APPARATUS FOR GENERATING UNIFORM MAGNETIC FIELDS WITH MAGNETIC WEDGES

Abstract: A magnetic structure generating a uniform magnetic field comprising at least one pair (12, 14) of magnetic wedges of identical shape arranged symmetrically about a first axis and abutting one another along a common edge. Each magnetic wedge has a first side and a second side which terminate at the common edge. The first wedge of each pair is positioned to define a first angle between the first side and the first axis and a second angle between the second side and the first axis, and is uniformly magnetized in a direction relative to a second axis (perpendicular to the first axis) which is a function of the sum of the first and second angles. The second wedge of each pair is magnetized such that the absolute values of components of the magnetization along the first and second axes are equal to components of the magnetization of the first wedge along the first and second axes, and one of the magnetization components of the second wedge is opposite to the magnetization component of the first wedge.

Description: APPARATUS FOR GENERATING UNIFORM MAGNETIC FIELDS WITH MAGNETIC WEDGES COPYRIGHT NOTICE A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyright rights whatsoever. RELATED APPLICATIONS AND PATENTS This application is related to copending U.S. application, Serial No. 08/613,756. Related patents include U.S. Patent No. 5,495,222 for "Open Permanent Magnet Structure for Generating Highly Uniform Field;" U.S. Patent No. 5,475,355 for "Method and Apparatus for Compensation of Field Distortion in a Magnetic Structure Using Spatial Filter;" U.S. Patent No. 5,428,333, for "Method and Apparatus for Compensation of Field Distortion in a Magnetic Structure;" U.S. Patent No. 5,278,534 for "Magnetic Structure Having a Mirror;" U.S. Patent No. 5,285,393 for "Method for Determination of Optimum Fields of Permanent Magnet Structures with Linear Magnetic Characteristics;" U.S. Patent No. 5,412,365 for "High Field Magnets for Medical Applications;" U.S. Patent No. 5,162,771 for "Highly Efficient Yoked Permanent Magnet;" U.S. Patent No. 5, 107,239 for "Hybrid Permanent Magnets;" U.S. Patent No. 5,119,057 for "Optimum Design of Two-Dimensional Permanent Magnets;" and U.S. Patent No. 4,990,083 for "Yokeless Permanent Magnet Structure and Method of Construction". All of the above patents and applications are hereby incorporated by reference into this application. BACKGROUND OF THE INVENTION The invention disclosed herein relates generally to magnetic structures, such as structures used in medical applications of nuclear magnetic resonance ("NMR") imaging . More particularly, the present invention relates to magnetic structures formed from one or more pairs of magnetic wedges having configurations and remanences designed to generate substantially uniform fields within a region of interest. The structures may be closed, partially open, or totally open. A desirable feature of a magnet for NMR imaging is open access to the patient. An open magnet design should preferably be flat and provide maximum access to the patient by separating the imaging region from the source of the magnetic field which can be either a superconductive or a permanent magnet. One problem facing the designer of a flat magnetic structure is the confinement of the field to the region of interest. Another problem is designing a structure with reasonably small dimension of the flat surface relative to the size of the imaging region because large and powerful magnetic structures are usually necessary to generate the required field within a region large enough to satisfy the requirements of practical applications Schemes have been proposed for the development of open superconductive structures. Because of the high degree of design flexibility offered by permanent magnets, open structures based on the use of permanently magnetized materials are often preferable in spite of their field strength limitations compared to superconductive magnets. As described in U.S. Patent No. 5,495,222, issued to the same inventors as those identified herein, permanent magnets may be used in accordance with the recently developed methodology described therein in the design of partially open magnet configurations. The present invention provides new approaches to magnet design that provide for structures which may be closed, partially open, or totally open, and which minimize the amount of magnetic material needed to generate a strong, uniform field within a region of interest. Further background information on magnetic structures and the use of permanent magnetic structures to generate highly uniform fields is found in the patents hsted above and at least in the following additional sources, each of which is hereby incoφorated herein by reference: [1] Abele, M., Structures of permanent magnets. John Wiley and Sons, Inc., New York, 1993. [2] Jensen, J.H., Abele M.G., Maximally Efficient Permanent Magnet Structures. Journal of Applied Physics 79(2), 1157-1163, January 15, 1996. [3] Abele MG, Jensen JH, Rusinek H, Open Hybrid Permanent Magnet. Technical Report No. 29, Department of Radiology. New York University Medical Center, March 15, 1995. [4] Jensen JH, Abele M.G., Effects of Field Orientation on Field Uniformity in Permanent Magnet Structures. Journal of Applied Physics 76(10), 6853-6855, 1994. [5] Abele M.G., Jensen JH, Rusinek H., Linear Theory of Pole Piece Design in Permanent Magnets. Proceedings of XIII International Workshop on Rare-Earth Magnets and Applications. CAF Manwaring, DGR Jones, AJ Williams and IR Harris, Eds, University of Bύmingham, Edgbaston, United Kingdom, pp.167-176, 1994. [6] Abele M.G., Generation of Highly Uniform Fields with Permanent Magnets (invited paper). Journal of Applied Physics 76(10), 6247-6252, 1994. [7] Abele M.G., Rusinek H., Field Computation in Permanent Magnets with Linear Characteristics of Magnetic Media and Ferromagnetic Materials. Technical Report No. 24, Department of Radiology, New York University Medical Center, August 15, 1991. SUMMARY OF THE INVENTION It is an object of the present invention to provide an improved apparatus for generating a substantially uniform magnetic field within a region of interest. It is another object of the present invention to provide a totally open magnetic structure which is useful for medical imaging. It is another object of the present invention to reduce or compensate for distortions in a magnetic field generated by a totally open magnetic structure. It is another object of the present invention to rninimize the amount of magnetic material needed to generate a strong uniform magnetic field within a region of interest. It is another object of the present invention to provide improved magnetic structures for NMR imaging. The above and other objects are achieved by a magnetic structure generating a substantially uniform magnetic field within a region of interest comprising at least one pair of magnetic wedges of substantially identical shape arranged symmetrically about a first axis and abutting one another along a common edge. Each magnetic wedge has a first side and a second side which terminate at or near the common edge. A first wedge of each pair of wedges is positioned to define a first angle between the first side and the first axis and a second angle between the second side and the first axis. The first wedge is uniformly magnetized in a direction relative to a second axis (perpendicular to the first axis) which is a function of the sum of the first and second angles. A second wedge of each pair of wedges is magnetized such that the absolute values of components of the magnetization along the first and second axes are equal to components of the magnetization of the first wedge along the first and second axes, respectively, and such that one of the magnetization components of the second wedge is opposite to the respective magnetization component of the first wedge. With this structure, the region of interest containing the substantially uniform field is between the first sides of the pair of wedges, the second sides of the pair of wedges, or both. To minimize or eliminate the surface charge density along the third side of each wedge, the third side of each wedge is preferably parallel to the direction in which the wedge is magnetized. The magnetization directions may be configured to provide a purely transverse field (generally parallel to the second axis) or a purely longitudinal field (generally parallel to the first axis) in the region of interest. For a transverse field, the first wedge is magnetized at an angle relative to the second axis equal to the sum of the first and second angles minus 90°. The second wedge is magnetized such that the magnetization component along the second axis is equal to the magnetization component of the first wedge along the second axis and the magnetization component along the first axis is equal and opposite to the magnetization component of the first wedge along the first axis. For a longitudinal field, the first wedge is magnetized at an angle relative to the second axis equal to the sum of the first and second angles. The second wedge is magnetized such that the magnetization component along the first axis is equal to the magnetization component of the first wedge along the first axis and the magnetization component along the second axis is equal and opposite to the magnetization component of the first wedge along the second axis. To help close the flux of the magnetic field the magnetic structure may comprise a high magnetic permeability yoke positioned between either the first or second sides of the at least one pair of wedges and an exterior region, whereby magnetic flux is substantially zero in the exterior region. The yoke may extend along and abut the first or second sides of the at least one pair of wedges, or may not abut the first or second sides of the at least one pair of wedges, leaving a space between the sides and the yoke into which may be positioned a filter structure as described herein. The first and second wedges may assume a variety of shapes such as spherical wedges or prismatic wedges, but are preferably triangular in cross section. To increase the magnitude of the magnetic field intensity, a plurality of pairs of wedges may be provided, each pair being arranged symmetrically about the first axis. The common edges of the pairs of wedges are preferably in an adjacent relationship to one another, and the angular widths between the first and second sides of the plurality of wedges are preferably substantially equal. The wedges in one pair of the plurality of pairs of wedges may have a shape which is substantially different than or identical to the shape of the wedges in another pair of the plurality of pairs of wedges. The multiple pairs of wedges preferably abut one another, so that the first sides of at least one pair of wedges abut the