[Springer Series in Materials Science] Magneto-Science Volume 89 || Fundamentals of Magnetic Field Effects

1 Fundamentals of Magnetic Field Effects

Magnetic energy is smaller than thermal energy or electric energy. The magnetic energy of an electron spin of 1 Bohr magneton in a field of IT corresponds to thermal energy of 0.67 K or electric energy of 58 //V. Furthermore, thermal disturbance reduces magnetic energy in nonferromagnetic systems. It is 12.5 mJ mol" in 1 T at 300 K for a paramagnetic system. This is about 10" the thermal energy of 2.5 kJ mol" at the same temperature. Consequently, it does not seem that magnetic field effects (MFEs) occur at ordinary or elevated temperatures at which materials are processed. However, a variety of MFEs were found first in selected systems and later in popular systems by utilizing appropriate mechanisms based on quantum mechanics, electromagnetism and magnetic properties of materials.

In this chapter, we first introduce the quantum mechanical or electromagnetic origins of MFEs associated with the summary of the quantities and units in magnetism. The fundamental mechanism of magneto-thermodynamic effects, magnetic force effects and quantum mechanical effect (dynamic spin chemistry) are then described. Recent developments in magnet technology have helped the discovery of these MFEs. In many cases, MFEs are enhanced in proportion to the square of the magnetic field strength. This chapter also describes the science and technology of high magnetic field generation.

1.1 Basis of Magneto-science

1.1,1 Origins of Magnetic Field Effects

Magnetic field effects (MFEs) are classified according to the time-space properties of fields (steady or time-varying, and homogeneous or gradient ones) as listed in Table 1-1-1. This classification depends on the MFEs themselves, not on practical applied fields. If, for instance, a magnetic effect occurs in a much smaller region compared to the spatial variation of


Table 1-1-1 Classification of magnetic field effects

Year

1974

1975 1976

1981 1985

1987

1988 1991

1991

1994

1994 1996

1996

2004

Magneti

Steady field

Time-varying field

ic field Effect

Homogeneous field Quantum effect (Zeeman effect) Magneto-thermodynamic effect Magnet ic torque Lorentz force

Gradient field Magnet ic force (Faraday force) Alternating field Eddy current High-frequency field Energy injection

Table 1-1-2 Epoch-making findings of magnetic field effects

Investigator

A. Matsuzaki, S. Nagakura R. Aogaki Y. Tanimoto, H. Hayashi J.Torbet T. Kakeshita

M. Yamaguchi

S. Ozeki I. Mogi

E. Beaugnon, R. Toumier I. Yamamoto

S. Ueno N. Hirota, K. Kitazawa T. Kimura, M.Yamato S. Nakabayashi

Effect

Magnetic fluorescence quenching MHD effects in electrolysis MFE on photoreaction

Magnetic alignment of fibrin Magnetic field-induced martensitic transformation MFE on chemical equilibrium

Magnetic adsorption Mre on the deposition of metal leaves Diamagnetic levitation

MFE on electrode potential

Moses effect Magnetic Archimedes effect

Magnetic alignment of crystalline polymers MFE on the refractive index of water

Mechanism

Quantum effect

MHD Quantum effect

Magnetic alignment Magneto-thermodynamic effect Magneto-thermodynamic effect

MHD

Magnetic force

Magneto-thermodynamic effect Magnetic force Magnetic force

Magnetic alignment

Ref

1

11 2

8 4

3

13 12

5

14

6 7

9

15

an applied gradient field, it is regarded as a homogeneous field. When a magnetic effect proceeds and terminates within a shorter period than the duration of an applied pulsed field, it can be considered to be steady.

Time-varying magnetic fields can induce large MFEs via the induction of eddy current or the injection of energy. We, however, focus our main interest on the MFEs induced by steady fields, both homogeneous and gradient, in this volume. Below, we briefly introduce the origins of MFEs referring to the epoch-making findings listed in Table 1-1-2.

A. Quantum Effect In the quantum theory, the particle which possesses spin S is accompanied by magnetic moment. This magnetic moment interacts with magnetic field B by the Zeeman effect. This interaction is expressed by the Hamiltonian for an electron.

H = -gll^S^B (1)

where g is the g-factor (g = 2 for the electron spin) and JUB is the Bohr

1.1 Basis of Magneto-science 3

magneton. The z-component of the spin is 5 = -1/2 and 1/2 when the z-axis is along the magnetic field. Consequently, the state of the electron spin is split into two energy levels in the magnetic field.

Frequently, radicals with unpaired electrons appear during a chemical reaction. Magnetic fields can influence the kinetics or the yields of some chemical reactions due to the radicals.'^^ In the early stage of a photo-induced radical reaction, two radicals make a radical pair at the total spin with the singlet state 5 = 0 or the triplet state S =1 while the energy level of the singlet state is not split by a magnetic field and the triplet state is split into three energy levels assigned by 5,= - 1 , 0, 1. The yields of final reaction products depend on the singlet or the triplet state in the early stage of the reaction. Hence, both states convert to each other at a certain probability by magnetic fields. As a consequence, the yields of the final products vary with the strength of magnetic fields. This is called the radical pair mechanism of the MFE and was clearly verified in the photodecomposition of dibenzoyl peroide. ^

B. Magneto-thermodynamic Effect Every substance has the magnetic free energy Gm more or less under the influence of a magnetic field. If the magnetic free energy is considerably different between the reactant and the product sides in a reaction, the reaction tends to proceed towards the side with the lower magnetic free energy. This is called the magneto-thermodynamic effect. The same situation can occur in first-order phase changes also. When all components which take part in the reaction or phase change are paramagnetic or diamagnetic, the change in magnetic free energy is written

AGm=--ÂxB' (2)

where Ax is the change in susceptibility per reaction or phase change and /lo is the permeability of a vacuum. (The exact expression is given in section 1.2.) The value of AGm is on the order of 1 J mol" even when B =10 T. This is much smaller than the thermal energy of 2.5 kJ mol"' at room temperature by a factor of about 10 ^ Therefore, the magneto-thermodynamic effect cannot be detected even under high magnetic fields of 10-30 T which can be supplied by current magnets. However, if ferromagnefic components take part in a reacfion or phase change, the change in magnetic free energy becomes so large that it is several or several tens of percent of the thermal energy because of a large change in magnetization AM.

AGm = -AMB (3)

This was first exemplified by the MFE on the chemical equilibrium of ferromagnetic metal hydride-hydrogen reactions: Applying magnetic fields to the ferromagnetic LaCosH^ and hydrogen system induces increase in the

4 1 Fundamentals of Magnetic Field Effects

equilibrium hydrogen pressure, which is a measure of the chemical equilibrium.^^ The MFE on a phase change was found in the magnetic field-induced martensitic transformations in ferrous alloys.'*^

C. Magnetic Force If a magnetic body is positioned in a gradient field, the body is acted on by a magnetic force, called the Faraday force, as

F^ = VM^ = V^B^ (4) az fJ.Q dz

where V is the material volume and M is the magnetization. The second equality folds on a paramagnetic or diamagnetic substance with the susceptibility x- This force is very weak for usual diamagnetic substances in ordinary fields. Recently, however, we can enhance the product B(dB/dz) with the aid of high field magnets. Consequently, some diamagnetic substances, such as water and organic compounds, can be levitated in air by applied fields which range up to 20 T or more. ^ Thus new functional materials can be synthesized under micro-gravity in laboratories on Earth, not in space.

Meanwhile, the free surface of water is deformed by magnetic fields of several Tesla. This is termed the Moses effect. ^ Furthermore, the diamagnetic levitation and the Moses effect can be enhanced by using a counter paramagnetic liquid. This is called the magnetic Archimedes effect analogous to the usual Archimedes effect. ^

D. Magnetic Torque and Alignment The magnetic free energy of a magnetic body, including paramagnetic and diamagnetic bodies, is expressed by

Gm = -VM' B = -VMB cos 6 (5)

where 6 is the angle between the magnetization M and the field B. As a consequence, the body undertakes the magnetic torque T as

T = -^^ = VMB^me (6) de

Here, the torque acts around the axis perpendicular to both M and B. This torque is very weak for a single diamagnetic molecule. But the magnetic torque becomes capable of overcoming the thermal disturbance if a molecular assembly is formed with A^=10' -10 molecules. Consequently, the molecular assemblies are aligned by magnetic fields and show anisotropics in macroscopic properties. This magnetic alignment was first indicated in the biological material of fibrin fibers ^ and later in a variety of substances, such as crystalline polymers^^ and organic gels.' ^

E. Lorentz Force and MHD An electric charge q which moves with the velocity v in the magnetic field

1.1 Basis of Magneto-science 5

B is acted on by the Lorentz force F as

F = qvXB (7)

For fluid matter, liquid and gaseous, the interacting force / between an electric current density i and the magnetic field B is given by

f=ixB (8)

This is the base of magnetohydrodynamics (MHD). Every chemical reaction inevitably includes some ions in the process. The ions are forced to move perpendicular to both the current and the field. This is typical in electrochemical reactions. The MHD effect was first investigated in macroscopic systems of electrolysis,^'^ and more recently for microscopic systems in relation to the morphology of electrochemical products.'^^

1.1.2 Quantities and Units in Magnetism

Magneto-science is related not only to magnetism but also to various scientific and technological fields. We must simultaneously treat different kinds of quantities, e. g. mechanical, thermal, electric, chemical and others. Unfortunately, there are diverse definitions of the quantities and unit systems in magnetism. Original papers in magnetism are mostly described in CGS units even now. However, SI units (MKS units) are now widely used in other fields. As a result, we encounter serious difficulties frequently in nomenclature and numerical calculations. The best and only way to overcome this difficulty is to conform to SI units in magneto-science. Let us reconfirm the basic quantities in magnetism and their units.

A. Fields in Magnetism The magnetic flux density B [T = Wb m" ] involves contributions by the magnetic field strength H [A m"'] and magnetization M [A m"' = J T"'m"^] while brackets [ ] denote SI units.

B=HoH + idoM (9)

where jUo is the permeability of a vacuum (//o= 4;r x 10" H m"'). When an isolated body is placed in the external field He, the demagnetization field Hd is produced in the body. This originates in the continuity of magnetic flux density B.

H = He^Hci (10)

Eventually,

B = fioHe + ldoHd+HoM (11)

It is noted that the quantity I = JUQM [T = Wb m" ] is defined as the magnetic polarization in SI units.


B. Susceptibility and Permeability The magnetization curve shows the H-dependence of magnetization Af. The ferromagnet has a nonhnear, nonsingle value dependence called the hysteresis curve. In other substances, the magnetization is proportional to the magnetic field strength.

M = xH (12)

where x [absolute number] is the (volume) susceptibility. The susceptibility is positive and negative for paramagnetic and diamagnetic substances, respectively.

