Patent Publication Number: US-10321552-B2

Title: Undulator magnet array and undulator

Description:
TECHNICAL FIELD 
     The present invention relates to an undulator magnet array, and to an undulator incorporating an undulator magnet array. 
     BACKGROUND ART 
     In an undulator used to extract synchrotron radiation from an electron beam in a synchrotron radiation facility, there is provided a pair of magnet arrays disposed parallel to and opposite each other to produce a periodic magnetic field, and by undulating electrons that travel between the pair of magnet arrays at a speed close to that of light, intense synchrotron radiation is generated. The periodic magnetic field can be produced with permanent magnets or electromagnets. To obtain synchrotron radiation with a short wavelength in the X-ray region in particular, however, the magnetic field needs to have a period of the order of several centimeters or less, and with electromagnets, it is impossible to produce a magnetic field with sufficient intensity. For this reason, most undulators adopt permanent magnets. 
       FIG. 22  shows an example of an undulator magnet array used in a conventional undulator. In  FIG. 22 , the undulator magnet array  901  has a first magnet array  910  and a second magnet array  920 . The magnet arrays  910  and  920  each contain four magnets  930  per period λ u , and the magnetization direction (indicated by arrows inside the magnets  930 ) of the magnets  930  contained in each magnet array changes, in the plane through the magnet arrays  910  and  920 , by 90° from one magnet to the next (Patent Document 1 and Non-Patent Document 1). Such an undulator magnet array  901  is called, for example, a Halbach magnet array. The electrons traveling between the magnet arrays  910  and  920  at a speed close to that of light are undulated by the action of the periodic magnetic field produced by the magnet arrays  910  and  920  to emit light (synchrotron radiation) with a wavelength λ given by the following equation.
 
λ(λ u   ,B,E )=130λ[1+(93.37λ u   B ) 2 /2] /E   2  
 
     In the above formula, λ is the wavelength of the synchrotron radiation in nanometers (called the fundamental wavelength), λ u  is the spatial period of the magnet arrays in meters, B is the amplitude of the magnetic field in tesla, and E is the energy of electrons in gigaelectronvolts. 
     In a synchrotron radiation facility, the energy of electrons is fixed, and the spatial period is determined during the designing of an undulator; thus, to allow selection of a particular wavelength during the operation of the synchrotron radiation facility, the magnetic field amplitude has to be adjustable. The magnetic field amplitude can be adjusted easily within a certain range by varying the interval, called gap, between the magnet arrays  910  and  920 . 
     LIST OF CITATIONS 
     Patent Literature 
     Patent Document 1: Japanese Patent Application published as No. 2012-160408 
     Non-Patent Literature 
     Non-Patent Document 1: K. Halbach, “Permanent Magnet Undulators”, J. Physique, C1 (1983) 211 
     SUMMARY OF THE INVENTION 
     Technical Problem 
     Inconveniently, the attractive force between magnet arrays containing several hundred to several thousand strong magnets amounts to several tons, and under this attractive force, the above-mentioned adjustment needs to be performed with an accuracy of several micrometers. Accordingly, an undulator uses, all over it, structural materials and driving mechanisms with extremely high rigidity, resulting in a total weight exceeding ten tons. Moreover, distributing loads requires a large number of components, necessitating complex structures and high production and assembly accuracy. All this leads to increased cost and time required for the manufacture, transport, and installation of an undulator. 
     An object of the present invention is to provide an undulator magnet array and an undulator that contribute to reduction of magnetic forces that act on magnet arrays. 
     Means for Solving the Problem 
     According to one aspect of the present invention, an undulator magnet array includes a first magnet array and a second magnet array that are disposed parallel to each other with an interval in between so as to lie opposite each other, and the magnetization direction of magnets contained in the first magnet array and the magnetization direction of magnets contained in the second magnet array change, in the plane through the first and second magnet arrays, periodically along the magnet arrangement direction of the respective magnet arrays. Here, the first magnet array is formed by coupling together a first shifted magnet array and a first reference magnet array each containing a plurality of magnets, and the second magnet array is formed by coupling together a second reference magnet array and a second shifted magnet array each containing a plurality of magnets. Moreover, the first shifted magnet array is disposed opposite the second reference magnet array, the first shifted magnet array being shifted relative to the second reference magnet array by a predetermined shift amount in a predetermined direction parallel to the magnet arrangement direction as compared with in a reference state where the amplitude of the periodic magnetic field produced by the first and second magnet arrays is maximized, and the second shifted magnet array is disposed opposite the first reference magnet array, the second shifted magnet array being shifted relative to the first reference magnet array by the shift amount in the predetermined direction as compared with in the reference state. 
     The above-mentioned shifting helps reduce the magnetic force that acts between the first and second magnet arrays. The first and second magnet arrays may each be composed of a shifted magnet array, which is shifted, and a reference magnet array, which is unshifted; deposing these in the manner described above permits the cancelling-out of a magnetic force that acts in the magnet arrangement direction as well. 
     Advantageous Effects of the Invention 
     According to the present invention, it is possible to provide an undulator magnet array and an undulator that contribute to reduction of magnetic forces that act on magnet arrays 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram showing an outline of an undulator magnet array according to a first embodiment of the present invention; 
         FIG. 2  is a diagram showing a reference configuration of the undulator magnet array according to the first embodiment of the present invention; 
         FIG. 3A  and  FIG. 3B  are diagrams illustrating the angle of the magnetization direction of magnets; 
         FIG. 4  is a diagram showing a first improved configuration of the undulator magnet array according to the first embodiment of the present invention; 
         FIG. 5  is a diagram showing the configuration of an undulator magnet array for comparison with the undulator magnet array of the first improved configuration; 
         FIG. 6  is a diagram showing a second improved configuration of the undulator magnet array according to the first embodiment of the present invention; 
         FIG. 7  is a diagram illustrating a procedure for fabricating the undulator magnet array of the second improved configuration; 
         FIG. 8  is a diagram illustrating a procedure for fabricating the undulator magnet array of the second improved configuration; 
         FIG. 9  is a diagram showing a third improved configuration of the undulator magnet array according to the first embodiment of the present invention; 
         FIGS. 10A, 10B, and 10C  are diagrams showing the gap-dependence of the magnetic forces that act between the upper and lower magnet arrays in the first embodiment of the present invention; 
         FIG. 11  is a diagram showing a fourth improved configuration of the undulator magnet array according to the first embodiment of the present invention; 
         FIG. 12  is a diagram showing the results of simulations according to the first embodiment of the present invention; 
         FIG. 13  is a diagram showing the results of simulations according to the first embodiment of the present invention; 
         FIG. 14  is a diagram showing the phase-dependence of the magnetic force that acts between the upper and lower magnet arrays and the phase-dependence of the amplitude of the magnetic field produced by the upper and lower magnet arrays in a second embodiment of the present invention; 
         FIG. 15  is a diagram showing the configuration of an undulator magnet array with M=8 according to a third embodiment of the present invention; 
         FIG. 16  is a diagram showing the configuration of another undulator magnet array with M=8 according to the third embodiment of the present invention; 
         FIG. 17  is a diagram showing the configuration of an undulator magnet array with M=2 according to the third embodiment of the present invention; 
         FIG. 18  is a diagram showing the configuration of another undulator magnet array with M=2 according to the third embodiment of the present invention; 
         FIG. 19  is a side view of an undulator according to a fourth embodiment of the present invention; 
         FIG. 20  is a front view of the undulator according to the fourth embodiment of the present invention; 
         FIG. 21  is a plan view of a synchrotron radiation facility according to the fourth embodiment of the present invention; and 
         FIG. 22  is a diagram showing the configuration of a conventional undulator magnet array. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, examples embodying the present invention will be described specifically with reference to the accompanying drawings. Among the diagrams referred to, the same parts are identified by the same reference numerals, and in principle no overlapping description of the same parts will be repeated. In the present description, for the sake of simple description, symbols and other designations referring to information, signals, physical quantities, components, and the like are occasionally used alone, while omitting or abbreviating the names of the information, signals, physical quantities, components, and the like that correspond to those symbols and other designations. 
