theoretical investigation on site preference of...

LJournal of Alloys and Compounds 319 (2001) 62–73www.elsevier.com/ locate / jallcom

Theoretical investigation on site preference of foreign atoms in rare-earthintermetallics

a , a a,b c*Y. Wang , J. Shen , N.X. Chen , J.L. WangaInstitute of Applied Physics, Beijing University of Science and Technology, Beijing, 100083, China

bDepartment of Physics, Tsinghua University, Beijing, 100084, ChinacState Key Laboratory of Magnetism, Institute of Physics, Chinese Academy of Sciences, Beijing, 100080, China

Received 22 November 2000; accepted 4 January 2001

Abstract

This paper adopts the lattice inversion method in acquiring the interatomic potentials related to rare-earth and transition metals andcharacterizes the site occupancies of Fe and Co in Y(Co,Fe) , Y (Co,Fe) , Y (Fe,Co,M) , and YTi(Fe,Co) by combining these5 2 17 3 29 11

potentials with computer simulation. The results of this research are in good agreement with experimental data. The substitution behaviorsand the energy variation introduced by alloying elements M (M5V, Mo, Cu, Ni) in YTi(Fe,M) are also calculated. A special importance11

is laid on the statistical analysis of the structure of YTi(Fe,Co) . This research is the first to present the phonon density of states of 1:1211

structures, and it also carries out a qualitative analysis featuring the coordination and the relevant potentials. 2001 Published byElsevier Science B.V.

Keywords: Rare earth intermetallic compounds; Theory and modeling; Site occupancy

1. Introduction require parameterized fittings. The substitution behaviorshave also been deduced from empirical laws, such as

For many iron rare-earth based compounds, replacing atomic radius, atomic magnetic moment and formationiron with cobalt can effectively enhance the magnetic enthalpy [1,7]. In some cases, the site distribution can beproperties, especially the Curie temperature and saturation derived inversely from the knowledge of structure–mag-magnetization [1]. For instance, much attention has been netism correlation. All these come down to the effect ofdirected to RFe Ti for its high T and anisotropy field. local atomic environment on the substitution. It is a11 C

However, the relatively lower saturation magnetization shortcut to explore site preference from the viewpoint oflimits the maximum magnetic energy product. The substi- energy through computer simulation, which combinestution of Co for Fe increases its magnetization and T at interatomic potentials with different crystal structures.C

the cost of the anisotropy [2–6]. Therefore, many efforts Methods of the kind have been widely applied to the studyhave been dedicated to the study of the relation between of metals and intermetallics. It is the structural complexitythe Co occupation in the 3d sublattice and the anisotropy and particularly, the difficulties in acquiring potentials ofin Y–Ti–Fe–Co system. Similar investigations have also rare-earth compounds that frustrates attempts to adopt thisbeen conducted on R(Co,Fe) [7,8], R (Co,Fe) [9] and method.5 2 17

R (Fe,Co) M [10], in which the determination of Co This paper applies the lattice inversion formula to some3 292x x

site preference is a must. Experimentally, due to the small virtual crystal structures, such as face centered cubic (fcc)atomic weight difference between Fe and Co, X-ray yttrium, acquiring the potentials involved with rare-earthpowder diffraction (XRD) is incapable of distinguishing and transition metals. Disregarding magnetic effect, withbetween the two sites. Techniques involved in previous these effective potentials, our research is directed to the

¨studies are mainly neutron diffraction, Mossbauer spec- site occupancies of Fe and Co in Y(Co,Fe) , Y (Co,Fe) ,5 2 17

troscopy and nuclear magnetic resonance, all of which Y (Fe,Co,M) and YTi(Fe,Co) , as well as some other3 29 11

elements in Y–Fe–Co–M. As is believed, it initiates a newpattern to explore the structures of complex materials.*Corresponding author.