In an anisotropic medium, the susceptibility must be expressed by a symmetrical second-rank tensor as

MC Ml

Mi =

'Xn I2I

.Xi'

Xn Z22 X32

Xn' X23

x».

' / / l

IHi

[H. iX'J=Xji) (13)

Mi'

Ml

Mi =

'Xi 0

.0

0 X2

0

0" 0

XK

• / / l

/ / 2

[Hi

where the subscripts 1, 2 and 3 denote the jc, y and z axes in the Cartesian coordinate, respectively. ^ In general, the symmetrical second-rank tensor posses the principal axes. That is, the tensor is represented by a simple form when it refers to the principle axes.

(14)

where X\^ X2 and X3 are the principal components of the susceptibility tensor. The three principal components are equal for an isotropic substance (;f 1= X2= Xr)^ two of them are equal but the third is different for a uniaxial substance (xi= Xi^X^ ^"d all of them are different from each other for a biaxial substance iXx^X^^X^)-

The permeability // [H m ' ] is defined by the following equation.

B=îH (15)

In an anisotropic medium, the permeability is a symmetrical second-rank tensor [ju,y] like the susceptibility tensor.

C. Magnetic Quantities of Materials The total magnetization Mxox [J T"'] of a system is given by

Mtot=jdvM = VM (16) V

Here the second equality holds when the magnetization is homogeneous in the system volume V [m^]. In materials science, the magnetization is frequently measured in mass magnetization a [J T"' kg''] or in molar magnetization Mmoi [J T"' mol"']. The magnetization possessed by an atom, molecule, ion or electron is expressed by the magnetic moment fi [J T"'].

The susceptibility x is usually defined by Eq.(12); alternatively, we

1.1 Basis of Magneto-science

Table 1-1-3 Units in magnetism

Quantity

Permeability of a vacuum /io Permittivity of a vacuum Co Magnetic flux density B Magnetic field strength H Magnetization M

Magnetic flux 0 Magnetic moment jn Bohr magneton JIB

Volume magnetization M Mass magnetization a

Molar magnetization A/moi Volume susceptibility

X. = M/H Mass susceptibility X. = o/H

Molar susceptibility ;tmol = M m o l / H

Volume susceptibility K. = M/(ôH)

Mass susceptibility K-w = cr/(/io/f)

Molar susceptibility KTmol = MraoxliUoH)

Permeability jU = BIH Relative permeability

/is = M//^ Demagnetization factor N

Inductance L

SI units [MKS]

471X 1 0 ' H m ' 8.855 X 10"" F m ' T = W b m ' A m - ' J T ' m-' = A m ' B = ^QH + ÔM

Wb = Tm-J T ' = A m-9.2733 X 10--'J T-

J T- ' m-' = A m-' J T- ' kg-'

J T- ' m o l ' [a.n.]^

m^ kg-'

m^ m o l '

J T- ' m-'

J T - k g - '

J T - ' m o l '

H m ' [a.n.]

[a.n.] Hd = -NM H

CGS units [cgsemu]

1 1 in [cgsesu] G Oe erg Oe cm-^ B=H + 4nM Mx = G cm' erg O e '

' 9.2733 X 10--' erg O e ' erg O e ' cm""* erg O e ' g '

erg O e ' m o r ' erg Oe"- cm-^

erg Oe-' g '

erg Oe ' -mo l '

erg Oe"' cm"^

erg Oe"- g '

erg Oe"^ m o l '

[a.n.] [a.n.]

[a.n.] Hd = -4KNM

cm

Conversion [cgsemu]-^[MKS]

1 G = 1 0 - ' T 1 Oe = 79.58 A m ' 1 e r g O e c m - ' = l O ' J T ' m - ^

1 M x = lO-'Wb l e r g O e - ' = 10- ' JT- '

1 e r g O e - ' c m - ' = 1 0 ' J T - ' m - ' 1 erg O e ' g ' = 1 J T ' k g ' (1 [cgsemu] = 1 [MKS]) 1 erg O e ' m o l ' = lO' ' J T ' mol ' ' 1 erg Oe"' cm"^ = 47c[MKS]

1 erg Oe"' g ' = 47i x 10"' m^ kg"'

1 erg Oe"^ m o l ' = 47C X 10^ m^ m o l ' 1 erg Oe-^ cm"^ = 10' J T- ' m"^

l e r g O e - ' g - ' = 1 0 ' J T - ' k g - '

1 erg Oe- ' m o l ' = 10 J T ' ' mol"'

1 [cgsemu] = 47t X 10-' H m"' 1 [cgsemu] = 1[MKS]

1 [cgsemu] = 1 [MKS]

l c m = 1 0 - ' H

t [a.n.]: absolute number.

can introduce a different definition of the susceptibility K as

M = K(iiioH) = KBO, (K = //jUo) (17)

where ô simply represents /UQH. This susceptibility K [J T~^ m'^] is very convenient to use because of (1) easier numerical conversion from CGS units, which are traditionally adopted in magnetism, to MKS units and (2) more understandable physical meaning of units compared to the usual susceptibility x- For instance, the mass susceptibility x

1 erg Oe~^g~^(emu) in CGS units is converted to x'w= 10 J T~^kg"', but x^ = 4KX 10" m^ kg ^ Moreover, using the susceptibility K and the field 5o= jtxo// possibly gets rid of the eyesore of Ho from electromagnetic formulae. The Faraday force presented by Eq.(4) is simplified as

F, = VKB^ (18) dz

where B = JHQH [T] . Table 1-1-3 lists the conversion between CGS units and MKS units. D. Demagnetization Field The demagnetization field appears more or less in every magnetized body.


even a paramagnetic or diamagnetic one. In general, the demagnetization field depends on the figure of the body and the distribution of the external field. But the demagnetization field is uniform and proportional to the magnetization when an ellipsoidal body is located in a uniform field, i. e.,

Hd=^-NM (19)

where N [positive absolute number] is called the demagnetization factor. Its value is numerically calculated for general ellipsoids and exactly formulated for ellipsoids of revolution, which approximate the figure of a sample in many experiments.^^ ' ^

Usually, it is allowable to treat the magnetic field strength H as being equal to the external field H^ in nonferromagnetic substances. Let us consider the influence of the demagnetization field on the susceptibility exactly. Substituting Eqs.(lO) and (19) for Eq.(12) leads to

= ;t{//e-N;t(//e-^M)} = --- = ;f/fe{i + (-^;f)+(-N;f)'+---} = Y^//e^^^^

Therefore, the effective susceptibility x^ff against the external field //e is given by

Xcff=Y^^(\-mX (21)

The demagnetization factor A depends on the principal axes of an ellipsoid while N\+N2+N3=l. Consequently, the effective susceptibility depends on the direction for an ellipsoidal body with isotropic susceptibility.

References

1. A. Matsuzaki, S. Nagakura, Chem. Lett., 1974, 679. 2. Y. Tanimoto, H. Hayashi, S. Nagakura, H. Sakuragi, K. Tokumaru, Chem. Phys. Lett.,

41,267(1976). 3. M. Yamaguchi, H. Nomura, I. Yamamoto, T. Ohta, T. Goto, Phys. Lett., A 126, 133

(1987). 4. T. Kakeshita, K. Shimizu, S. Funada, M. Date, Acta Met., 33, 1381 (1985). 5. E. Beaugnon, R. Toumier, Nature, 349, 470 (1991). 6. S. Ueno, M. Iwasaka, IEEE Trans. Mag., 30, 4698 (1994). 7. Y. Ikezoe, N. Hirota, J. Nakagawa, K. Kitazawa, Nature, 393, 749 (1998). 8. J. Torbet, J.-M. Freyssinet, G. Hudry-Clergeon, Nature, 289, 91 (1981). 9. H. Sata, T. Kimura, S. Ogawa, M. Yamato, E. Ito, Polymer, 37, 1879 (1996).

10. Y. Matsumoto, I. Yamamoto, M. Yamaguchi, Y. Shimazu, F. Ishikawa, Jpn. J. AppL Phys., 36, L\391 (1991).

11. R. Aogaki, K. Fueki, T. Mukaibo, Denki Kagaku, 43, 504 (1975). 12. I. Mogi, S. Okubo, Y. Nakagawa, J. Phys. Soc. Jpn., 60, 3200 (1991). 13. S. Ozeki, H. Uchiyama, J. Phys. Chem., 92, 6485 (1988). 14. I. Yamamoto, H. Tega, M. Yamaguchi, Trans. Mat. Res. Soc. Jpn., 18B, 1201 (1994). 15. H. Hosoda, H. Mori, N. Sogoshi, A. Nagasawa, S. Nakabayashi, J. Phys. Chem. B,

1.2 Magnetic Energy and Magneto-thermodynamic Effects 9

108, 1461 (2004). 16. J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford (1985). 17. J. A. Osbom, Phys. Rev., 67, 351 (1945). 18. E. C. Stoner, Phil. Mag. Ser., 7, 36, 803 (1945).

1.2 Magnetic Energy and Magneto-thermodynamic Effects

In this section, we deduce the MFEs of the thermodynamic properties of chemical reactions and phase changes on the basis of electromagnetism and thermodynamics. As a result, systematic formulae are derived for the magnetic field-induced changes in the thermodynamic properties, such as the equilibrium constant, heat of reaction, critical temperature of phase change, etc.

1.2.1 Magnetic Energy and Free Energy

A. Magnetic Energy When a system undergoes an infinitesimal magnetic change, the magnetic work into the system is expressed by

dWrna,=ldvHdB ( 1 ) V

where V is the volume of the system, v the volume element, H the magnetic field strength and dB the infinitesimal change in the magnetic flux density*^ Here, the magnetic flux density B is contributed by the external field He, the demagnetization field Hd and the magnetization M as written by Eq.( l l ) in section 1.1. As a consequence, the magnetic work is decomposed into

dWmag =jdvidoHcdHc-^jdviJoHcdM ^ ^ (2)

= jdv//o//edHe+Jdv)UoHddHd+jJv/ioHdM V V V

Because the first term on the far right hand-side is the magnetic work required to produce the magnetic field in a vacuum, it can be eliminated when working in the field of materials science. ^ The second term is the magnetic work required to produce the demagnetization field. The third term is that required to magnetize the substance, which is determined by the magnetization curve M vs. ^ as a thermodynamic property. Therefore, when we are interested in the thermodynamic properties of a substance, the magnetic energy is written by

dUmag=\dviHoHdM (3)

On the other hand, when dealing with the magnetic energy of a real body of magnetic substance, we must include the term of the demagnetization field.


d^Lg=jdv//oêdM (4) V

B. Free Energy and Chemical Potential In this chapter, we employ Eq.(3) as the magnetic energy to investigate the thermodynamic behavior of a system. ^ We assume, for simplicity's sake, that the magnetic field strength H and the magnetization M are homogenous in the whole system. Thus,

d Lmag = j d vjUo d M = ^QH d M tot (5) V

where Mtot is the total magnetization of the system as defined by Eq.(16) in section 1.1. Then the first law of thermodynamics includes the magnetic term as

/=i

where U is the internal energy, S entropy, V volume, T temperature, P pressure and A i, A2, ..., An and jUi, jU2, ..., jUn are the mole numbers and chemical potential of components 1, 2, ..., n, respectively. The internal energy L is a function of the extensive parameters of 5, V, TVi, A2, .., Nn and Mtot.