second sides of another pair of wedges. Alternatively, there may be space between each pair of wedges. To close the flux in an exterior region on the opposite side of the magnetic structure as the region of interest, a high magnetic permeabihty yoke may be positioned between corresponding sides of one of the pairs of wedges and an exterior region. The magnetic structure may be closed, partially open, or totally open. For a closed magnet the magnetic structure would comprise one or more additional magnetic elements connected to the pair(s) of wedges, the one or more additional magnetic elements defining with the wedges a cavity in which the region of interest is situated. For a partially open magnet, the structure may contain additional elements which partially enclose with the wedges a cavity, but which cavity is open at at least one side thereof. For a totally open magnet, the region of interest is unbounded by any magnetic element other than the wedges. Means may be provided for compensating for distortions in the substantially uniform magnetic field produced by the magnetic wedges. The means may comprise at least one filter structure positioned adjacent to either the first or second sides of a pair of wedges. The filter structure comprises one or more filter elements, and may be positioned between the high magnetic permeabihty yoke and the pair of wedges. The means may also comprise a pair of dipole distributions of uniform moments positioned adjacent to either the first or second sides of a pair of wedges, the dipole distributions being positioned symmetrically about the first axis. If the dipole distributions are oriented in opposite directions along the first axis, they compensate for distortions in any component of the magnetic field along the second axis. If the dipole distributions are oriented in the same direction along the first axis, they compensate for distortions in any component of the magnetic field along the first axis. Two sets of dipole distributions may be employed to compensate for distortions in both components of the field. The compensation means may comprise at least one pair of elements positioned at or near ends of first or second sides of a pair of wedges. The elements may comprise a pair of substantially identical ferromagnetic pole pieces positioned symmetrically about the first axis, or at least one pair of substantially identical magnetized elements positioned symmetrically about the first axis. BRIEF DESCRIPTION OF THE DRAWINGS The invention is illustrated in the figures of the accompanying drawings which are meant to be exemplary and not limiting, in which like references refer to like or corresponding parts, and in which: Figs. 1-3 are schematic diagrams of cross sections of embodiments of magnetic structures of the present invention with a single pair of wedge magnets; Figs. 4A-4C are vector diagrams that provide the values of and H within wedges for magnetic structures of embodiments of the present invention; Fig. 5 is a graph showing plots of the intensities H t and H e in the limit r → O versus the orientation a„,; Fig. 6 is a schematic diagram of a cross section of one embodiment of a magnetic structure of the present invention generating a longitudinal field; Fig. 7 is a schematic diagram of a cross section of one embodiment of a magnetic structure of the present invention having a high permeability yoke; Fig. 8 is a schematic diagram of the structure of Fig. 7 shown with a filter structure; Fig. 9 is a table of coefficients of the field expansion as a series given in Eq. (2.2) of the wedge structures of Figs. 7 and 8; Fig. 10A is a schematic diagram showing the equipotential lines of a wedge magnet of the structure of Fig. 7; Fig. 1 OB is a schematic diagram showing the equipotential lines of a wedge magnet of the structure of Fig. 8; Fig. 11 is a graph showing the effect of an increasing number of filter elements on the intensity of the magnetic field; Fig. 12 is a graph showing the potential of components of a filter with three active filter elements (no = 3); Fig. 13 is a schematic diagram showing the equipotential lines of a wedge magnet of one embodiment of the present invention having a single active filter element (n 0 = 1); Fig. 14 is a graph showing field intensity versus distance on the x axis for three different structures of the present invention; Fig. 15 is a schematic diagram showing equipotential lines for a hybrid wedge magnet in accordance with one embodiment of the present invention; Fig. 16 is a schematic diagram of a rectangular magnetic block; Fig. 17 is a graph of the field intensity and gradient along the x axis of the rectangular block of Fig. 16; Fig. 18 is a schematic diagram showing the equipotential lines generated by a rectangular magnetic block; Figs. 19 is a schematic diagram of wedge magnets of one embodiment of the present invention with a high permeabihty yoke extending along the interior sides of the wedges; Fig. 20 is a schematic diagram of the wedge magnets of Fig. 19 with K= l; Fig. 21 is a schematic diagram of wedge magnet structures of embodiments of the present invention with a high permeabihty yoke extending along the exterior sides of the wedges along the line x = 0; Fig. 22 is a graph of field intensity along the x axis for wedge magnetic structures with and without filter structures; Figs. 23 and 24 are schematic diagrams showing equipotential lines of a portion of the wedge magnet of Fig. 21 without and with a filter structure; Fig. 25 is a schematic diagram showing equipotential lines of a single wedge magnet; Fig. 26 is a graph of the field intensity component along the x axis and its first and second derivatives generated by a single wedge magnet; Fig. 27A is a schematic diagram showing the boundaries of two imaging regions near a magnetic structure of one embodiment of the present invention containing a single wedge magnet pair; Fig. 27B is a schematic diagram showing the boundaries of two imaging regions near the magnetic structure of Fig. 27B further containing magnetic dipoles; Fig. 28 is a graph of field intensity component along the x axis in the neighborhood of point P 0 ι in Figs. 27A and 27B; Fig. 29 is a schematic diagram showing equipotential lines of a single wedge magnet of the present invention having a high permeability medium placed along its exterior side; Fig. 30 is a graph of the field intensity component along the J axis for the wedge magnet of Fig. 29; Fig. 31 is a schematic diagram of a closed hexagonal magnetic structure of one embodiment of the present invention; Figs. 32A and 32B are schematic diagrams of single wedge pair magnetic structures of embodiments of the present invention with magnetic elements as partial terminations; Fig. 33 is a schematic diagram of a single wedge pair magnetic structure of an embodiment with ferromagnetic pole pieces; Figs. 34, 35A and 35B are schematic diagrams of two wedge pair magnetic structures of embodiments of the present invention; Fig. 36 is a schematic diagram showing equipotential lines of two magnetic wedges of the structure of Fig. 35A; Fig. 37 is a graph of field intensity for the double wedge magnet of Fig. 35A; Fig. 38 is a graph of field intensity of a single wedge magnet for two values of cc f ; Fig. 39 is a schematic diagram showing equipotential lines of two magnetic wedges of the structure of Fig. 35B; Fig. 40 is a schematic diagram for a double wedge pair magnetic structure of one embodiment of the present invention with the transformation of an equipotential line into an interface with a high magnetic permeability material; Fig. 41 is a vector diagram for the computation of field components for the double wedge structure of Fig. 40; Fig. 42 is a schematic diagram of a closed magnetic structure of one embodiment of the present invention incoφorating the double wedge structure of Fig. 40; Fig. 43 is a schematic diagram of a magnet having pole pieces and rectangular magnetic components with geometries similar to the geometries of the structure of Fig. 42; Fig. 44 is a graph of field distribution along the x axis (curve (a)) and > axis (curve (b)) for the magnetic structure of Fig. 43; Fig. 45 is a schematic diagram for the y > 0 region of a triple wedge pair magnetic structure of an embodiment of the present invention; Figs. 46A, 46B, and 46C are vector diagrams for the computation of field components in the three magnetic wedges of Fig. 45; Fig. 47 is a schematic diagram of a portion of a closed magnetic structure of an embodiment of the present invention incoφorating the triple wedge pair magnetic structure of Fig. 45; Fig. 48 is a schematic diagram of a magnet having pole pieces and rectangular magnetic components with geometries similar to the geometries of the structure of Fig. 47; Fig. 49 is a graph of field distribution along the x axis (curve (a)) may axis (curve (b)) for the magnetic structure of Fig. 48; and Fig. 50 is a perspective three-dimensional view of a single wedge pair magnetic structure of one embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The magnetic structures of preferred embodiments of the present invention having single or multiple pairs of wedge magnets will be described in the following sections with reference to the drawings in the figures. Section 1 contains an analysis of the properties of a two dimensional magnetic structure and, in particular, of the distribution of magnetization required to generate an arbitrarily assigned orientation of the field within a region of interest. Section 2 contains a description of the achievement of a uniform field within the region of interest including the design of filter structures that cancel the dominant spatial harmonics of the field distortion. Section 3 contains an analysis of the effect of the relative orientation of the wedges on the properties of the region of interest, and a description of single wedge pair embodiments of the present invention incoφorated closed and open magnetic structures. Sections 4 and 5 contain a description of magnetic structures of preferred embodiments having multiple wedge magnets and terminations for such wedge magnets. Section 6 extends the two dimensional formulation of previous sections to an embodiment of a three dimensional wedge structure of finite dimensions. The magnetic structures described herein have potential use in dedicated scanners wherein the region of interest is the imaging area and is close to the surface of a body, for instance the breast or the spine area of the human body. For example, the magnetic structure 10 of Fig. 50 contains a pair of identical magnetized wedges 12 and 14 arranged symmetrically about the x axis. The wedges 12 and 14 have a common edge 16 extending along the z axis. Wedges 12 and 14 have sides 18 and 20, respectively, which are effectively the "interior" sides of the structure 10 and sides 22 and 24, respectively, which are effectively the "exterior" sides of the structure 10. The wedges 12 and 14 are magnetized in order to generate a magnetic field within a region of interest between the interior sides 18 and 20, which is the imaging region for a part of the human body. As described in detail herein, the magnetic field can be generated at a desired strength and uniformity by varying the orientation, size and number of wedges, and by adding filter structures, dipole distributions, and other structural elements. A yoke 26 made of high magnetic permeabihty material extends along the exterior sides 22 and 24 to close the flux. Each particular application determines the geometrical constraints of the imaging region and, as a consequence, provides the input of the design of the magnet configurations and appropriate filter structures as described herein. 