WO1988008126A1

WO1988008126A1 - Magnetic field generating apparatus

Info

Publication number: WO1988008126A1
Application number: PCT/GB1988/000286
Authority: WO
Inventors: Ian Leitch Mcdougall; Ian Jesse Brown; Michael Begg
Original assignee: Oxford Magnet Technology Limited
Priority date: 1987-04-15
Filing date: 1988-04-14
Publication date: 1988-10-20
Also published as: EP0345300A1; JPH02502967A

Abstract

Magnetic field generating apparatus comprises a magnetic field generator (1, 2) for generating a magnetic field in a working volume (8) in which a component of the magnetic field (Hz) along an axis (4) and with respect to an origin defined by the magnetic field generator is defined in terms of a field expansion formula (I), where r is the radius of the sphere of interest, rO is the radius of a reference sphere upon which the components of the field expansion (a) are known, F) are spherical polar coordinates defining the position of the point of interest upon the surface of the sphere, (b) are Associated Legendre polynomials of order n and degree m, and (c) are the values of the field distortions as measured on a reference sphere of radius rO. A magnetic field modification system (6 - 7) comprising a number of ferromagnetic members is arranged to cancel at least one of the non-zero order terms and at least one of the non-zero degree order terms in the field expansion formula.

Description

MAGNETIC FIELD GENERATING APPARATUS The invention relates to magnetic field generating apparatus. Magnetic field generating apparatus is used in a wide variety of applications including, for example, magnetic resonance imaging (MRI), magnetic resonance spectroscopy and the like. In these applications it is important to generate within a predetermined region a magnetic field of high homogeneity. That is there is substantially no variation in field strength throughout the region. The degree of homogeneity which is generally acceptable is a variation in field strength of less than 15ppm within a sphere of diameter 50cm. In the past, magnetic field generating assemblies have been constructed from sets of electrical coils, particularly superconductive coils. These assemblies have generally generated fields of satisfactory homogeneity but require a considerable volume of space due to the physical size of the coils required and, in the case of superconductive coils the need to position the coils within a cryostat.

In some cases, even with relatively large structures, unsatisfactory field variations still occur. In the past, correction of these distortions has been achieved using small resistive electrical coils (or shimming coils) positioned around the bore of the apparatus. This has a number of disadvantages including the additional power requirement. In accordance with the present invention, magnetic field generating apparatus comprises magnetic field generating means for generating a magnetic field in a working volume; and a magnetic field modification system comprising one or more ferromagnetic members arranged about an axis extending through the working volume to reduce axisymmetric and non-axisymmetric variations in the magnetic field such that the homogeneity of the magnetic field within the working volume is improved.

It has been found that it is possible to integrate passive ferromagnetic members into an active magnetic field generator in such a way that not only axisymmetric distortions but also non-axisymmetric distortions in the field are reduced or even cancelled.

In one arrangement, the magnetic field generating means generates a substantially non-homogeneous magnetic field within the working volume, the magnetic field modification system cooperating with the magnetic field generating means to modify the magnetic field so that the magnetic field within the working volume is substantially homogeneous.

We have found that this arrangement leads not only to an increase in the degree of homogeneity of the field within the predetermined volume but also achieves a significant decrease in the amount of space required for the apparatus for a negligible increase in weight.

It should be understood that by a "non-homogeneous magnetic field" we mean a field in which the field strength varies by more than just a few ppm. For example the variation could be in the order of a few percent. Thus, in this arrangement the magnetic field modification system does not simply correct for small tolerance problems in the field generated by the magnetic field generating means but makes a substantial contribution to the field homogeneity. Indeed the magnetic field generating apparatus could be designed initially .by considering the magnetic field generating means (e.g. electrical coils) and the modification system in an integrated manner.

The ferromagnetic members are preferably formed from grain orientated ferro-magnetic material since the grain orientation reduces problems due to rotation of the magnetic field vector within the elements.

