MMagnetic Tip Trap System
Oki Gunawan, ∗ Jason Kristiano,
1, 2 and Hendra Kwee
2, 3 IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA Simetri Foundation, Tangerang, Indonesia, 15334 Surya College of Education, Tangerang, Indonesia, 15115 (Dated: June 14, 2019)We report a detailed theoretical model of recently-demonstrated magnetic trap system based ona pair of magnetic tips. The model takes into account key parameters such as tip diameter, facetangle and gap separation. It yields quantitative descriptions consistent with experiments such as thevertical and radial frequency, equilibrium position and the optimum facet angle that produces thestrongest confinement. We arrive at striking conclusions that a maximum confinement enhancementcan be achieved at an optimum facet angle θ max = arccos p / PACS numbers: 04.60.Bc
Various electromagnetic trap systems play importantrole in physics for their ability to trap and isolate par-ticles or matter that have produced many applicationsand discoveries. Examples are Penning trap [1, 2], opticaldipole trap or optical tweezer [3–5], magneto optic trap[6, 7] and various diamagnetic traps [8–12]. For diamag-netic trap systems, high field-gradient product ( B ∇ B )is necessary to achieve trapping or levitation [13]. A newapproach is to use magnetic tip geometry as recentlydemonstrated by O’Brien et al. [14]. The tip geome-try maximizes B ∇ B at the trapped object that leads tostronger field confinement, thus high frequency and highquality factor ( Q ). This characteristics is of high interestfor research that explores macroscopic limits of classicalmechanics and quantum mechanics. Such magnetic trapalso offers interesting alternative to optical trap as thelatter can lead to excessive heating and encounters insta-bilities in vacuum [15]. The ability to achieve high mag-netic field gradients in a localized position using magnetictip is also useful for other applications such as for nuclearmagnetic resonant imaging [16] and magnetic force mi-croscopy [17].Currently, there is strong interest in magnetic trap sys-tem for various applications such as precision gravime-try [18], study of displacement and velocity of Brownianparticle [19], gas temperature measurement [20], and re-search that explores the boundaries of the classical andquantum systems. For example, trapped nanodiamondcan be used to investigate the quantum mechanical prop-erties such as superposition of states [21, 22], control ofelectron spin of nanodiamond nitrogen-vacancy centersand to observe the electron spin resonance properties [23].Such a trap could also serve an important role to testquantum mechanical properties of gravity [24, 25].In the recent demonstration of a magnetic tip trapO’Brien et al. uses two cylindrical magnets with sharp-ened tips and a microdiamond as the trapped object [14].The tips are separated by a gap d = 2 a as shown in Fig. 1(a). The trapping occurs due to diamagnetic re-pulsion that balances the gravity of the diamond andthe cylindrical symmetry that produces a stable potentialconfinement in three dimension. The study in Ref. [14]has reported many important physical characteristics ofthe trap such as the vertical and radial trap frequency,damping factor and maximum field confinement at cer-tain facet angle. However the detailed field and potentialdistribution of the magnetic trap has not been presented.In this report we present a theoretical model that pro-vides analytical solution of the magnetic field of the trapalong its principal axis ( z ). The model leads to rich de-scription such as equilibrium height, axial and radial os-cillation frequency, the optimum facet angle of the tip toachieve the ”confinement enhancement” and the criticalgap beyond which such effect no longer applies. Beyondthe recent interest in various magnetic traps, this modelalso serves as a new elementary example of a simple mag-netic trap system based on conical tip geometry. Thisadds to the collection of various type of diamagnetic trapsystems that have been known in physics [8–12, 26, 27]. FIG. 1. (a) Geometry of the magnetic tip trap with a trappeddiamond near the center. (b) Field distribution along the z -axis for the ”Reference magnetic trap” with θ = 35 ◦ (see text). a r X i v : . [ phy s i c s . c l a ss - ph ] J un We present a theoretical model of a pair of magnetictips system as shown in Fig. 1(a). Each magnetic tipconsists of a cylindrical segment of semi-infinite lengthand a conical segment. The magnet has a uniform vol-ume magnetization M parallel to the cylindrical axis,however the two tips have opposing magnetization. Thecylindrical segment has a radius R and the conical tip hasa facet angle θ [28]. The magnetic trap has a gap opening d = 2 a . A diamagnetic object such as a diamond beadcan be trapped or levitates near the center of the trap at equilibrium position z .We first consider the magnetic field along the principalaxis z due to the upper magnetic tip. We can calculatethe magnetic field by integrating the field contributionsdue to bound surface current K b = M × ˆn all aroundthe conical and the cylindrical segments, where ˆn is thenormal of the surface element (see Supplementary Mate-rial (SM) A for detailed calculations). Interestingly thisleads to a closed-form solution. The magnetic field dueto the upper magnetic tip is given as: B ( z ) = − µ M " cos θ [1 − sin θ arctanh(sin θ )] + cos θ sin θ arctanh R + ( a − z ) sin θ cos θ p R + ( a − z ) cos θ + R ( a − z ) sin 2 θ ! + z − a − R tan θ p R + ( a − z + R tan θ ) + ( a − z ) sin θ cos θ − R cos 2 θ sin θ p R + ( a − z ) cos θ + R ( a − z ) sin 2 θ ˆz . (1)By exploiting the symmetry of the problem, the totalmagnetic field due to both tips is given as: B T ( z ) = B ( z ) − B ( − z ). An example of total magnetic field plotalong the z axis (for R = 1 mm and a = 15 µ m and θ = 35 ◦ ) is given in Fig. 1(b). We observe that the fielddistribution near the center of the trap is approximatelylinear which leads to a harmonic potential trap. The totalpotential of the trapped object due to both magneticinteraction and the gravity per unit volume is given as: U T ( z ) = − χ µ B T ( z ) + ρgz, (2)where ρ is the density of the trapped object, χ is themagnetic susceptibility, g is the gravitational accelera-tion, and µ is the magnetic permeability in vacuum. Wenote that for a spherical diamagnetic object we shouldreplace χ/ χ/ (2 + χ ) [29], however the former isa good approximation for very small χ as in the case ofmany diamagnetic materials (with exception of supercon-ductor where χ = − M = 10 A / m, and microdi-amond as the trapped object with χ = − . × − [30]and ρ = 3513 kg / m [31], R = 1 mm and a = 15 µ m. Werefer to this setup as the ”Reference magnetic trap” inthis study. For further analysis, we define the feature sizeof the magnetic trap which is given as λ = | χ | µ M /ρg which indicates the ”strength” of the magnetic trap. Forthe reference magnetic trap here we have λ = 805 µ m.We now perform the analysis in the limit of a verystrong trap (i.e. λ (cid:29) a ) where the diamagnetic objectwill be trapped near the center ( z ∼ U T ( z ) ≈ k z z /
2, where k z is the ”spring constant” given as: k z = ∂ U T /∂z . Wecould obtain a very compact expression for the potential”spring constant” k z (per unit volume of the trapped ob-ject) in the strong magnetic trap limit where z ∼ k z = − χµ M R cos θ ( a + R tan θ ) a ( R + a cos θ + aR sin 2 θ ) . (3)We show that this theoretical model provides rich de-scriptions of the magnetic trap characteristics that yieldsreasonable agreement with the experimental observation[14]. First, the model allows us to calculate the a naturalfrequency of the vertical oscillation of the trapped ob-ject: f z = p k z /ρ/ π , which yields f z = 348 Hz. This iswithin the range of the reported frequency of f z = 323 Hzto 411 Hz, at pressure of 760 Torr and 0.16 Torr respec-tively [14]. Note that the observed trapped frequencyat room pressure (760 Torr) is lower due to significantdamping effect.Second, we can calculate the equilibrium position ofthe