111) surfaces [37], those areas are not 226 studied further. FeGa magnetic domain patterns in our study are analogous to Fe-3%Si investigated 227 by MFM [43]. These domain patterns differ from maze-like domain patterns observed in FeGa 228 polycrystals by MFM [44], and cellular domain structure in the arc-melted Fe-19%Ga polycrystal 229 by the Bitter method [45]. 230  231 Fig. 4 Magnetic domain patterns of directionally solidified (Fe0.83Ga0.17)100−xAgx samples in the slab plane at 232 room temperature, (a) x=0, (b) x=0.2 by MFM. The horizontal direction is parallel to RD and vertical direction is 233  12  parallel to TD. (c) Internal domain structure. (d) illustrates that the flux is transported away from the surface, either 234 to the opposite surface or towards the neighboring domains. 235 In summary, based on independence of 𝜆𝑠(=  𝜆∥ − 𝜆⊥) on magnetic domain distribution, a 236 paradigm is proposed, combining ODF with magnetostriction tensor of single crystals. This 237 paradigm not only quantifies the effects of texture on magnetostriction, but also accurately predicts 238 λ100 in a trace amount of Ag-doped FeGa alloy. Given that ODFs and magnetostriction tensors of 239 phases are obtained, based on the rule of mixture relationship, our paradigm can be extended to 240 magnetostriction prediction in rare earth element-doped FeGa alloys and even other types of 241 magnetostrictive materials. Our findings may accelerate the development of highly 242 magnetostrictive alloys for sensor and actuator applications.  243  244 SUPPLEMENTARY MATERIAL 245 See supplementary material for the macrostructure investigated by optical microscopy. 246  247 Acknowledgment 248 This work was financially supported by the National Key R&D Program of China 249 (2022YFE0109500, 2021YFB3501401), Natural Science Basic Research Program of Shaanxi  250 (Program No. 2023-JC-ZD-24), the Space Science and Application Project of China Manned 251 Space Engineering (Grant No. KJZ-YY-NCL10), the Key R&D Project of Shaanxi Province 252 (2023GXLH-006), the Key R&D Project of Shaanxi Province (2022GXLH-01-07), Innovation 253 Capability Support Program of Shaanxi (Nos. 2018PT-28, 2017KTPT-04), the Fundamental 254 Research Funds for the Central Universities (China) and the World-Class Universities 255 (Disciplines), the Characteristic Development Guidance Funds for the Central Universities (China).  256  257 The authors have no conflicts to disclose. 258  259  13  The data that support the findings of this study are available from the corresponding author upon 260 reasonable request. 261 References 262 [1] KARACA H E, KARAMAN I, BASARAN B, et al. Magnetic field and stress induced martensite 263 reorientation in NiMnGa ferromagnetic shape memory alloy single crystals [J]. Acta Mater, 2006, 264 54(1): 233-45. 265 [2] CLARK A E, SAVAGE H T. Magnetostriction of rare earth-Fe2 compounds under 266 compressive stress [J]. J Magn Magn Mater, 1983, 31-34: 849–51. 267 [3] JIANG L, ZHANG G, YANG J, et al. Research on microstructure and magnetostriction of 268 Fe83Ga17Dyx alloys [J]. J Rare Earths, 2010, 28: 409–12. 269 [4] JIANG L, YANG J, HAO H, et al. Giant enhancement in the magnetostrictive effect of FeGa 270 alloys doped with low levels of terbium [J]. Appl Phys Lett, 2013, 102(22): 222409. 271 [5] LI J, XIAO X, YUAN C, et al. Effect of yttrium on the mechanical and magnetostrictive 272 properties of Fe83Ga17 alloy [J]. J Rare Earths, 2015, 33(10): 1087–92. 