f 2 1 U-Vvi o J cosech

In document A Mössbauer study of spin relaxation of ⁵⁷Fe ions (Page 104-110)

c X c (4-28) Orbach

Now T can be e v a l u a t e d u s i n g t h e f a c t t h a t (mean l i f e o f e x c i t e d s t a t e ) x ( e n e r g y l i n e w i d t h ) - R o r , e n e r g y w i d t h = R x ( t r a n s i t i o n p r o b a b i l i t y ) H e n c e ,

(r )

c c+a so t h a t , T nm V Orbach

3

Ac 27TpV5

L » ]

3 rA ^ c 27TpV5

k

R (T ) + c J c+a n ' m ' 4 - 2 8 ) , o n e

f

3

2lTpV5 I \ c (N +1) nm (N , ,+1) n ' m ' 7 c^c+b e x p ( c/ kT) e x p ( ^ c / k T ) -1 < a | 2 p v ^ b > 1 r n n 1 n,m < c | 2 p v* |b > l m • n' m n ' n ’ ( 4 - 29 ) , and p u t t i n g t h e v a l u e o f T rA e x p ( ^ c / k T ) - l * G(c) ( 4 - 3 0) where G(c) I < a | 2 pn B V | c ^ ^ I < c | ^ Pn , B ^ ; | b > 2 f ( 4- 3 ! ) n, m n ’m'

The summation o v e r o t h e r u p p e r l e v e l s ( l i k e | c > ) h a s been t a k e n t o f i n a l l y

o b t a i n t h e r e l a x a t i o n r a t e from a l l such p a r t i c i p a t i n g l e v e l s .

One comment r e g a r d i n g t h e v a l u e o f t h e s h o u l d be made h e r e .

The l i n e w i d t h c a l c u l a t e d e n t i r e l y from t h e t r a n s i t i o n p r o b a b i l i t y e x p r e s s i o n

i s q u i t e v a l i d when t h e r e a r e no o t h e r f a c t o r s , s uch as d i p o l a r s p i n - s p i n

i n t e r a c t i o n , which may c a u s e f u r t h e r b r o a d e n i n g o f t n e e n e r g y w i d t h . Hence

t h e above e x p r e s s i o n ( 4 - 3 0 ) i s s u i t a b l e f o r d i l u t e s y s t e m s .

F u r t h e r , i f o t h e r r e l a x a t i o n p a t h s a r e o p e r a t i v e , t h e n Tc w i l l

be l a r g e r and

Orbach

9b.

the upper limit of the relaxation rate.

3

It is also important to note that the term in expression (4-30) occurs as a result of assuming the Debye model for the phonon density of states.

4.5. Phonon Spectrum of MgCO

It is worth-while at this stage to examine the applicability of the Debye model of the phonon spectrum to the above calculations of relaxation rates. The assumption of the simple Debye model for the phonon density of states may not be well justified over the entire energy range of the phonons taking part in the relaxation process. Very little is known about the detailed phonon spectrum of MgCO.,, but the isomorphic CaCO., has been widely studied. Plihal (1973)(94) has calculated both the phonon dispersion and the phonon density of states curves for CaCO^. At low energies up to about 100 cm ^ the phonon density of states curve for CaCO., (Fig. 4.4) looks Debye-like, then drops in the region of about 125 cm and further rises to a peak value at the energy of the lower Raman-active E normal mode (about 155 cm V At higher energies the density of states decreases sharply and remains fairly small, except for a sharp peak at about 300 cm *.

Raman vibronic spectra of a number of rhombohedral carbonates, including both CaCO, and MgCO^, have been observed by Rutt and Nicola (1974)

-1

(95) (9b)

The E mode at g

(Fig. 4.5), and also by Plihal and Schaack (1970)

155 cm * in CaCO, is Raman active and the corresponding mode in MgCO^ is at 212cm An important point is that this mode corresponds to the major peak in the

phonon spectrum. Therefore, as a first approximation it may be assumed that the low energy part of the phonon density of states curve in MgCO., is similar to that calculated by Plihal for CaCO,, but stretched by a factor of (212/155). The higher energy Raman-active E i modes for CaCO and MgCO^ are at about 281 and 329 cm * respectively, and they fall in the region of another peak in the phonon density of states curve.

F i g u j c 4 . 4. For CaCO^ c r y s t a l t he c a l c u l a t e d one-phonon d e n s i t y D in (numbers o f f r e q u e n c i e s ) / 5 cm"1 .

(Taken from M . P l i h a l , phys . s t a t . so 1 . (b) 56, 4 9 5 , 1973)

Table 1. 2. < Jhserved Raman frequencies lem ).

Nearest- neighbour ( onipotind 1 x ter nal Internal Com bination distance

1> 19 E. A i. L-, A, -i- 1i ■ i (A) ( ;i( O, 155 2X1 71 1 1085 14 3 5 1748 2 356 C d C O , 1 58 271 712 1084 1388 1718 2 288 C o m , (301) (1090) — 2 109 I c< (), 194 299 731 1088 17 38 2 142 M g f O , 212 329 739 1084 1445 1763 2 101 Nit o , 235 343 736 1089 1428 1731 2 071 / n( O , 194 302 726 1090 1406 1735 2 1 10 M n C O , i s t 289 721 1088 1417 1729 2 195 345 ---n ' ^ - r v l r Z n C O — — M q C O 271 ( ^ ___

f' i gure 4 . 5. Ihc Observed spectra at low frequency (external modes). A. II, C denote gas dis­ charge light.

(Taken from - Rutt and N i c o l a , J . P h y s . C : v o l 7, 4 5 4 4 , 1 9 7 4 ) .

98.

For energies up to about 212 cm the application of the

Debye model for the phonon density of states in MgCO^ thus seems fairly reasonable, except that for energies around 190 cm * the density of states may be significantly smaller than predicted by the Debye model

(when considering Figure 4.4 modified for MgCOg),

Therefore, it is expected that the Debye model will give a reasonable description of the long wavelength phonons in which one is interested for the calculation of the Raman relaxation rates at low temperatures. When considering the Orbach process, however, all the phonons should be considered and the term (^c/ft)3 in expression (4-30) should be replaced by an equivalent quantity that is proportional to the density

of states of the phonons of energy Ac . So quite possibly the levels

ip,- and could make significantly smaller contribution to the Orbach

relaxation rate than predicted on the Debye model, and also in general any other higher energy level should make a fairly small contribution to the Orbach rate unless it happens to coincide with the high energy

peak in the phonon density of states curve.

4.6. Velocity of Acoustic Phonons in MgC07

---O

The calculation of the spin-lattice relaxation rate requires the knowledge of the velocity of acoustic phonons (i.e. the velocity of

sound) in the crystal. In the absence of any data on the direct measure­

ment of the velocity of sound in MgCO^ crystal, its value may be estimated

from the Debye temperature of the lattice. An estimate of the

can be o b t a i n e d by the e x t r a p o l a t i o n of the high tem p e r a t u r e M ö s s b a u e r i somer s hifts to 0°K in the classical limit. An advan t a g e of this m e t h o d is that it gives the e f f e c t i v e Debye t e m p e r a t u r e for Fe ions in MgCO , r ather than that of the host lattice. For a Debye solid the zero-point m o t i o n

(35)

shift (i.e. the isomer shift due to zero-p o i n t motion) is given by

(S(ZPM)

The m e a s u r e d val u e s of the isomer shifts at d i f f e r e n t t e m p e r a t u r e s h ave b een p l o t t e d in Figure 4.6. T he e x t r a p o l a t i o n o f the high t e m p e r a t u r e data to 0 K gives 6

(ZPM) 0 .080 ± 0.005 mm/sec. w h i c h c o r r e s p o n d s to the D ebye t e m p e r a t u r e 0^ = 290 ± 20 K.

The Debye t e m p e r a t u r e can a lso be e s t i m a t e d from the p h o n o n s p e c t r u m o f MgCO^. The D e b y e - W a l l e r factor for the M ö s s b a u e r event d e p e n d s on an ave r a g e o ver the p h o n o n spectrum, and is stro n g l y w e i g h t e d tow a r d s the low f r e q u e n c y e n d ^ 7 ^ . If p(u)) be the d e n s i t y of the

p h o n o n states, then the D e b y e - W a l l e r factor d epends on the integral of p(coD/^ the jow t e m p e r a t u r e limit and on the integral of at h i g h t e m p e r a t u r e s ^ 7^. C o n s e q u e n t l y , the D e b y e - W a l l e r factor, and h e n c e

the s p ectral a r e a (or r e c o i l l e s s fraction), is d e t e r m i n e d more e f f e c t i v e l y by low e n e r g y p h o n o n s r a t h e r than by high e n e r g y ones. Thus the M ö s s b a u e r a b s o r p t i o n a rea will be less sens i t i v e to p h o n o n s at e n e r g i e s g r e a t e r

than a bout 212 cm ^ (low e n e r g y E mode) and for p r a c t i c a l p u r p o s e s the

8*

e f f e c t i v e Debye t e m p e r a t u r e at Fe sites m ay be a s s u m e d to b e about 212 cm (0p - 3 0 5 K ) . T his a s s u m p t i o n is in good agre e m e n t w ith the Debye

t e m p e r a t u r e e s t i m a t e d from the zero- p o i n t m o t i o n of the Fe-ion.

W h e n F e ^ + is s u b s t i t u t e d for Mg^ + in Mg C O ^ the vibrat i o n a l -1

p r o p e r t i e s o f the Fe sites m ay c hange to some extent, as the Fe ion is c o n s i d e r a b l y h e a v i e r than the Mg ion. It is to be n o t e d that the low

1 0 0.

0

100

200

300

In document A Mössbauer study of spin relaxation of ⁵⁷Fe ions (Page 104-110)