Demagnetizing Factors for Cylinders

Fluxmetric (ballistic) and magnetometric demagnetizing factors Nf and N , for cylinders as functions of susceptibility x and the ratio y of length to diameter have been evaluated. Using a one-dimensional model when y 2 10, Nf was calculated for 1 5 x < Q) and N,,, was calculated for x + 00. Using a two-dimensional model when 0.01 5 y 5 50, an important range for magnetometer measurements, N , and Nf were calculated for -1 5 x < 00. Demagnetizing factors for x < 0 are applicable to superconductors. For x = 0, suitable for weakly magnetic or saturated ferromagnetic materials, Nf and N,,, were computed exactly using inductance formulas.

magnetic poles at each end of a cylinder [ 191. This simplistic model could be used only for long uniformly magnetized cylinders, and the results deviated significantly from experimental data on ferromagnetic samples. During the 1920's and 1930's, there were several theoretical papers on Nf for material with constant susceptibility x. The results were given as functions of x and the length-todiameter ratio y. These used one-dimensional models with approximations as needed to suit the computational techniques of the time. The first systematic theoretical calculation of N' for the high susceptibility case was done by Wurschmidt [20], [21]. He calculated Nf of cylinders using a one-dimensional model in which the cylinder had side surface poles and point end poles. He used Taylor expansions for the magnetization and the demagnetizing field at the midplane. The calculation was complicated, and he completed it only for the case y = 50 and x --+ W. For the y and x dependence of Nf, he gave qualitative results using the first few terms of the expansion. A similar approach with simpler expressions was used by Neumann and Warmuth [22], who calculated Nf for x + 00 a n d y 2 10.
To obtain the susceptibility dependence of Nf, Stablein and Schlechtweg [23] used a quadratic approximation and two linear differential equations. The model was improved by substituting uniform end-surface poles for point end poles. Their results included 30 values of Nf for 10 I y 5 500 and 12.56 I x c W . An extension of the y region to 0 was achieved by Warmuth [24]- [26], who fitted existing data and extrapolated graphically using the demagnetizing factor N of ellipsoids as a reference. The values of Nf calculated from the one-dimensional models were consistent with the data of ballistic measurements on soft magnetic materials. Bozorth and Chapin [27] compiled the results, which were later plotted in Bozorth's book [28].
To obtain axial demagnetizing factors more accurately, especially for short cylinders, two-dimensional calculations are needed. The simplest case is x = 0, where N , and Nf as functions of y can be derived analytically. The approximation x = 0 applies to diamagnets, paramagnets, and saturated ferromagnets. N,,, for 25 values of y from 0.2 to 1000 were obtained accurately to four significant figures by Brown [29] from a calculation of self-inductance [30] and listed as a table in Brown's book [31]. Crabtree [32] obtained the same values for the average demagnetizing factor by integration of the local field over the cylindrical volume. Brown's method to cylinders of polygonal cross section [33], and KaczCr and Klem extended it to hollow cylinders [34]. Nf for x = 0 was calculated exactly by Joseph [35]. Approximate values for N , and Nf for x = 0, accurate for large y, were calculated by Vallabh Sharma using uniformly magnetized volume elements [36]. Sato and Ishii [37] obtained a simple expression to approximate N, for x = 0. Chen and Li [38], [39] obtained Nf for x = 0 using magnetostatic potential calculations.
The susceptibilities x = -1 and x -+ 00 correspond to perfectly diamagnetic and ideally soft ferromagnetic materials, respectively. N , and Nf for these susceptibilities were first treated by Taylor for perfectly conducting cylinders [40], [41]. He developed a method introduced by Smythe that expressed charge densities on the side and ends in terms of a set of orthogonal polynomials, and expanded the electrostatic potential at the cylinder center [42]- [44]. Taylor calculated electric and magnetic polarizabilities for conducting cylinders for 0.25 I y 5 4 in both the longitudinal and transverse directions. N,(m) can be deduced from his electric polarizability results because of the analogy between electrostatics and magnetostatics. Because his calculation for magnetic polarizability was for a uniform quasi-static but nonpenetrating applied field, N,( -1) can also be deduced from his results. According to Taylor, his convergence error was less than 0.1 % for the longitudinal direction.
