1900:, this current creates and oscillating magnetic flux which can be coupled into the bore of the LGR. This magnetic flux, in turn, induces circumferential currents along the inner wall of the LGR. The induced current, once again by Faraday's law, creates an approximately uniform oscillating magnetic field in the bore of the LGR. A second coupling loop, connected to a receiver, can be used to detect the magnetic flux produced by the LGR. Alternatively, using a
1978:
1986:
420:
20:
1693:
2348:
1701:
2399:
response of the superconductor alters in the inductance of the LGR and, therefore, its resonant frequency. As described below, tracking the change in the resonant frequency as the temperature of the sample is changed allows one to deduce the temperature dependence of the magnetic penetration depth.
2019:
Multi-loop, multi-gap LGRs have also been developed. The simplest of these is the two-loop, one-gap LGR. In this case, magnetic field lines form closed loops by passing through each of the bores of the LGR and the currents on the inner walls propagate in opposite directions - clockwise in one bore
1736:
measurements without requiring additional electromagnetic shielding. In the toroidal geometry the two ends of a cylindrical LGR are joined to form a completely closed structure. In this case, the magnetic field is completely confined within the bore of the resonator and there is no radiative power
264:
One advantage of the LGR is that it produces regions of uniform electric and magnetic fields that are isolated from one another. A uniform electric field exists within the slit of the LGR and a uniform magnetic field exists within the bore of the resonator. The uniform magnetic field makes the LGR
1817:
The toroidal LGR is particularly convenient when characterizing the electromagnetic properties of liquid samples or particles suspended in a liquid. In these cases, the bore of the toroidal LGR can be partially filled with the liquid sample without requiring a special sample holder. This setup
265:
a good source of microwave magnetic fields in ESR experiments. Furthermore, because the electric and magnetic fields are isolated from one another, one can use the LGR to independently probe the electric and magnetic properties of materials. For example, if the gap of the LGR is filled with a
2144:
times greater than that of the conventional one-loop, one-gap LGR having the same bore and gap dimensions. It is also worth noting that, since magnetic field lines pass from one bore to the other, radiative power losses are strongly suppressed and the resonator maintains a high quality factor
1643:
1683:
waveguide mode is well above the resonant frequency of the LGR, the magnetic field lines will be prevented from propagating into free space. The presence of the electromagnetic shield will alter the resonant frequency and quality factor of the LGR, but typically by only a few percent.
4926:
1962:, the magnitude of the reflection coefficient will fall below one as power is transferred into the LGR. The coupling between the external circuits and the LGR can be tuned by adjusting the relative positions and orientations of the coupling loop and LGR. At critical coupling,
3747:
4484:
55:) is an electromagnetic resonator that operates in the radio and microwave frequency ranges. The simplest LGRs are made from a conducting tube with a narrow slit cut along its length. The LGR dimensions are typically much smaller than the free-space wavelength of the
845:
5972:
3067:
4138:
resonator. The ultrahigh quality factors are obtained by coating the LGR surfaces with a superconducting material, such as a lead-tin alloy. The resonator is then cooled below the superconducting transition temperature of the coating using a bath of
1766:
circuit. In general, the effective capacitance, inductance, and resistance of the toroidal LGR will differ from the expressions given above for the cylindrical LGR. However, in limit that the radius of the torus is large compared to the bore radius
3193:
1822:. Alternatively, if the liquid sample is nonmagnetic, the entire toroidal LGR can be submerged in the liquid (or gas). In this case, the dielectric properties of the sample only modify the effective capacitance of the resonator and the changes in
3960:
2653:
4782:
1935:) as a function of microwave frequency. Far away from resonance, the magnitude of the reflection coefficient will be close to one since very little power is coupled into the LGR at these frequencies. However, near the resonance frequency
363:
can be used to fully determine the complex permittivity of the dielectric material. Likewise, if the bore of the LGR is filled with a magnetic material, the effective inductance of the LGR will be modified and the resulting changes in
1715:
In some applications requiring high quality factors, the electromagnetic shielding provided by a concentric circular waveguide surrounding a cylindrical LGR can be bulky and awkward to work. A toroidal LGR can be used for
1288:
1002:
The effective resistance of a LGR can be estimated by considering the length of conductor through which the current travels and the cross-sectional area available to it. The relevant conductor length is the circumference
5262:, on the other hand, alters the imaginary part of the impedance and modifies the resonant frequency. Written in terms of the empty-resonator resonant frequency and quality factor, the above impedance can be expressed as
3450:
1532:
1149:
2757:
5828:
2314:
7064:
Hyde, James S.; Froncisz, Wojciech; Sidabras, Jason W.; Camenisch, Theodore G.; Anderson, James R.; Strangeway, Robert A. (2007). "Microwave frequency modulation in CW EPR at W-band using a loop-gap resonator".
3846:
997:
4822:
5565:
652:
3547:
2952:
486:
4296:
7274:
Pereg-Barnea, T.; Turner, P. J.; Harris, R.; Mullins, G. K.; Bobowski, J. S.; Raudsepp, M.; Liang, Ruixing; Bonn, D. A.; Hardy, W. N. (2004). "Absolute values of the London penetration depth in YBa
6214:
2682:
of its surface. Therefore, the effective volume of the resonator bore is reduced by an amount equal to the volume from which the magnetic flux has been excluded. This excluded volume is given by
903:
3272:
1495:. The suppression of the quality factor is due to radiative power loss from magnetic field lines that extend out of LGR bore and into free space. An order-of-magnitude estimate of the effective
4638:
4387:
2466:
1219:
741:
3355:
Instead, the strategy is to track the changes in frequency as the sample temperature varies (while keeping the LGR at a fixed temperature). The absolute penetration depth can be expressed as
5200:
4040:
5756:
5699:
5233:
4814:
6947:
Wood, Robert L.; Froncisz, Wojciech; Hyde, James S. (1984). "The loop-gap resonator. II. Controlled return flux three-loop, two-gap microwave resonators for ENDOR and ESR spectroscopy".
3965:
This final expression shows how the LGR shifts in resonant frequency can be used to determine the temperature dependence of the magnetic penetration depth in a superconducting sample.
4209:
This section describes how LGRs can be used to determine the electromagnetic properties of materials. When there are no materials filling either the gap or bore of the resonator, the
5868:
1424:
5726:
5669:
5260:
3539:
3198:
Generally, it is not possible to use LGR frequency-shift measurements to determine the absolute value of the penetration depth because it would require knowing the sample thickness
2960:
1379:
1337:
7108:
Sidabras, Jason W.; Mett, Richard R.; Froncisz, Wojciech; Camenisch, Theodore G.; Anderson, James R.; Hyde, James S. (2007). "Multipurpose EPR loop-gap resonator and cylindrical TE
6358:
6247:
5860:
4116:
4078:
4710:
3098:
5642:
3854:
3350:
2879:
2495:
1904:(VNA), a single coupling loop can be used to both inject a signal into the LGR and measure its response. The VNA can measure the ratio of the forward and reflected voltages (
516:
6328:
6274:
4332:
4199:
1078:
2568:
1493:
1450:
3299:
6029:
5595:
4514:
4359:
2142:
1034:
2013:
1673:
1174:
is the resistivity of the conductor. The effective capacitance, inductance, and resistance then lead to a simple expression for the expected quality factor of the LGR
7217:
6301:
6002:
4726:
2680:
2560:
2369:
1704:
Photograph of the two halves of a copper toroidal loop-gap resonator. Also visible are an inductive coupling loop suspended within the resonator bore and a so-called
682:
3541:
is the change in penetration depth as the sample temperature is increased above the base temperature. One can, therefore, express the change in penetration depth as
2118:
1054:
204:
4561:
4171:
1929:
1888:. The exposed centre conductor is then bent into a loop and short-circuited to the outer conductor. The opposite end of the coaxial cable is connected to either a
4668:
3500:
2522:
2245:
2173:
2065:
1960:
1847:
1812:
1792:
1524:
1172:
713:
580:
389:
341:
294:
181:
41:
261:
circuit. This circuit model works well provided the dimensions of the resonator remain small compared to the free-space wavelength of the electromagnetic fields.
2850:
1764:
945:
259:
231:
6585:
Bobowski, Jake S. (2013). "Using split-ring resonators to measure the electromagnetic properties of materials: An experiment for senior physics undergraduates".
1969:
It is also possible to capacitively couple electric fields into and out of the gap of the LGR using suitably-fashioned electrodes at the end of a coaxial cable.
7035:
6918:
6049:
5615:
4534:
4379:
4230:
4136:
3473:
3319:
3216:
3090:
2899:
2820:
2800:
2780:
2337:
2213:
2193:
2085:
2038:
1867:
1734:
1036:
of the conductor's inner surface. The depth that the current penetrates into the inner surface of the LGR bore is determined by the electromagnetic skin depth
733:
560:
540:
409:
361:
314:
158:
135:
112:
7331:
Dubreuil, J.; Bobowski, J. S. (2019). "Ferromagnetic resonance in the complex permeability of an Fe3O4-based ferrofluid at radio and microwave frequencies".
59:
at the resonant frequency. Therefore, relatively compact LGRs can be designed to operate at frequencies that are too low to be accessed using, for example,
2662:, when a superconducting sample is place in the bore of a LGR, the magnetic flux is expelled from the interior of the sample to within a penetration depth
2351:
The
Meissner effect in a platelet superconducting single crystal. The blue lines represent an applied magnetic field. Magnetic fields penetrate a depth
1989:
Photograph of an aluminum two-loop, one-gap LGR. Also visible are an inductive coupling loop suspended within the right-hand bore of the resonator and a
1460:
In practice, the measured quality factor a cylindrical LGR, without additional electromagnetic shielding, will be much less than the predicted value of
1638:{\displaystyle R_{\mathrm {r} }\approx {\frac {\pi }{6}}{\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}\left({\frac {\omega _{0}r_{0}}{c}}\right)^{4}\,,}
1230:
3361:
2148:
The multi-loop, multi-gap LGRs with more than two loops have more than one resonant mode. If the central bore is singled out as having inductance
1086:
2688:
5772:
2253:
7017:
Rinard, G. A.; Quine, R. W.; Eaton, S. S.; Eaton, G. R.; Froncisz, W. (1994). "Relative
Benefits of Overcoupled Resonators vs Inherently Low-
6698:
1499:
can be made by treating the LGR as a conducting loop. In the limit that the wavelength of the radiation is much larger than the loop radius
138:
1981:
Designs of some multi-loop, multi-gap LGRs. Top: Two-loop, one-gap LGR. Middle: Three-loop, two-gap LGR. Bottom: Five-loop, four-gap LGR.
6863:
Madsen, Sabrina L.; Bobowski, Jake S. (2020). "The
Complex Permeability of Split-Ring Resonator Arrays Measured at Microwave Frequencies".
909:. Therefore, the resonant frequency of the LGR is determined from its geometry and is, to first approximation, independent of its length.