E-mail address: [email protected] (Y. Wang). Emphasizing the statistical analysis of the structure of

0925-8388/01/$ – see front matter 2001 Published by Elsevier Science B.V.PI I : S0925-8388( 01 )00909-4

Y. Wang et al. / Journal of Alloys and Compounds 319 (2001) 62 –73 63

YTi(Fe,Co) , this is the first attempt to present the phonon where the inversion coefficients I(n) are given by11

density of states (DOS) of 1:12 structures. Qualitativeb ks d21analysis is carried out using the coordination and the ]]S F GDO I n r b 5 d (5)s d k1b ns db(n)ub(k)relevant potentials. Section 2 is a brief introduction to the

lattice inversion method, Section 3 relates the potentialI(n) relates only to the geometrical structure of a crystalacquisition and the calculation procedure, Section 4 con-but not to the element type. This inversion method can alsotains the results and Section 5 the calculation and analysisbe used for the acquisition of interatomic pair potentialsof phonon DOS.between distinct atoms. With the contribution of theinteratomic potentials between identical atoms subtractedfrom total cohesive energy, the partial cohesive energy of

2. Lattice inversion method the distinct atoms is obtained, and then the interatomic pairpotentials between distinct atoms can be determined via

The acquisition of interatomic potentials prepares the lattice inversion [12,13].groundwork for this research. Acting on the lattice inver-sion method developed by Chen [11], we can do withoutcomplicated fitting and parameter adjustment when obtain-

3. Acquisition of effective potentials and theing the interatomic potentials. This method is summed asmethodologyfollows: it assumes that the cohesive energy per atom in a

perfect crystal can be expressed as the sum of pairIn principle, interatomic potentials can be acquiredpotentials, i.e.

entirely within the framework of first-principle (FP), i.e.`1 acquiring F(x) by inverting FP calculated cohesive energy]E x 5 O r n F b n x (1)s d s d s ds d0 02 E(x), without dealing with any external data input orn51

presupposed function form. As for the particular systemwhere x is the nearest-neighbor distance, F the pair

discussed in this paper, the treatment is different in view ofpotential function, r (n) the nth neighbor coordination0 the difficulties that current FP methods encounter when itnumber and b (n)x the distance of the nth neighbor from0 comes to dealing with iron and cobalt. Many of the densitythe reference atom, b (1)51. Extend the series hb (n)j to a0 0 functional methods, such as ASW [14,15] and planewavemultiplicative closed semigroup hb(n)j, in which, for any

pseudopotentials [16], lead to a relative magnitude intwo integers m and n, there exists an integer k so that

cohesive energy which is inconsistent with experimentalb(k)5b(m)b(n). Thus

data (see the last column in Table 1 for example). Since` potentials related to Fe and Co play the most important1

]E x 5 O r n F b n x (2)s d s d s s d d role in the competing occupation of Fe and Co, it is not2 n51advisable to use interatomic potentials derived from inac-

where curate prediction about the cohesive energy. Therefore, weemploy the experimental data listed in Table 1 to obtain21r b b n b n [ b ns df s d g s d s dh j0 0 0 the potentials between identical atoms through an inversionr n 5 (3)s d H0 b n [⁄ b ns d s dh j0 process as discussed in Section 2.

As mentioned, some virtual structures are devised forIn the above extension, we have to insert some virtualthis research. The hcp structure of Ti, Co and Y are alllattice point in r(n). Then the pair potential F(x) can bereplaced by isometric fcc, from which the inversionobtainedcoefficient I(n) is constructed. According to our ex-

`

periences, pair potentials obtained, respectively from hcpF x 5 2O I n E b n x (4)s d s d s s d dn51 and fcc are almost similar to each other although the

Table 1Experiment data used in the inversion method [17]

Elements Crystal structures and lattice constants E B Calculated Ecoh coh11 2 a(eV/at.) (10 N/m ) (eV/at.)

˚ ˚a , c (A) a (A) used in (0 K, 1 atm.) (room temperature)0 0 0

the inversion

Y hcp (3.65, 5.73) fcc (5.090) 4.37 0.366 4.46 (4.91)Fe bcc (2.87) bcc (2.870) 4.28 1.683 6.62 (6.39)Ti hcp (2.95, 4.68) fcc (4.132) 4.85 1.051 5.90 (6.36)Co hcp (2.51, 4.07) fcc (3.538) 4.39 1.914 5.82 (6.36)

a Calculation results of this work using ASW [14,15]. In parenthesis are calculation results from LAPW [20].

