Research: Science & Education
Magnetic Susceptibility Measurements of Solid Manganese Compounds with Evan’s Balance Z. S. Teweldemedhin, R. L. Fuller, and M. Greenblatt Department of Chemistry, Rutgers, The State University of New Jersey, Piscataway, NJ 08855-0939 Magnetism is a property of all materials that contain electrically charged particles. It is a universal phenomenon that manifests itself in every entity surrounding us because the atom, which is the fundamental unit of any material, is the source of electrical charges (1). A moving electrical charge (i.e., electric current) induces a magnetic field in a material. In an atom, the magnetic field is due to the coupled orbital and spin magnetic moments associated with the motion of electrons. The orbital magnetic moment is due to the motion of electrons around the nucleus, whereas the spin magnetic moment is due to the precession of the electrons [i.e., it does not involve movement of their centers of gravity (1)] about their own axes. The resultant of the orbital and spin magnetic moments of the constituent atoms of a material gives rise to the observed magnetic properties. Knowledge about the magnetic properties of materials is an important aspect in the study and characterization of substances. Unlike many other commonly employed analytical methods, magnetic property characterization is a nondestructive technique. In technology, certain magnetic materials continue to play a vital role in changing our world and shaping our future innovations. Their applications range from electrical and radio engineering to the electronics industry. They are widely used in information storage and recording devices, and with the advent of high-temperature superconducting materials in recent years, there is a promising future for their use in magnetically levitated transportation. The principle behind this type of transportation is the repulsion of magnetic field by a material in the superconducting state (Meissner effect). Theory Often, magnetic properties of materials are studied by applying a magnetic field and measuring the induced magnetization in the materials (1–5). The magnetic induction, B, that a substance experiences when placed in an applied external magnetic field, H, is given by the expression: B = H + 4πM
(1)
where M is the magnetic moment of the compound per unit volume, or the magnetization. The volume susceptibility of the compound, χv, is defined as: χv = M/H
χM = C/T
(3)
where C is the Curie constant. The variation of the inverse molar magnetic susceptibility (χM–1 ) with temperature is linear, with a slope equal to 1/C. The value of C is related to the number of unpaired electrons (i.e., the effective magnetic moment, µ eff ) present in the compound
(2)
The two most important responses observed are characterized as diamagnetic and paramagnetic moment. In diamagnetic materials (as well as materials in the superconducting state), all the electrons of the atoms in the materials are paired [e.g. H2(g), NaCl(s)] and the resultant magnetic moment is zero. The external magnetic field induces a current whose associated magnetic field, called a diamagnetic moment, is directed opposite to the applied field (1). 906
The diamagnetic susceptibility is negative relative to the applied magnetic field and independent of temperature, and the magnitude is usually small [~10–6 cgs units (1, 2)]. All inert gases and most organic compounds are examples of diamagnetic materials. Unlike diamagnetic materials, substances that exhibit superconductivity are only diamagnetic below a certain critical temperature (T c, the transition temperature to the superconducting state). The magnitude of the diamagnetic susceptibility of such substances is very large (i.e., below Tc, B = 0 in eq 1 for a superconductor; thus H = –4πM, χv = –1/4π, and χv is orders of magnitude larger than in normal diamagnetic materials) and varies somewhat with temperature. In contrast, the constituent atoms of the paramagnetic materials have unpaired electrons that give rise to a net resultant magnetic moment, called a paramagnetic moment. Generally, at room temperature, the individual magnetic moments of these substances are randomly oriented with no net magnetization (Fig. 1). However, in an applied external magnetic field, each will tend to align in a direction parallel to the external magnetic field. The magnitude of the susceptibility of paramagnetic substances is in the range of 10–3 to 10–6 cgs units and is positive and independent of the field (1, 2). Typical paramagnetic compounds include gaseous compounds like molecular oxygen and nitric oxide (NO), vapors of alkali metals and certain salts of transition and rare-earth metals. The alignment of the magnetic moments of a paramagnetic substance under an applied magnetic field is opposed by the thermal motion of the magnetic ions, which tends to randomize the moments. Hence the observed paramagnetic susceptibility of a substance increases with decreasing temperature since the effect of thermal motion is minimized at lower temperatures. This inverse relationship of the susceptibility of a paramagnetic substance to temperature is given by Curie’s law,
Figure 1. Random orientation of magnetic moments in paramagnetic material (where the applied field, H = 0).
