Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys

Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys

In imitation of f.c.c.-Ni—Fe alloy, the statistical-thermodynamic approach is applied for quantitative analysis of the thermal and composition—magnetic moment fluctuations’ influence on the kinematic diffuse-scattering intensity of radiations (X-rays or thermal neutrons) in magnetic alloys with the...

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Datum:2012
Hauptverfasser: Bokoch, S.M., Tatarenko, V.A., Vernyhora, I.V.
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Veröffentlicht: Інститут металофізики ім. Г.В. Курдюмова НАН України 2012
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spelling irk-123456789-983362016-04-12T03:02:39Z Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys Bokoch, S.M. Tatarenko, V.A. Vernyhora, I.V. In imitation of f.c.c.-Ni—Fe alloy, the statistical-thermodynamic approach is applied for quantitative analysis of the thermal and composition—magnetic moment fluctuations’ influence on the kinematic diffuse-scattering intensity of radiations (X-rays or thermal neutrons) in magnetic alloys with the atomic short-range order (SRO). На прикладі стопу ГЦК-Ni—Fe застосовано статистично-термодинамічний підхід до чисельної аналізи впливу термічних і концентраційно-магнетних флюктуацій на інтенсивність кінематичного дифузного розсіяння випромінення (Рентґенових променів або теплових невтронів) у атомарно невпорядкованих (з близьким порядком (БП)) магнетних стопах. На примере сплава ГЦК-Ni—Fe применён статистико-термодинамический подход к численному анализу влияния термических и концентрационно-магнитных флюктуаций на интенсивность кинематического диффузного рассеяния излучений (рентгеновских лучей или тепловых нейтронов) в атомно неупорядоченных (с ближним порядком (БП)) магнитных сплавах. 2012 Article Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys / S.M. Bokoch, V.A. Tatarenko, I.V. Vernyhora // Успехи физики металлов. — 2012. — Т. 13, № 3. — С. 269-302. — Бібліогр.: 88 назв. — англ. 1608-1021 PACS numbers: 05.10.Ln, 61.05.cf,61.72.Bb,64.60.De,75.30.Et,75.40.-s, 75.50.Bb http://dspace.nbuv.gov.ua/handle/123456789/98336 en Успехи физики металлов Інститут металофізики ім. Г.В. Курдюмова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description In imitation of f.c.c.-Ni—Fe alloy, the statistical-thermodynamic approach is applied for quantitative analysis of the thermal and composition—magnetic moment fluctuations’ influence on the kinematic diffuse-scattering intensity of radiations (X-rays or thermal neutrons) in magnetic alloys with the atomic short-range order (SRO).
format Article
author Bokoch, S.M.
Tatarenko, V.A.
Vernyhora, I.V.
spellingShingle Bokoch, S.M.
Tatarenko, V.A.
Vernyhora, I.V.
Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys
Успехи физики металлов
author_facet Bokoch, S.M.
Tatarenko, V.A.
Vernyhora, I.V.
author_sort Bokoch, S.M.
title Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys
title_short Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys
title_full Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys
title_fullStr Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys
title_full_unstemmed Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys
title_sort statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic f.c.c.-ni-fe alloys
publisher Інститут металофізики ім. Г.В. Курдюмова НАН України
publishDate 2012
url http://dspace.nbuv.gov.ua/handle/123456789/98336
citation_txt Statistical thermodynamics of the substitutional short-range atomic order and kinematics of the diffuse scattering of radiations in (para)magnetic F.C.C.-Ni-Fe alloys / S.M. Bokoch, V.A. Tatarenko, I.V. Vernyhora // Успехи физики металлов. — 2012. — Т. 13, № 3. — С. 269-302. — Бібліогр.: 88 назв. — англ.
series Успехи физики металлов
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AT tatarenkova statisticalthermodynamicsofthesubstitutionalshortrangeatomicorderandkinematicsofthediffusescatteringofradiationsinparamagneticfccnifealloys
AT vernyhoraiv statisticalthermodynamicsofthesubstitutionalshortrangeatomicorderandkinematicsofthediffusescatteringofradiationsinparamagneticfccnifealloys
first_indexed 2025-07-07T06:23:49Z
last_indexed 2025-07-07T06:23:49Z
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fulltext 269 PACS numbers: 05.10.Ln, 61.05.cf,61.72.Bb,64.60.De,75.30.Et,75.40.-s, 75.50.Bb Statistical Thermodynamics of the Substitutional Short-Range Atomic Order and Kinematics of the Diffuse Scattering of Radiations in (Para)Magnetic F.C.C.-Ni—Fe Alloys S. M. Bokoch1,4, V. A. Tatarenko2, and I. V. Vernyhora2,3 1Institute for Advanced Materials Science and Innovative Technologies, Department of Materials Design and Technology, 15 Sauletekio Ave., LT-10224 Vilnius, Lithuania 2G. V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine, Department of Solid State Theory, 36 Academician Vernadsky Blvd., UA-03680 Kyyiv-142, Ukraine 3Institute for Applied Physics, N.A.S. of Ukraine, Department of Modelling of Radiation Effects and Microstructure Transformations in Constructional Materials, 58 Petropavlivska Str., UA-40030 Sumy, Ukraine 4NIK Electronics, 34 Lesi Ukrayinky Blvd., UA-01601 Kyyiv, Ukraine In imitation of f.c.c.-Ni—Fe alloy, the statistical-thermodynamic approach is applied for quantitative analysis of the thermal and composition—magnetic moment fluctuations’ influence on the kinematic diffuse-scattering intensity of radiations (X-rays or thermal neutrons) in magnetic alloys with the atomic short-range order (SRO). Within the temperature—concentration (T—c) domains of macroscopically ferromagnetic and paramagnetic states of f.c.c. alloy, the relation for the diffuse-scattering intensity distribution over the quasi-wave vectors (including the Bragg’s ‘fundamental’ point), depending on the total ‘mixing’ energies of atoms, are obtained within the scope of (i) the self- consistent-field (SCF) and mean-SCF (MSCF) approximations as well as (ii) the simplest approximation by ‘interpolation’. The 2D patterns of (001)*-type dif- fuse-scattering intensity distribution over a reciprocal space as well as the cor- responding distributions (local configurations) of Fe and Ni atoms over the f.c.c.-lattice sites are modelled by a statistical Monte Carlo technique using the available experimental data on the Warren—Cowley SRO parameters extracted. Taking into account the magnetic (‘exchange’) interactions of atoms within the Успехи физ. мет. / Usp. Fiz. Met. 2012, т. 13, сс. 269—302 Оттиски доступны непосредственно от издателя Фотокопирование разрешено только в соответствии с лицензией © 2012 ИМФ (Институт металлофизики им. Г. В. Курдюмова НАН Украины) Напечатано в Украине. 270 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA statistical thermodynamics of alloys with the atomic SRO, it is possible to clari- fy the significant (T—c)-deviations of the diffuse-scattering intensity from its values corresponding to classical relation by the Krivoglaz—Clapp—Moss (KCM) formula with constant interatomic-interaction parameters. The principal possi- bilities and validity of using the generalized form and main assumptions of the KCM formula under an analysis of the diffuse-scattering intensity in magnetic alloys are ascertained. One of the intriguing findings of a given work belongs to revealing a possibility for simple and quite accurate estimation of the total ‘mixing’ energies, including their magnetic and nonmagnetic contributions, by means of the experimental SRO intensities for magnetic alloys obtained with X- ray diffraction techniques only applied for synchrotron radiation instrumenta- tion instead of conventional neutron scattering techniques, which are common- ly used as a powerful probe in physics of magnetic materials. Obtained analyti- cal and computational results on diffuse scattering characterization of (pa- ra)magnetic f.c.c.-Ni—Fe alloys are in a decent fit with all the reliable X-ray and thermal neutron diffraction data collected over years. На прикладі стопу ГЦК-Ni—Fe застосовано статистично-термодинамічний підхід до чисельної аналізи впливу термічних і концентраційно-магнетних флюктуацій на інтенсивність кінематичного дифузного розсіяння випромі- нення (Рентґенових променів або теплових невтронів) у атомарно невпоряд- кованих (з близьким порядком (БП)) магнетних стопах. У рамках (а) на- ближень самоузгодженого (СУП) та середнього самоузгодженого (ССУП) полів і (б) найпростішої «інтерполяційної» апроксимації одержано вираз для розподілу інтенсивности дифузного розсіяння по квазихвильових век- торах (в тому числі й для Бреґґової «фундаментальної» точки) залежно від повних енергій «змішання» атомів ГЦК-стопу у температурно-концентра- ційних (T—c) областях його макроскопічно феромагнетного та парамагнет- ного станів. З використанням статистичної методи Монте-Карло та присту- пних експериментальних даних про параметри БП Уоррена—Кавлі змоде- льовано 2D-картини розподілу інтенсивности дифузного розсіяння у пло- щинах типу (001)* оберненого простору та відповідне розміщення (локальні конфіґурації) атомів Fe й Ni по вузлах. Врахування магнетних («обмінних») взаємодій атомів у статистичній термодинаміці стопів з атомним БП прояс- нює значні T—c-відхили інтенсивности дифузного розсіяння від значень, що відповідають клясичному виразу для неї за формулою Кривоглаза—Клеппа— Мосса (ККМ) зі сталими параметрами міжатомових взаємодій. З’ясовано принципові можливості й обґрунтованість використання узагальненої фор- ми та основних припущень формули ККМ для аналізи інтенсивности дифу- зного розсіяння у магнетних стопах. Одним з результатів даної роботи є встановлення можливости простого та достатньо точного оцінювання пов- них енергій «змішання» із врахуванням магнетних і немагнетних внесків за експериментальними БП-інтенсивностями для магнетних стопів, визначе- ними із застосуванням лише Рентґенових дифракційних методик для синх- ротронної міряльної апаратури замість традиційних методик невтронного розсіяння, яких зазвичай використовують для діягностики магнетовпоряд- кованих матеріялів. Представлені аналітичні та розрахункові результати стосовно дифузного розсіяння у (пара)магнетних стопах ГЦК-Ni—Fe узго- джуються з надійними даними мірянь дифракції Рентґенових променів і теплових невтронів, яких було накопичено впродовжбагатьох років. ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 271 На примере сплава ГЦК-Ni—Fe применён статистико-термодинамический подход к численному анализу влияния термических и концентрационно- магнитных флюктуаций на интенсивность кинематического диффузного рассеяния излучений (рентгеновских лучей или тепловых нейтронов) в атомно неупорядоченных (с ближним порядком (БП)) магнитных сплавах. В рамках (а) приближений самосогласованного (ССП) и среднего самосогла- сованного (СССП) полей и (б) простейшей «интерполяционной» аппрокси- мации получено выражение для распределения интенсивности диффузного рассеяния по квазиволновым векторам (в том числе и для брэгговской «фундаментальной» точки) в зависимости от полных энергий «смешения» атомов ГЦК-сплава в температурно-концентрационных (T—c) областях его макроскопически ферромагнитного и парамагнитного состояний. С исполь- зованием статистического метода Монте-Карло и доступных эксперимен- тальных данных о параметрах БП Уоррена—Каули смоделированы 2D- картины распределения интенсивности диффузного рассеяния в плоскостях типа (001)* обратного пространства и соответствующее пространственное размещение (локальные конфигурации) атомов Fe и Ni на узлах. Учёт маг- нитных («обменных») взаимодействий атомов в статистической термодина- мике сплавов с атомным БП проясняет значительные T—c-отклонения ин- тенсивности диффузного рассеяния от значений, соответствующих класси- ческому выражению для неё по формуле Кривоглаза—Клэппа—Мосса (ККМ) с постоянными параметрами межатомных взаимодействий. Выяснены принципиальные возможности и обоснованность использования обобщён- ной формы и основных предположений формулы ККМ для анализа интен- сивности диффузного рассеяния в магнитных сплавах. Одним из результа- тов данной работы является установление возможности простого и доста- точно точного оценивания полных энергий «смешения» при учёте магнит- ных и немагнитных вкладов на основе экспериментальных БП-интенсивно- стей для магнитных сплавов, определённых с применением только рентге- новских дифракционных методик для синхротронной измерительной аппа- ратуры вместо традиционных методик нейтронного рассеяния, которые обычно используются для диагностики магнитоупорядоченных материа- лов. Представленные аналитические и численные результаты о диффузном рассеянии в (пара)магнитных сплавах ГЦК-Ni—Fe согласуются с надёжны- ми данными измерений дифракции рентгеновских лучей и тепловых нейтронов, накопленными за многие годы. Keywords: f.c.c.-Ni—Fe alloys, interatomic interactions, magnetic impurity interactions, short-range atomic ordering, diffuse scattering, statistical ther- modynamics, Monte Carlo simulation. (Received 7 March, 2012; in final version 19 September, 2012) 1. INTRODUCTION Nowadays, metallic alloys take one of the leading places in materials science and advanced technological applications due to their strength, heat resistance, electro- and thermal conductivities, specific optical 272 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA and magnetic properties, an occurrence of superconductive state, etc. The whole spectrum of these physical properties is closely related with the microscopic (crystal-lattice symmetry, point defects, static and dy- namic crystal-lattice imperfections, etc.), mesoscopic (nanoscale for- mations, defects clusters, etc.) and macroscopic (phase composition, size, shape and spatial distribution of inclusions, etc.) structures of the system at issue [1—7]. On the other hand, the majority of the equilibri- um microscopic structural states and the nonequilibrium processes con- trolled by kinetics of both the structural and magnetic phase transfor- mations are determined mostly by the interatomic-interaction energies (including magnetic (‘exchange’) ones), namely, appropriate inter- change (‘mixing’) energies of atoms, in such mixed systems. Undoubt- edly, it is evident that the exhaustive knowledge and detailed under- standing of both the microscopic parameters for (para)magnetic intera- tomic-interaction energies and the configuration part of the thermody- namical potentials (such as the Helmholtz free energy) of certain (dis)ordered phases are crucial for the self-consistent determination of the ‘nature’ and precise prediction of many thermodynamic and kinetic properties of solid alloys. Thus, due to its importance, many researchers have been attracted to this field, and, as a result, a set of outstanding experimental findings [8, 9] as well as theoretical ones [1—7, 10—61] have appeared for the last decades. Moreover, both the classical (semi-phenomenological) theoret- ical descriptions [10—19, 29—61] and ab initio electronic structure cal- culations (based on quantum-mechanical first principles) [20—28] were found to be remarkably fruitful. It is also worth noting that all these theoretical achievements were stimulated by many impressive results of experimental works on an ‘elastic’ interaction of radiations (X-rays or thermal neutrons) with disordered condensed matter [6—9] (for de- tails, see also references therein). Consequently, to date, one of the unique experimental methods, which enables to extract theoretically the quantitative information about ‘mixing’ energies of atoms in al- loys, is referred to as the ‘elastic’ diffuse scattering of radiations from disordered solid solutions [6—9]. The interrelation between the diffuse scattering intensities in atomically disordered solid solutions and the ‘mixing’-energies’ Fourier components has been proposed theoretically for the first time by Krivoglaz (1957) [10] and later by Clapp and Moss (1966—1968) [11—13]. Nowadays, this famous relation is widely known as the KСМ formula. Using the KCM formula in its original form and within the scope of its main approximations (independence of ‘mixing’ energies on concentration and temperature), it is possible to evaluate the ‘natural’ temperature and concentration dependences of the diffuse scattering intensities (see below) or evaluate the ‘mixing’ energies on the basis of known experimental data on diffuse scattering intensities. However, in many practical cases, the situation becomes complicated ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 273 due to the presence of several anomalies in the experimental diffuse scattering patterns such as the concentration- and temperature-in- duced diffuse intensity splitting [8] near the Lifshitz high-symmetry points within the first Brillouin zone (1 st BZ) in reciprocal space of a host crystal lattice, their ‘unusual’ temperature and concentration de- pendences, etc. In order to explain these features, a number of improved approximations were developed, in particular, considering the many- body interatomic-interactions’ effects, interatomic correlation phe- nomena, etc. In fact, as will be shown below, the absence of a proper un- derstanding of some above-mentioned diffuse scattering anomalies was the main reason for numerous critiques addressed to the conventional ‘pairwise’ interatomic-interaction models in mixed solids as well as to the KCM approximation for some metallic alloys. To our knowledge, the most fruitful theoretical methods are as follow: Tahir-Kheli method [14], cluster variation method (CVM) [15—17], spherical model (SM) [18, 19], Onsager cavity field (OCF) method [20, 21] and other advanced first-principles approaches [22—28], inverse Monte Carlo (IMC) [29] method and its linearized (LIMC) [30] version, Vaks—Zein—Kamyshen- ko cluster-field (CF) approach [31—33], Tokar—Masanskii—Grishchenko theory (based on the ‘gamma’ expansion method (GEM)) [34—37], alpha- expansion (AE) methods including high-temperature methods (HTM) [38—41]. Also, a series of quite new approaches was developed on the basis of the so-called ‘ring’ approximation [42—55] and recently on the basis of the self-consistent-field (SCF) and mean-SCF (MSCF) approxi- mations’ approach (taking into account strong interrelations between magnetic and atomic subsystems of an alloy) [56—61]. All these meth- odologies for evaluations of the interatomic ‘mixing’ energies (diffuse intensities) can be naturally subdivided into two groups: (i) the recipro- cal-space representation methods (i.e. in k-space) [10—14, 18—28, 42— 61] and (ii) the direct-space representation approximations (i.e. in r- space) [15—17, 29—41]. It is clear that the main disadvantage of the lat- ter methods is the restriction of the extent of interatomic-interaction energies to the finite number of coordination shells due to considering of a limited number of Warren—Cowley SRO parameters, α(rlmn) (where l, m, and n denote the conventional Miller’s indices). This shortcoming disappears, if one employs the methods developed for the infinite radi- us of interaction energies (i.e. within the Fourier representation). In case of alloys with a short-range interatomic-interaction nature, for example, such as the ‘exchange’ magnetic interaction, the results of both k-space approach and r-space one are almost identical. In a given work, in imitation of f.c.c.-Ni1−cFec alloy, a simple and ac- curate as well as physically understandable statistical-thermodynamics model of the atomic SRO states for (para)magnetic substitutional alloys with two magnetic constituents is developed (Sec. 2). The model is based on classical statistical-mechanics approximations only (i.e. SCF + MSCF 274 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA [56—61]), which assume the ‘pairwise’ interatomic and magnetic inter- actions solely, neglecting the interatomic correlation and many-body interaction effects. Along with such analytic description, as an alterna- tive, we also consider an ‘interpolating’ approximation, assuming the (3rd order) polynomial temperature dependence of the total ‘mixing’- energies’ Fourier components in the (T—c)-domain of a macroscopically ferromagnetic alloy. Thus, within the scope of both considerations, the relations for the diffuse scattering intensity are obtained, and, as a re- sult, similar ones can be applied in general to any magnetically (dis)ordered substitutional solid solutions with the atomic SRO. The main results and their discussion are presented in Sec. 3. In particular, here, the Monte Carlo (MC) simulations are carried out, using the liter- ature data on the experimentally obtained values of Warren—Cowley SRO parameters for disordered f.c.c.-Ni—Fe alloy (at different composi- tions and annealing temperatures). In addition, the modelled local atomic configurations in a real space and their Fourier transforms (dif- fuse scattering intensity patterns in a reciprocal space) are thoroughly analysed. In addition, both the rigorous statistical-thermodynamics and ‘interpolating’ approximations are applied for calculations of the (T—c)-dependences of the diffuse scattering intensities for two domi- nant quasi-wave vectors k within the 1 st BZ, namely, superstructural kX(001) point and ‘fundamental’ Bragg’s kΓ(000) one for disordered (para)magnetic f.c.c.-Ni—Fe solid solution. In conclusion of Sec. 3, the available models and theories, which were applied before for the esti- mation of interatomic-interaction energies and diffuse scattering pat- terns for f.c.c.