Magnetic levitation force between a superconducting bulk magnet and a permanent magnet

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1 Mgneti levittion fore between sueronduting bulk mgnet nd ermnent mgnet Wng C Y He L Meng C Li R S Hn Z X Go Dertment of Physis Key Lbortory for Artifiil Mirostruture nd Mesosoi Physis Peking University Beijing 87 P. R. Chin Abstrt The urrent density ( ) ρ in disk-shed sueronduting bulk mgnet nd the mgneti levittion fore exerted on the sueronduting bulk mgnet by ylindril ermnent mgnet re lulted from first riniles. The effet of the sueronduting rmeters of the sueronduting bulk is tken into ount by ssuming the voltge-urrent lw = ( ) n E E nd the mteril lw B = µ H. The mgneti levittion fore is dominted by the remnnt urrent density ( ρ ) ' whih is indued by swithing off the lied mgnetiing field. High ritil urrent density nd flux ree exonent my inrese the mgneti levittion fore. Lrge volume nd high set rtio of the sueronduting bulk n enhne the mgneti levittion fore further. PACS number(s): 74.6.w 74.5.H 74.5.Ld. Introdution Sueronduting mgnets ly n imortnt rt in the mgnetilly levitted trin system. The mgneti field generted by disk-shed sueronduting bulk mgnet () n inrese with its nd volume. A lrge light rre erth (LRE) - B Cu O 7-x (e.g. Nd Sm Eu or Gd) is believed to be ble to tr mgneti field more thn 5 T t 77 K [-]. Therefore the ossibility of high ritil temerture for the mgnetilly levitted system hs been investigted [4-5]. The lultion of the hystereti fore between suerondutor nd ermnent mgnet () hs been reorted [6]. In suh lultions however there is no tred mgneti field in the suerondutor. We onsider

2 tht the suerondutor hs been mgnetied before is led under the suerondutor so the suerondutor ts s n with tred mgneti field nd then lulte the mgneti levittion fores between the nd the. In rtie the sueronduting disk n be ooled below T in n xil lied uniform mgneti indution B then the lied mgneti field is swithed off nd the mgneti flux eses from the disk. This is lled the ese roess. The lultion of the urrent density in the sueronduting disk will be divided into three stes with different lied mgneti fields. irst we onsider the surfe sreening urrent density ' indued just fter swithing off the xil lied uniform mgneti field suh tht B&. Seond the remnnt urrent density ' generted fter B & = nd B = whih retes the tred mgneti field of the sueronduting disk. inlly the urrent density generted by the non-uniform mgneti field of the. The result obtined in the revious ste is tken s the initil urrent density in the next ste. It is diffiult to lulte the bove urrent densities ' nd ' in sueronduting disk diretly so we suggest method tht shuns suh diffiulty. We onsider nother roess tht the mgneti flux enetrtes into the sme sueronduting disk. After the sueronduting disk is ooled in ero mgneti field below its ritil temerture we strt t time t = with B = nd then swith on n xil lied uniform mgneti field suh tht B&. The surfe sreening urrent is indued. As soon s B & = nd B = onst the sueronduting disk is in the xil lied uniform mgneti field. During the enetrtion of the mgneti flux into the disk the urrent density genertes. This is lled the enetrtion roess. In the bove ese nd enetrtion roesses the disk the uniform mgneti field nd the flux motion re the sme so the grdient of mgneti field hs the sme vlue but in oosite diretion tht is = ' nd = '. The nd n be lulted from the eqution of motion for urrent density in sueronduting disk.