     First Embodiment 
     A first embodiment of the present invention will be described.  FIG. 1  shows the structure of an undulator magnet array  1  according to the first embodiment. The undulator magnet array  1  has two magnet arrays  10  and  20  disposed parallel to each other with an interval in between so as to lie opposite each other. The magnet arrays  10  and  20  are each composed of a plurality of magnets  30  arranged in a straight line. The magnets  30  are permanent magnets such as neodymium magnets. Unless otherwise stated, the magnets  30  that constitute the magnet arrays  10  and  20  all have the same shape and the same size, and in each magnet array, the plurality of magnets  30  are arranged at a constant pitch. 
     In this embodiment, for the sake of concrete explanation, three mutually perpendicular axes are assumed, namely X-axis, Y-axis, and Z-axis. Z-axis is parallel to the arrangement direction of the magnets  30  in each magnet array. The arrangement direction of the magnets  30  is the same between the magnet arrays  10  and  20 . Y-axis is parallel to the direction connecting between the magnet arrays  10  and  20 . The plane parallel to both X-axis and Y-axis will be called XY-plane, the plane parallel to both Y-axis and Z-axis will be called YZ-plane, and the plane parallel to both Z-axis and X-axis will be called ZX-plane. YZ-plane is the plane through the magnet arrays  10  and  20 . More precisely, the center of all the magnets  30  constituting the magnet arrays  10  and  20  lies on YZ-plane. 
     Moreover, what are meant by “up”, “down”, “left”, “right”, and similar terms are defined as follows. The left-right direction is parallel to Z-axis, the positive side along Z-axis being the “right” side, the negative side along Z-axis being the “left” side. The up-down direction is parallel to Y-axis, the positive side along Y-axis being the “up” side, the negative side along Y-axis being the “down” side. Furthermore, of two given magnet arrays, the one located relatively above the other will be called the upper magnet array (example of first magnet array), and the one located relatively below the other will be called the lower magnet array (example of second magnet array). Specifically, here, the magnet array  10  is considered to be located above the magnet array  20 . Accordingly, in the following description, the magnet arrays  10  and  20  are occasionally called the upper magnet array  10  and the lower magnet array  20  respectively. 
     Of any given magnet such as the magnets  30 , the direction pointing from the S pole to the N pole of the magnet will be called its magnetization direction (the direction in which it is magnetized). Though not clear from  FIG. 1  (see, for example,  FIG. 2 ), the magnetization direction of the plurality of magnets  30  contained in the magnet array  10 , and likewise the magnetization direction of the plurality of magnets  30  contained in the magnet array  20 , changes, in the YZ-plane, that is, in the plane through the magnet arrays  10  and  20 , periodically along the direction parallel to Z-axis. The period (spatial period) of the change of the magnetization direction of the magnets  30  in each magnet array ( 10 ,  20 ) is represented by λ u . The period λ u  is common to the magnet arrays  10  and  20 , and is, for example, about several tens of mm (millimeters). 
     The magnetic field produced by the magnet arrays  10  and  20  will be called the undulator magnetic field B. It should be noted that what is called the undulator magnetic field B here is, of the entire magnetic field produced by the magnet arrays  10  and  20 , that component which is perpendicular to the electron beam axis on the electron beam axis. The electron beam axis is the axis of the electron beam that travels between the magnet arrays  10  and  20 . The electron beam axis is parallel to Z-axis, and runs through the middle between the magnet arrays  10  and  20 . The direction and magnitude of the undulator magnetic field B changes periodically along Z-axis, and in  FIG. 1 , the periodic change of the undulator magnetic field B is depicted by a curve between the magnet arrays  10  and  20 . The void between the magnet arrays  10  and  20  in the Y-axis direction is called the gap. 
     Basic Configuration 
     In this embodiment, peculiar magnet arrangements in the undulator magnet array  1  will be described. First, with reference to  FIG. 2 , as a reference example of the undulator magnet array  1 , the configuration of an undulator magnet array  1   REF  will be described. The undulator magnet array  1   REF  includes magnet arrays  10   REF  and  20   REF  as a reference example of the magnet arrays  10  and  20 . It should be noted that, in  FIG. 2  as well as in any diagram referred to later that shows an undulator magnet array, only part of all the magnets  30  constituting it are illustrated. 
     The undulator magnet array  1   REF  is a Halbach magnet array, and contains four magnets  30  per period λ u . In any diagram including  FIG. 2  that show magnets (such as the magnets  30 ), their magnetization direction is indicated by arrows inside the magnets (such as the magnets  30 ). 
     Referring to  FIG. 3A , assume a reference vector that is parallel to Y-axis and that, starting at the origin of Y-axis, points to the positive side along Y-axis; then the angle of a vector representing a magnetization direction to the reference vector will be called the angle of the magnetization direction. The angle of a magnetization direction is 0° or more but less than 360°. Accordingly, if a given magnetization direction coincides with the direction pointing from the negative to the positive side along Y-axis, the angle of the magnetization direction is 0°; if a given magnetization direction coincides with the direction pointing from the positive to the negative side along Y-axis, the angle of the magnetization direction is 180°. Here, with respect to the reference vector, the angle of a magnetization direction is counted clockwise in the plane of  FIG. 2 . That is, if a given magnetization direction coincides with the direction pointing from the negative to the positive side along Z-axis, the angle of the magnetization direction is 90°; if a given magnetization direction coincides with the direction pointing from the positive to the negative side along Z-axis, the angle of the magnetization direction is 270°. 
     In the undulator magnet array  1   REF , the number M of magnets  30  present in one period λ u  is four, and thus, in each of the magnet arrays  10   REF  and  20   REF , the magnetization direction of the magnets  30  changes in YZ-plane by 90° (=360°/M) from one magnet to the next along the direction parallel to Z-axis. Here, the direction of the change is opposite between the upper and lower magnet arrays  10   REF  and  20   REF . In the following description, as shown in  FIG. 3B , a magnet  30  of which the angle of the magnetization direction is θ is designated as a θ magnet  30 . The width of a magnet  30  in the Z-axis direction is represented by the symbol “d”. 
     In the magnet array  10   RFF , from left to right, 0°, 90°, 180°, and 270° magnets  30  are arranged in this order, in repeated cycles, four successive magnets  30  forming a magnet array corresponding to one period. Accordingly, with the center of the 0° magnet  30  in the magnet array  10   REF  taken as the origin of Z-axis, the centers of the 0°, 90°, 180°, and 270° magnets  30  in the magnet array  10   REF  are located at the positions away from the origin rightward in the Z-axis direction by distances of (i×4)×d, (i×4+1)×d, (i×4+2)×d, and (i×4+3)×d respectively (where i is an integer). 
     In the magnet array  20   REF , from left to right, 0°, 270°, 180°, and 90° magnets  30  are arranged in this order, in repeated cycles, four successive magnets  30  forming a magnet array corresponding to one period. Accordingly, with the center of the 0° magnet  30  in the magnet array  20   REF  taken as the origin of Z-axis, the centers of the 0°, 270°, 180°, and 90° magnets  30  in the magnet array  20   REF  are located at the positions away from the origin rightward in the Z-axis direction by distances of (i×4)×d, (i×4+1)×d, (i×4+2)×d, and (i×4+3)×d respectively. 
     Here, at the positions opposite the 0°, 90°, 180°, and 270° magnets  30  in the magnet array  10   REF , the 0°, 270°, 180°, and 90° magnets  30  in the magnet array  20   REF  are disposed respectively. More specifically, the straight line connecting between the center position of the i-th magnet in the magnet array  10   REF  and the center position of the i-th magnet in the magnet array  20   REF  is parallel to Y-axis, the first to fourth magnets in the magnet array  10   REF  being 0°, 90°, 180°, and 270° magnets  30  respectively, the first to fourth magnets in the magnet array  20   REF  being 0°, 270°, 180°, and 90° magnets  30  respectively. 
