Terence I. Quickenden' and Robert C. Marshall University of Queensland Brisbane, Australia, 4067
I
I
Magnetochemistry in SI Units
The conversion of magnetochemical formulas from the frequently used CGS-EMU system of units to the International System of Unih (SI) (1-4) contains several traps for the unwary. It is the aim of this paper to provide the basic information which will enable this conversion to be carried out simply and with the minimum of confusion. A survey of the literature indicates that most magnetochemists use the CGS-EMU system of units in it9 unrationalized form, although in occasional instances (5), rationalized CGS-EMU units have been used. Bleaney and Bleauey (6) and also Mulay (7) have set out a few magnetochemical formulas in MKSA units, hut their treatments are by no means comprehensive. The SI system is itself a development of the rationalized MKSA system, and the relationship between the two systems has been discussed by McGlashan (4). The tables presented here, include the commonly used systems of units, hut for simplicity, the Gaussian system and the CGS systems based on four basic quantities, have not been considered. For a detailed description of these systems, the reader is referred to references (1-4). There are three major difficulties which are often involved when magnetic formulas are converted from one systcm of units to another. Firstly, pa, the permeability of a vacuum, is often omitted from formulas expressed in the unrationalized CGS-EMU system because in this system it is equal to one and has no dimensions. However, in the SI system, po has a dimensioned value of 477 X lo-' kg m s-? A-2 and must be inserted in the appropriate position (see Table 1) when formulas in the unrationalized CGS system are convertcd to SI units. It is desirable to check any such conversion by means of a dimensional analysis. A second difficulty relating to the conversion of M, is deformulas arises because the magnetizati~n,~ fined in two different ways by different workers. If B is the magnetic flux density and H is the magnetic field strength, then in one case M is defined by either of the equations
+ 4rrM
B =
(unrationalized)
(1)
(rationalized)
(2)
or B
3
@OH +M
whereas in the other case, M is defincd by either of the equations B
=
pa(H
+ 4rrM)
(unratioualized)
(3)
'
Present address: Department of Physicd and Inorganic Chemistry, University of Western Australia, Nedlands, 6009, Western Aust,ralia. 20ften referred to as the Intensity of Magnetization, I, in non-SI systems.
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Journal of Chemicol Education
or B
=
+
@+(H M ) (rationalized)
(4)
In eqns. (1) and (2), B and M are regarded as similar quantiti~swhile in eqns. (3) and (4), H and M are regarded as similar quantities. The distinction is of no consequence in the unrationalized CGS-EMU system where po is dimensionless and equal to one. However, in the other unit systems shown in Table 1, the distinction is important because po has a non-unit numeric value, and in some cases has dimensions. I n the SI system, eqn: (4) provides the definitionof M. Rationaheation must also he taken into account when converting units and formulas from one system to another. Many of the quantities in rationalized and unrationlized systems differ (5) by a dimensionless factor of 4a. Thus in goingfrom anunrationalized to a rationalized system, H is replaced by 4aH, pa is replaced by p0/4a, P is replaced by p/4a, and B is unchanged. If M is defined by eqn. (I), M is replaced by M/4a on rationalization, whereas if M is defined by eqn. (3) M remains unchanged on rationalization. The effect of rationalization on common magnetic formulas and quantities is shown in Tables 1and 2. The purpose of rationalization is to ensure that factors of 2a and 4a do not appear in electromagnetic formulas unless the formulas refer to situations which involve circular or spherical symmetry.
