JPH0774820B2 - Magnetic field gradient measuring method using search coil and search coil - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION [Industrial field of application] The present invention is used in accelerators, electron storage rings, and the like.
The present invention relates to a magnetic field gradient measuring method using a search coil and a search coil for measuring a magnetic field gradient of a polar electromagnet with high accuracy.

[Conventional technology]

Main electromagnets used in the accelerator include a two-pole electromagnet for deflecting charged particles and a four-pole electromagnet for converging charged particles.
The quadrupole electromagnet has a function of giving a charged value deviated from the central orbit from a convergence value proportional to the magnitude of the deviance to converge the charged particles near the central orbit.

FIG. 2 shows a sectional view of the quadrupole electromagnet. Where 1 is the magnetic pole,
2 is a coil, 3 is the center of a quadrupole electromagnet, and 4 is a bore radius. The electrons traveling in the z direction through the quadrupole electromagnetic scale receive a focusing force in the x direction and a diverging force in the y direction. Convergence in the y direction is performed by a quadrupole electromagnet arranged at the next stage having a 90 ° magnetic pole position different from that of the quadrupole electromagnet.

The convergence force Fx that acts on the electron with the charge −e and the velocity v passing through the coordinate (x, 0) on the x-axis is given by the following equation.

Fx = −evBy (x, 0) where By (x, 0) is the y-direction component of the magnetic field at coordinates (x, 0). In an ideally worn quadrupole electromagnet, By (x, 0) is proportional to x. That is, the magnetic field gradient B'y (x, 0) is Becomes Therefore, the convergence force Fx is proportional to the deviation from the center.

In the case of a quadrupole electromagnet, it is desirable that the error from the predetermined constant value of the magnetic field gradient B′y (x, 0) be as small as possible within the determined range of x in order to obtain the convergent force as designed. So, when making a 4-pole electromagnet,
In order to evaluate its performance, it is necessary to measure the magnetic field gradient B'y (x, 0) with high accuracy.

As a method for measuring the magnetic field gradient B'y (x, 0), there is a method using a search coil. FIG. 3 shows an example of a box-shaped racetrack-shaped search coil. D is the width of the search coil 5 in the x direction, L is the height in the y direction, and M is the length in the z direction. As shown in the figure, the search coil 5 is one in which the coil is wound around a frame body having a hollow inside, and changes in the magnetic field passing through the hollow are measured. That is, with the quadrupole electromagnet excited to a constant current value, the search coil is moved along the x-axis,
The spatial variation of the magnetic field, that is, the magnetic field gradient B'y (x, 0) is measured by electromagnetic induction. FIG. 4 shows a sectional view of the search coil 5 of FIG. 3 in the xy plane. In the figure, 3 is the center of the quadrupole electromagnet, and 6 is the center of the search coil.
x ₀ and 7 are windings of the search coil. The search coil output voltage when the search coil 5 is moved at a constant speed in the x direction is proportional to the integral value of the y direction component By of the magnetic field in the cross section D × L. When the position of the search coil is represented by its center coordinate x ₀ , the magnetic field gradient ′ y (x, 0) derived from the search coil output as a function of the position x ₀ is not the magnetic field gradient at x ₀ but the center of the search coil. Is the average magnetic field gradient in the cross section D × L of the search coil when is at x ₀ . In an ideal quadrupole electromagnet having only a quadrupole magnetic field component,
This average is in agreement with the value at x ₀ . This is because the y-direction component By (x, y) of the quadrupole magnetic field is proportional to x and does not depend on y,
Therefore, the average magnetic field in the cross section D × L of the search coil is equal to the magnetic field at the center coordinate x ₀ .

[Problems to be Solved by the Invention]

The problems with the conventional magnetic field measuring method and search coil using this type of search coil are as follows.