Typically, the magnetic field generating means will generate a solenoidal field defining an axis coincident with the axis of the magnetic field modification system.

An example of this is apparatus suitable for MRI in which the axis would lie in the patient direction.

A non-axisymmetric modification system has the advantage of reducing fabrication costs due to the need for less stringent manufacturing tolerances.

Typically, the elements will comprise flat sheets of magnetisable material which are rolled onto a cylindrical surface.

The invention is also applicable to the correction of relatively small distortions in the magnetic field comparable with those corrected by conventional shimming techniques. In these arrangements, the magnetic field modification system preferable comprises an array of magnetisable elements positioned around the working volume.

We have found that it is possible to devise a magnetic field modification system, the parameters for which can be relatively easily determined using a mini or possibly microcomputer. This reduces the cost of the assembly considerably. it also enables many more terms in the field expansion to be compensated for.

Typically, the magnetisable elements will be positioned within the magnetic field generating means.

Preferably, the magnetic field modification system comprises a non-magnetic support on which the magnetisable elements are mounted.

For example, the support could comprise a number of sets of locations around the working volume. Removable magnetisable elements can then be positioned in particular ones of the locations so as to effect minor modifications to the magnetic field.

Typically, the sets, of locations will each be arranged in the form of a linear array. Typically, the magnetic field generating assembly defines a cylindrical or tubular bore and the magnetisable elements are positioned on a substantially common radius around the axis of the bore.

This arrangement is particularly suitable for use with conventional magnets since the magnetic field modification system can be mounted as a unit into the bore of the already existing magnet.

In certain applications problems could arise from any non-uniformity in the direction of the magnetisation (vector quantity) in the magnetisable elements. One solution to this problem is to provide each magnetisable element as a grain oriented laminated structure. The benefit of using oriented laminations is that the magnetic suseptability is anisotropic with a pronounced favourable direction of magnetisation in the "rolled direction". This affect, combined with the natural tendency of a magnetic material to magnetise in the direction of predominant geometric aspect, leads to a far more uniform direction of magnetisation of the elements. This results in a better agreement between calculated and measured effects and hence added reliability in element design.

It should also be noted that in all these arrangements the magnetic field modification system is a passive system and does not require its own power source. This should be contrasted with previous arrangements which have involved the use of additional electrical coils.

In general, the magnetic field generating means will comprise one or more electrical coils, particularly superconducting coils, but could also be provided by a permanent magnet system.

Some examples of magnetic field generating apparatus according to the invention will now be described with reference to the accompanying drawings, in which:-

Figure 1 illustrates the variables used in the mathematical analysis set out below;

Figure 2 is a cross-section through a quadrant of a magnet assembly having one example of a modification system;

Figure 3 illustrates schematically part of a modification system of the type shown in Figure 2;

Figure 4 illustrates the part of the modification system shown in Figure 3 in a form suitable for mathematical analysis;

Figure 5 is a schematic, perspective view of a second example of the apparatus;

Figure 6 illustrates graphically an example of the variation in the direction of the magnetic field among the surface of a cylinder supporting iron shims in the Figure 5 example; and,

Figure 7 illustrates graphically the magnetisation properties of typical grain oriented steel with varying magnetisation direction and at various angles to the rolling direction.

One method of determining the configuration of a magnetic field modification system for use with magnetic field generating means such as a superconducting coil system is to represent the magnetic field within the volume mathematically. This can be done in a variety of ways including for example Fourier Bessel expansions, numerical variance on a surface or in a volume of arbitrary shape, or prolate and oblate spherical expansions 'for elliptical regions. In the present case, we will represent the magnetic field in terms of a Fourier-Legendre series which has the form: ∞ n

Hz = ∑ (r/r₀)n ∑ P^m _n (Cosθ) { C^m _n CosmΦ + S^m _n Sinmφ} (1) n=0 m=0 where : r is the radius of the sphere of interest, r₀ is the radius of a reference sphere upon which the components of the field expansion (C^m _n, S^m _n) are known,

