- Email: [email protected]
Thermoelectric power of transition solutes in liquid tin
Volume 41A, number 1
PHYSICS LETTERS
28 August 1972
THERMOELECTRIC POWER OF TRANSITION SOLUTES IN LIQUID TIN S. TAMAKI* and N.E. CUSACK University of East Anglia, Norwich, UK Received 4 July 1972 Thermoelectric powers of Fe, Ni and Co dissolved in liquid Sn are measured and discussed in relation to resistivity and susceptibility by reference to the concept of virtual bound state.
For a dilute alloy of solute 1 in solvent 0 the increment of resistivity is: =
~ ~x~y where L~p~ =Acv~2(If i~p =Acv~2(a(f
(1)
2)
2
2??]
K3F(k)dK
‘
0
and f is the scattering amplitude expressible by phase shifts [1,21 Previous work [31showed that I at. % og Ni, Co, Fe in liquid Sn increased the resistivity by L~p 1 to 2ii~2cm. This is several times larger than for nontransition solutes. The corresponding enhancement of magnetic susceptibility has been explained by Tamaki [4] in terms of a virtual bound state whose energy centre approached EF more closely in the order Ni, Co, Fe. Suggested by these results, the following basic assumption will be made here, namely, that L~pand the corresponding thermoelectric power increment, ~S, are dominated by the 1 = 2 phase shift of the impurity atoms and, as a first approximation, eq. (1) may be used with all 27’S other than ~2 ignored. It follows that h.p, spy, and 2kF)sin2’ri L~p (20~rhc/ze 2(EF) (2) .
—~
~‘
Permanent address: Niigata University, Japan.
z~.S 2o1~’~13T AOckF. [2 Sin?72
di~ 2 sin F
~~pS /p °
.
(3)
°
From eq. (3) z~Sis approximately proportional to A 0. This is a testable consequence though not investigated here. From the observed ~S, the known properties of the pure solvent, and 272 obtained from eq. (2),
2kF
*
F
1 —f0I
1f~+f’f0—2If0l2)), c = at. fraction of solute, a a(K) the pure solvent structure factor,A = 3h(4d2e2m2)1, ~2 = volume per atom. The remaining symbols have their usual meanings; in particular (F(K)) = (4k4)~ F JC
where z is the number of conduction electrons per atom in the solvent. This quantity does not contain the structure factor. [3 lnp/3EIE it follows that Using S = (~2k~T/3Ie~)
d~ 2/dEcan be found from eq. (3). The samples were prepared by vacuum melting of the pure components and measured in fused silica sample holders using the “small ~T’ method. The counter electrode was the platinum component of the platinum-platinum 13% rhodium thermocouples used for measuring the junction temperatures. The solid wires were separated from the sample by thin molybdenum foils in the manner used in previous work [5]. Various samples were observed in argon atmospheres or vacuo over 800 to I 1000C. Typical results are shown in the figures and table 1. In general L~sSincreases with impurity concentration but dz~S/dTis almost zero. The latter result can be seen to be consistent with eq. (3) as follows. txp and therefore ‘~2are very little temperature dependent and the most temperature-dependent factors are therefore T and AOkF. Over the range of temperature, their temperature variation cancels to a considerable extent and the actual sign of dL~.S/dTis therefore sensitive to details not sufficiently well known to predict numerically. 41
Volume 4lA, number I
PHYSICS LETTERS
28 August 1972
Table I p
0
Vd(EF)
EF. E0
(eV)
(eV~’
(eV)
Zd Pure Sn
(p12cm) 62.0
0
Nil at ~ Co I at ~
62.65 64.5
0.65 1.5
8.68 ir/lO 7.94 rr/I 0
—1.0 2.2
8.7 7.9
0.76 0.98
0.71 1.18
1.77 1.30
le I at ~
64.0
2.0
7.57410
-2.7
7.6
1.07
1.41
1.12
-
--
212(E)
____________________
600
700
800
-2
900
1000 ~C
—~O~--°i-o~ °~
-4
° ~-~-
-
arecot f(E0-
=
“d~1~= ~10
—-~----~—--~-—---~
Fe 1 at ~,o
—
L )
=
iord172 ~
=
+ z~
0.._._0_Fe2at0/~
0
=
(l0/ir~)sin(zdir/10)
zd(IOI?T)7)2(LI..). —12
~ *4~-~--Fe4at~/~
—~
-14
Fig. 1. Temperature dependence of the thermoelectric power of dilute alloys of iron in liquid tin. -o-, ~x-,-~- different samples. Nickel and cobalt show a similar small dependence ~ at!. of impurities 0 2
4
6
—2 ° —6 —8
-12
0N1(O)
Here. E~,~, fld(EF) are, respectively, the centre, width parameter, and density of states of the virtual bound state. Zd, derived from the Friedel sum rule, is the number of electrons localised in the virtual .