Using a similar approach with simpler base functions,  [46]. The fact that the side and endpole densities have basically a 6-'13 dependence, where 6 is the distance from the corner, was used to construct the set of polynomials. To estimate their error, Templeton and Arrott calculated the root-mean-square deviation of the normalized potential from 0 and found it to be less than 0.3 1 % [45]. Compared to an approximate formula with 8 adjustable parameters, the deviations of their 12 computed Nf(oo) values were less than 0.25 %. The work was based on their earlier magnetostatic analysis of the magnetization process in soft ferromagnetic cylinders with constant end-pole densities [47], [48]. The details of the calculation were published by Aharoni. who also calculated the self-energy of cylinders [49], and more generally, cylinders with nonuniform magnetization [50]. For susceptibilities other than 0, -1, and 00, different techniques have been used. Archer and Guancial [5 11 and Fawzi et al. [52] calculated the distribution of magnetization and magnetic field in long cylinders with large susceptibilities using volume and boundary integral equations. Using experimental resistance network analogs, Okoshi [53] obtained Nf for x -, 00, and Yamamoto and Yamada [54] obtained Nf and N,,, for large x.
Several papers have treated demagnetizing factors at points. Joseph and Schlomann [55] solved for local demagnetizing factors in uniformly magnetized cylinders and used a series expansion to account for nonuniform magnetization. Kraus [56] determined the complete local demagnetizing tensor for uniformly magnetized cylin-ders. Brug and Wolf [57] calculated the magnetization distribution in disks and obtained the local demagnetizing factor for materials that undergo phase transitions.
In Zijlstra's book [58], Nf and N , are plotted. These types of graphs and tables appear in other books on magnetism and magnetic materials, and they are widely used, sometimes inappropriately, in magnetic measurements of ferromagnetic, ferrimagnetic, weakly magnetic, and superconducting materials. However, there remain some problems. For x = 0, the most accurate case, the number of y values for N , and Nf is insufficient for accurate interpolation. For x # 0, almost all books give results obtained before 1950, and there are no data for x < 0. For long cylinders (y > lo), there is a lack of data on the x dependence of Nf, and there are no data on N,. For short cylinders (y < lo), there are even less data, and those that exist have large errors because they were obtained by extrapolation. In summary, there is no complete picture for the y and x dependence of Nf and N,.
In this paper, we calculate Nf and N, for a complete range of y and x. Susceptibility x is traditionally assumed to be constant in the material and is therefore defined as M / H , where M is the magnetic moment per unit volume and H is the internal magnetic field. For the case x = 0, in which the magnetization is uniform, we give 61 exact inductance calculations of N , and Nf for lop5 I y 5 lo3. For x # 0, more elaborate methods are used. For y > 10, the variation of magnetization across the radius of the cylinder is negligible at the midplane, and we calculate Nf as a function of y and x (-1 I x < 00) based on the one-dimensional model of Stablein and Schlechtweg [23]. Unlike them, we use Taylor expansions for M ( z ) , calculate the demagnetizing field Hd(z) directly at 25 points along the axis, and obtain more accurate results. The model is also applicable to N , for x -, 00. For 0.01 I y I 50, a two-dimensional finite element method is used that takes into account the variation of magnetic pole density along the side and ends of the cylinder. Values of N , and Nf are given for -1 I x < 00.

FLUXMETRIC AND MAGNETOMETRIC DEMAGNETIZING
FACTORS The demagnetizing correction is nontrivial for samples in open magnetic circuits. An exact correction can be obtained only for ellipsoids [4], [59], [60], where both the magnetization M and the demagnetizing field Hd are uniform under a uniform applied field H,. If the three principal ellipsoid axes coincide with the x , y , and z axes, the internal field is where N is the demagnetizing tensor, with N, + N,, + Nz = 1.