3755:
4921:{\displaystyle Z_{\varepsilon }=R+j\omega L+{\frac {1}{j\omega \left(\varepsilon ^{\prime }-j\varepsilon ^{\prime \prime }\right)C}}\,.}
137:. At, or near, the resonance frequency, a circumferential current is established along the inner wall of the resonator. The effective
7199:
Hardy, W. N.; Bonn, D. A.; Morgan, D. C.; Liang, Ruixing; Zhang, Kuan (1993). "Precision measurements of the temperature dependence of
2371:
into the crystal. In a single-crystal superconductor, there is a unique penetration depth for each of the crystallographic directions.
953:
1884:
into and out of the LGR. The coupling loops are made by first removing a length of outer conductor and dielectric from a semi-rigid
6635:
Bobowski, Jake S.; Clements, Aaron P. (2017). "Permittivity and
Conductivity Measured Using a Novel Toroidal Split-Ring Resonator".
5268:
3742:{\displaystyle \Delta \lambda _{a}(T)=\lambda _{a}(T)-\lambda _{a}(T_{0})=-{\frac {V_{\mathrm {r} }}{ab}}{\frac {1}{f_{0}}}\left\,.}
592:
2904:
438:
6684:
5767:
4536:. The impedance measurement is most easily done using the vector network analyzer (VNA) to measure the reflection coefficient
76:
4479:{\displaystyle {\frac {Z}{QR}}={\frac {1}{Q}}+j\left({\frac {\omega }{\omega _{0}}}-{\frac {\omega _{0}}{\omega }}\right)\,.}
4238:
2376:
6057:
853:
3233:
840:{\displaystyle f_{0}\approx {\frac {1}{2\pi }}{\frac {1}{\sqrt {LC}}}={\frac {c}{2\pi r_{0}}}{\sqrt {\frac {t}{\pi w}}}\,,}
4569:
2562:, a small change in the effective volume of the resonator bore will result in a change in the resonant frequency given by
2411:
1897:
1180:
1794:, the capacitance, inductance, and resistance of the toroidal LGR are approximated by the expressions above if one takes
1675:
due to the resistivity of the LGR conductor. The radiative losses can be suppressed by placing the LGR inside a circular
2175:, then one of the resonant modes is one in which all of the magnetic flux from each of the external loops of inductance
6680:
4937:
2020:
and counterclockwise in the other. The equivalent circuit, neglecting losses, is a parallel combination of inductors
3984:
6717:
Bobowski, Jake S.; Nakahara, Hiroko (2016). "Design and characterization of a novel toroidal split-ring resonator".
6411:
Froncisz, W.; Hyde, James S. (1982). "The loop-gap resonator: a new microwave lumped circuit ESR sample structure".
6384:
Hardy, Walter N.; Whitehead, L. A. (1981). "Split‐ring resonator for use in magnetic resonance from 200–2000 MHz".
5731:
5674:
5208:
1737:
loss. The toroidal LGR consists of two halves that are bolted together along the outer diameter of the structure.
685:
519:
6825:
Rinard, G. A.; Quine, R. W.; Eaton, S. S.; Eaton, G. R. (1993). "Microwave
Coupling Structures for Spectroscopy".
4787:
5967:{\displaystyle Z_{\mu }=R+j\omega \left(\mu ^{\prime }-j\mu ^{\prime \prime }\right)L+{\frac {1}{j\omega C}}\,.}
3062:{\displaystyle \delta f_{0}={\frac {f_{0}}{2V_{\mathrm {r} }}}\left(V_{\mathrm {s} }-2ab\lambda _{a}\right)\,.}
2380:
4720:
Now suppose that the gap of resonator has been completely filled with a dielectric material that has complex
2881:. In the above expression, it has been assumed that the microwave magnetic field is applied parallel to the
1384:
5704:
5647:
5238:
3505:
1901:
1342:
1296:
68:
6333:
6222:
5833:
3188:{\displaystyle \lambda _{a}={\frac {c}{2}}-{\frac {V_{\mathrm {r} }}{ab}}{\frac {\delta f_{0}}{f_{0}}}\,.}
1932:
235:. Since the LGR current is a maximum at the resonant frequency, the equivalent circuit model is a series
4083:
4045:
3955:{\displaystyle \Delta \lambda _{a}(T)=-{\frac {V_{\mathrm {r} }}{ab}}{\frac {\Delta f_{0}(T)}{f_{0}}}\,.}
4721:
906:
56:
4673:
5620:
3324:
2855:
2648:{\displaystyle \delta f_{0}=-{\frac {f_{0}}{2}}{\frac {\delta V_{\mathrm {r} }}{V_{\mathrm {r} }}}\,.}
2471:
494:
7305:
7240:
7168:
7121:
7074:
6991:
6956:
6834:
6796:
6736:
6654:
6604:
6459:
6420:
4210:
1708:
1496:
88:
6306:
6252:
4489:
A measurement of the frequency dependence of the impedance of an empty LGR can be used to determine
4304:
4176:
2375:
Loop-gap resonators have been used to make precise measurements of the electrodynamic properties of
1059:
7374:
5235:
enhances the effective resistance of the LGR and, therefore, lowers its quality factor. A nonzero
1893:
1696:
Drawing of a toroidal LGR with a section cut out so as to expose the bore and gap of the resonator.
1463:
1429:
1224:
where, for a good conductor, the electromagnetic skin depth at the resonance frequency is given by
6777:
Bobowski, Jake S. (2018). "Probing split-ring resonator permeabilities with loop-gap resonators".
3277:
7340:
7295:
6872:
6786:
6726:
6644:
6594:
6007:
5573:
4777:{\displaystyle \varepsilon _{\mathrm {r} }=\varepsilon ^{\prime }-j\varepsilon ^{\prime \prime }}
4492:
4337:
2123:
1963:
1877:
1006:
269:
material, the effective capacitance of the LGR will be modified which will change the frequency
6249:
contributes additional dissipation which lowers the quality factor of the filled resonator and
1992:
1651:
423:
Cross-sectional view of a cylindrical loop-gap resonator with the critical dimensions labelled.
7256:
7202:
7137:
7090:
6752:
6694:
6526:
6477:
6279:
5980:
2665:
2527:
2354:
660:
2090:
1039:
189:
7350:
7313:
7248:
7176:
7129:
7082:
7046:
6999:
6964:
6929:
6882:
6842:
6804:
6744:
6662:
6612:
6558:
6516:
6508:
6467:
6428:
6393:
4539:
4149:
2823:
1907:
1889:
60:
7157:"Split‐ring resonators for measuring microwave surface resistance of oxide superconductors"
6446:
Twig, Ygal; Sorkin, Anton; Cristea, David; Feintuch, Akiva; Blank, Aharon (December 2017).
4646:
3478:
2500:
2218:
2151:
2043:
1938:
1825:
1797:
1770:
1502:
1157:
691:
565:
367:
319:
272:
166:
26:
4563:
from an inductively-coupled LGR. The impedance and reflection coefficient are related by
2659:
2395:. In these experiments, a superconducting sample is placed inside the bore of a LGR. The
1977:
947:
circuit, the quality factor, which determines the sharpness of the resonance, is given by
206:
of the conductor used to make the LGR. It is, therefore, possible to model the LGR as an
6551:
Bobowski, Jake S. (July 22–24, 2015). M. Eblen-Zayas; E. Behringer; J. Kozminski (eds.).
6497:"Uniform Field Loop-Gap Resonator and Rectangular TEU02 for Aqueous Sample EPR at 94 GHz"
2829:
1743:
924:
238:
210:
7309:
7244:
7172:
7125:
7078:
6995:
6960:
6838:
6800:
6740:
6658:
6608:
6463:
6424:
7020:
6903:
6690:
6521:
6496:
6495:
Sidabras, Jason W.; Sarna, Tadeusz; Mett, Richard R.; Hyde, James S. (September 2017).
6034:
5600:
4519:
4364:
4215:
4121:
3974:
3458:
3304:
3201:
3075:
2884:
2805:
2785:
2765:
2322:
2198:
2178:
2070:
2023:
1852:
1719:
718:
545:
525:
394:
346:
299:
143:
120:
97:
64:
2901:-axis of the sample. Since the presence of the superconductor reduces the LGR volume,
71:(ESR) experiments, and precision measurements of electromagnetic material properties (
7368:
7003:
6968:
6552:
6432:
5766:
Next, suppose that the bore of a LGR is filled with a magnetic material have complex
4143:
1885:
1881:
1985:
72:
6982:
Hyde, James S.; Froncisz, W.; Oles, T. (1989). "Multipurpose loop-gap resonator".
2379:. Most notably, a LGR was used to reveal the linear temperature dependence of the
419:
19:
4118:. Measuring such small changes in relative frequency requires an extremely high-
7252:
2396:
918:
207:
184:
161:
92:
7354:
7317:
5701:
of the material filling the gap of the LGR. This analysis gives the values of
1283:{\displaystyle \delta \approx {\sqrt {\frac {2\rho }{\mu _{0}\omega _{0}}}}\,,}
7086:
6886:
6666:
6512:
4140:
2195:
is shared with the central loop. For this mode, the resonant frequency of an
1819:
266:
115:
6276:
shifts the resonant frequency. A measurement of the frequency dependence of
6004:
into its real and imaginary components and writing the impedance in terms of
3973:
In a d-wave superconductor, the penetration depth typically changes by a few
3445:{\displaystyle \lambda _{a}(T)=\lambda _{a}(T_{0})+\Delta \lambda _{a}(T)\,,}
2015:
array of planar split-ring resonators in the left-hand bore of the resonator.
1144:{\displaystyle R_{\rho }\approx \rho {\frac {2\pi r_{0}}{\delta \,\ell }}\,,}
6563:
6554:
Using Split-Ring
Resonators to Measure Complex Permittivity and Permeability
1676:
7260:
7141:
7094:
7050:
6846:
6756:
6530:
6481:
6808:
2752:{\displaystyle V_{\mathrm {ex} }\approx ab\left(c-2\lambda _{a}\right)\,,}
1692:
582:
is the length of the resonator. The resonator bore acts as a single-turn
411:
can be used to extract the complex permeability of the magnetic material.
7300:
5823:{\displaystyle \mu _{\mathrm {r} }=\mu ^{\prime }-j\mu ^{\prime \prime }}
2309:{\displaystyle f_{0}\approx {\frac {1}{2\pi }}{\sqrt {\frac {n}{LC}}}\,,}
583:
4173:
have been achieved using copper LGRs coated with lead-tin and cooled to
1700:
3274:
at low temperature. Therefore, to use the LGR measurement to determine
2383:, characteristic of a d-wave superconductor, in a single crystal of YBa
1740:
Like the cylindrical LGR, the toroidal LGR can be modelled as a series
7286:
measured by zero field ESR spectroscopy on Gd doped single crystals".
7181:
7156:
7133:
6933:
6748:
6616:
6472:
6447:
6397:
2347:
1818:
allows one to characterize the magnetic properties of, for example, a
3978:
1056:. Therefore, the cross-sectional area through which charge flows is
5617:
are known before hand, a measurement of the frequency dependence of
735:
resonator, the resonant frequency is, to an approximation, given by
7345:
6877:
6791:
6731:
6649:
6599:
6448:"Surface loop-gap resonators for electron spin resonance at W-band"
3841:{\displaystyle \Delta f_{0}(T)=\delta f_{0}(T)-\delta f_{0}(T_{0})}
2319:
where it has been assumed that all loops have the same inductance
1869:
can be used to determine the complex permittivity of the sample.