64 Y. Wang et al. / Journal of Alloys and Compounds 319 (2001) 62 –73

semigroup b(n) of hcp is much more complicated, and can view of the above-mentioned difficulties, the Fe–Cobe commuted when we do not concentrate on the phase- potential is acquired with simplified geometrical averagingsensitive properties. The original inverted results are a [18,19]series of points, which, for the convenience of sequence ]] ]] ]D 5 D D , R 5 R R , g 5 ggprocess, are rendered into a fitting using Morse function 0ij 0i 0j 0ij 0i 0j ij i jœ œ œ

2 2a (R2R )0F r 5 D [u 2 2u], u 5 e , g 5 2R a (6)s d It is noteworthy that the limitations of DFT in predicting0 0

cohesive energy of Fe and Co also exist in other theoreticalIn Fig. 1 and Table 2 are the fitting results. work [20]. Compared with experimental data, results ofDifferent from the FP potentials in previous work [11], cohesive energy given by DFT are larger in absolute value

this research adopts semiempirical effective potentials. In and inconsistent in relative values, owing to the problems

Fig. 1. Related potential curves in Y–Fe–Ti (a) and Y–Fe–Co (b) systems.


Table 2 initial structure alloyed with ternary or quaternary ele-Part of the potential parameters acquired from the inversion method ments. The contents of different alloying elements are

˚Potential types R (A) D (eV) g obtained from their random distribution in the parent0 0

compounds. All these configurations are relaxed and theY–Y 4.5329 0.3150 6.7554Fe–Fe 2.8589 0.4287 7.8589 obtained energy and lattice parameters are averaged on atTi–Ti 3.3951 0.4301 7.3695 least 20 sample units. The averaged energy is adopted as aCo–Co 2.7808 0.4451 7.9613 criterion for site preference. The calculation unit ofY–Fe 3.5999 0.3673 7.2863 YTi(Fe,Co) is a 23234 cell (416 atoms in total)11Y–Ti 3.9230 0.3681 7.0558 expanded from ThMn unit cell (see Fig. 2). Only one12Y–Co 3.5504 0.3744 7.3336 concentration is simulated for (YCo M ) (M5Ti,4.875 0.125 8Fe–Ti 3.1155 0.4294 7.6103

Fe, Ni, Cu, Mo, Al, B), hexagonal Y Co Fe and2 16Fe–Co 2.8196 0.4368 7.9099Y Fe M (M5Co, Ni, Cu, Ti, V, Mo).Ti–Co 3.0726 0.4376 7.6597 3 28

arising from the application of current band theory to thetreatment of 3d electrons and magnetism. Generally speak- 4. Substitution behaviors of alloying elements in rare-ing, two-body potentials have drawbacks in that they lead earth transition metal compoundsto a false Cauchy relation. Nevertheless, pair potentialsreproduce cohesive energy, lattice constant, bulk modulus 4.1. Atomic occupancy in YCo M52x x

and some other simple properties very well. It is feasible toemploy pair potentials for getting the site preference RM , R M , and R M can be derived from the12 2 17 3 29

information from complex structure, as is justified by the ordered replacement of certain proportion (1 /2, 1 /3, 2 /5,good agreement between our theoretical calculations and respectively) of R atoms in CaCu structure by a couple of5

experiments. Due to the limitation of FP calculations, the M atoms, the so-called dumbbells. So the 1:5 structure isacquisition of cross potentials is simplified for expedience. the first target of our investigations. Table 3 lists theIn principle, cross potentials in alloys can be obtained calculated lattice constants and corresponding experimentaldirectly from available cohesive energy by means of the data of RCo (R5Y, Nd, Sm). In the hexagonal 1:55

strict and concise inversion formula. structure, transition metals are seated in 2c and 3g sites.The conjugate gradient method is used to relax the Because Fe and Co bear similarities in many aspects of

their properties, it is not easy to identify the differences intheir cohesive properties by means of effective potentials.Whereas the inverted potentials obtained by the inversionmethod correctly define the site selectivity of Fe, Ni, Al, B,Ti and Mo in Co based compounds (see Table 4). Theentry of Fe and Ni into Co sublattice brings about onlyslight lattice relaxation, which can be expected from theresemblance in potential curves. According to our calcula-tions, B replacing 2c is preferable to B replacing 3g, whileAl, which is in the same group with B, prefers to the 3gsite. Both are in accordance with the experimental evi-dence that R–Co–B can be synthesized into ordered alloyssuch as R Co B and R Co B [23], while11m 513m 2m m11 5m13 2

Al has different site preference and no such ordered alloyscan be formed [22]. The preference of Cu in 2c contradicts