     In any diagram including  FIG. 2  that shows magnets (such as the magnets  30 ), an attractive force that acts on a pair of magnets is indicated by a dot-filled arrow. In  FIG. 2 , attractive forces are indicated exclusively for those pairs of magnets which have a magnetization direction of 0° and those pairs of magnets which have a magnetization direction of 180° (the same applies to  FIG. 4  and the like referred to later). In any diagram including  FIG. 2  that shows an undulator magnet array, an overall force that acts on the upper and lower magnet arrays is indicated by a hollow arrow. In the undulator magnet array  1   REF  presented as the reference, a strong attractive force acts between the magnet arrays  10   REF  and  20   REF . 
     First Improved Configuration 
     A first improved configuration of the undulator magnet array will be described. An undulator magnet array  1   A  shown in  FIG. 4  according to the first improved configuration is an example of the undulator magnet array  1 , and includes an upper magnet array  10   A  and a lower magnet array  20   A  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper magnet array  10   A  itself is configured similarly to the upper magnet array  10   REF  described previously, and the lower magnet array  20   A  itself is configured similarly to the lower magnet array  20   REF  described previously. 
     However, in the undulator magnet array  1   A , as compared with the undulator magnet array  1   REF , the upper magnet array is disposed, relative to the lower magnet array, with a leftward displacement corresponding one-fourth of the period λ u . In the following description, for the sake of convenient description, displacing by a distance corresponding to one w-th of the period λ u  is occasionally described as shifting by one w-th of the period (where w is a real number). In the undulator magnet array  1   A , as compared with the undulator magnet array  1   REF , the upper magnet array is shifted leftward by one-fourth of the period relative to the lower magnet array. 
     Accordingly, at the positions opposite the 0°, 90°, 180°, and 270° magnets  30  in the magnet array  10   A , the 90°, 0°, 270°, and 180° magnets  30  in the magnet array  20   A  are disposed respectively. More specifically, the straight line connecting between the center position of the i-th magnet in the magnet array  10   A  and the center position of the i-th magnet in the magnet array  20   A  is parallel to Y-axis, the first to fourth magnets in the magnet array  10   A  being 0°, 90°, 180°, and 270° magnets  30  respectively, the first to fourth magnets in the magnet array  20   A  being 90°, 0°, 270°, and 180° magnets  30  respectively. 
     In any diagram including  FIG. 4  that shows magnets (such as the magnets  30 ), a repulsive force that acts on a pair of magnets is indicated by a solid black arrow. In  FIG. 4 , repulsive forces are indicated exclusively for those pairs of magnets  30  which contain 0° and 180° magnets  30  (the same applies to  FIG. 5  and the like referred to later). 
     The magnetic circuit formed by the configuration of  FIG. 4  is considered to be the middle one between the magnetic circuit formed by the configuration of  FIG. 2  and the magnetic circuit formed by the configuration of  FIG. 5 . In the configuration of  FIG. 5 , as compared with the undulator magnet array  1   REF , the upper magnet array is shifted leftward by one-half of the period (that is, by the distance corresponding to one-half of the period λ u ) relative to the lower magnet array, with the result that repulsive forces act between the upper and lower magnet arrays. 
     In the magnetic circuit formed by the configuration of  FIG. 2  (that is, in the undulator magnet array  1   REF  as the reference), the amplitude of the undulator magnetic field B (that is, the amplitude of the Y-axis component of the undulator magnetic field B of which the direction and magnitude change in the Z-axis direction) is maximized, and in the magnetic circuit formed by the configuration of  FIG. 4  (that is, in the undulator magnet array  1   A ), the amplitude of the undulator magnetic field B is smaller than that in  FIG. 2 . In the magnetic circuit formed by the configuration of  FIG. 5 , the amplitude of the undulator magnetic field B is zero. 
     In the magnetic circuit formed by the configuration of  FIG. 4 , the upper and lower magnet arrays do not receive any force that acts in the up-down direction. This helps make compact the driving mechanism for varying the gap. This, however, comes at a cost: the upper magnet array as a whole receives a strong rightward force while the lower magnet array as a whole receives a strong leftward force, with the result that the structural components that support the upper and lower magnet arrays are acted on by strong forces in the left-right direction. That is, while size reduction is possible in the driving mechanism for varying the gap, high rigidity is required in the structural components (such as the base  170  in  FIG. 19  referred to later) that support the upper and lower magnet arrays. 
     Second Improved Configuration 
     Now, as a second improved configuration, an undulator magnet array  1   B  as shown in  FIG. 6  will be considered. The undulator magnet array  1   B  according to the second improved configuration is an example of the undulator magnet array  1 , and includes an upper magnet array  10   B  and a lower magnet array  20   B  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper and lower magnet arrays  10   B  and  20   B  are configured basically similarly to the upper and lower magnet arrays  10   REF  and  20   REF  respectively. 
     However, in the undulator magnet array  1   B , as compared with the undulator magnet array  1   REF  as the reference, the left half of the upper magnet array is shifted leftward by one-fourth of the period (that is, by the distance corresponding to one-fourth of the period λ u ) relative to the lower magnet array, and in addition the right half of the lower magnet array is shifted leftward by one-fourth of the period (that is, by the distance corresponding to one-fourth of the period λ u ) relative to the upper magnet array. As a result of not the upper magnet array but the lower magnet array being shifted in the right half, the periodicity of the undulator magnetic field B is preserved. 
     With reference to  FIGS. 7 and 8 , the configuration of the undulator magnet array  1   B  will be described.  FIGS. 7 and 8  show a procedure for fabricating the undulator magnet array  1   B  starting with the undulator magnet array  1   REF . In the undulator magnet array  1   REF , the left and right halves of the upper magnet array  10   REF  will be called the magnet arrays  310  and  320  respectively, and the left and right halves of the lower magnet array  20   REF  will be called the magnet arrays  360  and  370  respectively. 
     First, as shown in  FIG. 7 , starting with the undulator magnet array  1   REF , the left-half magnet array  310  of the upper magnet array  10   REF  is shifted leftward by one-fourth of the period relative to the lower magnet array  20   REF . In  FIG. 7 , the magnet array  310  after being so shifted is depicted as a magnet array  310   a . As a result of the shifting, between the left-half magnet array  310   a  of the upper magnet array and the right-half magnet array  320  of the upper magnet array, a void corresponding to one-fourth of the period λ u , is left as a to-be-interpolated region  330 . In this to-be-interpolated region  330 , a magnet  30  having the same magnetization direction as the magnet  30  in an interpolation-source region  340  is disposed for interpolation. The magnet disposed for interpolation is depicted as a magnet  330   a . The interpolation-source region  340  is the region located between the left end of the magnet array  320  and the position displaced rightward from that left end by a distance corresponding to λ u /4. The magnet array composed of the magnet array  310   a  and the magnet  330   a  will be identified as a magnet array  310   b . Coupling together the magnet arrays  310   b  and  320  at their boundary plane, referred to as the coupling plane BD 1 , forms the upper magnet array  10   B . 
     On the other hand, as shown in  FIG. 8 , starting with the undulator magnet array  1   REF , the right-half magnet array  370  of the lower magnet array  20   REF  is shifted leftward by one-fourth of the period relative to the upper magnet array  10   REF . In  FIG. 7 , the magnet array  370  after being so shifted is depicted as a magnet array  370   a . Shifting the magnet array  370  leftward causes a magnet  30  near the left end of the magnet array  370  to have to overlap a magnet  30  near the right end of the magnet array  360 ; this is coped with by removing, out of the magnets  30  in the magnet array  370 , the magnet  30  which the above shifting causes to have to overlap one in the magnet array  360 . Coupling together the magnet arrays  360  and  370   a  at their boundary plane, referred to as the coupling plane BD 2 , forms the lower magnet array  20   B . 