bound 2?7 state near the impurity atom. Since ~p gives sin 7, not sin ~ ~7 has been chosen between ir and ~ir in order to make zd physica1l~more reasonable, i.e. comparable with the number of d electrons in the free transition metal atom. ~ is not exactly the same for Fe, Ni, Co but is about 1eVThus for all (EF--EO) varies from 1.77 to 1.12 eV. thethree. thermoelectric and resistivity increments are explained and reconciled with the
\
susceptibilities referred to above if the centre of the virtual bound state approaches E 1: from below in the \
\~
x-~0~
14 Fe(6)
.
manner shown in table I, and it is therefore tentatively
concluded that this model is adequate for the electronic behaviour of Fe, Ni and Co in dilute solution in liquid Sn.
Fig. 2. Concentration dependence of the thermoelectric power of liquid tin alloys with 3-d transition metals impurities at 1000°C
Values of ~
are giben in table 1 and deductions
can be made from them with the help of the following equation from Klein and Heeger [6] : 42
References Ill T. Faber and J. Ziman, Phil. Mag. 11(1965)153. 121 J.M. Dickey, A. Meyer and W.H. Young, Phys. Rev. 160 (1967) 490. 131 S. Tamaki, J. Phys. Soc. Jap. 25 (1968) 1596. 14] S. Tamaki, J. Phys. Soc. Jap. 25 (1968) 1602. ~sj V.F1.C. Crisp, P.W. Kendall and N.E. Cusack, J. Phys. C., metal. Phys. Supp., No.1 (1970) 102. [6] A.P. Klein and A.J. Heeger, Phys. Rev. 144 (1966)
PHYSICS LETTERS
28 August 1972
THERMOELECTRIC POWER OF TRANSITION SOLUTES IN LIQUID TIN S. TAMAKI* and N.E. CUSACK University of East Anglia, Norwich, UK Received 4 July 1972 Thermoelectric powers of Fe, Ni and Co dissolved in liquid Sn are measured and discussed in relation to resistivity and susceptibility by reference to the concept of virtual bound state.
For a dilute alloy of solute 1 in solvent 0 the increment of resistivity is: =
~ ~x~y where L~p~ =Acv~2(If i~p =Acv~2(a(f
(1)
2)
2
2??]
K3F(k)dK
‘
0
and f is the scattering amplitude expressible by phase shifts [1,21 Previous work [31showed that I at. % og Ni, Co, Fe in liquid Sn increased the resistivity by L~p 1 to 2ii~2cm. This is several times larger than for nontransition solutes. The corresponding enhancement of magnetic susceptibility has been explained by Tamaki [4] in terms of a virtual bound state whose energy centre approached EF more closely in the order Ni, Co, Fe. Suggested by these results, the following basic assumption will be made here, namely, that L~pand the corresponding thermoelectric power increment, ~S, are dominated by the 1 = 2 phase shift of the impurity atoms and, as a first approximation, eq. (1) may be used with all 27’S other than ~2 ignored. It follows that h.p, spy, and 2kF)sin2’ri L~p (20~rhc/ze 2(EF) (2) .
—~
~‘
Permanent address: Niigata University, Japan.
z~.S 2o1~’~13T AOckF. [2 Sin?72
di~ 2 sin F
~~pS /p °
.