1. Analysis of Magnetized Wedges The magnetic structure of Fig. 50 can be analyzed by considering a cross section of the structure taken along the z axis. The basic component of the wedge magnet is the two-dimensional structure of Fig. 1. The structure consists of two identical wedges 12 and 14 of magnetic material symmetrically arranged with respect to the axis x with a common edge 16 as shown in Fig. 1. The wedges are located in a non magnetic medium. Assume an ideal demagnetization characteristic B= J+ μ 0 H, (1.1) where B, H are the magnetic induction and intensity respectively, and μo is the permeabihty of a vacuum. The orientations of the four surfaces 18, 22, 24, and 20 of the wedges 12 and 14 are given respectively by the angles α,, , α 3 = 2π - a 2 , a 4 =2π - a (1.2) The magnetization J of the wedges 12 and 14 induces a uniform surface charge density σ h on each of the four interfaces that have the axis z in common. The induced surface charge densities are σ, = J 0 cos(α, - θ), σ 2 = J 0 cos(α 2 - θ), σ i = -σ 2 , σ = -σ x , (1.3) where θ is the angle between the remanence J of the wedge in the region y > 0 and the axis y. At a point P, the surface charge densities σ h generate an intensity of the magnetic field H h H„ = - ln 2πμ 0 π - a h + where r, ψ are the polar coordinates of point P, as indicated by Fig. 1, r 0h is the radial dimension of interface Sj, between wedges and external medium, ηj, is the coordinate of P in the frame of reference of the interface defined by the unit vectors, xj,, ηj,, as indicated in Fig. 1. Sides (P1P2) and (P3P4) are assumed to be parallel to the remanence of the two wedges. Assume r 0 j, -> «>. The two wedges 12 and 14 generate a uniform field if the charges σj, satisfy the condition ∑σ h τ h = (1.5) A=l Eq. 1.5 has two particular solutions: σ 4 = -σ, , σ 3 = -σ 2 , σ, sin α, + σ 2 sin 2 = 0 (1-6) and σ 4 = σ, , σ 3 = σ 2 , σ, cosα, + σ 2 cosα 2 = 0. (1.7) When condition (1.6) is satisfied, the remanence J in the region y > 0 (in the wedge 12) is oriented at an angle θi relative to the axis>> and JΛ-y) = -J x (y\ J y (-y) = Jy{y). (1.9) Eq. 1.9 corresponds to the distribution of magnetization shown in Fig. 2 (the thick arrows within the wedges in the figure representing the magnetization). Under condition (1.6), the components of intensity H x in the region -a < a < a (1.10) are where Jo is the magnitude of J, and and In the region a 2 <a<2π-a 2 , (114) the intensity H e is also parallel to the axis y, and oriented in the opposite direction of H; as indicated in Fig.2. Its magnitude is given by ^ 0 (H,-H.) = AJ 0 , (1.15) which is independent of the orientation α,„ of the wedges. The constant K given by Eq. (1.13) depends on the angle of the wedges only, and it attains its maximum K=\ for 2 - λ =^-. (1.16) In the limit «-=f (1-17) the wedges are symmetric with respect to the x = 0 plane. The remanence is parallel to the axis and Eqs. (1.11) and (1.15) yield Under condition (1.7) the remanence J in the region y > 0 (in the wedge 12) is oriented at an angle θ 2 relative to the axis y 0, = α. +α 2 (1.19) and Eq. (1.20) corresponds to Ihe distribution of magnetization shown in Fig. 3. In the region (1.10) the components of intensity H; are: In the region (1.14) the intensity H e is also parallel to the axis x and oriented in the opposite direction of H and its magnitude is given by the same equation (1.15). Eqs. (1.7), (1.11), (1.19) and (1.21) show that a rotation of J by +π/2 results in a rotation of vectors H;, H e by -π/2 [1]. By virtue of results (1.11) and (1.21) any combination of remanences satisfying conditions (1.8), (1.9) and (1.19), (1.20) may be used to generate a uniform field of arbitrary orientation. The boundary conditions across the surfaces of the wedges are not violated by the addition of a magnetic field generated by other sources. In particular, in the ideal system of infinitely large dimensions r 0 j„ the addition of a uniform field intensity equal and opposite to either H e or Hi, confines the field to either region (1.10) or region (1.14) respectively. Either intensity H c or H x can be eliminated by assuming that either sides s 2 , s 3 (22 and 24) or sides Sj, S4 (18 and 24) are interfaces between the wedge material and a medium of infinity magnetic permeability 26. As one skilled in the art will recognize, the medium 26 need not directly abut the sides of the wedges, but may be positioned at a distance from the sides and still close the flux of the magnetic field generated by the wedges. For example, see Fig. 32A. With the condition H e = 0 the vector diagrams of Fig. 4A-4B show vectors B, H within the wedge 12 in the region^ > 0 in the two areas (1.8), (1.9) and (1.19), (1.20). In both cases vector B is parallel to interface s 2 . The sohd lines in Fig. 4A correspond to H e = 0 and the dashed lines correspond to Hi = 0. Consider the solution H, = 0 (1.22) in the limit r 0 j, - <»• Two particular cases are of interest. Assume first α 2 = (1.23) In this case the magnetic induction B within the wedges reduces to B = μH (1.24) i.e., in thejjv > 0 region the remanence is peφendicular to the h=l interface (side 18) and the intensity H within the wedges is oriented parallel to the x axis with a magnitude This is shown in the vector diagram of Fig. 4C derived from the diagram of Fig. 4A for the case (1.23). The second case is . =γ (1.26) which, by virtue of the vector diagram of Fig. 4A, yields B = 0. μ 0 H = -J (1.27) Thus in the case (1.26) the wedges do not carry a flux of the magnetic induction and the remanences of the wedges are peφendicular to the h=2, h=3 interfaces (sides 20 and 22). In the case of finite dimensions r 0h , the field generated by the wedges is no longer uniform. Fig. 1 shows the truncation of the wedges along segments (PιP 2 ) and (P3P4) symmetrically arranged with respect to the axis JC. If the wedge truncations are parallel to the remanence, σj,, h = 1...4, are the only charges induced by J and the field generated at each point is given by the sum of Eqs. (1.4). Consider first condition (1.6). In the limit r « r 0h , Eqs. 1.4 yield an intensity H; on the JC axis with components H^ O, H^ iH^ —KI^, (1.28) π r*0 S O where (Hjy) 0 is the magnitude of the intensity at JC - +0 (as x converges to 0 from the positive side, i.e., x > 0) and i.e., the endpoints of sides Sj, S2 are located on a circle of diameter^ and center at JC = 0, y = yo/2. At JC = 0 the intensity suffers a discontinuity given by Eq. (1.15). However, the derivative of Hj y with respect to JC dixly. J 0 > π ^ V) is continuous at JC = 0, and is independent of the orientation α,-, of the wedges 12 and 14. Length .yø is the normalization factor of the wedge dimensions. The values of H;, H« in the limit r -> 0 are plotted in Fig. 5 versus orientation α * ,. The plottings of Fig. 5 are computed within the range of distance r/r 0h = 10 '2 . A particular geometry of the wedge structure is 0: 2 = π/2, i.e., the case of S2 and S3 (sides 22 and 24) located on the plane JC = 0. By virtue of Eq. (1.18) the remanence is peφendicular to Sj (side 18) and foτy 0 = ∞ the induction B is everywhere parallel to the axis , y. Consider now condition (1.7). The truncation of the wedges along segments (P1P2) and (P3P4) parallel to the remanence yields the geometry shown in Fig 6. Again in the limit r « roj,, Eqs. (1.4) yield an intensity H on the axis JC of components (1.31) where Ηjx is the magnitude of the intensity at JC - • +0. The two radial dimensions r 0 ι, r 02 satisfy the equation -^- = -^ 2 — -= χ 0 ) (1.32) cos , cosα 2 where x 0 is the new normalization factor of the wedge dimensions. Again the derivative of the intensity with respect to JC is continuous at JC = 0, and its value is given by d 0 H te K ( — ~ — ) = - — (tanα, + tanα 2 ). (1.33) d(x l x 0 ) J 0 π Thus in the second case (1.7) the derivative of the intensity is a function of the orientation of the wedges. In the limit α 2 = π/2, Eq. (1.33) reduces to d(x *lr m ) .(!- J 0 -.) = _£ π .L 4-d(x±ly- 0 )A J 0 (1 .34) In the limit α 2 = π/2, the remanence is parallel to Sj, and by virtue of Eq. (1.7) no surface change is induced on interfaces Sj, s 4 whose radial dimensions become infimtely large. As a consequence the field generated by the wedges reduces to the field of a uniform charge distribution on the plane surface of dimension 2r<) 2 , in the JC = 0 plane. At a distance r » Γ OJ ,, the field generated by the wedges reduces to the field generated by a uniform dipole distribution located on the z axis with a dipole moment per unit length p oriented either in the .y or JC direction. In the case defined by condition (1.6) the dipole moment is p^ lA^ cosθ , (1.35) where Aj is the area of each wedge cross section. In the case defined by condition (1.7) the dipole moment p is By virtue of Eqs. (1.7) and (1.19), if Aj=A 2 , p \ and p 2 have the same magnitude independent of the orientation of J. 2. Generation of a Uniform Field 2.1 Transverse Field - Field parallel to the y axis The preceding section has presented the basic properties of the wedge structures of the present invention. As explained above, the magnetic field can be confined to either side of the wedge structure by short-circuiting the other side with a high magnetic permeabihty material. In a general case 0- 2 ≠ π/2, the field in the region 0-2 < α < 2π-ct2 can be eliminated by assuming that s 2 , s 3 are the interfaces between the wedges and a μ = 00 material. In the particular case 0. 