     By combining together the upper and lower magnet arrays  10   B  and  20   B  with the coupling planes BD 1  and BD 2  shown in  FIGS. 7 and 8 , respectively, aligned at the same position in the Z-axis direction, the undulator magnet array  1   B  is formed. The shifting of the magnet arrays  310  and  370  leaves the upper magnet array to protrude one-fourth of the period beyond the lower magnet array at the left and right ends of the undulator magnet array; such protrusions can be trimmed off in fabricating the undulator magnet array  1   B . 
     In the magnetic circuit formed by the undulator magnet array  1   B  in  FIG. 6 , in each of the magnet arrays  10   B  and  20   B , the forces acting in the left-right direction are canceled out. Thus, while compressive and tensile forces in the Z-axis direction act only on those structural components which hold the magnet arrays integrally (such as the magnet array beams  110  and  120  in  FIG. 19  referred to later), no magnetic force acts on those structural components which support the upper and lower magnet arrays (such the base  170  in  FIG. 19  referred to later). Thus, as compared with in a case where the undulator magnet array  1   REF  as the reference is used, it is possible to reduce the rigidity required in the structural components that support the upper and lower magnet arrays, and hence to greatly reduce the weight of an undulator as a whole. 
     Starting with the undulator magnet array  1   REF , of the upper magnet array, the part that is shifted leftward relative to the lower magnet array will be called the upper shifted magnet array (example of first shifted magnet array), and the unshifted part will be called the upper reference magnet array (example of first reference magnet array). Likewise, starting with the undulator magnet array  1   REF , of the lower magnet array, the part that is shifted leftward relative to the upper magnet array will be called the lower shifted magnet array (example of second shifted magnet array), and the unshifted part will be called the lower reference magnet array (example of second reference magnet array). In the example in  FIGS. 7 and 8 , the magnet arrays  310   b  and  320  correspond to the upper shifted and reference magnet arrays respectively, and the magnet arrays  370   a  and  360  correspond to the lower shifted and reference magnet arrays respectively. In the example in  FIG. 6 , the magnet arrays  10 S B  and  10 R B  correspond to the upper shifted and reference magnet arrays respectively, and the magnet arrays  20 S B  and  20 R B  correspond to the lower shifted and reference magnet arrays respectively. 
     The upper magnet array  10   B  is formed by coupling together the upper shifted and reference magnet arrays  10 S B  and  10 R B , and the lower magnet array  20   B  is formed by coupling together the lower reference and shifted magnet arrays  20 R B  and  20 S B . In  FIG. 6 , bold-line arrows schematically indicate how forces act on where the upper shifted and reference magnet arrays are coupled together and where the lower reference and shifted magnet arrays are coupled together (the same applied to  FIG. 9  and the like referred to later). 
     The upper shifted magnet array is disposed opposite the lower reference magnet array (the same applies to any later-described undulator magnet array that includes an upper shifted magnet array and a lower reference magnet array). That is, in the undulator magnet array  1   B , at the positions opposite the 0°, 90°, 180°, and 270° magnets  30  in the upper shifted magnet array  10 S B , the 90°, 0°, 270°, and 180° magnets  30  in the lower reference magnet array  20 R B  are disposed respectively. More specifically, the straight line connecting between the center position of the i-th magnet in the upper shifted magnet array  10 S B  and the center position of the i-th magnet in the lower reference magnet array  2 R B  is parallel to Y-axis, the first to fourth magnets in the upper shifted magnet array  10 S B  being 0°, 90°, 180°, and 270° magnets  30  respectively, the first to fourth magnets in the lower reference magnet array  20 R B  being 90°, 0°, 270°, and 180° magnets  30  respectively. 
     The lower shifted magnet array is disposed opposite the upper reference magnet array (the same applies to any later-described undulator magnet array that includes a lower shifted magnet array and an upper reference magnet array). That is, in the undulator magnet array  1   B , at the positions opposite the 0°, 270°, 180°, and 90° magnets  30  in the lower shifted magnet array  20 S B , the 270°, 0°, 90°, and 180° magnets  30  in the upper reference magnet array  10 R B  are disposed respectively. More specifically, the straight line connecting between the center position of the i-th magnet in the lower shifted magnet array  20 S B  and the center position of the i-th magnet in the upper reference magnet array  10 R B  is parallel to Y-axis, the first to fourth magnets in the lower shifted magnet array  20 S B  being 0°, 270°, 180°, and 90° magnets  30  respectively, the first to fourth magnets in the upper reference magnet array  10 R B  being 270°, 0°, 90°, and 180° magnets  30  respectively. 
     Third Improved Configuration 
     A third improved configuration of the undulator magnet array will be described. An undulator magnet array  1   C  shown in  FIG. 9  according to the third improved configuration is an example of the undulator magnet array  1 , and includes an upper magnet array  10   C  and a lower magnet array  20   C  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper and lower magnet arrays  10   C  and  20   C  are configured basically similarly to the magnet arrays  10   REF  and  20   REF , and to them, the technique of shifting magnet arrays described previously in connection with the second improved configuration is applied. Moreover, in the third improved configuration, the upper magnet array  10   C  includes a plurality of upper shifted magnet arrays and a plurality of upper reference magnet arrays, and the lower magnet array  20   C  includes a plurality of lower reference magnet arrays and a plurality of lower shifted magnet arrays. 
     The configuration of the undulator magnet array  1   C  will now be described more specifically. The upper magnet array  10   C  is formed by coupling together a plurality of upper shifted magnet arrays  10 S C  and a plurality of upper reference magnet arrays  10 R C , and the lower magnet array  20   C  is formed by coupling together a plurality of lower shifted magnet arrays  20 S C  and a plurality of lower reference magnet arrays  20 R C . 
     In the upper magnet array  10   C , the upper shifted and reference magnet arrays  10 S C  and  10 R C  are disposed alternately. That is, in the upper magnet array  10   C , one upper shifted magnet array  10 S C  is disposed between one upper reference magnet array  10 R C  and another upper reference magnet array  10 R C  (assuming that the upper shifted magnet array  10 S C  of interest is not located at an end of the upper magnet array  10   C ), and one upper reference magnet array  10 R C  is disposed between one upper shifted magnet array  10 S C  and another upper shifted magnet array  10 S C  (assuming that the upper reference magnet array  10 R C  of interest is not located at an end of the upper magnet array  10   C ). 
     In the lower magnet array  20   C , the lower shifted and reference magnet arrays  20 S C  and  20 R C  are disposed alternately. That is, in the lower magnet array  20   C , one lower shifted magnet array  20 S C  is disposed between one lower reference magnet array  20 R C  and another lower reference magnet array  20 R C  (assuming that the lower shifted magnet array  20 S C  of interest is not located at an end of the lower magnet array  20   C ), and one lower reference magnet array  20 R C  is disposed between one lower shifted magnet array  20 S C  and another lower shifted magnet array  20 S C  (assuming that the lower reference magnet array  20 R C  of interest is not located at an end of the lower magnet array  20   C ). 
     Suppose the Z-axis coordinate (position on Z-axis) of the left end of the upper magnet array  10   C  and the Z-axis coordinate (position on Z-axis) of the left end of the lower magnet array  20   C  are equal. Then, the j-th upper shifted magnet array  10 S C  from the left end of the upper magnet array  10   C  is disposed opposite the j-th lower reference magnet array  20 R C  from the left end of the lower magnet array  20   C  (where j is an integer). That is, at the positions opposite the 0°, 90°, 180°, and 270° magnets  30  in the j-th upper shifted magnet array  10 S C , the 90°, 0°, 270°, and 180° magnets  30  in the j-th lower reference magnet array  20 R C  are arranged respectively. More specifically, the straight line connecting between the center position of the i-th magnet in the j-th upper shifted magnet array  10 S C  and the center position of the i-th magnet in the j-th lower reference magnet array  20 R C  is parallel to Y-axis, the first to fourth magnets in the j-th upper shifted magnet array  10 S C  being 0°, 90°, 180°, and 270° magnets  30  respectively, the first to fourth magnets in the j-th lower reference magnet array  20 R C  being 90°, 0°, 270°, and 180° magnets  30  respectively. 
