CONTENTS
A. Introduction
B. Magnetic Susceptibility - the Measured Quantity
a. The Gouy Method
b. The Faraday Method
c. The NMR Method of Evans
References

All matter is either attracted or repelled by a magnetic field. Compounds which are repelled by a magnetic field are said to be diamagnetic, and all matter exhibits some diamagnetism arising from polarization of the inner electron cores of all atoms. Compounds which contain one or more unpaired electrons are attracted by a magnetic field and are said to be paramagnetic . The size of paramagnetic moments is several orders of magnitude greater than that of diamagnetic moments, so that a compound containing one or more unpaired electrons will normally exhibit paramagnetic behavior.

Paramagnetic moments arise from the fact that electrons possess the property of spin (s = 1/2, m_s = ±1/2). If a compound contains one or more unpaired electrons then it will have a net spin S = ås. (Some examples are given in Table I.)

This property of "spin" gives rise to a magnetic moment m

Magnetic moments are not measured directly. Instead it is the volume magnetic susceptibility k or c_v which is measured. The sample feels a magnetic flux, B, due not only to the external field H but also to its volume magnetic susceptibility:

Although c_v is the measured quantity, it is not terribly useful for studying the magnetic behavior of many substances, particularly solids. Therefore, two other types of susceptibilities are more generally reported:

Units are not generally reported with these susceptibilities, though they are usually calculated in cgs units and have dimensions

c_v - cm^-3

c_g - gm^-1

c_M - mole^-1

The magnitude and sign of c_M for diamagnetic and paramagnetic systems is given below in Table II.

The molar susceptibilities of a number of common organic solvents are given in Table III.

Other diamagnetic susceptibilities may be estimated using Pascal constants. (See, for example, ref. 3 or 7 for a list of Pascal constants and how they are used. Ref. 5 gives a list of Pascal constants for ionic species.)

Paramagnetic compounds, as mentioned above, include both diamagnetic and paramagnetic contributions to the total susceptibility:

where c_M' is the paramagnetic contribution to the susceptibility. Pascal constants can be used to correct for the diamagnetism of ligands in inorganic compounds. (c_M' is larger in magnitude than c_M(meas) since c_M(dia) is negative in sign.) Once the corrected c_M' is obtained, the magnetic moment of a paramagnetic compound is calculated from the relation

Since m_eff is (usually) independent of temperature, It is thus clear that c_M' is a function of temperature; many interesting types of magnetic behavior may be observed by studying c_M' as a function of T. (See ref. 1-4, esp. ref. 3.)

Three methods of measuring magnetic susceptibilities are discussed below. There are additional methods, and the reader is referred to ref. 2 and 8 for discussion of these.

The reader is referred to references 2 and 5 for detailed descriptions of the method. In the Gouy method, the sample, packed carefully and uniformly into a tube of uniform cross-section, is weighted in the absence and then in the presence of a known magnetic field. (Figure Ia) The tube used is called a Gouy tube, and there are several varieties. Some of the better Gouy tubes are double-ended (Figure Ib), which allows correction for the diamagnetic susceptibility of air contained in the tube when it is weighed empty (see below). Gouy tubes are not obtainable commercially, but are made by the investigator to meet the specifications of his/her particular apparatus and experiments. An inside diameter of 3 mm is usually the minimum employed; 8 mm i.d. is the usual maximum (larger diameter tubes are more useful for liquid samples). A convenient top for a Gouy tube is a small-bore standard taper joint (Figure Ib) with a glass loop in the stopper. Pyrex tubing is acceptable for most measurements.

Figure Ia	Figure Ib

In the absence of specifically-constructed Gouy tubes, thin-walled NMR tubes may be used for fairly accurate measurements. However, in this case the tube should be centered between the poles in such a way that the bottom of the straight portion of the tube, rather than the rounded tip, is at the center of the pole gap. Alternatively, a flat- bottomed NMR tube may be used, if available.