Θ,Φ are spherical polar coordinates defining the position of the point of interest upon the surface of the sphere, P^m _n are Associated Legendre polynomials of order n and degree m, and

C_n ^m, S^m _n are the values of the field distortions as measured on a reference sphere of radius r₀,

The values of the coefficients C^m _n and S^m _n are determined by plotting the magnetic field over the periphery of a number of disks positioned through the body of the sphere. The disks chosen are perpendicular to the Z axis which, in the case of MRI lies along the patient. The radii of the disks are chosen so that all plotted points lie upon the required spherical surface. If we refer to Figure 1 we see that on each disk of plotted points both r and θ are constant. The field variation is hence simplified to the following: ∞

H_z = ∑ {A^m CosmΦ + B^m SinmΦ} m=0

This is a simple Fourier series. We hence see that the variation around each plotted disk is determined only by the degree ('m') of the field contaminants involved while the amplitude of the variation around the disk is determined by both the degree and the order of the contaminants.

If we were to consider the values of the contaminant coefficients when expressed on various spheres of differing radii then the coefficients scale as follows:

Here C^m _n (r₁ ) is the value of C^m _n on a sphere radius r₁ and

C^m _n (r₂) on a sphere radius r₂.

We can then make the following general statement concerning the degree and order of the field contaminants.

1. The variation in the value of the coefficients when determined upon spheres of differing radii is dependent only upon the Order of the contaminant.

2. The variation around any disk perpendicular to the Z direction is dependent only upon the Degree of the contaminant.

3. The variation in the amplitude of the variation around each disk, when we consider different disks on the spherical volume is dependent on both the Order and the Degree.

We have found that it is possible to design a number of configurations using passive ferromagnetic materials such as iron which cancel not only non-zero order terms (n ≠ 0) but also non-zero non-axisymmetric (m ≠ 0) degree terms.

In the first arrangement shown in Figure 2, two pairs of coils 1, 2 are provided, one of each pair being shown in the drawings. The coils of each pair are symmetrically arranged about a central plane 3 of the magnet and are coaxial with the cylindrical axis 4 of the bore of the magnet (not shown). Radially inwardly of each coil is positioned a respective iron shim 6, 7. Each iron shim is fabricated from a grain oriented material in order to reduce variations in the magnetic field vector and the shims are shaped so as to produce in conjunction with the coils 1, 2 a substantially homogenous magnetic field within a spherical working volume 8 at the centre of the magnet. The shims 6, 7 may or may not be axisymmetric and may comprise single plates (curved around the axis 4) or stacks of such plates.

Typically, the coils 1, 2 and shims 6, 7 will be positioned within a cryostat to enable the coils to perform superconductively.

Figure 3 illustrates a generalised form of shim 9 rolled onto a mounting cylinder 13 of non-magnetic material. These non-axisymmetric arrangements are useful for correcting large mis-placement errors in coil positions or stress distortions to support structures. The shims are fabricated from irregular shapes cut from sheets of grain oriented material and comprise steel.

A method for obtaining the dimensions of these shims will now be outlined with reference to Figure 4. Initially, certain assumptions will be made.

(i) The shim is magnetised in the axial (2) direction and remains so when iron is added or subtracted, (ii) Radial thickness is small compared with the means radius; the contaminants An En or sampled field values will be directly proportional to thickness.

(iii) The (Macro) shims can be modelled either on magnetized arcs (Φ direction, transverse) to be described below or as magnetised bars (Z direction, axial) although, axial magnetisation (M_z) must be assumed. (iv) The mathematics below describes shims cut from iron sheets and wrapped around a cylindrical former or made by removing iron from a magnetised tube providing assumptions (i) and (ii) apply. Extension to shims on conical surfaces or any surface of revolution is possible (two components of magnetisation M_z,

M_r must be considered).