     The j-th lower shifted magnet array  20 S C  from the left end of the lower magnet array  20   C  is disposed opposite the j-th upper reference magnet array  10 R C  from the left end of the upper magnet array  10   C . That is, at the positions opposite the 0°, 270°, 180°, and 90° magnets  30  in the j-th lower shifted magnet array  20 S C , the 270°, 0°, 90°, and 180° magnets  30  in the j-th upper reference magnet array  10 R C  are arranged respectively. More specifically, the straight line connecting between the center position of the i-th magnet in the j-th lower shifted magnet array  20 S C  and the center position of the i-th magnet in the j-th upper reference magnet array  10 R C  is parallel to Y-axis, the first to fourth magnets in the j-th lower shifted magnet array  20 S C  being 0°, 270°, 180°, and 90° magnets  30  respectively, the first to fourth magnets in the j-th upper reference magnet array  10 R C  being 270°, 0°, 90°, and 180° magnets  30  respectively. 
     In the example in  FIG. 9 , shifted and reference magnet arrays alternate every period of arrangement of the magnets  30 . That is, in the example in  FIG. 9 , the magnet arrays  10 S C ,  10 R C ,  20 S C , and  10 R C  each contain magnets  30  corresponding to one period, that is, only four magnets  30 . Accordingly, a comparison between  FIGS. 6 and 9  leads to the following observation: the four magnets  30  from the right end of the upper shifted magnet array  10 S B  in  FIG. 6  form the upper shifted magnet array  10 S C  in  FIG. 9 ; the four magnets  30  from the right end of the lower reference magnet array  20 S B  in  FIG. 6  form the lower reference magnet array  20 R C  in  FIG. 9 ; the four magnets  30  from the left end of the upper reference magnet array  10 R B  in  FIG. 6  form the upper reference magnet array  10 R C  in  FIG. 9 ; and the four magnets  30  from the left end of the lower shifted magnet array  20 S B  in  FIG. 6  form the lower shifted magnet array  20 S C  in  FIG. 9 . 
     Although in the example described above, one upper shifted magnet array, one upper reference magnet array, one lower shifted magnet array, and one lower reference magnet array are each composed of magnets  30  corresponding to one period, each of those magnet arrays may instead be composed of magnets  30  corresponding to a plurality of periods (for example, ten periods). 
     With the third improved configuration, not only is it possible to obtain the effects and benefits of the second improved configuration, but it is in addition possible to disperse the compressive and tensile forces that act on the structural components (such as the magnet array beams  110  and  120  in  FIG. 19  referred to later) that hold the magnet arrays integrally, and also to reduce the rigidity required in those structural components. 
     Fourth Improved Configuration 
     A fourth improved configuration of the undulator magnet array will be described. Through the approaches described previously in connection with the first to third improved configurations, ideally, the attractive and repulsive forces between the upper and lower magnet arrays balance out completely. In reality, however, owing to the magnetic permeability of actual magnets  30  not being equal to one, and also owing to the upper and lower magnet arrays being divided into shifted and reference magnet arrays, the attractive and repulsive forces do not completely balance out, leaving either an attractive force or a repulsive force to appear depending on the gap (see  FIG. 10C ). This brings about a state where the curve of the dependence of the attractive force on the gap (see  FIG. 10A ) and the curve of the dependence of the repulsive force on the gap (see  FIG. 10B ) do not match. 
     One possible solution is to fine-adjust the attractive and repulsive forces by shifting the upper or lower shifted magnet arrays, which are already shifted by one-fourth of the period, further leftward or rightward from its shifted position by a minute amount. In the fourth improved configuration, such shifting by a minute amount is approximated by rotating the magnetization direction. 
     An undulator magnet array  1   D  shown in  FIG. 11  according to the fourth improved configuration is an example of the undulator magnet array  1 , and includes an upper magnet array  10   D  and a lower magnet array  20   D  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper magnet array  10   D  is formed by coupling together a plurality of upper shifted magnet arrays  10 S D  and a plurality of upper reference magnet arrays  10 R D , and the lower magnet array  20   D  is formed by coupling together a plurality of lower shifted magnet arrays  20 S D  and a plurality of lower reference magnet arrays  20 R D . 
     By replacing the upper shifted and reference magnet arrays  10 S C  and  10 R C  in the upper magnet array  10   C  in  FIG. 9  with upper shifted and reference magnet arrays  10 S D  and  10 R D  respectively, the upper magnet array  10   D  in  FIG. 11  is formed. By replacing the lower shifted and reference magnet arrays  20 S C  and  20 R C  in the lower magnet array  20   C  in  FIG. 9  with lower shifted and reference magnet arrays  20 S D  and  20 R D  respectively, the lower magnet array  20   D  in  FIG. 11  is formed. 
     In the magnet arrays  10 R C  and  20 S C  in  FIG. 9 , the 0°, 90°, 180°, 270° magnets  30  can be replaced with (360°−Δφ), (90°−Δφ), (180°−Δφ), and (270°−Δφ) magnets  30  respectively, and the magnet arrays  10 R C  and  20 S C  having undergone the replacement are the magnet arrays  10 R D  and  20 S D  respectively. In the magnet arrays  10 S C  and  20 R C  in  FIG. 9 , the 0°, 90°, 180°, 270° magnets  30  can be replaced with (0°+Δφ), (90°+Δφ). (180°+Δφ), and (270°+Δφ) magnets  30  respectively, and the magnet arrays  10 S C  and  20 R C  having undergone the replacement are the magnet arrays  10 S D  and  20 R D  respectively. The symbol Δφ represents a predetermined positive angular amount, which is smaller than 90° and is typically a minute angular amount close to 0°. Rotating the magnetization direction by Δφ exerts an effect equivalent to shifting by a minute amount as mentioned above. 
     The magnetization direction of the magnets  30  is designed to suit the range of the gap in actual use, and by forming the undulator magnet array  1   D  based on the results of such designing, while the range of the gap is limited, it is possible to bring the attractive and repulsive forces sufficiently close to zero. 
     Simulations 
     Next, the details and results of simulations performed with some of the undulator magnet arrays described above will be presented. In these simulations, it was assumed that the period λ u  was 18 mm (millimeters) and the total length of the undulator magnet array (its length in the Z-axis direction) was 4.5 m (meters). It was further assumed that the variable range of the gap was 3 to 9 mm, and that the residual flux density and the relative magnetic permeability of the permanent magnets constituting the magnets  30  were 1.2 T (tesla) and 1.06 respectively. 
       FIG. 12  shows the results of a simulation (computation) on the attractive force in an undulator magnet array across different gaps. In  FIG. 12 , the horizontal axis is a scale of the gap, and the vertical axis is a scale of the attractive force between the upper and lower magnet arrays. Here, a negative attractive force is actually a repulsive force. The solid sequential line  410   REF  represents the results of computation of the attractive force between the upper and lower magnet arrays in the undulator magnet array  1   REF  in  FIG. 2 . The solid sequential line  410   B  represents the results of computation of the attractive force between the upper and lower magnet arrays in the undulator magnet array  1   B  in  FIG. 6 . The broken sequential line  410   C  represents the results of computation of the attractive force between the upper and lower magnet arrays in the undulator magnet array  1   C  in  FIG. 9 . 
     With a gap of 3 mm, as compared with the undulator magnet array  1   REF , which exhibited an attractive force of about 5.4 tf (tons-force), the undulator magnet array  1   B  exhibited an attractive force of about 220 kgf (kilograms-force), achieving reduction of the attractive force down to the order of the weight of the undulator magnet array itself. The undulator magnet array  1   C , while exhibiting a repulsive force exceeding an attractive force in a region of larger gaps, exhibited a magnetic force as low as about 400 kgf in terms of absolute value. The undulator magnet arrays  1   REF ,  1   B , and  1   C , all exhibited so weak a force in the left-right direction as to be regarded as zero. 