trapped object that yields z = − ρg/k z = − . µ m,which is small compared to the gap d = 30 µ m, in otherwords the object (microdiamond) remains near the cen-ter of the trap as observed [14]. We note that, theoret-ically, stable levitation always exists irrespective of thegap due to the magnetic field characteristics that divergesnear the tip as shown in Fig. 1(b). When the gap islarge, the object will levitate lower until it is balancedby the diamagnetic repulsion force which is proportionalto B dB/dz . However due to the finite size, the diamag-netic object will eventually touch the lower tip when thegap is large.Third, the spring constant k z or the vertical trap fre-quency f z depend on the half gap separation a , the mag-net cylindrical radius R and the facet angle θ . For a given FIG. 2. (a) The equilibrium height( z ) dependence on the half-gap ( a )and the facet angle ( θ ) for magnetictrap with ”strength” λ = 805 µ m.The white dashed curve is the theo-retical optimum facet angle (Eq. 4)and the black circles are the max-imum point for z /a . The star isthe data point for the ”Referencemagnetic trap”. (b) The trap fre-quency ( f z ) dependence with re-spect to half-gap ( a ) and facet angle( θ ) for λ = 805 µ m. (c) The equi-librium height ( z ) vs. half-gap ( a )with various magnetic trap strength( λ = 10 , , and 0 . R ). (d) Thecritical half-gap ( a c ) beyond whichthe confinement enhancement nolonger applies, plotted with respectto the magnetic trap feature length λ (at θ max =35.3 ◦ ). The theory(Eq. 5) fits well for λ /R < a and R , maximum frequency can be achieved at an op-timum facet angle θ max . We can calculate this optimumfacet angle which is given as (see SM B): θ max = arccos s R (2 a + 2 √ aR + 3 R )4 a + 4 a R + 9 R (4)We note that in the strong magnetic trap limit ( λ > a )the optimum facet angle depends only by geometricalfactor of the magnetic tip i.e. a/R and none of thephysical properties of the magnet or the trapped object( M, χ, ρ ). For the ”Reference magnetic trap” here, weobtain θ max = 34 . o which is quite close to reportedvalue based on numerical calculation of boundary inte-gral method for flat facets tips θ max = 28 ◦ [14]. In thelimit of very small gap ( a (cid:28) R, λ ), Eq. 4 reduces to avery simple expression: θ max = arccos p / . ◦ .Fourth, the experimental study also reported fre-quency of horizontal oscillation mode which is half thatof vertical frequency i.e. f x = f y = f z / f r = f z / ∇ · B = 0) and curl-free ( ∇ × B = 0) we could relatethe linear coefficients of the axial and radial magneticfield ([32], pg. 4) as: B z ( r, z ) = a + a ( z − z ) + ... and B r ( r, z ) = b r + b r ( z − z ) + ... . This yields b = a / f r /f z = p k r /k z = p ( b /a ) = 1 / a, R, θ, χ, ρ, M ) to the trap characteristics. We per-form numerical calculation to obtain the vertical trapfrequency which represents the strength of the trap con-finement. First we start with the Reference magnetictrap ( λ = 805 µ m) and calculate the equilibrium posi-tion ( z ) of the trapped object in the magnetic trap bynumerically solve for dU T /dz = 0, and then we calculatethe spring constant k z at that position ( z ). The equi-librium position z as a function of half-gap a and facetangle θ is plotted in Fig. 2(a). We observe that for a con-stant a value, we achieve maximum z at θ ≈ ◦ (blackcircles). This maximum behavior apparently applies toall values of a/R which implies that the magnetic trapalways yields a higher levitation position z at an opti-mum facet angle θ max even when the gap is large. Nextwe study z behavior with respect to half-gap a at varyingstrength of magnetic trap ( λ ) as shown in Fig. 2(c). Weobserve that for a very strong magnetic trap (e.g. where λ = 10 R ) z becomes higher than the others ( λ = 1and 0 . R ). It is near zero (near the center of the trap)until it drops off at higher gap value at a/R > . z in Fig. 2(a) we now numeri-cally calculate the trap frequency f z = p k z ( z ) /ρ/ π asa function of a , θ and z as plotted in Fig. 