DE4136834C2 - Magnetic field correction device - Google Patents

Description

The invention relates to magnetic field correction devices, used for example in electromagnets, around a uniform field for a magnetic To generate nuclear magnetic resonance.

Fig. 8 is a schematic perspective view of a known magnetic field correction device, which is given, for example, in JP-OS 52-193230. An inner cylinder ( 10 ) made of a non-magnetic material, to which rod-shaped magnetic elements ( 2 ) adhere, is arranged within a coil ( 3 ). The magnetic field generated by the coil ( 3 ) is not sufficiently uniform, as is required within the uniform field area ( 4 ). The magnetic elements ( 2 ) are thus attached to the inner cylinder ( 10 ) in order to modify the spatial distribution of the magnetic field and to improve the uniformity of the magnetic field within the uniform field region ( 4 ).

In the known device, the magnetic elements ( 2 ) are attached to the inner cylinder ( 10 ) at arbitrarily selected positions and the change in the magnetic field distribution within the uniform field area ( 4 ) is determined experimentally. An arrangement of the magnetic elements ( 2 ) is selected by the person skilled in the art on the basis of his experience in order to achieve the required uniformity within the uniform field area ( 4 ).

The above-mentioned magnetic field correction device has the following disadvantages. The dimensions, the number and the positions of the magnetic elements ( 2 ) are mainly determined by the experience of the assembly engineer. There are no quick and fixed rules for determining the dimensions, the number and the positions of the magnetic elements ( 2 ). Thus the arrangements of the magnetic elements ( 2 ) are not necessarily optimized and the quality of the magnetic field fluctuates from one device to another. Furthermore, the magnetic elements ( 2 ) are sometimes attached at positions outside the area in which the designer of the magnetic field correction device desires to place the magnetic elements ( 2 ) in order to avoid interference with other particles of the device.

A formula for developing the magnetic field H _z in a device generating a magnetic field is known from the publication WO 88/08126. Also described is a system consisting of a number of ferromagnetic parts to remove certain terms in the equation that are different from the zero order. The development formula contains terms that represent measured field distortions.

The article "Shimming Solenoidal Magnets" by John H. Coupland in Nuclear Instruments and Methods in Physics Research A256 (1987) pp. 174-179 describes theoretical Basics of trimming solenoid magnets. It will be a mathematical expression for the magnetic field of a long Trimmings specified. However, this document contains no evidence of the arrangement and dimensioning of Trimming elements in order to target specific error orders remove.

From US Patents 4,698,611 and 4,803,433 there are arrangements known for trimming magnets, but neither of them Notes on the dimensioning or arrangement of the used trim elements to give certain Correct error orders of a magnetic field.

The published patent application DE-40 21 345 A1, which was only published after the priority date of the present application, but which itself has an older seniority, describes a passive compensation arrangement for the homogenization of a magnetic field. In such a compensation arrangement, passive compensation elements, which consist of two elements, which are of different lengths and have different end faces, are arranged in non-magnetic holders on the inner wall of a support cylinder of a solenoid 3 . In order to make terms of first and higher order disappear in a component of the magnetic field in a certain region of the magnetic field, a certain installation angle is selected on the one hand, and on the other hand the compensation elements are arranged symmetrically with respect to the plane perpendicular to the magnetic axis. By zeroing two development terms in the mathematical expression for the magnetic field component, certain end face angles of the compensation elements are determined and an end face ratio of the elements is derived therefrom.

The object of the invention is a To create correction device by means of which the Arrangement and dimensions of magnetic elements used for Correction of inhomogeneities in a magnetic field is used will result in a simple way.

This object is achieved by the in the claims designated magnetic field correction devices solved.  

Fig. 1 is a cross sectional view of a solenoid device, which is equipped with an inventive magnetic field correction device;

Fig. 2 is a longitudinal sectional view of the device of Fig. 1;

Fig. 3 is a schematic cross-sectional view showing the circumferential position of a trim magnetic element (magnetic shim) (hereinafter also referred to as a magnetic element);

Fig. 4 is a schematic longitudinal sectional view of a Trimmagnetelementes;

Fig. 5 is a perspective view of a trim magnetic element consisting of two rod components;

Fig. 6 is a graph showing the values of a function g indicates the (alpha) relative to a final angle (alpha);

Fig. 7 is an illustration of an example of the circumferential angular positions of the non-magnetic tubes for receiving the trim magnetic elements; and

Fig. 8 is a schematic perspective view of a known magnetic field correction device.