2 = π/2, this transformation does not apply to condition (1.7), because the field generated by the wedges 12, 14 is peφendicular to the plane JC = 0 and the surface charge σ 2 , σ 3 induced by J on the plane x = 0 are canceled by the μ = 00 material. Thus the field generated by J would reduce to the field generated by the surface charges induced on a truncation of the wedges required to keep the radial dimensions of sj, S 4 finite. If the wedge structure is intended to generate a uniform field within a given distance T \ from the common edge, Eqs. (1.30) and (1.33) provide the order of magmtude of the dimensions of the wedges necessary to achieve an assigned degree of uniformity within such a distance. Assume the particular case 0- 2 = π/2 in the absence of the high magnetic permeabihty yoke 26. If δH; is the maximum value of the field distortion that can be tolerated within distance r 1} Eq. (1.30) indicates that the order of magnitude of y 0 must satisfy the condition Thusyo must be several orders of magnitude larger than the assigned dimension n in order to achieve a uniformity 5H/(Hi) 0 of the order of 10 " or better. The achievement of such a highly uniform field with practical wedge dimensions is outside the range of traditional shimnaing and requires a modification of geometry and magnetization of the structure. The technique described herein is based on a theory of field correction developed by the inventors, based on the elimination of the spatial harmonics of the field distortion in magnets designed to generate the uniform fields. To discuss the field correction technique, consider the two-dimensionaL cross- sectional geometry of Fig. 7 that corresponds to the α 2 = π/2 case where s 2 and s 3 (sides 22 and 24) coincide with the plane surface of a μ = ∞ material 26, and assume the distribution of the remanence defined by condition (1.6). The , y dimension of the μ = ∞ surface 26 (schematically represented by the heavy line in Fig. 7) is assumed to be large compared to the 2y 0 dimension of the wedge structure. As shown in Fig. 7, consider a cylindrical surface of radius r 0 coaxial with the axis z, and assume r 0 <yo- By virtue of the geometrical and magnetization symmetry within the cylinder of radius ro the potential Φ(r,α) of the field distortion generated by the finite dimension .yø of the wedges 12, 14 can be expanded in the series ir,a) = ∑g n (-f sm2na (2.2) «=1 r 0 where g„ are the amplitudes of the harmonics of the field distortion on the cylindrical surface r = ro. The harmonics can be compensated by assuming a surface distribution of dipole moment density p s (y) on the plane JC = 0, oriented peφendicular to the plane. Because of symmetry the magnitude of p s satisfies the condition The distribution of p s extends on the plane JC = 0, outside the interval 2y 0 . The potential Φ s (r, α) generated by p s in the region r ≤ r 0 can be expanded in a series like Eq. (2.2), and its value on the cylinder of radius r =y 0 is The harmonics of the field distortion are compensated if Φ, = -Φ(r 0 , ). (2.5) Thus the distribution of p s must satisfy the system of integral equations *j"o (2.6) (n = 1,2,3,....) The fower order harmonics are the dominant components of the field distortion. Thus the system of Eqs. (2.5) can be limited to the lower values of n, and the distribution of ρ s (y) can be confined to a finite interval^ -y t contained within 0 . The simplest example is the compensation of the fundamental harmonic n = 1, without introducing other harmonics, in which case p s (y) must satisfy Eqs. (2.6) where As a consequence p s (y) must oscillate between positive and negative values within the interval^ -y { . A solution of Eqs. (2.6) is obtained by dividing interval y e -y t in a number «ø of intervals, each having a uniform dipole moment density p s j,. Eqs. (2.6) reduce to R+l (-1) nπμ 0 • 1- ^OΓ -Λ-, )/>,, =#•. (2.8) Λ = 1,2,...« 0 where the intervals are numbered starting from the outside boundary .y =y e . The optimization of the division of_y e -yι yields the dimensions = -, exp , (h = 1,2,3,...) (2.9) Thus the total interval^ -y t satisfies the equation For no - oo the radial integral y e - y t would diverge. As a consequence a finite dimension^ - y t can be used only for the compensation of a finite number of harmonics. The structure defined by Eqs. (2.8) and (2.9) is an active filter 28 that can be implemented by inserting magnetized material between the μ = ∞ plane 26 located on the plane JC = 0 and a number 2n 0 of magnetically insulated μ = ∞ strips 28a, 28b, 28c of dimensions given by Eqs. 2.9 as indicated in Fig. 8. The insulated strips acquire the potentials Φ,„ = ^. (2.11) In a traditional magnet design an active filter consisting of the surface distribution of dipole movements p s can be transformed into a passive filter by transforming any equipotential surface that encloses all the dipoles into the surface of a μ = ∞ body. This is the principle of the design of the pole pieces of a traditional magnet, where the active filters are located on the interface between the magnetic material and the magnet cavity. In contrast, the filter structure 28 depicted by Eqs. (2.5) is located on the yoke 26 of the wedge magnet. The outermost strip of the filter (h=l) is the major component of the filter structure 28 that compensates for the rate of decrease of the magnitude of Hi as the distance r from point O increases. If Hi is oriented in the positive direction of the axis y, as indicated in the schematic of Fig. 8, the region^ > 0 is a region of negative values of Φ and, for h = 1 the potential Φ s j, of the outermost strip must satisfy the condition Φ . , < 0. (2.12) Thus one expects a positive sign of Φ ώ , and the sign of Φ srJ alternates from negative to positive from odd numbered to even numbered strips. A zero equipotential surface of the field generated by the wedges 12, 14 and the filter components 28 is found that encloses the yoke 26 and the even numbered strips. The odd numbered strips are outside the Φ = 0 surface. Thus the even numbered strips are the only components of the filter 28 that can be replaced by a modified profile of the yoke 26. The odd numbered strips and in particular the h = 1 outermost strip must be implemented either as active components or as a combination of active magnetic material and passive ferromagnetic materials. This property characterizes the difference between the filter structure 28 applied to the yoke 26 and the filter structure applied to the pole pieces of traditional magnets, where the fundamental harmonic of the field distortion is canceled by a filter component that can always be replaced by a modified geometry of the pole pieces. The geometry shown in Fig. 8 corresponds to a value K = 0.3 which yields the wedge angle a 2 - a 1 = τd2 - a 1 = 17.46° (2.13) As shown in the figure, assume an n 0 - 3 filter designed to cancel the first three harmonics, and consider the region contained within a cylinder of radius r 0 = .3y 0 . The coefficients g n of expansion (2.2) of the field generated by the wedges 12, 14 with a dimension of the yoke 26 large compared to yo, are shown in the table in Fig. 9, and the equipotential lines in the y > 0 region (with K = 0.3) are shown in Figs. 10A and 10B with (Fig. 10B) and without (Fig. 10A) the filter structure 28. The dimensions of the strips are chosen according to Eq. (2.9). The coefficients of the expansion of the total field generated by the wedges and the filter are shown in the table in Fig. 9, wherein the values provided under part or column 1 represent coefficients without the filter and those under part or column 2 represent coefficients with a n 0 = 3 filter structure. One observes that the elimination of the first three harmonics is achieved at the cost of an increase by orders of magnitude of the amplitude of the higher order harmonics. Also hsted in the table in Fig. 9 are the values of g„(yi/ro) 2n computed on a reference cylinder of radius >>/ = 0.35 r 0 . The effect of the increase of the amplitude of the higher order harmonics on the field within the region of interest is compensated by the factors (r/ro) 2n that decreases rapidly with n if the dimension of the imaging region is selected to be sufficiently smaller than . With the n 0 = 3 filter structure 28 depicted in Fig. 8 a field uniformity is achieved within a radial distance / • / « 0.2 y 0 . The cylindrical surface of radius r is indicated in the schematic of Fig. 7. Thus with the compensation of the first three harmonics the dimension of the wedge structure necessary to achieve a uniform field within a given dimension rj is reduced by orders of magmtude compared to the dimension of the uncorrected structure given by Eq. (2.1). The effect of an increasing number n 0 of filter components on the intensity close to the origin O is shown in Fig. 11 where the intensity is plotted versus JC for n 0 = 1, 2, 3 and compared to the intensity generated without filters. The potential of the three filter components is shown in Fig. 12. To reduce the magnitude of the sandwich required to generate Φ sl , the active outermost strip of the filter can be replaced by an hybrid structure with a.μ = ∞ component whose geometry is determined by an equipotential line of the combined fields of the wedge 12 and the filter 28. To illustrate the procedure, consider the case where the active filter reduces to a single element designed to cancel the gradient of the field at JC = 0. The equipotential lines are shown in Fig. 13, and the field intensity at.y = 0 is plotted versus x in Fig. 14. Shown in Fig. 15 is the transformation of the active filter element built into the yoke 26 into a sandwich of magnetic material between the yoke 26 and a.μ = ∞ component whose external profile approximates an equipotential line of the field of Fig. 14 outside the wedge 12. The interface between the μ - ∞ component of the hybrid filter and the magnetic material is a plane surface parallel to die JC = 0 plane, and the remanence of the magnetic material of the hybrid structure of Fig. 15 is parallel to J and its magnitude is J 2 ~ 3.2J 0 (2.15) Eq. (2.15) corresponds to what could be the remanence of a rare-earth hybrid filter element of a ferrite wedge magnet. Fig. 14 shows the intensity on the axis x that results from the transformation of the active filter into the hybrid structure of Fig. 15. 