2(b). Firstwe observe a behavior similar to Fig. 2(a) for small gap( a/R < . θ max due to the confinement enhance-ment effect of the magnetic tip. This plot resembles theplot from numerical computation in Fig. 1(b) of Ref. [14].However we also observe an interesting behavior, for alarge gap beyond a ”critical value” a c this confinementenhancement effect no longer applies, i.e. when we plot f z vs. θ at constant a - there is now a minimum near θ max .For the magnetic trap with very large gap ( a > a c ), toachieve high frequency one can use very low facet angle(no tip) or very high (very sharp tip) which are not de-sirable. In practice we want to use smaller gap to achievehigher frequency, but not too small to provide some spacefor the trapped object and to allow optical detection.Finally we study the dependence of this critical half-gap a c with respect to varying strength of magnetic trapby repeating the analysis in Fig. 2(a) and (b) at differ-ent value of λ . The results is shown in Fig. 2(d). Wecalculate a c using numerical calculations by finding thevalue a where ∂ f z /∂θ = 0 whose data are shown asred points in Fig. 2(d). We observe a reasonable trendthat the critical half-gap a c increases monotonically withincreasing strength of the magnetic trap (or λ ).Furthermore we have also attempted to derive the the-oretical relationship of this critical gap as a function of λ using series expansion of k z as a function of λ up to the4th order (see SM D). We arrive at a simple relationship: a c R = λ (cid:16)p . R + 3 . λ − λ (cid:17) . R + 3 . λ (5)This relationship allows us to quickly estimate the criti-cal gap below which the confinement enhancement effectstill applies in this magnetic tip trap. However, we alsonote that for strong magnetic trap ( λ /R >
2) discrep-ancy occurs between the theoretical model and the nu-merical calculation due to higher order of a that mustbe considered. We also note that for practical purpose,the optimum facet angle θ max = arccos p /
3, which iscalculated in the limit of strong trap, applies very well tomost situation where a < a c .In closing, compared to other existing electromagnetictrap systems known in physics [1–12], this magnetic tipsystem presents a new, simple, elementary type of mag-netic trap based on conical geometry. The model that wehave developed provides rich theoretical understandingof the system that will help advance further developmentand applications. ∗ [email protected][1] F. M. Penning, Physica (Utrecht) , 873 (1936).[2] L. S. Brown and G. Gabrielse, Rev. Mod. Phys. , 233(1986).[3] A. Ashkin, IEEE J. Quant. Elect. , 841 (2000).[4] S. Chu, J. Bjorkholm, A. Ashkin, and A. Cable, Phys.Rev. Lett. , 314 (1986). [5] S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, andA. Ashkin, Phys. Rev. Lett. , 48 (1985).[6] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E.Pritchard, Phys Rev Lett , 2631 (1987).[7] W. D. Phillips, Rev. Mod. Phys. , 721 (1998).[8] M. D. Simon, L. O. Heflinger, and A. K. Geim, Am. J.Phys. , 702 (2001).[9] I. F. Lyuksyutov, D. G. Naugle, and K. D. D. Rath-nayaka, Appl. Phys. Lett. , 1817 (2004).[10] O. Gunawan, Y. Virgus, and K. Fai Tai, Appl. Phys.Lett. , 062407 (2015).[11] J. F. Hsu, P. Ji, C. W. Lewandowski, and B. D’Urso,Sci. Rep. , 30125 (2016).[12] J. P. Houlton, M. L. Chen, M. D. Brubaker, K. A. Bert-ness, and C. T. Rogers, Review of Scientific Instruments , 125107 (2018).[13] M. D. Simon and A. K. Geim, J. Appl. Phys. , 6200(2000).[14] M. O’Brien, S. Dunn, J. Downes, and J. Twamley, Appl.Phys. Lett. , 053103 (2019).[15] B. R. Slezak, C. W. Lewandowski, J.-F. Hsu, andB. DUrso, New J. Phys. , 063028 (2018).[16] H. J. Mamin, M. Poggio, C. L. Degen, and D. Rugar,Nat. Nanotech. , 301 (2007).[17] D. Rugar, R. Budakian, H. Mamin, and B. Chui, Nature , 329 (2004).[18] M. T. Johnsson, G. K. Brennen, and J. Twamley, Sci.Rep. , 37495 (2016).[19] T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, Sci-ence , 1673 (2010).[20] J. Millen, T. Deesuwan, P. Barker, and J. Anders, Nat.Nanotech. , 425 (2014).