-Ni—Fe alloy, are critically analysed. The summary and perspectives of a given work are presented in Sec. 4. Finally, we would like to emphasize that, here, we will not discuss the diffuse scattering intensities’ distribution over the reciprocal space in details as well as the ‘mixing’ energies’ Fourier components symmetry behaviour near the ‘fundamental’ Bragg’s kΓ(000) point, i.e. in the vicinity of a centre of the 1 st BZ, although, the suggested models are valid for such a recip- rocal-space region as well. Due to the special importance of mentioned information for quantitative interpretation of X-ray or thermal-neu- tron small-angle scattering data, such an analysis will be done exhaust- ively and separately in the forthcoming publication. 2. STATISTICAL-THERMODYNAMICS MODEL OF THE SUBSTITUTIONAL ATOMIC SHORT-RANGE ORDER AND KINEMATICS OF THE DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe-TYPE ALLOYS Within the scope of the SCF approximation [1—7] (at temperature, T, higher than the (Kurnakov) order—disorder phase transformation point, ТK), the equilibrium relation between the diffuse scattering in- ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 275 tensities and the ‘mixing’ energies’ Fourier components for a binary substitutional alloy is defined by the famous Krivoglaz—Clapp—Moss formula [10—13] as follows: ∝ α ≅ + − β  SRO tot ( ) ( ) 1 (1 ) ( ) D I c c w k k k , (1) where ( )α k is the k-th Fourier component of the SRO parameters (or so-called diffuse scattering intensity, ISRO(k), in dimensionless units [Laue units]); β = (kBT) −1 is proportional to the inverse absolute temper- ature; c is the relative concentration of an alloying element; kB is the Boltzmann constant; D is the normalization factor defined as [10—13]: stst st 1 1u.c. tot1 1 1u.c. tot 1 1 1 ( ) 1 (1 ) ( ) 1 1 . 1 (1 ) ( ) BZBZ BZ D d N c c w N c c w − ∈∈ − ∈     ≅ α =   Ω + − β       =  + − β    kk k k k k k    (2) In Eq. (2), the integration is carried out over the volume Ω of the 1 st BZ. Replacing the integral by the sum leads to the summation that should be carried over all the Nu.c. points of quasi-continuum with quasi-wave vec- tors k belonging to the 1 st BZ. Note that, for many alloys, the coefficient D ≈ 1 with accuracy of about 3% (this statement is also true for f.c.c.- Ni—Fe alloy; for details, see analysis in Sec. 3). One should note that, in all theories and approximations [10—61], the factor D is defined by Eq. (2) with distinctions only in the definition of the so-called total ‘mix- ing’-energies’ Fourier components of an alloy, tot ( )w k (see below). In Eqs (1) and (2), the total ‘mixing’-energies’ Fourier components, tot ( )w k , and the atomic SRO parameters’ Fourier components, ( )α k (i.e. diffuse scattering intensities, ISRO(k)), are defined by correspond- ing quantities in a direct space by means of the conventional Fourier transform as follow [1—9]: ′= − = − ⋅ tot tot ( ) ( ) exp( )w w i r R R k r k r , ′= − α = α − ⋅ ( ) ( ) exp( )i r R R k r k r ,(3a) where, taking into account the intrinsic self-consistency of Eqs (1) and (2), the following constraints should be satisfied [1—9]: α = ≡( ) 1r 0 (i.e. ∈ α ≡  st u.c. 1 ( ) BZ N k k ), (3b) = ≡ indirect ( ) 0w r 0 (i.e. ∈ ≡  st indirect 1 ( ) 0 BZ w k k ). In fact, the second formula in Eq. (3b) means the gauge condition of 276 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA the lack of indirect self-action of atoms in alloys. Furthermore, in the second formula of Eq. (3a), the quantities α(r) define the so-called Warren—Cowley SRO parameters determined as follow [1—9]: α = − − = −( ) 1 ( ) (1 ) 1 ( ) ,AB BAP c P cr r r (3c) where 1 − c (c) is the average concentration of A (B) kind atom, P AB(r) (PBA(r)) is the ‘two-particle’ probability of finding of A (B) kind atom sep- arated by a distance r = R − R′ from B (A) kind atom located at the origin (r = 0) in a substitutional solid solution A1−cBc based on the Bravais lattice. In Eq. (3a), wtot(r) is the total ‘mixing’ energy presented in terms of a certain coordination shell with a radius-vector r and defined through the ‘pairwise’ interatomic ‘mixing’ energies as follows [1—9]: tot tot tot tot ( ) ( ) ( ) 2 ( )AA BB ABw W W W= + −r r r r , (4) where tot ( )Wαβ r are the energies of a total ‘pairwise’ interaction between the atoms (α, β = А, В) located at the sites R and R′ at the distance r from each other. As it was shown in [56—61], the total ‘mixing’ energy for f.c.c.-Ni—Fe alloys can be presented as a sum of three microscopic energy contributions, namely, ‘direct’ short-range interaction, e.g., (i) ‘electrochemical’ or (ii) magnetic (for instance, Ising- or Heisenberg- type ‘exchange’) interactions, and (iii) indirect interaction, i.e. ‘strain- induced’ one (which is long-range and quasi-oscillating in a real space). Note that the ‘electrochemical’ interactions are usually referred to as the ‘atom—atom’ or ‘ion—ion’ ‘effective’ interactions arising between the atoms (ions) located at the sites of a rigid (non-relaxed) crystal lat- tice. This interaction has an electromagnetic nature matched with ex- change-correlation effects. The ‘strain-induced’ interaction arises be- cause of interference of the local static distortion fields of a host crystal lattice due to introduction of alloying atoms. In other words, this con- tribution takes into account semi-phenomenologically the atomic-sizes’ mismatch effects in an alloy. An exhaustive theoretical analysis of all these ‘mixing’ energy contributions for f.c.c.-Ni—Fe alloys and most salient literature in this matter can be found elsewhere [59, 60]. By the analogy with Refs [10—55], but based on the SCF and MSCF approximations [56—61], one can immediately develop the simple mod- el for calculation of the kinematic diffuse-scattering intensity for sub- stitutional (para)magnetic f.c.c.-Ni—Fe alloys with two magnetic con- stituents. Thus, in general case, such a part of the scattered-radiation intensity is caused by both the composition and magnetic-moments’ fluctuations in an alloy. Along with the rigorous model, the simple in- terpolating approach can also be suggested. Thus, according to Eqs (1) and (2), one should calculate the total ‘mixing’ energies’ Fourier com- ponent, tot ( )w k , as a function of temperature, T, and composition, c. Let us consider briefly the statistical-thermodynamics model of ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 277 such an alloy. As shown in [56—61], the total configuration-dependent part of free energy of (dis)ordered f.c.c.-Ni—Fe alloy is as follows: α α= + − + Σtot at mag at mag( ) conf conf conf conf conf { }F U U T S S , where Uconf and Sconf are the configuration internal energies and entropy, respectively, for both atomic (at) and magnetic (mag) interacting subsystems of an alloy (α = Ni, Fe). According to [56—61], the configuration entropy of the magnetic subsystem (i.e. moments with the spin numbers sα = 1/2, 1, 3/2, 2 or 5/2, …) in the long-range ordered magnetic states such as the ferromagnetic or ferrimagnetic ones can be presented within the scope of the correlationless (MSCF) approximation as follows [56—65]: α α α α α α α α α       ≅ + − −              mag( ) conf u.c. 1 1 ln sh 1 ln sh ( ) 2 2 B sS N k c y y y B y s s . (5) Here, ( )sB y α α is the conventional Brillouin function defined as [62—64]: α α α α α α α α        = + + −               1 1 1 1 ( ) 1 cth 1 cth 2 2 2 2 sB y y y s s s s , (6) yα ≡ (sαH α eff)/(kBT) is the magnetic-to-thermal-energy ratio correspond- ing to the Weiss intracrystalline ‘molecular’ field α β αβ β≅ − μ Γ σΣ eff B H g (within the MSCF approximation with coefficients {Γαβ}), σα is the spe- cific spontaneous magnetization of α-th magnetic subsystem (α = Ni, Fe), g is the Landé factor, μB is the Bohr magneton. The configuration entropy for atomic subsystem of a binary alloy within the (Т—с)-domain of disordered state (with SRO only) can be presented in the correlationless ‘regular solid solution’ form [1—7]: [ ]≅ − + − −at conf u.c. ln (1 ) ln(1 )BS N k c c c c . (7) Within the scope of the correlationless approximation, configuration internal energies of magnetic and atomic interacting subsystems in the atomic SRO state of f.c.c.-Ni—Fe alloy have the forms as follow [56—61]: ≅ σ + − σ + + − σ σ     mag 2 2 2 2 2 2 conf u.c. FeFe Fe Fe NiNi Ni Ni FeNi Fe Ni Fe Ni 1 ( ) ( )(1 ) 2 2 ( ) (1 ) , U N J c s J c s J c c s s 0 0 0 (8) ≅ Δ + at 2 conf 0prm u.c. prm 1 ( ) 2 U U N w c0 , (9) respectively. In Eq. (8), ( )J ′αα 0 is the ‘exchange’-integrals’ Fourier component (α, α′ = Ni, Fe) corresponding to k = 0. In Eq. (9), ΔU0prm is the configuration-dependent part of the internal energy, which is a linear function of the relative substitutional-atoms’ concentration, c (Fe in 278 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA f.c.c. Ni or Ni in f.c.c. Fe); prm ( )w 0 is the ‘paramagnetic’ ‘mixing’- energies’ Fourier component within the (T—c)-domain of a paramagnetic alloy (i.e. above the Curie temperatures, where the replacement tot prm ( ) ( )w w→k k  is valid; for details, see below). Finally, considering the above-mentioned approximation of the total configuration-depen- dent part of free energy of an alloy, tot conf F , and taking into account Eqs (5)—(9), one can immediately obtain a relation, e.g., for (para)magnetic f.c.c.-Ni1−cFec alloy with the atomic SRO only in the following form: ( ) ( ) Δ ≅ + + σ + − σ +   + − σ σ + + − − −       − + σ − σ              tot 0prm 2 2 2 2 2 2 2conf prm FeFe Fe Fe NiNi Ni Ni u.c. u.c. FeNi Fe Ni Fe Ni B B Fe Fe Fe Fe Fe Fe 1 ( ) ( ) ( )(1 ) 2 2 ( ) (1 ) ln 1 ln 1 1 1 ln sh 1 ( ) ln sh ( ) 2 2 UF w c J c s J c s N N J c c s s k T c c c c k T c y y s s 0 0 0 0    − σ σ +            + − + σ − σ − σ σ                Fe Fe Fe Ni Ni Ni Ni Ni Ni Ni Ni Ni ( ) 1 1 (1 ) ln sh 1 ( ) ln sh ( ) ( ) . 