3 . Modeling A. Configurtion We onsider disk-shed with rdius nd thikness b levitted over o-xil ylindril with rdius R nd thikness d. The enter of the is tken s the origin of the ylindril oordinte system ( ρφ ) shown in ig.. B. Bsi eqution for urrent density We use the method resented by E. H. Brndt [6-8] to lulte the urrent density in sueronduting disk. The key issue is to find n eqution of motion for the urrent density ( rt ) inside the disk. Using the Mxwell eqution = H nd onsidering B = µ H A = nd B = (ssuming no urrent soures i.e. no ontts) we obtin the eqution of motion for the urrent density: µ = B= A = A where B ( B +B ) B nd ( ) = = A = A A. As usul the dislement urrent whih ontributes only t very high frequenies is disregrded in this eddy-urrent roximtion. We desribe the suerondutor by the mteril lw = µ B H nd the voltge-urrent lw ( ) ( ) n E = E. B = µ H is good roximtion when the flux density B nd the ritil sheet urrent b re lrger thn the lower ritil field B everywhere inside the sueronduting disk [7]. This requirement is often stisfied in the mgneti levittion mesurement. The voltge-urrent lw E( ) E ( ) n = is tully result of the urrent deendene of the tivtion energy U( ) = U ( ) by using the Arrhenius lw = υ = υ ( ) log the rmeters E = Bυ n U kt B = nd υ υ ( U kt) E B B ex U kt B. So we obtin = ex B whih is the effetive vortex veloity. σ ( = n ) is the flux ree exonent. Beuse of the xil symmetry the urrent density eletri field E nd vetor otentil A (defined by = A B ) hve only one omonent ointing long the imuthl diretion φˆ ; thus = ( ρ ) φ ˆ = E( ρ ) φ ˆ E A. Sine B = µ H we hve nd = A( ρ ) φ ˆ µ = A where A = A A φ is the vetor otentil generted by the urrent density in the disk nd A φ is the vetor otentil of lied mgneti field. The solution of this Poisson

4 eqution in ylindril geometry is with r = ( ρ ) nd r ' = ( ρ ' ') ( ) b ρ = µ ρ' ' ( rr ') ( r ' ) + () b A d d Q A φ yl. The integrl kernel ( ') = ( ρρ ' ') Q rr f () with ( ρρ ' ') f = π dφ ρ'osφ π ( ' ) + ρ + ρ' ρρ'osφ ρ ' = k K ( k ) E( k ) πk ρ () where k = 4 ρρ ' ( ρ + ρ' ) + ( ' ) (4) K nd E re the omlete elliti integrls of the first nd seond kind resetively. To obtin the desired eqution of motion for ( t ) ρ we exress the indution lw E = B & = A & in the form E = A. & The guge of A ( A =) to whih n rbitrry url-free vetor field my be dded resents no roblem in this simle geometry. Knowing the mteril lw E = E ( ) n we obtin A& = E( ). This reltion between A & nd llows us to eliminte either A or from Eq. (). After eliminting A we obtin ( r ) = µ ' ( rr ') ( r ' ) & ( ρ' ') E t drq & t A (5) S This imliit eqution for the urrent density ( t) yl φ r ontins the time derivtive & under the integrl sign. In the generl se of nonliner E( ) the time integrtion of Eq. (5) hs to be erformed numerilly. or this urose the time derivtive should be moved out from the integrl to obtin & s n exliit funtionl of. This inversion my be hieved by tbulting the kernel ( ') r Qyl rr on D grid i r j nd then inverting the mtrix Q ij to obtin Q whih is the tbulted reirol kernel ( ') ij Q rr. The eqution of motion for the imuthl urrent density yl ( t ) ρ then reds 4

5 where Q ( r ) b = µ ρ' ' ( ') ( ) ( ' ') rr b yl + φ ρ & & t d dq E A (6) is the reirol kernel whih is defined by dρ' d' Q ( rr ') Q( r' r'' ) = δ( r r '') (7) b b C. Current density in or n xil lied uniform mgneti indution B = B ˆ we hoose the vetor otentil A φ ρ = B. At time t = with B = nd = we swith on the lied field suh tht B& nd then the surfe sreening urrent is esily indued. Immeditely fter tht t time t =+ ε the mgneti field nd urrent density inside the disk re still ero sine they need some time to diffuse into the onduting mteril. Therefore t t =+ ε the eletri field E( ) is lso ero nd thus the first term in Eq. (6) vnishes. Wht remins is the lst term whih should be the time derivtive of the surfe sreening urrent. This surfe sreening urrent is thus [7] b ρ ( ) ( ) r t = H t dρ ' d' Q ( ') b yl rr (8) The surfe sreening urrent lsts only very short time. As soon s &B = nd B = onst the motion for the urrent density in the disk must be desribed by ( ) b µ r ρ' ' ( ') ( ) b yl rr & t = d dq E (9) Eq. (9) is esily time integrted by strting with ( ) ( ρ ) ( ρ ) ( ρ ) t = t + dt = t + & t dt. ρ t = nd then utting After bout s the distribution of the urrent density in the disk rehes its sturted vlue then the sturted vlue dereses with time. We tke ' = nd the urrent density t t =8 s fter swithing off the lied mgneti field s the initil urrent density ( ρ t ) '( ρ t 8 s) mgnet is led t ( ) = = in next ste nd the ermnent +. + is the initil distne nd is the minimum distne between the to surfe of the nd the bottom surfe of the. D. The ermnent mgnet Beuse of the xil symmetry of the system only the ross setion of the system is onsidered with the xis 5

