- Email: [email protected]
Explicit temperature dependence of the Lorenz number in nearly ferromagnetic metals
Solid State Communications,
Vol. 13, pp. 1707—1711, 1973.
Pergamon Press.
Printed in Great Britain
EXPLICIT TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER IN NEARLY FERROMAGNETIC METALS MT. Béal-Monod Laboratoire de Physique des Solides*, Université Paris—Sud—Centre d’Orsay 91405—Orsay France and D.L. Mills Physics Department, University of California of Irvine Irvine, California 92664, U.S.A. (Received 3 July 1973; in revised form 6 September 1973 by P. G. de Gennes)
The temperature dependence of the Lorenz number 2inatnearly ferromagnetic very low tempera. metals explicitly found to vary likeA CT temperature ture andis to reach the Sommerfeld value + at BT+ very high with a very slow (1 + cst/T)’ law. These analytic expressions explain previous computed results in agreement with experiments.
THE LORENZ number of nearly magnetic metals exhibiting strong exchange enhancement has been worked out previously by one of us.1 This paper (M) presented only computed results and compared them to experimental observations.2 The conclusions of (M) were the following: the Lorenz number rises sharply when T increases from 0°Kand reaches its saturation, the Sommerfeld value, very slowly. Since these results were obtained through a computer no simple power law could be worked out although it was noticed that the saturation seemed to be approached
to calculate the high temperature behavior of the electrical resistivity. For convenience, we take the same notations that those used in (M) in the following. The expression of the Lorenz number given in (M) (formula 14) is: L (1)
=
S
ir2 kB
.
2
when L 5 is the Sommerfeld value, Ls
=
—
and
slowly.
~ f dx x~n(x) [1 + n(x)] [j3~(x r) ~ j3~(x r)J —
The purpose of the present note is simply to find explicit power laws both at low and high temperatures for that Lorenz number. We first show that the complicated integrals involved in (M) at low temperature can be calculated analytically so that the temperature dependence can be easily extracted. Furthermore, we extend theorythe of(M) at very high temperature taking into the account temperature dependence of the spin fluctuation susceptibility already used elsewhere3
2 iT
_____________________________________ j dxx2n(x) [I + n(x)J 13 3(xr) °
(2) with (1 1~m(xr)
=
J
tM ~2(~ ~m+! d~ + ~2)2+x2r2
_1)(m.4)/2 2)m
(uksTII
0
(3) These formulas are appropriate to the paramagnon model with T~the Fermi energy Cd of the electrons responsible for the paramagnons, but they do not assume T~T 51=(l —l)Ed. .
*
Laboratoire associé au CRNS. 1707
)
1708
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER
For convenience, we rewrite (2) as:
where 2 Imx(q,w,T) is twice the imaginary part of the parainagnon propagator x = x°I(l—I~°) with
3N1—N2 2D ir
with
(4)
EM
2 dE 14
(5)
~b f ~ d~14
(6)
N 1 N 2
=
f E
a
=
0
0
4d.~I
EM
D=bf
~
2
(7)
S
2(l+~2)+x2r2 °°x2~dxn(x)[l+n(x)] (8) o ~ expressed in terms of the low temperature expression for the paramagnon susceptibility as explained in (M). =
=
will first use (13 a)and to we explicit the later low temperature behavior of L/L8 will use on formula (13 b) to examine the high temperature regime.