The quadrupole electromagnet actually used has higher-order magnetic field components other than the quadrupole magnetic field component, that is, 12 poles, 20 poles, 28 poles, etc., and these magnetic field components have y coordinate dependence. Also, it has a non-linear dependency on the x coordinate. Therefore, in an actual quadrupole electromagnet having these high-order magnetic field components, the average value of the magnetic field in the cross section D × L of the search coil does not match the value of the magnetic field at the center position of the search coil. Therefore, the magnetic field gradient'y based on the search coil output
(X ₀ , 0) is the true magnetic field gradient B'y at the position (x ₀ , 0)
Not the same as (x ₀ , 0).

The high-order magnetic field components other than the quadrupole magnetic field component in the actual quadrupole electromagnet are several% at most within the bore radius of FIG.
And the difference between the value of the magnetic field at the center position of the average value of the magnetic field and search coil in a search coil of the cross-section D × in L if somewhat smaller size of the search coil is reduced 10 ^-3
Since there is no particular problem in measuring the order accuracy, this difference has not been a problem in the past.

However, when charged particles must be held in a predetermined orbit for a long time like an electron storage ring, a quadrupole electromagnet with a particularly high magnetic field accuracy is required. ^-High accuracy of about ⁴ orders is required. However, the conventional search coil that does not consider the higher-order magnetic field components other than the quadrupole magnetic field component has a problem that the magnetic field gradient measurement cannot be performed with such high accuracy. Further, if the cross section D × L of the search coil is made small in order to improve the measurement accuracy, there are problems that the manufacturing accuracy of the search coil dimensions cannot be ensured, the output signal becomes small, and the S / N ratio deteriorates.

An object of the present invention is to eliminate the above-mentioned conventional problems and to measure a magnetic field gradient of a quadrupole electromagnet having a high-order magnetic field component other than the quadrupole magnetic field component with high accuracy and a magnetic field gradient measuring method using a search coil. To provide a coil.

[Means for Solving the Problems]

To achieve the above object, according to the present invention, the gradient B′y of the magnetic field By in the y direction, which is the direction perpendicular to the x axis, which is the axis of symmetry in the pole of the quadrupole electromagnet, is measured. In a magnetic field gradient measuring method using a box-shaped racetrack-shaped search coil having a width D in the x direction, a height L in the y direction, and a length M in the z direction, the magnetic field By is defined as a quadrupole component in the Cartesian coordinate system x, y. And the sum of each higher-order component, whereby the error between the true magnetic field and the average magnetic field in the search coil is calculated by measuring the center coordinates (x ₀ , 0) of the search coil, the horizontal width D in the x direction, and the y direction. Height L
And the magnetic field gradient is measured using a search coil having D / L such that the magnitude of this error is minimized at the point of maximum coordinate in the x direction within the measurement range.

The shape of the search coil is preferably such that D / L is approximately 1.

[Action]

In this way, the error caused when measuring the magnetic field gradient of the magnetic field including higher-order components other than the 4-pole component with the search coil having a finite magnitude is correctly evaluated, and the magnitude of this error is measured within the measurement range. By determining the shape of the search coil so as to be the minimum or less than a predetermined limit value and measuring the magnetic field gradient using the search coil, it is possible to measure the magnetic field gradient with high accuracy. The principle will be described below.

If the scalar potential V (x, y) of the magnetic field is expressed by the sum of the higher-order magnetic field components, the following equation (1) is obtained.

The value obtained by partial differentiation with respect to y, that is, the y-direction component of the magnetic field is expressed by the following equation.

Here, μ ₀ is the magnetic permeability. Therefore, the y-direction component of the magnetic field at the point x ₀ on the x-axis to be measured can be expressed by the following equation (3).

Further, the y-direction component of the magnetic field in the search coil having the width D and the height L centered on the point x ₀ is as follows (4).

Therefore, with the point x ₀ as the center and between the average magnetic field and the true magnetic field in the search coil having the above size,
An error ε ⁽ⁿ⁾ (x ₀ ) represented by the equation (5) is generated.

From this equation, it is possible to obtain the error value when the shape of the search coil, that is, D / L and the point x ₀ for measuring the magnetic field gradient are given.

Since the high-order magnetic field component is rapidly attenuated as the absolute value of x becomes small as described later, the above error is evaluated at the maximum x coordinate x _m of the measurement range. Then, the shape of the search coil is determined so that the error ε (x _m ) is the minimum or less than a predetermined limit value.