[21] Z.-q. Yin, T. Li, X. Zhang, and L. Duan, Phys. Rev. A , 033614 (2013).[22] M. Scala, M. S. Kim, G. W. Morley, P. F. Barker, andS. Bose, Phys. Rev. Lett. , 180403 (2013).[23] T. M. Hoang, J. Ahn, J. Bang, and T. Li, Nat. Comm. , 12250 (2016).[24] D. Kafri, J. M. Taylor, and G. J. Milburn, New J. Phys. , 065020 (2014).[25] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht,M. Toros, M. Paternostro, A. A. Geraci, P. F. Barker,M. S. Kim, and G. Milburn, Phys. Rev. Lett. ,240401 (2017).[26] M. V. Berry and A. K. Geim, Eur. J. Phys. , 307(1997).[27] O. Gunawan and Y. Virgus, J. Appl. Phys. , 133902(2017).[28] The experiment in Ref. [14] uses magnetic tip with fourflat facets. We have also modeled such system and foundthat our conical tip model is a very good approximation.[29] A. Zangwill, Modern electrodynamics (Cambridge Uni-versity Press, 2013).[30] J. Heremans, C. H. Olk, and D. T. Morelli, Phys. Rev.B , 15122 (1994).[31] J. A. Dean, Lange’s handbook of chemistry upplementary Material: A Magnetic Tip Trap System
Oki Gunawan, Jason Kristiano, and Hendra KweeJune 14, 2019
A. Magnetic Field Calculation
We will calculate the magnetic field along the principal axis ( z ) of a pair of magnetic tipsas shown in Fig. 1(a) in the main text. First we consider a single magnetic tip as shown inFig. S1(a). The magnetic tip system can be modeled as a semi-infinite cylindrical section and aconical tip. The magnet has a uniform magnetization M with direction along the principal axispointing toward the tip. The tip is located at z = a , the conic has radius R and ”facet angle” θ with respect to horizontal, as shown in Fig. S1.First consider the conical tip that has bound surface current element at z = u due to themagnetization. The surface current forms a current loop with radius r = ( u − a ) cot θ , thatproduces magnetic field: d B c = µ r dI ( r + ∆ z ) / ( − ˆz )= µ u − a ) cot θ dI [( u − a ) cot θ + ( u − z ) ] / ( − ˆz ) . (1)Figure S1: (a) A magnetic tip trap model. (b) Current loop on the cone surface.The surface current element on the conical surface can be calculated as: K b = M × ˆn where ˆn is the surface normal. The current element is given as: dI = K b dS , where K = M sin θ and dS = du/ sin θ . As a result, the current element is equal to dI = M du . The magnetic fieldproduced by the conical tip is given by: B c ( z ) = − µ M Z a + R tan θa ( u − a ) cot θ du [( u − a ) cot θ + ( u − z ) ] / ˆz . (2)1 a r X i v : . [ phy s i c s . c l a ss - ph ] J un he integral yields analytical result: B c ( z ) = − µ M ˆz " ( a − z ) sin θ cos θ − R cos 2 θ sin θ p R + ( a − z ) cos θ + R ( a − z ) sin 2 θ − cos θ sin θ arctanh(sin θ )cos θ sin θ arctanh R + ( a − z ) sin θ cos θ p R + ( a − z ) cos θ + R ( a − z ) sin 2 θ ! − sin θ . (3)Next we calculate the magnetic field due to the cylindrical sheath, which is straightforward.Consider a bound surface current element at z = u with radius R and current dI = M du , as inFig. S2. The magnetic field produced by magnet sheath is given by: B s ( z ) = − µ M ˆz Z ∞ a + R tan θ R du [ R + ( u − z ) ] / = − µ M ˆz " z − a − R tan θ p R + ( a + R tan θ − z ) (4)Figure S2: Magnetic field produced by the cylindrical sheath.Total magnetic field due to the conical tip and the sheath is given as: B ( z ) = − µ M ˆz (cid:2) cos θ [1 − sin θ arctanh(sin θ )]+cos θ sin θ arctanh " R + ( a − z ) sin θ cos θ p R + ( a − z ) cos θ + R ( a − z ) sin 2 θ + z − a − R tan θ p R + ( a − z + R tan θ ) + ( a − z ) sin θ cos θ − R cos 2 θ sin θ p R + ( a − z ) cos θ + R ( a − z ) sin 2 θ . (5)Finally, by exploiting the symmetry of the problem, the total magnetic field of the upperand lower magnetic tip is given as: B T ( z ) = B ( z ) − B ( − z ) . (6)2 . Trap Frequency and Optimum Facet Angle The total trap potential per unit volume of trapped-object along z can be approximated asharmonic potential due to magnetic interaction plus gravitational term: U T ( z ) = 12 k z z + ρgz (7)where k z is the ”spring constant” per unit volume: k z = ∂ U M ∂z (8)and U