2 2 y c y y y s s (10) Based on Eq. (10), one can see that tot conf F can be transformed immedi- ately into its classical form [1—7] within the (T—c)-domain of a para- magnetic alloy (i.e. above the Curie temperatures, where the replace- ment tot prm conf conf F F→ is valid too). Using Eq. (10) and considering condi- tions when the derivatives of free energy, tot conf Fe F∂ ∂σ and tot conf Ni F∂ ∂σ , are equal to 0 for thermodynamically equilibrium state, the set of tran- scendental equations is obtained for determination of the specific spontaneous magnetizations, σFe and σNi, of both magnetic subsystems: { } { }   σ ≅ − − σ + − σ −    σ ≅ − σ + − σ        Ni Fe 2 2 Ni NiNi Ni Ni FeNi Fe Ni Fe 2 2 Fe FeFe Fe Fe FeNi Fe Ni Ni 1 ( )(1 ) ( ) (1 ) , (1 ) (11) 1 ( ) ( ) (1 ) . s B s B B J c s J c c s s c k T B J c s J c c s s ck T 0 0 0 0 In Eq. (10), the specific combination of prm ( )w 0 and ( )J ′αα 0 gives the so-called total ‘mixing’ energies’ Fourier component of an alloy for ‘fundamental’ Bragg’s Γ(000) point, and thus, for any quasi-wave vec- tor within the 1 st BZ, it can be presented as follows [56—61]: FeFe tot prm mag chem si mag chem FeFe 2 2 2 2 si NiNi Ni Ni FeFe Fe Fe NiFe Ni Fe Ni Fe ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) . (12) w w w V w V J s J s J s s ≈ + ≅ ϕ + + ≅ ϕ + + + σ + σ − σ σ k k k k k k k k k k k           ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 279 In Eq. (12), chem ( )ϕ k and si ( )V αα k are the Fourier components of ‘electro- chemical’ ‘mixing’ energies and substitutional-atoms’ (α—α) ‘strain- induced’ interaction energies, respectively. As a result, from (12), one can see that the total ‘mixing’ energies’ Fourier components are the temperature and concentration dependent quantities (due to the contri- butions of prm ( , , )w c Tk and mag ( , , )w c Tk ; for details, see an analysis in [60]) and uniquely determine the diffuse-scattering intensities of radi- ations, ISRO(k,c,T) (1), for magnetic binary substitutional alloys at presence of composition—magnetic moments’ fluctuations. Thus, omit- ting the ‘weak’ temperature dependence of ‘paramagnetic’ ‘mixing’ en- ergies, prm ( , , )w c Tk [60], the pronounced temperature dependence of tot ( , , )w c Tk (12) at constant composition arises mostly due to the tem- perature dependences of spontaneous magnetizations for both magnetic subsystems, σFe(c,T) and σNi(c,T), explicitly defined by Eq. (11). Note that, within the paramagnetic domain of a magnetic-alloy phase dia- gram, Eq. (12) corresponds immediately to the classical KCM- approximation constraint with the (T—c)-independent value of tot ( )w k . In order to express the total ‘mixing’ energies in the explicit form as a function of temperature, one can consider a simplest interpolation mod- el for approximate estimation. Thus, according to [65] (see also a set of quite similar representations in [62—64]), the temperature dependence of the spontaneous magnetization in Eqs (11) for α-th (α = Ni, Fe) sub- system of atomically disordered alloys based on a cubic Bravais lattice at certain composition can be presented by the approximate relation: α α   σ ≅ + κ −    1 1 C C T T T T . (13) Here, κα is the adjustable parameter, which depends on a concentra- tion. Substituting Eq. (13) into (12), we obtain immediately as follows: ≅ + + + 2 3 tot 0 1 2 3 ( ) ( ) ( ) ( ) ( )w a a T a T a Tk k k k k . (14) One can easily find that the wave-vector-dependent coefficients, а0(k), а1(k), а2(k), а3(k), entered into Eq. (14), are defined as follow: FeFe 2 2 0 chem si FeFe Fe NiNi Ni NiFe Ni Fe 1 2 2 1 FeFe Fe Fe NiNi Ni Ni NiFe Ni Fe Ni Fe 2 2 2 2 FeFe Fe Fe Fe NiN ( ) ( ) ( ) ( ) ( ) 2 ( ) , ( ) ( ) (2 1) ( ) (2 1) 2 ( ) ( 1) , ( ) ( ) ( 2 ) C C a V J s J s J s s a T J s J s J s s a T J s J − − = ϕ + + + − = κ − + κ − − − κ + κ −  = κ − κ + k k k k k k k k k k k k          2 2 i Ni Ni Ni NiFe Ni Fe Ni Fe Ni Fe 3 2 2 2 2 3 FeFe Fe Fe NiNi Ni Ni NiFe Ni Fe Ni Fe ( ) ( 2 ) 2 ( ) ( ) , ( ) ( ) ( ) 2 ( ) . C s J s s a T J s J s J s s−  κ − κ − − κ κ − κ − κ   = − κ + κ − κ κ  k k k k k k     (15) 280 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA Next, within the scope of the MSCF approximation, the magnetic phase-transition temperature (Curie temperature), TC, entered into Eq. (15), is determined by the ‘exchange’ ‘integrals’ of magnetic interac- tions, ( )J ′αα 0 , and the concentration of Fe atoms, c, in a homogeneous atomic SRO state exclusively in accordance with the formula [56—60]: { ( ) } 1 B Ni Ni NiNi Fe Fe FeFe 2 Ni Ni NiNi Fe Fe FeFe 1 2 2 Ni Ni Fe Fe NiFe (6 ) (1 ) ( )(1 ) (1 ) ( ) (1 ) ( )(1 ) (1 ) ( ) 4(1 ) (1 ) ( ) (1 ) . CT k s s J c s s J c s s J c s s J c s s s s J c c −≅ − + − + + − − + − − + + + + + −  0 0 0 0 0      (16) Thereby, combining of Eqs (14), (15) and (16), we arrive at the same conclusion as in the case of above-mentioned SCF + MSCF model, namely, that, in the magnetic (T—c)-domain of an alloy, the total ‘mix- ing’ energies’ Fourier components are the (T—c)-dependent quantities. Moreover, as can be seen from Eq. (14), such dependences at constant composition can be approximated by the polynomial temperature de- pendences of the 3 rd order, where the parameters, а0(k), а1(k), а2(k), а3(k), in accordance with Eq. (15), are defined through the microscopic energy contributions of different natures. Finally, within the scope of both ‘mixing’ energies’ Fourier components’ representations for tot ( )w k (i.e. in accordance with Eqs (12) and (14)), the diffuse scattering intensity of radiations, I(k) ∝ α(k), is still determined by the KСМ formula (1) with an appropriate definition of the normalization factor D (2). 3. RESULTS AND DISCUSSION 3.1. Monte Carlo Simulations of the Atomic SRO States in (Para)Magnetic F.C.C.-Ni—Fe Alloys In order to model the local atomic configurations and corresponding diffuse scattering intensity patterns for f.c.c.-Ni—Fe alloys, the statis- tical MC simulation method [66—68] is used. The method is based on the most suitable MC algorithms, which have been developed before, with regard to the SRO modelling for binary substitutional alloys, and more details about these techniques can be found elsewhere [69—78]. Thus, the used algorithm is as follows: (i) the initial random configura- tion of Ni and Fe atoms (with respective concentrations, cNi = 1 − c and cFe = c; cNi + cFe = 1) is generated over the sites of f.c.c. crystallite lattice with a linear size of L = 100 unit-cell parameters (the total number of sites within the crystallite corresponds to 100×100×100×4 positions); (ii) two different Ni and/or Fe atoms are chosen randomly and ex- ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 281 changed; (iii) the residual parameter R is calculated in accordance with the following formula: = α − α α model exp 2 exp 2( ) ( ) lmn lmn lmn lmn lmn R , (17) where model lmnα , exp lmnα are the Warren—Cowley SRO parameters for mod- elled crystallite and experimental data, respectively; (iv) if the parame- ter R decreases, the atoms are left at the ‘new’ positions, and otherwise, the exchange is rejected, the atoms are returned to the ‘old’ lattice sites. The steps (ii)—(iv) are repeated until R reaches some minimum value, e.g., 10 −(3—6). Within the described algorithm, the exchange of at- oms is realized between arbitrary-distant Ni and Fe atoms. Further- more, the modelled alloy is considered without site vacancies that is the reasonable assumption only for analysis of the equilibrium SRO states with or without a long-range order (LRO) of the atomic spatial distribu- tion. The periodic free-surface boundary conditions are also applied. The 2D (001)*-type diffuse scattering intensity patterns are calcu- lated using the conventional Fourier transformation (3) for both sets of Warren—Cowley SRO parameters, namely, experimental and mod- elled ones, exp lmnα and model lmn α , respectively. The MC simulations are carried out for Permalloy ( 62Ni0.765Fe0.235 [79] or Ni0.775Fe0.225 [81, 82]), Elinvar (Ni0.535Fe0.465 [81]) and Invar (Fe0.698 62Ni0.302 [80] or Fe0.632Ni0.368 [83]) compositions of f.c.c.-Ni—Fe alloys isothermally annealed at temperatures above and below the Cu- rie points, TC. In Tables 1 and 2, the experimental Warren—Cowley SRO parame- ters, exp lmnα , taken from [79—83], where both the thermal neutrons with the Borie—Sparks—Gragg separation technique [79, 80] and the anoma- lous X-ray scattering methods with the 3λ separation technique [81— 83] were used, are presented along with the SRO parameters calculated for modelled crystallites, model lmnα , at Permalloy, Elinvar and Invar com- positions and given isothermal annealing temperatures, Ta. Then, using data from Tables 1 and 2 and Fourier transform (3), we have calculated the 2D diffuse-scattering intensity patterns in (001)*- type plane of a reciprocal space for f.c.c.-Ni—Fe alloys, ISRO(k,c,T) (Figs 1 and 2). As can be seen, the 2D isodiffuse intensity distributions, ISRO(k,c,T), depend significantly on the (T—c)-region, where f.c.c.-Ni— Fe alloys with the atomic SRO are considered. From Figs 1 and 2, one can see that, in spite of the insignificant dif- ference between the absolute values of both experimental and modelled diffuse intensity distributions, the 2D diffraction patterns for modelled crystallites in their reciprocal-space representation details are in overall decent agreement with the experimentally measured ones [79—83]. As can also be seen from Figs 1 and 2, the maximum of the diffuse scattering intensities is localized at the Х(100)-type superstructural 282 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA points, kX, of a reciprocal space (with the insignificant elongation in (1 ky 0)-type directions, which is probably due to the interference of the ‘itinerant’ electron waves with the static concentration waves when kF ≅ kX, where kF is the Fermi wave number) that shows the preferential formation of the homogeneous atomic SRO states locally ordered like to the L12-type superstructure unit cells. TABLE 1. The Warren—Cowley SRO parameters, exp lmnα (left columns), extract- ed from the neutron elastic diffuse scattering experiments [79, 80] for single crystals of f.c.c.-Ni—Fe alloys with various compositions, c, and at different annealing temperatures, Ta, as well as the modelled SRO parameters, model lmn α (right columns). The superscript ‘ 62’ denotes an atom of 62Ni isotope. N o . s h e ll lm n exp lmnα | model lmnα 62Ni0.765Fe0.235 808 K [79] 62Ni0.765Fe0.235 873 K [79] 62Ni0.765Fe0.235 958 K [79] Fe0.698 62Ni0.302 743 K [80] 0 000 0.932 1 1.1 1 0.988 1 1.0214 1 1 110 −0.111 −0.1305 −0.0868−0.0957 −0.0946 −0.1310 −0.0201 −0.0282 2 200 0.136 0.1388 0.100 0.1123 0.0848 0.0933 0.0451 0.0562 3 211 −0.006 −0.0361 −0.0024−0.0135 −0.00302 −0.0193 0.0046 0.0017 4 220 0.052 0.0685 0.0200 0.0427 0.0183 0.0259 0.0042 0.0148 5 310 −0.022 −0.0452 0.0140 −0.0029 −0.00823 −0.0363 0.0003 −0.0030 6 222 0.031 0.0260 0.0175 0.0168 0.0067 0.0057 0.0042 0.0021 7 321 −0.014 −0.0113 −0.0026 0.0078 −0.00205 −0.0085 0.0008 0.0077 8 400 0.022 −0.0029 0.0084 0.0091 0.00612 −0.0165 0.0037 0.0034 9a 330 −0.013 −0.0376 – – −0.00073 −0.0192 – – 9b 411 −0.013 −0.0177 – – −0.0034 −0.0133 – – 10 420 0.001 0.0199 – – 0.00392 0.0049 – – 11 332 0.019 −0.0090 – – −0.00093 −0.0087 – – 12 422 0.005 0.0192 – – 0.0041 0.0104 – – 13a 431 0.016 −0.0009 – – −0.0013 −0.0043 – – 13b 510 0.016 −0.0192 – – −0.00032 −0.0236 – – 14 521 0 0.0039 – – 0.000785 0.0057 – – 15 440 −0.007 −0.0167 – – 0.00162 −0.0131 – – 16a 433 −0.007 −0.0112 – – −0.00059 −0.0103 – – 16b 530 −0.007 −0.0098 – – −0.0025 −0.0098 – – 17a 442 0.008 0.0134 – – 0.00121 0.0089 – – 17b 600 0.008 −0.0374 – – 0.000418 −0.0348 – – 18a 532 −0.008 −0.0206 – – −0.00105 −0.0186 – – 18b 611 −0.008 0.0026 – – −0.00461 0.0027 – – 19 620 −0.011 0.0055 – – 0.00346 0.0018 – – 20 541 0.002 0.0027 – – 0.000276 −0.0016 – – 21 622 0.012 0.0097 – – 0.00151 0.0063 – – 22 631 0.006 −0.0002 – – −0.00202 −0.0032 – – 23 444 – −0.0144 – – −0.00127 −0.0124 – – ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 283 TABLE 2. The Warren—Cowley SRO parameters, exp lmnα (left columns), extract- ed from the X-ray anomalous diffuse scattering experiments [81—83] for sin- gle crystals of f.c.c.