992:{\displaystyle Q\approx {\frac {1}{R}}{\sqrt {\frac {L}{C}}}\,.}
91:. The slit along the length of the resonator has an effective
5560:{\displaystyle {\frac {Z_{\varepsilon }}{QR}}=\left+j\left\,.}
2120:, then the resonant frequency of the two-loop, one-gap LGR is
2497:
is the volume of the LGR bore. Since the resonant frequency
647:{\displaystyle L=\mu _{0}{\frac {\pi \,r_{0}^{2}}{\ell }}\,,}
6345:
6342:
6315:
6261:
6234:
6231:
6155:
6112:
6109:
5926:
5923:
5907:
5815:
5812:
5796:
5743:
5740:
5713:
5686:
5683:
5656:
5509:
5506:
5478:
5462:
5376:
5373:
5345:
5329:
5326:
5247:
5220:
5217:
5146:
5143:
5115:
5099:
5026:
5023:
4995:
4979:
4976:
4898:
4895:
4879:
4769:
4766:
4750:
1966:
is achieved and the reflection coefficient approaches zero.
2947:{\displaystyle \delta V_{\mathrm {r} }=-V_{\mathrm {ex} }}
481:{\displaystyle C=\varepsilon _{0}{\frac {w\,\ell }{t}}\,,}
1080:. Combining these results gives an effective resistance
2145:
without requiring additional electromagnetic shielding.
160:
that limits this current is, in part, determined by the
63:. These structures can have very sharp resonances (high
4291:{\displaystyle Z=R+j\omega L+{\frac {1}{j\omega C}}\,,}
6209:{\displaystyle {\frac {Z_{\mu }}{QR}}=\left+j\left\,.}
1892:
or a receiver. In the case of a signal generator, an
1679:. Provided that the cutoff frequency of the lowest TE
898:{\displaystyle c=1/{\sqrt {\varepsilon _{0}\mu _{0}}}}
7205:
7023:
6906:
6336:
6309:
6282:
6255:
6225:
6060:
6037:
6010:
5983:
5871:
5836:
5775:
5734:
5707:
5677:
5650:
5623:
5603:
5576:
5271:
5241:
5211:
4940:
4825:
4790:
4729:
4676:
4649:
4572:
4542:
4522:
4495:
4390:
4367:
4340:
4307:
4241:
4218:
4179:
4152:
4124:
4086:
4048:
3987:
3857:
3758:
3550:
3508:
3481:
3461:
3364:
3327:
3307:
3280:
3267:{\displaystyle \lambda _{a}\approx 100~\mathrm {nm} }
3236:
3204:
3101:
3078:
2963:
2907:
2887:
2858:
2832:
2808:
2788:
2768:
2691:
2668:
2571:
2530:
2503:
2474:
2414:
2357:
2325:
2256:
2221:
2201:
2181:
2154:
2126:
2093:
2073:
2046:
2026:
1995:
1941:
1910:
1855:
1828:
1800:
1773:
1746:
1722:
1654:
1535:
1505:
1466:
1432:
1387:
1345:
1299:
1233:
1183:
1160:
1089:
1062:
1042:
1009:
956:
927:
856:
744:
721:
694:
663:
595:
568:
548:
528:
497:
441:
397:
370:
349:
322:
302:
275:
241:
213:
192:
169:
146:
123:
100:
29:
6865:
IEEE Transactions on
Microwave Theory and Techniques
6637:
IEEE Transactions on
Microwave Theory and Techniques
4633:{\displaystyle S_{11}={\frac {Z_{0}-Z}{Z_{0}+Z}}\,,}
2461:{\displaystyle L=\mu _{0}V_{\mathrm {r} }/\ell ^{2}}
1214:{\displaystyle Q\approx {\frac {r_{0}}{\delta }}\,,}
4784:. In this case, the effective capacitance becomes
4334:. Re-expressed in terms of the resonant frequency
3301:to within 10%, one would have to know the value of
7231:: Strong evidence for nodes in the gap function".
7211:
7029:
6912:
6352:
6322:
6295:
6268:
6241:
6208:
6043:
6023:
5996:
5966:
5854:
5830:. In this case, the effective inductance becomes
5822:
5750:
5720:
5693:
5663:
5636:
5609:
5589:
5559:
5254:
5227:
5194:
4920:
4808:
4776:
4704:
4662:
4632:
4555:
4528:
4508:
4478:
4373:
4353:
4326:
4290:
4224:
4193:
4165:
4130:
4110:
4072:
4034:
3954:
3840:
3741:
3533:
3494:
3467:
3444:
3344:
3313:
3293:
3266:
3210:
3187:
3084:
3061:
2946:
2893:
2873:
2844:
2814:
2794:
2774:
2751:
2674:
2647:
2554:
2516:
2489:
2460:
2363:
2331:
2308:
2239:
2207:
2187:
2167:
2136:
2112:
2079:
2059:
2032:
2007:
1954:
1923:
1861:
1841:
1806:
1786:
1758:
1728:
1667:
1637:
1518:
1487:
1444:
1418:
1373:
1331:
1282:
1213:
1166:
1143:
1072:
1048:
1028:
991:
939:
897:
839:
727:
707:
676:
646:
574:
554:
534:
510:
480:
432:The capacitance of the gap of the LGR is given by
403:
383:
355:
335:
308:
288:
253:
225:
198:
175:
152:
129:
106:
35:
7155:Bonn, D. A.; Morgan, D. C.; Hardy, W. N. (1991).
4931:Separating the real and imaginary terms leads to
715:is the inner radius of the LGR bore. For a high-
316:of the resonance. Measurements of the changes in
5195:{\displaystyle Z_{\varepsilon }=\left+j\left\,.}
3218:very precisely. For example, in fully doped YBa
6772:
6770:
6768:
6766:
4080:platelet sample in a LGR with a bore volume of
4035:{\displaystyle \Delta f_{0}/f_{0}\sim 10^{-10}}
1814:to be equal to the circumference of the torus.
6712:
6710:
6630:
6628:
6626:
6580:
6578:
6576:
6574:
6546:
6544:
6542:
6540:
2408:The inductance of the LGR can be expressed as
87:Loop-gap resonators (LGRs) can be modelled as
6858:
6856:
6820:
6818:
6360:at the resonant frequency of the filled LGR.
5758:at the resonant frequency of the filled LGR.
5751:{\displaystyle \varepsilon ^{\prime \prime }}
5694:{\displaystyle \varepsilon ^{\prime \prime }}
5228:{\displaystyle \varepsilon ^{\prime \prime }}
4670:is the output impedance of the VNA (usually,
8:
7194:
7192:
4809:{\displaystyle \varepsilon _{\mathrm {r} }C}
114:and the bore of the resonator has effective
7333:Journal of Magnetism and Magnetic Materials
7037:Resonators for Pulsed Magnetic Resonance".
1648:and can be much larger than the resistance
23:A cylindrical loop-gap resonator of length
3502:is the experimental base temperature, and
2822:are the sample dimensions along the three
7344:
7299:
7204:
7180:
7022:
6905:
6876:
6790:
6730:
6686:Introduction to Electrodynamics (4th ed.)
6648:
6598:
6562:
6520:
6471:
6341:
6335:
6314:
6308:
6287:
6281:
6260:
6254:
6230:
6224:
6202:
6186:
6180:
6169:
6160:
6154:
6126:
6117:
6108:
6091:
6067:
6061:
6059:
6036:
6015:
6009:
5988:
5982:
5960:
5942:
5922:
5906:
5876:
5870:
5862:and the impedance of the LGR is given by
5842:
5841:
5835:
5811:
5795:
5781:
5780:
5774:
5739:
5733:
5712:
5706:
5682:
5676:
5655:
5649:
5628:
5622:
5602:
5581:
5575:
5553:
5537:
5531:
5518:
5505:
5487:
5477:
5461:
5455:
5440:
5431:
5404:
5398:
5385:
5372:
5354:
5344:
5325:
5319:
5302:
5278:
5272:
5270:
5246:
5240:
5216:
5210:
5188:
5168:
5155:
5142:
5124:
5114:
5098:
5092:
5048:
5035:
5022:
5004:
4994:
4975:
4969:
4945:
4939:
4914:
4894:
4878:
4857:
4830:
4824:
4816:and the impedance of the LGR is given by
4796:
4795:
4789:
4765:
4749:
4735:
4734:
4728:
4681:
4675:
4654:
4648:
4626:
4611:
4593:
4586:
4577:
4571:
4547:
4541:
4521:
4500:
4494:
4472:
4456:
4450:
4439:
4430:
4409:
4391:
4389:
4366:
4345:
4339:
4314:
4306:
4284:
4266:
4240:
4217:
4186:
4178:
4157:
4151:
4123:
4102:
4094:
4085:
4064:
4056:
4047:
4023:
4010:
4001:
3995:
3986:
3948:
3940:
3920:
3910:
3893:
3892:
3886:
3865:
3856:
3829:
3816:
3791:
3766:
3757:
3735:
3721:
3708:
3683:
3663:
3654:
3637:
3636:
3630:
3615:
3602:
3580:
3558:
3549:
3516:
3507:
3486:
3480:
3460:
3438:
3423:
3404:
3391:
3369:
3363:
3334:
3326:
3306:
3285:
3279:
3256:
3241:
3235:
3203:
3181:
3173:
3162:
3152:
3135:
3134:
3128:
3115:
3106:
3100:
3077:
3055:
3044:
3021:
3020:
3001:
3000:
2986:
2980:
2971:
2962:
2934:
2933:
2916:
2915:
2906:
2886:
2864:
2863:
2857:
2831:
2807:
2787:
2767:
2745:
2734:
2697:
2696:
2690:
2667:
2641:
2632:
2631:
2619:
2618:
2608:
2597:
2591:
2579:
2570:
2542:
2535:
2529:
2508:
2502:
2480:
2479:
2473:
2452:
2443:
2436:
2435:
2425:
2413:
2356:
2324:
2302:
2285:
2270:
2261:
2255:
2220:
2200:
2180:
2159:
2153:
2127:
2125:
2104:
2092:
2072:
2051:
2045:
2025:
1994:
1946:
1940:
1915:
1909:
1896:is established in the coupling loop. By
1854:
1833:
1827:
1799:
1778:
1772:
1745:
1721:
1659:
1653:
1631:
1625:
1609:
1599:
1592:
1578:
1568:
1561:
1551:
1541:
1540:
1534:
1510:
1504:
1477:
1471:
1465:
1431:
1405:
1404:
1392:
1386:
1363:
1362:
1350:
1344:
1323:
1304:
1298:
1276:
1266:
1256:
1240:
1232:
1207:
1196:
1190:
1182:
1159:
1137:
1130:
1119:
1106:
1094:
1088:
1066:
1061:
1041:
1020:
1008:
985:
973:
963:
955:
926:
887:
877:
871:
866:
855:
833:
816:
807:
791:
773:
758:
749:
743:
720:
699:
693:
668:
662:
640:
628:
623:
618:
612:
606:
594:
567:
547:
527:
502:
496:
474:
464:
458:
452:
440:
396:
375:
369:
348:
327:
321:
301:
280:
274:
240:
212:
191:
168:
145:
122:
99:
28:
2346:
1984:
1976:
1699:
1691:
418:
18:
7112:cavity for aqueous samples at 94 GHz".