(v) The macro shim design problem is posed as a non-linear constrained optimisation problem, minimise field contaminants or maximise a function of field quality.

Express the lower and upper shim boundaries ⍕ _L (Z')⍕_U (Z') (Figure 4) as functions satisfying:

a) At end points A, C: ⍕_L (Z' = A) = ⍕_L (Z' = C) = 0 (2) ⍕_U (Z' = A) = ⍕_U (Z' = C) = 0 (3) b) ⍕_U (Z') - ⍕_L (Z') ≥ 0 i.e positive in (4) zero angular width

The boundaries can be defined by trigometric series satisfying equations (2) and (3).

Examples:

NG ⍕_L (Z') = ∑ G_K sin {kπ(Z' - A) / (C - A)} (5) k=1

NH ⍕_U (Z') = ∑ H₁ sin {kπ(Z' - A) / (C - A)} (6)

1=1 The shape of the shim is determined by NG + NH, coefficients G_k, H₁, the system optimisation variables, and magnetic field quality by the contaminants A^m _n, B_n ^m in a spherical volume or functions of simple field values within an imaging volume of arbitrary shape.

From assumptions (i) and (ii) the contaminants A^m _n,

B^m _n are:

Msat = Saturation magnetisation θ' = polar angle a' = Mean cylinder radius μ' = cos(θ')

T = Cylinder thickness δ_m = 1 if m = 0

R_o = Imaging sphere radius δ_m = 0 if m ≠ 0

The Z component of field at any point in the imaging volume can be expressed in cylindrical polar coordinates through a Bessel function transformation.

From equations (7) and (8) via the Fourier - Legendre series, the mean field and variance

can be formed.

The design problem is posed as a constrained non-linear optimisation problem, minimise σ_B ² _Z subject to physical constraints, e.g. positive or zero angular width,

The magnitude of the corrections introduced by the iron shims 6, 7 are relatively large (for example up to a few percent). The example shown in Figures 5 - 7 is intended to generate much smaller corrections within a magnetic field generating system which itself generates a very nearly homogenous field. Typically, in this second case, the degree of correction will be in the order of a few ppm.

The magnetic field generator shown in Figure 5 comprises a number of superconducting coils (not shown) positioned within a cryostat 21 which is schematically illustrated in Figure 5. These coils define a cylindrical bore 22 within which a substantially homogenous magnetic field is generated within a spherical region at the centre of the bore 22.

Due to manufacturing tolerances and enviromental effects, the magnetic field within this spherical region can be slightly distorted which is very significant when the magnet is used for magnetic resonance imaging and the like. To deal with this, a magnetic field modification system 24 is slid within the cylindrical bore 22. The assembly 24 comprises an inner cylinder 25 of a non-ferromagnetic material around the outside of which are positioned three bands 26 of non-magnetic material, spaced apart in the axial direction. The inner surface of each band 26 has 40 elongate grooves 27 spaced around the circumference of the cylinder 25. Each set of aligned grooves 27 is adapted to receive an elongate tray 28 of a non-magnetic material which can be slid through the grooves and extends along the full length of the cylinder 25 at a substantially constant radial distance from the axis of the bore 22. Each tray 28 has 11 pockets 29 positioned at predetermined locations along its length.

Into each pocket 29 may be positioned one of more identical shim pieces constructed from laminations of a grain oriented (ie rolled) steel. In one example, each steel or iron piece has dimensions of 50m m and a common thickness. Different thicknesses of n can then be generated in each pocket by stacking a suitable number of the shim pieces into the pocket. Since there are 11 pockets 29 on each tray 28 and a total of 40 trays, this gives a total of 440 individual predefined locations for the shim pieces arranged around the circumference of the cylinder 25.

The volume of iron at each location is calculated by a mini-computer using a sequential quadratic programming

(SQP) algorithm in which the search direction is the solution of the quadratic programming (QP) problem.