M is the magnetic energy potential per unit volume given as: U M ( z ) = − χB T ( z )2 µ . (9)Using B T calculated from Eq. 6, we can obtain k z in a surprisingly compact form: k z = − χµ M R cos θ ( a + R tan θ ) a ( R + a cos θ + aR sin 2 θ ) . (10)Condition dk z /dθ = 0 implies the optimum facet angle θ max is obtained for θ that satisfies: ddθ (cid:20) cos θ ( a + R tan θ ) ( R + a cos θ + aR sin 2 θ ) (cid:21) = 0 . (11)By using η = a/R , the equation which is satisfied by θ max is:(9 + 4 η + 4 η ) cos θ max − (12 + 8 η ) cos θ max + 4 = 0 . (12)This yields solution for θ max as: θ max = arccos s η + 2 √ η )9 + 4 η + 4 η . (13)In the limit of small gap ( a (cid:28) R ), the optimum facet angle reduces to a very simple expres-sion: θ max ≈ arccos p / . ◦ . (14) C. Magnetic Trap with Cylindrical Symmetry
We will try to find relationship between the axial and radial frequency of an object trapped in amagnetic trap with cylindrical symmetry. First we consider the divergence-free property of themagnetic field: ∇ · B = 0 or equivalently the Gauss Law for magnetic field H B · dA = 0. For amagnetic field with cylindrical symmetry, the flux at the bottom of the cylinder is [32]: − Z B z (2 πr ) dr (15)and the flux at the top of the cylinder is: Z ( B z + dB z )(2 πr ) dr. (16)3t the cylindrical sheath, the magnetic flux is: B r (2 πr ) dz. (17)Total of the three component should be zero, so we have: − Z B z (2 πr ) dr + Z ( B z + dB z )(2 πr ) dr + B r (2 πr ) dz = 0 (18) B r ( r, z ) = − r Z ∂B z ∂z r dr (19)A general form of B z near z = z and r = 0 up to second order is given by: B z ( r, z ) = b + b ( z − z ) + b ( z − z ) + b r + b r + b ( z − z ) r. (20)Now we can calculate (19): B r ( r, z ) = − r (cid:20) b r + b ( z − z ) r + 13 b r (cid:21) + c. (21)Constant c is zero because boundary condition B r (0 , z = 0). Since ∇ × B = 0, both the axialand radial magnetic field must satisfy: ∂B r ∂z − ∂B z ∂r = 0 , (22) − b r − b − b r − b ( z − z ) = 0 . (23)To make the equation consistent for all values of r and z , the coefficients must be b = 0, b = 0,and b = − b /
2. Finally, we get both the axial and radial magnetic field up to second order: B z ( r, z ) = b + b ( z − z ) + b (cid:20) ( z − z ) − r (cid:21) , (24) B r ( r, z ) = − b r − b ( z − z ) r. (25)Now we can calculate the ratio of the radial and the axial frequencies: f r /f z = p k r /k z . Thespring constant for the axial component: k z = ∂ U T /∂z ∝ ∂ B z ( r = 0 , z ) /∂z ∝ b (26)The spring constant for the radial component: k r = ∂ U T /∂r ∝ ∂ B r ( r, z = z ) /∂r ∝ ( b / (27)Therefore, we have: f r /f z = ( b / /b / D. Critical Gap
In order to find the critical gap, i.e. the gap beyond which there is no more confinementenhancement effect, we can expand the magnetic energy up to the fourth-order: U M ( z ) ≈ − χ µ M ( αz + βz ) (28)4 = 32 R cos θ ( a cos θ + R sin θ ) a ( a + 2 R + a cos 2 θ + 2 aR sin 2 θ ) (29) β = 8 R cos θ ( a cos θ + R sin θ )3 a ( a + 2 R + a cos 2 θ + 2 aR sin 2 θ ) (cid:2) a (60 a + 47 a R + 28 R ) cos θ + a (60 a − a R − R ) cos 3 θ + a (12 a − R ) cos 5 θ +2 R (44 a + 49 a R + 16 R ) sin θ + 3 a R (44 a + 21 R ) sin 3 θ + a R (44 a − R ) sin 5 θ (cid:3) (30)The equilibrium height of diamond z is the solution of: U T ( z ) ≈ − χ µ M αz + ρg = 0 , (31)so we can express z in terms of λ = | χ | µ M /ρg and α : z = 4 λ α . (32)The spring constant up to leading order of z is: k z ( θ, z ) = − χ µ M ( α + 6 βz )= − χ µ M (cid:18) α + 96 βλ α (cid:19) . (33)Confinement enhancement exists if there is an optimum angle θ max where the spring constant k z becomes maximum. However there is a critical gap (or half-gap a c ) beyond which there is no θ between 0 and 90 ◦ that yields maximum k z . The condition for the critical gap is: ∂ k z ∂θ = 0 , (34)evaluated at optimum angle θ max = 35 . ◦ . The relation between a c , R , and λ to satisfy thatcondition is: − . a + 16 . aR + 1295 . R + 0 . λ ) R λ = 0 . (35)Hence, the solution for critical gap is: a c R = λ ( p . R + 3 . λ − λ )153 . R + 3 . λ ..