The free energy G is a function of the intensive parameters of 7, P, H (in place of the extensive parameters 5, V, Mtot in U) and the extensive parameters NuNi, .... An. This is obtained by the Legendre transformation.

G=U-TS + PV-HQHM,oi (7)

As a matter of fact, its infinitesimal variation of G is given by

dG = -SdT+VdP + HidN,-HoM,oxdH (8)

The total magnetization is the sum of the magnetization of the constituent components.

n

Mtot = \dvM = ^ N,mi (9) i i=\

where mi is the molar magnetization of the /-th component. Accordingly, the infinitesimal variation in free energy takes the form

dG = -SdT-\-VdP+J^lLi,dNi-J^HoNimidH (10)

Because two kinds of the second-rank partial derivatives in relation to H and Ni are equal to each other," ^

l^ = -,om. ( „ ) Integrating this equation gives the chemical potential as

jUi = Hi'''^-\-liir^ = jn;'^ - j Hom.dH (12) H=0

The first term is the nonmagnetic chemical potential as a function of T and


P and the second term is the magnetic chemical potential as a function of r, P and H. These are totally called the magnetochemical potential analogous to the electrochemical potential in the electrochemical system. Thus, the infinitesimal variation in free energy is

dG = - 5 d r + VdP + J(Ai/'^+/x/'"^)dM-X/ioMm,dif (13)

Eqs. (12) and (13) are basic relations to investigate the MFEs on the thermodynamic properties of chemical reactions and phase changes.

1.2.2 Chemical Equilibrium in Magnetic Fields

A. Condition of Equilibrium When a system achieves equilibrium under the influence of constant temperature r, pressure P and magnetic field strength //, the free energy G must be minimum for a hypothetical change in mole numbers M (/ =1, 2..., n\

m ôNi JT.P.N,*N,.H , , ..

\dNi JT.P,NJ:^N,M

Meanwhile, a chemical reaction is generally written by

where v, and A, are the stoichiometric coefficient and the molecular formula of the i-ih component while v, is negative for the reactants (/=1, ...,/?) and positive for the products (/ = /7+l, ..., n). The changes in mole numbers can be represented by one parameter, the extent of reaction ^ as

dM=V/d(^, (/ = l,2,-s«) (16)

Consequently, the conditions for equilibrium Eq. (14) are rewritten into

dG

^S )T,P,H

Taking into account Eq. (13), the first condition for the chemical equilibrium Eq. (17) gives the relation

i{îr^îr)v.=^ (18)

By defining the nonmagnetic free energy change g^^^ and the magnetic free energy change g^""^ per hypothetical unit reaction (d( =1 mol), this becomes


^(o,+^(.)^0, fg"" = Xv,/i,'°', g""' = Xv,/i,""'l (19) V /=i i=\ J

This means that the magnetic free energy change g^""^ counterbalances the nonmagnetic free energy change g " at chemical equilibrium under the influence of a magnetic field.

If all components are paramagnetic or diamagnetic, the magnetization is proportional to the magnetic field strength.

m, = ;f,// = {x^ I lUo){HoH) = K,B (20)

where Ki [J T~ mol"^] is the molar susceptibility of the /-th component. Thus,

^- 1 1 1 n

^ • , r- | / /0/ /=Iv,X=-JfiÎ /=1 //=o ^ '=1 ^ '=1

^^'"^=-iv, j HoX^HdH = --n,H'±va^=--B'±v^K^ (2i)

In the case of ferromagnetic components, the magnetization is not a simple function of the magnetic field strength. However, we can assume that the magnetization is saturated up to the saturation magnetization of m^'\ [J T" mol"*] even in low fields. Therefore,

g('")=-Jv, J î,m^r^H = -ii,H±vM''=-Bt,yM'' (22) /=i //io '=1 '=1

B. Direction and Gibbs Phase Rule of Reactions We investigate the second condition of chemical equilibrium Eq. (17). This relation is transformed into the following inequality after some mathematical derivation.'*^

This inequality means that an increase in magnetic field shifts the reaction towards the direction in which the total magnetization of the system is increased, and vice versa.

Let us suppose a system which is composed of c chemical reactions, n components and m phases under the influence of temperature r, pressure P and magnetic field strength H. This system has « -i- 3 intensive parameters of r, P, H, î, ^2, ..., An while they satisfy m Gibbs-Duhem relations and c equations of chemical equilibrium Eq. (17). Therefore, the number of independent intensive parameters, the degree of freedom, is

F = « + 3 - m - c (24)

This is the Gibbs phase rule of chemical reactions under the influence of a magnetic field. For instance, the liquid-gas phase change of a substance A has the degree of freedom F = 2 because there is no reaction (c = 0)


between the unchanged component A (« =1) in the two phases (m = 2). Alternatively, we can assume that there is one reaction (c = 1) between two components of Auquid and Agas (n = 2) in the respective phases (m = 2). Accordingly, the phase change occurs on the equilibrium plane in the three-dimensional T-P-H space.

1.2.3 Magnetic Field Effects on Thermodynamic Quantities

A. Equilibrium Constant First, we assume an ideal gas reaction in which the nonmagnetic chemical potential of the i-ih component is given by

ju/'^=jU,*+/?nn(/?/P*) (25)

where P* is the standard pressure (ordinarily P*=latm) and /i,* is the standard chemical potential at temperature T and standard pressure P*. Accordingly, the condition of the chemical equilibrium of Eq. (18) is rewritten to

J^Vi{Hi^-\-Hi''"') = -RTj^Vi\n(Pi /P*) = -RT\nK, (26)

where Kp is the (pressure) equilibrium constant which is a function of 7, P and H. Comparing the equilibrium constant in a magnetic field and that in zero field leads to

pi'") n

ln^p""-ln^pi''i=-^^—, iKp=YliPi/P*r') (27) RT j=i

where Kp " and Kp ^^^ are the equilibrium constant in the magnetic field and that in zero field, respectively. Below, we use similar notations for other quantities.

For instance, suppose that one component (/ =g) is gaseous and the others are in a solid or liquid phase in a reaction. In this case, the equilibrium P is dominated by the pressure of the gaseous component. If every component is paramagnetic or diamagnetic, the MFE on the equilibrium pressure is expressed by

ln(p[î/p(0])^. =J?i_^v,7C, (28) 2RT i=i

Similarly, for a ferromagnetic system,

ln(pf//I/pf01)v. Â.fy.îs^ (29) RTtt

In the case of an ideal solution reaction, the nonmagnetic chemical potential is given by

iU/° = /*+/?rin(jc,/x*) (30)

where ;c* is the standard concentration and /i,* is the standard chemical potential at temperature T and standard concentration JC*. Thus, the MFE on the (concentration) equilibrium constant Kx is expressed by equations similar to Eq. (27) while P//P* is replaced by JC//;C*.


B. Electromotive Force An electrochemical system is composed of two half cells. Each half cell is a reaction like Eq. (15), consisting of two phases, the a and the p phases in which the components (/=1, 2, ..., r) and the components (/ = r+1, r+2,..., s) are included, respectively. The chemical potential of the i-ih component includes the nonelectromagnetic, the electronic and the magnetic terms as

/// =/z/'^ +/i/^^ +^/'"^ =/i, *+/?rin(jc, /x*) + z/Fâ +/i/"^ (31)

where jc* is the standard concentration, jc/ and z, are the concentration and the valence of the /-th component, respectively, F is the Faraday constant and *Fa is the electric potential for the a phase (*Pp for the p phase). In this case, the equilibrium condition Eq. (18) is replaced by

t,{iî'''-^^r-^nry,=o 02)

This leads to the electromotive force E of the half cell. ^

£ = „-«Fp=^^v,(/x,*+/?nn(;c,/jc*) + /i/'"0 (33) fq i=i

where Fq is the total charge transfer by the unit reaction (d<^=l mol). r s

q = - X ' '= X ' ' (34) (a phase) ((5 phase)

Thus, the magnetic field-induced change in the electromotive force is

Fq i=\ Fq

where g^""^ is the magnetic free energy change for the half cell. Now we go on to the full cell, which is composed of two electrodes

and electrolyte solution. That is, one half cell consists of one electrode (Phase a) and the solution (Phase p) and the other one consists of the solution (Phase p) and the other electrode (Phase y). The entire reaction is expressed by Eq. (15) when the two electrodes do not contribute to it. Applying Eq. (35) to both half cells and adding them gives the MFE on the electromotive force between the two electrodes.

A£ = £[/^]_£[o]^(^^^^)[/ / i_(^^_^yoi^_Lf ^^y.)^_L^(-) (36) Fql^x Fq

Where n is the number of constituent components in the solution, g"""^ is the magnetic free energy change for the reaction and Fq is the total charge contributing to the reaction. This is rewritten to the following equation for the paramagnetic or diamagnetic system.

A£ = ^ l ; v , 7 C , (37) 2Fq /=i

It is easy to adapt this consideration to the MFE on the electromotive force in gradient magnetic fields.^^


C. Heat, Free Energy and Entropy of Reaction A chemical reaction is accompanied by the heat of reaction (the enthalpy change) Aff, the free energy of reaction (the free energy change) ACf and the entropy of reaction (the entropy change) A5^ when it occurs between the components at the standard pressure P*, temperature T and magnetic field strength H. These three thermodynamic quantities are possibly a function of T and H and they are related to each other as

AH = AG^TAS (38)

where the superscript ^ is omitted from the notation for simplification. The free energy of reaction is connected to the equilibrium constant.

AG = -RT\nKp (39)

Consequently, the magnetic field-induced change in AG is obtained by

^G^î - AG^'^ = -RT{\n K,'"' - In ^p^^ ) = g'^' (40)

Furthermore, the heat of reaction is expressed by the van' t Hoff relation.