/2’
Back to (13 a), we first remark that: n(w/T) [I + n (wIT)] = e’~
(9)
Therefore the temperature dependence of ~ is fully contained in ‘2 and 14. again for convenience, we go back to the definition of x r:
r2 a /
(e(~T — 1)2
W ~,
lTcikdf T
siT,
=
/1
=
~
j
T2
an(w/r
i)=
;
ar (14)
So (13 a) gives:
a
I
12
(10)
p2T2n-!a75
°°w2’~dwn(w/T) X2+w2 (15)
Now we consider separately 12 and 14, 14
—
We will write
a
i (12)
=
$
a
1
oow3dwn(wfT) x2 + w2
_
P
where
F1 o2k~i?2
—
\
0
w2
+ x2
~
so that:
—
(00
wdwn(w/r)1
00
I4=~~~~J wdwn(w/T)_X21
2
X2+w
0
was~M defined in formula (9) ~ to is the maximum ~: value of =~ 2.corresponding With these notations, toof the(M). maximum back (8), value oneof gets: °°
‘2n =
w2’~dw n(w/T) [1 + n(w/T)]
x2 + w2
2T2~~’
p
(16) p2T 1 aT a 00 w x2 dw+n(wIT) w2 (17) The first integral in (16) is trivial going back to the ‘2
=
variable x = w/T: $wdw n(w/T)
=
T2j
X
dx
iT2
=
______
(13 a) 0
at low temperature.
0
So we get:
More generally, if T is not restricted to very low ternperature compared to ~d one has using (10), (11), (12) and q = i~,one has (I —T~3 2 kd I ‘2n = T lrNd(O) c2q 7’2P1+1
6
iT2 A2 3p2 T2 T2 ‘2 (19) given by(17). Now in (17), we get put: =
with ‘2
(18)
ex_l
—
+~ =
2lTt T~Z
=
(2ir T)
2ir 1.zT
f
0
2”~n(wfT) [I w
+n(w/T)]
I)
\3/2
when T stands for kB times the temperature. One also writes: ~2(l + ~2)2 = p2 ~2 (11)
=
1
w aT(e’~
=
=
X =
+ e~T_
~—
e(4T
(1~Ty~2 b= (aksT”2)3
(1_1)-3/2
a—
2~[f(Ek, T) ~f(~+q, T)] I(Ek.q~k”~). f(ek,T) is the Fermi distribution. 2 Im x reduces to A(q,w) [formula (5) of(M)] at low temperature (f(ek,T) ~f(Ek, 0)] where it leads to (13 a). We ~
Low temperature regime (T ( Cd).
0
12n
Vol. 13, No. 10
2lm~(q.w.T)dw (13b)
(20)
j
Vol. 13, No. 10
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER
= —— P2TaTJ
a
So that L(T=0)
tdt (e2~’—1)(t2+z2) (21a)
____________
/ 1 a~J,(z)\ Il+ rar~ 2z 2p2T2\ 2z az (21b) by definition of the digamma function4 i,(/(z). Then:
12
=
1 2p2
00
a
1 12
/4
1
/
1
_Ilnz___~r(z))=_
——~
iT2 ~(~+2Z2+Z_~3
=
L~
5
(1+E2)2
a
3+12— b
$
6M
______
(1 + E2)2
0
(27b)
=
arctg EM ~-~-~dE az /
—
az
~i(z)
az
=
B 2~ —
1 iT2 ;i + -~-
z -+
00
z —~ 0
A of remarks about (27 b): thatcouple paper)
(23b)
(i) it is easy to check that when the Stoner enhancement (1 —1)~increases, EM increases too,
for T< Ed. When the temperature T lies far below the spin fluctuation temperature T,, defined by formula (4) of(M) as: (24)
~d
One can reasonably assume that z ~ I for T< T,, and use (23 a) for values of E not too close to zero; but near E ~ be0 (i.e. < T/T~,) formula (23 b) which must used.i~Actually ititisisthat last region which will give the dominant temperature dependence oft. At T= 0 one has directly: EM (3a_~_~)dE 4 b ~ 2 (1 +E2)2 12 a CM dE = 5 ii bE2( I + E2 )2
ç
~M
$
dE (I +E2)
2
E2dE 5 (I + E2)2
$
C~
0
°
I
(23 a)
B 4 ((23b) allows to check 2~are Bernouilhi number. that the the integrals involved in (22) converge when -÷ 0). Formula (22) with (20) and (23 a) (23 b) in (I) gives the low temperature variation of the Lorenz number
T8~= (1—1)
ik~
1 —T 12k~1 —i 3k~i—T The limit (27 a) was the one found previously by Schriempf, Schindler and Mills5 formula (2. 11) of
with z given by (20) 4 Now, note here two useful asymptotic behaviors of thewe trigamma function a~p(z)/az:
Iz + 2z
EM _____
E2_ 4U2k214kST
=
EM
I+
with (22)
00