2.2 Longitudinal Field - Uniform field parallel to the x axis Consider now the wedge structure that satisfies condition (1.7) and assume the two-dimensional geometry Λ, = — - arcsin , α 2 = — , (2.16) where the remanences of the two wedges are parallel to sides Sj and s (18 and 20). Assume a finite dimension ro2 of sides s 2 , S 3 (22 and 24). By virtue of Eq. 1.32 one has roi = ro4 = 00. In the absence of any ferromagnetic material, the intensity H x on the axis JC within a distance |x| « r 02 is As with the transverse field described above, in the ideal case ro2 = ∞, either H^ or H^ are eliminated by assuming that either cti, 0 4 or 0, 2 , α 3 are the interfaces between the wedges and a μ = °o material and the intensity on either side of the magnet becomes r*0 Compare a wedge structure 2 that satisfies condition (1.7) with the schematic of Fig. 16, also intended to generate a field oriented in the direction of the axis JC. The rectangular block 2 of magnetic material is magnetized in the direction of the axis JC and the plane JC = 0 is assumed to be the surface 4 of a μ - ∞ material. The dimension of the block 2 on the y direction is 2. ø and the field is computed outside the block on the JC axis, infinitely close to the surface of the block 2, as a function of its length xr As shown in Fig. 17, the field intensity outside the block vanishes for JC 0 = 0 and its asymptotic value forjcø → oo is LtmiH,)^ = - (2.19) *a→°° ^ 2μ 0 Also plotted in Fig. 17 is function by virtue of (1.31) the asymptotic value of G is LimG(—) = +- (2.21) '0 π The equipotential lines of the field generated by the block 2 of magnetized material are shown in Fig. 18. Eq. 2.18 shows that for large values of K, the wedges generate an intensity larger than the maximum value (2.19) achieved with a single block. Assume the schematic of Fig. 19 where the sides α = αj, α = 0 are the interfaces with the μ = 00 material. In the limit ro 2 = r 03 = 00 and K = 1 the geometry of Fig. 19 transforms into the schematic of Fig. 20, where the magnetic material is magnetized in the direction of the axis x, and the μ = 00 material 26 is confined to the axis JC in the region JC > 0. With infinite dimensions of the magnetic material 26, the intensity is identically zero in the region JC > 0 and the flux of B = J = μo H e is parallel to the JC axis everywhere. A finite dimension of sides S 2 , S 3 (22, 24) in the schematic of Fig. 19 yields a non zero field within the magnetic material. Thus, because of the truncation of the wedges, the presence of the μ = ∞ material generates a singularity of the intensity at JC =y = 0 that cannot be compensated by a filter structure located outside a given distance from 0. Consider the schematic of Fig. 21 where the plane JC = 0 is the surface of the μ = ∞ material. As shown in the figure, the wedges are truncated on the plane JC = JC 0 , and as a consequence, the field is generated by the uniform surface charge induced by J on the surface of the truncation. The equipotential lines in the structure of Fig. 21 are shown in Fig. 23. The effect of the charges induced by J on the surface JC = JC 0 in reducing the intensity of the field on the axis JC is quite apparent in Fig. 23. The field is not uniform and the same approach for the elimination of the spatial harmonics described above can be followed in the schematic of Fig. 21 by developing a filter structure on the plane x = 0. Consider a cylindrical surface of radius r coaxial with die axis z, and assume that r 0 is smaller than jcø/(tan αj). The potential of the field distortion generated by the truncation of the wedges satisfies the condition Φ(y) = Φ(-.y), (2.22) and Φ=0 at x = 0. (2.23) Because of conditions (2.22), (2.23) the potential generated in the region |α| < 0 can be expanded in the series Assume that the filter structure extends on the plane JC = 0 over a finite interval beyond [y| > r 0 . Then, as with the transverse field described above, the computation of the filter structure is limited to the cancellation of the dominant, lower order harmomcs. Within the surface r = r 0 the value of the potential generated by the distribution of dipole moment p s on the JC = 0 plane is Φ < ^—-∑H)^-®^ ' 2 " +1 cos(2 M + l)α, (2.25) where p s (y) is the solution of the system of equation Again, Eq. (2.26) can be solved by replacing the continuous distribution of p s (g) with a stepwise distribution on a number of intervals, each carrying a uniform dipole moment density. Eq. (2.26) transforms to 2 ~^ ∑n+l 'S -2„-l _ v -2»-l >) Ό - « (2n + l)πμ 0 ° r h * " ' P * g " ' (2.27) w = l,2,...« 0 As an example, Fig. 22 shows the distribution of H x on the JC axis with and without a single strip filter designed to cancel the n = 1 harmonic (K = 0.5, Xoly 0 = 0.55735). The equipotential lines without and with the filter are shown in Figs. 23-24, respectively. In the schematic of Fig. 21, the field perturbation is a function of K only, and JC is the geometrical normalization factor of the magnetic structure. The field distortion in Fig. 21 coincides with the field generated by a strip of uniform charge density + K Jo of dimension 2jcø tanα, at a distance JC 0 from the μ = α> plane. The filter structures described herein compensate for the field distortion generated by the finite dimensions of the wedges 12, 14. Additional causes of field distortion are the tolerances of magnetization and fabrication that generate additional harmomcs, usually smaller than the harmonics due to the magnet geometry. One skilled in the art will recognize that the compensation of the additional harmonics due to the tolerances, or shimming of the magnet, may be accomplished by a correction of the strength of the individual elements of the filter structure. The possibility of performing the adjustment from the outside, through the yoke of the magnet, is another characteristic feature of the wedge structure of the present invention. 3. Defining the Imaging Region 3.1 Properties of the Region of Interest The imaging area for the magnetic structure is in the region of interest, where the field generated by the wedges attains its maximum value. The region of interest is close to the origin O of the magnetic structure on either concave or convex side of the wedge system. The plotting of the equipotential lines in Fig. 25 indicates that the concave side provides a region with higher field intensity and better field uniformity. An essential characteristic of the field configuration of Fig. 25 is the closed Φ = 0 equipotential line that passes through the saddle point S located on the JC axis. At point S the field intensity is zero, and the Φ = 0 line separates the region of interest from the external region where the field generated by the wedges is essentially the field of the linear dipole distribution given by Eq. (1.35). The plotting of H y versus JC at , y = 0 and its first and second derivatives with respect to JC are presented in Fig. 26 that corresponds to £ = 0.3, α, = 50' . (3.1) By virtue of Eq. (1.13) the angle of the wedge (angular width) is « 2 - α, « 17.46° . (3.2) Fig. 26 shows that the field generated by the truncated wedge 12 exhibits a sign reversal at the origin (In Fig. 26, K = 0.3, a t = 50°, and r 0h = 4, with plots restricted to the JC axis within the range ±10 r 0h , and the vertical axis normalized to Jo/μo). The field discontinuity at JC = 0 approximates well the ideal value of Eq. (1.15) KJo/μ 0 of the discontinuity ofthe infinite wedge. The derivatives of H y are continuous at x = 0; thus H y decreases at the same rate with the distance from 0 on both sides ofthe wedge system. The uniformity ofthe field generated by the wedges 12, 14 can be measured by selecting a point Po as the imaging center and computing the variation Δ|H| of the magnitude ofthe field |H| in the region surrounding Po. The imaging region is defined as the area where where C is an arbitrary constant. Assume for instance points Poi, Po e located on the axis JC at distances ±1 from 0 respectively, and consider the wedge geometry of Fig. 25 with radial dimensions r, = r 4 = 4.0 (3.4) Fig. 27A (with radial dimensions ofthe wedge rj = 4) shows the boundaries ofthe two regions surrounding Po,, Poe that correspond to the value ofthe constant C C = 0.02 (3.5) As shown in Fig. 27A the imaging regions are narrow strips that approximately follow the contours ofthe wedge boundaries. Obviously, the geometry ofthe regions depicted in Fig. 27 A is the result ofthe large gradient of |H| generated by the relatively small radial dimensions (3.4) ofthe wedges 12, 14. Consider point P OJ to be the center ofthe imaging region. The gradient of |H| at Poi can be compensated by a system ofthe two linear dipole distributions 30 of uniform moments per unit length p + , p. located at * = *o = . y = ±y k ( 3 - 6 ) and oriented in opposite directions along the axis JC, as indicated in Fig. 27B. If the dipole moments have equal magnitude p, they generate on the axis JC an intensity ofthe magnetic field of components Figure 27B illustrates the boundaries ofthe imaging regions for the wedge magnet with compensating dipoles 30. While the effect on the geometry ofthe imaging region about Poi is quite significant, it is negligible in the region around point Po e - At Po;, Η y vanishes and, as a consequence, the dipoles 30 do not affect the value ofthe field intensity generated by the wedges at Poi. The dipoles 30 generate a gradient dH y 2p ^-'-- i- (38) Eq. (3.8) provides the value ofthe dipole moment p per unit length necessary to compensate for the gradient ofthe field generated by the wedges 12, 14 at P OJ . The effect of the correction of the gradient at point Poi on the distribution of the field along the jc-axis is shown in Fig. 28, where the coordinate yj, has been selected as y h = tan a 2 * 2.41. (3.9) As described above, the field in either region (1.10) or (1.14) can be eliminated in practice by "short circuiting" one ofthe two regions by placing a medium of infinite magnetic permeabihty at either the interfaces h=2, h=3 (sides 22, 24) or the interfaces h=l, h=4 (sides 18, 20). Figure 29 shows the equipotential lines resulting from the short circuit of interfaces h=2, h=3, and the distribution ofthe field along the axis JC is shown in Fig. 30. In the limit |x| - 0 the field intensities on both sides ofthe wedges satisfy the conditions in agreement with Eq. (1.15). The short circuit of interfaces h=2, h=3 eliminates the saddle point S in Fig. 25. Both intensities Hi, H e on the axis JC are oriented in the same direction, and the small value of H e found in the JC < 0 region is the residual fringe field resulting from the finite radial dimension ofthe wedge system. 