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A dispersion function of paramagnetic relaxation
Fang, P . H . 1958
Physica X X I V 970-974
A DISPERSION
FUNCTION OF PARAMAGNETIC RELAXATION
P. H. FANG National Bureau of Standards, Washington, U.S.A.
Synopsis An empirical dispersion function for paramagnetic relaxation is proposed which is intended to describe experimental data for the magnetic susceptibilities of some alums. This function gives an asymmetric frequency spectrum of the complex susceptibilities, thus an i m p r o v em e n t over the function of C a s i m i r and D u P r 6 in analyzing the experimental data. The characteristics of the distribution" of the relaxation time of this function are discussed.
I. Introduction. In a recent work on the paramagnetic relaxation of some alums at very low temperatures, G o r t e r and his group in Leiden observed a relaxation dispersion of the complex susceptibility (c.s.) 1). These authors mentioned that since the observed frequency spectrum of the susceptibility is asymmetrical the relaxation.behavior cannot be described b y the ColeCole function 3). In this paper an empirical dispersion function will be proposed which describes adequately the experimental results. Based on this new function, the relaxation time will be determined and its characteristics will be discussed. 2. A n empirical dispersion /unction. The experimental data of G o r t e r et al. 1) is suggestive of the function proposed b y Cole and D a v i d s o n s) to describe the complex permittivity (c.p.) of some supercooled liquids. The important difference can be seen in the Argand diagram where the asymptotic straight line occurs on the high frequency side for the Cole-Davidson function and occurs on the low frequency side for the data of G o r t e r et al. This is illustrated in Figs. l a and l d. It is obvious that b y some simple mathematical transformations one can obtain the desired function of l d from the function of l a, as indicated in Figs. l b and l c. The Cole-Davidson function for c.p. is given below 8): ~* = s' - is" = ~
+ (s0 - s~)(1 + i~p0)-~
(1)
where p0 is the relaxation time (here vp0 replaces o~T0of Cole-Davidson with o~ = 2~v), 0 < fl < 1, and the other symbols have their usual meaning. To change the side where the asymptotic straight line occurs, the following transformations are introduced s0d'
(d-e~)
-- 9 7 0 -
-+.Zb' -* Zb"
(2)
A DISPERSION FUNCTION OF PARAMAGNETIC RELAXATION
971
which give Zb* = Zb'
--
iZb" = Z0 -- (X0 -- Zoo)(1 + ivpo)-a
(3)
where Xb* is c.Sl of Fig. lb. However, the sense of the increasing frequency along the curve of Fig. lb is opposite to that of Fig. l d. In order to reverse this sense another transformation, T, is made 4)
T:
i~p0 - ~ (i,po) - z
Zb* -+ Zc*
(4)
where Zc* is shown in Fig. l c, and is given explicitly b y zo* = xo -
(zo -
FfG. Io TYPICAL COLE- DAVISON ¢~o 6o E'
zoo)(1 +
1/i,,p)-a.
(s)
FIG. Id TYPICAL GORTER etol
x.
£,, Eq(211
,.
i], x',
Xb
FIG. Ib
X¢
FIG. I¢
This gives a function which describes the typical data of G o r t e r et al., as shown in Fig. l d, with Z* -- Zc*. (6) When we express the c.s. in a normalized form with Z0 = 1 and denote I -Zoo b y F', then X' = 1 -- F ' cos~ 7t cos/5t ;~" = F ' cos~ I sin/5t
(7)
where 1 = cot -1 vpo. The symbol F' in expression (7) is similar to the F of C a s i m i r and D u P r 4 5) (which is discussed in reference 1)) ; and like F, can be associated with the specific heat of the intrinsic magnetic moment of the material. Numerical calculations were made based on Eq. (5) to fit the experimental results given in Fig. 2 and Fig. 3 for two samples of alums. In the calculation the value of the parameter F' was determined from the experimental value of zoo. The value of/5 could have been determined from the angle between the asymptotic straight line and the z'-axis. However, due to the scarcity of data in the low frequency region, /5 was determined from the following relation: cos/3+1 (~/2(1 +/5)) = (Z"max/F')expt. (8)
972
P.H. FANG
T h e derivation of this relation will be discussed in section 3. .I i
0
.2 i
.3 i
A ~
.5 ~
.6 i
.7 i
.8 i[
.9 i
tO
.3
v
4
~ FIG, 2
TI
1.641eK
OILUTEO F e - 4 1 ALUU. o
EXPERIMENT
--
J:3
THEORY
N c , S3 ?S Oe p
, .sz
X m • .o)s
O'
2
.3
.4
.~
.6
7
.' ~
.E~
" .9
÷~'\
•
> °
SO
/
/ /
.4
V
~ FIG. 3
T,
2.042"
K
POWDEREO PRESSED S A M P L E o - -
EXPERIMENT
FeAI
THEORY
l:O Hc • 3 3 7 5 0 e
ALUM.