-Ni1−cFec alloys at different annealing temperatures, Ta, as well as the modelled SRO parameters, model lmn α (right columns). N o . s h e ll lm n exp lmnα | model lmnα Ni0.775Fe0.225 1273 K [81] Ni0.775Fe0.225 1273 K [82] Ni0.535Fe0.465 1273 K [81] Fe0.632Ni0.368 753 K [83] 0 000 1.00121 1 0.9901 1 1.00002 1 1.0025 1 1 110 −0.10821 −0.13664−0.1557−0.1518−0.07665−0.1396 −0.0583 −0.0797 2 200 0.11948 0.12729 0.1288 0.1363 0.06463 0.10073 0.0522 0.0675 3 211 −0.00472 −0.02733−0.0165−0.0326 −0.0022 −0.0189 −0.0032 −0.0107 4 220 0.03077 0.05169 0.0217 0.0547 0.00371 0.01819 0.0002 0.0152 5 310 −0.01792 −0.05157−0.0224−0.0542 −0.0100 −0.0436 −0.0062 −0.0186 6 222 0.01305 0.00634 0.0165 0.0087 0.00371 −0.0084 – – 7 321 −0.00752 −0.01405−0.0123−0.0158 −0.0039 −0.0109 – – 8 400 0.01734 −0.00332 – – 0.00711 −0.0028 – – 9a 330 0.00052 −0.03033 – – −0.0022 −0.0193 – – 9b 411 0.00461 −0.00766 – – 0.00077 −0.0106 – – 10 420 0.00483 0.01855 – – 0.00128 0.00355 – – 11 332 −0.00322 −0.00897 – – −0.00077−0.0090 – – 12 422 0.00302 0.01203 – – −0.000370.00142 – – 13a 431 −0.00251 −0.00426 – – −0.00085−0.0054 – – 13b 510 −0.00231 −0.03124 – – 0.00026 −0.0251 – – 14 521 −0.00259 −0.00172 – – −0.00114−0.0001 – – 15 440 0.002310 −0.01563 – – 0.00077 −0.0130 – – 16a 433 −0.00422 −0.01156 – – 0.00035 −0.0066 – – 16b 530 −0.00171 −0.00793 – – −0.00026−0.0073 – – 17a 442 – 0.00199 – – – −0.0037 – – 17b 600 0.00873 −0.03226 – – −0.00379−0.0214 – – Moreover, by moving from Permalloy (of L12-Ni3Fe type) through Elinvar (of L10-NiFe type) to Invar (of L12-Fe3Ni type) composition re- gions, the diffuse scattering intensities for the superstructural point kX, ISRO(kX,c), decrease significantly. It can be explained by the signifi- cant decrease of values of the ‘mixing’ energies’ Fourier components, prm ( , )Xw ck , with increasing of Fe atoms’ concentration (see Table 3 and Eqs (12) and (14) in line with [59, 60]). Using the Warren—Cowley SRO parameters of modelled crystallites, for instance, for f.c.c.- 62Ni0.765Fe0.235 Permalloy (Table 1) annealed at Та = 958, 873 or 808 K, the local atomic configurations of Ni and Fe at- oms are reconstructed (Fig. 3). Due to the L12(0)-type symmetry of LRO superstructures of f.c.c.-Ni—Fe alloys and for the ease of further anal- ysis, the atomic distributions are presented in (001)*-type 2D sections 284 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA in a direct space. From Fig. 3, one can see that the number of LRO regions ordered by the L12-type superstructure increases with decreasing temperature when changing from PMS (paramagnetic state of atomic SRO) region to FMS (ferromagnetic state of atomic SRO) one. As shown in [84], the dif- fuse scattering intensities significantly deviate from those predicted by the classical KCM formula (1) that confirms the conclusion made in [56— 61] about the essential role of the magnetic contribution into the total ‘mixing’ energies of an alloy within the FMS region (see analysis below). a b c d Fig. 1. The 2D diffuse-scattering intensity patterns in (001)*-type plane of a reciprocal space, ISRO(k,c,T) in [Laue units]: (a) 62Ni0.765Fe0.235@808 K [79], (b) 62Ni0.765Fe0.235@873 K [79], (c) 62Ni0.765Fe0.235@958 K [79], (d) Fe0.698 62Ni0.302@743 K [80]. The modelled data correspond to the MC simulation results. ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 285 3.2. (T—c)-Dependence of the Atomic SRO States in (Para)Magnetic F.C.C.-Ni—Fe Alloys As shown recently in [60], the ‘paramagnetic’ ‘mixing’ energies Fouri- er components, prm ( )w k , calculated for quasi-wave vectors k along the high-symmetry Δ(kΓ → kX) direction within the 1 st BZ are the concen- tration-dependent functions in accordance with the approximant: 2 prm 0 1 2 ( , ) ( ) ( ) ( )w c K K c K c≈ + +k k k k (18) a b c d Fig. 2. The 2D diffuse-scattering intensity patterns in (001)*-type plane of a reciprocal space, ISRO(k,c,T) in [Laue units]: (a) Ni0.775Fe0.225@1273 K [81], (b) Ni0.775Fe0.225@1273 K [82], (c) Ni0.535Fe0.465@1273 K [81], (d) Fe0.632Ni0.368@753 K [83]. The modelled data correspond to the MC simulation results. 286 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA with the fitting parameters, K0(k), K1(k) and K2(k), presented in Table 3. The ‘exchange’ ‘integrals’ Fourier components for magnetic interac- tions, ( )J ′αα k , which were calculated within the scope of the MSCF ap- proximation [58—60] for various possible spin numbers for Ni and Fe atoms in f.c.c.-Ni—Fe alloys, are presented in Table 4. As evidently from Table 4, the ‘exchange’ ‘integrals’ Fourier components, NiNi ( )J k and a b c d Fig. 3. The temperature dependence of Ni (○) and Fe (●) atomic configurations in (001)*-type crystallographic plane for f.c.c.- 62Ni0.765Fe0.235 Permalloy ob- tained by the MC simulations: (a) initial random configuration; (b), (c) and (d) modelled atomic SRO configurations at Ta = 958, 873 and 808 K, respectively. The chosen temperatures correspond to the respective annealing tempera- tures reported in [79]. The atomic L12-Ni3Fe type LRO clusters are clearly vis- ible and marked by dark areas. ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 287 NiFe ( )J k , correspond to ferromagnetic interactions, and FeFe ( )J k corre- sponds to the antiferromagnetic character. In spite of the recently ap- peared and debatable point of view about the completely ferromagnetic interactions in f.c.c.-Ni—Fe alloys, our result is in an overall agreement with many experimental and theoretical findings (see Refs [56—60] and references therein) as well as with the conception of f.c.c. γ-Fe itself. As shown by means of the MC simulations above, the atomic SRO states of f.c.c.-Ni—Fe alloys depend significantly on the (T—c)-region where a certain solid solution is considered. Using the concentration dependence of the ‘paramagnetic’ ‘mixing’ energies’ Fourier components, prm ( , )w ck , for two quasi-wave vectors k within the 1 st BZ, kX and kΓ, (Table 3) and the temperature—concentra- tion dependence of the magnetic ‘mixing’ energies’ Fourier compo- nents, mag ( , , )w c Tk (according to (11) and (12)), one can calculate the (T—c)-dependences of the diffuse SRO intensity, ISRO(k,c,T), by means of the KСМ formula (1). Here, all calculations are done for three cases: (і) tot prm ( ) ( ) constw w≡ =k k  [10—13], (іі) tot prm ( , ) ( , )w c w c≡k k  (Table 3) [60], and (ііі) tot prm mag ( , , ) ( , ) ( , , )w c T w c w c T≡ +k k k   (12). The results of such calculations are shown in Fig. 4. In Figs 4, a, b, one can see that, for both wave-vectors, the function ISRO(k,c,T) has the well-known form predicted by the KCM theory [10— 13], assuming that tot ( ) constw =k . A maximum (minimum) of the dif- fuse-scattering intensities at the kX (kΓ) point lies at the concentration c = 1/2 and decreases (increases) with increasing T, obeying a trivial law. TABLE 3. The coefficients K0(k), K1(k) and K2(k) (in [eV]) entered into (18) for estimation of the concentration-dependent ‘paramagnetic’ ‘mixing’ ener- gies Fourier components for quasi-wave vectors k within the 1 st BZ, namely, kХ(100), Δ(kΓ → kX) and kΓ(000), for f.c.c.-Ni1−cFec alloys [60]. k K0(k) K1(k) K2(k) Approximation X(0 0 1) −0.414 0.450 – 1st order X(0 0 1) → Γ(0 0 0) 0.855 −2.177 2.087 2nd order Γ(0 0 0) 0.843 −2.339 2.344 2nd order TABLE 4. The ‘exchange’ (magnetic) interaction energies’ Fourier compo- nents (in [meV]) for two quasi-wave vectors k within the 1 st BZ, kХ(100) and kΓ(000), depending on local spin numbers of Ni and Fe atoms in f.c.c.-Ni1−cFec alloys (c ∈ [0, 1]) [58—60]. sNi sFe NiNi ( )J 0 FeFe ( )J 0 NiFe ( )J 0 NiNi ( )XJ k FeFe ( )XJ k NiFe ( )XJ k 1/2 1/2 −215.9 274.6 −517.6 72.0 −91.5 172.5 1/2 1 −215.9 103.0 −316.9 72.0 −34.3 105.6 1/2 3/2 −215.9 54.9 −231.5 72.0 −18.3 77.2 288 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA a b c d e f Fig. 4. The (T—c)-dependences ISRO(k,c,T) (in [Laue units]) at the high-sym- metry points within the 1 st BZ for f.c.c.-Ni—Fe alloys: superstructural X(001) and Bragg’s ‘fundamental’ Γ(000); (a) prm ( ) X w k = −0.3 eV and (b) prm ( )w Γk = = 0.5 eV [10—13]; (c) prm prm ( ) ( , ) X X w w c=k k  and (d) prm prm ( ) ( , )w w cΓ Γ=k k  (Ta- ble 3) [60]; (e) tot ( , , )Xw c Tk and (f) tot ( , , )w c TΓk (12). In (e), (f), dotted lines are the Curie points’ curve of the magnetic phase transitions, TC(c), calculat- ed in accordance with (16). The calculation results according to (14) are not presented because they are almost the same as for (12). ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 289 In the second case shown in Figs 4, c, d, when tot prm ( , ) ( , )w c w c≡k k  , a maximum (minimum) of the diffuse-scattering intensity, ISRO(k,c,T), is shifted to the composition of c ≅ 1/3 for kX or c ≅ 1/4 for kΓ, respec- tively, due to some competition of the addendums, prm (1 — ) ( , )с с w cβ k and 1, in denominator of the KCM formula (1). A special attention should be given to the third case presented in Figs 4, e, f, where the total ‘mixing’ energies’ Fourier components are (T—c)- dependent functions, tot ( , , )w c Tk (12). As can be seen, the relation SRO SRO ( ) ( ) ( ) ( ) C C X XT T T T I T c I T c < > ∂ ∂ ∂ > ∂ ∂ ∂k k is satisfied and again indi- cates that, within the FMS (T—c)-domain, the atomic L12-type SRO state is stabilized with respect to its PMS region. Apparently, taking into account the magnetic interactions does not change the total dif- fuse-scattering intensity pattern but only increases its absolute value (for both kX and kΓ) within the FMS state of an alloy. Let us now analyse the diffuse scattering from f.c.c.- 62Ni0.765Fe0.235 solid solution, using the statistical-thermodynamics model presented in Sec. 2. The diffuse-scattering patterns from such an alloy were studied in details experimentally by the elastic diffuse scattering of thermal neutrons [79, 84, 85]. The data obtained in those studies were repeatedly used in devel- oping and testing of different theoretical models and approximations for the evaluation of interatomic-interaction energies [30—36, 79]. Table 5 shows the total ‘mixing’ energies for f.c.c.