6 diretion hosen s the symmetry xis of the nd the nd ρ (rdil diretion) rllel to the surfes of the nd the. The to surfe enter of the rohes nd reedes from the s ( ω ) s = sin t + b+ () where frequeny ω reresents the seed t whih the rohes nd reedes from the. Exerimentlly unertinty will be used when the touhes the nd therefore the limit = should be voided. In this lultion we hoose =. s the minimum distne between the nd the. or this onfigurtion the vetor otentil of the hs only one omonent long the φ diretion nd n be derived by integrting the vetor otentil of irulr urrent loo with rdius R long the thikness d s ( + s+ d ) + R + ρ ρr osφ + ( + s+ d ) π rem π () + s+ R + ρ ρ R φ + ( + s) B Aφ ( ρ ) = R osφln dφ os where B rem is the remnnt indution of the. The rdil indution Bρ = Aφ n then be written s i ( ) Brem R Bρ ( ρ ) = ki K( ki ) E( ki ) π ρ i = k i () where K nd E re omlete elliti integrls of the first nd seond kind resetively. And k 4ρR = i = i ( R + ρ ) + ( + s+ id ) () E. The in non-uniform mgneti field After the is led under the we del with the in non-uniform mgneti field. The eqution of motion for urrent density in is µ = A. The solution of this Poisson eqution in ylindril geometry ( ) b = ' ' ( rr ') ( r ') (4) A ρ µ dρ d Q b nd & d d Q E A (5) b ( ρ ) = ρ' ' ( ') ( ') ( ' ') rr + b φ ρ rr & µ Eq. (5) is esily time integrted by strting with ( ρ t ) ( ρ t 8 s ) ( ρ ) ( ρ ) ( ρ ) t = t + dt = t + & t dt. = = nd then utting 6

7 . The levittion fore As the urrent density ( t) ρ nd the rdil mgneti indution B ρ inside the hve been derived the vertil levittion fore long the -xis n be redily obtined s b = π ρ ( ρ ) ( ρ ) (6) d d B b ρ. Results nd disussions Now we onsider the vlue of the lied mgneti field H in field ooling. or short ylinders in the Ben limit we hve n exliit exression for the field of full enetrtion H i.e. the vlue of the lied mgneti field H t whih the enetrting flux nd urrent fronts hve rehed the seimen enter [7]. At H > H the urrent density in the Ben model does not hnge nymore nd the tred mgneti field reted by the urrent density will not hnge either. or ylinders with rbitrry set rtio b/ the field of full enetrtion is or H = b ln + + b b b= H = b ln( + ) nd B µ b ln( ) (7) = +. Tyilly =.5 m nd 4 = A/m then B = 5.58 T. The tyil vlue of B rem for NdeB rre-erth ermnent mgnet is bout.5 T nd the tyil mgneti field lied to mgnetie the melt-textured-grown YBCO disk is ~ T t 77 K so it is resonble to tke B B =. nd H H =.. We tke E = µ = = ω =. σ = b= R = d = rem =.5 =. B B =. nd H H =. in the following lultion unless seil delrtion. rem A. Surfe sreening urrent ig. shows the rofile of the surfe sreening urrent density ( ) ρ lulted by Eq. (8). The thikness of this urrent-rrying lyer deends on the hoie of the omuttionl grid in the disk. The lyer thikness my be redued nd the reision of the omuted surfe sreening urrent n be enhned by hoosing non-equidistnt grid whih is denser ner the disk surfe. An rorite hoie of suh non-equidistnt grid = ( ρ ) r is i i i 7

