Passive Shimming by Eliminating Spherical Harmonics Coefficients of All Magnetic Field Components Generated by Correction Iron Pieces

Transcription

1 Title Paive Shimming b Eliminating Spherial Harmoni Corretion Iron Piee Author()Noguhi, So; Kim, SeokBeom; Hahn, Seungong; Iwaa, CitationIEEE Tranation on Magneti, 50(2): Iue Date Do URL Right 2014 IEEE. Peronal ue of thi material i permit an urrent or future media, inluding reprinting/re reating new olletive work, for reale or reditr thi work in other work. Tpe artile (author verion) File Information MAS_revied.pdf Intrution for ue Hokkaido Univerit Colletion of Sholarl and Aa

2 CMP Paive Shimming b Eliminating Spherial Harmoni Coeffiient of All Magneti Field Component Generated b Corretion Iron Piee So Noguhi 1, SeokBeom Kim 2, Seungong Hahn 3, and Yukikau Iwaa 3 1 Graduate Shool of Information Siene and Tehnolog, Hokkaido Univerit, Sapporo , Japan 2 Graduate Shool of Natural Siene and Tehnolog, Okaama Univerit, Okaama , Japan 3 Frani Bitter Magnet Laborator, Maahuett Intitute of Tehnolog, Cambridge, MA 02139, USA When a magnet generating highl homogeneou magneti field i deigned, a himming i required. The himming i uuall performed o that the magneti field onl in the ai diretion i ompenated b iron piee and/or oil. The ompenation i ommonl ahieved b eliminating the oeffiient of the pherial harmoni epanion of the magneti field generated b the main magnet. Some paper howed the oeffiient of the pherial harmoni epanion in onl the -diretion for paive himming. However, reentl ome magnet generate a magneti field tilted from the -ai, uh a a magi-angle-pinning NMR/MRI. It i, therefore, neear to eliminate the oeffiient of the pherial harmoni epanion in the - and -diretion. Inde Term Magneti moment, MRI, NMR, paive himming, pherial harmoni epanion. I. INTRODUCTION IT i neear to ompenate a magneti field around the enter with himming when it i required to generate highl homogeneou magneti field, uh a MRI and NMR. The paive himming, one of the himming method, i to ompenate the magneti field uing ome piee of iron [1 [3. Sine the magneti field homogeneit of a few PPM i required around the magnet enter, the pherial harmoni epanion of the magneti field i uuall emploed [4, [5. A magnet for MRI or NMR i ommonl aiall mmetri. Therefore, onl the -omponent of the magneti moment of orretion iron piee ued for paive himming i taken into aount o that onl the -omponent of the magneti field i homogenied [4. The paper [5 preented the pherial harmoni oeffiient of the -omponent of the magneti field generated b all the omponent of magneti moment beaue the open MRI magnet generate the aiall ammetri magneti field. In addition, the onfiguration of the orretion iron piee wa optimied in order to eliminate the pherial harmoni oeffiient. Reentl a newl developed magnet generate a magneti field tilted from the -ai for a magi-angle-pinning NMR [6. Hene, the oeffiient of the pherial harmoni epanion of all the magneti field omponent, that are generated b all the omponent of the magneti moment, are trongl deired to be eliminated b the paive himming. Conequentl, we