The sample tube should be etched about two inches from the top and filled to that mark reproducibly. (Note: It is essential in the Gouy method that the sample be at least an inch and preferably two or three inches longer than the radius of the poles of the magnet.) The sample, if solid, should be finely ground and packed uniformly into the tube by adding small amounts of sample and tamping it as uniformly as possible with a glass rod of slightly smaller diameter than the bore of the Gouy tube, or by tapping the tube on a table after each addition of solid.

The Gouy tube is suspended from the balance by means of a thread or chain. A thread is acceptable for room temperature measurements, but for low-temperature work a fine-mesh silver chain, obtained from any jewelry store, should be used.

It is much easier to make relative magnetic susceptibility measurements on a given apparatus than to try to calibrate the magnetic field, measure the area of the sample tube absolutely, etc. Therefore, the method usually employed is to measure the weight change observed at a certain filled setting for a standard compound and use it to calibrate the apparatus. Several standard compounds are:

This is prepared⁹ by heating 28 g of CoSO₄.7 H₂O and 30 g NH₄SCN and 50 ml distilled H₂O to boiling (I). In a separate container (II), heat 27 g HgCl₂ + 300 ml distilled H₂O to boiling, filter, reheat to boiling. Add I to II with vigorous stirring. (Be careful to avoid excessive splattering and bumping.) Boil 1-2 minutes longer with vigorous stirring. Wash several times by decantation with distilled water. Filter, dry at about 120^o for several hours. Yield = 15 g. The only problem with this compound is that it packs poorly into Gouy tubes because it adheres to everything it touches.

The weight percent NiCl₂ must be determined by chemical analysis.¹⁰

in the region around 20 ^oC. The distilled water should be freshly boiled before use. Water is a particularly useful calibrant for diamagnetic samples.

The Gouy method requires weighing the sample in the absence and presence of a magnetic field, as is described in ref. 2 and 5. Therefore, one must turn the magnet on and off several times.

Note:  Before turning on the magnet, remove your watch to avoid magnetization, and turn on the cooling water bath and circulator.

1.   After turning on the cooling water, check to see that the voltage adjustment dial on the power supply is set at zero.

2.   Turn on the switch on the power supply. A pilot light indicates operation.

3.   Set the magnet current at the desired setting, as shown by the current reading on the power supply current meter. Consult instruction manual or laboratory instructor for reasonable current settings for any particular magnet gap. (Attainable magnetic field per ampere/coil is dependent on magnet gap; the smaller the gap, the higher the magnetic field per ampere/coil.)

4.   When the sample is to be weighed with the magnet on, it must be centered in the pole gap by readjusting the balance table, if necessary. (To operate the apparatus most efficiently, all parts of it should be bolted rigidly to the floor so that there is no possibility of moving the sample out of the center of the field.) The bottom of the tube should be placed in the center of the pole gap by rolling the magnet into position and lengthening or shortening the chain, as necessary.* Be sure that the sample is centered between the poles before raising the magnet! Also be sure that the glass jacket which surrounds the Gouy tube is centered so that it will not be harmed by raising the magnet.

*If you are using an NMR tube, center the bottom end of the flat portion of the tube and disregard the rounded tip.

5.   After the sample has been weighed with the magnet on, it is desired to turn off the magnet. A good technique is to weigh the sample with the magnet on, then re-set the current to the original setting, re-weigh, etc. You should allow a minute or two for the magnet (and thus the balance) to reach equilibrium after each current adjustment. Each weighing should be repeated 3-5 times and the average weight used in the calculation.

6.   Before turning off the magnet power switch, be sure that the current has been turned to zero.

7.   When all measurements are completed and the magnet has been turned off, shut off the cooling bath and circulator before leaving.

Experimental Procedure

This is described in great detail by Figgis and Lewis² and concisely by Angelici.⁵ Copies of these books are available in the library. The procedure and calculations are summarized below:

The measurements necessary for calculating the magnetic susceptibility of a solid or liquid sample are:

1.   weight of empty tube, field off

2.   weight of empty tube, field on

3.   weight of tube filled to the line with water, field off

4.   weight of tube filled to the line with sample, field off

5.   weight of tube filled to the line with sample, field on

6.   weight of tube filled to the line with calibrant, field off

7.   weight of tube filled to the line with calibrant, field on

Calculations

For more detail, see ref. 2 and 5. Briefly,

where B is the calibration constant, obtained from the measurements done on the calibrant and its known c_g. The volume susceptibility of air is 0.029 x 10^-6.