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by the following expression: 1 1 1 µ eff = (8χM T) / 2 = (8C) /2 = g [S (S + 1)] /2 = 1/ 2 [n (n + 2 )]
(4)
where Landé’s factor, g, ≈ 2.0 for an electron or ion with no orbital contribution to the magnetic moment, S is the resultant spin quantum number (the total spin angular momentum of all the unpaired electrons in the system), and n is the number of unpaired electrons in the system (i.e., S = n/2). Equation 4 is generally valid for most compounds of the first-row transition metals, where the orbital contribution to the magnetic moment is completely quenched and only the spin contribution, S, need be considered. The µ eff for these systems where Curie’s law is obeyed then becomes independent of temperature (2). This will be discussed in more detail subsequently. Paramagnetic materials, besides having unpaired electrons, contain paired electrons in the inner (closed) shells of the constituent atoms. The presence of these paired electrons makes diamagnetism an inherent property of all materials. Thus the magnetic moment measured is in fact the sum of both paramagnetic (positive quantity) and the associated diamagnetic (negative quantity) moment. Since the presence of an intrinsic magnetic moment in a substance results in a large paramagnetic moment, diamagnetic effects are often neglected in calculations. However, if the paramagnetic and diamagnetic moments of the substance under investigation are of comparable magnitude, where accurate measurements are desired, corrections due to the diamagnetic contributions are made to the measured magnetic moments. So far we have considered paramagnetic substances in which there are no significant interactions between the magnetic moments of the constituent atoms. How-
(a)
(b)
(c)
Figure 2. Types of magnetic ordering. (a) Ferromagnetic. (b) Antiferromagnetic. (c) Ferrimagnetic.
ever, magnetic moments of individual atoms in many substances do interact with each other in several different ways. In these materials, Curie’s law is not obeyed. Often, the magnetic behavior of these substances is best described by a Curie–Weiss law that takes into account, among other things, the interactions among the individual magnetic moments: χM = C/(T – θ)
(5)
where θ is the Weiss constant (or paramagnetic Curie temperature). The magnitude of θ is related to the strength of exchange correlations between the magnetic moments. Interactions of magnetic moments in condensed systems (e.g., solids) will in most cases lead to different types of magnetic ordering, characteristic of the substance. This is shown in Figure 2, where the net spin on each atom is represented by an arrow aligned “with” or “against” the applied external magnetic field. The type of magnetic interactions present in a particular substance is primarily determined by the nature of the constituent ions and chemical bonding. The magnetic interactions between the individual magnetic moments result in a net stabilization energy. The two most common types of magnetic ordering or interactions are called ferromagnetism and antiferromagnetism. In ferromagnetic substances, the magnetic moments of adjacent atoms are aligned parallel to one another below a certain critical temperature (known as the Curie temperature, TC) as shown in Figure 2a. This type of magnetic ordering is characterized by a spontaneous magnetization of the substance below TC even in the absence of an external magnetic field. Iron, nickel, cobalt, and some rare-earth metals, as well as some compounds and alloys of these elements, are typical examples that show ferromagnetic ordering. In antiferromagnetic substances, the individual magnetic moments are aligned antiparallel to one another below a certain critical temperature (known as Néel temperature, T N ). This interaction gives no net magnetization when the neighboring atoms have magnetic moments of identical magnitude as shown in Figure 2b. Most compounds of the transition elements, such as MnO, MnF 2 , FeCl 2 ; elements such as chromium and α-manganese; and some rare-earth metals such as cerium, praseodymium, neodymium, samarium, and europium, display antiferromagnetic