AH = /? r '—lnA:p

Thus, the magnetic field-induced change in AH is given by

Finally, using Eq. (38) gives

AH = RT'—\nK, (41) dT '

A//t^] - Aîo] ^ (m) _ 7 A ^ ( - ) (42) dT

A5t^]_A5i«]=_A^(- ) (43) oT

The formulae of Eqs. (41)-(43) are applicable to a solution or solid reaction. It is noted that Eq. (43) is common to the case of magnetic refrigeration (heat)." ^

D. Phase Change Two phases coexist in a first-order phase change. For instance, a liquid-gas phase change is

Aiiquid = Agas (44)

Accordingly, this phase change is expressed by v i = - l and V2=l in Eq. (15). All of the formulae obtained above can be applied to this phase change. We assume that the system is paramagnetic or diamagnetic with the molar susceptibility of KI for the liquid (Phase 1) and K2 for the gas (Phase 2). Eq. (21) yields the magnetic free energy change per unit mole reaction.

^ ( - ) = - 1 5 ^ ( ^ 2 - ^ i ) = -^^ 'AfC, {ki=Xi^lô,B = ôH) (45)

Eventually, using Eq. (28) gives the MFE on the equilibrium pressure in the gas phase.


pm I pio] ^ expf ^ ^ ^ 1 == 1 + -^ (46) \ IRT ) IKT

Because the degree of freedom is 2 for the phase change under the influence of magnetic fields, there are two independent parameters among temperature 7, pressure P and magnetic field H, If the pressure is fixed, the phase equilibrium is determined by T and H, That is, the critical temperature of the phase change is a function of magnetic field ff at a constant pressure.

We have one of the Maxwell relations for a magnetic system.

a(/io//) _ a^

ar aMtc (47)

Meanwhile, the change in free energy AG is zero on the equilibrium plane in relation to Eq. (38).

TAS = AH, (A5=52-5, ,A/ / = / / 2 - / / i ) (48)

Using the above two equations leads to dJUoH)^ AH

dT TAm where AH and Am are the heat of enthalpy and the molar magnetization change from Phases 1 to 2. This is the magnetic Clapeyron-Clausius equation. If the two phases are paramagnetic or diamagnetic with an isotropic susceptibility of KI or K2, it is

Am = m2-mi=K2B-K\B = BAK (50)

Accordingly,

i d r = - ^ d 5 (51) T AH ^ ^

Eventually, the magnetic field-induced change in the critical temperature is given by

AT R^ AK

^ i r = - : r 7 7 7 ' (A^ = ^ i - ^ 2 =(;iri-; ir2)/Aô) (52) To 2AH

where we reasonably assume that the magnetic field-induced changes in AH and the critical temperature To are small. References

1. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford (1984).

2. E. A. Guggenheim, Thermodynamics, North-Holland, Amsterdam (1977). 3. M. Yamaguchi, I. Yamamoto, Dynamic Spin Chemistry (S. Nagakura, H. Hayashi, T,

Azumi, eds.), pp. 131-151, Kodansha, Tokyo (1998). 4. I. Yamamoto, K. Ishikawa, S. Mizusaki, Y. Shimazu, M. Yamaguchi, F. Ishikawa, T.

Goto, T. Takamasu, Jpn. J. AppL Phys., 41, 416 (2002). 5. M. Yamaguchi, I. Yamamoto, F. Ishikawa, T. Goto, S. Miura, J. Alloys Compounds,

253-254, 191 (1997).

1.3 Effects of Magnetic Orientation, Magnetic Force and Lorentz Force 17

6. I. Yamamoto, M. Fujino, M. Yamaguchi, F. Ishikawa, T. Goto, S. Miura, J. Alloys Compounds, 293-295, 251 (1999).

1.3 Effects of Magnetic Orientation, Magnetic Force and Lorentz Force

1,3.1 Magnetic Orientation

A. Magnetic Orientation Due to Magnetic Susceptibility Anisotropy^^ Usually a molecule has anisotropic magnetic susceptibilities. For example, diamagnetic susceptibility of benzene parallel to the molecular plane is different from that perpendicular to the plane. This anisotropy arises from the magnetic induction of a ring current when the magnetic field is applied. Therefore, the magnetic energy is orientation-dependent and energetically the molecule undergoes molecular orientation to the most stable direction. However, the energy of a molecule is negligibly small compared with the thermal energy at room temperature, and, as a result, the magnetic orientation of a single molecule is only expected to occur at very low temperatures.

In contrast to the magnetic energy of a molecule, aggregates, which have an ordered structure like a crystal, can obtain anisotropic magnetic energies larger than the thermal energy at room temperature. This is because the anisotropic magnetic energy of an aggregate increases with increasing volume, whereas its translational thermal energy, randomizing its orientation, is constant, regardless of volume. For the purpose of simplicity, we assume here that an aggregate is composed of n moles of molecules, has a cylindrical shape and has molecular diamagnetic susceptibilities ;fmoii and ;fmoiii, which are the susceptibilities perpendicular and parallel to the cylindrical axis, z (see Fig. 1-3-1). Then the magnetic energy £:mag of the aggregate in a magnetic field, //, is given,

£n,ag(0,H) = -(1 / l)îon iXnôu cos' e-f XrôXL sin' e)H^

= ~ ( l / 2 ) / l o n ( ; f m o I l + ( ; i : m o l H - ; f m o l l ) C O s ' 0 ) / / ' ( 1 )

= -(1 / 2)îQn {XrnoXL H- A;|fnioi cos' e)H^

where e is the angle between the magnetic field and the cylindrical axis, z, and Axmoi (= Xmom - Xmou) is the anisotropic magnetic susceptibility. Fig.l-3-2 shows schematically the angular dependence of mag {0, H) when Xmoin < Xmou < 0. Putting the anisotropic magnetic energy A£mag {H) = £ mag(0, H) -£mag (^/2, //), the aggregate undergoes magnetic orientation when ACmag {H) becomes larger than the thermal energy.

It is notable that AEm^g(H) is dependent on n. This means that the size of an aggregate is a very important factor in magnetic orientation. An aggregate with small Axmoi can undergo magnetic orientation when it is large in size, i.e., large «, or under a higher magnetic field, compared with


H 4 ^

4

Fig. 1-3-1 A cylindrical aggregate in a magnetic field. //, magnetic field; z, a cylinder-axis of aggregate; 0, angle between z and //. Xmoiii, molar magnetic susceptibility of an aggregate parallel to z; ;jfmoii, molar magnetic susceptibility of an aggregate perpendicular to z.

-pU/iX^yiH'rZ

0 n/2 n

Angle, e

Fig. 1-3-2 Angular dependence of magnetic energy of a diamagnetic cylindrical aggregate with Xmoin < ;tmoi±< 0, shown in Fig. 1-3-1. 0 is the angle between the cylinder-axis, z, and magnetic field, H.

one with large A%moi. Since the anisotropic magnetic energy of an aggregate depends on n, A magC//) of an aggregate can exceed the translational energy simply by increasing its volume.

Angular dependence of aggregates obeys the Boltzmann distribution law. When the orientation is limited in a two-dimensional plane, the probability P(6, H, T) dO that the z axis of an aggregate orients at the angles between 0 and 0 + d0 at // and temperature T is given,

P(0, HJ)(\e = exp[-£niag {e. H)/kT]de/ j j exp[-£n.ag (0, H) /kT]de (2)

where k is the Boltzmann constant. Fig. 1-3-3 shows schematically two distribution curves P(6,H, T) of aggregates when Xmom < Xmon < 0. Keeping H and T constant, aggregates showing curve (b) have a larger | A moi I value compared to those showing curve (a). Keeping AXmoi and T constant, aggregates showing curve (b) are placed in a higher magnetic field, compared to those showing curve (a). Keeping AXmoi and H constant, aggregates showing curve (b) are placed at a temperature lower than those


Fig. 1-3-3 Angular dependence of probability P(0,//, T) of aggregates with Xmoin < Xtno\± < 0, shown in Fig. 1-3-1.

showing curve (a). Therefore it is possible to obtain AXmoi from analysis of

Experimentally, P(e, H, T) is estimated from the number of aggregates within angles G^Ô, In this case, the width of angles, 2A0, affects the quality of P{6,H,T)P Even using the same experimental data, a higher field is apparently required for the magnetic orientation when A0 is small. A lower field is apparently required for the orientation, when A0 is large. Furthermore, crystal shape and the direction of magnetic field (horizontal or vertical) also affect P(^,//, 7), since gravity also influences the stability of oriented aggregates. - ^

The degree of magnetic orientation is also expressed by using the order parameter S{H, T), when the orientation is in a two-dimensional plane.

5(//,r) = Jj2cos^0-l]P(^,//,r)d0 (3)

S(H,T) changes from -1 to 1. S(H,T) = 1 when all aggregates are oriented parallel (0 = 0) to /f, S(H, T) = -\ when they are oriented perpendicular {6 = nil) to the field, and S{H, 7) = 0 for randomly oriented aggregates.

Magnetic susceptibility of a material is given by the second-rank symmetric tensor. When the axes of the tensor are chosen correctly, the tensor becomes diagonal and the magnetic susceptibility of a material is presented by using three susceptibilities, Xu X2 and x^ along the principal axes, jci, X2 and JC3, respectively. In the case of crystals, their magnetic axes are sometimes correlated to their crystal symmetry, as shown in Table 1-3-1."^ When a crystal is tetragonal, for example, its magnetic axes coincide with the crystal axes and it has two magnetic susceptibilities. So once the crystal system of a crystal is known, one can easily estimate the orientation of a crystal in a magnetic field.

Thus far, the relationship between anisotropic magnetic energy and


Table 1-3-1 Relationship between magnetic axes and crystal axes^

Crystal system

Cubic

Tetragonal

Hexagonal

Trigonal

Orthorhombic

Monoclinic Triclinic

Physical property

Isotropic

Uniaxial

Uniaxial

Uniaxial

Biaxial

Biaxial Biaxial

Relation between magnetic axes and crystal axes

All magnetic axes coincide with crystal axes ^^-^2"^ X^ All magnetic axes coincide with crystal axes X^=X2^Xy xy axis coincides with hexagonal axis X^=Xi^X^ xy axis coincides with trigonal axis X^^Xi^Xy All magnetic axes coincide with crystal axes X2 axis coincides with crystal axis b No correlation with crystal axis

t X^^ >t2, and x^ are magnetic susceptibilities for magnetic axes, jci, Xi, and JC3, respectively. a, b and c are crystal axes.

angular distribution of aggregates has been discussed. Anisotropic magnetic susceptibility is also related to the motion of an aggregate suspended in viscous medium in a magnetic field. ^ When it is suspended in viscous medium in a magnetic field, as shown in Fig. 1-3-1, an aggregate receives the magnetic torque T{e, //),

T(e,H) = -Êm.MH)/ê = (\/2)^lonAx(sin2e)H' (4)

Thus the aggregate rotates to minimize its magnetic energy. The rotational motion of the aggregate is given by balancing the magnetic torque and the hydrodynamic torque,

Lde/dt = -T(d,H) = -(\/2)HonAxisin2e)H

where t is the time. The left-hand side of Eq. (5) is the hydrodynamic torque on the

aggregate through the surrounding viscous medium when the aggregate rotates at an angular velocity of d6/dt. The parameter L depends on the volume and the shape of the aggregate. The solution to Eq. (5) is

tan 6 = tan do exp (-t/r) (6)

(5)

Here Go is the initial angle between the aggregate and the magnetic field and T is the orientation relaxation time, which is defined as

r-'=(V/L)iUoAx^H' (7)

where V is the volume of the aggregate, L the parameter representing the shape of the aggregate, and Ax^ the anisotropic volume magnetic susceptibility (VAx^ = n AXmoi). In the case of a sphere with radius a, L = 8 nrja^ and V = (4/3) ;ra\ where rj is the viscosity of medium. Then Eq. (7) becomes

T-'=HOAX.H'/6TI (8)

Thus AXv is obtained from the orientation relaxation time, r. In the case of


an ellipsoid, the general formula of L was derived by Jeffery. ^ Therefore, we can determine AXv (Axmoi) from r, when L can be obtained analytically.