For the determination of the magnetic susceptibility of solid samples the Faraday method is particularly appealing. It requires very small amounts of solid in comparison to the Gouy method, and no sample tube packing errors are involved. In this method a small sample is suspended between two like faces which have been specially designed to provide a constant

across the sample, as shown in Figure II.

The force on the sample, F_x, is then given by: F_x= mc_gH(MH)/¶x

A sample container consisting of a tiny quartz bucket is used. Several milligrams of calibrant or sample are required. A common calibrant is HgCo(SCN)₄, discussed in part A.

Figure II

Positioning of the sample in the region of constant H(¶H)/¶ x is very important, and replacing it in the identical position for sample and calibrant is crucial. c_g is determined directly in this method:

where B = calibration constant

s = sample

b = boat

This and other⁸ methods are based on the fact that the position of a given proton resonance signal is dependent upon the bulk susceptibility of the medium in which the molecule containing the proton is found. The susceptibility of a solute (B) in a solvent (A) can be determined by adding a small amount of some inert substance (C) which contains protons which give rise to a sharp singlet. The method involves dissolving a known weight of B in a known volume of a stock solution A + C; this solution is then placed inside a special small-bore NMR tube which is in turn placed exactly in the center of a regular-sized NMR tube which contains the stock solution of A + C only. The NMR spectrum of C will show two signals for the singlet, one from the inner(sample) tuber and one from the outer (reference) tube. For the geometry of a superconducting magnet,¹⁴ the separation between these two signals is given by:

[For the geometry of an electromagnet, the 4p/3 factor becomes 2p/3.^12,13

where d_ref = chemical shift of reference peak C, in ppm

d_sample = chemical shift of sample peak C, in ppm

c_gsample = gram susceptibility of the sample solution

c_gref = gram susceptibility of the reference solution

This can be written in terms of gram susceptibility to allow calculation of the c_g of the solute (B) as:

where c_g(B) = gram susceptibility of B

m = grams B per liter of solution or mg B per mL of solution

c_g(A+C) = gram susceptibility of the stock solution

(This is usually simply c_g(A) when the concentration of C is small. c_g of the solvent (A) can be calculated from the values of c_M in Table III)

d_o = density of the reference solution (A + C)

d_s = density of the sample solution (A + B + C)

The third term is usually negligible. It can, however, be easily determined if the weight of the solution (per ml) is measured when the solution is made up. The density of the solvent can be found in the handbook or can also be measured.

The above method gives only the magnitude of c_g. The sign must be supplied by the experimenter (i.e., you must know whether the sample is paramagnetic or diamagnetic). It is not difficult to determine the sign, since the magnitude itself will tell whether the sample is diamagnetic or paramagnetic. In addition, most paramagnetic samples will show broadened NMR signals for the sample solutions. For this reason, it is wise not to have the concentration of B too high if B is paramagnetic, in order to avoid extreme broadening of its NMR signal.

It should also be noted that d, in units of ppm, implies a factor of 10^-6 on the right side of the equation (11).

In order to observe two separate proton signals from C the solution must be of high enough concentration to produce a measurable separation. It is well worth the time it takes to estimate the expected susceptibility and calculate the weight of sample needed per milliliter to produce a measurable separation. Reasonable separations are 10-30 Hz on a 60 MHz NMR spectrometer and correspondingly larger separations at higher fields (e.g., 50-150 Hz on a 300 MHz spectrometer), although most diamagnetic samples will give much smaller separations. On a pulsed NMR spectrometer, only four transients need to be collected (for phase cycling purposes), and the power level should be set VERY low! This is a VERY strong sample! Note that if you want to lock the spectrometer frequency to a deuterium signal, the solvent must be deuterated!