+
arctgE~
(_!_~~+L~S!~))dE 2 4z 2 az
2+~
(27a)
_______
a~~(z)) (2lc)
J(~_~.E2)E2(!+2z2+z_..2z3
b~
d~
6M
=
—
Then (4) becomes:
S
1709
(26)
L(T 0)/Ls decreases toward zero. This is exactly what=was indicated on the computed curves of (M) (Fig. 4). At the limit, L(T = 0) 0 and if the electrical resistivity still varies like T2, then in order to have i~ 0, the thermal resistivity should be practically constant. This seems to be observed in Ni 3 Al.° -~
-~
(ii) EM is proportional to the ratio kslkd. When this ratio is small so that EM <~ or
(k~/k~)<3(1 —1)/i,
i.e. when the wavelength of the S electron is much larger than the correlation length of the spin 2 can be dropped compared to I fluctuin the ations then E denominators of the integrals in (27 a) and then, L(T = 0)/L 8 becomes independent of the enhancement and 2<3(1 —T) (28) L(T= 0) 5 (k~ Ls = j-~ for I In that case, the q term in the denominator of Im x can be dropped and one there recovers the low ternperature scaling in T/T~~ found by Kaiser and Doniach,7 scaling which breaks down in general [when (28) is not fulfilled which is usually the case] as discussed in(M).
1710
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER
Now let us examine the first term in Tin (22); we get approximately: 2IrMT /1 /l 3a + 2.~2+~ —2Z~ç— 0 ~3
$
E21—
p
iT2\\ +—)
6 /
Vol. 13, No. 10
d~
C1 + C2/T, C2 being positive or negative depen3 ding on the value I and involving/rn of ks/kd. It was shown can that integrals overof energy x(q,w,T) be evaluated, asymptotically, very simply using the Krarners-Kronig relations and the f-sum rule. If we use the same tricks here, we find: J 4~q2
2) T .~(T)—A(0)--—F—+0(T 247 k 8f F= iT~I~ cM 4 ks arctg EM —
T-’O (29) (30) EM
I + E~ Therefore L
1
L
—.~
~—
L 5
l+~(o)_cL+or2 T
IL (1+—
T\ F—J
Ls T~o\ L5 T=0 T81/
8~
2), T-~0 (31) + ~T Note that the temperature dependence of the electrical resistivity p is essentially given by the denominator of (29) which varies like T2 + 0(T4), therefore according to (31), the thermal resistivity W is such that WT varies like the numerator of(29), is like T2 + 0(T3); then W varies like T + 0(T2).
T-~°° (33) 2 CSt 11+1——-— I _k~ q ‘2 ~ ET2 \ 3T 6T the same cst. is involved in ‘2 and /4 and will disappear in the Lorenz number. Then one gets with (1), (4), (5 —8): 5k~ T-~°° (34) 2T 3ir L I Ls 1 + T-’°° (35) ~—
—
3172 T
Some remarks about (35): (i) the T-~oo limit of L/Ls is I as it should be since there one reaches a classical regime and L must be equal to L 8. Then since p is cst too, the thermal resistivity W should reach zero like p liT. (ii)Ihas disappeared in that high temperature 00
High temperature regime(T~Cd). 2 L/L Experimentally 5 increases slowly toward most likely at high We will that section formula (13temperature. b) together with the use highinternperature dependence of~(q,w,T) which plays an important role in that regime. It has been shown elsewhere3 that the high T Curie type behavior of x makes the electrical resistivity to behave like
asymptotic behavior 2l (iii) or finally, at extremely 7’, lOesI3ir is larger equal except to 1. Therefore L/Lshigh varies very slowly with a weak linear law and even at high T, L/Ls is not very close to its saturation. This could explain why the computed curves of (M) for L/Ls vs T indicate a slow temperature Variation.