According to the numerical calculation results, the suitable value of D / L in the size range of the actual quadrupole electromagnet is in the range of 1.0 to 2.0, and in particular, for the most common size of the quadrupole electromagnet, D / L1 is specified. The shape of the search coil can measure the magnetic field gradient with sufficiently high accuracy.

〔Example〕

Four poles of the scalar potential V (x, y) of the magnetic field in Eq. (1), 1
When the sum of 2-pole, 20-pole and 28-pole components is expressed in the Cartesian coordinate system of x and y,
It becomes (6).

V (x, y) = 2a ₂ xy + a ₆ (6xy ⁵ −20x ³ y ³ + 6x ⁵ y) + a ₁₀ (10xy ⁹ −120x ³ y ⁷ + 252x ⁵ y ⁵ −120x ⁷ y ³ + 10x ⁹ y) + a ₁₄ (14xy ¹³ −364x ³ y ¹¹ ＋ 2002x ⁵ y ⁹ −3432x ⁷ y ⁷ ＋ 2002x ⁹ y ⁵ −364x ¹¹ y ³ ＋ 14x ³ y) (6) where the term a ₂ is a 4-pole magnetic field component, and the term a ₆ is 12 poles. Magnetic field component,
The term a ₁₀ is the 20-pole magnetic field component, and the term a ₁₄ is the 28-pole magnetic field component. From equation (6), the y-direction component of the magnetic field By (x, y) is
It becomes an expression.

By (x, y) = − μ ₀ [2a ₂ x + a ₆ (30xy ⁴ −60x ³ y ² ＋ 6x ⁵ ) + a ₁₀ (90xy ⁸ −840x ³ y ⁶ + 1260x ⁵ y ⁴ −360x ⁷ y ² ＋ 10x ⁹ ) + a ₁₄ (182xy ¹² −4004x ³ y ¹⁰ ＋ 18018x ⁵ y ⁸ −2402x ⁷ y ⁶ ＋ 10010x ⁹ y ⁴ −1092x ¹¹ y ² ＋ 14x ¹³ )] (7) By using Eq. When calculated, it becomes equal to zero. That is, as described above, the quadrupole magnetic field component does not become an error factor even in the case of a search coil having a finite size, and the error factor is a high-order magnetic field component of 12 poles or more.

When ε ⁽ⁿ⁾ (x ₀ ) is calculated for n = 6 which is a 12-pole magnetic field component, the following equation (8) is obtained.

The position of the search coil is standardized by x ₀ / L, and ε ⁽⁶⁾ (x ₀ ) ＝
If 0, then Eq. (8) becomes an algebraic equation for D / L,
D / L can be calculated. That is, the value of D / L of the search coil that makes the error due to the 12-pole magnetic field component zero can be obtained as a function of the standardized measurement position x ₀ / L.

Similarly, the relationship between D / L and x ₀ / L that makes the error due to the 20-pole magnetic field component and the 28-pole magnetic field component zero can be obtained. It should be noted that magnetic field components higher than the 28-pole magnetic field component have a much smaller abundance ratio than the 4-pole magnetic field component, and are therefore ignored.

Figure 1 shows the result of numerical calculation. As is clear from the figure, the value of D / L at which the error is zero differs for each magnetic field component and for each position on the x ₀ / L axis. However, when x ₀ / L is large, the value of D / L that makes the error zero converges to 1.