In the drawings, the same reference numerals are used same or corresponding parts or sections.

With reference to the accompanying drawings now preferred embodiments of the invention described.

Fig. 1 shows a cross-sectional view of a solenoid device, which is equipped with an inventive magnetic field correction device, and FIG. 2 1 is a longitudinal sectional view of the apparatus of FIG..

An inner cylinder ( 10 ) is arranged coaxially within a cylindrical coil ( 3 ), which generates the main magnetic field in it. A number of non-magnetic tubes ( 1 ) made of non-magnetic material are attached to the inner surface of the inner cylinder ( 10 ) in such a way that the magnetic elements ( 2 ) can be selectively inserted into and removed from the respective non-magnetic tubes ( 1 ). For simplification, FIGS. 1 and 2 show only the non-magnetic tubes ( 1 ) and the magnetic elements ( 2 ) for correcting the positive linear X-component error field. As will be described below, the field is made homogeneous within a uniform field region ( 4 ) by means of the fields generated by the magnetic elements ( 2 ).

According to the invention, the parameters (the number, the circumferential and axial positions, the cross-sectional areas, etc.) of the magnetic elements are determined in accordance with the invention in order to satisfy certain predetermined relationships. Fig. 3 is a schematic cross-sectional view indicating the circumferential position of a Trimmagnetelementes, and Fig. 4 is a schematic longitudinal sectional view of a Trimmagnetelementes. In FIGS. 3 and 4 has a rod-shaped Trimmagnetelement (2) a cross-sectional area (A) and is in a radius (a) relative to the Z-axis disposed at an angle (psi) relative to the X-axis. The two ends of the trim magnet element ( 2 ) form a larger angle (alpha I) (which is referred to as the "outer end angle") and a smaller angle (alpha II) (which is referred to as the "inner end angle") with the Z axis, which coincides with the central axis of the coil ( 3 ). The XY plane intersects the coil ( 3 ) in its center, so that the origin (0, 0, 0) coincides with the center of the uniform field area ( 4 ). The position of an arbitrary point (P) within the uniform field area ( 4 ) is represented by the polar coordinates (r, theta, phi).

The Z component (Bz) of the magnetic field generated by a trim magnetic element ( 2 ) at point (p) is thus represented as follows:

in which
K is a constant which is determined by the magnetic properties of the trim magnetic element,
Epsilon m is the Neumann coefficient, which is equal to 1 for m = 0 and which is otherwise equal to 2; and
P ^m n represents the assigned legend threshing polynomial of the nth degree and mth order.

The relationship between the expressions in the Polar coordinate expression (1) and that in Cartesian Coordinates will be up to n = 2 in the below Table 1 stated:

TABLE 1

In Table 1 above, the integers correspond to n and m that of equation (1) above.

On the other hand, the Z component (Bcz) of the magnetic field generated by the coil ( 3 ) in the uniform field region ( 4 ) is expressed up to the second-order terms as follows:

B _cz = B ₀ + A ₁ X + A ₂ Y + A ₃ ZX + A ₄ ZY + A ₅ XY + A ₆ (X ² - Y ² ) (2)

where (B ₀ ) represents the magnetic field at the origin (0,0,0), and the expression (A1X) represents, for example, the linear or error component of the first order along the X-axis.

According to one embodiment of the invention, the error elements are compensated up to the second order according to equation (2) and eliminated by means of the magnetic elements ( 2 ), so that the magnetic field is made substantially uniform within the uniform field region ( 4 ). The expressions (A1X, A2Y, A3ZX etc.) are thus compensated for by means of the magnetic field generated by the magnetic elements ( 2 ). The arrangement (positions and dimensions) of the magnetic elements ( 2 ) for compensating for the X component error (A1X) is described below, for example.