derived the pherial harmoni oeffiient of all the magneti field omponent to all the magneti moment omponent, and then we tried the paive himming of the magi-angle-pinning NMR. In the paive himming, the onfiguration of the orretion iron piee are optimied b the miro geneti algorithm (µga) in order to the pherial Manuript reeived June 29, Correponding author: S. Noguhi ( noguhi@i.it.hokudai.a.jp). harmoni oeffiient of all the omponent. II. COEFFICIENTS OF SPHERICAL HARMONICS A. Spherial Harmoni Coeffiient b Magneti Moment The magnet flu dφ generated b the magneti moment dm at point Q a hown in Fig. 1(a) i preented b dφ = dm ( ) 1 Q (1) 4πµ 0 R where µ 0 and R are the permeabilit of free pae and the ditane between the point Q and P [7, repetivel. Uing the pherial harmoni funtion, 1/R i given a 1 R = { (n m)! ε m (n + m)! r n r0 n+1 P m n (o α) P m n (o θ) o [m(ψ ϕ)} (2) where ε m i the Neumann fator, P m n i the aoiated Legendre funtion, r and r 0 are the ditane from the origin to the point P and Q, and α, θ, ψ, and ϕ are the angle hown in Fig. 1(a), repetivel. The magneti field db m i epreed a db m = µ 0 P dφ. (3) Here, the dimenion of the orretion iron piee with magnet moment dm = (dm ρ, dm ϕ, dm ) in lindrial omponent i given in Fig. 1(b). The magneti field B m generated b the orretion iron piee i obtained b the integration of the volume, a follow: 2 ρ2 ψ2 ( ) dm 1 B m = 4π P Q ρdψdρd. (4) R 1 ρ 1 ψ 1 Here, it i aumed that the magneti moment dm i ontant in a orretion iron piee. The following equation i derived

3 CMP from alulating (4). B m = where (n, m) D (n, m) [r n Pn m (o θ) {C (n, m) o mψ + D (n, m) in mψ} (5) = i (n, m) D (n, m) +i (n, m) D (n, m) [ C + i (n, m) D(n, m) (6) and i, i, and i are the unit vetor in the -, -, and - diretion, repetivel. The detail of the oeffiient C and D annot be given here beaue of no pae [8. φ r θ α ψ P (a) R r 0 dm Q jd in Fig. 1(a). The following equation i derived from (2) and (7): jd { (n m)! r n da = µ 0 ε m Pn m (o α) 4πr 0 (n + m)! P m n (o θ) o [m(ψ ϕ)} (8) The magneti field generated b the urrent B i B = dadv (9) v where v i the volume of the magnet. Finall, we an obtain uing (7) (9) B = [r n Pn m (o θ) where (n, m) D (n, m) {C (n, m) o mψ + D (n, m) in mψ}(10) = i (n, m) D (n, m) +i (n, m) D (n, m) r n 0 [ C + i (n, m) D(n, m). (11) The oeffiient C and D of the ring oil are repreented in [7. It i, however, neear to ompute the oeffiient C and D b a numerial integration when the magnet onfiguration i aiall ammetri, uh a a dipole magnet. When the main magnet generate a tilted magneti field, the pherial harmoni oeffiient an be tranformed uing the method preented in [ dm dm φ dm ρ C. Paive Shimming For ompenating the magneti field with paive himming, [ (n, m) C [ (n, m) 0 D + (n, m) D = (12) (n, m) 0 ψ1 ψ 2 (b) ρ 1 ρ 2 with n 1 ha to be atified. In the paive himming, the onfiguration of the orretion iron piee ha to be optimied in order to eliminate the oeffiient C (n, m) and D (n, m) b adding D (n, m) and D (n, m). The magneti moment dm i omputed b a numerial imulation method, uh a a finite element method or a