(2b)
If the applied field is along one of the principal axes, we have H = Ha + Hd = H , -N M , (3) where N is called the demagnetizing factor. In SI units, 0 5 N I 1. In cylindrical samples, which are commonly used in magnetic measurements, the demagnetizing field is not uniform, and two kinds of susceptibility-dependent demagnetizing factors are defined.
If the sample is located in a uniform applied field Ha along its axis, the fluxmetric (or ballistic) demagnetizing factor Nf is defined as the ratio of the average demagnetizing field to the average magnetization at the midplane perpendicular to the axis. The magnetometric demagnetizing factor N , is defined as the ratio of the average demagnetizing field to the average magnetization of the entire sample [58]: (4) Nf and N , are functions of the ratio y of cylinder length to diameter and the susceptibility x of the material. For ferromagnetic or ferrimagnetic materials, this x should be regarded as an effective x , similar to the differential susceptibility dM/dH at the corresponding magnetic state.
In [58], the definition of N , is limited to x = 0.

INDUCTANCE CALCULATIONS
Brown [29] showed how N , could be determined using a self-inductance calculation in which a uniformly magnetized cylinder was modeled as a solenoid. In fact, both N , and Nf may be calculated using the mutual inductance of two model solenoids of the same diameter. N , is obtained when the solenoids have the same length, and the problem reduces to the self-inductance calculation. Nf is obtained when the length of one of the model solenoids approaches 0, and the problem is that of the mutual inductance of a solenoid and a single-turn loop located at its midplane. In this section we calculate exact values of N , and Nffor x = 0 and a wide range of y.
A. Formulas for Inductance An exact formula for the self-inductance L, of a thin solenoid of length 21, radius a, and number of turns n is [611 (6) where F(k,) and E(k,) are the complete elliptic integrals of the first and second kind of modulus k,, which is defined by (7) and po is the permeability of vacuum.
An exact formula for the mutual inductance L, of the same thin solenoid and a coaxial single-turn loop of the same radius at its midplane is [62] Cohen [63] derived an exact general formula for the mutual inductance of two concentric coaxial thin solenoids (denoted by subscripts 1 and 2). We have used it successfully for these calculations, as an alternative to (6) and (8) where the modulus k, is defined by When x = 0, a cylinder in an axial field has a uniform magnetization M . An ideal thin solenoid carrying current I through n turns over a length 21 is equivalent, with respect to the B' field, to a longitudinally magnetized cylinder coincident with it [29]. Thus the cylinder can be modeled as a solenoid with the same M , and its average Hd can be obtained from M and average B' using (10). We take the solenoid as having one turn (n = l ) , so For N f , we obtain the average B' at the midplane from the flux a0 in the one-turn secondary loop of radius a: The average demagnetizing field is The definition of mutual inductance is The final expression for the fluxmetric demagnetizing factor is Equations (15) and (16) have been derived, by direct integration rather than inductance formulas,' by Joseph [35].

C. Results
Values of N,(x = 0) and Nf(x = 0) as functions of y (= Z/a) computed using (6), (15), (8), and (16) are given in Table I. For N,, the data agree with those given by Brown [29], [31]. For Nf, the data agree with those obtained by Joseph [35] and by Chen and Li [38]. In Table  I we also give N for ellipsoids of revolution with longitudinal axes 21 and transverse axes 2a calculated from well-known formulas [4], [59], [60].

IV. ONE-DIMENSIONAL MODEL FOR LONG CYLINDERS
A. Calculation of M Assume that a cylinder of length 21 and diameter 2a is located in a uniform applied field Ha along the z axis, as shown in Fig. 1. The material has constant susceptibility x, which leads to at any point inside the cylinder. Since V * B = 0, the volume magnetic pole density, proportional to V -M, equals 0 inside the cylinder; that is, all poles are on the surface.