As can be seen from equation (1), the number of magnetic field components generated by a trim magnetic element ( 2 ) is infinite. However, since the mounting radius (a) of the trim magnetic element ( 2 ) is larger than (r) (a <r), the factor: (r / a) ⁿ that appears in equation (1) is negligibly small if (n ) is sufficiently large. Thus, the expressions of equation (1) alone are of practical importance for small values of (n, m). Only the radial components of (Bz) (namely the expressions for (m ≠ 0)) are considered here. Thus, the following expressions B ^nm are considered: (n and m corresponding to those of equation (1)): B ¹¹ , B ²¹ , B ²² , B ³¹ , B ³² , B ³³ , B ⁴¹ , B ⁴² , B ⁴³ , B ⁴⁴ , B ⁵¹ , B ⁵² , B ⁵³ , B ⁵⁴ .

As indicated in Table 1 above, the B ¹¹ component corresponds to the X component in the Cartesian coordinates. The goal here is to compensate for the error expression (A1X). Thus, the above expressions (B ^nm ) except B ^{11 must} disappear. To achieve this, the positions and dimensions of the magnetic elements ( 2 ) are determined in accordance with the basic idea of the invention as follows. First, according to FIG. 1, eight magnetic elements ( 2 ) are arranged along the circumference in such a way that their circumferential angles (psi) with respect to the X axis are:

ψ = (π / 2) (± (1/2) ± (1/3) ± (1/4)), (3)

so that m = 2, 3 and 4

Σ cos m (Φ-ψ) = 0

where the summation is over the eight values of psi which is given by equation (3) above become.  

Thus, the components B ^nm disappear for m = 2, 3, 4, namely B ²² , B ³² , B ³³ , B ⁴² , B ⁴³ , B ⁴⁴ , B ⁵² , B ⁵³ , B ⁵⁴ . Only the expressions B ^nm for m = 1 remain: B ¹¹ , B ²¹ , B ³¹ , B ⁴¹ and B ⁵¹ , which are to disappear with the exception of the first expression B ¹¹ .

The expressions B ²¹ and B ⁴¹ disappear by selecting the end angles of the magnetic elements ( 2 ) accordingly. If the sum of the inner end angle (alpha I) and the outer end angle (alpha II) of the trim magnet element ( 2 ) is equal to pi (= 3.14):

αI = π - αII (4)

so applies

Thus B ²¹ and B ⁴¹ disappear.

Finally, the expressions B ³¹ and B ^{51 should} disappear. Thus, the magnetic elements ( 2 ) each consist of two bars, as shown in Fig. 5, and their cross-sectional areas (A1, A2) and inner end angle (alpha 1, alpha 2) are selected such that they satisfy the relationships listed above, the outer end angles being determined by equation (4) above. The lengths (L1, L2) of the two rod-shaped components of the magnetic elements are determined by the inner and outer end angles and the fastening radius (a) of the magnetic elements ( 2 ) (see FIG. 4).

The expressions B ³¹ and B ⁵¹ for the trim magnet element ( 2 ) in FIG. 5 satisfy the following proportions:

B ³¹ ∝ A ₁ [P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ ] + A ₂ [P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ] (5)

B ⁵¹ ∝ A ₁ [P ¹ ₆ (cosα ₁ ) sin ⁷ α ₁ ] + A ₂ [P ¹ ₆ (cosα ₂ ) sin ⁷ α ₂ ] (6)

Let a function g (alpha) of alpha be determined by:

g (α) P ¹ ₆ (cosα) sin ⁷ α / P ¹ ₄ (cosα) sin ⁵ α = P ¹ ₆ (cosα) sin ² α / P ¹ ₄ (cosα) (7)

Thus for the proportionality relationship (6) we get:

B ⁵¹ ∝ A ₁ g (α ₁ ) [P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ ] + A ₂ g (α ₂ ) [P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ] (7a)

Figure 6 shows the values of g (alpha) with respect to the angle (alpha). From Fig. 6 it can be seen that given R for (alpha 1, alpha 2) there are values such that

R = g (α ₁ ) = g (α ₂ ), α ₁ ≠ α ₂ (8)

Substituting R of equation (8) into equation (7a) results in:

B ⁵¹ ∝ R {A ₁ [P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ ] + A ₂ [P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ]} (7b)