REFERENCES 1.
MILLS D.L.,J. Phy~.Chem. Sol. 34, 679 (I973).(note that two misprints appeared in formulas (10 a) and (10 b) : ~/12 should replace ~/6 and ks should be equal to (1/2)kd, furthermore. in formula (l4b) one should read —133(X r)/6 instead of—I3 3(~r)/3).
2. 3.
The experimental situation together with the corresponding references are given in (M) JULLIEN R., BEAL-MONOD M.T. and COQBLIN B..Phys. Rev. Len. 30, 1057 (1973).
4. 5. 6.
See for instance in Table of Integrals, Series and Products by GRADSHTEYN IS. and RYZHIK 1.M.. Academic Press, New York (1965). SCHRIEMPF J.T., SCHINDLER A.!. and MILLS DL., P/n’s. Rei’. 187. 959 (1969). SCHRIEMPF J.T., MAC INNES W.M. and SCHINDLER A.!.. paper BD 8. Bull. Am. P/n’s. Soc. 17, 256 (1972).
7.
KAISER A.B. and DONIACH S. Jut. J. Magn. 1. 11(1970).
Vol. 13, No. 10
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER On montre que Ic nombre de danstemperature les métaux presque 2 aLorenz três basse et atteintmagnétiques Ia valeur de vane comnie A + BT + CT Sommerfeld a très haute temperature avec une variation très lente en (1 + cst/T)~ Ces rCsultats analytiques expliquent des rCsultats numeriques antCrieurs en accord avec l’expCrience. -
1711
Vol. 13, pp. 1707—1711, 1973.
Pergamon Press.
Printed in Great Britain
EXPLICIT TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER IN NEARLY FERROMAGNETIC METALS MT. Béal-Monod Laboratoire de Physique des Solides*, Université Paris—Sud—Centre d’Orsay 91405—Orsay France and D.L. Mills Physics Department, University of California of Irvine Irvine, California 92664, U.S.A. (Received 3 July 1973; in revised form 6 September 1973 by P. G. de Gennes)
The temperature dependence of the Lorenz number 2inatnearly ferromagnetic very low tempera. metals explicitly found to vary likeA CT temperature ture andis to reach the Sommerfeld value + at BT+ very high with a very slow (1 + cst/T)’ law. These analytic expressions explain previous computed results in agreement with experiments.
THE LORENZ number of nearly magnetic metals exhibiting strong exchange enhancement has been worked out previously by one of us.1 This paper (M) presented only computed results and compared them to experimental observations.2 The conclusions of (M) were the following: the Lorenz number rises sharply when T increases from 0°Kand reaches its saturation, the Sommerfeld value, very slowly. Since these results were obtained through a computer no simple power law could be worked out although it was noticed that the saturation seemed to be approached
to calculate the high temperature behavior of the electrical resistivity. For convenience, we take the same notations that those used in (M) in the following. The expression of the Lorenz number given in (M) (formula 14) is: L (1)
=
S
ir2 kB
.
2
when L 5 is the Sommerfeld value, Ls
=
—
and
slowly.