B. Magnetic Orientation Due to Shape Anisotropy'^ Another mechanism of magnetic orientation is shape anisotropy. In this mechanism, an aggregate with isotropic magnetic susceptibility can also undergo magnetic orientation only if its shape is anisotropic. Its mechanism is slightly complicated. When a diamagnetic substance is placed in a magnetic field //, as shown in Fig. 1-3-4, it is magnetized and magnetic poles are induced at its two ends, and because of these magnetic poles, a magnetic field, called the demagnetizing field, //d, is induced.

^ / jUo=/ /ex+M + //d (9)

where B is the magnetic flux density, //ex the external magnetic field and M the magnetization. In the case of a diamagnetic substance (x < 0), //a is the same direction as //ex, whereas the direction of Af is opposite to //ex as,

M = ;t://ex/(l + A^Z)<0 (10)

and

H, =-NM = -NxH,. / (l-^ NX)- -NxH,A\- Nx)^ -NxH,.>0 (11)

where A is the number of induced magnetic poles. The demagnetizing field is proportional to N. In Fig. 1-3-4, two orientations of a diamagnetic material are shown. The number of magnetic poles in orientation (b) is larger than that in (a), //d in orientation (b) is larger than that in (a). Therefore, orientation (b) is more unstable in energy than orientation (a), since the //d of (b) is larger than that of (a). As the demagnetizing field is dependent on the shape of the material and the magnetizing direction, as schematically shown in Fig. 1-3-4, the magnetic energy of the material

//ex

(b)

Fig. 1-3-4 Schematic model for magnetic orientation due to shape anisotropy. //ex, external magnetic field; B, magnetic flux density; M, magnetization; N and 5, induced magnetic pole; //d, demagnetizing field.


depends on the shape and its magnetizing direction. As a result, a material of isotropic magnetic susceptibility undergoes magnetic orientation to the direction where its magnetic energy is minimum. Few examples of magnetic orientation due to shape anisotropy have been reported.^'^^ However, this mechanism seems to be adapted only when their orientations cannot be explained by their anisotropic magnetic susceptibility. Therefore, further theoretical and experimental analysis of this mechanism is urgently required. This mechanism is of general importance, since the orientation of aggregates occurs simply when they have anisotropy in shape, regardless of magnetic susceptibility.

Examples of magnetic orientation are given in Chapter 5.

1.3.2 Magnetic Force

Magnetic force arises from the interaction of the magnetism of a material and a magnetic field. When a magnetic field is nonuniform, a substance receives the magnetic force Fmag/

Fmag = -dEmag /dz = jloX^dHldz (12)

where £mag is the isotropic magnetic energy, x the isotropic magnetic susceptibility, and dHldz the gradient of H in the z-direction. Characteristics of magnetic force are summarized in Table 1-3-2. The force is dependent on if. It is linear and parallel or anti-parallel to that of dHldz. The force repels diamagnetic materials (j < 0) to a lower magnetic field, whereas it attracts paramagnetic materials Of > 0) to a higher field. The direction of the force is invariant when the magnetic field direction is changed to the opposite one.

Magnetic force is very important as a mechanical effect of magnetic fields. For example, one can move the aqueous solution of paramagnetic ions in water simply by using a conventional permanent magnet (0.2-0.3 T). ^ It is possible to separate paramagnetic ions ^ and paramagnetic particles. ^

When a vessel containing water is placed in a horizontal high magnetic field gradient, water is split into two parts, because water (diamagnetic) is repelled out from the magnetic field. ^^ This phenomenon is called the Moses effect.

Table 1-3-2 Comparison of magnetic force and Lx)rentz force

Magnetic force Lorentz force

Interacting physical Magnetic susceptibility Electric charge quantity

Type of MF^ Nonhomogeneous Homogeneous MF dependence Quadratic {tf) Linear (//) Direction of force Parallel or anti-parallel to MF Perpendicular to MF

Linear Circular Inversion of MF Invariant Inverted

t MF = magnetic field.


One can levitate many materials, water, plastics and etc., by using a vertical high magnetic field gradient.'^^ Suppose that a diamagnetic material, whose mass and mass magnetic susceptibility are m and j , is placed in a vertical magnetic field. It is levitated when the magnetic force to the material, which is opposite to gravity, is equal to gravity as a whole,

{mx/Ho)BdB/dz = mg (13)

where z is in the vertical direction. When the condition given by Eq. (13) is held at every infinitesimal place in the material, the material is said to be placed in pseudo-microgravity. One can levitate a water droplet (diamagnetic) by applying a vertical magnetic force of 1360 T m~ against gravity. It is also possible to prepare pseudo-hypergravity by applying the magnetic force to the same direction to gravity.

Many examples of magnetic force are discussed in Chapter 2.

1.3.3 Lorentz Force

The Lorentz force arises from the interaction of a moving electric charge and a magnetic field. When it moves in a magnetic field, an electrically charged particle receives a Lorentz force FL,'^

FL=qvxB (14)

where q is the electric charge of a particle, v the velocity of the particle, and B the magnetic flux density. Characteristics of the Lorentz force are Usted in Table 1-3-2. This force is proportional to B (= /doH). It induces a torque on the particle in the plane perpendicular to B, and induces rotational motion. The direction of the torque is reversed when the direction of B is reversed. One can differentiate the effects of the Lorentz force and the magnetic force using Table 1-3-2.

In solution, ions and charged particles cannot move alone due to collision with the solvent and other solutes. As a result, the Lorentz force induces convection of the solution. This mechanism is called the magnetohydrodynamics (MHD) mechanism. It is important in not only electrochemical reactions where electric current moves in solution but also in processes in solution where ions move to a specific reaction zone such as the solid/liquid interface.

Generally speaking, the speed of convection induced by the Lorentz force in an electrochemical reaction can be estimated from the following equation, "

aii/ar + (iiV)ii = -(l/p)VP + vV'ii + ( l /p) /xB (15)

where u is the velocity of the solution, t the time, p the density of solution, P the pressure, v the dynamic viscosity of solution, i the electric current and B the magnetic flux density. Analysis of Eq. (15) is reported elsewhere.^^^ The solufion of Eq. (15) depends on the experimental


conditions. In the case of anodic oxidation of iodine ions, for example, the speed of convection is experimentally obtained to be about 16 mm s~ at 0.6 T. ^ MHD-induced convection is quite fast and very significantly affects transportation from bulk solution to a reaction zone.

The direction of convection induced by the Lorentz force is also strongly affected by the experimental conditions. The force can induce locally small vortexes, i.e., the micro-MHD effect.' ^ With the aid of the Lorentz force, right- and left-handed helical tubes and twisted ones can be prepared selectively.' ^

Examples of the effects of the Lorentz force are presented in Chapter 3.

References

1. K. Kitazawa (supervised), S. Ozeki, Y. Tanimoto, M. Yamaguchi (eds.), Jikikagaku (Magneto-Science), IPC, Tokyo (2(X)2) (in Japanese).

2. Y. Tanimoto, R. Yamaguchi, Y. Kanazawa, M. Fujiwara, Bull. Chem. Soc. Jpn., 75, 1133(2002).

3. M. Fujiwara, T. Chidiwa, Y. Tanimoto, J. Phys. Chem. B, 104, 8075 (2000). 4. A. Weiss, H. Witte, Magnetochemie, Chapt. 5, Verlag Chemie, Weinheim (1973) (in

German). 5. T. Kimura, M. Yamato, Y. Koshimizu, M. Koike, T. Kawai, Langmuir, 16, 858

(2000). 6. G. B. Jeffery, Proc. R. Soc. London, A102, 161 (1922). 7. T. Sugiyama, M. Tahashi, K. Sassa, S. Asai, JSIJ International, 43, 855 (2003). 8. A. Katsuki, I. Uechi, Y. Tanimoto, Bull. Chem. Soc. Jpn., submitted. 9. Y. Ito, Kagaku To Kyoiku (Chemistry and Education), 38, 86 (1990).

10. M. Fujiwara, D. Kodoi, W. Duan, Y. Tanimoto, J. Phys. Chem. B, 105, 3343 (2001). 11. T. Ohara, S. Mori, Y. Oda, Y. Wada, O. Tsukamoto, Denkigakkai Ronbunshi, 116B,

979(1995). 12. M. Iwasaka, S. Ueno, J. Appl. Phys., IS, 7177 (1994). 13. E. Beaugnon, R. Toumier, Nature, 349, 470 (1991). 14. R. Aogaki, K. Fukui, T. Mukaibo, Denki Kagaku, ^3, 5QA {1915). 15. I. Uechi, M. Fujiwara, Y. Fujiwara, Y. Yamamoto, Y. Tanimoto, Bull. Chem. Soc.

Jpn., IS, 2378 (2002). 16. R. Aogaki, Proceedings of Symposium on New Magneto-Science '00, p.27, NRIM,

Japan Science & Technology, Omiya (1999). 17. I. Uechi, A. Katsuki, L. Dunin-Barkovskiy, Y. Tanimoto, J. Phys. Chem. B, 108, 2527

(2004).

1.4 Dynamic Spin Chemistry

1.4.1 What Is Modern Spin Chemistry?

Classical magnetochemistry deals mainly with macroscopic magnetic properties in chemistry,'^ whereas spin chemistry emphasizes more a microscopic view of magnetic phenomena. ^ In other words, spin chemistry

1.4 Dynamic Spin Chemistry 25

can be regarded as a modem version of magnetochemistry. At present, spin chemistry is more or less broadly defined as a research field specializing in magnetic phenomena in chemistry, including synthesis of molecular magnets based on organic compounds, magnetic field effects on chemical reactions and application of magnetic resonance spectroscopy to chemical phenomena.^^ Dynamic aspects of modern magnetochemistry may be called dynamic spin chemistry, which is chiefly concerned with chemical reactivity, dynamics and kinetics of magnetic phenomena in chemistry.