Table 1. Unrationalized CGS-EMU (PO = 1)
Description of formula
B
++
+
poH 4rM B = pa(H 4rrM) H 4rM = H 4rM M M Magnetic susceptibility x = X -
Magnetic flux density
=
=
+
F/1 = ZI,13/d Deflnition of the ampere Relative permeability P. = P hH Farce developed in the Fara= mxnH d m method hl Force developed in the Gouy = method m x m ( H ~%H W Magnetic moment of an atom pat, = (xMT)'/z in Bohr magnetons = 2.8279fw~T)'h 1 Magnetic energy term from = - SdVHdB Maxwell's equations 6 4r . .) 1 Magnetic energy inside a long = -_ (1 4rx)HXV solenoid (5) &r Zusx~H' Effect of a magnetic field on K, an equilibrium constant - = exp Ko 2RT (10)
,,
(Nz2)"' +
By using Table 2, the numeric value of a magnetic quantity in the SI system can be obtained from the corresponding values in other unit systems. Similarly, Table 1 enables the conversion of common magnetochemical formulas to the SI system. In this table, the columns have been subdivided according to the two alternative definitions of M. This distinction is important when converting to the SI system from systems other than the unrationalized CGS-EMU system. Following common practice, pa has been omitted from all formulas listed under the unrationalized CGS system, because in this case it is equal to one and is dimensionless. The van Vleck (8)equation XM =
N.,C[ W , ' ~ ' Z / ~-TZ W P ~exp ( - W < * / ~ T ) exp (-W."/kT)
here will encourage magnetochemists to make the transition to SI units. One of us (R. C. M.) acknowledges receipt of a Commonwealth Postgraduate Scholarship.
Appendix: Symbol
B
d
(5)
i
which is not included in Table 1, is sometimes used for calculating molar magnetic susceptibilities. It is correct, as written, for use with unrationalized CGS-EMU units, but if SI units are substituted, the right hand side of the equation must be multiplied by ro (= 47 X 10" kgm s-= A-?). To assist magnetochemists who commonly use reference materials of known susceptibility when making magnetic measurements, a list of such materials and their susceptibilities in SI units, has been included in Table 3. The symbols used in Tables 1 and 2 are listed with their names a t the end of this paper. All symbols except the molar magnetic susceptibility X M , the mass magnetic susceptibility x,, and the effective magnetic moment pelf of an atom, are in accordance with the SI recommendations. In the case of the three exceptions, no SI recommendations have been made, and the terminology used has no officialstanding. The authors hope that the information presented
Symbols Used in this Paper Qwntity magnetic flux density, magnetic induction distance force magnetic field strength electric current Boltzmann constant equilibrium constant in zero magnetic field equilibrium constant in a magnetic field length mass magnetization molar mass Avogadro constant gas constant thermodynamic temperature, absolute temperature volume
Name of SI Unit tesla or weber per square meter
Symbol for SI Unit T or Wb m-l
meter newton ampere per meter ampere joule per kelvin
m
meter kilogram ampere per meter kilogram per mole reciprocal mole joule per mole kelvin kelvin
cubic meter
Magnetochemical Formulos in the SI and Other Unit Systems
Rationalized CGS-EMU (m = 4 ~ )
Unrationalized MKSA ka m s-*A-')
(ro = 10'
Rationabed MKSA (ua = 4 2 X 10-7 ke ms-ZA-")
SI
B = M + M
K-" = KO
ex*
(Int. System of Units) B=m(H+M)
2 r Z v ~ x ~ H ~ RT Volume 49, Number 2, February 1972
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115
Literature Cited
joule mass fraction of NiClz the energy of weve function i, in zero magnetic field the first order Zecman effect coefficient the second order Zeeman effect coefficient permeability permeability of a vacuum Bohr magneton magnetic moment of an atom in Bahr magnetons relative permeability stoichiometrie coefficient of substance B 3.141.59... density
(1) I S 0 Recommendation R 31. "Part V: Quantities and units of eketricity and magnetism" (1st ed.), International Orpanisstion for Standardization. 1965. (2) Report of t h e Commission on Symbola. Terminology, and Unita. IUPAC, prepared by M. L. MoGlashan, Pure A p p l . Chem., 21, 1
joule
,,o-n>
4r
ma kg-'
1
kg m ssPA-a
105/4r A m+/Oc
Wb m-'/G
Journal of Chemical Educafion
oersted
m3kg-'/(cmJg-1)
kg m S - ~ A - ~1
Am-'
1/ 4 r
A m-1
1
Wb m-%
1
Wb m+
1
1
1
1
1