<.''
We further assume for this one-dimensional model that .
For a section of cylinder of length dz at z , is the radial component of M at the side surface. Substituting a ( z ) = poMr(z) gives, on the side surface, the surface magnetic pole density   On the end planes of the cylinder, we have uniform surface magnetic pole densities: From (3) This is a general equation relating the expansion coefficients M2r of magnetization, the applied field H,, the susceptibility x, and the position z for a cylinder of length 21 and diameter 2a. In our problem, H,, I, a, and x are given, and the n + 1 coefficients M,, are unknown. We can choose n + 1 positions, z = zO, zI, * * * , z,, and get a set of n + 1 linear equations. M2, (i = 0, 1, -* , n) are then obtained by solving these equations simultaneously.

N , = -( H d ) / ( M ) = H a / ( M )
where the ( ) brackets denote the volume average, and   "The row for x = 0 is comparable to data for Nf(0) in Table I  Earlier results for y > 10 exist for Nf but not for N,.
Wurschmidt's result for y = 50 and x + 00 is 3.4% smaller than our result [20] results for x + 00 (Table 11) are the most accurate, and our data agree within 1.1 % . Compared with the exact results for x = 0 in Table I, the data in Table I1 have errors o f 0 . 7 % , 0.2%, and0.0% f o r y = 10, 20, a n d y L 50.

A . Calculation of Surface Pole Density
The magnetization, in general, varies throughout the cylinder in both the radial and axial directions. A calculation of demagnetizing factors for a short cylinder (y < 10) must take this variation into account, especially near the comers, where the magnetization sharply diverges for susceptibilities far from 0. As in the one-dimensional model, we assume that the cylinder consists of material with constant x, so the demagnetizing fields are completely specified by the surface pole density.
To obtain the distribution of poles on the surface for a specified susceptibility, we divide the surface of the cylinder into a set of nonoverlapping elements of uniform pole density. With the cylindrical symmetry there is no azimuthal dependence of the pole distribution, and we are therefore able to use elements in the shape of rings about the central axis. We solve the set of equations that describes the interaction between these rings of surface poles and use the resulting distribution to obtain the magnetometric and fluxmetric demagnetizing factors.
The method of dividing the surface of a magnetized body into a set of interacting elements of uniform pole density has been used before for the calculation of the magnetic fields of rectangular bodies. Ruehli and Ellis [65] assumed a constant susceptibility, and Normann and Mende [66] used a field dependent magnetization with the assumption that the volume pole distribution is negligible. Both of these studies were interested primarily in the field and magnetization distributions rather than the demagnetizing factors. A method that involves dividing the volume into uniformly magnetized elements was used by Brug and Wolf [57] for the case of thin disks that undergo magnetic phase transitions. They used a demagnetizing matrix that was derived by Hegedus, Kadar, and Della Torre [67], [68] for interacting volume elements in cylindrical geometries. Volume elements have also been used by Soinski [69] for rectangular and ring-shaped samples. A method of obtaining demagnetizing fields in bodies of arbitrary shape was presented by Vallabh Sharma [36] using rectangular volume elements. Templeton and Arrott [45] used the principle that the magnetic potential is 0 at each point on a grid inside a body with infinite susceptibility. They calculated the demagnetizing factors of cylinders and bars and later extended this to the case of a material that saturates [70].
The following method solves for the pole distribution at the surface of a cylinder for an arbitrary value of the susceptibility. The surface is divided into rings of area