In one method the system is considered as a systematic means of placing pieces of ferromagnetic material in such a way that their placement causes a reduction in the deviations, from some specified constant value, of the principal magnetic flux density vector, B_p, whose source is the main field generating apparatus over an arbitrarily defined volume of space. With reference to a right handed cylindrical coordinate system, whose origin is fixed relative to the main field generating apparatus, these deviations may occur in the azimuthal, axial and radial directions.

Over the specified volume of space where a reduction of the deviation of the principal magnetic flux density is required a measure of the "goodness" of the flux density is required. This can be accomplished by

which defines the variance of B_ over volume V. G, the measure of goodness will fall to zero when B --does not deviate at all from some defined mean value B over volume V. In order to make use of equation (9) it is necessary to split B in equation (9) into two parts Bop and ΔB . These two parts represent the field due to the main field generating apparatus and the modification to the main field by the placement of ferromagnetic objects in the main field.

Bop is assumed to be a known function of the three orthogonal coordinates and is in practice measured. ΔB is the function necessary to correct the deviations of Bop from the specified constant magnetic flux density B necessary to correct the deviations of Bop from the specified constant magnetic flux density Bp. The sources of ΔB are the ferromagnetic objects.

The amount of ΔB produced by a quantity of ferromagnetic material placed in a magnetic field may be calculated from the following standard subset of Maxwells equation namely ⍕ = 1 ∫ M.∇(1/R) dτ (10)

4π

H = - ∇⍕ (11)

B = μ_oH (12)

Where the S.I. system of units is assumed and the quantities have the following meaning: ⍕ - is the magnetic scalar potential M - is the magnetisation vector H - is the magnetic field intensity vector B - is the flux density vector μ_o - is the permeability of free space. τ - denotes integration over the volume of the ferromagnetic object R - is the distance from a point in or on the ferromagnetic object to a point in the volume where the correction to Bo is required.

V - is the standard gradient operator. Finally ΔBp is the projection of equation 12 onto the principal axis. For the present purposes the following simplifying assumptions are usually necessary. 1. The magnetisation vector M has only one component namely the principal one, Mp 2. The magnetisation vector, Mp is known.

With these assumptions equations (10), (11) and (12) can be used together with equation (9) to define an optimisation procedure in which the thicknesses of the shim pieces at the locations in the mechanical apparatus are chosen in such a way that equation (9) is made acceptably small over the specified volume, V, of space. In an alternative method the field expansion (1) given above can be simplified for a given axial position (ie fixed Cosθ) so that the field distortion produced by N pieces of iron placed on the surface of a cylinder is fixed, and that for given order "n" the expansion reduces to

n N pieces

Hzn = Σ ∑ {C^m Cos m Φ_i + S^m Sin m Φ _i} (13) m = 0 i = 1

which is simply a finite Fourier series that may be expressed as

Npieces n Npieces

Hzn = ∑ A⁰ + ∑ ∑ 2 Re (D^-me^jmΦi) (14) i = 1 - m = 1 i = 1 with

which is just the complex form of the Fourier series.

Examination of equation (14) reveals the possibility of producing "pure shims" if appropriate azimuthal (Φ) distributions of iron are used.

For example consider the creation of a shim that produces no m=2 (ie mainly m=1) using two pieces of iron at angular positions Φ₁ and Φ₂.

The condition that this distribution produces no m=2 may be derived from (14) by substituting m=2 in the series and summing over the number of objects present. This gives the condition for no m=2 to be

e^j2Φ1 + e^j2Φ2 = 0 = e^j2Φ1 (1 + e^j2(Φ ₂ ^{- Φ} ₁ ⁾) or e^j2(Φ ₂ ^{- Φ} ₁ ⁾ = - 1 ,₁₅₎ which gives 2(Φ₂ - Φ₁) = π or

The foregoing demonstrates that two iron objects separated by 90° can produce no m=2 contaminant. By examination of (14) it is also clear that this distribution can produce no m=6, 10, 14 ... either, although some m=0 will be produced.