- 62Ni0.765Fe0.235 Permalloy esti- mated as ‘effective’ ones in certain approaches with using the different approximations [30, 33, 36, 79]. The theoretical data are presented for several isothermal-annealing temperatures, Та, in accordance with [79, 85]. As can be seen from Table 5, the values of total ‘mixing’ energies sig- nificantly differ, depending on the models used for their calculation. For ease of further analysis, we have recalculated the Fourier components of those real-space energies. Thus, using the data shown in Tables 3 and 4 as well as the ‘mixing’ energies for different coordination shells in a real space, wtot(R — R′,T), from Table 5 and applying the Fourier transform (3), the ‘mixing’ energies’ Fourier components, tot(prm) ( )w k , for quasi-wave vectors k within the 1 st BZ, namely, kX, Δ(kX → kΓ) and kΓ, are calculated. For each set of ‘mixing’ energies, the temperature of the absolute loss of stability with respect to the formation of concentration waves with the quasi-wave vector kX is calculated, according to the formula [4—7]: inst tot (1 ) ( )X BT c c w k− − k . (19) The results of such calculations are presented in Table 6. From Table 6, one can immediately see that, from all instability temperatures, Tinst, on- ly those, which are calculated using the data extracted from [33] ( 1CFM: Ta = 745 K, 780 K), [36] ( 1GEM: for all thermal treatments) and in our model, are agreed reasonably with the experimental value of the order— 290 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA disorder phase transformation temperature, TK ≈ 771—773 K. The worse results (with deviation of ±150 K) are obtained for the interaction ener- gies taken from [33] ( 1CFM: Ta = 808 K, 958 K; 2,3CFM: Ta = 780 K) and for [36] (SM: for all thermal treatments). Tinst calculated by using the da- ta from other references poorly match the experimental value of TK. The values of the ‘mixing’ energies obtained using the KCM [79] and LIMCM [30] approaches (see Tables 5 and 6) are underestimated com- paring with the СFM [33], SM and GEM [36] data as well as with the present-work results and, therefore, give significantly lower values of the instability temperatures (Table 6). One can also conclude that the main reason of such a disagreement is caused by the use of incorrect values of the Warren—Cowley SRO parameters, α(rlmn), calculated in [79] (especially for the FMS (T—c)-domain), neglecting the magnetic contribution and its temperature and concentration dependences (see, e.g., Eqs (12) or (14) along with (1) and (2)). It should also be mentioned that the assumption made in [34—37] about separating the interatomic interaction into two contributions, the short-range and long-range ones (with consecutive expansion by a small parameter ‘γ’), is quite reasonable; however, at the same time, it is not so evident as in the suggested models, where the total ‘mixing’ energies are given in the form of (12) or (14). In such cases, the total ‘mixing’ energies consist of both the ‘short-range’ (‘electrochemical’ and magnetic) and ‘long- range’ (‘strain-induced’) energy contributions in their explicit forms. 3.3. Kinematics of the Diffuse Scattering of Radiations in a (Para)Magnetic F.C.C.-Ni0.765Fe0.235 Permalloy Let us consider another application of the models developed in Sec. 2 by way of illustration of detailed calculations of the diffuse scattering in- tensities for f.c.c.- 62Ni0.765Fe0.235 Permalloy. With such a goal, at the be- ginning, we calculate the temperature dependence of the spontaneous magnetizations for both interacting magnetic subsystems, σFe(T), σNi(T), using the SCF + MSCF approximations (11) and the interpolation scheme (13) (with the admissible adjustable parameters, κNi, κFe). The results of such calculations are shown in Fig. 5, where one can see that the results on the temperature dependence of spontaneous magnetiza- tions obtained in both approximations in accordance with Eqs (11) and (13) are in a good agreement with each other within the whole temperature in- terval (0, TC). Slight disagreements for Ni-subsystem magnetization, σNi(T), including its nonphysical value > 1, appeared at T ∈ (0, 450 K), is caused by the inaccuracy of Eq. (13). Thus, using the values of σFe(T), σNi(T) (Fig. 5) and the ‘paramagnetic’ ‘mixing’ energies, prm ( , )w ck (Ta- ble 3), as well as the ‘exchange’ ‘integrals’, ( )J ′αα k (Table 4), one can cal- culate the T-dependence of total ‘mixing’ energies’ Fourier components, tot ( , , )w c Tk , in accordance with Eqs (12) and (14) for c = 0.235. T A B L E 5 . T h e t o ta l ‘m ix in g ’ e n e r g ie s, w to t( R − R ′,T ), c a lc u la te d u si n g k n o w n a p p r o x im a ti o n s b a se d o n t h e r a d ia ti o n d if - fu se -s c a tt e r in g d a ta f o r s in g le c r y st a l o f f. c .c .- 6 2 N i 0 .7 6 5 F e 0 .2 3 5 P e r m a ll o y [ 7 9 , 8 5 ] a t v a r io u s a n n e a li n g t e m p e r a tu r e s, T a . w to t( R − R ′,T ), m e V l m n 1 1 0 2 0 0 2 1 1 2 2 0 3 1 0 2 2 2 3 2 1 4 0 0 3 3 0 4 1 1 N o . o f s h e ll 1 2 3 4 5 6 7 8 9 a 9 b |R − R ′|/ a 0 ≈ 0 .7 1 1 ≈ 1 .2 2 ≈ 1 .4 1 ≈ 1 .5 8 ≈ 1 .7 3 ≈ 1 .8 7 2 ≈ 2 .1 2 ≈ 2 .1 2 *958 K K C M 2 4 .5 −6 .6 8 2 .4 1 .6 6 −0 .4 3 0 .3 2 0 .1 7 0 .6 7 −0 .3 1 0 .4 1 L IM C M 3 2 .0 −6 .9 3 .4 2 .1 – – – – – – 2 L IM C M 3 3 .9 −5 .3 4 .1 1 .7 – – – – – – 1 C F M 6 1 .8 −1 3 .7 5 .7 4 .1 −1 .0 0 .7 0 .4 1 .6 −0 .7 – S M 5 8 .3 −1 5 .9 5 .8 4 .0 −1 .0 0 .7 0 .4 1 .6 −0 .7 0 .9 1 G E M 6 1 .5 −1 3 .4 5 .8 – – – – – – – 2 G E M 6 2 .0 – – – – – – – – – 808 K K C M 2 0 .5 −9 .7 2 0 .8 2 0 .2 7 0 .1 7 −0 .8 8 0 .4 2 −0 .1 −0 .4 1 0 .1 3 8 1 IM C M 2 8 .1 −1 0 .6 0 .5 −1 .0 – – – – – – 2 L IM C M 2 9 .5 −9 .1 2 .6 0 .0 – – – – – – 1 C F M 5 9 .5 −2 1 .9 2 .2 0 .8 0 .5 −2 .3 1 .2 −0 .3 1 .2 – ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 291 C o n ti n u a ti o n o f T A B L E 5 . S M 5 5 .8 −2 6 .5 2 .2 0 .7 0 .4 −2 .3 1 .2 −0 .3 −1 .2 0 .3 1 G E M 5 9 .5 −2 0 .7 2 .2 – – – – – – – 2 G E M 6 0 .2 – – – – – – – – – K C M 2 1 .3 −9 .7 1 .4 7 0 .9 −0 .7 5 −0 .6 3 0 .5 2 −0 .3 1 0 .2 8 −0 .2 4 1 L IM C M 3 2 .3 −9 .4 2 .6 0 .8 – – – – – – 2 L IM C M 3 2 .9 −7 .6 2 .8 0 .3 – – – – – – 780 K 1 C F M 5 7 .1 −2 0 .5 3 .6 2 .3 −1 .9 −1 .6 1 .3 −0 .8 0 .7 – 2 C F M 6 6 .1 −1 9 .7 3 .8 2 .5 −1 .9 −1 .6 1 .3 −0 .8 0 .7 – 3 C F M 6 8 .7 −1 9 .4 3 .9 2 .5 −1 .9 −1 .6 1 .3 −0 .8 0 .7 – S M 5 3 .6 −2 4 .4 3 .7 2 .2 −1 .8 −1 .6 1 .3 −0 .8 0 .7 −0 .7 1 G E M 5 7 .8 −1 7 .8 3 .7 – – – – – – – 2 G E M 5 8 .6 – – – – – – – – – 776 K 1 L IM C M 3 3 .8 −1 1 .1 3 .7 2 .2 – – – – – – 2 L IM C M 3 4 .9 −8 .3 4 .3 1 .3 – – – – – – 745 K 1 C F M 6 1 .3 −1 7 .7 4 .2 1 .4 −2 .2 −2 .2 0 .0 9 −1 .8 −0 .7 – S M 5 7 .2 −2 1 .8 4 .2 1 .2 −2 .2 −2 .2 0 .0 8 −1 .8 −0 .7 −0 .4 1 G E M 6 1 .8 −1 4 .7 4 .3 – – – – – – – 2 G E M 6 2 .7 – – – – – – – – – * w to t( r ) ≡ w p r m (r ). a 0 i s t h e e q u il ib r iu m l a tt ic e p a r a m e te r . K C M i s t h e K r iv o g la z — C la p p — M o s s a p p r o x im a ti o n [ 7 9 ]. L IM C M i s t h e l in e a r iz e d i n v e r s e M o n te C a r lo m e th o d [ 3 0 ]: 1 L IM C M w it h 4 S R O p a r a m e te r s α (r ) fr o m [ 7 9 ]; 2 L IM C M w it h 2 8 S R O p a r a m e te r s α (r ) fr o m [ 7 9 ]. C F M i s t h e c lu s te r v a r ia ti o n f ie ld m e th o d [ 3 3 ]: 1 C F M w it h n n 3 W = 0 m e V ; 2 C F M w it h n n 3 W = − 1 7 .2 m e V ; 3 C F M w it h n n 3 W = − 2 1 .5 m e V ( n n 3 W i s t h e t h r e e -p a r ti c le e n e r g y v a lu e f o r n e a r e s t n e ig h b o u r s ). S M i s t h e s p h e r ic a l m o d e l [3 6 ]. G E M i s t h e ‘ g a m m a ’ e x p a n s io n m e th o d [ 3 6 ]: 1 G E M a n d 2 G E M w it h Σ s = Α α2 a n d Σ s = Α α2 + Β α3 , r e s p e c ti v e ly . 292 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 293 TABLE 6. The total (‘effective’) ‘mixing’ energies’ Fourier components, tot ( )w k (in [meV]), calculated using the data for f.c.c.- 62Ni0.765Fe0.235 Permalloy from Ta- bles 3, 4 and 5 for quasi-wave vectors k within the 1 st BZ: near the ‘fundamental’ Γ(000) point, Δ(k → kΓ); at the ‘fundamental’ Γ(000) point precisely, kΓ; for the superstructural X(100) point, kX. The absolute instability temperatures, Tinst (in [K]), calculated with (19), using the values of eff tot ( ) X w k , are presented too. Theory Ta, K Energies 958 808 780 776 745 KCM, [79] eff tot ( )w Γ→k k 338.1 225.6 241.1 – – eff tot ( ) X w k −135.6 −158.8 −152.8 – – Tinst 282.9 331.3 318.7 – – 1LIMCM, [30] eff tot ( )w Γ→k k 449.4 273.6 403.2 454.2 – eff tot ( ) X w k −171.4 −192.0 −196.8 −205.0 – Tinst 357.5 400.5 410.5 427.6 – 2LIMCM, [30] eff tot ( )w Γ→k k 493.8 361.8 420.0 487.8 – eff tot ( ) X w k −179.8 −193.4 −196.0 −208.2 – Tinst 375.0 403.4 408.8 434.3 – 1CFM, [33] eff tot ( )w Γ→k k 847.4 708.8 683.8 – 661.7 eff tot ( ) X w k −306.2 −425.6 −378.6 – −377.6 Tinst 638.7 887.8 789.7 – 787.7 2CFM, [33] eff tot ( )w Γ→k k – – 803.8 – – eff tot ( ) X w k – – −409 – – Tinst – – 853.1 – – 3CFM, [33] eff tot ( )w Γ→k k – – 839.2 – – eff tot ( ) X w k – – −418.4 – – Tinst – – 872.7 – – SM, [36] eff tot ( )w Γ→k k 815.0 640.4 605.2 – 575.4 eff tot ( ) X w k −314.6 −441.2 −385.2 – −384.9 Tinst 656.2 920.297 803.5 – 802.8 1GEM, [36] eff tot ( )w Γ→k k 796.8 642.6 675.6 – 756.6 eff tot ( ) X w k −372.8 −379.8 −367.6 – −369.8 Tinst 777.6 792.2 766.8 – 771.4 2GEM, [36] eff tot ( )w Γ→k k 744.0 722.4 703.2 – – eff tot ( ) X w k −248.0 −240.8 −234.4 – – Tinst 517.3 502.3 488.9 – – present work prm ( )w Γ→k k 458.7 458.7 458.7 458.7 458.7 prm ( )w Γk 500.0 500.0 500.0 500.0 500.0 prm ( ) X w k −308.0 −308.0 −308.0 −308.0 −308.0 * mag ( )w Γk 0 61.4 89.8 93.8 123.6 mag ( ) X w k 0 −20.5 −29.9 −31.3 −41.2 tot ( )w Γ→k k 458.7 520.1 548.5 552.5 582.3 tot ( )w Γk 500.0 561.4 589.8 593.8 623.6 tot ( ) X w k −308.0 −328.5 −337.9 −339.3 −349.2 Tinst 745 745 745 745 745 TK = 771 K is the experimental Kurnakov temperature value reported in [84]. The title abbreviations of theoretical methods are in Table 5. * For ‘exchange’ interactions, which are the ‘direct’ and limited to the 1 st coordination shell only, mag mag ( ) ( )w wΓ Γ= →k k k  . Because the estimated Curie point of f.c.c.- 62Ni0.765Fe0.235 is ТС = 862 K, for energies calculated at Та = 958 K, it is necessary to replace eff eff tot prm ( ) ( )w w→k k  . 294 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA Next, substituting the dependence tot ( , , )w c Tk into the KСМ formu- la (1) (assuming D ≡ 1 in (2)), it is possible to calculate the temperature dependence of the diffuse scattering intensities, ISRO(k,c,T), for quasi- wave vectors k within the 1 st BZ such as kX, Δ(kX → kΓ) and kΓ. The re- sults of such calculations and the recalculated data (using the ‘mixing’ energies taken from [30, 33, 36, 79] and the data presented in Tables 5 and 6) are shown in Fig. 6 in comparison with the experimental results reported in [79, 84, 85]. From Fig. 6, one can see that the temperature dependence of the dif- fuse scattering intensity (in accordance with both approximations (12) and (14)) for the dominant superstructural point kХ, ISRO(kХ,T) are in an excellent agreement with the experimental data reported in [79, 84, 85]. It is also visible that the ‘non-analyticity’ of the interatomic- interaction energies’ Fourier component at k = kΓ (see Refs in [58, 60]) leads to the difference in the diffuse scattering intensities for the qua- Fig. 5. The temperature dependence of the spontaneous magnetization for α- th (α = Ni, Fe) subsystem, σα(T), calculated for f.c.c.