8 obtined by tking ρ ρ( ) ( ) ( k N ρ ) = u = u u nd then tbulting = L. We divide into N rts in the similr wy yielding D grid of different oints should be onsidered. In this lultion we tke N 4. u = L on equidistnt grids = ( ) uk k N ρ N = N ρ N. The weights t A very interesting feture shown in ig. is tht the sreening urrent density in the side surfe of the disk is muh higher thn ritil urrent density eseilly on the brims of the to nd bottom surfes of the disk. Inside the disk ( ) ρ is lmost ero. The sreening urrent lsts only very short time. If the ooling of the surfe of the disk is idel the dissition het nnot drive the disk to norml stte so the disk is still in sueronduting stte. B. Current rofiles in As soon s B & = nd onst B = the surfe sreening urrent density ( ) shows the rofiles of the urrent density ( ρ ) lulted by Eq. (9). ( ) ridly dereses to sturted vlue.85 t 44 s ( ) ρ in the whole disk rehes the sme vlue of.6 ρ ridly dereses. ig. ρ t the side surfe of the disk t =.5 s however ( ρ ) inside the disk inreses. After bout. The distribution of ( ) ρ will kee this lteu feture in future however the height of the lteu will derese with time very slowly. At ( ) ρ dereses to.5. t = 8 s C. Current rofiles in the As soon s with B.B rem = is o-xilly led under the t ( ) mgneti field is lied. The initil urrent density in tkes ( t ) + n xil non-uniform ρ = 8 s different from tht one in suerondutor without tred mgneti field in whih se the initil urrent density is ero. When the rohes the with the initil urrent density ( ρ t ) = ( ρ t = 8 s) ( ρ ) t the brims of both the to nd bottom surfes of the disk inreses but the inrese t the to surfe is smller. or exmle t t = 4. s ( ρ ) t the brims of the bottom surfe inreses to.7 while ( ) ρ t the brims of the to surfe inreses to.55. As the is moving loser in most volume of the ( ρ ) still equls to ( ) ' ρ nd in rest volume t 8

9 the brims of the bottom surfe ( ρ ) beomes ( ρ ) of. ( ) ' whih is determined by the non-uniform mgneti field ' ρ rehes new lteu of.75 t the minimum distne between nd. When the is moving wy from the ( ) ' ρ is reversed to.75 t the mximum distne between nd. U to this moment the motion of the omletes one yle. D. Comrison of levittion fores The vertil levittion fore (solid lines) between nd n in ig. 5() shows tyil hystereti behvior. Severl interesting fetures muh different from (broken lines) between nd suerondutor without tred mgneti field re esily observed. irst s soon s is led t ( ) + = but > inditing n initil reulsive fore. Seond is ttrtive s the is moving wy from eseilly for higher rtio of Brem B. As the is moving wy from the however lwys shows reulsive feture. There is no ttrtive fore in the roess. Third is muh lrger thn tyilly s the is t its equilibrium osition of ( ) = +. The mgneti levittion fore is determined by the urrent density indued by the non-uniform mgneti field of the. The mgneti levittion fore however is dominted by the urrent density ' whih is indued by swithing off the lied mgnetiing field. When the remnnt indution of the inreses to B B = both levittion fores rem nd enhne. ig. 5(b) shows the first two yles of nd. The ttrtive rt of extends to longer rnge however still kees reulsive. E. Effet of the set rtio b on ig. 6 shows the vertil mgneti levittion fore versus the distne s for different set rtios b of the sueronduting disk for one yle. The whole urve moves to higher vlue of with the inrese of the set rtio b. The inset of ig. 6 shows the mximum levittion fore s funtion of the set rtio b. It n be seen tht for smll set rtio b the mximum levittion fore inreses with b ridly nd sturtes t b different from for b. Tehnilly sueronduting disk with its dimeter 9