The expression within the brackets {} of equation (7b) is identical to the right side of equation (5). As mentioned above, it can be seen from Fig. 6 that at a particular selected value for R, two particular angles (alpha 1, alpha 2) that satisfy equation (8) above for that value of R can be determined. For such a pair (alpha 1, alpha 2) the expressions B ³¹ and B ⁵¹ are proportional to each other because:

B ³¹ ∝ A ₁ [P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ ] + A ₂ [P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ]

B ⁵¹ ∝ R {A ₁ [P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ ] + A ₂ [P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ]}

Both expressions B ³¹ and B ⁵¹ thus disappear if the cross-sectional areas (A1, A2) satisfy the following relationship:

A ₁ [P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ ] + A ₂ [P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ] = 0 (9)

or

A ₂ / A ₁ = -P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ / P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ (9a)

Equation (9a) gives the cross-sectional ratio of the two rod components of each magnetic element ( 2 ).

A specific numerical example is now given. It is assumed that the designer of the magnetic field correction device intends to limit the range for the end angles of the magnetic elements ( 2 ) to a range within 140 °. If the value of R = g (alpha) is chosen according to FIG. 6 with R = -0.5, then the inner end angles of the respective rod components of a trim magnet element ( 2 ) which satisfy the above equation (8): alpha 1 = 40, 1 ° and alpha 2 = 66.5 °. The outer end angles corresponding to these are:
pi - 40.1 ° = 139.9 ° and pi - 66.5 ° = 113.5 °, which are within the 140 ° intended by the designer of the magnetic field correction device. Furthermore, the cross-sectional area ratio of the two rod components according to FIG. 5 is determined by the above equation (9a) as follows: A2 / A1 = 0.1336, so that the two expressions B ³¹ and B ⁵¹ disappear. Since the value for R can be chosen freely, the design freedom of the magnetic field correction device is guaranteed.

By arranging the magnetic elements ( 2 ) at circumferential angles Psi = (pi / 2) (± 1/2 ± 1/3 ± 1/4) ± pi / 4: with respect to the X axis according to FIG. 1, a negative linear ( or first order) X component (A1X <0) is generated. The angles (Psi) are increased by (pi) such that Psi → Psi + pi, or Psi = (pi / 2) ((± 1/2) ± (1/3) ± (1/4)) + pi applies, then a positive linear X component (A1X <0) is generated.

Similarly, by arranging the magnetic elements ( 2 ) at circumferential angles, Psi = (pi / 2) (± 1/2) ± (1/3) ± (1/4) + pi / 2 with respect to the X axis, becomes a negative one linear Y component (A2Y <0) generated. Furthermore, if the angles Psi are each increased by pi in such a way that Psi = (pi / 2) (± (1/2) ± (1/3) ± (1/4)) + 3 pi / 2, a positive linear becomes Y component (A2Y <0) generated. Thus, the linear expression or first order expression in equation (2) of any polarity can be compensated for and eliminated.

According to Table 1, the second order expressions (A3ZX, A4ZY) in the equation (2) correspond to the output expression B ^{21 of} the magnetic elements ( 2 ), while the expressions (AsXY and As (X ² -Y ² ) correspond to the expression B ²² It is shown by documentation similar to the above that the second order error components can be compensated and eliminated by the magnetic elements ( 2 ), each of which consists of two rod components (with a cross-sectional area (A1, A2), end angles (alpha) and circumferential angles (Psi ) with respect to the X axis), be arranged such that the end angles and the circumferential angles (Psi) satisfy specific relationships.

In particular, the first and second order error expressions with respect to X and Y are eliminated: (1) the non-magnetic tubes are attached at circumferential angles (pi / 2) (± 1/2 ± 1/3 ± 1/4), (pi / 2) ( ± 1/2 ± 1/3 ± 1/4) + pi, (pi / 2) (± 1/2 ± 1/3 ± 1/4) + (1/2) pi or (pi / 2) (± 1/2 ± 1/3 ± 1/4) + (3/2) pi arranged to accommodate respective magnetic elements ( 2 ) there; (2) If these magnetic elements are inserted into the respective non-magnetic tubes, then a sum of the two end angles of each of the two rod-shaped components of the magnetic elements with respect to an origin point at the center of a uniform field area is equal to pi; and (3) when the magnetic elements are inserted into the respective non-magnetic tubes, the cross-sectional areas (A1, A2) and the inner end angles (alpha 1, alpha 2) relative to the projection of the two rod-shaped components of the magnetic elements satisfy the following equation:

(A ₂ / A ₁ ) = -P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ / P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ ,

where P ¹ _{4 is} an associated Legendresch polynomial, and alpha 1 and alpha 2 two specific angles, so that g (alpha 1) = g (alpha 2), where g (alpha) = P ¹ ₆ (cos alpha) sin ² alpha / P ¹ ₄ (cos alpha), where P ¹ ₆ and P ¹ _{4 are} associated Legendresches polynomials.