ordering (1). Compounds that exhibit the ferromagnetic or antiferromagnetic type of magnetic ordering often obey the Curie–Weiss law (eq 5) at temperatures well above the transition temperatures (ordering temperature, TC or TN). Ferrimagnetism is another prevalent type of ordering of the magnetic moments in certain compounds. In a ferrimagnetic substance, although the individual magnetic moments of adjacent atomic particles are aligned antiparallel, they are not of equal magnitude (i.e., when the type and coordination of interacting neighboring atoms are different) (Fig. 2c). Hence, these substances exhibit a finite spontaneous magnetization below the ordering temperature. Transition metal compounds having the spinel-type structure—for example, Fe3O4 (FeO·Fe2O3)— are well studied examples of ferrimagnetic substances. Recently, a series of solid solutions with composition YxGd3-xFe 5O 12 (where 0 ≤ x ≤ 3) were prepared to demonstrate the trend in the ferrimagnetic properties of these compounds (6). It is customary to measure experimentally the magnetic susceptibility of a substance as a function of its
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Research: Science & Education weight, χg: χg = χv /ρ
(6)
where χg is the mass susceptibility of the compound and ρ is the density. The mass susceptibility of the compound is related to the molar susceptibility, χM: χM = χg·MW/Z
(7)
where MW is the molecular weight of the compound and Z is the number of moles of magnetic ions per formula weight of the compound. The molar magnetic susceptibility of a compound having isolated magnetic ions with no magnetic interactions (assuming total quenching of orbital moments) is given (2, 3) by : χM = [NAβ2g2S(S+1)]/(3kT)
(8)
In the above expression, NA is Avogadro’s number, k is the Boltzmann constant, and β is a constant unit called the Bohr magneton (BM). Note that eq 8, obtained from theoretical consideration of paramagnetism, is similar to that shown by Curie’s law, which was derived from experimental observations. As discussed above, in condensed systems (e.g. liquids and solids), generally there are some magnetic interactions between the magnetic ions such that a modified Curie’s law known as the Curie-Weiss law is applied: χ M = [NAβ2g2S(S+1)]/[3k(T – θ)]
µ2 N Aβ 2 µ 2eff ≈ 0.125 eff T–θ 3k T – θ
(10)
This equation allows us to calculate the effective magnetic moment of an ion, µ eff, which can be rewritten as: µ eff = 2.828[ χM (T – θ)]1/2
(11)
where T is the temperature in Kelvin. In substances with interacting magnetic moments and where the orbital contribution to the magnetic moments and spin-orbit coupling are significant, the molar susceptibility is given by: χM=
N Aβ 2 g 2 J (J + 1) 3k (T – θ)
Experimental Procedure The samples used are: MnO (ROC/RIC MN-15, Analytical reagent), α-MnS (Johnson Matthey Cat. #12835), MnTiO3 (Johnson Matthey Cat. #13133), MnO2 (Fisher Cat. #M-108), α-Mn2O3 (ROC/RIC MN-50, Analytical reagent), and KMnO4 (Fisher Cat. #P-279). The mass susceptibility of each compound is measured directly using a magnetic susceptibility balance (MSB, Johnson Matthey model). Given the values of θ for each compound (7, 8), the effective magnetic moment of the manganese ion is found from the measured values of the molar susceptibility using eq 11. Instrumentation
(9)
Equation 9 can be further simplified by defining g2S(S+1) as µ 2eff and combining all of the constants to give:
χM=
Spin-orbit coupling destroys the degeneracy of the ground state defined by the combinations of L and S such that each level is distinguished by the different values of J, which ranges from L+S to L–S (2). However, only that value of J that forms the ground state given by L-S for less than half-filled shells or L+S for more than half-filled shells is used in eq 13. In the presence of a magnetic field, each J state is further split into 2J+1 components, a phenomenon known as the Zeeman effect. The effect of these splittings is to modify the energy of all components of the ground state. This phenomenon accounts for the small temperatureindependent paramagnetic (TIP) contribution to the observed susceptibility in compounds such as KMnO4 (2).