10 roximtely equl to its hlf thikness b my be otimum for mgneti levittion; further inrese of its thikness will only inrese its weight without enhning the levittion fore signifintly.. Effet of sueronduting rmeters on Two sueronduting rmeters my influene. One is the flux ree exonent σ relted to the inning otentil. Usul vlue for σ in onventionl suerondutors is 5 for high T suerondutors σ is extrolted to be ~6 [9]. Smller σ mens lower inning otentil or higher temerture. The effet of σ for σ = 65 on the mgneti levittion fore versus s is shown in ig. 7. The whole urve moves to higher vlue of with the inrese of σ. The other sueronduting rmeter whih drstilly influenes the mgneti levittion fore is the ritil urrent density of the suerondutor. The lulted results of versus s for different ritil urrent densities re lotted in ig. 8. The inset of ig. 8 shows the deendene of the mximum levittion fore s funtion of the. In the revious lultion [7] the deendene of the mximum levittion fore between suerondutor without tred mgneti field nd s funtion of ritil urrent density is liner t very low ritil urrent density nd sturtes t high ritil urrent density. A fitting to the lulted dt results in. = (8) Muh different from the revious lultion the deendene of the mximum levittion fore between suerondutor with tred mgneti field nd s funtion of ritil urrent density inreses with α. A fitting to the lulted dt results in (<?? < ) (9) α whih is shown s solid line in the inset of ig. 8.

11 4. CONCLUSIONS The remnnt urrent density ( ρ t ) nd the urrent density ( t) ' ρ in the with tred mgneti re lulted from first riniles. rom the derived urrent density inside the disk the mgneti levittion fore between the nd the hs been determined. The suerondutor is desribed by the mteril lw B H nd the flux ree is desribed by the voltge-urrent lw ( ) n = µ E = E. The mgneti levittion fore is determined by the urrent density indued by the non-uniform mgneti field of the ; the mgneti levittion fore however is dominted by the remnnt urrent density ( ρ ) ' whih is indued by swithing off the lied mgnetiing field. High ritil urrent density nd the flux ree exonent my inrese the mgneti levittion fore. Lrge volume nd set rtio of the sueronduting disk n enhne the mgneti levittion fore further. The mgnetition of the sueronduting disk my effetively enhne the levittion fore between the sueronduting disk nd the. ACKNOWLEDGMENTS This work ws suorted by the Ntionl Siene oundtion of Chin (N 744) nd the Ministry of Siene nd Tehnology of Chin (Projet No. NKBRS-G999646). Referenes:. Yoo S I Ski N Tkihi H Higuhi T Murkmi M 994 Al. Phys. Lett Yoo S I ujimoto H Ski N Murkmi M 997. Alloys Comd Ikut H Mse A Miutni U Yngi Y Yoshikw M Itoh Y Ok T 999 IEEE Trns. Al. Suerond. 9 () ujimoto H Kmijo H Higuhi T et l. 999 IEEE T Al. Sueron H. ujimoto H. Kmijo PHYSICA C 4 59.

12 6. Brndt E H 996 Phys. Rev. B 54 (6) Brndt E H 998 Phys. Rev. B 58 () Qin M Li G Liu H K nd Dou S X Brndt E H Phys. Rev B 66 () Lirson B M Sun Z Geblle T H Besley M R nd Brvmn C 99 Phys. Rev. B 4 (): 45. igure tions: ig. : Configurtion of levitted over. The enter of the is tken s the origin of the ylindril oordinte system. ig. : Profile of the surfe sreening urrent density ( ) ig. : Current rofiles ( ) ρ lulted by Eq. (8). ρ in the sme disk t different moments.

13 ig. 4: Current rofiles ( ) ρ in the sme disk for different time in the first yle. ig. 5: Comrison of nd with different remnnt indution of the in the first two yles. () B B =. (b) B B =. rem rem ig. 6: The vertil mgneti levittion fore versus the distne s for different set rtios b of the sueronduting disk for one yle. Inset shows the mximum levittion fore s funtion of b the solid line is guide for eyes only. ig. 7: The vertil mgneti levittion fore versus the distne s t hosen rmeters for different ree exonents σ. Inset shows the mximum levittion fore s funtion of σ the solid line is guide for eyes only. ig. 8: The vertil mgneti levittion fore s funtion of the distne s for different ritil urrent densities. Inset shows the mximum levittion fore s funtion of the solid line is fit with mx. α ig.

14 ig. 4

15 ig. 5

16 ig.4 6

17 ig. 5 ig. 6 7

18 ig. 7 ig. 8 8