• Xoo "
.SS .025
3. Determination o/ relaxation time. In the work of G o r t e r et al., the relaxation time p0 was determined by two procedures: (a) The value of vp0 was set equal to 1 for the frequency where :g' --= ½(I + -t-X~), the point where Z' has an ir~flection in the equations o~ CasimirD u Pr~. (b) The value of vp0 was set equal to 1 for the frequency where ;g" had its maximum. They found a large difference in the values of p from the different procedures. This difference can be explained by the fact that these procedures of calculation did not take the asymmetry into consideration. According to Eq. (5)," at the inflection point, the value of vp0 should be found by solving the following equation sin(1 + fl)tdi. + sin(3 + /~)~dis + 2fl sin 1dis cos (2 + /~)tai8 = 0. (9)
cot -1 pvois. This equation is obt£ined by setting the second logarithmic derivative of ~:' (the dispersion part), equal to zero. On the otherhand, at the maximum of :g" (the absorption part), the following equation holds where
~.d18 =
~ b ~ = ~/2(1 + / ~ )
(IO)
973
A D I S P E R S I O N F U N C T I O N OF P A R A M A G N E T I C R E L A X A T I O N
where labs = cot-lp0Vabs. The values of v-1 determined from the dispersion or absorption are identical only if fl = 1, as in the case of Casimir and Du Pr~. However, if Eqs. (9) and (10) are taken into account, Vdis is not equal to vabs, but those equations result in the same value of relaxation time po. For the two cases given in Figs. 2 and 3 the redetermined value of the relaxation time, po, is given in table I. TABLE I p in units of 10 -8 s Casimir-Du Pr6
sample
pabs
Fig. 2 Fig. 3
48.4 31.3
[
P
pdis
from present work
60.3 32.0
22.0 20.2
B y using Eq. (10), the m a x i m u m of %" is obtained: (11)
Z"max = F' cos~ +1 (~/2(1 + fl))
which results in Eq. (8).
4. The distribution /unction o/ the rdaxation time, D F R T . The D F R T , denoted by g~o), of the new dispersion function is obtained in the following manner: 1 -- F'(1 + 1/ivpo)-fl = f ~ (g(p)/(1 + ivp)) d l o g p . (12) Note t h a t this definition of g(p) is different from the usual definition where the integration is taken over dp instead of d log p s). We adopt this definition in order to compare g(p) with D F R T of Cole-Davidson, where this definition is used. g~o) is then obtained by using the well-known Stieltjes transtorm v).
(sinfl~/~)(po/(p--po))/~ for p > po
g(p) = F ' = 0
for p < p0.
1
]
Two important characteristics of g(p) will be discussed below: (1) The continuous distribution function, g(p), has a minimum, or cut-off time p. This is just the opposite of the Cole-Davidson function which is given by 3), F(p)=
sin~fl~ --(
po-P -p
= 0
)a for p < po
/
(14)
for p > po.
That means F(p) has a maximum, or cut-off time. F r o m g(p), one can calculate the fraction / of processes with relaxation time larger than pl, i.e.
fl~/z~) (PO/(P -- po)) d log (p/p0) 1 =F'(sinfl~/~)Bm/p.(fl, l--fl) for p l > p 0 /
/(pl/pO) ----fpl~po (F' sin
= 1 for pl < p o ,
(15)
974
A D I S P E R S I O N F U N C T I O N OF PARAMAGNETIC R E L A X A T I O N
where B is the incomplete beta-function of pO/pl order. Therefore, the spectrum of / is a mirror image of that of Cole-Davidson, b y rotating at p = p0. A further understanding of the mechanism of the paramagnetic relaxation might be obtained from the comparison between the molecular model of Cole-Davidson with the lattice interactions. (2) We have interpreted p0 as the relaxation time. According to the usual picture of the distribution function, po should be the point where g(p) has a point of concentration. We see that po in both g(p) and F(p) does not have this property. However, when fl -~ ! they both become Debye distribution, and p0 becomes the relaxation time of Debye. Therefore, p0 does have the significance of a relaxation time. From the physical point of view, it is necessary to interpret the deviation from the Debye distribution. This deviation is intimately associated with the parameter 8. In conclusion, based on the resemblance between the Cole-Davidson function and the paramagnetic relaxation data of G o r t e r et al., a phenomenological function is proposed which takes the a s y m m e t r y described b y G orter et al. into account. Use of this function eliminates the inconsistencies that arise when the experimental data are analyzed with the Casimir- Du Pr6 function. Two samples of data drawn from figures of reference 1) are analyzed and the results seem to be satisfactory. It would be interesting to test whether this proposed function is applicable to all other data on the paramagnetic relaxation of alums *).