If we now insist that the "pure shims" produce no m=3 as well as no m=2 then this can be achieved by using the no m=2 distribution as the basic building block.

Consider now 4 iron objects, 4 being the next multiple of 2, arranged so as to produce no m=2 and no m=3. The conditions for this may be written down from (3) as

e^{j2 Φ}1 + e^j2Φ2 + e ^{j2 Φ}3 + e^j2Φ4 = 0 (16) e^j3Φ1 + e^j3Φ2 + e^j3Φ3 + e^j3Φ4 = 0 (17) which becomes

e^j2Φ1(1 + e^j2(Φ ₂ ^{- Φ} ₁ ⁾) + e^j2Φ3(1 + e^j2(Φ ₄ ^{- Φ} ₃ ⁾) = 0(18)

e^j3Φ1(1 + e^j3(Φ ₂ ^{- Φ} ₁ ⁾) + e^j3Φ3(1 + e^j3(Φ ₄ ^{- Φ} ₃ ⁾) = 0(19)

If we now apply the condition that no m=2 is produced, to equation (18), then this requires that

and

(20) and (21) can now be substituted into (19) to give:

e^j3Φ1 (1+e^j3(Φ ₃ ^{- Φ} ₁₎) (1 + e^j3π/2) = 0 (22) which is fulfilled when

e^j3(Φ _{3 -} ^Φ1) = - 1 or (23)

Condition (23) will allow no m=3 , 9, 15, 21 ... to be produced, whilst conditions (20) and (21) ensure that no m=2, 6, 10, 14, 18 ... is produced. This means that the combined shim will produce no m=2 , 3, 6, 9, 10, 14, 15, 18, 21. The process outlined above may be continued until an "effectively pure" m=1 shim is produced. By "effectively pure" we mean that the first values of m ≠ 1 produced is of such a high degree that it is rendered insignificant. The commonly used "pure shims" are given below and their angular positions are summarised in Table 1. ("Polarity" indicates the sign of the term to be reduced)

Table 1 Angular positions for pure shim terms

Degree Polarity Angular Positions Φ m=1 + ±7.5°, ±37.5°, ±52.5° ±97.5°

±172.5°, ±142.5°, ±127.5°, +82.5° 2 + ±7.5, ±37.5, ±142.5, ±172.5

±82.5, ±52.5, ±97.5, ±127.5 3 + 0°, ±120°

±60°, 180° 4 + 0°, ±90°, 180° - ±45°, ±135°

Degree Polarity Angular Positions Φ m=1 + -172.5°, -7.5°, 37.5°, 52.5°,

82.5°, 97.5°, 127.5°, 142.5° -142.5°, -127.5°, -97.5°, -82.5°. 52.5°, -37.5°, 7.5°, 172.5° 2 + 7.5, 37.5, 52.5, 82.5, -97.5.