~ f dx x~n(x) [1 + n(x)] [j3~(x r) ~ j3~(x r)J —
The purpose of the present note is simply to find explicit power laws both at low and high temperatures for that Lorenz number. We first show that the complicated integrals involved in (M) at low temperature can be calculated analytically so that the temperature dependence can be easily extracted. Furthermore, we extend theorythe of(M) at very high temperature taking into the account temperature dependence of the spin fluctuation susceptibility already used elsewhere3
2 iT
_____________________________________ j dxx2n(x) [I + n(x)J 13 3(xr) °
(2) with (1 1~m(xr)
=
J
tM ~2(~ ~m+! d~ + ~2)2+x2r2
_1)(m.4)/2 2)m
(uksTII
0
(3) These formulas are appropriate to the paramagnon model with T~the Fermi energy Cd of the electrons responsible for the paramagnons, but they do not assume T~T 51=(l —l)Ed. .
*
Laboratoire associé au CRNS. 1707
)
1708
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER
For convenience, we rewrite (2) as:
where 2 Imx(q,w,T) is twice the imaginary part of the parainagnon propagator x = x°I(l—I~°) with
3N1—N2 2D ir
with
(4)
EM
2 dE 14
(5)
~b f ~ d~14
(6)
N 1 N 2
=
f E
a
=
0
0
4d.~I
EM
D=bf
~
2
(7)
S
2(l+~2)+x2r2 °°x2~dxn(x)[l+n(x)] (8) o ~ expressed in terms of the low temperature expression for the paramagnon susceptibility as explained in (M). =
=
will first use (13 a)and to we explicit the later low temperature behavior of L/L8 will use on formula (13 b) to examine the high temperature regime.
/2’
Back to (13 a), we first remark that: n(w/T) [I + n (wIT)] = e’~
(9)
Therefore the temperature dependence of ~ is fully contained in ‘2 and 14. again for convenience, we go back to the definition of x r:
r2 a /
(e(~T — 1)2
W ~,
lTcikdf T
siT,
=
/1
=
~
j
T2
an(w/r
i)=
;
ar (14)
So (13 a) gives:
a
I
12
(10)
p2T2n-!a75
°°w2’~dwn(w/T) X2+w2 (15)
Now we consider separately 12 and 14, 14
—
We will write
a
i (12)
=
$
a
1
oow3dwn(wfT) x2 + w2
_
P
where
F1 o2k~i?2
—
\
0
w2
+ x2
~
so that:
—
(00
wdwn(w/r)1
00
I4=~~~~J wdwn(w/T)_X21
2
X2+w
0
was~M defined in formula (9) ~ to is the maximum ~: value of =~ 2.corresponding With these notations, toof the(M). maximum back (8), value oneof gets: °°
‘2n =
w2’~dw n(w/T) [1 + n(w/T)]
x2 + w2
2T2~~’
p
(16) p2T 1 aT a 00 w x2 dw+n(wIT) w2 (17) The first integral in (16) is trivial going back to the ‘2
=
variable x = w/T: $wdw n(w/T)
=
T2j
X
dx
iT2
=
______
(13 a) 0
at low temperature.
0
So we get:
More generally, if T is not restricted to very low ternperature compared to ~d one has using (10), (11), (12) and q = i~,one has (I —T~3 2 kd I ‘2n = T lrNd(O) c2q 7’2P1+1
6
iT2 A2 3p2 T2 T2 ‘2 (19) given by(17). Now in (17), we get put: =
with ‘2
(18)
ex_l
—
+~ =
2lTt T~Z
=
(2ir T)
2ir 1.zT
f
0
2”~n(wfT) [I w
+n(w/T)]
I)
\3/2
when T stands for kB times the temperature. One also writes: ~2(l + ~2)2 = p2 ~2 (11)
=
1
w aT(e’~
=
=
X =
+ e~T_
~—
e(4T
(1~Ty~2 b= (aksT”2)3
(1_1)-3/2
a—
2~[f(Ek, T) ~f(~+q, T)] I(Ek.q~k”~). f(ek,T) is the Fermi distribution. 2 Im x reduces to A(q,w) [formula (5) of(M)] at low temperature (f(ek,T) ~f(Ek, 0)] where it leads to (13 a). We ~
Low temperature regime (T ( Cd).