B. Calculation of N, and Nj
After obtaining the surface pole distribution, there are two ways to calculate the demagnetizing factors, based on the two equations where SI is the surface consisting of the top half of the cylinder, sj is the area of thejth ring, and the sum is over the rings on the top end-plane and on the side surface above the midplane. To calculate N,, a series of cross sections corresponding to each side-surface ring is considered, and the volume-averaged M, is calculated from a weighted average over these cross sections.
The second method, using (37b), requires a calculation of ( Hd ) . We use (32a), with i denoting an interior point of the cylinder. Again, the average is taken over the midplane for Nf and over the volume for N,.
Since the first method involves surface flux calculations, while the second involves interior field calculations, we refer to them as the surface and the volume methods, respectively. Table I11 gives some examples of the results obtained from both methods. We see from this table that the results do not agree, especially for large y or large x. The differences are even larger for N,. The source of the disagreement is a systematic error that is due almost entirely to the finite number of elements with which the surface pole density is calculated.
The field produced by a magnetic pole is very sensitive to the distance r between the pole and the point at which the field is considered, with an r P 2 dependence. In the two-dimensional model, the division of the surface of the cylinder into rings will produce a discretization error because Hd is calculated in the center of a region of Uniform pole density. This error is mainly due to the division of the side surface; at the center of each ring on this surface, the normal component of Hd is produced by poles on the ring itself with significant contribution from poles on adjacent rings. For the end planes, the poles on the same (39c) ( 3 9 4 We assume that the error in the surface pole density calculation is not too large so that
Solving (40a) for N gives the interpolation/extrapolation equation We use (40b) and the results (up to 5 significant digits) from the two methods to obtain the final Nf and N,. They are presented in Tables IV and V and Figs. 5, 6, and 7.
The entries in Table I11 provide data for illustrative computations. Values for y I 10 were obtained with the 74side-ring calculation. The values for y = 20 and 50 are from the 132-side-ring calculation, which gives a smaller discretization error for these cases. As an example of the worst case, Nf for x = 10 000 and y = 20 is 55.46 X lop4 when computed with 74 side rings, compared to 53.85 x lop4 for 132 side rings.
With the corrected Nf, , we can correct M, and Hd using (39b) and (39d) and compare them with M z s and Hdt, calculated from (39a) and (39c). We find that, for all Nf and N,, the discretization error is less than l o % , giving a demagnetizing factor error of < l %. Exceptions are when y = 20 and x 2 100 for the 74-side-ring calculation, and when y = 50 and x 2 100 or y = 10 and x = 10 000 for the 132-side-ring calculation. Thus (39e) is rather accurate in most cases and the interpolation/extrapolation approach is valid.
A comparison can be made with less-general published results obtained for specific values of x . The results agree with the exact self-inductance calculations in Table I  The results of Nf for y 1 10 can be compared with our one-dimensional calculation (Table 11). The deviations of the one-dimensional from the two-dimensional results are -1.2%, -1.2%, 0.6%, 1.8%, 0 . 7 % , and -4.2% f o r y = 10 and x = 10 000, 100, 10, 1, 0, and -1, respectively. For y = 20 and 50 the maximum deviation reduces to 1.2% and 1.9%, respectively. Such deviations are due mainly to the approximations made in the one-dimensional model. N s x ) / ( 1 + N,,x).

A . Nf,, f o r x = 0, 00, and -1
In ellipsoids, a uniform applied field produces a magnetization and a demagnetizing field that are both uniform. In cylinders, the magnetization and the demagnetizing field are both nonuniform except in two cases. For x = 0, the magnetization is uniform but the demagnetizing field is not. For x + 03, the demagnetizing field is uniform (and exactly opposite to the applied field) but the magnetization is not. For x = -1, the nonuniform magnetization and nonuniform demagnetizing field in the cylinder combine to exactly cancel the applied field so that the flux density B = 0. In these three cases, demagnetizing factors can be calculated more accurately by introducing electromagnetic scalar potentials. Moreover, there are some simple relations among N,, Nmy, and NmZ, the magnetometric demagnetizing factors along the three principal orthogonal axes. For x = 0, a cylinder has N,,, Nmy, and N,, according to the more general theorem of Brown and Morrish [72]- [74]. As examples from Table I, cylinders with y = 0.22, 0.80, and 1.6 are equivalent to ellipsoids with y = 0.30, 0.90, and 1.6. A cylinder with y = 0.9065 is equivalent to a sphere ( N = 0.3333) [36]. A cube is also equivalent to a sphere according to the theorem [75]- [77]. Experimentally, for weakly magnetic and saturated ferromagnetic materials, these two shape-isotropic geometries can be used as alternatives to spheres. However, the theorem of Brown and Monish on the equivalence of a body of arbitrary geometry (including a cylinder) and a possible ellipsoid cannot lead to an a priori estimate of the value of N , ( x = 0) except for a body with center symmetry, such as a cube. For cylinders, the transverse magnetometric demagnetizing factors are (42) Equations (41) and (42) are valid for N , only when x = 0. For x > 0, the sum is less than 1, while for x < 0, it is greater than 1. If Nf is considered, rather than N,,,, the sum is always less than 1. N f ( x -+ 00) and N,(x -, 03) are rather close to N for ellipsoids. We can fit the Nf(03) and N , (03) versus y data in Table I1  Taylor calculated the anisotropic electric (CY) and magnetic (0) polarizabilities of conducting cylinders [40], [4 13. In Taylor's terminology, "conducting" means "without field penetration," so electrically E = 0 and magnetically B = 0. Therefore we can relate the longi-  Fig. 7. N , (solid curves) and N,(dashed curves)  There is a relation between and at, [41], creases with increasing y. In fluxmetric ferromagnetic measurements, this rule tells us that can be used for dM/dH larger than a minimum value that depends on y. For y = 10, 100, and 1000, the minimum values are 200, 5 X lo4, and lo6.   N x ) . Therefore, with increasing y, the minimum x to satisfy N f ( x ) > 0 . 9 9 N f ( w ) increases. From Table I1 we can deduce this minimum x to be k / N f ( w ) with 15 < k < 17.  Fig. 8(a) and 8(c).