magneti moment method. Fig. 1. (a) Magneti moment dm at point Q generate magneti flu dφ at point P and (b) dimenion of a piee of iron with dm = (dm ρ, dm ϕ, dm ) in the lindrial omponent. B. Spherial Harmoni Coeffiient b Current The urrent denit j with length of d generate the following magneti vetor potential da: da = µ 0 jd 4πR where R i the ditane between the point P and Q, replaing the magneti moment dm with the urrent denit element (7) III. PASSIVE SHIMMING A. Magi-Angle-Spinning NMR The paive himming i applied to a pair of dipole magnet of the magi-angle-pinning NMR [6. Fig. 2 how the dipole magnet with the tilt of deg. in the - plane, the enter magneti field i 0.3 T, and it dimenion are referred in [6. Table I how the pherial harmoni oeffiient of the magneti field generated b the dipole magnet, however n 4. For paive himming, the µga [10 optimie the onfiguration of the orretion iron piee. Conidering the mmetr, the deign region of the orretion iron piee, a hown in Fig. 2, i divided into 2 (radiall) 36 (irumferentiall)

4 CMP D i(n, m) + D i (n, m) ). (13) Region of Iron Piee Configuration where w(n) i the following weight funtion. w(n) = 10 2n (14) B deg. 360 The weight funtion i deided aording to the deired bore ie of the homogenied magneti field. When whole the iron piee are abent, the initial objetive funtion F = The magneti moment dm of ever iron piee i omputed b the magneti moment method in the lindrial oordinate. Dipole Magnet B. Paive Shimming Reult Current Current unit: mm Fig. 3 how the optimied onfiguration of the orretion iron piee, and Table II repreent the pherial harmoni oeffiient of the magneti field affeted b the orretion iron piee. The objetive funtion optimied i F = Fig. 4 and 5 how the abolute value of the nonero pherial harmoni oeffiient with n = 2 and 4, repetivel. The oeffiient of n = 2 are redued adequatel, however, thoe of n = 4 are not. Fig. 2. Dipole magnet for magi-angle-pinning NMR. The onfiguration of the iron piee i optimied b the µga. The quarter region of the onfiguration i onidered due to the mmetr, and it i divided into 2 (radiall) 36 (irumferentiall) 18 (heigtwie) egmental element. TABLE I SPHERICAL HARMONICS COEFFICIENTS OF DIPOLE MAGNETS Coeffiient omponent omponent omponent C (0, 0) C (1, 0) C (1, 1) C (2, 0) C (2, 1) C (2, 2) C (3, 0) C (3, 1) C (3, 2) C (3, 3) C (4, 0) C (4, 1) C (4, 2) C (4, 3) C (4, 4) D (1, 1) D (2, 1) D (2, 2) D (3, 1) D (3, 2) D (3, 3) D (4, 1) D (4, 2) D (4, 3) D (4, 4) (heightwie) egmental element, and the µga deide whether the iron piee eit in a ubdiviion or not, like a topolog optimiation [11. The objet funtion F to be minimie i F = 4 i=,, n=1 m=0 w(n) ( Ci (n, m) + Ci (n, m) Fig. 3. The optimied onfiguration of orretion iron piee b µga. The blue-olored piee are in the inner laer, and the red-olored one in the outer. Table III how the inhomogeneit of the magneti field within 10 mm phere to the dipole magnet with and without paive himming. The inhomogeneit i dratiall improved to 4.3 ppm. In thi optimiation, the ie of the iron piee i too large to redue all the pherial harmoni oeffiient largel.