273 [6] ZHAO X, TIAN X, YAO Z, et al. The origin of large magnetostrictive properties of rare earth 274 doped Fe-Ga as-cast alloys [J]. J Magn Magn Mater, 2020, 514: 167289. 275 [7] MENG C Z, JIANG C B. Magnetostriction of a Fe83Ga17 single crystal slightly doped with 276 Tb [J]. Scr Mater, 2016, 114: 9-12. 277 [8] ENGLER O, RANDLE V. Introduction to texture analysis: Macrotexture, microtexture, and 278 orientation mapping [M]. 2nd ed. Boca Raton: CRC Press, 2010. 279 [9] BARUA R, TAHERI P, CHEN Y, et al. Giant Enhancement of Magnetostrictive Response in 280 Directionally-Solidified Fe83Ga17Erx Compounds [J]. Materials, 2018, 11(6): 1039. 281 [10] ZHOU C, LIU Y, CHEN K, et al. Improved magnetostriction in Galfenol alloys by 282 aligning crystal growth direction along easy magnetization axis [J]. Sci Rep, 2020, 10(1): 20055. 283 [11] HE Y, COEY J M D, SCHAEFER R, et al. Determination of bulk domain structure and 284 magnetization processes in bcc ferromagnetic alloys: Analysis of magnetostriction in Fe83Ga17 285 [J]. Phys Rev Mater, 2018, 2(1): 014412. 286 [12] LI X, BAO X, LIU Y, et al. Tailoring magnetostriction with various directions for 287 directional solidification Fe82Ga15Al3 alloy by magnetic field heat treatment [J]. Appl Phys Lett, 288 2017, 111(16): 162402. 289 [13] WU D, XING Q, MCCALLUM R W, et al. Magnetostriction of iron-germanium single 290 crystals [J]. J Appl Phys, 2008, 103(7). 291 [14] WANG H, WU R. First-principles investigation of the effect of substitution and 292 surface adsorption on the magnetostrictive performance of Fe-Ga alloys [J]. Phys Rev B, 2019, 293 99(20): 205125. 294 [15] WU Y, CHEN Y, MENG C, et al. Multiscale influence of trace Tb addition on the 295 magnetostriction and ductility of 〈100〉 oriented directionally solidified Fe-Ga crystals [J]. Phys Rev 296 Mater, 2019, 3(3): 033401. 297 [16] CAO K, DAI Z, ZHANG R, et al. Study on Magnetic Properties and the Nature of 298 Phase Transition of Laves Phase Rare Earth-Transition Metal Compounds Tb1-xDyxCo1.95 [J]. Journal 299 of Liaocheng University (Nat Sci Ed), 2024, 37(1): 29. 300  14  [17] HIELSCHER R, SCHAEBEN H. A novel pole figure inversion method: specification of 301 the MTEX algorithm [J]. J Appl Crystallogr, 2008, 41(6): 1024–37. 302 [18] LEE M C, LIN C Y, WANG S H, et al. Soft-Magnetic Fe-Based Nano-Crystalline Thick 303 Ribbons [J]. IEEE Trans Magn, 2008, 44(11): 3836-8. 304 [19] EVANS P G, DAPINO M J. Efficient model for field-induced magnetization and 305 magnetostriction of Galfenol [J]. J Appl Phys, 2009, 105(11): 113901. 306 [20] LI J H, GAO X X, ZHU J, et al. Ductility, texture and large magnetostriction of Fe–307 Ga-based sheets [J]. Scr Mater, 2010, 63(2): 246–9. 308 [21] MAINPRICE D, HIELSCHER R, SCHAEBEN H. Calculating anisotropic physical 309 properties from texture data using the MTEX open-source package [J]. Geol Soc Lond Spec Publ, 310 2011, 360(1): 175-92. 311 [22] SZPUNAR J A. Influence of crystal orientation distribution on anisotropy of 312 magnetostriction and magnetocrystalline energy in Fe‐Si steel [J]. J Appl Phys, 1984, 55(6): 2133-313 5. 314 [23] BUNGE H J. Texture and Magnetic Properties [J]. Textures Microstruct, 1989, 11: 315 727138. 316 [24] FEDERICO S, CONSOLO G, VALENTI G. Tensor representation of magnetostriction 317 for all crystal classes [J]. Math Mech Solids, 2019, 24(9): 2814-43. 