It should be noted that since a difference in chemical shift is to be used in the calculation, it is not necessary to measure the absolute chemical shift. For greatest accuracy the spectrum of C should be expanded to a 1-2 ppm scale, and the frequency separation (or chemical shift difference) measured with the cursor.

The special NMR tubes can be purchased from Wilmad Glass Co., Inc., Buena, NJ (Catalogue No. 517, 518 or 519). They are specially- constructed coaxial tubes, available either as a single unit or as separable pieces (Figure III). The price is about $20-30 per unit.

Figure III

The solvent (A) and inert substance (C) should be chosen to fulfill the requirements of solubility (A) and inertness (C). Several combinations might be:

There is no need to use deuterated solvents for this technique if you are using a continuous-wave NMR spectrometer. However, if you are using a pulsed NMR spectrometer, a deuterated solvent will allow field- frequency lock that will prevent magnet drift, and it will also allow the acquired pulse to contain mainly the free induction decay of the inert substance rather than mainly that of the solvent. It is obvious that A and C must be chosen so that proton resonances from A will not interfere with the singlet of C which is to be observed. This is the reason that the combination pyridine-toluene rather than pyridine-benzene was suggested above. It is not wise to use a solvent peak (A) as the proton resonance to be observed and used in the calculation, since there are cases in which solvent interactions with a paramagnetic solute produce contact or dipolar shifts in addition to the expected susceptibility shift.¹⁴

The accuracy of susceptibilities determined by this NMR method is very dependent on the separation (d_ref - d_sample) and how accurately it can be measured; it is also obviously dependent upon the accuracy to which the solutions have been prepared! In some cases it may be wise to standardize the solution if B is hygroscopic. In general, the NMR method is less accurate than the Gouy method. Its advantages are that it is quite simple and rapid, and it requires only small amounts of sample as compared to the Gouy method. For best accuracy and precision the Gouy method is probably preferable for solutions and Faraday for solids.

1. Cotton, F. A.; Wilkinson, G. Advanced Inorganic Chemistry; Wiley-Interscience: New York, 1980; Ed. 4, pp. 625-628. (Not in later editions!)

2. Figgis, B. N.; Lewis, J. In Modern Coordination Chemistry; Lewis, J.; Wilkins, R. G., Eds.; Interscience: New York, 1960; especially pp. 400-417.

3. Kahn, O. Molecular Magnetism; VCH Publishers: Weinheim, 1993.

4. Drago, R. S. Physical Methods for Chemists; Saunders College Publishing: Orlando, FL, 1992; Chapter 11, pp. 469-491.

5. Angelici, R. J. Synthesis and Technique in Inorganic Chemistry ; University Science Books: N.Y., 1986; Ed. 2, pp. 46-55.

6. Figgis, B. N.; Lewis, J. "Magnetochemistry," in Technique of Inorganic Chemistry; Jonassen, H. B.; Weissberger, A., Eds.; Interscience: New York, 1965; Vol. IV. Pp. 139-205 deal with theory; pp. 209-211 with the NMR method; pp. 212-219 with the Faraday method; pp. 219-223 with the Gouy method.

7. Emsley, J. W.; Feeney, J.; Sutcliffe, L. H. High Resolution Nuclear Magnetic Resonance Spectroscopy; Pergamon Press: Oxford, 1965; Appendix C.

8. Pople, J. A.; Schneider, W. G.; Bernstein, J. H. High Resolution Nuclear Magnetic Resonance; McGraw-Hill: New York, 1959, pp. 18, 19.

9. Li, N. C.; Scruggs, R. L.; Becker, E. D. J. Am. Chem. Soc. 1962, 84, 4650.

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11. Nettleton, H.; Sugden, S. Proc. Roy. Soc. A 1939, 173, 313.

12. Evans, D. F. J. Chem. Soc. 1959, 2003.

13. Deutsch, D. L.; Poling, S. M. J. Chem. Ed. 1969, 46, 167.

14. Sur, S. K. J. Magn. Reson. 1989, 82, 169.

15. Gerlach, D. H.; Holm, R. H. Inorg. Chem. 1969, 11 , 2292.