(3)
°
From eq. (3) z~Sis approximately proportional to A 0. This is a testable consequence though not investigated here. From the observed ~S, the known properties of the pure solvent, and 272 obtained from eq. (2),
2kF
*
F
1 —f0I
1f~+f’f0—2If0l2)), c = at. fraction of solute, a a(K) the pure solvent structure factor,A = 3h(4d2e2m2)1, ~2 = volume per atom. The remaining symbols have their usual meanings; in particular (F(K)) = (4k4)~ F JC
where z is the number of conduction electrons per atom in the solvent. This quantity does not contain the structure factor. [3 lnp/3EIE it follows that Using S = (~2k~T/3Ie~)
d~ 2/dEcan be found from eq. (3). The samples were prepared by vacuum melting of the pure components and measured in fused silica sample holders using the “small ~T’ method. The counter electrode was the platinum component of the platinum-platinum 13% rhodium thermocouples used for measuring the junction temperatures. The solid wires were separated from the sample by thin molybdenum foils in the manner used in previous work [5]. Various samples were observed in argon atmospheres or vacuo over 800 to I 1000C. Typical results are shown in the figures and table 1. In general L~sSincreases with impurity concentration but dz~S/dTis almost zero. The latter result can be seen to be consistent with eq. (3) as follows. txp and therefore ‘~2are very little temperature dependent and the most temperature-dependent factors are therefore T and AOkF. Over the range of temperature, their temperature variation cancels to a considerable extent and the actual sign of dL~.S/dTis therefore sensitive to details not sufficiently well known to predict numerically. 41
Volume 4lA, number I
PHYSICS LETTERS
28 August 1972
Table I p
0
Vd(EF)
EF. E0
(eV)
(eV~’
(eV)
Zd Pure Sn
(p12cm) 62.0
0
Nil at ~ Co I at ~
62.65 64.5
0.65 1.5
8.68 ir/lO 7.94 rr/I 0
—1.0 2.2
8.7 7.9
0.76 0.98
0.71 1.18
1.77 1.30
le I at ~
64.0
2.0
7.57410
-2.7
7.6
1.07
1.41
1.12
-
--
212(E)
____________________
600
700
800
-2
900
1000 ~C
—~O~--°i-o~ °~
-4
° ~-~-
-
arecot f(E0-
=
“d~1~= ~10
—-~----~—--~-—---~
Fe 1 at ~,o
—
L )
=
iord172 ~
=
+ z~
0.._._0_Fe2at0/~
0
=
(l0/ir~)sin(zdir/10)
zd(IOI?T)7)2(LI..). —12
~ *4~-~--Fe4at~/~
—~
-14
Fig. 1. Temperature dependence of the thermoelectric power of dilute alloys of iron in liquid tin. -o-, ~x-,-~- different samples. Nickel and cobalt show a similar small dependence ~ at!. of impurities 0 2
4
6
—2 ° —6 —8
-12
0N1(O)
Here. E~,~, fld(EF) are, respectively, the centre, width parameter, and density of states of the virtual bound state. Zd, derived from the Friedel sum rule, is the number of electrons localised in the virtual .
bound 2?7 state near the impurity atom. Since ~p gives sin 7, not sin ~ ~7 has been chosen between ir and ~ir in order to make zd physica1l~more reasonable, i.e. comparable with the number of d electrons in the free transition metal atom. ~ is not exactly the same for Fe, Ni, Co but is about 1eVThus for all (EF--EO) varies from 1.77 to 1.12 eV. thethree. thermoelectric and resistivity increments are explained and reconciled with the
\
susceptibilities referred to above if the centre of the virtual bound state approaches E 1: from below in the \
\~
x-~0~
14 Fe(6)
.
manner shown in table I, and it is therefore tentatively
concluded that this model is adequate for the electronic behaviour of Fe, Ni and Co in dilute solution in liquid Sn.
Fig. 2. Concentration dependence of the thermoelectric power of liquid tin alloys with 3-d transition metals impurities at 1000°C
Values of ~
are giben in table 1 and deductions
can be made from them with the help of the following equation from Klein and Heeger [6] : 42
References Ill T. Faber and J. Ziman, Phil. Mag. 11(1965)153. 121 J.M. Dickey, A. Meyer and W.H. Young, Phys. Rev. 160 (1967) 490. 131 S. Tamaki, J. Phys. Soc. Jap. 25 (1968) 1596. 14] S. Tamaki, J. Phys. Soc. Jap. 25 (1968) 1602. ~sj V.F1.C. Crisp, P.W. Kendall and N.E. Cusack, J. Phys. C., metal. Phys. Supp., No.1 (1970) 102. [6] A.P. Klein and A.J. Heeger, Phys. Rev. 144 (1966)