-127.5, -142.5, -172.5 97.5, 127.5, 142.5, 172.5, -7.5, -37.5, -52.5, -82.5

3 + -90°, 30°, 150° - -150°, -30°, 90°

4 + -112.5°, -22.5°, 67.5°, 157.5° - -157.5, -67.5°, 22.5°, 112.5°

Example 1 Pure 1st degree contaminant Stage 1: Firstly let us consider two pieces of iron positioned to produce no contribution to 2nd degree terms. In order to attain this the iron pieces must be separated by π/2 (90°). We use this array as a building block in stage two. stage 2: Using the same principle as before. Two arrays of objects separated by π/3 (60°) will produce no contribution to 3rd degree terms. If we therefore use the two objects generated in stage one as our object array then we achieve a 4 object building block that can produce no contribution to 2nd or 3rd degree contaminants Stage 3: Finally, if we take two arrays of objects separated by π/4 (45°) then the net contribution to 4th degree terms is zero. Using the 4 object building block in Stage 2 then we achieve an 8 object arrangement that can produce no contribution to 2nd, 3rd or 4th degree terms. Example 2 Pure 2nd degree contaminant Stage 1: Firstly let us consider two pieces of iron separated by π (180°). These iron pieces will produce no contaminants of degree 1, 3 , 5, 7 , 8, 11 ... Stage 2: If we use two arrays of objects separated by π/4 (45°) then no net 4th degree term can result. If we use the two objects generated in Stage 1 as our object array, then we produce a 4 object array that can produce no contaminant of degree 1, 3 or 4. Dealing with m=0 contaminants

So far only m>0 contaminants have been dealt with. Normally, there are also m=0 terms that require shimming; these terms will be made larger by the placement of iron on the cylinder to shim out the non-axisymmetric, m>0 terms. The m=0 terms can be shimmed out by placing rings of iron around the shim tube. However in practice it is not necessary to use rings since it is quite feasible to approximate a ring with a number of discrete pieces. To illustrate just how few pieces are required consider 8 discrete pieces of iron placed at 0° ± 45° ±90° ±135° and 180° then consideration of equation (14) shows that the only non-zero values of m that can be produced are m=8, 16, ... These contaminants are such high order that their distortion of the field on the sphere of interest should be negligible. A table of suitable positions for the iron for shimming m=0 is given in Table 2. It is found in practice that the minimum number of pieces required to ensure that no unwanted distortions are produced is 12. However in some designs a smaller number of pieces may have to be used in order that the approximate ring can be accurately modelled by the available iron pieces.

Table 2 Recommended "Axisymmetric" angular distributions of iron pieces

No of iron Angles (Φ) 1st Non Zero

Pieces

8 0°, ±45°, ±90., ±135°, 180° 8

12 0°, ±30°, ±60°, ±90°, ±120°, 12

±150°, 180°

16 0°, ±22.5°, ±45°, ±67.5°, 16

±90°, ±112.5°, ±135°, ±157.5°

24 +7.5°, ±22.5°, ±37.5°, ±52.5°, 24

±67.5°, ±82.5°, ±97.5°, ±112.5°,

±142.5°, ±157.5°, ±172.5°

36 0°, ±7.5°, ±22.5°, ±30°, ±37.5°, 12

±52.5°, ±60°, ±67.5°, ±82.5°,

±90°, ±97.5°, ±112.5°, ±120°,

±127.5°, ±142.5°, ±150°, ±157.5°,

±172.5, 180.

It may be recalled that the distortion produced by a piece of iron is given by the Fourier/Legendre series of equation (1). The C^m _n and S^m _n contained in this expression either correspond to measured values or to values produced by iron objects necessary to cancel out the- measured contaminants. However the C^m _n is multiplied by a Cos mΦ term whilst the S^m _n term is weighted by Sin mΦ. This might suggest at first glance that the two different types would require different procedures to shim them. In practice this is not so since rotation of a cosine term by 90°/m gives the equivalent sine term. Negative values C^m _n and S^m _n do not require special treatment either since rotation of a positive term by

180°/m gives the equivalent negative term.

An example of a particular arrangement will now be described.

A series of tables can be produced which detail the iron thicknesses required at each axial location to produce "pure" terms for an extensive list of contaminants. A few special points are of note: 1. The iron thicknesses specified assume that each iron piece is 50 × 50mm.

2. The magnetic field is assumed at 0.5 Tesla.

3. For the terms of degree greater than 0 the thicknesses assume that a 50 × 50mm piece of iron is positioned at all the angular positions required to produce a term of that given degree.