0
12n
Vol. 13, No. 10
2lm~(q.w.T)dw (13b)
(20)
j
Vol. 13, No. 10
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER
= —— P2TaTJ
a
So that L(T=0)
tdt (e2~’—1)(t2+z2) (21a)
____________
/ 1 a~J,(z)\ Il+ rar~ 2z 2p2T2\ 2z az (21b) by definition of the digamma function4 i,(/(z). Then:
12
=
1 2p2
00
a
1 12
/4
1
/
1
_Ilnz___~r(z))=_
——~
iT2 ~(~+2Z2+Z_~3
=
L~
5
(1+E2)2
a
3+12— b
$
6M
______
(1 + E2)2
0
(27b)
=
arctg EM ~-~-~dE az /
—
az
~i(z)
az
=
B 2~ —
1 iT2 ;i + -~-
z -+
00
z —~ 0
A of remarks about (27 b): thatcouple paper)
(23b)
(i) it is easy to check that when the Stoner enhancement (1 —1)~increases, EM increases too,
for T< Ed. When the temperature T lies far below the spin fluctuation temperature T,, defined by formula (4) of(M) as: (24)
~d
One can reasonably assume that z ~ I for T< T,, and use (23 a) for values of E not too close to zero; but near E ~ be0 (i.e. < T/T~,) formula (23 b) which must used.i~Actually ititisisthat last region which will give the dominant temperature dependence oft. At T= 0 one has directly: EM (3a_~_~)dE 4 b ~ 2 (1 +E2)2 12 a CM dE = 5 ii bE2( I + E2 )2
ç
~M
$
dE (I +E2)
2
E2dE 5 (I + E2)2
$
C~
0
°
I
(23 a)
B 4 ((23b) allows to check 2~are Bernouilhi number. that the the integrals involved in (22) converge when -÷ 0). Formula (22) with (20) and (23 a) (23 b) in (I) gives the low temperature variation of the Lorenz number
T8~= (1—1)
ik~
1 —T 12k~1 —i 3k~i—T The limit (27 a) was the one found previously by Schriempf, Schindler and Mills5 formula (2. 11) of
with z given by (20) 4 Now, note here two useful asymptotic behaviors of thewe trigamma function a~p(z)/az:
Iz + 2z
EM _____
E2_ 4U2k214kST
=
EM
I+
with (22)
00
+
arctgE~
(_!_~~+L~S!~))dE 2 4z 2 az
2+~
(27a)
_______
a~~(z)) (2lc)
J(~_~.E2)E2(!+2z2+z_..2z3
b~
d~
6M
=
—
Then (4) becomes:
S
1709
(26)
L(T 0)/Ls decreases toward zero. This is exactly what=was indicated on the computed curves of (M) (Fig. 4). At the limit, L(T = 0) 0 and if the electrical resistivity still varies like T2, then in order to have i~ 0, the thermal resistivity should be practically constant. This seems to be observed in Ni 3 Al.° -~
-~
(ii) EM is proportional to the ratio kslkd. When this ratio is small so that EM <~ or
(k~/k~)<3(1 —1)/i,
i.e. when the wavelength of the S electron is much larger than the correlation length of the spin 2 can be dropped compared to I fluctuin the ations then E denominators of the integrals in (27 a) and then, L(T = 0)/L 8 becomes independent of the enhancement and 2<3(1 —T) (28) L(T= 0) 5 (k~ Ls = j-~ for I In that case, the q term in the denominator of Im x can be dropped and one there recovers the low ternperature scaling in T/T~~ found by Kaiser and Doniach,7 scaling which breaks down in general [when (28) is not fulfilled which is usually the case] as discussed in(M).
1710
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER
Now let us examine the first term in Tin (22); we get approximately: 2IrMT /1 /l 3a + 2.~2+~ —2Z~ç— 0 ~3
$
E21—
p
iT2\\ +—)
6 /
Vol. 13, No. 10
d~
C1 + C2/T, C2 being positive or negative depen3 ding on the value I and involving/rn of ks/kd. It was shown can that integrals overof energy x(q,w,T) be evaluated, asymptotically, very simply using the Krarners-Kronig relations and the f-sum rule. If we use the same tricks here, we find: J 4~q2
2) T .~(T)—A(0)--—F—+0(T 247 k 8f F= iT~I~ cM 4 ks arctg EM —
T-’O (29) (30) EM
I + E~ Therefore L
1
L
—.~
~—
L 5
l+~(o)_cL+or2 T
IL (1+—
T\ F—J
Ls T~o\ L5 T=0 T81/
8~
2), T-~0 (31) + ~T Note that the temperature dependence of the electrical resistivity p is essentially given by the denominator of (29) which varies like T2 + 0(T4), therefore according to (31), the thermal resistivity W is such that WT varies like the numerator of(29), is like T2 + 0(T3); then W varies like T + 0(T2).
T-~°° (33) 2 CSt 11+1——-— I _k~ q ‘2 ~ ET2 \ 3T 6T the same cst. is involved in ‘2 and /4 and will disappear in the Lorenz number. Then one gets with (1), (4), (5 —8): 5k~ T-~°° (34) 2T 3ir L I Ls 1 + T-’°° (35) ~—
—
3172 T
Some remarks about (35): (i) the T-~oo limit of L/Ls is I as it should be since there one reaches a classical regime and L must be equal to L 8. Then since p is cst too, the thermal resistivity W should reach zero like p liT. (ii)Ihas disappeared in that high temperature 00
High temperature regime(T~Cd). 2 L/L Experimentally 5 increases slowly toward most likely at high We will that section formula (13temperature. b) together with the use highinternperature dependence of~(q,w,T) which plays an important role in that regime. It has been shown elsewhere3 that the high T Curie type behavior of x makes the electrical resistivity to behave like
asymptotic behavior 2l (iii) or finally, at extremely 7’, lOesI3ir is larger equal except to 1. Therefore L/Lshigh varies very slowly with a weak linear law and even at high T, L/Ls is not very close to its saturation. This could explain why the computed curves of (M) for L/Ls vs T indicate a slow temperature Variation.
REFERENCES 1.
MILLS D.L.,J. Phy~.Chem. Sol. 34, 679 (I973).(note that two misprints appeared in formulas (10 a) and (10 b) : ~/12 should replace ~/6 and ks should be equal to (1/2)kd, furthermore. in formula (l4b) one should read —133(X r)/6 instead of—I3 3(~r)/3).
2. 3.
The experimental situation together with the corresponding references are given in (M) JULLIEN R., BEAL-MONOD M.T. and COQBLIN B..Phys. Rev. Len. 30, 1057 (1973).
4. 5. 6.
See for instance in Table of Integrals, Series and Products by GRADSHTEYN IS. and RYZHIK 1.M.. Academic Press, New York (1965). SCHRIEMPF J.T., SCHINDLER A.!. and MILLS DL., P/n’s. Rei’. 187. 959 (1969). SCHRIEMPF J.T., MAC INNES W.M. and SCHINDLER A.!.. paper BD 8. Bull. Am. P/n’s. Soc. 17, 256 (1972).
7.
KAISER A.B. and DONIACH S. Jut. J. Magn. 1. 11(1970).
Vol. 13, No. 10
TEMPERATURE DEPENDENCE OF THE LORENZ NUMBER On montre que Ic nombre de danstemperature les métaux presque 2 aLorenz três basse et atteintmagnétiques Ia valeur de vane comnie A + BT + CT Sommerfeld a très haute temperature avec une variation très lente en (1 + cst/T)~ Ces rCsultats analytiques expliquent des rCsultats numeriques antCrieurs en accord avec l’expCrience. -
1711