MODELING OF THE MAGNETIC FIELD NEAR FLAT-COIL WITH A VIEW TO IMPROVING THE DETECTION EFFICIENCY OF FINE PECULIARITIES OF SUPERCONDUCTIVITY

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1 PROCEEDNGS OF THE YEREVAN STATE UNVERSTY Physical and Mahemaical Sciences 202,, p Ph ys i cs MODENG OF THE MAGNETC FED NEAR FAT-CO WTH A VEW TO MPROVNG THE DETECTON EFFCENCY OF FNE PECUARTES OF SUPERCONDUCTVTY B. K. KURGHNYAN Cene on Supeconduciviy and Scienific nsumenaion, Chai of Solid Sae Physics YSU, Amenia Modelling of a RF magneic field s configuaion nea he fla coil face of he unnel diode based single-laye fla-coil-oscillao (SFCO) has been done ha seves as a senso in one of mos of effecive mehods fo sudy of supeconduciviy (he SFCO echnique). Paiculaly, he disibuion of he measuing magneic field of fla coil is invesigaed. An analyical fomula has been deived, which allows o deemine he paallel and pependicula o coil face componens of he measuing RF magneic field a any poin nea he coil. The esuls of calculaions ae aimed a an achievemen of he opimum sensiiviy of he SFCO echnique duing he expeimenal sudy of fine peculiaiies of supeconduciviy ha ae impoan fo undesanding he phenomenon. Keywods: SFCO echnique (a single-laye fla-coil-oscillao mehod), opimizaion of expeimens, Fulde Feell akin Ovchinnikov Supeconduciviy. noducion. n he eseach insumenaion based on he single-laye flacoil-oscillao (SFCO) devices [] he fequency and ampliude of low powe unnel diode (TD) hamonic oscillao seve as measuing paamees [2]. The physical quaniies duing he invesigaion of supeconduciviy by means of his echnique [3] ae deemined boh by he disoion of he disibuion of magneic field lines of he measuing MHz fequency field of fla coil, and by he enegy absopion of he same field [5], as a esul of which he fequency o/and ampliude of he oscillao ae changed. So, in he famewoks of his mehod he knowledge of magneic line configuaion of he measuing RF magneic field of he single-laye fla coil senso is equied and modeling of he disibuion of magneic field lines is ugen. So, a measuemens by he use of he SFCO mehod, in ode o ge opimal condiions fo is applicaion, i is necessay o know he magneic field disibuion nea he fla coil ha is he esonance cicui of he oscillao. The field disibuion has been expeimenally invesigaed in deail in [4]. n his pape we have pesened he heoeical invesigaion of he disibuion of measuing magneic field nea fla coil and have compaed wih available expeimenal daa in [4]. Theoy. The calculaions ae based on he well known fomula in elecodynamics fo deiving he magneic veco poenial [6] Bilo86@yahoo.com

2 54 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p J A dv. () c R The calculaions have been made in he spheical coodinae sysem wih he oigin in he cene of he single-laye fla coil. is eviden fom he symmey of fomula () in his coodinae sysem ha only he componen A of he veco poenial of magneic field of he fla coil seving as a senso of he oscillao diffes fom zeo, since he veco of cuen in his coil is dieced along his axis. This componen could be wien in he following fom J 2 A R sin drd d, (2) c R whee is he adius veco of he poin, whee he field is being calculaed, c is he speed of ligh, J is he cuen pe uni aea of he wie, R is he adius veco of a poin on he wie and he inegaion is ealized ove he enie space. will be assumed in wha follows ha he cuen poducing he measuing RF magneic field flows on he suface of fla coil uns and so is pependicula o he adius veco of he poin on he wie. n case of such a close appoximaion o he ealiy, i is easily obained wih he help of expession (2) ha cos d ( x ) ho A 2 ( x cos ) d cosd d c ( x ) 2 c 2 2, (3) ( x ) 2 x cos whee is he angle beween and x. Fom popeies of egende special funcions follows ha ( x cos )(min(, x )) 2 A P (cos ) (cos ), P d (4) 2 c (max(, x )) ( ) whee is he angle beween he coil s axis and R. Noe ha as sin cos, (5) 2 2 x 2xcos i.e. cos is a funcion only of ( is he angle beween he wie s adius and he un adius). Hence, based on he above calculaions, i is easy o wie A as P (cos ) A Q, (6) c ( ) whee he following noaion was used: 2 x sin x cos P, d x 2x cos if x, Q (7) 2 sin x cos P, d x x 2x cos if x.