-Ni0.765Fe0.235 alloy with- out the atomic LRO; σα (1) is extracted by solving Eqs (11), σα (2) is extracted with use of (13) (κNi ≈ 3/4, κFe ≈ 1/2). In calculations, the ‘exchange’ ‘integrals’ Fourier components, ( )J ′αα k , are chosen for values of sNi = 1/2, sFe = 3/2 (Table 4). Note that, at the Kurnakov point precisely, TK, the functions σα (1,2)(T) should demonstrate the jumps, Δσα (1,2)(TK), due to the jump of the atomic LRO parameter, Δη(TK), and, in such a case, the functions σα (1,2)(T ∈ (0, TK)) must be calculated in accordance with the more general SCF + MSCF equations with respect to (11), which can be found elsewhere [56—61]. Since here, we are in- terested in consideration of the (meta)stable FMS region only, such a reasona- ble correction in the σα (1,2)(T) functions may be omitted for T ∈ (TK, TC) (and even for the quenched ‘high-temperature’ SRO state fixed at T ∈ (0, TK)). Here, FML, FMS and PMS are the ferromagnetic atomic LRO, ferromagnetic atomic SRO and paramagnetic atomic SRO regions, respectively. The estimat- ed Curie temperature, TC, is equal to 862 K. ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 295 si-wave vectors kΓ and Δ(kX → kΓ). (Taking into account such a differ- ence is crucially important for the correct interpretation of the dif- fraction data on isostructural and spinodal decompositions in solid so- lutions.) The obvious disagreement between the calculated temperature de- pendence of the diffuse scattering intensities for ‘fundamental’ Bragg’s (structural) reflection, ISRO(kΓ,T), and the estimated depend- ences ISRO(k → kΓ,T) (obtained by means of the extrapolation of exper- imental data taken from Refs [79, 84, 85]) is most likely due to the ex- perimental difficulties appeared in the measurement and during sepa- ration of the diffuse scattering intensity (because of the strong coher- ent scattering as well as the scattering by phonons, magnons, and other nonlocalized excitations) in the vicinity of the ‘fundamental’ Γ(000) point. In addition, the diffuse scattering intensities for all quasi-wave vectors significantly deviate from the KСМ prediction (at assumption of tot ( ) constw =k ), when the temperature decreases from PMS to FMS regions. At the same time, the results of the suggested models and the classical KCM approach [10—13] are matched completely within the a b Fig. 6. The temperature dependence of the diffuse scattering intensities, ISRO(k,T) in [Laue units], for f.c.c.- 62Ni0.765Fe0.235 Permalloy, for quasi-wave vectors k within the 1 st BZ: kX, Δ(kX → kΓ) and kΓ. The data in [30, 33, 36, 79] are obtained theoretically; ■–the experimental data on ISRO(k) from [84] (for 808 K from [85], for 873 and 958 K from [79]). nKCM (n = 1, 2, 3)–the results of a given work: 1KCM–a classical approximation with an assumption of tot prm ( ) ( ) constw w≡ =k k  [10—13], 2KCM and 3KCM are based on Eq. (12) and Eq. (14), respectively. As assumed here, prm ( ) X w k = −0.30 eV, prm ( )w Γk = 0.50 eV, and prm ( ) X w Γ→k k = 0.46 eV (see also Tables 3 and 6) [60]. Here, the re- sults of other theories should be addressed to KCM [79], 1LIMCM [30], 2LIMCM [30], SM [36], 1GEM [36], 2GEM [36], 1CFM [33], 2CFM [33], and 3CFM [33]. Tinst is the absolute stability-loss temperature: Tinst (1,2,3) = 626, 745, 755 K. The data, for which Tinst > 745 K, are not shown for lucidity. The esti- mated Curie temperature, TC, is equal to 862 K (16). 296 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA PMS region for the alloy at issue (T > TC). 3.4. Constant D in the KCM Formula for F.C.C.-Ni—Fe Alloys Finally, let us analyse in this Section the constant D (2) entered into the KCM formula (1). In the theoretical works [30—36] and experi- mental studies [79, 84, 85], much attention was paid to the analysis of the normalization constant D (2). In particular, the main conclusion was about the strong concentration and temperature dependences of D, D(T,c), and the significant deviation of D from D ≡ 1 with decreasing temperature. As a result, the authors claim that the KCM formula (1) cannot be used adequately for the analysis of f.c.c.-Ni—Fe alloys at temperatures close to the Kurnakov points and within the FMS region. Their argumentation is based solely on the fact that the KCM approach is adequate if and only if the condition is satisfied as follows: tot ( ) / ( ) 1X Bw k Tk  . However, as can be seen from Fig. 6, there is no necessity in such a conclusion. Let us consider this point in detail fit- ting with observed data. Using the total ‘mixing’ energies’ Fourier components’ representa- tion in the form of (12) with the factor D (2) for f.c.c.-Ni—Fe alloys, one can carry out the integration over the 1 st BZ in order to obtain the (T— c)-dependence of D = D(T,c). The results are shown in Fig. 7. For such a calculation, three cases are considered as follow: (i) prm ( ) constw =k (as original KCM assumption [10—13]), (ii) prm prm ( ) ( , )w w c=k k  (based on Table 3) [60], and (iii) tot tot ( ) ( , , )w w c T=k k  (based on Eqs (12) or (14) in line with Tables 3 and 4). As can be seen from Fig. 7, the constant D deviates slightly from the value D = 1 within the whole (T—c)-domain of f.c.c.-Ni—Fe alloys with the atomic SRO. There is only the exception near the Invar composition region (cFe ≅ 0.65). This unconventional behaviour is likely due to the competition between the (T—c)-dependent energy contributions such as Eq. (12) (within the PMS region) multiplied by the concentration— temperature factor c(1 − c)β and 1 in denominator in Eq. (2). As a re- sult, within the Invar composition region, it is necessary to consider the correction of D ≠ 1 into the KCM formula (1). In any event, the con- clusion about the constant D reported in [30—37, 79, 84, 85] does not correspond to validity because of fact that the factor D tends strongly to D = 1 within the FMS region, compared with the РMS one (Figs 7, b, c). Moreover, in this case, the ratio prm ( ) / ( ) 1Bw k Tk  [10—13] is not violated, and the ratio tot ( ) / ( )Bw k Tk can be even compared with 1. 4. CONCLUSIONS In a given work, within the scope of the SCF and MSCF approximations ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 297 [56—61], on an example of detailed consideration of disordered f.c.c.- Ni—Fe alloy with the atomic SRO, the statistical-thermodynamics model of the substitutional atomic SRO states for binary (pa- ra)magnetic alloys based on the cubic Bravais crystal lattice with two magnetic constituents is improved. In addition, a simple approxima- tion representing the total ‘mixing’ energies’ Fourier components as a polynomial temperature-dependent function of 3 rd order within the magnetic (T—c)-domain of an alloy is suggested. As a result, the inter- polating relations for the (T—c)-dependence of the diffuse scattering intensities are obtained on the basis of the classical KCM formula and representing the total ‘mixing’ energies’ Fourier components as a sum of three energy contributions of different natures, namely, the ‘short- range’ and ‘direct’ ‘electrochemical’ and ‘exchange’ (magnetic) inter- actions as well as the ‘long-range’ and ‘indirect’ ‘strain-induced’ in- teraction. In both cases, we have suggested the explicit form for (T—c)- dependence of the total ‘mixing’ energies (Eqs (12) and (14)) for f.c.c.- Ni—Fe alloys. Using the statistical MC simulation technique and the Warren— a b c Fig. 7. The (T—c)-dependence of the normalization ‘constant’, D (namely, D = D(c,T)), in Eq. (2) calculated at several assumptions: (a) prm ( ) constw =k [10—13], (b) prm prm ( ) ( , )w w c=k k  (Table 3) [60], (c) tot tot ( ) ( , , )w w c T=k k  (see Eqs (12) or (14) and Tables 3 and 4). Here, c ≡ cFe. The function TC(cFe) is calcu- lated according to Eq. (16). 298 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA Cowley SRO parameters extracted from the radiation diffuse scatter- ing experiments [79—83], the local atomic configurations (and their Fourier transforms, i.e. diffuse scattering patterns) for f.c.c.-Ni—Fe alloys within the Permalloy, Elinvar and Invar compositions are ob- tained at temperatures above and below the Curie points, TC. As shown, the diffuse scattering intensity at the superstructural quasi- wave vector, kX(001), significantly depends on temperature and com- position of an alloy. This phenomenon testifies on the essential role of the alloy magnetism in thermodynamics of the atomic SRO states. Within the scope of suggested models, we have calculated the (T—c)- dependence of the diffuse scattering intensities for two preferential quasi-wave vectors within the 1 st BZ, namely, for the high-symmetry points–superstructural kX(001) and ‘fundamental’ kΓ(000). As a re- sult, as shown unambiguously, taking into account the magnetic (‘ex- change’) interactions in the statistical thermodynamics of disordered alloys leads to the significant deviation of the diffuse scattering inten- sity from its classical form governed by the KCM formula, which as- sumes the (T—c)-independence of the total ‘mixing’ energies. Such a deviation is mainly caused by the strong (T—c)-dependence of the mag- netic ‘mixing’-energies’ contribution (Eqs (12) and (14)) within the magnetic domain of an alloy phase diagram as well as its strong influ- ence on the atomic subsystem and vice versa. Furthermore, we have carefully verified the accuracy of suggested models on an example of the prediction of the diffuse scattering inten- sities for f.c.c.- 62Ni0.765Fe0.235 Permalloy across its magnetic and para- magnetic domains with the atomic SRO. The results were compared with the most reliable data of the neutrons’ diffuse scattering experi- ments [79, 84, 85] as well as with the theoretical results reported be- fore [30—36]. As a result, we have found an excellent agreement of both approximate models reported here with the experiments (Fig. 6). Finally, within the scope of the suggested models, the normalization factor D, which enters into the KCM formula, has been estimated with- in the whole (para)magnetic (T—c)-domain of f.c.c.-Ni—Fe alloys with the atomic SRO. As shown, the constant D does not significantly devi- ate from the value D = 1 for the whole (T—c)-domain of (para)magnetic f.c.c.