7039:Journal of Magnetic Resonance, Series A
6827:Journal of Magnetic Resonance, Series A
6379:
6377:
6375:
6373:
6369:
4205:Measuring permittivity and permeability
1419:{\displaystyle f_{0}=1\,\mathrm {GHz} }
5721:{\displaystyle \varepsilon ^{\prime }}
5664:{\displaystyle \varepsilon ^{\prime }}
5255:{\displaystyle \varepsilon ^{\prime }}
3534:{\displaystyle \Delta \lambda _{a}(T)}
1374:{\displaystyle r_{0}=1\,\mathrm {cm} }
1332:{\displaystyle \omega _{0}=2\pi f_{0}}
6557:. BFY Proceedings. College Park, MD.
6353:{\displaystyle \mu ^{\prime \prime }}
6303:can be used to extract the values of
6242:{\displaystyle \mu ^{\prime \prime }}
5205:This expression shows that a nonzero
415:Resonant Frequency and Quality Factor
7:
5855:{\displaystyle \mu _{\mathrm {r} }L}
4111:{\displaystyle 1~\mathrm {cm} ^{3}}
4073:{\displaystyle 1~\mathrm {mm} ^{2}}
1880:loops are typically used to couple
542:is the thickness of the bore wall,
5843:
5782:
4797:
4736:
4696:
4187:
4098:
4095:
4060:
4057:
3988:
3913:
3894:
3858:
3759:
3638:
3551:
3509:
3416:
3338:
3335:
3260:
3257:
3136:
3022:
3002:
2938:
2935:
2917:
2865:
2701:
2698:
2633:
2620:
2481:
2437:
1542:
1412:
1409:
1406:
1367:
1364:
1339:. For an aluminum resonator with
14:
4705:{\displaystyle Z_{0}=50~\Omega )}
3352:which is typically not possible.
16:Type of electromagnetic resonator
7161:Review of Scientific Instruments
7114:Review of Scientific Instruments
6922:Review of Scientific Instruments
6900:Froncisz, W.; Oles, T. (1986). "
6719:Review of Scientific Instruments
6452:Review of Scientific Instruments
6386:Review of Scientific Instruments
5637:{\displaystyle Z_{\varepsilon }}
3345:{\displaystyle 10~\mathrm {nm} }
3072:Solving this expression for the
2874:{\displaystyle V_{\mathrm {s} }}
2490:{\displaystyle V_{\mathrm {r} }}
511:{\displaystyle \varepsilon _{0}}
4232:of the LGR can be expressed as
3092:-axis penetration depth yields
6323:{\displaystyle \mu ^{\prime }}
6269:{\displaystyle \mu ^{\prime }}
4699:
4327:{\displaystyle j={\sqrt {-1}}}
4194:{\displaystyle 1~\mathrm {K} }
3932:
3926:
3877:
3871:
3835:
3822:
3803:
3797:
3778:
3772:
3727:
3714:
3695:
3689:
3621:
3608:
3592:
3586:
3570:
3564:
3528:
3522:
3435:
3429:
3410:
3397:
3381:
3375:
2524:of the LGR is proportional to
2377:unconventional superconductors
2234:
2222:
1526:, the radiation resistance is
1073:{\displaystyle \delta \,\ell }
1:
7067:Journal of Magnetic Resonance
6984:Journal of Magnetic Resonance
6949:Journal of Magnetic Resonance
6501:Journal of Magnetic Resonance
6413:Journal of Magnetic Resonance
1488:{\displaystyle r_{0}/\delta }
1445:{\displaystyle Q\approx 3900}
7004:10.1016/0022-2364(89)90026-7
6969:10.1016/0022-2364(84)90214-2
6433:10.1016/0022-2364(82)90221-9
4381:, the impedance is given by
3294:{\displaystyle \lambda _{a}}
1426:the above analysis predicts
7253:10.1103/PhysRevLett.70.3999
6920:-band loop-gap resonator".
6779:Canadian Journal of Physics
6587:American Journal of Physics
6024:{\displaystyle \omega _{0}}
5590:{\displaystyle \omega _{0}}
4509:{\displaystyle \omega _{0}}
4354:{\displaystyle \omega _{0}}
2137:{\displaystyle {\sqrt {2}}}
2067:in series with capacitance
7391:
7355:10.1016/j.jmmm.2019.165387
7318:10.1103/PhysRevB.69.184513
2381:magnetic penetration depth
2343:LGRs and superconductivity
1973:Multi-Loop, Multi-Gap LGRs
1898:Faraday's law of induction
1029:{\displaystyle 2\pi r_{0}}
686:permeability of free space
520:permittivity of free space
185:electromagnetic skin depth
7087:10.1016/j.jmr.2007.01.002
6887:10.1109/TMTT.2020.2988672
6667:10.1109/TMTT.2016.2645147
6513:10.1016/j.jmr.2017.08.001
5644:can be used to determine
2008:{\displaystyle 2\times 2}
1668:{\displaystyle R_{\rho }}
586:with inductance given by
67:) making them useful for
7212:{\displaystyle \lambda }
6296:{\displaystyle Z_{\mu }}
6051:of the empty LGR yields
5997:{\displaystyle Z_{\mu }}
2675:{\displaystyle \lambda }
2555:{\displaystyle L^{-1/2}}
2364:{\displaystyle \lambda }
1711:placed in the LGR bore.
677:{\displaystyle \mu _{0}}
7233:Physical Review Letters
6564:10.1119/bfy.2015.pr.005
3981:, which corresponds to
2113:{\displaystyle L=L_{0}}
1902:vector network analyzer
1049:{\displaystyle \delta }
199:{\displaystyle \delta }
89:lumped-element circuits
69:electron spin resonance
7213:
7120:(3): 034701–034701–6.
7051:10.1006/jmra.1994.1090
7031:
6914:
6847:10.1006/jmra.1993.1265
6354:
6324:
6297:
6270:
6243:
6210:
6045:
6025:
5998:
5968:
5856:
5824:
5752:
5722:
5695:
5665:
5638:
5611:
5591:
5561:
5256:
5229:
5196:
4922:
4810:
4778:
4706:
4664:
4634:
4557:
4556:{\displaystyle S_{11}}
4530:
4510:
4480:
4375:
4355:
4328:
4292:
4226:
4195:
4167:
4166:{\displaystyle 10^{6}}
4146:. Quality factors of
4132:
4112:
4074:
4036:
3956:
3842:
3743:
3535:
3496:
3469:
3446:
3346:
3315:
3295:
3268:
3212:
3189:
3086:
3063:
2948:
2895:
2875:
2846:
2816:
2796:
2776:
2753:
2676:
2649:
2556:
2518:
2491:
2462:
2372:
2365:
2333:
2310:
2241:
2209:
2189:
2169:
2138:
2114:
2081:
2061:
2034:
2016:
2009:
1982:
1956:
1933:reflection coefficient
1925:
1924:{\displaystyle S_{11}}
1863:
1843:
1808:
1788:
1760:
1730:
1712:
1697:
1669:
1639:
1520:
1489:
1446:
1420:
1375:
1333:
1284:
1215:
1168:
1145:
1074:
1050:
1030:
993:
941:
899:
841:
729:
709:
678:
648:
576:
562:is the gap width, and
556:
536:
512:
482:
424:
405:
385:
357:
337:
310:
290:
255:
227:
200:
177:
154:
131:
108:
57:electromagnetic fields
44:
37:
7214:
7032:
6915:
6809:10.1139/cjp-2017-0436
6355:
6325:
6298:
6271:
6244:
6211:
6046:
6026:
5999:
5969:
5857:
5825:
5768:relative permeability
5753:
5723:
5696:
5666:
5639:
5612:
5592:
5562:
5257:
5230:
5197:
4923:
4811:
4779:
4722:relative permittivity
4707:
4665:
4663:{\displaystyle Z_{0}}
4635:
4558:
4531:
4511:
4481:
4376:
4356:
4329:
4293:
4227:
4196:
4168:
4133:
4113:
4075:
4037:
3957:
3843:
3744:
3536:
3497:
3495:{\displaystyle T_{0}}
3470:
3447:
3347:
3316:
3296:
3269:
3213:
3190:
3087:
3064:
2949:
2896:
2876:
2852:is the sample volume
2847:
2817:
2797:
2777:
2754:
2677:
2650:
2557:
2519:
2517:{\displaystyle f_{0}}
2492:
2463:
2366:
2350:
2334:
2311:
2247:-gap LGR is given by
2242:
2240:{\displaystyle (n-1)}
2210:
2190:
2170:
2168:{\displaystyle L_{0}}
2139:
2115:
2082:
2062:
2060:{\displaystyle L_{0}}
2035:
2010:
1988:
1980:
1957:
1955:{\displaystyle f_{0}}
1926:
1864:
1844:
1842:{\displaystyle f_{0}}
1809:
1807:{\displaystyle \ell }
1789:
1787:{\displaystyle r_{0}}
1761:
1731:
1703:
1695:
1670:
1640:
1521:
1519:{\displaystyle r_{0}}
1490:
1447:
1421:
1376:
1334:
1285:
1216:
1169:
1167:{\displaystyle \rho }
1146:
1075:
1051:
1031:
994:
942:
907:vacuum speed of light
900:
842:
730:
710:
708:{\displaystyle r_{0}}
679:
649:
577:
575:{\displaystyle \ell }
557:
537:
513:
483:
422:
406:
386:
384:{\displaystyle f_{0}}
358:
338:
336:{\displaystyle f_{0}}
311:
291:
289:{\displaystyle f_{0}}
256:
228:
201:
178:
176:{\displaystyle \rho }
155:
132:
109:
38:
36:{\displaystyle \ell }
22:
7203:
7021:
6904:
6334:
6307:
6280:
6253:
6223:
6058:
6035:
6008:
5981:
5869:
5834:
5773:
5762:Complex permeability
5732:
5705:
5675:
5648:
5621:
5601:
5574:
5269:
5239:
5209:
4938:
4823:
4788:
4727:
4716:Complex permittivity
4674:
4647:
4570:
4540:
4520:
4493:
4388:
4365:
4338:
4305:
4239:
4216:
4177:
4150:
4122:
4084:
4046:
3985:
3969:Experimental details
3855:
3756:
3548:
3506:
3479:
3459:
3362:
3325:
3321:with an accuracy of
3305:
3278:
3234:
3202:
3099:
3076:
2961:
2905:
2885:
2856:
2830:
2806:
2786:
2766:
2689:
2666:
2569:
2528:
2501:
2472:
2412:
2355:
2323:
2254:
2219:
2199:
2179:
2152:
2124:
2091:
2071:
2044:
2024:
1993:
1939:
1908:
1853:
1826:
1798:
1771:
1744:
1720:
1709:split-ring resonator
1652:
1533:
1503:
1497:radiation resistance
1464:
1430:
1385:
1343:
1297:
1231:
1181:
1158:
1087:
1060:
1040:
1007:
954:
925:
854:
742:
719:
692:
661:
593:
566:
546:
526:
495:
439:
395:
368:
347:
320:
300:
273:
239:
211:
190:
167:
144:
121:
98:
27:
7310:2004PhRvB..69r4513P
7245:1993PhRvL..70.3999H
7173:1991RScI...62.1819B
7126:2007RScI...78c4701S
7079:2007JMagR.185..259H
6996:1989JMagR..82..223H
6961:1984JMagR..58..243W
6839:1993JMagR.105..137R
6801:2018CaJPh..96..878B
6741:2016RScI...87b4701B
6659:2017ITMTT..65.2132B
6609:2013AmJPh..81..899B
6464:2017RScI...88l3901T
6425:1982JMagR..47..515F
4361:and quality factor
2845:{\displaystyle abc}
1894:oscillating current
1759:{\displaystyle LRC}
940:{\displaystyle LRC}
633:
428:Resonance frequency
296:and quality factor
254:{\displaystyle LRC}
226:{\displaystyle LRC}
7209:
7027:
6910:
6350:
6320:
6293:
6266:
6239:
6206:
6041:
6021:
5994:
5964:
5852:
5820:
5748:
5718:
5691:
5661:
5634:
5607:
5587:
5557:
5252:
5225:
5192:
4918:
4806:
4774:
4702:
4660:
4630:
4553:
4526:
4506:
4476:
4371:
4351:
4324:
4288:
4222:
4191:
4163:
4128:
4108:
4070:
4032:
3952:
3838:
3752:Finally, defining
3739:
3531:
3492:
3465:
3442:
3342:
3311:
3291:
3264:
3208:
3185:
3082:
3059:
2944:
2891:
2871:
2842:
2812:
2792:
2772:
2749:
2672:
2645:
2552:
2514:
2487:
2458:
2373:
2361:
2329:
2306:
2237:
2205:
2185:
2165:
2134:
2110:
2077:
2057:
2030:
2017:
2005:
1983:
1964:impedance matching
1952:
1921:
1878:Inductive coupling
1859:
1839:
1804:
1784:
1756:
1726:
1713:
1698:
1665:
1635:
1516:
1485:
1442:
1416:
1371:
1329:
1280:
1211:
1164:
1141:
1070:
1046:
1026:
989:
937:
895:
837:
725:
705:
674:
644:
619:
572:
552:
532:
508:
478:
425:
401:
381:
353:
333:
306:
286:
251:
223:
196:
173:
150:
127:
104:
49:loop-gap resonator
45:
33:
7288:Physical Review B
7182:10.1063/1.1142427
7134:10.1063/1.2709746
7030:{\displaystyle Q}
6934:10.1063/1.1138663
6913:{\displaystyle Q}
6749:10.1063/1.4940725
6700:978-0-321-85656-2
6617:10.1119/1.4823807
6473:10.1063/1.5000946
6398:10.1063/1.1136574
6195:
6175:
6132:
6099:
6081:
6044:{\displaystyle Q}
5958:
5610:{\displaystyle Q}
5546:
5525:
5446:
5413:
5392:
5310:
5292:
5181:
5162:
5061:
5042:
4912:
4695:
4624:
4529:{\displaystyle Q}
4465:
4445:
4417:
4404:
4374:{\displaystyle Q}
4322:
4282:
4225:{\displaystyle Z}
4185:
4131:{\displaystyle Q}
4092:
4054:
3946:
3908:
3669:
3652:
3468:{\displaystyle T}
3333:
3314:{\displaystyle c}
3255:
3211:{\displaystyle c}
3179:
3150:
3123:
3085:{\displaystyle a}
3009:
2894:{\displaystyle b}
2815:{\displaystyle c}
2795:{\displaystyle b}
2775:{\displaystyle a}
2639:
2606:
2332:{\displaystyle L}
2300:
2299:
2283:
2208:{\displaystyle n}
2188:{\displaystyle L}
2132:
2080:{\displaystyle C}
2033:{\displaystyle L}
1873:Coupling to a LGR
1862:{\displaystyle Q}
1729:{\displaystyle Q}
1619:
1585:
1584:
1559:
1274:
1273:
1205:
1135:
983:
982:
971:
893:
831:
830:
814:
786:
785:
771:
728:{\displaystyle Q}
638:
555:{\displaystyle t}
535:{\displaystyle w}
472:
404:{\displaystyle Q}
356:{\displaystyle Q}
309:{\displaystyle Q}
153:{\displaystyle R}
130:{\displaystyle L}
107:{\displaystyle C}
61:cavity resonators
7382:
7359:
7358:
7348:
7328:
7322:
7321:
7303:
7301:cond-mat/0311555
7271:
7265:
7264:
7218:
7216:
7215:
7210:
7196:
7187:
7186:
7184:
7167:(7): 1819–1823.
7152:
7146:
7145:
7105:
7099:
7098:
7061:
7055:
7054:
7036:
7034:
7033:
7028:
7014:
7008:
7007:
6979:
6973:
6972:
6944:
6938:
6937:
6919:
6917:
6916:
6911:
6897:
6891:
6890:
6880:
6860:
6851:
6850:
6822:
6813:
6812:
6794:
6774:
6761:
6760:
6734:
6714:
6705:
6704:
6681:Griffiths, David
6677:
6671:
6670:
6652:
6632:
6621:
6620:
6602:
6582:
6569:
6568:
6566:
6548:
6535:
6534:
6524:
6492:
6486:
6485:
6475:
6443:
6437:
6436:
6408:
6402:
6401:
6381:
6359:
6357:
6356:
6351:
6349:
6348:
6329:
6327:
6326:
6321:
6319:
6318:
6302:
6300:
6299:
6294:
6292:
6291:
6275:
6273:
6272:
6267:
6265:
6264:
6248:
6246:
6245:
6240:
6238:
6237:
6215:
6213:
6212:
6207:
6201:
6197:
6196:
6191:
6190:
6181:
6176:
6174:
6173:
6161:
6159:
6158:
6138:
6134:
6133:
6131:
6130:
6118:
6116:
6115:
6100:
6092:
6082:
6080:
6072:
6071:
6062:
6050:
6048:
6047:
6042:
6030:
6028:
6027:
6022:
6020:
6019:
6003:
6001:
6000:
5995:
5993:
5992:
5973:
5971:
5970:
5965:
5959:
5957:
5943:
5935:
5931:
5930:
5929:
5911:
5910:
5881:
5880:
5861:
5859:
5858:
5853:
5848:
5847:
5846:
5829:
5827:
5826:
5821:
5819:
5818:
5800:
5799:
5787:
5786:
5785:
5757:
5755:
5754:
5749:
5747:
5746:
5727:
5725:
5724:
5719:
5717:
5716:
5700:
5698:
5697:
5692:
5690:
5689:
5670:
5668:
5667:
5662:
5660:
5659:
5643:
5641:
5640:
5635:
5633:
5632:
5616:
5614:
5613:
5608:
5596:
5594:
5593:
5588:
5586:
5585:
5566:
5564:
5563:
5558:
5552:
5548:
5547:
5542:
5541:
5532:
5530:
5526:
5524:
5523:
5522:
5517:
5513:
5512:
5492:
5491:
5486:
5482:
5481:
5466:
5465:
5456:
5447:
5445:
5444:
5432:
5419:
5415:
5414:
5409:
5408:
5399:
5397:
5393:
5391:
5390:
5389:
5384:
5380:
5379:
5359:
5358:
5353:
5349:
5348:
5333:
5332:
5320:
5311:
5303:
5293:
5291:
5283:
5282:
5273:
5261:
5259:
5258:
5253:
5251:
5250:
5234:
5232:
5231:
5226:
5224:
5223:
5201:
5199:
5198:
5193:
5187:
5183:
5182:
5180:
5169:
5167:
5163:
5161:
5160:
5159:
5154:
5150:
5149:
5129:
5128:
5123:
5119:
5118:
5103:
5102:
5093:
5067:
5063:
5062:
5060:
5049:
5047:
5043:
5041:
5040:
5039:
5034:
5030:
5029:
5009:
5008:
5003:
4999:
4998:
4983:
4982:
4970:
4950:
4949:
4927:
4925:
4924:
4919:
4913:
4911:
4907:
4903:
4902:
4901:
4883:
4882:
4858:
4835:
4834:
4815:
4813:
4812:
4807:
4802:
4801:
4800:
4783:
4781:
4780:
4775:
4773:
4772:
4754:
4753:
4741:
4740:
4739:
4711:
4709:
4708:
4703:
4693:
4686:
4685:
4669:
4667:
4666:
4661:
4659:
4658:
4639:
4637:
4636:
4631:
4625:
4623:
4616:
4615:
4605:
4598:
4597:
4587:
4582:
4581:
4562:
4560:
4559:
4554:
4552:
4551:
4535:
4533:
4532:
4527:
4515:
4513:
4512:
4507:
4505:
4504:
4485:
4483:
4482:
4477:
4471:
4467:
4466:
4461:
4460:
4451:
4446:
4444:
4443:
4431:
4418:
4410:
4405:
4403:
4392:
4380:
4378:
4377:
4372:
4360:
4358:
4357:
4352:
4350:
4349:
4333:
4331:
4330:
4325:
4323:
4315:
4297:
4295:
4294:
4289:
4283:
4281:
4267:
4231:
4229:
4228:
4223:
4200:
4198:
4197:
4192:
4190:
4183:
4172:
4170:
4169:
4164:
4162:
4161:
4137:
4135:
4134:
4129:
4117:
4115:
4114:
4109:
4107:
4106:
4101:
4090:
4079:
4077:
4076:
4071:
4069:
4068:
4063:
4052:
4041:
4039:
4038:
4033:
4031:
4030:
4015:
4014:
4005:
4000:
3999:
3961:
3959:
3958:
3953:
3947:
3945:
3944:
3935:
3925:
3924:
3911:
3909:
3907:
3899:
3898:
3897:
3887:
3870:
3869:
3847:
3845:
3844:
3839:
3834:
3833:
3821:
3820:
3796:
3795:
3771:
3770:
3748:
3746:
3745:
3740:
3734:
3730:
3726:
3725:
3713:
3712:
3688:
3687:
3670:
3668:
3667:
3655:
3653:
3651:
3643:
3642:
3641:
3631:
3620:
3619:
3607:
3606:
3585:
3584:
3563:
3562:
3540:
3538:
3537:
3532:
3521:
3520:
3501:
3499:
3498:
3493:
3491:
3490:
3475:is temperature,
3474:
3472:
3471:
3466:
3451:
3449:
3448:
3443:
3428:
3427:
3409:
3408:
3396:
3395:
3374:
3373:
3351:
3349:
3348:
3343:
3341:
3331:
3320:
3318:
3317:
3312:
3300:
3298:
3297:
3292:
3290:
3289:
3273:
3271:
3270:
3265:
3263:
3253:
3246:
3245:
3217:
3215:
3214:
3209:
3194:
3192:
3191:
3186:
3180:
3178:
3177:
3168:
3167:
3166:
3153:
3151:
3149:
3141:
3140:
3139:
3129:
3124:
3116:
3111:
3110:
3091:
3089:
3088:
3083:
3068:
3066:
3065:
3060:
3054:
3050:
3049:
3048:
3027:
3026:
3025:
3010:
3008:
3007:
3006:
3005:
2991:
2990:
2981:
2976:
2975:
2953:
2951:
2950:
2945:
2943:
2942:
2941:
2922:
2921:
2920:
2900:
2898:
2897:
2892:
2880:
2878:
2877:
2872:
2870:
2869:
2868:
2851:
2849:
2848:
2843:
2824:crystallographic
2821:
2819:
2818:
2813:
2801:
2799:
2798:
2793:
2781:
2779:
2778:
2773:
2758:
2756:
2755:
2750:
2744:
2740:
2739:
2738:
2706:
2705:
2704:
2681:
2679:
2678:
2673:
2654:
2652:
2651:
2646:
2640:
2638:
2637:
2636:
2626:
2625:
2624:
2623:
2609:
2607:
2602:
2601:
2592:
2584:
2583:
2561:
2559:
2558:
2553:
2551:
2550:
2546:
2523:
2521:
2520:
2515:
2513:
2512:
2496:
2494:
2493:
2488:
2486:
2485:
2484:
2467:
2465:
2464:
2459:
2457:
2456:
2447:
2442:
2441:
2440:
2430:
2429:
2370:
2368:
2367:
2362:
2338:
2336:
2335:
2330:
2315:
2313:
2312:
2307:
2301:
2298:
2287:
2286:
2284:
2282:
2271:
2266:
2265:
2246:
2244:
2243:
2238:
2214:
2212:
2211:
2206:
2194:
2192:
2191:
2186:
2174:
2172:
2171:
2166:
2164:
2163:
2143:
2141:
2140:
2135:
2133:
2128:
2119:
2117:
2116:
2111:
2109:
2108:
2086:
2084:
2083:
2078:
2066:
2064:
2063:
2058:
2056:
2055:
2039:
2037:
2036:
2031:
2014:
2012:
2011:
2006:
1961:
1959:
1958:
1953:
1951:
1950:
1930:
1928:
1927:
1922:
1920:
1919:
1890:signal generator
1868:
1866:
1865:
1860:
1848:
1846:
1845:
1840:
1838:
1837:
1813:
1811:
1810:
1805:
1793:
1791:
1790:
1785:
1783:
1782:
1765:
1763:
1762:
1757:
1735:
1733:
1732:
1727:
1674:
1672:
1671:
1666:
1664:
1663:
1644:
1642:
1641:
1636:
1630:
1629:
1624:
1620:
1615:
1614:
1613:
1604:
1603:
1593:
1586:
1583:
1582:
1573:
1572:
1563:
1562:
1560:
1552:
1547:
1546:
1545:
1525:
1523:
1522:
1517:
1515:
1514:
1494:
1492:
1491:
1486:
1481:
1476:
1475:
1456:Radiative losses
1451:
1449:
1448:
1443:
1425:
1423:
1422:
1417:
1415:
1397:
1396:
1380:
1378:
1377:
1372:
1370:
1355:
1354:
1338:
1336:
1335:
1330:
1328:
1327:
1309:
1308:
1289:
1287:
1286:
1281:
1275:
1272:
1271:
1270:
1261:
1260:
1250:
1242:
1241:
1220:
1218:
1217:
1212:
1206:
1201:
1200:
1191:
1173:
1171:
1170:
1165:
1150:
1148:
1147:
1142:
1136:
1134:
1125:
1124:
1123:
1107:
1099:
1098:
1079:
1077:
1076:
1071:
1055:
1053:
1052:
1047:
1035:
1033:
1032:
1027:
1025:
1024:
998:
996:
995:
990:
984:
975:
974:
972:
964:
946:
944:
943:
938:
904:
902:
901:
896:
894:
892:
891:
882:
881:
872:
870:
846:
844:
843:
838:
832:
829:
818:
817:
815:
813:
812:
811:
792:
787:
778:
774:
772:
770:
759:
754:
753:
734:
732:
731:
726:
714:
712:
711:
706:
704:
703:
683:
681:
680:
675:
673:
672:
653:
651:
650:
645:
639:
634:
632:
627:
613:
611:
610:
581:
579:
578:
573:
561:
559:
558:
553:
541:
539:
538:
533:
517:
515:
514:
509:
507:
506:
487:
485:
484:
479:
473:
468:
459:
457:
456:
410:
408:
407:
402:
390:
388:
387:
382:
380:
379:
362:
360:
359:
354:
342:
340:
339:
334:
332:
331:
315:
313:
312:
307:
295:
293:
292:
287:
285:
284:
260:
258:
257:
252:
232:
230:
229:
224:
205:
203:
202:
197:
182:
180:
179:
174:
159:
157:
156:
151:
136:
134:
133:
128:
113:
111:
110:
105:
42:
40:
39:
34:
7390:
7389:
7385:
7384:
7383:
7381:
7380:
7379:
7365:
7364:
7363:
7362:
7330:
7329:
7325:
7285:
7281:
7277:
7273:
7272:
7268:
7230:
7226:
7222:
7201:
7200:
7198:
7197:
7190:
7154:
7153:
7149:
7111:
7107:
7106:
7102:
7063:
7062:
7058:
7019:
7018:
7016:
7015:
7011:
6981:
6980:
6976:
6946:
6945:
6941:
6902:
6901:
6899:
6898:
6894:
6862:
6861:
6854:
6824:
6823:
6816:
6776:
6775:
6764:
6716:
6715:
6708:
6701:
6679:
6678:
6674:
6634:
6633:
6624:
6593:(12): 899–906.
6584:
6583:
6572:
6550:
6549:
6538:
6494:
6493:
6489:
6445:
6444:
6440:
6410:
6409:
6405:
6383:
6382:
6371:
6366:
6337:
6332:
6331:
6310:
6305:
6304:
6283:
6278:
6277:
6256:
6251:
6250:
6226:
6221:
6220:
6182:
6165:
6150:
6149:
6145:
6122:
6104:
6090:
6086:
6073:
6063:
6056:
6055:
6033:
6032:
6011:
6006:
6005:
5984:
5979:
5978:
5947:
5918:
5902:
5901:
5897:
5872:
5867:
5866:
5837:
5832:
5831:
5807:
5791:
5776:
5771:
5770:
5764:
5735:
5730:
5729:
5708:
5703:
5702:
5678:
5673:
5672:
5651:
5646:
5645:
5624:
5619:
5618:
5599:
5598:
5577:
5572:
5571:
5533:
5501:
5497:
5496:
5473:
5469:
5468:
5467:
5457:
5451:
5436:
5430:
5426:
5400:
5368:
5364:
5363:
5340:
5336:
5335:
5334:
5321:
5315:
5301:
5297:
5284:
5274:
5267:
5266:
5242:
5237:
5236:
5212:
5207:
5206:
5173:
5138:
5134:
5133:
5110:
5106:
5105:
5104:
5094:
5088:
5078:
5074:
5053:
5018:
5014:
5013:
4990:
4986:
4985:
4984:
4971:
4965:
4958:
4954:
4941:
4936:
4935:
4890:
4874:
4873:
4869:
4862:
4826:
4821:
4820:
4791:
4786:
4785:
4761:
4745:
4730:
4725:
4724:
4718:
4677:
4672:
4671:
4650:
4645:
4644:
4607:
4606:
4589:
4588:
4573:
4568:
4567:
4543:
4538:
4537:
4518:
4517:
4496:
4491:
4490:
4452:
4435:
4429:
4425:
4396:
4386:
4385:
4363:
4362:
4341:
4336:
4335:
4303:
4302:
4271:
4237:
4236:
4214:
4213:
4207:
4175:
4174:
4153:
4148:
4147:
4120:
4119:
4093:
4082:
4081:
4055:
4044:
4043:
4019:
4006:
3991:
3983:
3982:
3971:
3936:
3916:
3912:
3900:
3888:
3861:
3853:
3852:
3825:
3812:
3787:
3762:
3754:
3753:
3717:
3704:
3679:
3675:
3671:
3659:
3644:
3632:
3611:
3598:
3576:
3554:
3546:
3545:
3512:
3504:
3503:
3482:
3477:
3476:
3457:
3456:
3419:
3400:
3387:
3365:
3360:
3359:
3323:
3322:
3303:
3302:
3281:
3276:
3275:
3237:
3232:
3231:
3229:
3225:
3221:
3200:
3199:
3169:
3158:
3154:
3142:
3130:
3102:
3097:
3096:
3074:
3073:
3040:
3016:
3015:
3011:
2996:
2992:
2982:
2967:
2959:
2958:
2929:
2911:
2903:
2902:
2883:
2882:
2859:
2854:
2853:
2828:
2827:
2826:directions and
2804:
2803:
2784:
2783:
2764:
2763:
2730:
2720:
2716:
2692:
2687:
2686:
2664:
2663:
2660:Meissner effect
2627:
2614:
2610:
2593:
2575:
2567:
2566:
2531:
2526:
2525:
2504:
2499:
2498:
2475:
2470:
2469:
2448:
2431:
2421:
2410:
2409:
2406:
2394:
2390:
2386:
2353:
2352:
2345:
2321:
2320:
2291:
2275:
2257:
2252:
2251:
2217:
2216:
2197:
2196:
2177:
2176:
2155:
2150:
2149:
2122:
2121:
2100:
2089:
2088:
2069:
2068:
2047:
2042:
2041:
2022:
2021:
1991:
1990:
1975:
1942:
1937:
1936:
1911:
1906:
1905:
1875:
1851:
1850:
1829:
1824:
1823:
1796:
1795:
1774:
1769:
1768:
1742:
1741:
1718:
1717:
1690:
1682:
1655:
1650:
1649:
1605:
1595:
1594:
1588:
1587:
1574:
1564:
1536:
1531:
1530:
1506:
1501:
1500:
1467:
1462:
1461:
1458:
1428:
1427:
1388:
1383:
1382:
1346:
1341:
1340:
1319:
1300:
1295:
1294:
1262:
1252:
1251:
1243:
1229:
1228:
1192:
1179:
1178:
1156:
1155:
1126:
1115:
1108:
1090:
1085:
1084:
1058:
1057:
1038:
1037:
1016:
1005:
1004:
952:
951:
923:
922:
915:
883:
873:
852:
851:
822:
803:
796:
763:
745:
740:
739:
717:
716:
695:
690:
689:
664:
659:
658:
614:
602:
591:
590:
564:
563:
544:
543:
524:
523:
498:
493:
492:
460:
448:
437:
436:
430:
417:
393:
392:
371:
366:
365:
345:
344:
323:
318:
317:
298:
297:
276:
271:
270:
237:
236:
209:
208:
188:
187:
165:
164:
142:
141:
119:
118:
96:
95:
85:
65:quality factors
25:
24:
17:
12:
11:
5:
7388:
7386:
7378:
7377:
7367:
7366:
7361:
7360:
7323:
7294:(18): 184513.