3 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p Now, poceeding fom an expession fo he veco poenial of he field and using he well known elaion H o A, we come o an expession ( A sin ) ( A ) H eˆ eˆ. (8) sin Afe coesponding mahemaical ansfomaions he final analyical fomula fo H is wien as H Q P (cos c ) (9) and fo H : ( A ) P (cos ) ( Q ) H. (0) c ( ) is easy o see ha 2 2 Q Q, if x 2x cos, () 2 2 ( ) Q, if x 2x cos. So, he mahemaical expession fo H ansfoms o H c P (cos ) Q ( ) , if x 2x cos, ( ), if x 2x cos. Analyical expessions (9) and (2) have been obained fo he componens of magneic field poduced by an x adius coil made fom adius wie. Howeve, fo fla single-laye coil seving as a senso fo he measuing oscillao used in ou expeimenal eseach of supeconduciviy, we have deal wih he Achimedean spial, which in case when he numbe of uns exceeded 0 may be consideed o a good appoximaion as a sysem of ighly enclosed coaxial ings. n his model he calculaions ae significanly simplified and he magneic field of coil is a veco sum of magneic fields poduced by consiuen ings. f we denoe he oal numbe of uns (ings) in he coil by N, he adius of he -h un by R, and Q of his un coespondingly by Q,, hen he expessions fo H and H ake he following fom: N H Q P (cos ), c (2), (9a) P (cos ) Q 2 2,, if R 2R cos, H c (2a) ( ) 2 2 ( ), if R 2R cos, whee and ae he pola coodinaes of he poin, whee he magneic field is calculaed. Now noe ha R Rhole (2 ), whee R hole is he adius of hole of he coil cene, and Q, depends only on he un numbe ( ), adius of he wie () and R hole. To simplify calculaions we will use he fac ha as only Q, is a funcion of R, hence, he summaion symbol efes only o Q, ems. Fuhemoe, in he case

4 56 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p of even -s, i is easy o show ha Q, ems ae zeo. Also we will use he N Q, Q noificaion (wih edesignaion of he fome Q ). To ge handy analyic expessions fo an expeimene we need o make some appoximaions based on he fom of coil as well as he chaaceisics of ou supeconduciviy eseach. A he pepaaion of coil i is impossible o avoid he echnological hole a is cene ha is equal o R hole =0.5 mm fo he coil a issue in he pesen pape. As he coil was wound using he wie of 2 0. mm diamee, so iespecive of he un numbe / R 0.. Hence, Q, is expandable ino a seies in small quaniy of / R, i being shown in ou subsequen calculaion ha in he expansion ino seies in / R one can confine o he 5-h em accuacy. n ou expeimenal eseach of supeconduciviy we have deal wih he componens of measuing magneic field sengh paallel (H x o H y ) and pependicula (H z ) o he coil plane ha ae expessed hough H and H (see (9a), (2a)). So, i is impoan o have analyic expessions jus fo H x (o H y ) and fo he H z componens. is easy o see ha H x H y H sin H cos (3) and fo he H z componen H z H cos H sin. (4) Resuls and Discussions. One of possible ways of esing he coil s field configuaion is he calibaion by moving diffeen diamee disk-shaped coppe plaes owads he coil s face up o he given disance and back [4]. Having in view he compaison of calculaed values of he hoizonal componens of field sengh H H ) wih coesponding expeimenal daa (see Fig., 2), we availed ( x y ouselves of he calibaion daa in [4]. Noe in his connecion ha in his wok he auhos used coppe plaes fo measuemens. n he inse o Fig. ae pesened he measued in [4] daa on he dependence of TD oscillao fequency on he disance of coope plae fom he coil suface ( h ).The sengh of measuing magneic field a evey poin of space can be calculaed using fomula (3) deived by us. Since a plae can be consideed as a sysem of balls ha ae a equal disances fom he coil suface, i makes sense o compae he expeimenal daa obained in [4] and pesened in he inse o Fig. wih he dependence of nea field value of magneic coil on he disance fom coil suface calculaed accoding o above elaion (3) fo he egion of field homogeneiy. Fig.. shows he gaph of calculaed dependence of H (o H ) componen of nea RF magneic field of fla coil on he x y eal disance disance h fom he coil a he lengh equal o half of he adius fom he coil axis (along he line sin R coil / 2 ). Noe ha fo popoions of coppe plaes ( )R coil used fo measuemens in [4] he measuing RF field is sill homogeneous (see he inse fom he cied wok o Fig. 2). n he inse o Fig. 2. ae measued daa [4] on he dependence of TD oscillao fequency shif on he diamee of coppe discs ha ae a fixed disance fom he coil suface (he fequency shif is nomalized by he aea of he plae).

Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p. 53 59. 57 Ab.Uni 20 00 80 60 40 20 0 3 5 7 9 3 5 7 9 hh, mm Fig.. H x (o H y, paallel o he coil face componens) componen vs.

he disance of coope plae fom he coil of coil 8 mm diamee [4]. The coninuous lines connecing he poins epesen he abovemenioned dependence fo vaious disances of he plaes fom he coil.

5 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p Ab.Uni hh, mm Fig.. H x (o H y, paallel o he coil face componens) componen vs. he disance (h) fom he coil suface, obained fom fomula (3). The paamees of calculaions ae sin = 5 mm, = 0.05 mm, R hole = 0.5 mm and N = 00 uns, R coil = 0 mm. nse: The oscillao fequency vs. he disance of coope plae fom he coil of coil 8 mm diamee [4]. The coninuous lines connecing he poins epesen he abovemenioned dependence fo vaious disances of he plaes fom he coil. As he conibuion of he inhomogeneous pa of he field o he fequency shif of oscillao was shown in [4] o be small compaed wih ha of he homogeneous pa of he field, ha gave us gounds fo making compaison of he obsevaional esuls obained in [4] wih ou calculaion esuls given in he same figue. 40 R coil =0 mm Ab.Uni R, R mm Fig. 2. The dependence of H x (o H y ) componens on he disance R fom he coil axis in he plane ha is a 5 mm disance fom he coil face, obained fom fomula (3). The paamees of calculaions ae cos = 5 mm, = 0.05 mm, R hole = 0.5 mm and N=00 uns, R coil = 0 mm. nse: The dependence of oscillao fequency shif on he disance fom he coil fo coppe discoid plaes of vaious adii, when he plaes ae bough fom infiniy down o he given disance [4] (he fequency shif is nomalized o he plae aea). Coninuous lines connec poins ha coespond o cases, whee plaes ae a he same disance fom he coil. Seen in Fig. 2 is he plo of calculaed dependence of H x (o H y ) componen of field on he disance (R) fom he coil axes in he plane ha is paallel o he coil face and is a 5 mm disance fom ha (i.e. in cos =5 mm plane). The qualiaive coincidence of esuls winesses ha he paial conibuion of he inhomogeneous pa of field o he fequency shif of oscillao is small compaed wih ha of homogenous pa. f insead of a plae a ball-shaped objec is moved paallel o he coil face, hen he fequency shif dependence on he ball disance fom he coil

58 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p. 53 59.

3 shows he dependence of fequency shif of he fla coilbased oscillao measued in [7] on he disance of ball-shaped paicle fom he coil axis a is moion in he plane paallel o he coil suface along is diamee.