5 CMP TABLE II SPHERICAL HARMONICS COEFFICIENTS OF IRON PIECES 1200 Dipole Magnet Abolute Value Coeffiient omponent omponent omponent C (0, 0) C (1, 0) C (1, 1) C (2, 0) C (2, 1) C (2, 2) C (3, 0) C (3, 1) C (3, 2) C (3, 3) C (4, 0) C (4, 1) C (4, 2) C (4, 3) C (4, 4) D (1, 1) D (2, 1) D (2, 2) D (3, 1) D (3, 2) D (3, 3) D (4, 1) D (4, 2) D (4, 3) D (4, 4) Dipole Magnet Iron Piee Total C(2,1) D(2,2) C(2,0) C(2,2) D(2,1) C(0,0) C(2,0) C(2,2) D(2,1) Non-ero Coeffiient Fig. 4. Abolute value of the non-ero pherial harmoni oeffiient with n = 2. However, it wa ueful to ompute the pherial harmoni oeffiient of all the omponent b all the magneti moment omponent, and the optimiation of the orretion iron piee onfiguration wa done. TABLE III INHOMOGENEITY OF MAGNETIC FIELD WITHIN 10 MM SPHERE w/o himming (ppm) with himming (ppm) -omponent omponent omponent All omponent IV. CONCLUSION We alulate the pherial harmoni oeffiient of all the omponent generated b all the magneti moment omponent. When a main magnet i tilted, it i neear to eliminate the pherial harmoni of all the omponent b paive himming. We have tried to ompenate the magneti Abolute Value Iron Piee Total C(4,1) C(4,3)D(4,2) D(4,4) C(4,0) C(4,2) C(4,4) D(4,1) D(4,3) C(4,0) C(4,2) C(4,4) D(4,1) D(4,3) Non-ero Coeffiient Fig. 5. Abolute value of the non-ero pherial harmoni oeffiient with n = 4. field of the tilted dipole magnet o that the onfiguration of the orretion iron piee wa optimied b the µga. In the future, it i neear to eliminate higher order oeffiient with a maller piee of iron. In addition, it i neear to emplo ative himming with paive one. The propoed method in thi paper i ueful in the imultaneou ative and paive himming. REFERENCES [1 A. Belov, V. Buhuev, M. Emelianov, V. Emelianov, V. Eregin, Y. Severgin, S. Sthevki, and V. Vailiev, Paive himming of the uperonduting magnet for MRI, IEEE Tran. Appl. Superond., vol. 5, no. 2, pp , [2 L. Feng, Z. Jianfeng, X. Lin, and S. Croier, A hbrid feild-harmoni approah for paive himming deign in MRI, IEEE Tran. Appl. Superond., vol. 21, no. 2, pp , Apr [3 Y. Zhang, D. Xie, B. Bai, H. S. Yoon, and C. S. Koh, A novel optimal deign method of paive himming for permanent MRI magnet, IEEE Tran. Magn., vol. 44, no. 6, pp , Jun [4 H. S. Lope, F. Liu, W. Weber, and S. Croier, Paive him and a himming approah for biplanar permanent magneti reonane imaging magnet, IEEE Tran. Magn., vol. 44, no. 3, pp , Mar [5 S. Kakugawa, H. Hino, A. Komura, M. Kiatmura, H. Takehima, T. Yatuo, and H. Taaki, Three-dimenional optimiation of orretion iron piee for open high field MRI tem, IEEE Tran. Appl. Superond., vol. 14, no. 2, pp , Jun [6 J. Voio, S. Hahn, D. K. Park, J. Ling, Y. Kim, J. Bauñán, and Y. Iwaa, Magi-angle-pinning NMR magnet development: field anali and prototpe, IEEE Tran. Appl. Superond., vol. 23, no. 3, p , Jun [7 F. Roméo and D. I. Hoult, Magnet field profiling: anali and orreting oil deign, Magneti Reonane in Mediine, vol. 1, pp , [8 S. Noguhi, F. Nobuama, S. B. Kim, S. Hahn, and Y. Iwaa, Spherial harmoni oeffiient of all magneti field omponent generated b iron piee, Pro. 19th COMPUMAG, PB5-14, Jul [9 M. A. Blano, M. Flóre, and M. Bermejo, Evaluation of the rotation matrie in the bai of real pherial harmoni, Journal of Moleular Struture, vol. 419, pp , [10 Y. Watanabe, K. Watanabe, and H. Igarahi, Optimiation of meander line antenna onidering oupling between nonlinear iruit and eletromagneti wave for UHF-band RFID, IEEE Tran. Magn., vol. 47, no. 5, pp , Ma [11 Y. Tominaga, Y. Okamoto, S. Wakao, and S. Sato, Binar-baed topolog optimiation of magnetotati hielding b a hbrid evolutionar algorithm ombining geneti algorithm and etended ompat geneti algorithm, IEEE Tran. Magn., vol. 49, no. 5, pp , Ma 2013.