Table 3Calculated lattice parameters and cohesive energy in RCo (R5Y, Nd,5

Sm). In parenthesis are the corresponding experiment data [21]

˚ ˚RCo a (A) c (A) a /c E (eV/at.)5 coh

Y 5.124 3.996 1.282 24.269(4.935) (3.964) (1.245)

Nd 5.094 4.029 1.264 24.010(5.028) (3.977) (1.264)

Sm 5.019 4.086 1.228 23.594(5.002) (3.964) (1.262)

Fig. 2. Crystal structure of ThMn -type compound.12


Table 4Atomic occupancies of M in (YCo M ) ; the left and right sides of4.875 0.125 8

‘→’ represent the site preferences evaluated from the unrelaxed energyand relaxed energy, respectively

M Preferred site Experiment

Ti 3g→3g 3g [7]Fe 3g→3g 3g [7]Ni 2c→2c 2c [7]Cu 2c→2c 3g [7]Mo 3g→3g 3gAl 2c→3g 3g [7,8,22]B 2c→2c 2c [8]

with experimental knowledge, although the reason isunknown yet.

As is well known, the stable binary R–Fe compoundswith CaCu structure do not exist. However, its kinship5

with 1:12, 2:17 and 3:29 cannot be ignored. Comparedwith two available related experiments, one for YFe [25]5

extrapolated from ternary compound and the other forSmFe film stabilized by oxygen [24], our results are5

tenable (see Table 5). The relatively large deviation in ratioa /c is attributed to the incapacity of pair potentials fordefining the anisotropic properties.

4.2. Y Co Fe and Y Fe M (M5Co, Ni, Cu, Ti, V, Mo)2 16 3 28

Fig. 3. Energy variation in Y Fe M (M5Co, Ti) with M occupying3 28

In 2:17, both experiments [9] and our calculations show different crystalline sites (‘unrelaxed’ represents the curves beforerelaxation, and ‘relaxed’ represents those after relaxation).that Fe preferentially occupies the dumbbell sites, 4f in

hexagonal Y Co and 6c in rhombohedral Y Co .2 17 2 17

Whereas in 1:12 and 3:29, Fe exhibits a tendency to repel the unfavorable sites for Co. These results agree very wellsuch sites; it is Ti, V, Mo that occupy these sites. In with the experiments. In this aspect,V and Mo resemble Ti,monoclinic 3:29 R–Fe, experiments show that Co in 8j site and Ni and Cu are like Co.(0.8,0.248,0.346) gives the largest occupancy fraction, Coin 2c site (0.5,0.5,0.5) in the next place [10]. So far the 4.3. YFe Ti, YCo Ti, YTi(Fe Co ) atomic11 11 112x x

attempt to completely replace Fe with Co is still underway. occupancy and structural analysisWith the concept that, in the quaternary compound such asY (Ti,Fe,Co) , the interaction between Ti and Co in small 4.3.1. YFe Ti, YCo Ti3 29 11 11

concentration is not significant, we use the hypothetical With cohesive energy as a criterion, Ti occupies 8i sites.model of Y Fe Co to study the substitution of Co in3 28

Y Fe characterized by C2/m space group (see Fig. 3).3 29 Table 6We can see that the preferential sites are Fe1(2c), Fe8(8j), Crystalline sites and the coordinates in calculated Y Fe compound after3 29

Fe11(4e) for Co and Fe2, Fe3, Fe6 for Ti (the corre- structure relaxation, which is characterized by C2/m space groupspondence between names and coordinates is shown in Atom Sites x y yTable 6). Further, Fe2, Fe3, Fe6 preferential for Ti are just

Y1 2a 0.000 0.000 0.000Y2 4i 0.404 0.000 0.815

Table 5 Fe1(Co) 2c 0.500 0.000 0.500Calculated lattice parameters in RFe (R5Y, Nd, Sm); the corresponding Fe2(Ti) 4i 0.862 0.000 0.7065

experiment data are shown in parenthesis Fe3(Ti) 4i 0.745 0.000 0.477Fe4 8j 0.802 0.218 0.091˚ ˚RFe a (A) c (A) a /c E (eV/at.)5 coh Fe5 8j 0.628 0.356 0.185

Y 5.192 4.089 1.270 24.198 Fe6(Ti) 4g 0.000 0.638 0.000(5.03) (4.17) (1.206) [25] Fe7 4i 0.113 0.000 0.728