       FIG. 13  shows the magnetic field distribution in the undulator magnet array with a gap of 3 mm under the simulation conditions (computation conditions) mentioned above. In  FIG. 13 , the horizontal axis is a scale of the position (z) in the Z-axis direction, and the vertical axis is a scale of the Y-axis component By of the undulator magnetic field B. The solid-line curve  420   REF  represents the variation of the magnetic field By against the variation of the position in the Z-axis direction in the undulator magnet array  1   REF , and the solid-line curve  420   B  represents the variation of the magnetic field By against the variation of the position in the Z-axis direction in the undulator magnet array  1   B . Though not shown in  FIG. 13 , the undulator magnet array  1   C  exhibited a curve approximately similar to the solid-line curve  420   B . 
     It is seen that, while the intensity of the magnetic field produced by the undulator magnet array  1   B  (or  1   C ) is about 75% of that produced by the undulator magnet array REF, the undulator magnet array  1   B  (or  1   C ) produced a magnetic field with sufficient intensity and satisfactory periodicity. It is known that, in a Halbach magnet array, the magnetic attractive force is proportional to the square of the intensity of the undulator magnetic field B. With the undulator magnet array  1   REF , even when the intensity of the undulator magnetic field B is reduced to about 75% of that in the state in  FIG. 13 , 50% or more of the original attractive force remains (for example, an attractive force of “5.4 tons×0.75 2 ” remains). Thus, even with consideration given to the drop of the magnetic field intensity resulting from the shifting of magnet arrays, the undulator magnet array  1   B  (or  1   C ) turns out to be superior. 
     Second Embodiment 
     A second embodiment of the present invention will be described. The second embodiment, as well as the third to fifth embodiments described later, is based on the first embodiment, and to the features of the second to fifth embodiments that go unmentioned in the following description, the description of the corresponding features in the first embodiment applies unless inconsistent. Features from different ones of the first to fifth embodiments may be combined together unless inconsistent. 
     The amount of leftward shift Δz (example of predetermined shift amount) of the upper shifted magnet array relative to the lower reference magnet array will be defined in terms of the phase φz. The amount of leftward shift of the lower shifted magnet array relative to the upper reference magnet array also equals Δz. The phase φz corresponding to one period λ u  equals 2π in radian notation. Therefore, the phase φz is given by “φz=2π×Δz/λ u ”. In the undulator magnet array  1   REF , no shifting as described above is involved, and thus, in the undulator magnet array  1   REF , “φz=0”. In the undulator magnet arrays  1   A ,  1   B ,  1   C , and  1   D  described above, the shift amount Δz equals one-fourth of the period λ u , and thus “φz=π/2”. In the undulator magnet array in  FIG. 5 , the shift amount Δz equals one-half of the period λ u , and thus “φz=π”. 
     In  FIG. 14 , the curve Fy represents the magnetic force in the Y-axis direction that acts between the upper and lower magnet arrays. In  FIG. 14 , the shift amount Δz by which the upper shifted magnet array is shifted rightward relative to the lower reference magnet array is assumed to be negative. The magnetic force Fy in the Y-axis direction is an attractive force when “−π/2&lt;φz&lt;π/2”, and is a repulsive force when “−π≤φz&lt;−π/2” or “π/2&lt;φz≤π”. The magnetic force Fy in the Y-axis direction as an attractive force is at its maximum when “φz=0”, and decreases toward zero as the phase φz increases or decreases from 0 toward π/2 or (−π/2). The magnetic force Fy in the Y-axis direction as a repulsive force is at its maximum when “φz=π”, and decreases toward zero as the phase φz decreases from π toward π/2 or increases from (−π) toward (−π/2). 
     In  FIG. 14 , the curve By AMP  represents the magnitude of the amplitude of the Y-axis component of the undulator magnetic field B. The magnitude By AMP  of the amplitude of the Y-axis component of the undulator magnetic field B is at its maximum when the phase φz equals 0, and decreases toward 0 as the phase φz increases or decreases from 0 to π or (−π). 
     The magnetic force Fy in the Y-axis direction and the magnitude By AMP  of the amplitude of the Y-axis component of the undulator magnetic field B do not rely on the polarity of the phase φz but is determined only by the absolute value of the phase φz, and accordingly the following description focuses on the range of “0≤φz≤π”. 
     The first embodiment places emphasis on the cancelling-out of the attractive and repulsive forces, and proposes an undulator magnet array ( 1   A ,  1   B ,  1   C , or  1   D ) such that “φz=π/2”. Instead, an undulator magnet array may be formed such that the phase φz equals neither zero nor π/2. Specifically, the undulator magnet array  1  may be formed by use of an arbitrary shift amount Δz that fulfills, for example, “0&lt;φz&lt;π”. 
     In generalized terms, an undulator magnet array  1   GN  (unillustrated) according to the second embodiment, which is an example of the undulator magnet array  1 , includes an upper magnet array formed by coupling together an upper shifted magnet array and an upper reference magnet array each composed of a plurality of magnets  30  and a lower magnet array formed by coupling together a lower reference magnet array and a lower shifted magnet array each composed of a plurality of magnets  30 , wherein the upper shifted magnet array is disposed opposite the lower reference magnet array, the upper shifted magnet array being shifted relative to the lower reference magnet array by a predetermined shift amount Δz in a predetermined direction (in the example in  FIG. 6 , in the leftward direction) parallel to the magnet arrangement direction as compared with in a reference state (hereinafter referred to as the maximized magnetic-field state, which occurs when φz=0) where the amplitude of the periodic magnetic field produced by the upper and lower magnet arrays is maximized, and the lower shifted magnet array is disposed opposite the upper reference magnet array, the lower shifted magnet array being shifted relative to the upper reference magnet array by a predetermined shift amount Δz in a predetermined direction (in the example in  FIG. 6 , in the leftward direction) parallel to the magnet arrangement direction as compared with in the maximized magnetic-field state. 
     The undulator magnet array  1   A ,  1   B ,  1   C , or  1   D  according to the first embodiment is a type of undulator magnet array  1   GN , and undulator magnet arrays according to the third and fourth embodiments described later also belong to the undulator magnet array  1   GN . 
     In the undulator magnet array  1   GN , the shift amount Δz is determined so as to fulfill “0&lt;φz&lt;π”. That is, the shift amount Δz is less than one-half of the period λ u  of the change of the magnetization direction of the magnets  30 . Thus, as compared with the undulator magnet array  1   REF  where “φz=0”, it is possible to reduce the magnitude of the magnetic force (attractive or repulsive force) that occurs between the upper and lower magnet arrays, and it is thus possible to obtain effects and benefits as mentioned in the description of, among others, the first and second improved configurations of the first embodiment. 
     However, with the phase φz close to 0 or close to π, it is difficult to obtain substantial benefits. Accordingly, for example, in the undulator magnet array  1   GN , it is preferable that the shift amount Δz be determined so as to fulfill “π/4≤φz≤3π/4”. That is, it is preferable that the shift amount Δz be one-eighth or more but three-eighths or less of the period λ u . 
     Typically, for example, as in the undulator magnet array  1   A ,  1   B ,  1   C , or  1   D  according to the first embodiment, in the undulator magnet array  1   GN , it is preferable that the shift amount Δz be determined so as to fulfill “φz=π/2”. That is, it is preferable that the shift amount Δz be one-fourth of the period λ u . This maximizes the effect of reducing the magnitude of the magnetic force that occurs between the upper and lower magnet arrays. Here, defining the shift amount Δz to be one-fourth of the period λ u  should be understood not to preclude allowing for a slight margin to accommodate an error. As mentioned in the description of the fourth improved configuration of the first embodiment, the gap-dependence of the attractive force and the gap-dependence of the repulsive force, even if the difference between them is slight, do differ from each other. To cope with that, the shift amount Δz may be deviated slightly from one-fourth of the period λ u . Even with this deviation, it is possible to consider the shift amount Δz substantially equal to one-fourth of the period λ u , and it is possible to consider that the deviation permits the shift amount Δz to be in the range of one-eighth or more but three-eighths or less of the period λ u . 