( i ) / M ( O ) and a ( z ) / a ( + Z ) in
Also, M ( z ) and &(z) are even functions of z , but a ( z ) iS an odd function of z.

E. x Dependence of N,,,/Nf
For x = lo5 (that is, x -+ w) we can see in Fig. 8(b) that Hd(z) is a constant equal to -Ha in the entire cylinder, which makes H very small (0 for x -, 00) and M ( z ) finite. M ( z ) is approximately parabolic as shown in Fig.  8(a) and as already pointed out by other authors [2 11, [27].
The average Hd is equal to -Ha. For  But because Hd (z) becomes rather z dependent and its absolute value at z = 1 is much larger than at z = 0 [ Fig.   8(b)], N , / N f is larger.
As x decreases to 0.001, both M ( z ) and Hd(z) remain nearly constant in Fig. 8(a) and 8(b). The reason is that M ( z ) is very small compared to Ha, so &(I) is even smaller than H,. The variation of the extremely small Hd(z) cannot be seen in Fig. 8(b), and the modification of the field by such a small Hd (z) causes an invisible change in M in Fig. 8(a). However, if we expand the scales [

F. x Dependence of Nf
To understand the susceptibility dependence of Nf for y > 1, we focus on the magnetic pole distribution shown in Fig. 8(c) and 8(f). For the largest x , a(z) varies with z almost linearly on the cylinder surface except for the regions close to the ends. This means that the magnetic poles are the most uniformly distributed on the cylinder, and Hd (0), which has a greater contribution from the poles in the central region than from the ends, is the largest.
Thus Nf is its largest for the highest x . When x is decreasing, the variation of u(z) is progressively greater in the end regions, while the magnetic pole density in the central region becomes gradually lower.
Therefore, Nf decreases with decreasing x . When x = 0, all the poles are at the ends, a(z) is 0, and Nf should be its smallest. In fact, Nf continues to decrease when x becomes negative. The reason is that, although a(x) remains 0 in a large central region for x < 0, a(z) for z close to 1 increases with decreasing x and its sign is opposite to that of a( + 1 ) [Fig. 8(c)]. The side poles produce a field at the center directed opposite to the field produced by the end poles, so that the demagnetizing effect of the end poles is partially compensated by the effect of the side poles.
This makes Nf a little lower than for x = 0. The side poles close to the ends, with signs opposite to those of the end poles, have a different effect on N,. They greatly increase the value of Hd in the end regions, so the volume-aver-aged N,,, increases with decreasing x even for negative x.
Finally, we have the largest value of N , / N f at x = -1 .
When y < 1, the susceptibility dependence of Nf is the opposite. We can explain this as follows. For oblate cylinders, the end poles are the main contributors to the demagnetizing field Hd at the midplane. For a given ( M , ) at the midplane, the end-pole density is smallest when x -+ 00 since, in this case, the poles are the most uniformly distributed on the entire surface. Therefore Nf is the smallest. A smaller positive x repels the poles to the end regions, which gives rise to a larger end-pole density and a larger N f . Although all the poles are distributed on the ends when x = 0, the pole density on the ends is not the largest. This is because, when x < 0, the end-pole density has a sign opposite to that of the side-pole density nearby, as can be seen from Fig. 4; thus the end poles are further enhanced. As a consequence, the end pole density increases continuously with decreasing x regardless of its sign, and Nf takes its largest value when x = -1.