3.2 Closed and Open Magnetic Structures The wedge systems described above can be integrated either into a magnetic structure that confines the field within a closed cavity, or it can provide the basis for the design of a totally open magnet in which the region of interest is bound only by the wedge magnets. One embodiment of an integration of a pair of magnetic wedges 12, 14 in a closed magnetic structure is shown in Fig. 31 as a two dimensional magnet with a hexagonal cross section cavity 31. The heavy lines are part ofthe external yoke that short circuits the external interfaces ofthe wedges 12, 14. Thus, no field is found outside the external boundary ofthe structure of Fig. 31 and a uniform field of intensity H 0 oriented along the axis y is generated in die cavity 31. Triangular transition components 32 of remanence J t are inserted between the wedges 12, 14 and the rectangular components 34 ofthe magnetic structure. The transition components 32 carry the flux ofthe magnetic induction that flows in the cavity 31 of the region 2(JC Ø -X / ). The value ofthe induction B t within the transition components 32 is where K is related to the remanence Jø of the other components by the equation μ H o κ 0 = (3.12) Thus the remanence Ji is given by In the limit π α, (3.14) ~ 2 one has J i ~ J0 > (3.15) i.e., the transition components 32 are just an extension ofthe rectangular components 34 ofthe structure. As described herein, the use of a high permeabihty yoke 26 or layer along one set of sides of a pair of wedges 12, 14 provides the ability to confine the field to one side of the structure as a totally open magnet. As explained above, a high degree of field uniformity can be achieved through the use of a large magnet. Two approaches can be followed to improve the flat, open magnet performance while minimizing its size. First a partial termination ofthe wedges can be included in the form of magnetized elements 36, as indicated in the schematics of Figs. 32A and 32B. The partial termination in Fig. 32A follows the logic defined in the structure of Figs. 27A and 27B in the particular case ct 2 = π\2. In the partial termination of Fig. 32B the magnetic induction B t is zero in the transition triangles 38 which are magnetized with a remanence J; peφendicular to this external boundary as dictated by Eq. (2.23). As explained above, the short circuit ofthe side ofthe wedge opposite to the imaging region does not require that the yoke 26 be attached to the wedges 12, 14 interfaces. As a consequence, a space can be created between wedges 12, 14 and yoke 26 where a filter structure 28 as described herein can be inserted to eliminate a given number of spatial harmomcs ofthe field. Figs. 32A and 32B are embodiments of active partial terminations where magnetized material is used to reduce the effect ofthe termination of die wedges 12, 14. An example of passive partial termination is shown in Fig. 33 where ferromagnetic components 40 are performing the same function ofthe structures of Fig. 32A and 32B. The two ferromagnetic components 40 are attached to the wedges 12, 14 and the interface between them and the wedges 12, 14 follows an equipotential surface ofthe ideal field inside the wedges in the limit of perfect termination. 4. Multiple Wedge Structures The previous sections describe a magnetic structure containing a single pair of magnetic wedges. As described below, the intensity ofthe magnetic field generated by the structure can be increased by increasing the number ofthe wedge pairs in the structure. Yokeless magnets are structures of magnetized material where the field is confined within the magnet without the need of a ferromagnetic yoke. If a magnet designed to generate a given field within its cavity is enclosed in another yokeless magnet designed to generate a field ofthe same magnitude and orientation, the total field intensity within the cavity doubles, as long as the magnetic material has an ideal linear characteristic with zero susceptibility. As a consequence, a multiplicity of concentric yokeless magnets may be used to generate strong fields in excess ofthe remanence ofthe material. The same approach can be extended to a multiplicity of wedges having the axis z as a common edge. Fig. 34 shows the schematic of a structure of two wedge magnet pairs 12, 14, and 42, 44 designed to generate a transverse field (H α = 0), in the presence of a μ - oo ferromagnetic wedge 26 defined by the angle 2ct f . The pairs of wedges 12, 14 and 42, 44, have common edges 16 which are at the same location or are close to one another. Looking in the region^ > 0, the wedges 12, 42 have identical angular width α ι,2 - \,\ - α 2.2 ~~ α 2,ι = arcsin K, (4.1) where «2.ι = « (4.2) Corresponding wedges 14, 44 also have identical angular width. Because of he presence ofthe ferromagnetic yoke 26, the intensity ofthe field generated by each wedge magnet is zero in the region α > α 2 ^, and the total intensity ofthe field in the region |α| < i is Obviously a single wedge of angular width 0. 2 , 2 -ci-1,1 would generate a field -^y-*- = sin(α 2>2 - α, , ) < 2K (4.4) Assume a total angle cc t = a m2 - a lΛ , (4.5) divided in a number m of wedges of identical angular width fm. The structure of multiple wedges generates a field with a total value K t where K s is the value generated by a single wedge structure of angular width α,. In the limit m = 00, Eq. (4.6) reduces to K, a "t. (4.7) K s sinα, ' Eq. (4.7) shows that a multiple wedge structure is effective for large values of ct t , i.e. in magnets designed to generate a strong field approaching or even exceeding the value of the remanence. Assume a structure of m wedges of identical angular width and identical normalization factor y 0 . By virtue of (1.30), the total intensity and its derivative with respect to x at r = 0 increase proportionally with m. Consequently in the limit r/y 0 « 1 the total intensity ofthe field generated by the m wedges satisfies the condition The presence of the μ = ∞ wedge does not change the value of Hj y at x = 0 but it affects its derivative of H; y at JC = 0. Within the angle 2 ct f of the μ - ∞ wedge, the potential generated by the multiple wedge magnets in the proximity of the x =y = 0 can be expanded in the series where r 0 is an arbitrary distance and rlr 0 « 1. Coefficients c„ depend upon the geometry ofthe multiple wedge magnet. The.y component ofthe intensity is i.e. the derivative Hj y with respect to x at y = 0 is The harmonics n ≥ 2 of (4.11) cancel at JC = 0, independent of the value of α f . The fundamental harmonic (n = 1) of (4.11) has a singularity at JC = 0 for αf > π/2 and it cancels for «, <§• (4-12) Thus the gradient ofthe field at x = 0 given by Eq. (4.8) in the absence of the μ = oo wedge is eliminated by the presence of a concave ferromagnetic wedge that satisfies condition (4.12) as shown in Fig. 38. As an example, consider the structures ofthe two wedge magnets shown in Figs. 35A and 35B. Each wedge magnet is designed for K = 0.5 which corresponds to a π/6 wedge angle. A single wedge magnet of angle π/3 would generate a value of K s K, ~ 0.866. (4.13) The field generated by the structures of Figs. 35A and 35B corresponds to a total value of K t = 1, which represents a 15% gain compared to a single π/3 wedge magnet. Fig. 36 shows the equipotential lines ofthe field generated by the structure of Fig. 35 A where the angle ct f of Jhe μ = oo wedge 26 is π/2 (anάy 0 = 1.0). The corresponding distribution of Hj y on the axis x is shown in Fig. 37 which also shows the intensities generated on the axis JC by the individual wedge magnets ofthe structure of Fig. 35A, in the presence ofthe ferromagnetic yoke. In agreement with Eq. 4.11, both wedge structures exhibit the same value ofthe derivative ofthe intensity with respect to x at x = 0, which coincides with the value give by Eq. (3.8). Fig. 39 shows the equipotential lines generated by the structure of Fig. 35B where the angle α f of the μ = oo wedge is 2π/3. Because ofthe symmetry ofthe two wedge magnets relative to the plane x = 0, the surface charge induced on the interface between the wedges vanishes and no discontinuity ofthe intensity is formed on the interface, as shown by Fig. 36. The multiple wedge approach can be extended to structures designed to generate longitudinal fields (Hi y = 0). However, for finite dimensions r 0 h, the presence of the μ = oo yoke results in a significant difference ofthe field properties in the two types of magnets. In the structure of Fig. 40, if the wedges 12, 42 are truncated symmetrically with respect to the axis x, the potential generated by the truncation satisfies the condition Φ= 0 at , y = 0. As a consequence no singularity ofthe intensity occurs at r = 0. A different situation is found in structures designed to generate a longitudinal field for an angle αf of the μ = ∞ wedge 26 larger than π/2. In this case a singularity ofthe field at r = 0 is induced by the field generated by the truncation ofthe wedges, and compensation of the singularity can be achieved only at the cost of reducing the effectiveness ofthe multiple wedge approach. As a consequence, although both conditions (1.6) and (1.7) can be used in the design of a wedge magnet, the following section will be confined to the implementation of multiple wedge structures for transverse fields. 5. Termination Of Multiple Wedge Structures The properties ofthe multiple wedge structures presented in the previous section assume ideal geometries with wedge dimensions large compared to the dimension of a region of interest close to the common edge ofthe wedges. A way of implementing these properties in practical applications is to truncate the structures along equipotential surfaces ofthe field found in the ideal geometries, and to assume that these surfaces become the interfaces between wedges and high magnetic permeabihty materials. The introduction of these ferromagnetic materials provides an efficient way of integrating the multiple wedges in structures designed to satisfy the requirements of specific applications. As an example, Fig. 40 shows the transformation ofthe basic schematic the two wedge structure of Fig. 35B that results from an arbitrary equipotential line Φ = Φo becoming the interface ofthe wedges with a μ = ∞ material. Assume that the equipotential interface extends to infinity. The values of J, H, B in the components of remanences Jj and J 2 in the y > 0 region ofthe magnet are provided by the vector diagram of Fig. 41 which corresponds to a total value K t , = 1.0, (5.1) obtained with an angular width of 30° ofthe wedges 12, 42. Vector B is the magnetic induction generated in the non magnetic medium of Fig. 40 by each wedge ofthe structure. As the vector diagram in Fig. 41 shows, the magnetic induction is zero in the wedge of remanence J 2 that satisfies the condition J 2 = -μ 0 H 2 = -μ 0 H x (5.2) The termination ofthe wedge of remanence J 2 and the non magnetic region of magnetic induction 2-5 depends upon the requirements of each apphcation. For instance, Fig. 42 shows the apphcation ofthe structure of Fig. 40 in a magnet 50 designed around a prismatic cavity 52 of hexagonal cross-section. Components 54 of remanence J3 satisfy the condition B = 0, (5.3) and form the transition between the wedges 42, 44 of remanence J 2 and rectangular blocks 56 of remanence J 4 that close the flux ofthe magnetic induction through the external yoke 58 represented by the heavy line that encloses the magnet 50. In the schematic of Fig. 42, both remanences J 3 , J have the same magnitude of Ji and J 2 . The dimensions ofthe rectangular blocks 56 are chosen according to the optimum operating point ofthe demagnetization characteristics ofthe material of remanence J 4 , i.e., i.e., the magnetic induction B 3 is half the induction within the cavity 52. The two ferromagnetic components 60 derived from the equipotential lines of Fig. 40 channel the flux from the cavity into the two rectangular blocks 56 of remanence J 4 and are the equivalent ofthe pole pieces of a traditional magnet designed to focus the field generated by the magnetic material into a smaller region ofthe magnet cavity. A schematic of such a traditional magnet with the same dimensions ofthe magnetic material of remanence J 4 and the same geometry ofthe pole pieces is shown in Fig. 43. The equipotential lines of the field in Fig. 43 illustrate the differences between the magnet of Fig. 42 derived from the double wedge structure