It was well known in the 1950's that ortho- and para-hydxogtn interconversion is catalyzed by paramagnetic substances and dependent on the strength of magnetic fields applied to reaction systems/^ In spite of many attempts to induce changes in reaction rates on application of external magnetic fields, very little reproducible and reliable data were obtained for chemical reactions in the solution phase until the 1970's. During the last three decades there has been substantial scientific activity in the field of dynamic spin chemistry, mainly dealing with magnetic field and magnetic isotope effects on chemical and biochemical reactions as well as chemically induced spin polarization.^^ A critical step in the development of dynamic spin chemistry was the discovery of chemically induced dynamic nuclear polarization (CIDNP) and its interpretation on the basis of the radical pair model. ' ^ The so-called radical pair mechanism has turned out to be a valuable key concept for systematically investigating magnetic field effects on chemical yields and kinetics.

One of the most important researches in spin chemistry is the magnetic field effect on dynamical behavior of excited molecules in the gas phase. In 1974 Matsuzaki and Nagakura found that the intensity of fluorescence from the diamagnetic excited singlet state (*A2) of CS2 was appreciably reduced in the presence of an external magnetic field.^^ Magnetic field quenching of emission has been observed for I2 in the triplet state and NO2 in the doublet state. It should be noted that magnetic quenching of CS2 has opened a new field of energy transfer of excited molecules in the gaseous state.

1.4.2 Theoretical Background for the Radical Pair Model

The basic principle of radical pair (RP) is introduced in this subsection. The spin Hamiltonian for an RP consisting of radicals A and B is given by the following equations ' ^:

/fRP=X2A5AZ+r2B5BZ+//ex d )

/ / e x = - y ( 2 5 A 5 B + l / 2 ) (2)

^ A= ÂMBÔ / {h I iK) + laAn /An (3)

QB=gBliiBBo/{h/2n)-\-laBmlBm (4)

26 I Fundamentals of Magnetic Field Effects

where subscripts A and B refer to radical A and radical B, respectively. Here, the direction of the external field Bo is taken as the z axis. The coefficients An and asm are the hyperfine coupling constants of the n-ih and m-th nuclei. The two spins precess around the direction of the external field Bo with the following angular frequencies:

0)^' = gAflsBo / (h / 27t)-^ZaAn MAn ( 5 )

COB" = SBUBBO I {hi 2K)+ 1. a^m M^m (6)

where MAO and M^m are the spin quantum numbers corresponding to h for the n-th and m-th nuclei, respectively. The superscripts a and b refer to the respective nuclear spin state of radicals A and B. The singlet (5) and triplet {Tn\ n= 1, 0, -1) states of RP can be given as follows:

\S>={\aAliB>-\asPA>]/^'2 (7)

ir+i>=laAaB> (8)

iro>={laAi8B> + laBi3A>}/V2 (9)

\U>=\PAPB> (10)

The exchange term, //ex, determines the energy separation between the singlet and triplet states (A£ = £$-£7= 27),

£=<SI/ /exl5>^y (11)

ET=<Ti\H,,\T,>=-J (12)

When the inter-radical distance between radicals A and B is denoted by r, dependence of 7 on r can approximately be given by the following expression:

J(r) = JoQxp(-^r) (13)

Since Jo is usually negative for RPs composed of neutral radicals, the r-dependence of the energies of the singlet and triplet states is schematically illustrated in Fig. 1-4-1 in the absence (a) and presence (b) of an external magnetic field. Fig. 1-4-2 shows the Zeeman splitting as a function of magnetic field strength. The most important matrix element responsible for the intersystem crossing in the presence of the static magnetic field is described by the following equation:

(5l//Rpiro) = (a)/-a)B')/2 = ab (14)

Kaptein^^^ has obtained the time evolution of the time-dependent wave function y/(t) for an RP during the S-To conversion by solving the Schrodinger equation, i(h/2n) dyA^tydt = //RP V<0. where y/it) is represented


Energy

Fig. 1-4-1 Potential energy curves for a radical pair as a function of interradical distance (r).

Energy

Magnetic field

Fig. 1-4-2 Zeeman splitting as a function of applied magnetic field strengths.

by the product of the electron-spin states I S > and I To> and nuclear-spin states lA ab>:

y/(t) = {Cs(t)\S>-^Cj(t)\To>}\N,,> (15)

When the two electron spins feel a strong exchange interaction (J) at the moment of pair production, there may be no mixing of S and To. Once two spins are separated by a sufficiently long distance (more than 1.0 nm) and feel negligibly small J for some extended time, the spins of A and B precess with their respective Larmor frequency. The difference between singlet and triplet states lies in only the phase of their spins in the vector description of the RP model. Consequently, the phase of the two spins may alternate as a function of time. In other words, the S and To states oscillate quantum mechanically. This type of S-To conversion is visualized in Fig. 1-4-3. This can be regarded as intersystem crossing which may occur in the nanosecond time range, and the rate for S-To mixing is usually of the order of 10' s-\


Radical pair Z \

External magnetic field

Slow / ^ relaxation ^

S-To mixing < •

Recombination

Slow relaxation ^

Products

Fig. 1-4-3 A simplified vector model for triplet-singlet spin conversion processes including slow spin-lattice relaxation.

1.4.3 Experimental Methods and Examples

A wide variety of experimental methods have been applied to dynamic spin chemistry. Above all time-resolved optical spectroscopy and magnetic resonance are powerful tools to detect and characterize transient species. It should be noted that optical spectroscopy excels in time resolution, whereas magnetic resonance has strong advantages in identifying and characterizing paramagnetic transient species. Therefore, combined use of time-resolved optical and magnetic resonance spectroscopy is quite fruitful in the kinetic and mechanistic studies of reactions involving radical pairs and biradicals.

Scheme 1-4-1 summarizes a typical reaction scheme for the creation of RP intermediates. Most of the reactions involving RPs so far investigated may fall into the three categories shown in Scheme 1-4-1. Laser flash photolysis or pulse radiolysis of solutions containing electron donors and acceptors may result in the formation of radical ion pairs, while photochemistry of carbonyl compounds often involves homolytic bond cleavage or hydrogen abstraction.

Scheme 1-4-2 summarizes a generalized model reaction involving radical pairs. There are two different pathways leading to end products. One is cage product formation, and the other escape product formation. On going from the zero field to a high field, the end product yields decrease or increase depending on the spin multiplicity of its precursor and the nature of reaction pathways. When the singlet-triplet spin conversion is dominantly induced by hyperfine interaction, the cage product formation from singlet-born separated radical pairs increases in the presence of


Typical examples of radical pair formation

(1) Electron transfer

D m

(2) Hydrogen abstraction

nm +

D m + m A

Triplet

(3) Homolytic bond cleavage

T^^T^ — > (t Scheme 1-4-1

X IB +

Triplet

Singlet precursor Triplet precursor

^[At I B ] < '" • ^ [ A t t B ]

Close radical pair Close radical pair

Singlet-triplet T

conversion [At I B ] <r- ~>^[At t B ]

Separated radical pair Separated radical pair

At + Bt

Escape radicals

+ R (Solvent)

A-A, B-B,A-B

A-R, B-R, etc.

Escape products

Scheme 1-4-2

external magnetic fields. Table 1-4-1(a) lists changes in end product formation in the presence of external magnetic fields when the spin conversion is dominated by the hyperfme coupling. Table 1-4-1 (b) lists changes in end product yields in the presence of magnetic fields when the singlet-triplet conversion is dominated by the Zeeman interaction (A^ mechanism). Because B\/2 values {vide infra) commonly observed for organic radical pairs containing light atoms are usually smaller than 10 mT, the magnetic field effects induced by the hyperfine coupling (HFC) mechanism show some saturation below 50 mT. The saturation phenomenon is characteristic of the HFC mechanism, while the magnetic


Table 1-4-1 Changes in radical pair reaction yields induced by singlet-triplet spin conversion

(a) Hyperfine coupling mechanism in the presence of magnetic fields

Spin multiplicity of precursors

Singlet Triplet

(b) Ag Mechanism in

Spin multiplicity of precursors

Singlet Triplet

Cage product yields

Increase Decrease

the presence of magnetic fields

Cage product yields

Decrease Increase

Escape product yields

Decrease Increase

Escape product yields

Increase Decrease

Increase or decrease refers to changes observed on going from the zero field to high fields.

field effects due to Ag mechanism increase with increasing field strength without saturation.

The first CIDEP effect was observed in 1963 for H atoms during irradiation of liquid methane and methane-ethane mixture with 2.8 MeV electrons at ca. 100 K. ^ One of the two hyperfine lines for H atoms is emissive and the other absorptive. This clearly shows that the population differences of the two hyperfine levels have opposite signs as the result of complicated polarization effects ensuing from the initial creation of the radical pair consisting of • CH3 and • H, and subsequent recombination.

In 1967 CIDNP phenomena were independently reported by two research groups.''' ^^ The observation of CIDNP was made for the reaction of n-butyllithium with alkyl bromide and for thermolysis of benzoyl peroxide in cyclohexanone solution. These reports have provided a starting point for the vigorous development of CIDNP.

In a pioneering work published in 1969, Brocklehurst predicted that electron paramagnetic relaxation of radical ion pairs could affect the relative distribution of singlet and triplet recombination products.'^^ In 1974 Brocklehurst and coworkers reported magnetic field effects on the singlet/triplet ratio in the geminate recombination of radical ion pairs produced on pulse radiolysis in viscous squalene solutions containing fluorene. The solutions were irradiated with 50 ns pulses of 3 MeV electrons at doses of ca. 4 krad per pulse.'' ^

The reaction scheme can be summarized as follows: (1) ionization of solvent molecules (S): S -^ S + e~ (2) electron transfer from fluorene (Fl) to the solvent cation:

S + Fl -^ S + F r (3) formation of fluorene anion: Fl + e -^ Fl" (4) radical pair formation: F r + Fl' -^ ^[Fr Fl] (5) singlet-triplet spin conversion: ^[Fr Fl] -^ \FV Fl] (6) recombination of radical ion pairs and fluorescence emission:


^[Fr Fl] ->+^Fr+ F 1 - > 2 F 1 + /IVF

The fluorescence intensity from fluorene in the excited singlet state was enhanced on application of magnetic fields (< 0.5 T). The fluorescence intensity increased with increasing field strength (0-50 mT). The intensity exhibited a plateau in the region above 0.1 T. It should be noted that there may be saturation of magnetic field effects when the hyperfine interaction dominates the intersystem crossing. The observed results are in agreement with the theoretical prediction published in 1969 and can be explained in terms of the radical pair model. Because fluorene in the excited singlet state is a cage product, the yield of singlet-bom product increases when the hyperfine coupling mechanism is dominant (see Table 1-4-1(a)).