G. Error Transmission in Susceptibility Measurements
From the above analysis we see that, at present, the accuracy of N , and Nf can reach 1 % in general. To know if this accuracy is good enough for the purpose of magnetic measurements, we examine the influence of the error in N,,, or Nf on susceptibility measurements. We consider a cylinder consisting of material with constant x in a lon-   0.37, 0.17, and 0.06 for y = 1, 3, and 10; only a small part of the error in N,,, is transmitted to the final x result. In the third case, high-x materials are considered. To reduce the error due to the large a l , fluxmetric measurements should be made with long samples, since Nf < N,,, and Nf decreases with increasing y. To ensure that aI < 1 based on Table I1 and Fig. 2, y should be 12, 58,200, and 700, for x = lo2, lo3, lo4, and lo5, respectively.
The second term in (48) is the transmission error due to the measurement error of xe. The corresponding factor is a2 x/xe. For the first case where cyI is very small, a2 is very close to 1 since x = xe, so the error in x is almost the same as the xe measurement error. For our second case, from (47), we have a l + a2 = 1 . This is interesting because, when y is small so that N, + 1 and a1 + H . Application to Materials with x > 0 Most materials have x > 0, and our results can be used for demagnetization corrections of their magnetic measurements. For materials with linear or nearly linear magnetization curves, our Nf,, values are satisfactory. These include paramagnets, spin glasses, weakly magnetic materials, and iron-powder cores and ordinary ferromagnets in the initial and saturation states. However, even in these cases, some caution is required. We give an example below.
For magnetometric measurements of weakly magnetic materials (x < 0.01) only very small demagnetizing corrections are needed. However, such materials can also be measured by fluxmetric methods, as recommended by at least one measurement standard [79]. A source of error in fluxmetric measurements is if the sample diameter is less than that of the measurement coil. A large demagnetizing effect would occur in the measurement of M because the measured flux linkage is contributed not only by the M of the sample but also by the Hd within the coil volume produced by the sample's poles. Furthermore, fluxmetric measurements on weakly magnetic materials require many coil turns, which ensures that this error will arise. The error in x due to this effect can be as large as 30%, even if the requirements of [79] are followed [38], [64]. The magnetization curves of ferromagnetic materials are nonlinear, and it is difficult to assign to them specific x and Nf,,(x) values except in the initial state and when approaching saturation, as mentioned above. However, our results can still be used satisfactorily for long magnetically soft materials over a wide field range. We can regard x as the normal susceptibility x,, = M,/H,, where the subscript m denotes the maximum value at the endpoints of a symmetric magnetization loop, and use Nf(x).
To extract an unknown x from fluxmetric measurements of xe on samples with known y, one uses Nf. But a knowledge of x is required to select the appropriate Nf value. The known xe and y and the unknown x and Nfare related by (47) and Table I1 or Fig. 2, so the unknowns can be calculated simultaneously. An iterative process may be used. Nf is estimated based on the measured xe using Table I1 or Fig. 2, and x is calculated from xe and Nfusing (47). Then a better estimate of Nf is made. Since the differential susceptibility is field dependent in ferromagnets, this treatment involves some error. The resultant x is an effective susceptibility xeff. Its value is between x,, and an averaged differential susceptibility in the sample. We have xeff = x,, when H, --$ 0 or H , + 03, or when the sample has very large y and x,, and H, is close to H , (

