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MECHANICAL RELAXATION OF SUBSTITUTIONAL-INTERSTITIAL
(TETRAHEDRAL) ATOM COMPLEXES IN THE bcc LATTICE
M. Koiwa, S. Ishioka, G. Cannelli, R. Cantelli
To cite this version:
M. Koiwa, S. Ishioka, G. Cannelli, R. Cantelli. MECHANICAL RELAXATION OF SUBSTITUTIONAL-INTERSTITIAL (TETRAHEDRAL) ATOM COMPLEXES IN THE bcc LAT- TICE. Journal de Physique Colloques, 1981, 42 (C5), pp.C5-787-C5-792. �10.1051/jphyscol:19815122�.
�jpa-00220992�
CoLLoque C 5 , suppLkment au nOIO, Tome 4 2 , octobre 1981 page C5-787
MECHANICAL RELAXATION OF SUBSTITUTIONAL-INTERSTITIAL (TETRAHEDRAL) ATOM COMPLEXES I N THE bcc L A T T I C E
M. Koiwa, S. Ishioka, G. Cannelli and R. Cantelli X -X
The Research . I n s t i t u t e j"or Iron, SteeL and Other Metals, Tohoku ilniuersity, Sendai, Japan
" ~ s t i t u t o d i Acustica "O.M. Corb<noU, ConsigZio NazionaZe deLLe Ricerehe-Rome, I t a Ly
Abstract- The relaxation of a substitutional-tetrahedral interstitial atom pair in the bcc lattice is studied theoretically. Three relaxation modes can be cxcited mechanically, one by a tensilc stress along the <Ill> direction and the other two by the <lo02 direction. The kinetics o f the <loo> relaxation are examined in detail. Experimental results of internal friction studies on Kb-Ti-I1 and V-Ti-II alloys are discussed in the light of the present analysis.
1. Introduction: In internal friction studies o f Nb-l'i-H [1,2] and V - T i - H [ 3 ] alloys, new rclaxation peaks are found which are considered to be due to Ti-11 complexes. A similar effect has also been found by Kronmiiller and Vettcr [ 4 ] in magnetic after-effect measurements o n Fe-l'i-II and F e - T i - D alloys. llydrogen atoms are usually considered to occupy tetrahedral interstices in various bcc metals. In the present paper, therefore, we discuss the relaxation process of a substitutional atom - a tetrahedral interstitial atom (s-i) pair in the bcc lattice; the relaxation of the s-i(octahedra1) pair has been previously discusscd by one o f the present authors [ S ] .
2. Classification o f relaxation modes- A general theory o f the rclaxation of point dcfccts is discussed in detail by Nowick and Bcrry [ 6 ] . Although the mechanical relaxation modes associated with paircd point dcfects in cubic lattices have bcen derived and tabulated by Chang [7], the relaxation of the s - i (tetrahedral) pair has not been treated in his paper.
The s - i (tetrahedral) pair in the bcc lattice (Fig.1) is a <loo> monoclinic defect. Consider the defect oricntation in the figure. The principal axes ( x ' and 2 ' ) o f the A tensor are inclined with respect to the crystal axes:(x and z). Thc componcnts o f the tensor, Aij , in the crystal coordinate system arc given by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19815122
C5-788 JOURNAL DE PHYSIQUE
where Xi are the principal values.
By the standard method (see, e.g. [ 6 ] ) , the relaxation modes for the pair can be derived and are listed in Table 1. (we omitted ungerade modes.) Here we assumed that (1) the s. atom is immobile,
(2) i. atoms are confined to 24 interstices around s. atom, and (3) the i. atom jump only between nearest neighbour sites with a frequency v within one {100), and with v 1 from one {loo) to a neighbouring (100) plane, as shown in Fig.2. The numbering of sites is also indicated in this figure. Among the five types of relaxation in the table, only two of them, Teg and Eg , are excitable.
3. Relaxation process under uniaxial stresses.- All the essential information of the relaxation is given in Table 1, the derivation of which is based upon a group-theoretical consideration. In order to visualize the relaxation process it seems worthwhile to consider the problem from a somewhat different point of view; we shall examine the rearrangement of i. atoms under the application of the unidirectional stresses: <Ill> and <loo> tensile stresses.
3.1. The <Ill> case.- Under the <Ill> tensile stress, the interstices are classified into two groups of sites, 1 and 2 as shown in Fig.3a;
in each group the probability of occupation of sites by i. atoms is the same. We define the occupation probability, nl and n2 , that the
i. atom occupies any of sites in group 1 or 2. The rate equations for n1 and n2 are
These are expressed by a single equation for (nl -n2 ) , and the relaxation time is
The relaxation of the compliance is
where C o is the mole concentration of s - i pairs and vo the atomic volume of the matrix atom.
In the absence of substitutional impurity atoms, the symmetry of tetrahedral interstitials is tetragonal ( 0=0, X13=O ),and no relaxation occurs for the <Ill> tensile stress.
3.2. The <loo> case.- Under the <loo> tensile stress, the interstices are classified into three groups, 1 , 2 and 3, as shown in Fig.3b; we
the <Ill> case. The rate equations are
= - + v'n, ,
= - ( 2 u + v ' ) n , t ~ ~ l z , tu'n,,
f i 3 = - 2 r ) n , t 3 v n 2 , 1 (4)
These can be solved in a straightforward way, and we only list important quantities below. The relaxation times T+ and r - are
The normal coordinates are
u'nL - (2L) 2 D) n, t t z v - d + D)n, , (6)
The relaxation of the compliance for the two modes is
The sum of the two is independent of v and v ' ;
4. Numerical example.- The relaxation times for the three modes, one for the <Ill> case and two for the <loo> case are shown in Fig.4 as a function of the ratio of jump frequencies, v l / v . As mentioned before, the relaxation strength for the <Ill> case is considered to be rather small. Hereafter, we only discuss the ~ 1 0 0 , case.
For the sake of brevity, we assume that h e 2 = X33; this relation exactly holds for free hydrogen atoms in the matrix. The relaxation spectrum as a function of the angular frequency w can be written, from eqs. (5) and (7) as
The spectrum changes sensitively depending upon the ratio, v ' / v .
Figure 5 shows the spectra S(w) for various values of v ' / v .
(Here, wo is taken as w0'+ = 1 for v l / v = 1 . ) At v l / v =2, the peak is the highest. By increasing or decreasing the ratio, v l / v ,the peak decreases its height and broadens, and eventually splits into two peaks.
The relaxation strengths for the two components corresponding to
T+ and T - are shown in Fig.4. The peak for .r+ is always higher
than that for T- ; the strength ratio is 3 for extreme values
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of V ' / V . At V ' / V = 2 , the relaxation strength for the T - peak is
zero, and the spectrum is o f a single relaxation time.
5. Comparison with experimental results.- The relaxation peaks observed in K b - T i - H [2], and V - T i - H [3] alloys have been interpreted as due to T i - H complexes. The peaks are broader than those expected for the single relaxation process. In the previous analysis [2], it has been suggested that the broadness may be due to the presence of titanium-hydrogen complexes with different number o f hydrogen atoms:
T i - H , T i - H 2 , a - . . The present analysis suggests that the broadness could be, at least partly, o f an intrinsic nature; even for a simple complex T i - H , there exist three relaxation processes.
In the previous paper [2], two possibilities have been suggested for the origin o f subsidiary peaks at a lower temperature; (1) the reorientation of hydrogen atoms around T i - T i complexes, (2) a satellite arising also from Ti-I1 c o ~ p l e x e s which are responsible for the main peak. 'Jhe present analysis shows that the satellite does
<."
exist if two lump frequencies v and v ' , are considerably different.
With the assumption that X22=X33, however, the low temperature peak always has a larger relaxation strength than the higher temperature peak. Experimentally, the higher temperature peak is much larger for N b - T i - H idd V - T i - M alloys. Therefore, it seems reasonable to interpret the subsidiary peak as arising from Ti - f 1 complexes. A different conclusion may be derived if a diffcrent anisotropy is assumed for the A tensor of the defect pair.
The present calculation has been made on the assumption that hydrogen atoms are confined to 24 interstices around a titanium atom, and are not allowed to jump out to outside sites. A more realistic treatment should consider the possibility o f outward jumps.
References
G. Cannelli and R. Cantelli; Proc. 6th Int. Conf. on Internal Friction and Ultrasonic Attenuation in Solids, Tokyo (1977) p.491.
G. Cannelli, R. Cantelli and M. Koiwa; Phil. Mag. in press.
S. Tanaka and M. Koiwa; this issue.
H. Kronmuller and K. Vetter; Phys. Stat. Sol.(b) - 9 4 , 201 (1979).
El. Koiwa; Phil. Mag., 24, 8 1 (1971); - 24,107 (1971) ;=, 539
(1971); 24,799 (1971); - 25, 701 (1972).
A. S. Nowick and B. S. Berry; Anelastic Relaxation in Crystalline Solids (1972) Academic Press, Kew York.
R. Chang; J. Phys. Chem. Solids, 25, 1081 (1964).
interstitial atom p a i r in the bcc lattice
Representation Type normal coordinate* relaxation frequency mode
A Equilibrium s +s +sZ 0
1 g x Y
*2p, Inactive Px+Py+Pz 4v+2v '
Inactive
shear or
<Ill> tensile stress E <loo>
g
(2+2) eq. (7) es. (5)
tensile stress
c.. are the occupation probabilities of j i sites.
J 1
For the numbering of sites, see Fig. 2.
: The substitutional- : The nunbering o f interstitial a t o n edral interstices.
pair in the bcc lattice.
0: Metal atons
e : A substitutional atom o : A tetrahedral interstitial
ator:)
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< 1 1 I > C A S E
Fig. 3 : Classification of tetrahedral interstices under the application of uniaxial stresses:
a) the <Ill> tensile stress.
b) the <loo> tensile stress.
The interstices on each ring have the <100> CASE sane occupation probability.
w : Relaxation tines or t e three relaxational nodes, and the relaxation strengths for the two <loo>
relaxations (see eq. (g)) .
Fig. 5 : Spectra of the
<loo> relaxation for various values of v ' / v .