and a traditional magnet. The plotting ofthe field along axes x (curve (a)) and y (curve (b)) shown in Fig. 44 shows the non uniformity of the field in the gaps between the pole pieces ofthe traditional magnet. The field intensity in Fig. 44 is normalized to the value ofthe uniform intensity Ho = J o generated by the double wedge structure ofthe magnet of Fig. 42. The value Hy/Ho < 1 of the field at the center ofthe gap in Fig. 43 and the field non uniformity are the consequence ofthe fringe field outside the region of interest that characterizes traditional magnet designs. The termination of a wedge structure by means of ferromagnetic components confined by an equipotential surface ofthe field generated by the wedges can be extended to structures of multiple wedges with m > 2. An example is presented in Fig. 45 that shows the transformation of a triple wedge structure designed to generate a total value K t K t = 15, (5.5) The wedges- 12, 42, 62 have identical angular widths of 30°. The values of J, H, B in the three wedges 12, 42, 62 are provided by the vector diagrams of Figs. 46A, 46B, and 46C, respectively. Again, the three remanences J , J 2 , J 3 have the same magmtude J 0 and vector B is the magnetic induction generated by each wedge of the region x > 0 of the structure. As shown by the vector diagrams of Figs. 46A and 46B, the magnetic induction does not vanish in the region x < 0 of the structure. The induction B 2 = B in the wedge of remanence J 2 is oriented parallel to the axis y and the induction B3 is the wedge of remanence J 3 is oriented parallel to the yoke ofthe structure. The magnitude of -8 3 is equal to the magnitude of vector B. As indicated in Fig. 45 the equipotential line transformed into the boundary between the wedges 12, 42 of remanences Ji, J 2 and the μ = oo material is oriented at an angle O e j with respect to the axis y given by nα,, = -j= (5.6) i.e., O e i = 49.1°. The orientation α e3 ofthe equipotential lines in the medium of remanence J 3 is parallel to the external yoke, i.e., α e3 = 60°. The integration ofthe triple wedge structure into a magnet designed around the same prismatic cavity of Fig. 42 for the value of K t given by Eq. 5.5 is shown in structure of Fig. 47 where the remanences of all the components have the same magnitude Jo. The transition between the wedge 62 of remanence J 3 and a rectangular block 66 of remanence J5 is accomplished by a rectangular wedge 64 of remanence J 6 and the triangular region 68 of non magnetic material where the intensity, oriented in the negative direction ofthe axis , y, is s -i J '-i' (57) the magnetic reduction _9 4 in the triangle 64 of remanence J is B 4 = -B. (5.8) As in Fig. 42, the structure of Fig. 47 is enclosed by the external yoke 58 that channels the total flux ofthe induction carried by the rectangular component 66 of remanence 3$. Fig. 48 shows the schematic of a traditional magnet where the geometry ofthe pole pieces is identical to that ofthe ferromagnetic components ofthe structure of Fig. 44. The equipotential lines in Fig. 48 ofthe field generated by the same rectangular components of magnetic material of remanence J 5 , again illustrate the loss ofthe focusing effect ofthe pole pieces due to the fringe field in the traditional magnet. The plotting ofthe field along the x (curve (a)) and , y (curve (b)) axes ofthe magnet of Fig. 48 is shown in Fig. 49, where the intensity is normalized to the value Ho = 3J-J2μo generated by the triple wedge structure of Fig. 45. Applications ofthe multiple wedge structures to generate high fields range from closed permanent magnets to fully open magnets like the schematics illustrated in Figs. 35A and 35B. As described above, a single wedge magnet, in the presence of a μ = 00 plate, is limited to the generation of a field strength equal to the remanences of the magnetic material. This upper hmit (K = 1) is achieved with an angular width π/2 ofthe magnet wedge. The ability of superimposing linearly the field generated by single wedges of magnetic material with zero magnetic susceptibility, results in the structures of multiple wedges described herein that remove the K = 1 limit and enable the generation of fields exceeding the remanence. This interesting property ofthe magnets described above can be implemented in practical structures of rare-earth material capable of generating fields in the 1 - 2 Tesla range, far above the limits of traditional permanent magnet designs. The magnitude ofthe intensity is controlled by the angle ofthe wedges and by the selected magnetic material. Some ofthe numerical calculations presented herein have been performed at K = 0.3, which would correspond to a field of approximately 0.35 Tesla with an Nd.Fe.B alloy. The orientation ofthe field in the region of interest is controlled by the orientation ofthe remanence ofthe wedges, as described above. Although, in principle, wedge magnets can be designed to generate an arbitrary orientation, the orientation parallel to the surface ofthe yoke is the preferred solution that eliminates field singularities and makes it possible to achieve field strengths ofthe order ofthe remanence ofthe material. The high fields generated by these structures extend the use of permanent magnets to field levels normally achieved with superconductive magnets. The use of rare earth materials in the types of structures described herein makes permanent magnet technology apphcable to field levels above one Tesla. Conversely the same structures built with ferrite materials may replace expensive rare earth magnets to generate fields in the 0.4 T range. The generation of a uniform field in the open structures described herein is achieved at the cost of a lack of field confinements. As discussed above, asymptotically the field generated by single wedge magnets in the absence of ferromagnetic components behaves like the field of a dipole, irrespective ofthe orientation ofthe magnetization. The presence of a μ = oo flat plate that supports the wedges eliminates the dipole moment ofthe structures of some embodiments as described above. Likewise the dipole moment is canceled in single or multiple wedge structures without the μ = ∞ plate, as long as the structures designed to generate a field parallel to the x = 0 plane are symmetric with respect to the x = 0 plane. This property is of particular importance when implemented in a three-dimensional structure, described below, in which case the far field reduces to the field of a quadrupole, whose magmtude decreases with the fourth power ofthe distance from the center ofthe magnet, resulting in a substantial improvement ofthe field confinement. 6. Three-dimensional Wedge Structure Assume a wedge structure limited to the region between the two planes z = ± zo with the distribution of remanence defined by condition (1.6). The angle α2 = {π/2} is selected for the wedge geometry in Fig. 50 and the plane x = 0 is assumed to be the surface of a μ = oo material 26. The two-dimensional formulation of the filter structure described above can be extended to the structure in Fig. 50 by designing a filter that compensates for the spherical harmonics ofthe field distortion computed in a spherical frame of reference p, θ, ψ, where the origin O is selected on the axis z at the center of interval 2zo, p is the distance of a point P from O, θ is the angle between p and the axis x and ψ is the angle between the plane y = 0 and the plane formed by p and the axis x. On a sphere of radius p 0 and center O, the potential generated by the wedges 12, 14 can be expanded in the series <W.ψ) = ∑g u Pι (ξ)anJΨ, (6-1) where are the Legendre's associated functions ofthe first kind, and ξ = cosø, (6.2) coefficients gij are the amplitudes ofthe harmomcs ofthe field distortion. Φ(θ,ψ) satisfies the condition Because of Eqs. (6.3) and (1.6), the symmetry conditions ofthe structure of Fig. 50 limit the values of IJ to l= 2n, j = 2m+ ϊ. (6.4) As described above, the compensation ofthe harmonics is achieved by means of a surface dipole moment density p s distributed on the μ = ∞, x = 0 plane outside a circle of radius po. p s is oriented in the direction ofthe axis x and it generates a potential Φ s on the surface p= po where r is the radial distance of a point ofthe plane x = 0 from O, and (P/)O is the derivative of P/ with respect to ξ at ξ = 0 which is given by the equation p s (r, ψ) is zero within the circle of radius p 0 . The harmomcs ofthe field distortion are canceled if the distribution of p s (r,ψ) satisfies the integral equations (n= 1,2,3,... , w= 0,1,2,...). Again, if the compensation ofthe field distortion is limited to the lower order harmomcs, the radial distribution ofp s (r, ψ) can be confined to a finite interval r e - r t outside the circle of radius p 0 . A solution of Eqs. 6.7 is obtained by dividing the area between the circles of radii r lt r 2 into a number no of concentric rings, and by dividing the rings into a number mo of sectors, each having a uniform dipole moment density / *. As in the case ofthe two-dimensional filter described above, the optimization ofthe distribution of p(r, ψ) yields the same Eq. (3.9) for the ring dimensions. The rings are divided in sectors of equal angular width, and their number mo is chosen to be large enough to provide the required approximation ofthe angular distribution of p(r, ψ) without introducing additional higher order harmonics. As an example, consider a one ring filter designed to cancel the single harmonic / = 2, / = 1, whose potential within the sphere of radius po is The field defined by potential Φ 2 1 has ay component ofthe intensity that increases linearly with x , £2.1 H y.2.\ = ~ 3" (6.9) Po Thus, the fundamental harmonic (6.8) is caused by the gradient of H y in the y = 0 plane, as in the case ofthe two-dimensional wedge structure analyzed above. Assume that the single filter ring is confined between the concentric circles of radii r, , r e . The distribution of dipole moment/?, on the ring that cancels the gradient of H y due to harmonic (6.8) is p s = p 0 sin ψ (6.10) independent ofthe radial coordinate. By virtue of Eqs. (6.7), (6.8), the amplitude /? in Eq. 6.10 is given by In general, the basic properties ofthe two-dimensional filters described above, and in particular, the transformation of active filter elements into either passive or hybrid components applies to a multiple ring filter built into the μ = ∞ plane that supports the three-dimensional structure of Fig. 50. While the invention has been described and illustrated in connection with preferred embodiments, many variations and modifications as will be evident to those skilled in this art may be made without departing from the spirit and scope ofthe invention, and the invention is thus not to be limited to the precise details of methodology or construction set forth above as such variations and modification are intended to be included within the scope of die invention.