To eliminate the second-order expressions compared to Zx and Zy: (1) the non-magnetic tubes are arranged at the following circumferential angles (pi / 2) (± 1/2 ± 1/3 ± 1/4), (pi / 2) (± 1/2 ± 1/3 ± 1/4) + pi, (pi / 2) (± 1/2 ± 1/3 ± 1/4) + (1/2) pi or (pi / 2) (± 1 / 2 ± 1/3 ± 1/4) + (3/2) pi to accommodate respective magnetic elements ( 2 ) there; (2) when these magnetic elements are inserted into the respective non-magnetic tubes, the outer end angles of the two rod-shaped components of the magnetic elements are equal to each other; and (3) if the magnetic elements are inserted into the respective non-magnetic tubes, the cross-sectional areas (A1, A2) and the inner end angles (alpha 1, alpha 2) relative to the projection of the two rod-shaped components of the magnetic elements satisfy the relationship:

(A ₂ / A ₁ ) = -P ¹ ₅ (cosα ₁ ) sin ⁶ α ₁ / P ¹ ₅ (cosα ₂ ) sin ⁶ α ₂ ,

where P ¹ _{5 is} an associated legend threshing polynomial; and alpha 1 and alpha 2 are two specific angles such that g (alpha 1) = g (alpha 2), where g (alpha) = P ¹ ₇ (cos alpha) sin ² alpha / P ¹ ₅ (cos alpha), and P ¹ ₇ and P ¹ ₅ are assigned legend threshing polynomials.

To eliminate the second order errors compared to XY and X ² -Y ² : (1) the non-magnetic tubes are arranged at circumferential angles
(pi / 2) (± 1 ± 1/3 ± 1/4) + (1/4) pi, (pi / 2) (± 1/1 ± 1/3 ± 1/4) + (3/4) pi, (pi / 2) (± 1/1 ± 1/3 ± 1/4) or (pi / 2) (± 1/1 ± 1/3 ± 1/4) + (1/2) pi, um there to record the respective magnetic elements; (2) if the magnetic elements are inserted into the respective non-magnetic tubes, the sum of two end angles of each of the two rod-shaped components of the magnetic elements is pi; and (3) if the magnetic elements are inserted into the respective non-magnetic tubes, then the cross-sectional areas (A1, A2) and the inner end angles (alpha 1, alpha 2) relative to the projection of the two rod-shaped components of the magnetic elements are sufficient according to the following equation:

(A ₂ / A ₁ ) = -P ² ₅ (cosα ₁ ) sin ⁶ α ₁ / P ² ₅ (cosα ₂ ) sin ⁶ α _2,

where P ² _{5 is} an associated Legendre's polynomial; and alpha 1 and alpha 2 are two specific angles such that g (alpha 1) = g (alpha 2), where g (alpha) = P ² ₇ (cos alpha) sin ² alpha / P ² ₅ (cos alpha), and P ² ₇ and P ² ₅ are assigned legend threshing polynomials.

Table 2 below shows an example of the numerical values of the arrangement of the magnetic elements ( 2 ).

Table 2

Table 2 shows alternative values of the pairs of End angles specified for components ZX and ZY.

Thus, by making each of the magnetic elements ( 2 ) from two rod components and setting them to predetermined values, namely

• (1) the cross sectional area ratio (A1 / A2) of the two rod components of each of the magnetic elements ( 2 );
• (2) the end angles (alpha) of the two rod components of each of the magnetic elements ( 2 ); and
• (3) the circumferential mounting brackets (Psi) of the magnetic elements ( 2 ),

all error expressions up to the second order in the equation are compensated for and eliminated by means of the magnetic field generated by the magnetic elements ( 2 ). Thus, according to the invention, the non-magnetic tubes ( 1 ) are arranged at predetermined locations opposite the coil ( 3 ), so that the magnetic elements ( 2 ) are selectively inserted therein in order to compensate and eliminate the error expressions in the equation (2).