     As described previously with reference to  FIG. 7 , in the upper shifted magnet array in the undulator magnet array  1   GN , in a region (corresponding to the to-be-interpolated region  330  in  FIG. 7 ) within the distance corresponding to the shift amount Δz from the upper reference magnet array neighboring, on the right of, the upper shifted magnet array, the magnet  30  of which the magnetization direction is determined based on the magnetization direction of a magnet  30  in the upper reference magnet array is disposed. This helps preserve the periodicity of the undulator magnetic field B. More specifically, in the above-mentioned region (corresponding to the to-be-interpolated region  330  in  FIG. 7 ), the magnet  30  with the same magnetization direction as the magnet  30  in the predetermined region (corresponding to the interpolation-source region  340  in  FIG. 7 ) in the upper reference magnet array is disposed. The predetermined region here (corresponding to the interpolation-source region  340  in  FIG. 7 ) is the region between, of opposite ends of the upper reference magnet array, the end closer to the upper shifted magnet array (that is, the left end of the upper magnet array  320  in  FIG. 7 ) and the position displaced from that end rightward by the shift amount Δz. 
     Like the undulator magnet array  1   C  or  1   D  of the first embodiment, the undulator magnet array  1   GN  may include a plurality of upper shifted magnet arrays, a plurality of upper reference magnet arrays, a plurality of lower shifted magnet arrays, and a plurality of lower reference magnet arrays. In this way, it is possible to distribute the compressive and tensile forces that act on the structural components (such as the magnet array beams  110  and  120  in  FIG. 19  referred to later) that hold the magnet arrays integrally, and it is thus possible to reduce the rigidity required in those structural components. In that case, in the upper magnet array, the upper shifted and reference magnet arrays are coupled together alternately, and in the lower magnet array, the lower reference and shifted magnet arrays are coupled together alternately. 
     Third Embodiment 
     A third embodiment of the present invention will be described. Although the first embodiment assumes that the number M of magnets  30  present in one period λ u  is four, it is also possible to form an undulator magnet array  1  where M is other than four. As examples, undulator magnet arrays  1  where M=2 or M=8 will be described below, with no intention of excluding undulator magnet arrays  1  where M is other than 2, 4, and 8. 
       FIG. 15  is a diagram showing the configuration of an undulator magnet array  1   PA  formed with M=8. The undulator magnet array  1   PA  is an example of the undulator magnet array  1 , and includes an upper magnet array  10   PA  and a lower magnet array  20   PA  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper magnet array  10   PA  is formed by coupling together one upper shifted magnet array  10 S PA  and one upper reference magnet array  10 R PA . The lower magnet array  20   PA  is formed by coupling together one lower reference magnet array  20 R PA , which is disposed opposite the upper shifted magnet array  10 S PA , and one lower shifted magnet array  20 S PA , which is disposed opposite the upper reference magnet array  10 R PA . 
       FIG. 16  is a diagram showing the configuration of another undulator magnet array  1   PB  formed with M=8. The undulator magnet array  1   PB  is an example of the undulator magnet array  1 , and includes an upper magnet array  10   PB  and a lower magnet array  20   PB  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper magnet array  10   PB  is formed by coupling together a plurality of upper shifted magnet arrays  10 S PB  and a plurality of upper reference magnet arrays  10 R PB . The lower magnet array  20   PB  is formed by coupling together a plurality of lower reference magnet arrays  20 R PB  and a plurality of lower shifted magnet array  20 S PB . As in a case with M=4 (that is, as in the third improved configuration of the first embodiment), in the upper magnet array  10   PB , the upper shifted magnet arrays  10 S PB  and the upper reference magnet arrays  10 R PB  are disposed alternately, and in the lower magnet array  20   PB , the lower shifted magnet arrays  20 S PB  and the lower reference magnet arrays  20 R PB  are disposed alternately; in addition, the upper shifted magnet arrays  10 S PB  are each disposed opposite one of the lower reference magnet array  20 R PB , and the lower shifted magnet arrays  20 S PB  are each disposed opposite one of the upper reference magnet arrays  10 R PB . 
     In the undulator magnet array  1   PA  in  FIG. 15  and in the undulator magnet array  1   PB  in  FIG. 16 , M=8, and thus the period λ u  equals “M×d=8×d” (see  FIG. 3C ); in each of the upper shifted, upper reference, lower shifted, and lower reference magnet arrays, the magnetization direction of the magnets  30  changes, in YZ-plane, by 45° (=360°/M) from one magnet to the next along the direction parallel to Z-axis. However, the direction of the change is opposite between the upper and lower magnet arrays. 
     In the undulator magnet array  1   PA  in  FIG. 15  and in the undulator magnet array  1   PB  in  FIG. 16 , the shift amount Δz equals λ u /4, which corresponds to φz=π/2. However, as mentioned in connection with the second embodiment, the shift amount Δz in them may instead be set at other than one-fourth of the period λ u . In the undulator magnet array  1   PB  in  FIG. 16 , one upper shifted magnet array, one upper reference magnet array, one lower shifted magnet array, and one lower reference magnet array are each composed of magnets corresponding to one period, they may instead be each composed of magnets corresponding to a plurality of (for example, ten) periods. 
       FIG. 17  is a diagram showing the configuration of an undulator magnet array  1   QA  formed with M=2. The undulator magnet array  1   QA  is an example of the undulator magnet array  1 , and includes an upper magnet array  10   QA  and a lower magnet array  20   QA  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper magnet array  10   QA  is formed by coupling together one upper shifted magnet array  10 S QA  and one upper reference magnet array  10 R QA . The lower magnet array  20   QA  is formed by coupling together one lower reference magnet array  20 R QA , which is disposed opposite the upper shifted magnet array  10 S QA , and one lower shifted magnet array  20 S QA , which is disposed opposite the upper reference magnet array  10 R QA . 
       FIG. 18  is a diagram showing the configuration of another undulator magnet array  1   QB  formed with M=2. The undulator magnet array  1   QB  is an example of the undulator magnet array  1 , and includes an upper magnet array  10   QB  and a lower magnet array  20   QB  as an example of the upper and lower magnet arrays  10  and  20  respectively. The upper magnet array  10   QB  is formed by coupling together a plurality of upper shifted magnet arrays  10 S QB  and a plurality of upper reference magnet arrays  10 R QB . The lower magnet array  20   QB  is formed by coupling together a plurality of lower reference magnet arrays  20 R QB  and a plurality of lower shifted magnet array  20 S QB . As in a case with M=4 (that is, as in the third improved configuration of the first embodiment), in the upper magnet array  10   QB , the upper shifted magnet arrays  10 S QB  and the upper reference magnet arrays  10 R QB  are disposed alternately, and in the lower magnet array  20   QB , the lower shifted magnet arrays  20 S QB  and the lower reference magnet arrays  20 R QB  are disposed alternately; in addition, the upper shifted magnet arrays  10 S QB  are each disposed opposite one of the lower reference magnet array  20 R QB , and the lower shifted magnet arrays  20 S QB  are each disposed opposite one of the upper reference magnet arrays  10 R QB . 
     In the undulator magnet array  1   QA  in  FIG. 17  and in the undulator magnet array  1   QB  in  FIG. 18 , M=2, and thus the period λ u  equals “M×d=2×d” (see  FIG. 3B ); in each of the upper shifted, upper reference, lower shifted, and lower reference magnet arrays, the magnetization direction of the magnets  30  changes, in YZ-plane, by 180° (=360°/M) from one magnet to the next along the direction parallel to Z-axis. 