Fe8(Co) 8j 0.804 0.752 0.348Nd 5.165 4.123 1.253 23.942

Fe9 4i 0.288 0.000 0.084Sm 5.094 4.182 1.218 23.531 Fe10 8j 0.589 0.753 0.425

(4.831) (4.342) (1.113) [24] Fe11(Co) 4e 0.000 0.250 0.750


Although Co alone cannot stabilize RFe , the calculation competition when more than one element is added. In light12

shows that Co, when entering 8f sites, gives rise to lower of the above analysis, we preset Ti in the 8i site for itsenergy in YFe Co . However, Ti lowers the energy in a strong i site preference. The variation of energy is calcu-122x x

much larger degree than Co does, which indicates that Ti lated in five segments (Table 7). The results indicate thatstabilizes more than Co does (Fig. 4). Ti, V and Mo have Co occupies the f site when 1 # x # 4, j site when 4 , x #

the same substitution behaviors in Co-based as in Fe-based 8, and then i site (see Fig. 5). It is shown in neutroncompounds. All these results are in good agreement with experiments that Co preferentially enters j and f sites with

¨previous work [2,26]. a larger fraction at j site [5]. Mossbauer spectroscopy inSmFe Co Ti gives 8f site preference [27]. The cases of102x x

mixed occupancy in the crystalline sites are not taken into4.3.2. YTi(Fe Co ) consideration in our calculations. Different substitution112x x

Compared with ternary system, the substitution in behaviors of V, Mo, Ni and Cu in YTiFe M and Co in112x x

quaternary system is more complex, in that there may exist SmTiFe Co , NdTiFe Co are also calculated, with112x x 112x x

Fig. 4. Energy variation with ternary element content x in YFe Ti (a) and YFe Co (b).122x x 122x x


Table 7Calculation steps when dealing the Co site preference in YTiFe Co112x x

Parent compounds y x Possible sites Site preference

YTiFe Co (0, 3] [0,3] (3 /4)i, (3 /4)j, (3 /4)f f . j . i112y y

YTiFe Co (0, 1] (3, 4] (1 /4)i, (1 /4)j, (1 /4)f f . j82y 31y

YTiFe Co (0, 3] (4, 7] (3 /4)i, (3 /4)j j . i72y 41y

YTiFe Co (0, 1] (7, 8] (1 /4)i, (1 /4)j j42y 71y

YTiFe Co (0, 3] (8, 11] (3 /4)i i32y 81y

Fig. 5. Energy variation with quaternary element content x in YTiFe Co .112x x

the results listed in Table 8. All the quaternary additions distances caused by alloying elements exerts an influenceon magnetism. Table 10 presents a statistical analysis ofexcept Cu will decrease the energy of the parent ternarythe distances between different crystalline sites. The ordersystems. Therefore, among these elements, Cu may be theof the intrasite bond distances is d .d .latest candidate for alloying element in view of the phase Fei–Fei Fej–Fej

d , and the average distances between coordinatedstability. Table 9 lists the calculated lattice constants in Fef–Fef

transition metal atoms and each site are ordered as d .several ternary compounds. The differences between ex- Fei–T

d .d . Generally, the exchange interaction ofperiments and calculations all fall within 2%. Fej–T Fef–T˚bonds longer than 2.8 A can be ignored. Thus, only bonds

˚4.3.3. Variation of interatomic distances shorter than 2.8 A are listed in Ref. [2]. In fact, theCurie temperature in 1:12 compounds is mainly de- immediate neighbors to 8i sites fall in two groups: 2.39

˚termined by the exchange correlation interaction of T–T and 2.94 A (see Fig. 6); the negative interaction of the firstwithin transition metal sublattice, which closely relates to group complicates the case. As is clear in Table 10, thethe T–T interatomic distances. The variation of bond addition of Ti results in a lattice expansion as well as a

Table 8Table 9

Calculated site preferences of M in R(TiFe M) quaternary systems11 Calculated lattice parametersR(TiFe M) E (eV/at.) Site Energy11 coh ˚ ˚a (A) c (A)

preference variationi site j site f site Exp. Cal. Exp. Cal.