Fig. 7 is an illustration of an embodiment of the circumferential angular positions of non-magnetic tubes for receiving the Trimmagnetelemente, wherein the reference numeral (1 A) represent the non-magnetic tubes to compensate for the X and Y components; ( 1 B) the non-magnetic tubes for the ZX and ZY components; and ( 1 C) the non-magnetic tubes for the XY and X ² -Y ² components. The non-magnetic tubes ( 1 ) have lengths that correspond to the lengths of the magnetic elements ( 2 ) that are to be used there.

As can be seen from equation (1) above, the absolute output value of the correction terms (B ^mn ) (e.g. (B ¹¹ ) for (A1X)) can be set by selecting the total cross-sectional area of the magnetic elements ( 2 ) while the area ratio (A2 / A1) of the two rod components is maintained at the specific predetermined value. Thus, a number of magnetic elements with different total cross-sectional areas are provided (the lengths are determined by the end angles and the mounting radius in the manner indicated above and the cross-sectional area ratios are determined as indicated above), and those which compensate for the error members are selected and included in the non-magnetic ones Pipes ( 1 ) introduced. The non-magnetic tubes ( 1 ) thus have sufficient internal diameters to accommodate the magnetic elements ( 2 ) with the largest cross-sectional area, which are provided for insertion there.

Claims (3)