     In the undulator magnet array  1   QA  in  FIG. 17  and in the undulator magnet array  1   QB  in  FIG. 18 , the shift amount Δz equals λ u /4, which corresponds to φz=π/2. With M=2, the width in the Z-axis direction of magnets corresponding to one-fourth of the period λ u  equals one half of the width d of the magnet  30 . A magnet of which the width in the Z-axis direction equals d/2, that is, a magnet half the size of the magnet  30  will be identified as a magnet  30   a . Two magnets  30   a  correspond to one magnet  30 . In each of the upper shifted magnet arrays ( 10 S QA ,  10 S QB ) and the lower shifted magnet arrays ( 20 S QA ,  20 S QB ) in  FIGS. 17 and 18 , the shifting by λ u /4 results in partial use of magnets  30   a . The shift amount Δz in the undulator magnet array  1   QA  in  FIG. 17  and in the undulator magnet array  1   QB  in  FIG. 18  may, as mentioned in connection with the second embodiment, instead be set at other than one-fourth of the period λ u . In the undulator magnet array  1   QB  in  FIG. 18 , one upper shifted magnet array, one upper reference magnet array, one lower shifted magnet array, and one lower reference magnet array are each composed of magnets corresponding to one period, they may instead be each composed of magnets corresponding to a plurality of (for example, ten) periods. 
     Fourth Embodiment 
     A fourth embodiment of the present invention will be described.  FIG. 19  shows an undulator (undulating equipment)  100  according to the fourth embodiment. 
     The undulator  100  includes the following components: an undulator magnet array  1  including an upper magnet array  10  and a lower magnet array  20 ; a magnet array beam  110  which holds the upper magnet array  10  integrally; a magnet array beam  120  which holds the lower magnet array  20  integrally; a vacuum chamber  130  which keeps in a vacuum state the space that encloses the undulator magnet array  1  and the magnet array beams  110  and  120 ; a high-rigidity beam  140 U which is disposed over the vacuum chamber  130  and which supports the upper magnet array  10  and the magnet array beam  110  from above; a high-rigidity beam  140 L which is disposed under the vacuum chamber  130  and which supports the lower magnet array  20  and the magnet array beam  120  from below; a vacuum introduction coupler  150 U which couples together the high-rigidity beam  140 U and the magnet array beam  110  by use of shafts  151 U while keeping the vacuum state in the vacuum chamber  130 ; a vacuum introduction coupler  150 L which couples together the high-rigidity beam  140 L and the magnet array beam  120  by use of shafts  151 L while keeping the vacuum state in the vacuum chamber  130 ; a ball screw driving mechanism  160  which is a driving mechanism coupled to the high-rigidity beams  140 U and  140 L and which enables, by use of a ball screw, the high-rigidity beams  140 U and  140 L to move in the up-down direction; and a base  170  to which the ball screw driving mechanism  160  is fitted and which has a substantially L-shaped sectional shape. A vacuum state in the vacuum chamber  130  denotes a state close to a vacuum, and may be any state with a barometric pressure at least lower than the atmospheric pressure. 
       FIG. 20  is a front view of the undulator  100 . In  FIG. 20 , for the sake of convenient illustration, the vacuum chamber  130  is omitted. Moreover, in  FIG. 20 , of all the constituent elements of the vacuum introduction couplers  150 U and  150 L, only a plurality of shafts  151 U which physically couple together the high-rigidity beam  140 U and the magnet array beam  110  and a plurality of shafts  151 L which physically couple together the high-rigidity beam  140 L and the magnet array beam  120  are shown. 
     According to control signals from an unillustrated controller, the ball screw driving mechanism  160  can, by moving both the high-rigidity beams  140 U and  140 L individually in the up-down direction, or by moving one of the high-rigidity beams  140 U and  140 L in the up-down direction, vary the gap between the upper and lower magnet arrays  10  and  20 . More specifically, for example, the ball screw driving mechanism  160  can, by moving the high-rigidity beam  140 U upward and the high-rigidity beam  140 L downward by the same amount, increase the gap between the upper and lower magnet arrays  10  and  20 , and can, by moving the high-rigidity beam  140 U downward and the high-rigidity beam  140 L upward by the same amount, decrease the gap between the upper and lower magnet arrays  10  and  20 . Thus, it can be said that the undulator  100  is provided with a holder which holds the undulator magnet array  1  such that the gap (interval) between the upper and lower magnet arrays  10  and  20  is variable. The holder can be considered to include the magnet array beams  110  and  120 , the high-rigidity beams  140 U and  140 L, the vacuum introduction couplers  1500  and  150 L, the ball screw driving mechanism  160 , and the base  170 . 
       FIG. 21  is a plan view schematically showing a synchrotron radiation facility  200 . The synchrotron radiation facility  200  includes an electron gun  201 , a linear accelerator  202 , a synchrotron  203 , a storage ring  204 , and one or more beam lines  205 . In the storage ring  204 , near a base part of each beam line  205 , an undulator  100  is disposed. 
     Electrons e are emitted from the electron gun  201 , are accelerated to a speed corresponding to an energy of about 1 GeV (gigaelectronvolts) by the linear accelerator  202 , are then further accelerated to a speed corresponding to an energy of about 8 GeV by the synchrotron  203  using radio-frequency waves, and then enter the storage ring  204  at a speed close to that of light. 
     The electrons e circulate inside the storage ring  204  while maintaining their energy, and are undulated by the periodic magnetic field produced by the undulator magnet array  1  disposed inside the storage ring  204  to emit synchrotron radiation R. The synchrotron radiation R enters a beam line  205 , and is, inside the beam line  205 , used for various research and practical purposes. 
     As described above, with the technology according to the first to fourth embodiments, it is possible to greatly reduce the magnetic attractive force between the upper and lower magnet arrays. It is thus possible to reduce the rigidity required in the structural components that support the upper and lower magnet arrays, and also to simplify the structure (including a driving mechanism) of an undulator and to greatly reduce the weight of the undulator as a whole. Consequently, it is possible to greatly save cost and time related to the manufacture and installation of the undulator. 
     Modifications 
     The embodiments of the present invention allow for many modifications made as necessary within the scope of the technical concept set forth in the appended claims. The embodiments described above are merely examples of how the present invention can be implemented, and the senses of the terms used to define the present invention and its features are not limited to those in which they are used in the description of the embodiments given above. All specific values mentioned in the above description are merely examples, and can naturally be altered to different values. 
     Although the above description deals with cases where the present invention is applied to a pair of magnet arrays (upper and lower magnet arrays) disposed opposite each other, similar configurations may be applied to various types of undulator magnet arrays and undulators (undulating equipment). For example, a Figure-8 undulator or a Spring-8 helical undulator includes three sets of magnet arrays (upper and lower magnet arrays), each set comprising a pair of magnet arrays disposed opposite each other, and the present invention can be applied to each of those sets. For another example, an Apple II undulator includes two sets of magnet arrays (upper and lower magnet arrays), each set comprising a pair of magnet arrays disposed opposite each other, and the present invention can be applied to each of those sets. 
     Although in the above embodiments, the magnet arrays  10  and  20  are assumed to be disposed side by side in the up-down direction, they may instead be disposed side by side in any direction (for example, in the left-right direction) other than the up-down direction. 
     LIST OF REFERENCE SIGNS 
       1 ,  1   REF ,  1   A  to  1   D ,  1   PA ,  1   PB ,  1   QA ,  1   QB  undulator magnet array 
       10 ,  10   REF ,  10   A  to  10   D ,  10   PA ,  10   PB ,  10   QA ,  10   QB  upper magnet array 
       10 S B  to  10 S D ,  10 S PA ,  10 S PB ,  10 S QA ,  10 S QB  upper shifted magnet array 
       10 R B  to  10 R D ,  10 R PA ,  10 R PB ,  10 R QA ,  10 R QB  upper reference magnet array 
       20 ,  20   REF ,  20   A  to  20   D ,  20   PA ,  20   PB ,  20   QA ,  20   QB  lower magnet array 
       20 S B  to  20 S D ,  20 S PA ,  20 S PB ,  20 S QA ,  20 S QB  lower shifted magnet array 
       20 R B  to  20 R D ,  20 R PA ,  20 R PB ,  20 R QA ,  20 R QB  lower reference magnet array 
       30 ,  30   a  magnet 
       100  undulator