(12)
The different measurement techniques for the magnetic property of substances are based on the specific type of interactions between the substance and the external magnet. Typically, these experimental techniques involve measurement of either the force exerted on the sample moving in an inhomogeneous field of an applied stationary magnet, or change in the magnetic flux density observed when the sample is placed in a magnetic field. These techniques are respectively classified as force methods and induction methods (9). Gouy, Faraday, and torsion balances are typical examples of force methods of measuring magnetic property of substances. The widely used induction methods employ the vibrating sample magnetometer (VSM), SQUID, and ac induction techniques. The choice of instrumental technique for studying the magnetic properties of materials depends on the type and physical state of the sample as well as on the information and accuracy desired. The SQUID (Superconducting QUantum Interference Device) magnetometer has in recent years found SQUID
and the effective magnetic moment, µ eff , becomes µ eff = g[J(J+1)]1/2
Superconducting Wire
(13)
where J is the resultant total angular momentum of the ground state and the value of g is given by Landé’s equation: g=1+
S(S + 1) – L(L + 1) + J (J + 1) 2J (J + 1)
Pickup Loop Input Coil
Current I
-
Sample I+ I+
(14)
Magnetic Field H
The quantum number L is the resultant orbital angular momentum of all unpaired electrons present in the atom. For a completely filled orbital, L is zero.
908
I
-
Figure 3. Measuring the magnetization of a sample using SQUID magnetometer.
Journal of Chemical Education • Vol. 73 No. 9 September 1996
Research: Science & Education
Table 1. Magnetic Moments and Parameters for Certain Solid Manganese Compounds µeffb (lit.)
µeffc = (8χMT)1/2
θd(K)
µeffe = [8χM(T-θ)]1/2
KMnO4
Mn7+,
0
—
0.25
0
296
0.25
0.03
0
α-MnS
Mn2+, d5
5.92
5.96
3.60
-490
296
5.83
4.92
5
MnO
Mn2+, d5
5.92
5.92
3.3
-600
295
5.80
4.89
5
MnTiO3
Mn2+, d5
5.92
—
4.0
-303
296
5.69
4.78
5
α-Mn2O3
Mn3+,
d4
4.90
4.87
4.26
-121
296
5.05
4.15
4
MnO2
Mn4+,
d3
3.87
3.60
2.44
-430
296
3.82
2.95
3
aµ
dn
µsoa = [n(n+2)]1/2
Compound
Mx,
d0
T(K)
nf (calc)
n (theory)
so is
spin-only magnetic moment. bµ eff (lit.) is the effective magnetic moment obtained from literature. c µ is our experimental effective magnetic moment calculated assuming Curie behavior. eff dθ values obtained from the literature or when not available determined by a SQUID magnetometer in our laboratory. eThese are the experimentally determined values of µ eff with Curie–Weiss law. fThe number of unpaired electrons per manganese ion calculated from the values in column µ e using µ= [n( n+2)]1/2 . eff
wide application primarily with the discovery of hightemperature superconducting materials (T c > 77 K). A SQUID is a highly sensitive detector (“magnetic flux-tovoltage transducer” converting a tiny change in magnetic flux to voltage) that makes use of a superconducting material and its associated quantum-mechanical effects (10). The dc SQUID, operating at a steady current bias, consists of two half-rings of superconducting material connected at the two ends by an insulating layer (two Josephson junctions) to form a full ring. A magnetic field applied through the ring interferes with the quantized magnetic flux induced by the current applied across the SQUID. This quantum interference effect causes the maximum current (critical current) tunneling across the two junctions to oscillate between two values. As a result a periodic voltage develops across the SQUID that is highly sensitive to very small changes in the magnetic flux signal from samples whose susceptibility is as low as ~10–12 emu (11). Typically, the signal from the sample is coupled to the SQUID through the superconducting pickup loops (Fig. 3). During the measurement, the sample is transported upward by small increments repeatedly through the pickup loops (wound in opposite directions