YTiFe V 24.4279 24.4004 24.3825 i Decrease YFe Ti 8.537 [5] 8.649 4.805 4.86710 11

YTiFe Ni 24.3373 24.3436 24.3484 f Decrease YCo Ti 8.391 [5] 8.504 4.725 4.78310 11

YTiFe Cu 24.2662 24.2754 24.2786 f Increase YFe V 8.474 [3] 8.631 4.761 4.83910 10 2

YTiFe Mo 24.5351 24.4730 24.4476 i Decrease YFe Mo 8.563 [28] 8.724 4.810 4.81810 10 2

SmTiFe Co 23.9999 24.0006 24.0110 f Decrease SmFe Ti 8.568 [29] 8.621 4.798 4.86410 11

NdTiFe Co 24.2069 24.2129 24.2178 f Decrease NdFe Ti 8.577 [30] 8.643 4.805 4.87010 11


Table 10Statistical analysis of different Fe–Fe interatomic distances in relaxed crystalline structures. In parenthesis are the nearest neighbor distances in thecorresponding structures

i–i i–j i–f j–j j–f f–f

YFe 2.833 2.669 2.633 2.701 2.468 2.40912

(2.393)

YCo 2.776 2.608 2.585 2.675 2.421 2.36112

(2.343)

YFe Ti 2.883 2.714 2.631 2.668 2.498 2.43411

(2.417)aYTiFe Co ( f ) 2.878 2.708 2.629 2.666 2.497 2.43410

( j) 2.880 2.707 2.627 2.670 2.495 2.432

YFe Ti [2] 2.3900 2.6627 2.6359 2.7112 2.4681 2.381811

a In fact, the contribution of Fe–Co, Co–Co bonds to magnetism are not negligible; f, j represent the case of Co occupying the f and j sites, respectively.

decrease of j–j distance, and the addition of Co to YTiFe between the contributions from acoustic branch and11

optic branch, and the relative motion between nonequi-brings about an overall lattice contraction. The distributionvalent Fe sites also contributes a wide frequency band.of coordinated atoms in YFe shows that, of 8j, 8f and 8i,12

2. It is inferred from the partial DOS that the modes abovethe first two sites bear a greater similarity in their local8 Thz are excited by Fe alone: within the range of 8–9environments, which may account for the exhibition, moreTHz, the contributions from 8j and 8f sites, almostoften than not, of two kinds of substitution behaviors: 8iequal, are greater than that from 8i site; and above 9site and mixed 8j and 8f sites, respectively.THz, the modes related to 8i site vanish and thecontribution from 8f site is dominant.

3. High frequency modes from Y–Fe only exist in the5. Phonon DOS of hypothetical binary 1:12vicinity of 7.6 THz, and this small contribution corre-lates to the great atomic weight of Y and the large Y–FePhonon density of states (DOS) reflects the lattice

˚vibration properties, and the derived specific heat and interatomic distance (.3 A).Debye temperature are important thermodynamic parame-ters. Starting with the effective potentials and lattice The DOS of Fig. 7 is calculated by adding up 936 kdynamics theory, we have calculated the total DOS, as points in 1 /8 the first Brillouin zone of simple tetragonal

˚well as the partial DOS projected to different elements and lattice. The cut-off distance of force constants is set at 6 A.crystalline sites in YFe and YCo . It is estimated that With only nearest neighbor interactions considered12 12

˚(r 53.5 A), the calculated DOS is shown in Fig. 8,cutoff

1. The cut-off frequencies of the vibration modes are which also illustrates the main characters of vibrational10.74 THz (YFe ) and 10.80 Thz (YCo ), respective- modes. It shows that the nearest neighbor approximation is12 12

ly, with two peaks at 3 THz and 7.5 THz and two acceptable, justifying the qualitative analysis using coordi-local-mode-like small peaks above 8 THz. The DOS is nation numbers and potential curves. It can be deducedextended due to the relatively low symmetry and the from the potential curves that the force constant decreases

˚various interatomic distances. It is hard to distinguish monotonically within 3.35 A. Accordingly, the knowledge

Fig. 6. Atomic coordination and interatomic distances of 8i, 8j and 8f sites in YFe .12


Fig. 7. Calculated total phonon DOS and partial DOS projected to different crystalline sites in hypothetical YFe and YCo .12 12

Fig. 8. Phonon DOS of YFe with only nearest neighbor interactions taken into account.12


of d .d .d may well lead to the specula- from the shorter interatomic distances in YFe , comparedFei–Fei Fej–Fej Fef–Fef 12˚tion that 8i sites make a minimal contribution to high with the neighbor distance of 2.48 A in Fe, such as d i– i

˚ ˚ ˚frequency, while 8f sites make the most substantial. So far, (2.39 A), d (2.41 A), d (2.47 A), which serve tof–f j–f

the above calculation results have yet to be verified by strengthen the very high vibrational modes. Furthermore,experiments. Young’s modulus of YFe Ti have been the heavy rare-earth atoms, which confine the motion of Fe11

studied in Ref. [31] (the critical modulus E 5101.9 GPa at atoms, intensify the vibration.0

0 K), and Debye temperature given by 330 K was derivedfrom macroscale mechanics. According to our estimationof the temperature-dependent Debye temperature, which isderived from the calculated DOS, at 0 K this value of 6. Conclusion and discussionYFe is 483 K, and at 300 K the value is 337 K.12

Furthermore, for a test of the potentials in reflecting the To sum up, this research applies the lattice inversionphonon vibrational properties, the phonon spectrum and method to the acquisition of interatomic potentials relatedDOS for bcc-Fe and fcc-Co are calculated using the same to rare-earth and transition metals and characterizes the sitepotentials parameters as in Y–Ti–Fe–Co alloy systems occupancies of Fe and Co in Y(Co,Fe) , Y (Co,Fe) ,5 2 17

(see Figs. 9 and 10.). It agrees well with the experimental Y (Fe,Co,M) , and YTi(Fe,Co) . Our calculations show3 29 11

results [32,33], although the potential model is simple and that Fe atoms prefer to occupy 3g sites in Co based 1:5does not adopt any fitted parameters related to the vi- compounds and 4f sites in Co based hexagonal 2:17brational properties. The dispersion curve of fcc-Co re- compounds. In 3:29 compounds, Co atoms have a prefer-produces the experimental results very well except in the ence for 2d sites and one kind of 8j site. In YTi(Fe,Co)11

vicinity of maximum frequency. For bcc-Fe, as pair system, Co, Ni and Cu preferentially occupy f sites andpotentials will always exhibit, this model gives a much consequently, elevate the cohesive energy, while V and Molower transverse branches in [110] direction, which in turn prefer the i sites with the energy decreased as in thecauses an overestimated modes distribution in lower part ternary compounds. Co atoms in SmTi(Fe,Co) and11

of DOS. It lies in the inherent drawback of pair potentials NdTi(Fe,Co) have the similar substitution behaviors as in11

in dealing with the anisotropic properties. Although Fe YTi(Fe,Co) . Most of the above results are in good11

takes up a large proportion in YFe and rare-earth atoms agreement with experiments. On top of that, this paper12

only have a minor contribution, the difference in mode gives a statistical structural analysis of relaxed YTi(Fe,distribution is clear. The cut-off frequency is higher in Co) compounds.12

YFe than in Fe (9.14 THz for bcc Fe; in experiment, As the first to present the phonon density of states of12

9.26 THz [32]), but the main peaks in both low and high 1:12 structure, this research carries out a qualitativefrequency move toward the lower part. This maybe result analysis featuring the coordination and the relevant po-

Fig. 9. Calculated phonon density of states for bcc-Fe and fcc-Co.


Fig. 10. Calculated phonon dispersion curves for bcc-Fe and fcc-Co. Circles correspond to experimental data [32,33].

tentials, which enables a deeper insight into the cohesive calculation model employed in this research is deficient inproperties of different crystalline sites. In view of their tackling such problems as solubility and occupation frac-similarities in many aspects and the complexity in this tion, because it does not take the entropy into account. Itmagnetic system, the competition between Fe and Co can be expected that, when x equals 4 or 8 in YFe M ,122x x

occupations is subtle and thus poses a knotty problem; on mixed occupancy obviously has much larger entropy.the other hand, the method using inverted potentials is Besides, the statistics involved in this model are simplehighly effective in solving this problem, lending sound and direct. A more sophisticated method should includeproof to the applicability of simple interaction in treating more combinations of mixed sites and the competitioncohesive properties of alloying systems. The simplified mechanism between different alloying elements.


[14] V.L. Moruzzi, C.B. Sommers, Calculated Electronic Properties ofAcknowledgementsOrdered Alloys, World Scientific, New Jersey, 1995.

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