6 58 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p axis would no coincide wih heoeical esul since in he cenal pa of field he influence of H z componen of he coil on he ball becomes subsanial. The inse o Fig. 3 shows he dependence of fequency shif of he fla coilbased oscillao measued in [7] on he disance of ball-shaped paicle fom he coil axis a is moion in he plane paallel o he coil suface along is diamee. The cuve in Fig. 3 gives he calculaed dependence of H z componen (4) on he disance R fom he coil axis in he plane ha is paallel o he coil face a 5 mm disance fom ha (i.e. in cos =5 mm plane). A qualiaive compaison of hese cuves is jusified, because, as was menioned above, he pesence of ball nea he coil axis sufficienly disos he H z componen of magneic field. 60 Ab.Uni R mm Fig. 3. The calculaed dependence of H z componen on he disance R fom he coil axis in he plane ha is a 5 mm disance fom he coil face, obained by fomula (4). The paamees of calculaions ae cos = 5 mm, = 0.05 mm, R hole = 0.5 mm and N = 00 uns, R coil = 0 mm. nse: The measued dependence of oscillao fequency ( F) and ampliude A shifs on he disance fom he coil axis of ball shaped in (Sn) paicle wih ~ mm adius ha is moved a mm disance fom he coil suface along is diamee [7]. follows fom he esuls, obained ha when hin plae-like objecs ae bough neae o he fla coil, i would be moe coec o compae he value of oscillao fequency shif wih H x (o H y ) componen of he measuing field, wheeas in case of anslaion of meallic ball-shaped objecs paallel o he coil face he value of oscillao fequency shif should be compaed (in he fis ode of appoximaion) wih he value of H z componen of measuing field. So, we eceived evidence ha he esuls of ou calculaions qualiaively agee wih he obained expeimenal daa. Hence, we infe ha i is possible o model he fla coil echnique used by us fo eseach of fine peculiaiies of supeconduciviy by heoeical mehods discussed in his pape. The deducion and invesigaion of he dependence expessed by fomula (4) and pesened in Fig. 3 was impoan and up o dae since i enabled us o claify he possibiliy of disinguishing beween he magneic and nonmagneic egions of a supeconducing unique sae using he single laye fla coil as a novel ype pobe as was discussed in [3]. The calculaions show ha a definie value of R (see Fig. 3.) he value assigned o he magneic field sengh becomes negaive ha implied he evesal of field diecion. The evesal of magneic field is clealy seen in expeimens, whee he elecomoive foce inducion by RF magneic field of he coil is measued [8]. Conclusions. Fuhe impovemens in TD acivaed single-laye fla-coiloscillao echnique would ceae bee oppouniies fo sudying he phenomenon