I+ and I– to eliminate any interference from nearby magnetic sources) while the output voltage at each point is read from the SQUID detector. In this experiment, however, the method to be used in measuring the magnetic susceptibility of substances is the Evan’s method (12, 13). It is a more recent technique based on the principle of stationary sample but moving magnet. Here, the balance [Johnson Matthey Magnetic Susceptibility Balance (MSB)] measures the force that the sample exerts on a suspended permanent magnet—in contrast to the Gouy balance (14), which measures the equal and opposite force that a magnet exerts on the sample. Results and Discussion Table 1 is a summary of the results of measurements on a series of manganese oxides and a sulfide with various formal oxidation states of the Mn ion and different numbers of unpaired 3d electrons per Mn. The µeff at room temperature of manganese ions evaluated using the Curie law are significantly different from values calculated assuming spin-only magnetic moment. This implies that the magnetic moments are not isolated and do interact with adjacent centers, as might be expected if magnetic dilution is not adequate. Moreover, µ eff is temperature-
independent only if Curie’s law is obeyed (2). However, the µ eff determined experimentally, taking into account Curie–Weiss behavior (when magnetic interactions were present, as determined by independent experiment on a SQUID magnetometer in our laboratory) is within 1–4% of the µ so (spin only) and/or literature value of the effective moment in each case. Except for Mn3+ ion, the µ eff observed at room temperature, assuming the Curie–Weiss law is valid, gave a slightly smaller value for the number of unpaired electrons than expected theoretically. This suggests that the magnetic susceptibility of these compounds be measured at higher temperatures, which, using the Evan’s balance becomes impossible. However, this limitation is more than compensated by the relative ease and speed of handling the instrument in the first-year chemistry laboratory. Note that the small positive value of χM for KMnO 4 may be attributed to temperature-independent paramagnetism (due to the mixing in of the low-lying excited states) that dominates the diamagnetic effect of the paired electrons in the inner shell of the Mn7+ ion. Acknowledgment We would like to thank W. H. McCarroll and K. V. Ramanujachary for their helpful discussions and suggestions. We are also grateful for the financial support of the Dreyfus Foundation and the National Science Foundation (Grant USE-9150484). Literature Cited 1. Vonsovskii, S. V. In Magnetism; Hardin, R., Transl.; Ketter: Israel, 1974; Vol. 1. 2. Figgs, B. N.; Lewis, J. In Modern Coordination Chemistry; Lewis, J.; Wilkins, R. G., Eds.; Interscience: New York, 1960; Chapter 6. 3. Kittel, C. Elementary Solid State Physics: A Short Course; Wiley: New York, 1962; pp 251–304. 4. Carlin, R. L. Magnetochemistry; Springer: Berlin, 1986. 5. Drago, R. S. Physical Methods in Chemistry; Saunders: Philadelphia, 1977; Chapter 11. 6. Geselbracht, M. J.; Cappellari, A. M.; Ellis, A. B.; Rzeznik, M. A.; Johnson, B. J. J. Chem. Educ. 1994, 71, 696–703. 7. Konig, E. Landolt-Bornstein: Magnetic Properties of Transition Metal Compounds, Hellwege, K.-H.; Hellwege, A. M., Eds.; Springer: Berlin, 1966; Vol. 2. 8. Goodenough, J. B. Magnetism and the Chemical Bond; Wiley: New York, 1963. 9. O’Connor, C. J. Prog. Inorg. Chem. 1982, 29, 203–283. 10. Clarke, J. Sci. Am. 1994, 271(2), 46–53. 11. Hatfield, W. E. In Solid State Chemistry: Techniques; Cheetham, A. K.; Day, P., Eds.; Clarendon: Oxford, 1988; pp 128–130. 12. Magnetic Susceptibilty Balance Instructional Manual; Matthey: Wayne, PA; 1991. 13. Woolcock, J.; Zafar, A. J. Chem. Educ. 1992, 69, A176–A179. 14. Shoemaker, D. P.; Garland, C. W., Steinfeld, J. I. Experiments in Physical Chemistry; McGraw-Hill: New York, 1974; pp 422–434.
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