The first experimental demonstration of the crucial role of nuclear magnetic moments was presented by Sagdeev and coworkers in 1972.' ^ They showed that in the radical reacfion of pentafluorobenzyl chloride C6F5CH2CI with n-butyllithium C4H9Li, the rafio of C6F5(CH2)4CH3 to C6F5(CH2)2C6F5 in the product increased on the application of an external magnetic field. The field dependence of singlet-triplet transitions for radical pairs induced by hyperfine interaction (HFI) accounts for the observed magnetic field effects. The results described in ref. 15 can be interpreted through the hyperfine mechanism in terms of the radical pair model. The yield of cage product C6F5(CH2)4CH3 decreases while that of the escape product C6F5(CH2)2C6F5 increases in the presence of magnetic fields. Recently, Wakasa and Hayashi re-examined the organometallic reaction reported by Sagdeev and coworkers. ^ However, they were not able to reproduce magnetic field effects on the end product yield.

In 1976 Schulten, Weller and collaborators'^^ demonstrated the substantial contribution of the isotropic HFI and nuclear magnetic moments to the efficiency of a geminate recombination for radical ion pairs generated by laser photolysis. Weller and collaborators reported that magnetic field effects on the yield of pyrene in the excited triplet state formed after photolysis of solutions containing pyrene and anilines in polar solvents at room temperature.'^^ They observed by means of nanosecond laser photolysis that the transient absorption intensity of the excited triplet decreased in the presence of low magnetic fields. The triplet yield at 50 mT was about 80% of the value in the zero field in methanol. The observed results can be explained in terms of the radical pair model. In this case the singlet-triplet spin conversion is reduced in the low field region, which corresponds to the reduced yield of molecular excited triplet.

The reaction scheme is given below: (1) excitation of pyrene: pyrene + HVA -> 'pyrene* (2) electron transfer from D (anilines) to pyrene:

'pyrene* -1- D -> '[pyrene" + D""] (3) singlet-triplet spin conversion:'[pyrene'-i-D''] -^ ^[pyrene +D^] (4) formation of molecular excited triplet:


^[pyrene" + D""] -> " pyrene* + D Weller et al. have also proposed the B1/2 value, which is defined as the half field where half the maximum field effects are observed in the following equation:

where B^ and By are evaluated from hyperfine coupling constants observed for radicals X and Y.

One of the typical magnetic field effects on the end-product yields was reported by Tanimoto and collaborators in 1976.' ^ They observed a decrease in the yield of cage product, phenyl benzoate formed in the singlet-sensitized photolysis of benzoyl peroxide in the presence of high magnetic fields up to 4.3 T. In this case, the A^ or Zeeman mechanism dominates instead of the hyperfine coupling mechanism. The singlet-bom radical pair initially consists of two benzoyloxy radicals which undergo decarboxylation of benzoyloxy radicals, thereby forming a pair composed of phenyl and benzoyloxy radicals. The difference in g-factor for the latter pair is fairly large. The singlet-triplet conversion is enhanced under higher magnetic fields when the Ag mechanism dominates. They found that changes in cage product yields [y(B)-K(0)] are proportional to the square root of field strength B^'^, where Y and 0 refer to the yield and zero field, respectively. The observed results are in agreement with the prediction based upon the radical pair model.

In 1976 Buchachenko et al. succeeded in detecting magnetic field effects on the concentration of magnetically active heavy carbon ( ^C) in the photochemical decomposition of dibenzyl ketone in benzene and hexane. ^^ This is one of the earliest reports on magnetic isotope effects. Sagdeev, Molin and coworkers found magnetic isotope effects in the triplet sensitized photolysis of benzoyl peroxide. ^^ Namely, they recorded ^ C-NMR spectrum of the cage product, phenyl benzoate C6H5COOC6H5, and concluded that the increase in the amount of ^ C at the C-1 position of the phenoxy moiety was due to the increased recombination probabilities induced by hyperfine interaction of the radical pairs containing heavy carbon.

In 1976 Buchachenko and coworkers reported magnetic isotope effects for heavy carbon "C in photodecomposition of dibenzyl ketone (DBK) at room temperature. On photoexcitation DBK undergoes Norrish type I cleavage in the triplet manifold. ^ They found that triplet-singlet spin conversion of radical pairs consisting of phenylacetyl and benzyl radicals was more rapid for the pair containing the heavy carbon than for the pair with the normal one. In other words, radical pairs containing the heavy carbon recombine faster than the pairs with '^C, and therefore the heavy carbon was enriched at the central carbon atom. This work clearly shows that isotope enrichment is possible for elements composed of magnetic and nonmagnetic nuclei by use of magnetic isotope effects.

1.5 High-field Generation for Magnetic Processing 33

References

1. P.W. Selwood, Magnetochemistry 2nd. Ed., Interscience Publisher, New York (1956). 2. a) S. Nagakura, H. Hayashi, T. Azumi (eds.). Dynamic Spin Chemistry, Kodansha,

TokyoAViley-VCH, Weinheim (1.998), b) H. Hayashi, Introduction to Dynamic Spin Chemistry, World Scientific, Singapore (2(X)4).

3. a) H. Hayashi, K. Itoh, S. Nagakura, Bull. Chem. Soc. Jpn., 39, 199 (1966), b) K. Itoh, H. Hayashi, S. Nagakura, Mol. Phys., 17, 561 (1969).

4. G. L. Gloss, J. Am. Chem. Soc, 91, 4552 (1969). 5. a) R. Kaptein, I. J. Oosterhoff, Chem. Phys. Lett., 4, 195 (1969), 214, b) R. A.

Kaptein, J. Am. Chem. Soc, 94, 6251 (1972). 6. A. R. Lepley, G. L. Gloss (eds.). Chemically Induced Magnetic Polarization, Wiley,

New York (1973) 7. a) K. M. Salikhov, Yu. N. Molin, R. Z. Sagdeev, A. L. Buchachenko, Spin

Polarization and Magnetic Ejfects in Radical Reactions, Elsevier, Amsterdam; Akademiai Kiado, Budpest (1984), b) K. M. Salikhov, Magnetic Isotope Effect in Radical Reactions, Springer, Heidelberg (1996).

8. a) U. Steiner, T. Ulrich, Chem. Rev., 89, 51 (1989), b) K. A. MacLauchlan, Advanced EPR (A. J. Hoff ed.), Elsevier, Amsterdam(1989), Ghapt. 10, c) K. A. MacLauchlan, U. E. Steiner, Mol. Phys., 73, 241 (1991).

9. a) A. Matsuzaki, S. Nagakura, Chem. Lett., 679 (1974), b) ibid. Bull. Chem. Soc Jpn., 49,359(1976).

10. R.W. Fessenden, R. H. Schuler, J. Chem. Phys., 39, 2147 (1963). 11. H. R. Ward, R. G. Lawler, J. Am. Chem. Soc, 89, 5518 (1967). 12. J. Bargon, H. Fischer, U. Johnsen, Z. Naturforsch., 22, 1551 (1967). 13. B. Brocklehurst, Nature, 221, 921 (1969). 14. B. Brocklehurst, R. S. Dixon, E. M. Gordy, V. J. Lopata, M. J. Quinn, A. Singh, F. P.

Sargent, Chem. Phys. Lett., 28, 361 (1974). 15. a) R. Z. Sagdeev, K. M. Salikhov, T. V. Leshina, M. A. Kamkha, S. M. Shein, Yu. N.

Molin, Pis'ma Zh. Eksp. Teor. Fiz.. 16, 599 (1972), b) R. Z. Sagdeev, Yu. N. Molin, K. M. Salikhov, T. V. Leshina, M. A. Kamkha, S. M. Shein, Org. Mag. Resonance, 5, 603(1973).

16. M. Wakasa, H. Hayashi, Mol. Phys., 100, 1099 (2002), and references cited therein. 17. K. Schulten, H. Staerk, A. Weller, H.-J. Werner, B. Nickel, Z. Phys. Chem., N.F., 101,

371 (1976). 18. Y. Tanimoto, H. Hayashi, S. Nagakura, H. Sakuragi, K. Tokumaru, Chem. Phys. Lett.,

41,267(1976). 19. A. L. Buchachenko, E. M. Galimov, V. V. Ershov, G. A. Nikiforov, A. D. Pershin,

Dokl. Akad. Nauk SSSR, 228, 379 (1976). 20. R. Z. Sagdeev, T. V. Leshina, M. A. Kamkha, O. I. Belchenko, Yu. N. Molin, A. L

Rezvukhin, Chem. Phys. Lett., 48, 89 (1977) and references cited therein.

1.5 High-field Generation for Magnetic Processing

1.5.1 Calculation and Characterization of Magnetic Fields

For the case of a uniform-current-density solenoid coil, shown in Fig. 1-5-1, magnetic fields along the z axis, Bz (T), are calculated by the following


equation:

MiiY-^m -ir-Pm (1)

where a = ailax, p = b/a\, y = z/a\, /JQ = the permeability of free space (4;rxl0"^ NA"^), j = the current density (Am~ ), a\ = the coil inner radius (m), (32 = the coil outer radius (m), and b = the coil half height (m).

At a point far from the magnet center, p (p» ai, b), the magnet can be seen as a magnetic dipole. The magnetic fields Br and Bz decreased with p^ The following relation was shown:

BAO.p) = -2BAp.O) (2)

By using Eqs. (1) and (2), stray fields can be roughly estimated. A numerical calculation is necessary for magnetic fields around the

magnet except in the central zone. The fields in the sphere, r^ + z^ < a\^, shown in Fig. 1-5-1, can be expressed as a power series, involving Legendre polynominals.^^ For field calculations at any point, many computer programs are available. A study by Watanabe^^ lists all the source codes, so it is easy to use from the viewpoints of reproducibility and modification. It can be used for solenoids as well as for racetrack and other shapes.

When permanent magnets or ferromagnetic materials are present, numerical calculations based on a finite element method are required. If a commercial package is used for a calculation, understanding its accuracy and limitations is very important. It is strongly recommended that researchers confirm whether the calculation results make sense from a physical point of view.

For general applications, B, grad B, and grad (B^l are important as spatial distributions. The last term determines the magnetic force working

ir,z)

Fig. 1-5-1 Cross section of a solenoid coil.