Acknowledgements. I would like to acknowledge the help of Dr. A. I). F r a n k 1in in preparing this material for presentation, and the communication from Professor G o r t e r in regard to Dr. V a n d e r M a r e l ' s comment. Received 21-4-58
REFERENCES I) V a n d e r Marel, L. C., V a n d e n B r o c k , J. and G o r t e r , C. J., Commun. Kamerlingh Onnes Lab., Leiden No. 306a; Physica 23 (1957) 361. (this paper refers to the earlier work of this group). 2) Cole, K. S. and Cole, R. H., J. chem. Phys. 9 (1941) 341. 3) D a v i d s o n , D. W. and Cole, R. H., J. chem Phys. 19 (1951) 1484. 4) F a n g , P. H. (to be published). In this paper the same transformation has been used in the derivation of a new dispersion function for the dielectric relaxation which describes the experimental data of the slow relaxation of some ferrites, as well as some cases of low frequency dielectric loss. 5) C a s i m i r , H. B. G. and D u Pr6, F. K., Commun. Suppl. No. 85a; Physiea 5 (1938) 507. 6) See, for example, M a c d o n a l d , J. R. and B r a c h m a n , M. K., Rev. mod. Phys. 28 (1936) 392. 7) T i t c h m a r s h , E. C., Fourier Integrals (Oxford, 2nd. ed. 1948) p. 317.
*) Note added in proo]. The existence of the unique g(p) automatically quarantees that the c.s. g i v e n by eq. satisfies K r o n i g- K r a m e r s' relations.
Physica X X I V 970-974
A DISPERSION
FUNCTION OF PARAMAGNETIC RELAXATION
P. H. FANG National Bureau of Standards, Washington, U.S.A.
Synopsis An empirical dispersion function for paramagnetic relaxation is proposed which is intended to describe experimental data for the magnetic susceptibilities of some alums. This function gives an asymmetric frequency spectrum of the complex susceptibilities, thus an i m p r o v em e n t over the function of C a s i m i r and D u P r 6 in analyzing the experimental data. The characteristics of the distribution" of the relaxation time of this function are discussed.
I. Introduction. In a recent work on the paramagnetic relaxation of some alums at very low temperatures, G o r t e r and his group in Leiden observed a relaxation dispersion of the complex susceptibility (c.s.) 1). These authors mentioned that since the observed frequency spectrum of the susceptibility is asymmetrical the relaxation.behavior cannot be described b y the ColeCole function 3). In this paper an empirical dispersion function will be proposed which describes adequately the experimental results. Based on this new function, the relaxation time will be determined and its characteristics will be discussed. 2. A n empirical dispersion /unction. The experimental data of G o r t e r et al. 1) is suggestive of the function proposed b y Cole and D a v i d s o n s) to describe the complex permittivity (c.p.) of some supercooled liquids. The important difference can be seen in the Argand diagram where the asymptotic straight line occurs on the high frequency side for the Cole-Davidson function and occurs on the low frequency side for the data of G o r t e r et al. This is illustrated in Figs. l a and l d. It is obvious that b y some simple mathematical transformations one can obtain the desired function of l d from the function of l a, as indicated in Figs. l b and l c. The Cole-Davidson function for c.p. is given below 8): ~* = s' - is" = ~
+ (s0 - s~)(1 + i~p0)-~
(1)
where p0 is the relaxation time (here vp0 replaces o~T0of Cole-Davidson with o~ = 2~v), 0 < fl < 1, and the other symbols have their usual meaning. To change the side where the asymptotic straight line occurs, the following transformations are introduced s0d'
(d-e~)
-- 9 7 0 -
-+.Zb' -* Zb"
(2)
A DISPERSION FUNCTION OF PARAMAGNETIC RELAXATION
971
which give Zb* = Zb'
--
iZb" = Z0 -- (X0 -- Zoo)(1 + ivpo)-a
(3)
where Xb* is c.Sl of Fig. lb. However, the sense of the increasing frequency along the curve of Fig. lb is opposite to that of Fig. l d. In order to reverse this sense another transformation, T, is made 4)
T:
i~p0 - ~ (i,po) - z
Zb* -+ Zc*
(4)
where Zc* is shown in Fig. l c, and is given explicitly b y zo* = xo -
(zo -
FfG. Io TYPICAL COLE- DAVISON ¢~o 6o E'
zoo)(1 +
1/i,,p)-a.