7283:
7279:
7275:
7266:
7228:
7224:
7220:
7208:
7188:
7147:
7109:
7100:
7056:
7026:
7009:
6974:
6939:
6909:
6892:
6852:
6814:
6785:(8): 878–886.
6762:
6706:
6699:
6691:Addison-Wesley
6672:
6622:
6570:
6536:
6487:
6458:(12): 123901.
6438:
6403:
6392:(2): 213–216.
6368:
6367:
6365:
6362:
6347:
6344:
6340:
6317:
6313:
6290:
6286:
6263:
6259:
6236:
6233:
6229:
6217:
6216:
6205:
6200:
6194:
6189:
6185:
6179:
6172:
6168:
6164:
6157:
6153:
6148:
6144:
6141:
6137:
6129:
6125:
6121:
6114:
6111:
6107:
6103:
6098:
6095:
6089:
6085:
6079:
6076:
6070:
6066:
6040:
6018:
6014:
5991:
5987:
5975:
5974:
5963:
5956:
5953:
5950:
5946:
5941:
5938:
5934:
5928:
5925:
5921:
5917:
5914:
5909:
5905:
5900:
5896:
5893:
5890:
5887:
5884:
5879:
5875:
5851:
5845:
5840:
5817:
5814:
5810:
5806:
5803:
5798:
5794:
5790:
5784:
5779:
5763:
5760:
5745:
5742:
5738:
5715:
5711:
5688:
5685:
5681:
5658:
5654:
5631:
5627:
5606:
5584:
5580:
5570:Provided that
5568:
5567:
5556:
5551:
5545:
5540:
5536:
5529:
5521:
5516:
5511:
5508:
5504:
5500:
5495:
5490:
5485:
5480:
5476:
5472:
5464:
5460:
5454:
5450:
5443:
5439:
5435:
5429:
5425:
5422:
5418:
5412:
5407:
5403:
5396:
5388:
5383:
5378:
5375:
5371:
5367:
5362:
5357:
5352:
5347:
5343:
5339:
5331:
5328:
5324:
5318:
5314:
5309:
5306:
5300:
5296:
5290:
5287:
5281:
5277:
5249:
5245:
5222:
5219:
5215:
5203:
5202:
5191:
5186:
5179:
5176:
5172:
5166:
5158:
5153:
5148:
5145:
5141:
5137:
5132:
5127:
5122:
5117:
5113:
5109:
5101:
5097:
5091:
5087:
5084:
5081:
5077:
5073:
5070:
5066:
5059:
5056:
5052:
5046:
5038:
5033:
5028:
5025:
5021:
5017:
5012:
5007:
5002:
4997:
4993:
4989:
4981:
4978:
4974:
4968:
4964:
4961:
4957:
4953:
4948:
4944:
4929:
4928:
4917:
4910:
4906:
4900:
4897:
4893:
4889:
4886:
4881:
4877:
4872:
4868:
4865:
4861:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4833:
4829:
4805:
4799:
4794:
4771:
4768:
4764:
4760:
4757:
4752:
4748:
4744:
4738:
4733:
4717:
4714:
4701:
4698:
4692:
4689:
4684:
4680:
4657:
4653:
4641:
4640:
4629:
4622:
4619:
4614:
4610:
4604:
4601:
4596:
4592:
4585:
4580:
4576:
4550:
4546:
4525:
4503:
4499:
4487:
4486:
4475:
4470:
4464:
4459:
4455:
4449:
4442:
4438:
4434:
4428:
4424:
4421:
4416:
4413:
4408:
4402:
4399:
4395:
4370:
4348:
4344:
4321:
4318:
4313:
4310:
4299:
4298:
4287:
4280:
4277:
4274:
4270:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4221:
4206:
4203:
4189:
4182:
4160:
4156:
4127:
4105:
4100:
4097:
4089:
4067:
4062:
4059:
4051:
4029:
4026:
4022:
4018:
4013:
4009:
4004:
3998:
3994:
3990:
3970:
3967:
3963:
3962:
3951:
3943:
3939:
3934:
3931:
3928:
3923:
3919:
3915:
3906:
3903:
3896:
3891:
3885:
3882:
3879:
3876:
3873:
3868:
3864:
3860:
3837:
3832:
3828:
3824:
3819:
3815:
3811:
3808:
3805:
3802:
3799:
3794:
3790:
3786:
3783:
3780:
3777:
3774:
3769:
3765:
3761:
3750:
3749:
3738:
3733:
3729:
3724:
3720:
3716:
3711:
3707:
3703:
3700:
3697:
3694:
3691:
3686:
3682:
3678:
3674:
3666:
3662:
3658:
3650:
3647:
3640:
3635:
3629:
3626:
3623:
3618:
3614:
3610:
3605:
3601:
3597:
3594:
3591:
3588:
3583:
3579:
3575:
3572:
3569:
3566:
3561:
3557:
3553:
3530:
3527:
3524:
3519:
3515:
3511:
3489:
3485:
3464:
3453:
3452:
3441:
3437:
3434:
3431:
3426:
3422:
3418:
3415:
3412:
3407:
3403:
3399:
3394:
3390:
3386:
3383:
3380:
3377:
3372:
3368:
3340:
3337:
3330:
3310:
3288:
3284:
3262:
3259:
3252:
3249:
3244:
3240:
3227:
3223:
3219:
3207:
3196:
3195:
3184:
3176:
3172:
3165:
3161:
3157:
3148:
3145:
3138:
3133:
3127:
3122:
3119:
3114:
3109:
3105:
3081:
3070:
3069:
3058:
3053:
3047:
3043:
3039:
3036:
3033:
3030:
3024:
3019:
3014:
3004:
2999:
2995:
2989:
2985:
2979:
2974:
2970:
2966:
2940:
2937:
2932:
2928:
2925:
2919:
2914:
2910:
2890:
2867:
2862:
2841:
2838:
2835:
2811:
2791:
2771:
2760:
2759:
2748:
2743:
2737:
2733:
2729:
2726:
2723:
2719:
2715:
2712:
2709:
2703:
2700:
2695:
2671:
2656:
2655:
2644:
2635:
2630:
2622:
2617:
2613:
2605:
2600:
2596:
2590:
2587:
2582:
2578:
2574:
2549:
2545:
2541:
2538:
2534:
2511:
2507:
2483:
2478:
2455:
2451:
2446:
2439:
2434:
2428:
2424:
2420:
2417:
2405:
2402:
2392:
2388:
2384:
2360:
2344:
2341:
2328:
2317:
2316:
2305:
2297:
2294:
2290:
2281:
2278:
2274:
2269:
2264:
2260:
2236:
2233:
2230:
2227:
2224:
2204:
2184:
2162:
2158:
2131:
2107:
2103:
2099:
2096:
2076:
2054:
2050:
2029:
2004:
2001:
1998:
1974:
1971:
1949:
1945:
1918:
1914:
1874:
1871:
1858:
1836:
1832:
1803:
1781:
1777:
1755:
1752:
1749:
1725:
1689:
1686:
1680:
1662:
1658:
1646:
1645:
1634:
1628:
1623:
1618:
1612:
1608:
1602:
1598:
1591:
1581:
1577:
1571:
1567:
1558:
1555:
1550:
1544:
1539:
1513:
1509:
1484:
1480:
1474:
1470:
1457:
1454:
1441:
1438:
1435:
1414:
1411:
1408:
1403:
1400:
1395:
1391:
1369:
1366:
1361:
1358:
1353:
1349:
1326:
1322:
1318:
1315:
1312:
1307:
1303:
1291:
1290:
1279:
1269:
1265:
1259:
1255:
1249:
1246:
1239:
1236:
1222:
1221:
1210:
1204:
1199:
1195:
1189:
1186:
1163:
1152:
1151:
1140:
1133:
1129:
1122:
1118:
1114:
1111:
1105:
1102:
1097:
1093:
1069:
1065:
1045:
1023:
1019:
1015:
1012:
1000:
999:
988:
981:
978:
970:
967:
962:
959:
936:
933:
930:
914:
913:Quality factor
911:
890:
886:
880:
876:
869:
865:
862:
859:
848:
847:
836:
828:
825:
821:
810:
806:
802:
799:
795:
790:
784:
781:
777:
769:
766:
762:
757:
752:
748:
724:
702:
698:
671:
667:
655:
654:
643:
637:
631:
626:
622:
617:
609:
605:
601:
598:
571:
551:
531:
505:
501:
489:
488:
477:
471:
467:
463:
455:
451:
447:
444:
429:
426:
416:
413:
400:
378:
374:
352:
330:
326:
305:
283:
279:
250:
247:
244:
222:
219:
216:
195:
172:
149:
126:
103:
84:
81:
32:
15:
13:
10:
9:
6:
4:
3:
2:
7387:
7376:
7373:
7372:
7370:
7356:
7352:
7347:
7342:
7338:
7334:
7327:
7324:
7319:
7315:
7311:
7307:
7302:
7297:
7293:
7289:
7270:
7267:
7262:
7258:
7254:
7250:
7246:
7242:
7238:
7234:
7206:
7195:
7193:
7189:
7183:
7178:
7174:
7170:
7166:
7162:
7158:
7151:
7148:
7143:
7139:
7135:
7131:
7127:
7123:
7119:
7115:
7104:
7101:
7096:
7092:
7088:
7084:
7080:
7076:
7072:
7068:
7060:
7057:
7052:
7048:
7044:
7040:
7024:
7013:
7010:
7005:
7001:
6997:
6993:
6989:
6985:
6978:
6975:
6970:
6966:
6962:
6958:
6954:
6950:
6943:
6940:
6935:
6931:
6927:
6923:
6907:
6896:
6893:
6888:
6884:
6879:
6874:
6870:
6866:
6859:
6857:
6853:
6848:
6844:
6840:
6836:
6832:
6828:
6821:
6819:
6815:
6810:
6806:
6802:
6798:
6793:
6788:
6784:
6780:
6773:
6771:
6769:
6767:
6763:
6758:
6754:
6750:
6746:
6742:
6738:
6733:
6728:
6725:(2): 024701.