318 [25] CLARK A E, RESTORFF J B, WUN-FOGLE M, et al. Magnetostrictive properties of 319 body-centered cubic Fe-Ga and Fe-Ga-Al alloys [J]. IEEE Trans Magn, 2000, 36(5): 3238–40. 320 [26] RAMíREZ G A, RIFFO A E M, GóMEZ J E, et al. Magnetoelasticity of Fe1-xGax thin 321 films on amorphous substrates [J]. Phys Rev B, 2021, 104(6): 064403. 322 [27] HE Y, JIANG C, WU W, et al. Giant heterogeneous magnetostriction in Fe–Ga alloys: 323 Effect of trace element doping [J]. Acta Mater, 2016, 109: 177–86. 324 [28] CHEN Y, WANG J, YANG B, et al. Morphology evolution and enhanced 325 magnetostriction of (Fe0.81Ga0.19)99.9Tb0.1 crystals prepared by liquid metal cooling Bridgman 326 directional solidification [J]. J Alloys Compd, 2021, 856: 158166. 327 [29] YANG S, REN X. Noncubic crystallographic symmetry of a cubic ferromagnet: 328 Simultaneous structural change at the ferromagnetic transition [J]. Phys Rev B, 2008, 77(1): 329 014407. 330 [30] LOGRASSO T A, ROSS A R, SCHLAGEL D L, et al. Structural transformations in 331 quenched Fe–Ga alloys [J]. J Alloys Compd, 2003, 350(1): 95–101. 332 [31] ATULASIMHA J, FLATAU A B, SUMMERS E. Characterization and energy-based 333 model of the magnetomechanical behavior of polycrystalline iron-gallium alloys [J]. Smart Mater 334 Struct, 2007, 16(4): 1265-76. 335 [32] ZHAO L, TIAN X, YAO Z, et al. Enhanced magnetostrictive properties of lightly Pr 336 doped Fe83Ga17 alloys [J]. J Rare Earths, 2019, 38(3): 257-64. 337 [33] WU W, LIU J, JIANG C, et al. Giant magnetostriction in Tb-doped Fe83Ga17 melt-338 spun ribbons [J]. Appl Phys Lett, 2013, 103(26): 262403. 339 [34] RAFIQUE S, CULLEN J R, WUTTIG M, et al. Magnetic anisotropy of FeGa alloys [J]. 340 J Appl Phys, 2004, 95(11): 6939–41. 341 [35] RESTORFF J B, WUN-FOGLE M, CLARK A E, et al. Induced Magnetic Anisotropy in 342 Stress-Annealed Galfenol Alloys [J]. IEEE Trans Magn, 2006, 42(10): 3087-9. 343 [36] MUDIVARTHI C. Magnetic and structural characterization of iron-gallium using 344 Kerr microscopy and neutron scattering [D]; University of Maryland, College Park, 2010. 345 [37] HUBERT A, SCHäFER R. Magnetic domains: the analysis of magnetic 346 microstructures [M]. Heidelberg: Springer, 1998. 347  15  [38] MUDIVARTHI C, NA S-M, SCHAEFER R, et al. Magnetic domain observations in Fe–348 Ga alloys [J]. J Magn Magn Mater, 2010, 322(14): 2023–6. 349 [39] SCHäFER R, SCHINNERLING S. Tomography of basic magnetic domain patterns in 350 ironlike bulk material [J]. Phys Rev B, 2020, 101(21): 214430. 351 [40] BATISTA L, RABE U, HIRSEKORN S. Characterization of the magnetic micro- and 352 nanostructure in unalloyed steels by magnetic force microscopy [J]. AIP Conf Proc, 2013, 1511(1): 353 1180-7. 354 [41] HE Z, SHA Y, ZHAO Z, et al. The calculation of magnetic domain and 355 magnetostriction in stressed grain-oriented silicon steel [J]. J Appl Phys, 2020, 127(3). 356 [42] SHILLING J, HOUZE G. Magnetic properties and domain structure in grain-oriented 357 3% Si-Fe [J]. IEEE Trans Magn, 1974, 10(2): 195-223. 358 [43] SHIN S, SCHäFER R, COOMAN B C D. Grain Boundary Penetration by Lancet 359 Domains in Fe-3%Si Grain-Oriented Steel [J]. IEEE Trans Magn, 2010, 46(9): 3574-81. 360 [44] ZHOU J K, LI D D, LI J G. Magnetic force microscopy observation of undercooled 361 Fe81Ga19 magnetostrictive alloys [J]. J Phys D: Appl Phys, 2008, 41(20): 205405. 362 [45] TIANEN M N, SEGUIN D J, WANG Y U, et al. Cellular magnetic domains in Fe–Ga 363 alloys [J]. Appl Phys Lett, 2020, 117(3): 032401. 364  365