(s)
FIG. Id TYPICAL GORTER etol
x.
£,, Eq(211
,.
i], x',
Xb
FIG. Ib
X¢
FIG. I¢
This gives a function which describes the typical data of G o r t e r et al., as shown in Fig. l d, with Z* -- Zc*. (6) When we express the c.s. in a normalized form with Z0 = 1 and denote I -Zoo b y F', then X' = 1 -- F ' cos~ 7t cos/5t ;~" = F ' cos~ I sin/5t
(7)
where 1 = cot -1 vpo. The symbol F' in expression (7) is similar to the F of C a s i m i r and D u P r 4 5) (which is discussed in reference 1)) ; and like F, can be associated with the specific heat of the intrinsic magnetic moment of the material. Numerical calculations were made based on Eq. (5) to fit the experimental results given in Fig. 2 and Fig. 3 for two samples of alums. In the calculation the value of the parameter F' was determined from the experimental value of zoo. The value of/5 could have been determined from the angle between the asymptotic straight line and the z'-axis. However, due to the scarcity of data in the low frequency region, /5 was determined from the following relation: cos/3+1 (~/2(1 +/5)) = (Z"max/F')expt. (8)
972
P.H. FANG
T h e derivation of this relation will be discussed in section 3. .I i
0
.2 i
.3 i
A ~
.5 ~
.6 i
.7 i
.8 i[
.9 i
tO
.3
v
4
~ FIG, 2
TI
1.641eK
OILUTEO F e - 4 1 ALUU. o
EXPERIMENT
--
J:3
THEORY
N c , S3 ?S Oe p
, .sz
X m • .o)s
O'
2
.3
.4
.~
.6
7
.' ~
.E~
" .9
÷~'\
•
> °
SO
/
/ /
.4
V
~ FIG. 3
T,
2.042"
K
POWDEREO PRESSED S A M P L E o - -
EXPERIMENT
FeAI
THEORY
l:O Hc • 3 3 7 5 0 e
ALUM.
• Xoo "
.SS .025
3. Determination o/ relaxation time. In the work of G o r t e r et al., the relaxation time p0 was determined by two procedures: (a) The value of vp0 was set equal to 1 for the frequency where :g' --= ½(I + -t-X~), the point where Z' has an ir~flection in the equations o~ CasimirD u Pr~. (b) The value of vp0 was set equal to 1 for the frequency where ;g" had its maximum. They found a large difference in the values of p from the different procedures. This difference can be explained by the fact that these procedures of calculation did not take the asymmetry into consideration. According to Eq. (5)," at the inflection point, the value of vp0 should be found by solving the following equation sin(1 + fl)tdi. + sin(3 + /~)~dis + 2fl sin 1dis cos (2 + /~)tai8 = 0. (9)
cot -1 pvois. This equation is obt£ined by setting the second logarithmic derivative of ~:' (the dispersion part), equal to zero. On the otherhand, at the maximum of :g" (the absorption part), the following equation holds where
~.d18 =
~ b ~ = ~/2(1 + / ~ )
(IO)
973
A D I S P E R S I O N F U N C T I O N OF P A R A M A G N E T I C R E L A X A T I O N
where labs = cot-lp0Vabs. The values of v-1 determined from the dispersion or absorption are identical only if fl = 1, as in the case of Casimir and Du Pr~. However, if Eqs. (9) and (10) are taken into account, Vdis is not equal to vabs, but those equations result in the same value of relaxation time po. For the two cases given in Figs. 2 and 3 the redetermined value of the relaxation time, po, is given in table I. TABLE I p in units of 10 -8 s Casimir-Du Pr6
sample
pabs
Fig. 2 Fig. 3
48.4 31.3
[
P
pdis
from present work
60.3 32.0
22.0 20.2
B y using Eq. (10), the m a x i m u m of %" is obtained: (11)
Z"max = F' cos~ +1 (~/2(1 + fl))
which results in Eq. (8).