7 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p pediced in 964 by Fulde and Feell [9] and independenly by akin and Ovchinikov [0] (he so-called FFO effec). The fis obsevaion of his phenomenon was qualiaive []. Howeve, no deailed invesigaion of his phenomenon has been made ye due o he lack of insumenaion wih enough esoluion. n accodance wih esimaions made in [2 4], we hope ha in case of ceaion of a fla coil-based senso (micoscope) using mm coil as a pobe, i would be possible o aain enough spaial esoluion fo invesigaion of he above phenomena. n his connecion noe, ha lihogaphic coils of mm in diamee ae aleady available in make (usmicowaves.com). Wih such a SFCO based echnique, as an effecive-pobing elemen a hand, one may each he aim of deecion (and hen, he visualizaion) of he node sucue in FFO-sae [9 ] wih a spaial esoluion of abou 0. m. Tha makes he poblem of heoeical and expeimenal sudy of he magneic field configuaion nea he fla coil face up o dae. hank my eseach supeviso Pof. S.G. Gevogyan fo he poblem definiion, as well as fo pemanen inees o my sudy esuls. This sudy was suppoed by he Amenian Naional Foundaion of Science & Advanced Technologies (NFSAT) and he U.S. Civilian Reseach & Developmen Foundaion (CRDF) unde he Gan #UCEP 07/07. The sudy was paially suppoed also by sae souces of Amenia in fames of R&D pojec #D-07. Received R E FE R E N C E S. Gevogyan S.G., Kiss T. e al. // Rev. Sci. nsum., 2000, v. 7, p Gevogyan S.G., Movsesyan G.D. e al. // Rev. Sci. nsum., 998, v. 69, p Gevogyan S., Kiss T. e al. // Supecond. Sci. Technol., 200, v. 4, p Gevogyan S.G., Kiss T. e al. // Physica C: Supecond. & Appl., 200, v. 366, p Gevogyan S.G., Kiss T. e al. // Physica C: Supecond. & Appl., 200, v. 363, p andau.d., ifshiz E.M. Field Theoy. M.: Nauka, 988, p. 42 (in Russian). 7. Gevogyan S.G. and Azayan M.G. Poc. of he 5-h nenaional Confeence on Semiconduco Mico- & Nanoeleconics. Aghvean, Amenia, 2005, p Gevogyan S.G., Muadyan S.T., Kughinyan B.K., Qeobyan M.. // Poceedings of he YSU. Physical and Mahemaical Sciences, 20, 3, p Fulde P., Feell R.A. // Phys. Rev., 964, v. 35, 3A, p. A550 A akin A.., Ovchinikov Y.N. // JETP, 964, v. 47, 3 (9), p ; Sov. Phys. JETP, 965, v. 20, p Radovan H.A., Muphy T.P., Palm E.C., Toze S.W., Hall D. and Saao J.. // Jounal of ow Tempeaue Physics, 2003, v. 33, p ; NHMF Repos, 2003, v. 0 (3),. magne.fsu.edu 2. Gevogyan S., Shiinyan H. e al. // Nucl. ns. & Meh in Phy. Res. A, 2004, v. 520, p Kaapeyan G.H., Shiinyan H.G., Gevogyan S.G., Gevogyan V.G. Poc. of he V Conf. on Educaional, Scienific & Technol. Appl. in abvew Envionm. & Technol. of N., 2008, Novembe Moscow, Russia, v.,.0.2., p Gevogyan S.G., Shiinyan H.G., Kaapeyan G.H., Gevogyan G.S., Polyanskii A.A. // Poceedings of he YSU. Physical and Mahemaical Sciences, 2009, 2, p

8 60 Poc. of he Yeevan Sae Univ. Phys. and Mahem. Sci., 202,, p Բ. Կ. Կուրղինյան. Հարթ կոճի շրջապատի չափող դաշտի մոդելավորումը` գերհաղորդական երևույթի թույլ արտահայտված առանձնահատկությունների բացահայտումն արդյունավետ դարձնելու նպատակով էջ Իրականացվել է ԳՀ ուսումնասիրման արդյունավետ մեթոդներից մեկիª թունելային դիոդով ակտիվացվող հարթ կոճով, զգայուն ինքնագեներատորի տվիչը հանդիսացող միաշերտ, հարթ կոճի շրջապատի ռադիոհաճախային (ՌՀ) չափող էլեկտրամագնիսական դաշտի բաշխվածության մոդելավորում: Արտածվել է այդ դաշտի մագնիսական բաղադրիչը հաշվելու համար անալիտիկ արտահայտություն, ինչը թույլ է տալիս կոճը շրջապատող տարածության ցանկացած կետում հաշվել չափող ՌՀ դաշտի մագնիսական բաղադրիչ-ները` ինչպես կոճի մակերևույթին զուգահեռ, այնպես էլ դրան ուղղա-հայաց ուղղություններով: Հաշվարկների նպատակն է, ԳՀ երևույթի թույլ արտահայտված (բայց երևույթի ընկալման իմաստով կարևոր) առանձնահատկությունների, հարթ կոճի մեթոդով բացահայտման ընթացքում փորձնականորեն չափող տեխնիկայի օպտիմալ զգայունությանը հասնելը:

Two-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch

Two-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion