AN X-RAY DIFFRACTION STUDY OF THE THERMAL MOTION OF ATOMS IN β-TIN

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AN X-RAY DIFFRACTION STUDY OF THE THERMAL MOTION OF ATOMS IN β -TIN

B. Greenberg, G. Rothberg

To cite this version:

B. Greenberg, G. Rothberg. AN X-RAY DIFFRACTION STUDY OF THE THERMAL MO- TION OF ATOMS IN β-TIN. Journal de Physique Colloques, 1980, 41 (C1), pp.C1-145-C1-146.

�10.1051/jphyscol:1980137�. �jpa-00219706�

JOURNAL DE PHYSIQUE Colloque Cl

,

supplkment au n ^O 1 , Tome 41, janvier 1980, page (21-145

AN X-RAY DIFFRACTION STUDY OF ME

THERMAL

MOTION OF

ATOFIS

IN 6-TIN

B. Greenberg and G.M. Rothberg

Department of Materials and MetaZZurgicaZ Engineering, Stevens I n s t i t u t e of TechnoZogy, Hoboken, N e w Jersey 07030, USA.

Metallic 6-tin has a tetragonal crystal struc- t u r e , thus the mean square atomic displacement in the c-direction u i may d i f f e r from t h a t in the a-b plane ui. t~liissbauer e f f e c t determinations of these displacements i n the e a r l y 1960's [I-41 were used t o r e f i n e l a t t i c e dynamical models. There has been no experimental a c t i v i t y of t h i s kind since then on s i n g l e c r y s t a l s and very l i t t l e on polycrystal s I51 .

The subject merits f u r t h e r work f o r several reasons.

( 1 ) The e a r l y work on s i n g l e c r y s t a l s may have suf- fered from nonuniform thickness of absorbers, incorrect resonance absorption cross section oo, unaccounted f o r e f f e c t s of quadrupole s p l i t t i n g since both s p l i t sources and absorbers were used, and nonuniform source motion; however, usually not enough d e t a i l s are given f o r one t o be c e r t a i n of the presence o r absence of any of these possible sources of e r r o r . ( 2 ) Anharmonic e f f e c t s may be studied by carrying out measurements above room temperature. The one s i n g l e crystal Miissbauer experiment above room temperature did s u f f e r from some of the e r r o r s mentioned.

( 3 )

Since MGssbauer and x-ray determinations of Debye-Waller f a c t o r s present d i f f e r e n t experimental d i f f i c u l t i e s , the comparison of r e s u l t s may illuminate problems asso- ciated with the two techniques. No x-ray s t u d i e s have been made in B-tin of the temperature depend- ence of the thermal motion.

W report here r e s u l t s of an x-ray study of e thermal motions between 100 and

4500K. A

more corn-

ment will be publis'

ed

elsewhere and i s available in unpublished from 161. Here only r e s u l t s related t o M6qsbauer parameters will be presented. X-ray d i f f r a c t i o n experiments were performed on a s i n g l e c r y s t a l . The crystal face studied was prepared by spark c u t t i n g , mechanical polishing and then elec- tropol ishing t o remove a1 1 damaged material . ^The

face intercepted the e n t i r e x-ray beam. Integrated i n t e n s i t i e s measured by

8-28

scans were collected from the (004). (503) and (053) d i f f r a c t i o n planes using n i c k e l - f i l t e r e d copper

K'%

radiation with pulse height analysis t o eliminzte

n/2

harmonics of the c h a r a c t e r i s t i c radiation. The occurrence of simul- taneous (multiple) r e f l e c t i o n s may s i g n i f i c a n t l y a f f e c t the measured i n t e n s i t y of a Bragg peak;

therefore detailed calculations were made which confirmed t h a t s'imul taneous r e f l e c t i o n s were not occurring during these measurements.

The i n t e n s i t y I ' corrected f o r background and thermal d i f f u s e s c a t t e r i n g i s taken t o be I '

=

K l

Fo

1 ^{'e-" where}

^K

i s a coastant, Fo i s t h e x-ray s t r u c t u r e f a c t o r assuming no thermal motiqr~

and e-" i s the x-ray Debye-Ua:ler f a c t o r which accounts f o r the r e d u c t i o ~ in i n t e n s i t y of t h e Bragg maxima due t o 'chema? motion. In the quasi- harmonic approximation

( Q H )

thermal expansion i s t h e only anharmonic e f f e c t considered. By defining a reduced temperature

~ l e t e discussion of the x-ray aspects of t h e e x p e r i - 1 n I ' becomes a l i n e a r function of

T"

w i t h slope

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980137

C1-146 JOURNAL DE PHYSIQUE

-2W i n t h e Oebye model. Here yG i s t h e Grnneisen constant,

V-ro

i s volume a t a standard reference temperature, i n o u r case 293'K. and

Q[?]

⁼

+ [i]

2 ~ [ f ]

] ^,

where 0 i s t h e Oebye i n t e g r a l and O i s t h e Debye temperature, taken t o be 132OK.

For our temperature range T" i s i n s e n s i t i v e t o t h e value of O.

I n t h e QH approximation u2aT". Table

I

g i v e s r e s u l t s a t T" =293OK and a t 50 degree i n t e r v a l s from 100-450°K. Below 260°K t h e u 2 versus T" data a r e f i t w e l l w i t h a s t r a i g h t l i n e . Deviations grow as T" increases above 260°K i n d i c a t i n g t h e presence o f constant volume anharmonic e f f e c t s . To d e t e r - mine 2W a t T">260°K t h e value o f K I F ~ ~ ~ from t h e l i n e a r f i t was used. A q u a d r a t i c polynomial i n T"

f i t t h e data w e l l a t T">260°K and i s c o n s i s t e n t w i t h v a r i o u s models o f anharmonic thermal motion.

The (503) and (053) data d i d n o t agree w i t h i n t h e i r s t a t i s t i c a l u n c e r t a i n t i e s ; however, s i n c e we c o u l d n o t j u s t i f y f a v o r i n g one data s e t over t h e other, t h e two s e t s were averaged and t h e e r r o r s c o r r e s - pondingly increased. For t h i s reason r e s u l t s i n t h e a-b plane have f a i r l y l a r g e u n c e r t a i n t i e s . I n

2 2

t a b l e

I

t h e r e c o i l l e s s f r a c t i o n s f = e-k

,

where k2 = 146.34A-*. The p o l y c r y s t a l l i n e value fa, i s t h e angular average o f t h e d i r e c t i o n a l l y dependent r e c o i l le s s f r a c t i o n and was 'found t o be p r a c t i c a l l y i d e n t i c a l t o t h e v a l u e obtained from u i v =

1/3 [u:+2uil.

The fav agree w e l l w i t h d i r e c t l y measured p o l y c r y s t a l l i n e Mijssbauer values. The i n d i v i d u a l MGssbauer r e s u l t s f o r fz and f x may be s u b j e c t t o t h e e r r o r s a l r e a d y mentioned. W i t h i n t h e i r uncer- t a i n t i e s the two s e t s o f resu7ts f o r fZ and fx a l s o agree. The room temperature Massbauer v a l u e f o r f Z / f x = 0.68(4) i s an average o f t h r e e measurements.

The 295OK x-ray r e s u l t s o f F i e l d 171 a r e u i = 232(1 )x10-'A2, ux = 201 ( 1 ) ~ 1 0 - ~ A ~ , f, = 0.0335(5),

fx = 0.053(1), f z / f x = 0.64(2). The s i n g l e c r y s t a l MGssbauer r e s u l t s i n d i c a t e fz/fx decreases mono- t o n i c a l l y w i t h temperature. Our r e s u l t s i n d i c a t e t h i s r a t i o i s a minimum near room temperature and increases a t h i g h e r temperatures s i n c e t h e (503) and (053) data s e p a r a t e l y y i e l d t h i s behavior.

Our r e s u l t s a l s o show constant volume anharmonic e f f e c t s i n c r e a s e

: u

by about 8 percent a t 430°K and u i by about 20 percent.

Table I. X-Ray Results. Numbers i n parentheses show standard d e v i a t i o n i n l a s t d i g i t .

This research was supported by N a t i o n a l Science Foundation q r a n t s DMR-77-09911 and SER77-06938.

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