4. The distribution /unction o/ the rdaxation time, D F R T . The D F R T , denoted by g~o), of the new dispersion function is obtained in the following manner: 1 -- F'(1 + 1/ivpo)-fl = f ~ (g(p)/(1 + ivp)) d l o g p . (12) Note t h a t this definition of g(p) is different from the usual definition where the integration is taken over dp instead of d log p s). We adopt this definition in order to compare g(p) with D F R T of Cole-Davidson, where this definition is used. g~o) is then obtained by using the well-known Stieltjes transtorm v).
(sinfl~/~)(po/(p--po))/~ for p > po
g(p) = F ' = 0
for p < p0.
1
]
Two important characteristics of g(p) will be discussed below: (1) The continuous distribution function, g(p), has a minimum, or cut-off time p. This is just the opposite of the Cole-Davidson function which is given by 3), F(p)=
sin~fl~ --(
po-P -p
= 0
)a for p < po
/
(14)
for p > po.
That means F(p) has a maximum, or cut-off time. F r o m g(p), one can calculate the fraction / of processes with relaxation time larger than pl, i.e.
fl~/z~) (PO/(P -- po)) d log (p/p0) 1 =F'(sinfl~/~)Bm/p.(fl, l--fl) for p l > p 0 /
/(pl/pO) ----fpl~po (F' sin
= 1 for pl < p o ,
(15)
974
A D I S P E R S I O N F U N C T I O N OF PARAMAGNETIC R E L A X A T I O N
where B is the incomplete beta-function of pO/pl order. Therefore, the spectrum of / is a mirror image of that of Cole-Davidson, b y rotating at p = p0. A further understanding of the mechanism of the paramagnetic relaxation might be obtained from the comparison between the molecular model of Cole-Davidson with the lattice interactions. (2) We have interpreted p0 as the relaxation time. According to the usual picture of the distribution function, po should be the point where g(p) has a point of concentration. We see that po in both g(p) and F(p) does not have this property. However, when fl -~ ! they both become Debye distribution, and p0 becomes the relaxation time of Debye. Therefore, p0 does have the significance of a relaxation time. From the physical point of view, it is necessary to interpret the deviation from the Debye distribution. This deviation is intimately associated with the parameter 8. In conclusion, based on the resemblance between the Cole-Davidson function and the paramagnetic relaxation data of G o r t e r et al., a phenomenological function is proposed which takes the a s y m m e t r y described b y G orter et al. into account. Use of this function eliminates the inconsistencies that arise when the experimental data are analyzed with the Casimir- Du Pr6 function. Two samples of data drawn from figures of reference 1) are analyzed and the results seem to be satisfactory. It would be interesting to test whether this proposed function is applicable to all other data on the paramagnetic relaxation of alums *).
Acknowledgements. I would like to acknowledge the help of Dr. A. I). F r a n k 1in in preparing this material for presentation, and the communication from Professor G o r t e r in regard to Dr. V a n d e r M a r e l ' s comment. Received 21-4-58
REFERENCES I) V a n d e r Marel, L. C., V a n d e n B r o c k , J. and G o r t e r , C. J., Commun. Kamerlingh Onnes Lab., Leiden No. 306a; Physica 23 (1957) 361. (this paper refers to the earlier work of this group). 2) Cole, K. S. and Cole, R. H., J. chem. Phys. 9 (1941) 341. 3) D a v i d s o n , D. W. and Cole, R. H., J. chem Phys. 19 (1951) 1484. 4) F a n g , P. H. (to be published). In this paper the same transformation has been used in the derivation of a new dispersion function for the dielectric relaxation which describes the experimental data of the slow relaxation of some ferrites, as well as some cases of low frequency dielectric loss. 5) C a s i m i r , H. B. G. and D u Pr6, F. K., Commun. Suppl. No. 85a; Physiea 5 (1938) 507. 6) See, for example, M a c d o n a l d , J. R. and B r a c h m a n , M. K., Rev. mod. Phys. 28 (1936) 392. 7) T i t c h m a r s h , E. C., Fourier Integrals (Oxford, 2nd. ed. 1948) p. 317.
*) Note added in proo]. The existence of the unique g(p) automatically quarantees that the c.s. g i v e n by eq. satisfies K r o n i g- K r a m e r s' relations.