-400 -300 -200 -100 0 100 200 300 400

Position along :: axis / mm

Fig. 1-5-2 Distributions of B, grad B and grad {B^)I2 along the z axis. They were calculated for a cryocooler-cooled magnet, which generates 10 T in a 100-mm room-temperature bore. (Courtesy of Japan Superconductor Technology, Inc.)

on an object. Examples of the distributions along the z axis are shown in Fig. 1-5-2. It is noted that grad B and grad {B^)I2 become zero at the magnet center, where the maximum B is obtained. The optimal shapes for generating B, grad B, and grad {B^)I2 are slightly different even in a single solenoid coil. Ordinary magnets are designed from the viewpoint of a high ^ or a uniform B.

By a combination of long solenoid coils and compensation coils, a very uniform ^ in a limited space can be achieved. An NMR spectrometer generates a very uniform B with an error component of about 1 ppb in a sample volume. Temporal fluctuation also exists. The stability of a power supply is in the range of 10" h"' to 10" h ^ The persistent mode of a superconducting magnet, mentioned below, realizes a temporal stability of less than 10- h-^

1.5.2 How to Generate High Magnetic Fields

A. Permanent Magnets Permanent magnets are convenient when the required space is small or the required field is low. They do not require auxiliaries such as power supply or cooling system. As a sample can be positioned just on the surface of the magnet, a large grad B and a large grad {B^)I2 are easily obtained. The residual flux density of an Nd-Fe-B magnet is about 1.5 T. Using the magnetic cylinder proposed by Halbach," ^ fields of more than 3 T can be obtained with rare-earth magnets. By modifying the Halbach cylinder, a field of 5.16 T was achieved in a 2 mm gap. ^


A high-rc superconductor bulk can be used like a permanent magnet. It requires a coolant or a refrigerator to maintain the superconducting state and there must be a gap of a few millimeters to separate it from the sample for thermal insulation. The maximum field, however, is very high. A trapped field of 17.24 T was reported with a Y-Ba-Cu-O disk at 27 K. >

B. Superconducting Magnets Although a superconducting magnet requires liquid helium or a refrigerator to maintain its superconducting state, it provides an economical solution to generating high magnetic fields in a large space for a long period of time. Once a very large magnet is installed, its energy consumption is not particularly large. Superconducting magnets are employed in MRI and fusion devices.

A superconducting magnet is composed of two kinds of superconductors, with the exception of some demonstration magnets. For a magnet of less than 10 T, NbTi conductors are mainly employed. For fields of more than 10 T, a combinafion of NbTi and NbsSn conductors is indispensable. NbsSn conductors are three or more times expensive and more difficult to handle than NbTi conductors due to their brittleness. The highest field of a whole-body MRI magnet is 9.4 T at present. This value is mainly limited by the performance of NbTi conductors, suggesting that the field of 10 T is a very important threshold value for a superconducting magnet. By combining NbTi and Nb3Sn conductors, fields up to 21.9 T have been achieved. ^ As the critical field of NbiSn is about 25 T, the generation of fields above 25 T requires other superconductors, such as high-Tc superconductors. The highest field of superconducting magnets, 23.4 T, was accomplished using NbTi, Nb3Sn and Bi-2212 conductors.^^ Increasing the highest field will be possible mainly by developing and improving superconductors.

A superconducting magnet is superior in field stability. In some magnets, a switch made of a superconducting wire is connected in parallel. When a magnet is being excited, the switch is heated to maintain the normal state, and a power supply increases the current going through the magnet. At the regulated current, the switch is cooled to the superconducting state, and the current circulates in the superconducting circuit including the magnet. This is called a persistent-mode operation. A field decay rate of less than 10" h~' is realized in the persistent-mode operafion of a well-designed and fabricated magnet, such as an NMR magnet.

The improvements made to the cryocooler-cooled superconducting magnet are remarkable. Fig. 1-5-3 shows a schematic cross section. This magnet was realized by combining two technological inventions. One is the development of a high-performance cryocooler, and the other is a current lead made of high-Tc superconductors which possess both low


Support rod

Cryocooler Room-temperature bore

Current lead (copper)

High-rc current lead

Cooling stage

~4K

Superconducting coil

\i— Radiation shield

~40K

h- Vacuum chamber

Fig. 1-5-3 Schematic cross section of a cryocooler-cooled superconducting magnet

thermal conductivity and high electric conductivity. A cryocooler-cooled magnet is very suitable for magnetic processing because a long-term operation can be performed without the need to refill coolants. A magnet aiming at 19 T in a 52 mm room-temperature bore is now under construction.^^

C. Resistive Magnets In order to generate higher steady-state fields than those created with a superconducting magnet, a resistive magnet is employed. This is made of copper-alloy solenoid coils and operates in cooling water. Although several kinds of coil windings have been proposed and fabricated, a Bitter coil, named after its inventor, Francis Bitter, is commonly employed now. The coil is made of Bitter plates, which have cooling holes for water and a slit to pile them spirally with an electrical contact. A typical example of a Bitter plate and an assembly of the coil are shown in Fig. 1-5-4.

The highest field achieved with a resistive magnet is 33.1 T in a 32-mm room-temperature bore. ^ Compared with the superconducting magnet, the resistive magnet is good at changing magnetic fields at a fast rate. Since its current density is one order of magnitude larger than that of the superconducting magnet, a large grad B and a large grad (B^)/2 are easily obtained.

The disadvantage of a resistive magnet is its extremely large energy consumption. The 33.1 T resistive magnet consumed 17.2 MW. The consumption energy is changed to heat in the resistive magnet. Additional energy is required to cool the magnet with water. For operating a high-field resistive magnet, a large DC power supply and a large water-cooling system must be installed. Only five facilities, the National High Magnetic


For tie rod

w-For alignment rod \ ' Slit

Cooling hole

Superconducting magnet

Resistive magnet

Assembly of a Bitter coil

Fig. 1-5-4 Schematic illustration of a hybrid magnet operating at the Tsukuba Magnet Laboratory. Examples of a Bitter plate and its assembly are also illustrated. (Courtesy of Dr. Toshihisa Asano, Tsukuba Magnet Laboratory, National Institute for Materials Science)

Field Laboratory (U.S.A.), the Grenoble High Magnetic Field Laboratory (France), the High Field Magnet Laboratory (Netherlands), the Tsukuba Magnet Laboratory (Japan) and the High Field Laboratory for Superconducting Materials (Japan), provide resistive magnets for external users.

D. Hybrid Magnets A hybrid magnet consists of an outer superconducting magnet and an inner resistive magnet. A schematic illustration of a hybrid magnet is shown in Fig. 1-5-4. It is the most economical method for generating the highest fields in a steady state. While a superconducting magnet is good at generating magnetic fields in a large space with little energy consumption, a resistive magnet is free from the critical fields of superconductors.

At present, the highest steady-state field of 45.2 T was achieved with a hybrid magnet at the National High Magnetic Field Laboratory.'^^ Although the hybrid magnet is a very powerful system for the generation of high magnetic fields, it possesses the disadvantages of the resistive magnet and the superconducting magnet. Only three facilities, the National High Magnetic Field Laboratory (45.2 T), the Tsukuba Magnet Laboratory (37.9 T) and the High Field Laboratory for Superconducting Materials (31.1 T), operate hybrid magnets for external researchers.


E. Pulsed Magnets A pulsed magnet can generate much higher fields in a short period. A nondestructive pulsed magnet is cooled with liquid nitrogen or water. By discharging the energy in a capacitor bank to a magnet, high magnetic fields are generated in a moment. A maximum field of 80.3 T with a pulse duration of 8 ms has been reported.' ^ Using a large power supply system, quasi-steady-state fields can be generated. A pulsed magnet, which could generate a flat top field of 60 T in 100 ms, has been developed. ^^

Using the exploding single turn coil technique or the flux compression technique, fields of more than 100 T can be generated. A coil is broken after every field generation.

1.5.3 Some Issues Regarding Processing in High Magnetic Fields

Several kinds of interactions should be considered when processing in high magnetic fields. Attention to the attractive force to ferromagnetic materials is the most important. When materials with high electrical conductivity move in magnetic fields, an eddy current occurs, causing heat generation and a breaking force. The interaction between the currents and the magnetic fields must also be considered in sample heating or measurement wiring.

Almost no measurement apparatuses have been designed for use in high magnetic fields. It is well known that the thermoelectric voltage of thermocouples is significantly affected by magnetic fields at low temperatures. Confirmation will be necessary to prove that the phenomena used for the measurement method will not be affected.

1.5.4 Conclusions

Each magnetic processing may require unique spatial and temporal distribution of the magnetic fields. Recent progress in magnet technology makes this possible. However, some limitations might exist due to economic, technological and theoretical difficulties. It is very important to identify the type of magnetic field required for a specific kind of processing.

Magnet technology, as well as magnetic processing, is progressing rapidly. The latest results are presented at the International Conference on Magnet Technology, which is held every other year. Selected papers presented at the conference are published in IEEE Transactions on Applied Superconductivity. Such information is helpful for understanding how to achieve a required field with the available technology.

References

1. D. B. Montgomery, Solenoid Magnet Design, Robert E. Krieger Publishing, Chapt. VIII, Malabar (1980).

2. T. Watanabe, Fusion Research, 63, 482 (1990).


3. K. Halbach, Nucl Instr. Meth., 169, 1 (1980). 4. M. Kumada, E. I. Antokhin, Y. Iwashita, M. Aoki, E. Sugiyama, IEEE Trans. Appl

Superconduct., 14, 1287 (2004). 5. M. Tomita, M. Murakami, Nature, 421, 517 (2003). 6. T. Kiyoshi, S. Matsumoto, A. Sato, M. Yoshikawa, S. Ito, O. Ozaki, T. Miyazaki, T.

Miki, T. Hase, M. Hamada, T. Noguchi, S. Fukui, H. Wada, IEEE Trans. Appl. Superconduct., 15, 1330 (2005).

7. T. Kiyoshi, M. Kosuge, M. Yuyama, H. Nagai, H. Wada, H. Kitaguchi, M. Okada, K. Tanaka, T. Wakuda, K. Ohata, J. Sato, IEEE Trans. Appl. Superconduct., 10, 472 (2000).

8. T. Kurusu, M. Ono, S. Hanai, M. Kyoto, H. Takigami, H. Takano, K. Watanabe, S. Awaji, K. Koyama, G. Nishijima, K. Togano, IEEE Trans. Appl. Superconduct., 14, 393 (2004).

9. M. D. Bird, S. Bole, Y. M. Eyssa, B.-J. Gao, H.-J. Schneider-Muntau, IEEE Trans. Magn., 32, 2542 (1996).

10. J. R. Miller, IEEE Trans. Appl. Superconduct., 13, 1385 (2003). U . K . Kindo, Physica B, 294-295, 585 (2001). 12. J. Schillig, H. Boenig, M. Gordon, C. Mielke, D. Rickel, J. Sims, IEEE Trans. Appl.

Superconduct., 10, 526 (2000).

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