I. Remarks Concerning x < 0 and Nonuniform x
For normal diamagnetic materials with uniform x, values of N, ( x = 0) are more than adequate for experimental work. However, large negative values of x arise in ac magnetic measurements of normal conductors and both ac and dc measurements of superconductors, where bulk magnetic moments have their source in eddy currents and shielding supercurrents, respectively. These magnetic moments allow us to ascribe values of M , H , and x to these materials.
In an ideal type-I superconductor, x = -1 because B and the permeability po( 1 + x) both equal 0 at every point in the material. The same applies to a normal conductor in an ac field when the skin depth is negligible compared to its dimensions. Thus there is an equivalence between these cylinders and a normal perfectly diamagnetic cylinder. For these cases, our values of N,,, ( x = -1) and Nf( x = -1) can be used. We have verified this experimentally with a magnetometric low-field ac susceptibility measurement of a niobium cylinder (y = 1.033) below the critical temperature. The susceptometer was calibrated using the known demagnetizing factor and dipole field of a superconducting niobium sphere [80]. We accounted for a 0.4% volume decrease of both the standard sphere and the sample cylinder upon cooling to 4 K, and deduced a value of N, equal to 0.361 0.001 from (47), with the uncertainty based on the measured scatter in xe.
Our two-dimensional calculations give 0.3622. Thus we see that cylindrical superconducting standards for magnetic measurements for use at low fields and temperatures can be made using accurate values of N , (-1).
The results of this work have to be used more carefully for materials that do not have constant susceptibility. In these cases, an effective susceptibility should be found. For example, the M ( H ) curve of an ideal type-I1 superconductor in fields below the lower critical field H,, is linear, with x = -1. In the mixed state, M ( H ) increases when H > H,, and the effective x should be close to the differential susceptibility at each point, which is positive.
This causes a discontinuity in the value of N , above H,, , and a proper demagnetizing correction should take account of this effect. Similar caveats apply to the intermediate state of superconductors.
Normal conducting cylinders in ac fields have M ( H ) loops, and a complex susceptibility with a negative real part can be defined [81]. However, this susceptibility is due to eddy currents constrained by the skin effect, different from our model assumption of uniform susceptibility. Nf,, for x = -1 may be used in the limit of small skin depth. Otherwise, to obtain good results, y must be large enough so that only a small correction is needed; our Nf,, values for an effective x < 0 can be used. A similar case arises in hard superconductors, where a portion of the magnetization comes from penetrated supercurrents that follow the critical-state model [82].
Since most cases of magnetic measurements involve nonlinear magnetization curves, the demagnetizing correction using the factors calculated for constant x should be made cautiously. For this, a deep understanding of the magnetization process and the ,demagnetizing effect is most important.

APPENDIX
The demagnetizing field produced by a ring-shaped distribution of magnetic poles is calculated below. Results are presented both for the case where a ring has a finite width, as on the end-plane of a cylinder, or a finite height, as on the side surface of a cylinder. The method follows that of Gray [83], who calculated the field produced by a disk of charge. The pole density is taken to be uniform over the width or height of the ring.
The magnetic scalar potential at a point (rir z,) produced by a pole density U on the j t h ring is given by where (P, and as represent the potential due to a ring on the end-plane and side surface, respectively. The parameters rl and r2 are the inner and outer radii of the ring of poles on the end surface, and zI and z2 are the limits in the z direction of the ring on the side surface. The limits of integration are changed to give +e = p F(r, 4) dr d4 -j : ' F(r, 4) dr d4, (-43) +s = sp G(z, 4) dz d4 -G(z, 4) dz d4, (A4) where F(r, 4) and G(z, 4)  (AI 1) We have arbitrarily set zi = 0 for ease of notation and have included the factor z,/ 1 zj I when needed to account for the sign reversal that occurs when zj < zi. The integrals in theffactors are given in [84] and reduce to where a2 = r2 and b2 = ri, and + b2(2na2)Y1Mcy2, P2))z2/lz2l (a2 < b2), where a2 = ri and b2 is the radius of the cylinder. The factors f;?, f&, f&, and f i r are given by the same expressions except r2 is replaced by rl and z2 is replaced by zI .