Claims: WHAT IS CLAIMED IS: 1. A magnetic structure generating a substantially uniform magnetic field within a region of interest comprising: at least one pair of magnetic wedges of substantially identical shape arranged symmetrically about a first axis and abutting one another along a common edge, each magnetic wedge having a first side and a second side which terminate at or near the common edge; a first wedge of each pair of wedges being positioned to define a first angle between the first side and the first axis and a second angle between the second side and the first axis and being uniformly magnetized in a direction relative to a second axis which is a function ofthe sum ofthe first and second angles, the second axis being peφendicular to the first axis; a second wedge of each pair of wedges being magnetized such that the absolute values of components ofthe magnetization along the first and second axes are equal to components ofthe magnetization ofthe first wedge along the first and second axes, respectively, and such that one ofthe magnetization components ofthe second wedge is opposite to the respective magnetization component ofthe first wedge; and wherein the region of interest containing the substantially uniform field is between the first sides ofthe pair of wedges, the second sides ofthe pair of wedges, or both. 2. The magnetic structure of claim 1 wherein a third side of each wedge of a pair of wedges away from the common edge is parallel to the direction in which the wedge is magnetized. 3. The magnetic structure of claim 1 wherein the first wedge is magnetized at an angle relative to the second axis equal to the sum ofthe first and second angles minus 90°. 4. The magnetic structure of claim 3 wherein the second wedge is magnetized such that the magnetization component along the second axis is equal to the magnetization component ofthe first wedge along the second axis and the magnetization component along the first axis is equal and opposite to the magnetization component of the first wedge along the first axis, whereby the magnetic field is generally parallel to the second axis. 5. The magnetic structure of claim 1 wherein the first wedge is magnetized at an angle relative to the second axis equal to the sum ofthe first and second angles. 6. The magnetic structure of claim 5 wherein the second wedge is magnetized such that the magnetization component along the first axis is equal to the magnetization component ofthe first wedge along the first axis and the magnetization component along the second axis is equal and opposite to the magnetization component ofthe first wedge along the second axis, whereby the magnetic field is generally parallel to the first axis. 7. The magnetic structure of claim 1 comprising a high magnetic permeability yoke positioned between either the first or second sides ofthe at least one pair of wedges and an exterior region, whereby magnetic flux is substantially zero in the exterior region. 8. The magnetic structure of claim 7 wherein the yoke extends along and abuts the first or second sides ofthe at least one pair of wedges. 9. The magnetic structure of claim 7 wherein the yoke extends along but does not abut the first or second sides ofthe at least one pair of wedges. 10. The magnetic structure of claim 1 wherein the first and second wedges are triangular in cross section. 11. The magnetic structure of claim 1 wherein the at least one pair of wedges comprises a plurality of pairs of magnetized wedges, each pair being arranged symmetrically about the first axis. 12. The magnetic structure of claim 11 wherein the common edges ofthe pairs of wedges are in an adjacent relationship. 13. The magnetic structure of claim 11 wherein the angular widths between the first and second sides ofthe plurality of wedges are substantially equal. 14. The magnetic structure of claim 11 wherein the wedges in one pair ofthe plurality of pairs of wedges have a shape which is substantially different than the shape ofthe wedges in another pair ofthe plurality of pairs of wedges. 15. The magnetic structure of claim 11 wherein the wedges in one pair ofthe plurality of pairs of wedges have a shape which is substantially identical to the shape of the wedges in another pair ofthe plurality of pairs of wedges. 16. The magnetic structure of claim 11 wherein the first sides of at least one pair of wedges abuts the second sides of another pair of wedges. 17. The magnetic structure of claim 11 comprising a high magnetic permeability yoke positioned between corresponding sides of one ofthe pairs of wedges and an exterior region, which exterior region is on the opposite side ofthe magnetic structure as the region of interest. 18. The magnetic structure of claim 11 wherein the plurality of pairs of magnetized wedges comprises at least three pairs of magnetic wedges. 19. The magnetic structure of claim 1 wherein the magnetic structure is open at at least one side thereof. 20. The magnetic structure of claim 19 wherein the region of interest is unbounded by any magnetic element other than the wedges. 21. The magnetic structure of claim 1 comprising one or more additional magnetic elements connected to the at least one pair of wedges, the one or more additional magnetic elements defining with the wedges a cavity in which the region of interest is situated. 22. The magnetic structure of claim 1 comprising compensation means for compensating for distortions in the substantially uniform magnetic field. 23. The magnetic structure of claim 22 wherein the compensation means comprises at least one filter structure positioned adjacent to either the first or second sides of a pair of wedges. 24. The magnetic structure of claim 23 comprising a high magnetic permeability yoke positioned between either the first or second sides of a pair of wedges and an exterior region, the filter structure or structures being positioned between the yoke and the pair of wedges. 25. The magnetic structure of claim 22 wherein the compensation means comprises a pair of dipole distributions of uniform moments positioned adjacent to either the first or second sides of a pair of wedges, the dipole distributions being positioned symmetrically about the first axis. 26. The magnetic structure of claim 25 wherein the dipole distributions are oriented in opposite directions along the first axis and compensate for distortions in any component ofthe magnetic field along the second axis. 27. The magnetic structure of claim 25 wherein the dipole distributions are oriented in the same direction along the first axis and compensate for distortions in any component ofthe magnetic field along the first axis. 28. The magnetic structure of claim 22 wherein the compensation means comprises at least one pair of elements positioned at or near ends of first or second sides of a pair of wedges. 29. The magnetic structure of claim 28 wherein the at least one pair of elements comprises a pair of substantially identical ferromagnetic pole pieces positioned symmetrically about the first axis. 30. The magnetic structure of claim 28 wherein the pair of elements comprises at least one pair of substantially identical magnetized elements positioned symmetrically about the first axis. 31. A magnetic structure generating a substantially uniform magnetic field within a region of interest comprising: a pair of magnetic wedges of substantially identical shape positioned within a three-dimensional rectangular coordinate system consisting of peφendicular x, y, and z axes so that the two wedges are arranged symmetrically on either side ofthe x axis and abut one another along a common edge extending along the z axis, each magnetic wedge having a first side and a second side which terminate at or near the common edge; the first wedge ofthe pair of wedges being positioned to define a first angle aj between the first side and the x axis and a second angle α 2 between the second side and the x axis and being uniformly magnetized at an angle Qι relative to the y axis wherein π θθ i = a +a 2 - — ; the second wedge ofthe pair of wedges being magnetized such that components of the magnetization J along the x and y axes are related to x and.y axis components ofthe magnetization in the first wedge according to the formulas Jλ-y) = -J x (y), J y (-y) = Jy(y); and wherein the substantially uniform field is generally parallel to the y axis. 32. The magnetic structure of claim 31 further comprising a high magnetic permeability yoke positioned between either the first or second sides of the pair of wedges and an exterior region, whereby magnetic flux is substantially zero in the exterior region. 33. The magnetic structure of claim 30 wherein a third side of each wedge ofthe pair of wedges away from the common edge is parallel to the direction in which the wedge is magnetized. 34. The magnetic structure of claim 31 further comprising at least one additional pair of magnetized wedges, each additional pair being arranged symmetrically about the x axis, having an additional common edge which is adjacent to or coincident with the common edge ofthe pair of wedges, and being magnetized according to the same conditions as the pair of wedges. 35. The magnetic structure of claim 34 wherein the angular width between first and second sides of each wedge in the additional pair of wedges is substantially equal to the angular width between the first and second sides ofthe pair of wedges. 36. The magnetic structure of claim 34 wherein each wedge in the additional pair of wedges abuts the corresponding wedge in the pair of wedges. 37. A magnetic structure generating a substantially uniform magnetic field within a region of interest comprising: a pair of magnetic wedges of substantially identical shape positioned within a three-dimensional rectangular coordinate system consisting of peφendicular x, y, and z axes so that the two wedges are arranged symmetrically on either side ofthe x axis and abut one another along a common edge extending along the z axis, each magnetic wedge having a first side and a second side which terminate at or near the common edge; the first wedge ofthe pair of wedges being positioned to define a first angle aj between the first side and the x axis and a second angle α 2 between the second side and the x axis and being uniformly magnetized at an angle θi relative to the y axis wherein θθ x = a^ + a 2 ; the second wedge of the pair of wedges being magnetized such that components of the magnetization J along the x and " axes are related to x may axis components of the magnetization in the first wedge according to the formulas JΛ-y) = J x (y), Jy(-y) = -J y (y) ; and wherein the substantially uniform field is generally parallel to the x axis. 38. A magnetic structure generating a substantially uniform magnetic field within a region of interest comprising: a pair of magnetic wedges of substantially identical shape positioned within a three-dimensional rectangular coordinate system consisting of peφendicular x, y, and z axes so that the two wedges are arranged symmetrically on either side of a plane formed by the x and z axes and abut one another along a common edge extending along the z axis, each magnetic wedge having a first side and a second side which terminate at or near the common edge; a first wedge of each pair of wedges being positioned to define a first angle between the first side and the plane and a second angle between the second side and the plane and being uniformly magnetized in a direction relative to the y axis which is a function ofthe sum ofthe first and second angles; a second wedge of each pair of wedges being magnetized such that the absolute values of components ofthe magnetization along the x and y axes are equal to components ofthe magnetization ofthe first wedge along the x and y axes, respectively, and such that one ofthe magnetization components ofthe second wedge is opposite to the respective magnetization component ofthe first wedge; and wherein the region of interest containing the substantially uniform field is between the first sides ofthe pair of wedges, the second sides ofthe pair of wedges, or both. 39. The magnetic structure of claim 38 further comprising a high magnetic permeability yoke positioned between either the first or second sides ofthe pair of wedges and an exterior region, whereby magnetic flux is substantially zero in the exterior region. 40. The magnetic structure of claim 38 wherein a third side of each wedge ofthe pair of wedges away from the common edge is parallel to the direction in which the wedge is magnetized. 41. The magnetic structure of claim 38 further comprising at least one additional pair of magnetized wedges, each additional pair being arranged symmetrically about the plane, having an additional common edge which is adjacent to or coincident with the common edge ofthe pair of wedges, and being magnetized according to the same conditions as the pair of wedges. 42. The magnetic structure of claim 41 wherein the angular width between first and second sides of each wedge in the additional pair of wedges is substantially equal to the angular width between the first and second sides ofthe pair of wedges. 43. The magnetic structure of claim 41 wherein each wedge in the additional pair of wedges abuts the corresponding wedge in the pair of wedges.

Inventors: ABELE MANLIO G, JENSEN JENS, RUSINEK HENRY, ABELE, MANLIO, G., JENSEN, JENS, RUSINEK, HENRY
Applicants: UNIV NEW YORK, NEW YORK UNIVERSITY
Application year: 1997
Main classification: H01F7
IPC codes: A61B5, G01R33, H01F7