The magnetic field gradient measurement range in a general quadrupole electromagnet is naturally within the magnetic field use range, and the range corresponds to the bore radius 4 in FIG. In this, each component V ⁽ⁿ⁾ of the scalar potential of the magnetic field increases with the n-th power of the distance r from the center of the quadrupole electromagnet as shown in equation (1), and each component of the magnetic field By
⁽ⁿ⁾ changes with r to the (n-1) th power. The quadrupole magnetic field component, which is the basic component of the quadrupole electromagnet, is a scalar potential V ⁽²⁾
Is the second line of r, and the magnetic field By ⁽²⁾ changes with the first power of r. And the higher-order magnetic field component is 4 for r with respect to the quadrupole magnetic field component.
If the size is equal to or larger than the power, it attenuates toward the center of the 4-pole electromagnet. Therefore, in order to minimize the error in the magnetic field gradient measurement, it is appropriate to use a search coil having a small error in the maximum x coordinate x _m of the magnetic field gradient measurement range. In general, x _m is a value which is several times or more of L, so that it is preferable to measure the magnetic field gradient using a search coil of D / L = 1 from FIG.

As a special case, if the measurement range is narrow and it is necessary to measure the range with high accuracy, an appropriate value of D / L> 1 should be selected. Since the optimal D / L value that minimizes is different, the D / L value that minimizes the error is determined in consideration of the existence ratio of each magnetic field component, and the search coil with that D / L value is determined. To measure the magnetic field gradient. The abundance ratio of each high-order magnetic field component can be estimated by computer simulation or the like. The abundance ratio of each high-order magnetic field component within the bore radius of an average quadrupole electromagnet is several percent or less. From the abundance ratio, the coefficient of each component a _n
And the error ε (x _m ) can be evaluated from Eq. (5). Therefore, the value of D / L is determined so that this error ε (x _m ) is within the minimum or desired accuracy.

As an example, if you want to measure the magnetic field gradient near x _{0 /} L = 0.4 so that the error due to the 28-pole component of the magnetic field becomes almost zero,
From Fig. 1, it can be seen that D / L should be approximately 1.6.

〔The invention's effect〕

As described above, according to the present invention, the magnetic field of the quadrupole electromagnet is represented by the sum of the quadrupole component and each higher-order component, whereby the error between the true magnetic field and the average magnetic field in the search coil is calculated. , As a function of the center coordinates (x ₀ , 0) of the search coil, the lateral width D in the x direction and the height L in the y direction, and the magnitude of this error is at the point of the maximum coordinate in the x direction within the measurement range. By taking advantage of the maximum value, the magnetic field gradient is measured by the search coil having the D / L value that minimizes the value of this error, so that extremely high accuracy measurement is possible. Also, in the practical measurement range, using the D / L1 search coil enables highly accurate magnetic field gradient measurement with the smallest error even when the existence ratio of each multipole component to the quadrupole component of the magnetic field is unknown. Become.

[Brief description of drawings]

FIG. 1 shows the relationship between the width D of the search coil and the height L for zeroizing the error for each higher-order magnetic field component with respect to the standardized center position x ₀ / L of the search coil. FIG. 4 shows the shape of the search coil used in the magnetic field gradient measuring method according to the present invention.
FIG. 3 is a perspective view of the search coil, and FIG. 4 is a cross-sectional view of the search coil. 5: Search coil.

Claims (2)

[Claims] 1. A gradient B'of a magnetic field By in the y direction, which is a direction perpendicular to the x axis, which is the axis of symmetry in the pole of the quadrupole electromagnet.
In a magnetic field gradient measuring method using a box-shaped racetrack-shaped search coil having a width D in the x direction for measuring y, a height L in the y direction, and a length M in the z direction, the magnetic field By is set to the orthogonal coordinate system x, y. The error between the true magnetic field and the average magnetic field in the search coil is represented by the sum of the quadrupole component and each higher-order component in the search coil, and the center coordinate (x ₀ , 0) of the search coil and the width in the x direction. The magnetic field gradient is obtained by using a search coil having D / L such that the magnitude of this error is minimized at the point of the maximum coordinate in the x direction within the measurement range, as a function of D and the height L in the y direction. A magnetic field gradient measuring method using a search coil, which comprises measuring. 2. A gradient B'of a magnetic field By in the y direction, which is perpendicular to the x axis, which is the axis of symmetry within the pole of the quadrupole electromagnet.
A box-shaped racetrack-shaped search coil having a lateral width D in the x direction for measuring y, a height L in the y direction, and a length M in the z direction, characterized in that D / L is substantially 1 Search coil to do.