4. For the axisymmetric terms (degree 0) then the thicknesses are specified for a single iron piece at each axial location. Various strengths may hence be attained in the axisymmetric terms by using more iron pieces at each axial location. A table of the angular positions which can be used to produce a contaminant with no contribution to terms with degrees 1 to 4 is given in Table 3. 5. The "pure" shim designs for particular terms are pure only with respect to the non-axisymmetric degrees. With this in mind the non-axisymmetric terms should be treated before the axisymmetric terms. In this way the axisymmetric errors produced by the non-axisymmetric shims (m > 0) will be superposed on the axisymmetric terms already existing in the magnet.

Consider a 0.5 Tesla magnet whose field plot reveals contaminants with the following cosine terms of first degree: C₁ ¹ = -50ppm

C₂ ¹ = 5ppm

C₃ ¹ = -0.5ppm

C¹ ₄ = 0.2ppm

C₅ ¹ = 0.1ppm

The passive shims will be positioned upon a 920mm diameter cylinder. STAGE 1: Refer to Table 3 labelled "920mm Diameter

Shims. M = 1 shims". We will use 8 pieces of iron at each axial position. STAGE 2: Use the iron thicknesses in the table together with the required ppm to generate the iron required for each contaminant at each axial position (see Table 4).

To illustrate the procedure Table 4 (for a 920mm diameter cylinder for contaminants of degree M = 1) shows that a 0.0495 thick shim in location 1 (z = -660.6mm) will produce lppm of 0.5T field. Hence to remove 50ppm of the C¹ ₁ contaminant requires a shim thickness of 50 × 0.0495 = 2.475mm: this is the figure shown in Table 4 for this location and order (n=1). STAGE 3 : Sum the iron requirements at each axial location. Note that a negative thickness corresponds to a rotation of π (180°) about the cylinder axis. In the calculation of iron shim distributions to compensate a given field distortion, a significant source of error is produced by any non-uniformity in the direction of magnetisation of the shims. To illustrate this phenomena, the error in direction of magnetisation is plotted against position in Figure 6 for a typical MRI magnet. In order to reduce the largely incalculable effects of this rotation of the magnetisation vector, a grain oriented laminated structure is used for the iron shims. As can be seen from Figure 7, the benefit of using oriented laminations is that the magnetic susceptability is anisotropic with a pronounced favourable direction of magnetisation in the "rolled direction". Figure 7 shows a series of curves for different gives directions illustrating the variation cf DC magnetisation with induction field.

Claims

1. Magnetic field generating apparatus comprising- magnetic field generating means for generating a magnetic field in a working volume; and a magnetic field modification system comprising one or more ferromagnetic members arranged about an axis extending through the working volume to reduce axisymmetric and non-axisymmetric variations in the magnetic field such that the homogeneity of the magnetic field within the working volume is improved.

2. Apparatus according to claim 1, wherein the magnetic field generating means generates a substantially non-homogeneous magnetic field within the working volume, the magnetic field modification system cooperating with the magnetic field generating means to modify the magnetic field so that the magnetic field within the working volume is substantially homogeneous.

3. Apparatus according to claim 1, wherein the magnetic field modification system comprises an array of magnetisable elements positioned around the working volume.

4. Apparatus according to claim 3, wherein the array of magnetisable elements is positioned within the magnetic field generating means.

5. Apparatus according to claim 3 or claim 4, wherein the magnetic field modification system comprises a non-magnetic support member on which the magnetisable elements are mounted.

6. Apparatus according to claim 5, wherein the magnetizable elements are positioned in sets of locations.

7. Apparatus according to claim 6, wherein each set of locations is arranged in the form of a linear array.

8. Apparatus according to any of the preceding claims, wherein the ferromagnetic members are made of-iron.

9. Apparatus according to any of the preceding claims, wherein the ferromagnetic members are formed from grain orientated ferromagnetic material.

10. Apparatus according to any of the preceding claims, wherein the magnetic field generating means comprises one or more electrical coils.

11. Apparatus according to claim 10, wherein the or each coil is a superconducting coil.