-Ni—Fe alloys, and within the magnetic region of an alloy, such a requirement is satisfied rigorously, compared with the paramagnetic state of the system at issue. As a result, we demonstrate that, in case of magnetic alloys, the numerous criticisms of the KCM formula as such expressed by many researchers over the last several decades are ground- less for realistic simulation of interatomic interactions. Moreover, the correct consideration of magnetism (even within the scope of the sim- plified local magnetic-moments’ model only) in the statistical thermo- dynamics of magnetic alloys is still making the KCM approach as a unique theoretical tool for precise evaluations of the interatomic ‘mix- ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 299 ing’ energies in magnetically ordered mixed systems. Therefore, as may be concluded, in many practical cases, it is not necessary to overestimate the relevance of both the interatomic corre- lation and the many-body interaction effects in statistical physics of disordered magnetic materials. Almost all reliable experimental fea- tures of the diffuse scattering of radiations can be easily explained and reasonably described within the scope of the simplest ‘pairwise’ intera- tomic-interactions’ model solely and with using the conventional SCF and MSCF approximations, in particular, the classical KСМ formula within the simple representation of the total ‘mixing’ energies’ Fouri- er components in the form of (12) or (14). In conclusion, one may suggest the possible applications of the sug- gested models for ‘indirect’ evaluation of the magnetic ‘mixing’ ener- gies (or ‘exchange’ ‘integrals’) for atoms with magnetic moments in alloys by using the synchrotron X-ray scattering technique only in- stead of the conventional neutron scattering methods [86—88]. Thus, (i) by measuring the temperature dependence of the diffuse scattering intensity, ISRO(ks,T), within the paramagnetic and magnetic (T—c)- domains for the superstructural quasi-wave vector, ks, generating the dominant fluctuation concentration wave in a magnetic alloy with the atomic SRO, and (ii) by interpolating these data by the KCM formula (1), taking into account the representation by (12) or (14), it becomes possible to estimate theoretically a set of microscopic parameters of an alloy, including magnetic ‘mixing’ energies and corresponding ‘ex- change’ ‘integrals’. We believe that the presented study may shed a fresh insight into the physics of substitutional atomic-SRO phenomena in conventional and up-to-date (para)magnetic alloys, revising several established con- ceptions in the statistical thermodynamics of solid solutions. ACKNOWLEDGEMENTS Authors of a given article are grateful to Dr. R. V. Chepulskii (Samsung Electronics, U.S.A.), Prof. B. Schönfeld (ETH, Switzerland), Prof. J. B. Staunton (Warwick University, U.K.), Dr. G. E. Ice (Oak Ridge Nation- al Laboratory, U.S.A.), Prof. S. Hata (Kyushu University, Japan), and Dr. F. Bley (France) for providing their own publications, communi- cating important references and stimulating discussions about atomic ordering in metallic alloys. One of the co-authors (S.M.B.) would like to thank the members of the Groupe de Physique des Matériaux (Universi- té de Rouen, France) for their kind hospitality during his stay in 2006— 2008 at the Rouen University as well as the Institute for Advanced Ma- terials Science and Innovative Technologies (Vilnius, Lithuania) for partial financial support. Another co-author (I.V.V.) acknowledges the Groupe de Physique des Matériaux (Université de Rouen, France) too for 300 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA partial financial support of a given work. REFERENCES 1. M. A. Krivoglaz and A. A. Smirnov, The Theory of Order—Disorder in Alloys (Lon- don: Macdonald: 1965). 2. D. de Fontaine, Solid State Physics (Eds. H. Ehrenreich et al.) (New York: Aca- demic Press: 1979), vol. 34, p. 73. 3. F. Ducastelle, Order and Phase Stability in Alloys (New York: Elsevier: 1991). 4. A. G. Khachaturyan, Progr. Mater. Sci., 22: 1 (1978). 5. A. G. Khachaturyan, Theory of Structural Transformations in Solids (Mineola, NY: Dover Publications Inc.: 2008). 6. M. A. Krivoglaz, X-Ray and Neutron Diffraction in Nonideal Crystals (Berlin: Springer: 1996). 7. M. A. Krivoglaz, Diffuse Scattering of X-Rays and Thermal Neutrons by Fluctua- tional Inhomogeneities of Imperfect Crystals (Berlin: Springer: 1996). 8. B. Schönfeld, Progr. Mater. Sci., 44, No. 5: 435 (1999). 9. G. E. Ice and C. J. Sparks, Annu. Rev. Mater. Sci., 29: 25 (1999). 10. M. A. Krivoglaz, Zh. Eksp. Teor. Fiz., 32: 1368 (1957). 11. P. C. Clapp and S. C. Moss, Phys. Rev., 142, No. 2: 418 (1966). 12. P. C. Clapp and S. C. Moss, Phys. Rev., 171, No. 3: 754 (1968). 13. S. C. Moss and P. C. Clapp, Phys. Rev., 171, No. 3: 764 (1968). 14. R. A. Tahir-Kheli, Phys. Rev., 188, No. 3: 1142 (1969). 15. R. Kikuchi, Phys. Rev., 81, No. 6: 988 (1951). 16. A. Finel, Statistics and Dynamics of Alloy Phase Transformations. NATO Ad- vanced Studies Institute, Series B: Physics. Vol. 319 (Eds. P. E. A. Turchi and A. Gonis) (New York: Plenum: 1994), p. 495. 17. J. M. Sanchez, V. Pierron-Bohnes, and F. Meja-Lira, Phys. Rev. B, 51, No. 6: 3429 (1995). 18. R. Brout, Phase Transitions (New York: Benjamin: 1965). 19. I. R. Yukhnoskii and Z.A. Gurskii, Quantum Statistical Theory of Disordered Sys- tems (Kiev: Naukova Dumka: 1991) (in Russian). 20. B. L. Gyorffy and G. M. Stocks, Phys. Rev. Lett., 50, No. 5: 374 (1983). 21. J. B. Staunton and B.L. Gyorffy, Phys. Rev. Lett., 69, No. 2: 371 (1992). 22. J. B. Staunton, Rep. Progr. Phys., 57, No. 12: 1289 (1994). 23. J. B. Staunton, D. D. Johnson, and F. J. Pinski, Phys. Rev. B, 50, No. 3: 1450 (1994). 24. D. D. Johnson, J. B. Staunton, and F. J. Pinski, Phys. Rev. B, 50, No. 3: 1473 (1994). 25. F. J. Pinski, J. B. Staunton, and D. D. Johnson, Phys. Rev. B, 57, No. 24: 15177 (1998). 26. J. B. Staunton, S. S. A. Razee, M. F. Ling, D. D. Johnson, and F. J. Pinski, J. Phys. D: Appl. Phys., 31, No. 19: 2355 (1998). 27. D. A. Rowlands, J. B. Staunton, B. L. Gyorffy, E. Bruno, and B. Ginatempo, Phys. Rev. B, 72, No. 4: 045101 (2005). 28. D. A. Rowlands, A. Ernst, B. L. Gyorffy, and J. B. Staunton, Phys. Rev. B, 73, No. 16: 165122 (2006). 29. V. Gerold and J. Kern, Acta Metall., 35, No. 2: 393 (1987). 30. F. Livet, Acta Metall., 35, No. 12: 2915 (1987). ATOMIC ORDER AND DIFFUSE SCATTERING OF RADIATIONS IN F.C.C.-Ni—Fe 301 31. V. G. Vaks, N. E. Zein, and V. V. Kamyshenko, J. Phys. F: Metal Phys., 18, No. 8: 1641 (1988). 32. V. G. Vaks, N. E. Zein, and V. V. Kamyshenko, J. Phys.: Condens. Matter, 1, No. 11: 2115 (1989). 33. V. G. Vaks and V. V. Kamyshenko, J. Phys.: Condens. Matter, 3, No. 10: 1351 (1991). 34. V. I. Tokar, Phys. Letters A, 110, No. 9: 453 (1985). 35. T. A. Grishchenko, I. V. Masanskii, and V. I. Tokar, J. Phys.: Condens. Matter, 2, No. 21: 4769 (1990). 36. V. I. Tokar, I. V. Masanskii, and T. A. Grishchenko, J. Phys.: Condens. Matter, 2, No. 50: 10199 (1990). 37. I. V. Masanskii, V. I. Tokar, and T. A. Grishchenko, Phys. Rev. B, 44, No. 9: 4647 (1991). 38. I. Tsatskis, Phil. Mag. Lett., 78, No. 5: 403 (1998). 39. I. Tsatskis, J. Phys.: Condens. Matter, 10, No. 9: L145 (1998). 40. I. Tsatskis and E. K. H. Salje, J. Phys.: Condens. Matter, 10, No. 17: 3791 (1998). 41. I. Tsatskis, Phys. Letters A, 241, Nos. 1—2: 110 (1998). 42. R. V. Chepulskii, J. Phys.: Condens. Matter, 10, No. 7: 1505 (1998). 43. R. V. Chepulskii and V. N. Bugaev, J. Phys.: Condens. Matter, 10, No. 33: 7309 (1998). 44. R. V. Chepulskii and V. N. Bugaev, J. Phys.: Condens. Matter, 10, No. 33: 7327 (1998). 45. R. V. Chepulskii and V. N. Bugaev, J. Phys.: Condens. Matter, 10, No. 39: 8771 (1998). 46. R. V. Chepulskii and V. N. Bugaev, J. Phys. Chem. Solids, 59, No. 9: 1469 (1998). 47. R. V. Chepulskii, J. Phys. Chem. Solids, 59, No. 9: 1473 (1998). 48. R. V. Chepulskii and V. N. Bugaev, Solid State Commun., 105, No. 10: 615 (1998). 49. R. V. Chepulskii, J. Phys.: Condens. Matter, 11, No. 44: 8645 (1999). 50. R. V. Chepulskii, J. Phys.: Condens. Matter, 11, No. 44: 8661 (1999). 51. R. V. Chepulskii, Phys. Rev. B, 61, No. 13: 8606 (2000). 52. R. V. Chepulskii, J. Phys.: Condens. Matter, 14, No. 8: L193 (2002). 53. R. V. Chepulskii, J. B. Staunton, E. Bruno et al., Phys. Rev. B, 65, No. 6: 064201 (2002). 54. R. V. Chepulskii, Phys. Rev. B, 69, No. 13: 134431 (2004). 55. R. V. Chepulskii, Phys. Rev. B, 69, No. 13: 134432 (2004). 56. V. A. Tatarenko, T. M. Radchenko, and V. M. Nadutov, Metallofiz. Noveishie Tekhnol., 25, No. 10: 1303 (2003) (in Ukrainian). 57. V. A. Tatarenko and T. M. Radchenko, Intermetallics, 11, Nos. 11—12: 1319 (2003). 58. V. A. Tatarenko, S. M. Bokoch, V. M. Nadutov, T. M. Radchenko, and Y. B. Park, Defect and Diffusion Forum, 280—281: 29 (2008). 59. S. M. Bokoch and V. A. Tatarenko, Solid State Phenomena, 138: 303 (2008). 60. S. M. Bokoch and V. A. Tatarenko, Uspehi Fiziki Metallov, 11, No. 4: 413 (2010). 61. I. V. Vernyhora, V. A. Tatarenko, and S. M. Bokoch, ISRN Thermodynamics, 2012: ID 917836-1—11 (2012); doi:10.5402/2012/917836. 62. J. S. Smart, Effective Field in Theories of Magnetism (Philadelphia: W. B. Saun- ders Company: 1966). 63. D. Jiles, Introduction to Magnetism and Magnetic Materials (London: Charman & Hall: 1991). 64. A. Aharoni, Introduction to the Theory of Ferromagnetism (New York: Oxford 302 S. M. BOKOCH, V. A. TATARENKO, and I. V. VERNYHORA University Press: 2000). 65. S. V. Semenovskaya, phys. stat. sol. (b), 64, No. 1: 291 (1974). 66. S. M. Bokoch, M. P. Kulish, and T. D. Shatnii, Metallofiz. Noveishie Tekhnol., 26, No. 5: 627 (2004) (in Russian). 67. S. M. Bokoch, M. P. Kulish, V. A. Tatarenko, and T. M. Radchenko, Proc. of the 1 st International Conf. on Diffusion in Solids and Liquids—DSL-2005 (July 6—8, 2005, Aveiro), vol. 1, p. 57. 68. S. M. Bokoch, N. P. Kulish, and V. A. Tatarenko, Fundamental’nyye Problemy Sovremennogo Materialovedeniya, 4: 78 (2007) (in Russian). 69. S. H. Rahman, Acta Crystallogr. A, 49, No. 1: 56 (1993). 70. S. H. Rahman, Acta Crystallogr. A, 49, No. 1: 68 (1993). 71. S. H. Rahman and M. Rodewald, Acta Crystallogr. A, 51, No. 2: 153 (1995). 72. K. Osaka and T. Takama, Acta Mater., 50, No. 6: 1289 (2002). 73. S. Hata, S. Matsumura, N. Kuwano, and K. Oki, J. Surf. Analysis, 3: 401 (1997). 74. S. Hata, H. Fujita, C. G. Schlesier, S. Matsumura, N. Kuwano, and K. Oki, Mater. Trans. JIM, 39, No. 1: 133 (1998). 75. S. Hata, S. Matsumura, N. Kuwano, and K. Oki, Acta Mater., 46, No. 3: 881 (1998). 76. S. Hata, S. Matsumura, N. Kuwano, K. Oki, and D. Shindo, Acta Mater., 46, No. 14: 4955 (1998). 77. S. Hata, T. Mitate, N. Kuwano, S. Matsumura, D. Shindo, and K. Oki, Mater. Sci. Eng. A, 312, No. 1—2: 160 (2001). 78. U. D. Kulkarni, Acta Mater., 52, No. 9: 2721 (2004). 79. S. Lefebvre, F. Bley, M. Fayard, and M. Roth, Acta Metall., 29, No. 5: 749 (1981). 80. P. Cenedese, F. Bley and S. Lefebvre, Mat. Res. Soc. Symp. Proc., 21, 351 (1984). 81. X. Jiang, G. E. Ice, C. J. Sparks, L. Robertson, and P. Zschack, Phys. Rev. B, 54, No. 5: 3211 (1996). 82. G. E. Ice, C. J. Sparks, A. Habenschuss, and L. B. Shaffer, Phys. Rev. Lett., 68, No. 6: 863 (1992). 83. J. L. Robertson, G. E. Ice, C. J. Sparks, X. Jiang, P. Zschack, F. Bley, S. Lefebvre, and M. Bessiere, Phys. Rev. Lett., 82, No. 14: 2911 (1999). 84. F. Bley, Z. Amilius, and S. Lefebvre, Acta Metall., 36, No. 7: 1643 (1988). 85. S. Lefebvre, F. Bley, M. Bessiere, M. Fayard et al., Acta Crystallogr. A, 36, No. 1: 1 (1980). 86. G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering (Cam- bridge: Cambridge University Press: 1978). 87. S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter (New York: Oxford University Press: 1984), vols. 1, 2. 88. Neutron Scattering from Magnetic Materials (Ed. T. Chatterji) (Amsterdam: Elsevier: 2006).

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