WO1998022957A1 - Apparatus for generating uniform magnetic fields with magnetic wedges - Google Patents

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 (n₀ = 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 P₀ι 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+ μ₀H, (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,, , α₃ = 2π - a₂, a₄ =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, = J₀ cos(α, - θ), σ₂ = J₀ cos(α₂ - θ), σ_i = -σ₂, 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

2πμ₀

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 r₀j, -> «>. The two wedges 12 and 14 generate a uniform field if the

charges σj, satisfy the condition

∑σ_hτ_h = (1.5)

A=l Eq. 1.5 has two particular solutions:

σ₄ = -σ, , σ₃ = -σ₂ , σ, sin α, + σ₂ sin ₂ = 0 (1-6)

and

σ₄ = σ, , σ₃ = σ₂ , σ, cosα, + σ₂ cosα₂ = 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

a₂<a<2π-a₂, (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.) = AJ₀, (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 θ₂ relative to the axis y 0, = α. +α₂ (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 r₀j„ 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 s₂, s₃ (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 s₂. 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 r₀j, - <»• Two particular cases are of interest. Assume first

α₂ = (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. μ₀H = -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 (PιP₂) 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) ^πr*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. J₀ > 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 foτy₀= ∞ 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 r₀ι, r₀₂

satisfy the equation

-^- = -^²— -= χ_{0 )} (1.32) cos , cosα₂

where x₀ 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 ₀H_te K

( — ^~ — ) = - — (tanα, + tanα₂). (1.33) d(x l x₀) J₀ 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 α₂ = π/2, Eq. (1.33) reduces to

d(x *lr_m) .(!- J₀-.) ₌ _£ π .L 4-d(x±ly-₀)A J_{0 (1}.34)

In the limit α₂ = π/2, the remanence is parallel to Sj, and by virtue of Eq. (1.7) no

surface change is induced on interfaces Sj, s₄ 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=A₂, p_\ and p₂ 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 s₂, s₃ 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 σ₂, 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, S₄ 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 y₀

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 α₂ = π/2 case where s₂ and s₃ (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 2y₀ dimension of the wedge structure. As shown in Fig. 7, consider a

cylindrical surface of radius r₀ coaxial with the axis z, and assume r₀ <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 ^r0 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 2y₀. The potential

Φ_s(r, α) generated by p_s in the region r ≤ r₀ can be expanded in a series like Eq. (2.2),

and its value on the cylinder of radius r =y₀ is

The harmonics of the field distortion are compensated if

Φ, = -Φ(r₀, ). (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_sj,. Eqs. (2.6)

reduce to

R+l

(-1) nπμ₀ ^•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 2n₀ 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_sj, 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

a₂ - a₁ = τd2 - a₁ = 17.46° (2.13)

As shown in the figure, assume an n₀ - 3 filter designed to cancel the first three

harmonics, and consider the region contained within a cylinder of radius r₀ = .3y₀. 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 n₀ = 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 r₀.

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 n₀ = 3 filter structure 28 depicted in Fig. 8 a field uniformity

is achieved within a radial distance /^• _/ « 0.2 y₀. 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 n₀ of filter components on the intensity close to

the origin O is shown in Fig. 11 where the intensity is plotted versus JC for n₀ = 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

J₂ ~ 3.2J₀ (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 , α₂ = — , (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 s₂, S₃ (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| « r₀₂ is

As with the transverse field described above, in the ideal case ro2 = ∞, either H^ or H^

are eliminated by assuming that either cti, 0₄ or 0,₂, α₃ 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 JC₀ = 0 and its asymptotic value

forjcø → oo is

LtmiH,)^ = - (2.19)

*a→°° ^ 2μ₀

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 ro₂ = r₀₃ = 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 S₂, S₃ (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 = JC₀, 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 = JC₀ 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

r₀ 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| > r₀. 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 = r₀ 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)πμ₀ ° 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, Xoly₀ = 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 JC₀ 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/μ₀ 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, = r₄ = 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 H₀ 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 ^Ho κ₀ = (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 ct₂ =

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

^αι,2 - \,\ - ^α2.2 ^{~~ 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 > α₂^, 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 y₀. 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/y₀ « 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 r₀ is an arbitrary distance and rlr₀ « 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 (anάy₀ = 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 r₀h, 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 J₂ 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 J₂ that satisfies the condition J₂ = -μ₀H₂ = -μ₀H_x (5.2)

The termination ofthe wedge of remanence J₂ 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 J₂ and rectangular

blocks 56 of remanence J₄ 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 J₃, J have the same magnitude of Ji and J₂.

The dimensions ofthe rectangular blocks 56 are chosen according to the optimum

operating point ofthe demagnetization characteristics ofthe material of remanence J₄,

i.e.,

i.e., the magnetic induction B₃ 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 J₄ 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 J₄

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 , J₂, J₃ have the same magmtude J₀

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 B₂ = B in the wedge of remanence J₂ is oriented parallel to the axis y and the induction B3 is the

wedge of remanence J₃ is oriented parallel to the yoke ofthe structure. The magnitude

of -8₃ 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, J₂ and the

μ = oo material is oriented at an angle O_ej with respect to the axis y given by

nα,, = -j= (5.6)

i.e., O_ei = 49.1°. The orientation α_e3 ofthe equipotential lines in the medium of

remanence J₃ 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 J₃ and a rectangular block 66

of remanence J5 is accomplished by a rectangular wedge 64 of remanence J₆ 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 _9₄ in the triangle 64 of remanence J is

B₄ = -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 J₅, 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 p₀ and center O, the potential generated by the wedges 12, 14 can be

expanded in the series

<W.ψ) = ∑g_uPι (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 p₀. 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 p₀. A solution of Eqs. 6.7 is obtained by dividing the area between the circles of radii r_lt r₂ 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

^Hy.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 = p₀ 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 α₂ 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 +a₂ - — ;

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 α₂ 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^ + a₂ ;

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.