1. Magnetic field correction device for correcting errors of the 1st order within a hollow cylindrical coil ( 3 )
a number of non-magnetic tubes ( 1 ) which are attached to a cylinder ( 10 ) with a predetermined radius and are arranged coaxially to the coil ( 3 ) and which each extend along the axial direction of the coil ( 3 ) and at predetermined angles along a circumference of the cylinder ( 10 ); and
a number of rod-shaped magnetic elements ( 2 ) which are removably inserted into the non-magnetic tubes ( 1 ), each magnetic element ( 2 ) consisting of two rod-shaped components of a predetermined length and with a predetermined cross-sectional area ratio, so that the magnetic elements have predetermined end angles (α _I , α _II ) if they are inserted into the respective non-magnetic tubes ( 1 ),
the non-magnetic tubes ( 1 ) at circumferential angles (π / 2) (± 1/2 + ± 1/3 ± 1/4), (π / 2) (± 1/2 ± 1/3 ± 1/4) + π, (π / 2) (± 1/2 ± 1/3 ± 1/4) + (1/2) π or (π / 2) (± 1/2 ± 1/3 ± 1/4) + ( 3/2) π are arranged to receive respective magnetic elements ( 2 ) there;
and when the magnetic elements ( 2 ) are inserted into the respective non-magnetic tubes ( 1 ), a sum of the two end angles (α _I , α _II ) of each of the two rod-shaped components of the magnetic elements ( 2 ) with respect to an origin point at the center of a uniform field region is the same is π;
and when the magnetic elements are inserted into the respective non-magnetic tubes, the cross-sectional areas (A1, A2) and the inner end angles (α ₁ , α ₂ ) relative to the origin of the two rod-shaped components of the equation are sufficient:
(A2 / A1) = - P ¹ ₄ (cosα ₁ ) sin ⁵ α ₁ / P ¹ ₄ (cosα ₂ ) sin ⁵ α ₂ )
where P ¹ _{4 is} an associated Legendresch polynomial; and α ₁ and α _{2 are} two specific angles, so that
g (α ₁ ) = g (α ₂ ) ≠ 0, where g (α) = P ¹ ₆ (cosα) sin ² α / P ¹ ₄ (cosα) and P ¹ ₆ and P ¹ _{4 are} associated Legendesches polynomials. 2. Magnetic field correction device for correcting 2nd order errors within a hollow cylindrical coil ( 3 ) in the ZX and ZY components
a number of non-magnetic tubes ( 1 ) which are attached to a cylinder ( 10 ) with a predetermined radius and are arranged coaxially to the coil ( 3 ) and which each extend along the axial direction of the coil ( 3 ) and at predetermined angles along a circumference of the cylinder ( 10 ); and
a number of rod-shaped magnetic elements ( 2 ) which are removably inserted into the non-magnetic tubes ( 1 ), each magnetic element ( 2 ) consisting of two rod-shaped components of a predetermined length and with a predetermined cross-sectional area ratio, so that the magnetic elements have predetermined end angles (α _I , α _II ) if they are inserted into the respective non-magnetic tubes ( 1 );
where the non-magnetic tubes at the circumferential angles (π / 2) (± 1/2 ± 1/3 ± 1/4), (π / 2) (± 1/2 ± 1/3 ± 1/4) ± π, ( π / 2) (± 1/2 ± 1/3 ± 1/4) + (1/2) π or (π / 2) (± 1/2 ± 1/3 ± 1/4) + (3/2 ) π are arranged to receive the respective magnetic elements ( 2 ) there;
and when the magnetic elements (2) are inserted into the respective non-magnetic tubes (1), the outer end angle of the two rod-like components of the magnetic elements (2) are equal to each other; and
if the magnetic elements ( 2 ) are inserted into the respective non-magnetic tubes ( 1 ), the cross-sectional areas (A1 / A2) and the inner end angles (α ₁ , α ₂ ) relative to the origin of the two rod-shaped components of the magnetic elements satisfy the following equation:
(A2 / A1) = -P ¹ ₅ (cosα ₁ ) sin ⁶ α ₁ / P ¹ ₅ (cosα ₂ ) sin ⁶ α ₂ ,
where P ¹ _{5 is} an associated legend threshing polynomial; and α ₁ and α _{2 are} two specific angles, so that
g (α ₁ ) = g (α ₂ ) ≠ 0, and
g (α) = P ¹ ₇ (cosα) sin ² α / P ¹ ₅ (cosα), where P ¹ ₇ and P ¹ _{5 are} associated Legendresches polynomials. 3. Magnetic field correction device for correcting 2nd order errors within a hollow cylindrical coil ( 3 ) in the XY and (X ² -Y ² ) components
a number of non-magnetic tubes ( 1 ) which are attached to a cylinder ( 10 ) with a predetermined radius and are arranged coaxially to the coil ( 3 ) and which each extend along the axial direction of the coil ( 3 ) and at predetermined angles along a circumference of the cylinder ( 10 ); and
a number of rod-shaped magnetic elements ( 2 ) which are removably inserted into the non-magnetic tubes ( 1 ), each magnetic element ( 2 ) consisting of two rod-shaped components of a predetermined length and with a predetermined cross-sectional area ratio, so that the magnetic elements have predetermined end angles (α _I , α _II ) if they are inserted into the respective non-magnetic tubes ( 1 );
the non-magnetic tubes ( 1 ) at circumferential angles (π / 2) (± 1 ± 1/3 ± 1/4) + (1/4) π, (π / 2) ± 1 ± 1/3 ± 1/4) + (3/4) π, (π / 2) (± 1 ± 1/3 ± 1/4) or (π / 2) (± 1 ± 1/3 ± 1/4) + (1/2) π are arranged to receive respective magnetic elements ( 2 ) there;
and when the magnetic elements (2) are inserted into the respective non-magnetic tubes (1), a sum of the two end angle (α _I, α _II) of each of the two rod-shaped components of the magnetic elements (2) is equal to π;
and when the magnetic elements (2) are inserted into the respective non-magnetic tubes (1), the cross-sectional areas (A1, A2) and the inner end angle (α _1, α ₂₎ relative to the origin of the two rod-like components of the magnetic elements (2) the following equation is sufficient:
(A2 / A1) = -P ² ₅ (cosα ₁ ) sin ⁶ α ₁ / P ² ₅ (cosα ₂ ) sin ⁶ α ² ,
where P ² _{5 is} an associated Legendre's polynomial; and α ₁ and α _{2 are} two specific angles, so that
g (α ₁ ) = g (α ₂ ) ≠ 0, where g (α) = P ² ₇ (cosα) sin ² α / P ² ₅ (cosα), and P ² ₇ and P ² _{5 are} associated Legendesches polynomials.