6724:
6720:
6713:
6711:
6707:
6702:
6696:
6692:
6688:
6687:
6682:
6676:
6673:
6668:
6664:
6660:
6656:
6651:
6646:
6642:
6638:
6631:
6629:
6627:
6623:
6618:
6614:
6610:
6606:
6601:
6596:
6592:
6588:
6581:
6579:
6577:
6575:
6571:
6565:
6560:
6556:
6555:
6547:
6545:
6543:
6541:
6537:
6532:
6528:
6523:
6518:
6514:
6510:
6506:
6502:
6498:
6491:
6488:
6483:
6479:
6474:
6469:
6465:
6461:
6457:
6453:
6449:
6442:
6439:
6434:
6430:
6426:
6422:
6418:
6414:
6407:
6404:
6399:
6395:
6391:
6387:
6380:
6378:
6376:
6374:
6370:
6363:
6361:
6338:
6311:
6288:
6284:
6257:
6227:
6203:
6198:
6192:
6187:
6183:
6177:
6170:
6166:
6162:
6151:
6146:
6142:
6139:
6135:
6127:
6123:
6119:
6105:
6101:
6096:
6093:
6087:
6083:
6077:
6074:
6068:
6064:
6054:
6053:
6052:
6038:
6016:
6012:
5989:
5985:
5961:
5954:
5951:
5948:
5944:
5939:
5936:
5932:
5919:
5915:
5912:
5903:
5898:
5894:
5891:
5888:
5885:
5882:
5877:
5873:
5865:
5864:
5863:
5849:
5838:
5808:
5804:
5801:
5792:
5788:
5777:
5769:
5761:
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5736:
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5652:
5629:
5625:
5604:
5582:
5578:
5554:
5549:
5543:
5538:
5534:
5527:
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5502:
5498:
5493:
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5437:
5433:
5427:
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5381:
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5360:
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5350:
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5316:
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5288:
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5279:
5275:
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5264:
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5213:
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5000:
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4887:
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4212:
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4202:
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4145:
4144:liquid helium
4142:
4125:
4103:
4087:
4065:
4049:
4027:
4024:
4020:
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4011:
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2426:
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2202:
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2156:
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2129:
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2101:
2097:
2094:
2074:
2052:
2048:
2027:
2002:
1999:
1996:
1987:
1979:
1972:
1970:
1967:
1965:
1947:
1943:
1934:
1916:
1912:
1903:
1899:
1895:
1891:
1887:
1886:coaxial cable
1883:
1882:magnetic flux
1879:
1872:
1870:
1856:
1834:
1830:
1821:
1815:
1801:
1779:
1775:
1753:
1750:
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1723:
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1687:
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1606:
1600:
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1579:
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1267:
1263:
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1227:
1226:
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1197:
1193:
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1184:
1177:
1176:
1175:
1161:
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960:
957:
950:
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948:
934:
931:
928:
920:
917:For a highly
912:
910:
908:
888:
884:
878:
874:
867:
863:
860:
857:
834:
826:
823:
819:
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804:
800:
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434:
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421:
414:
412:
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376:
372:
350:
328:
324:
303:
281:
277:
268:
262:
248:
245:
242:
234:
220:
217:
214:
193:
186:
170:
163:
147:
140:
124:
117:
101:
94:
90:
82:
80:
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74:
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7332:
7326:
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7269:
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7070:
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6983:
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6385:
6219:Once again,
6218:
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5765:
5569:
5204:
4930:
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4488:
4300:
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3972:
3964:
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3454:
3354:
3197:
3071:
2761:
2657:
2407:
2374:
2318:
2147:
2018:
1968:
1876:
1816:
1739:
1714:
1705:
1688:Toroidal LGR
1647:
1459:
1292:
1223:
1153:
1001:
916:
849:
656:
490:
431:
263:
86:
77:permeability
73:permittivity
52:
48:
46:
6871:(8): 3547.
6643:(6): 2132.
6507:: 129–135.
5977:Separating
3977:per degree
2658:Due to the
2397:diamagnetic
919:underdamped
162:resistivity
93:capacitance
7375:Resonators
7346:1811.01168
7339:: 165387.
7073:(2): 259.
6990:(2): 223.
6955:(2): 243.
6878:2006.13861
6833:(2): 137.
6792:1711.05819
6732:1811.02630
6650:1901.00994
6600:1811.01449
6419:(3): 515.
6364:References
4141:superfluid
3848:, one has
1820:ferrofluid
267:dielectric
139:resistance
116:inductance
83:Background
7207:λ
7045:(1): 71.
6346:′
6343:′
6339:μ
6316:′
6312:μ
6289:μ
6262:′
6258:μ
6235:′
6232:′
6228:μ
6193:ω
6184:ω
6178:−
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6163:ω
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6120:ω
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4697:Ω
4600:−
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4463:ω
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4448:−
4437:ω
4433:ω
4343:ω
4317:−
4276:ω
4258:ω
4211:impedance
4025:−
4017:∼
3989:Δ
3975:ångströms
3914:Δ
3884:−
3863:λ
3859:Δ
3810:δ
3807:−
3785:δ
3760:Δ
3702:δ
3699:−
3677:δ
3628:−
3600:λ
3596:−
3578:λ
3556:λ
3552:Δ
3514:λ
3510:Δ
3421:λ
3417:Δ
3389:λ
3367:λ
3283:λ
3248:≈
3239:λ
3156:δ
3126:−
3104:λ
3042:λ
3029:−
2965:δ
2927:−
2909:δ
2732:λ
2725:−
2708:≈
2670:λ
2612:δ
2589:−
2573:δ
2537:−
2450:ℓ
2423:μ
2359:λ
2280:π
2268:≈
2229:−
2000:×
1802:ℓ
1677:waveguide
1661:ρ
1597:ω
1576:ε
1566:μ
1554:π
1549:≈
1483:δ
1437:≈
1317:π
1302:ω
1264:ω
1254:μ
1248:ρ
1238:≈
1235:δ
1203:δ
1188:≈
1162:ρ
1132:ℓ
1128:δ
1113:π
1104:ρ
1101:≈
1096:ρ
1068:ℓ
1064:δ
1044:δ
1014:π
961:≈
885:μ
875:ε
824:π
801:π
768:π
756:≈
666:μ
636:ℓ
616:π
604:μ
570:ℓ
500:ε
466:ℓ
450:ε
194:δ
171:ρ
31:ℓ
7369:Category
7261:10054019
7239:: 3999.
7142:17411204
7095:17267251
6928:: 1095.
6757:26931873
6683:(2012).
6531:28803092
6482:29289191
2468:, where
1706:extended
584:solenoid
7306:Bibcode
7241:Bibcode
7169:Bibcode
7122:Bibcode
7075:Bibcode
6992:Bibcode
6957:Bibcode
6835:Bibcode
6797:Bibcode
6737:Bibcode
6655:Bibcode
6605:Bibcode
6522:5793860
6460:Bibcode
6421:Bibcode
2215:-loop,
921:series
905:is the
684:is the
518:is the
233:circuit
7259:
7219:in YBa
7140:
7093:
6755:
6697:
6529:
6519:
6480:
4694:
4643:where
4301:where
4184:
4091:
4053:
4042:for a
3979:kelvin
3455:where
3332:
3254:
2802:, and
2762:where
2404:Theory
2087:. If
1154:where
850:where
657:where
491:where
7341:arXiv
7296:arXiv
6873:arXiv
6787:arXiv
6727:arXiv
6645:arXiv
6595:arXiv
1931:, or
1716:high-
7257:PMID
7229:6.95
7138:PMID
7091:PMID
6753:PMID
6695:ISBN
6527:PMID
6478:PMID
6330:and
6031:and
5728:and
5671:and
5597:and
4516:and
2954:and
2393:6.95
2040:and
1849:and
1440:3900
1381:and
1293:and
688:and
391:and
343:and
183:and
75:and
7351:doi
7337:489
7314:doi
7284:6+y
7249:doi
7177:doi
7130:doi
7110:011
7083:doi
7071:185
7047:doi
7043:108
7000:doi
6965:doi
6930:doi
6883:doi
6843:doi
6831:105
6805:doi
6745:doi
6663:doi
6613:doi
6559:doi
6517:PMC
6509:doi
6505:282
6468:doi
6429:doi
6394:doi
4712:).
3251:100
79:).
53:LGR
7371::
7349:.
7335:.
7312:.
7304:.
7292:69
7290:.
7278:Cu
7255:.
7247:.
7237:70
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4988:(
4967:(
4963:+
4960:R
4956:[
4952:=
4943:Z
4916:.
4909:C
4905:)
4888:j
4871:(
4864:j
4860:1
4855:+
4852:L
4846:j
4843:+
4840:R
4837:=
4828:Z
4804:C
4798:r
4759:j
4743:=
4737:r
4700:)
4688:=
4683:0
4679:Z
4656:0
4652:Z
4628:,
4621:Z
4618:+
4613:0
4609:Z
4603:Z
4595:0
4591:Z
4584:=
4575:S
4545:S
4524:Q
4502:0
4474:.
4469:)
4458:0
4441:0
4427:(
4423:j
4420:+
4415:Q
4412:1
4407:=
4401:R
4398:Q
4394:Z
4369:Q
4347:0
4320:1
4312:=
4309:j
4286:,
4279:C
4273:j
4269:1
4264:+
4261:L
4255:j
4252:+
4249:R
4246:=
4243:Z
4220:Z
4188:K
4181:1
4159:6
4126:Q
4104:3
4099:m
4096:c
4088:1
4066:2
4061:m
4058:m
4050:1
4012:0
4008:f
4003:/
3997:0
3993:f
3950:.
3942:0
3938:f
3933:)
3930:T
3927:(
3922:0
3918:f
3905:b
3902:a
3895:r
3890:V
3881:=
3878:)
3875:T
3872:(
3867:a
3836:)
3831:0
3827:T
3823:(
3818:0
3814:f
3804:)
3801:T
3798:(
3793:0
3789:f
3782:=
3779:)
3776:T
3773:(
3768:0
3764:f
3737:.
3732:]
3728:)
3723:0
3719:T
3715:(
3710:0
3706:f
3696:)
3693:T
3690:(
3685:0
3681:f
3673:[
3665:0
3661:f
3657:1
3649:b
3646:a
3639:r
3634:V
3625:=
3622:)
3617:0
3613:T
3609:(
3604:a
3593:)
3590:T
3587:(
3582:a
3574:=
3571:)
3568:T
3565:(
3560:a
3529:)
3526:T
3523:(
3518:a
3488:0
3484:T
3463:T
3440:,
3436:)
3433:T
3430:(
3425:a
3414:+
3411:)
3406:0
3402:T
3398:(
3393:a
3385:=
3382:)
3379:T
3376:(
3371:a
3339:m
3336:n
3309:c
3287:a
3261:m
3258:n
3243:a
3228:7
3226:O
3224:3
3220:2
3206:c
3183:.
3175:0
3171:f
3164:0
3160:f
3147:b
3144:a
3137:r
3132:V
3121:2
3118:c
3113:=
3108:a
3080:a
3057:.
3052:)
3046:a
3038:b
3035:a
3032:2
3023:s
3018:V
3013:(
3003:r
2998:V
2994:2
2988:0
2984:f
2978:=
2973:0
2969:f
2939:x
2936:e
2931:V
2924:=
2918:r
2913:V
2889:b
2866:s
2861:V
2840:c
2837:b
2834:a
2810:c
2790:b
2770:a
2747:,
2742:)
2736:a
2728:2
2722:c
2718:(
2714:b
2711:a
2702:x
2699:e
2694:V
2643:.
2634:r
2629:V
2621:r
2616:V
2604:2
2599:0
2595:f
2586:=
2581:0
2577:f
2548:2
2544:/
2540:1
2533:L
2510:0
2506:f
2482:r
2477:V
2454:2
2445:/
2438:r
2433:V
2427:0
2419:=
2416:L
2391:O
2389:3
2385:2
2327:L
2304:,
2296:C
2293:L
2289:n
2277:2
2273:1
2263:0
2259:f
2235:)
2232:1
2226:n
2223:(
2203:n
2183:L
2161:0
2157:L
2130:2
2106:0
2102:L
2098:=
2095:L
2075:C
2053:0
2049:L
2028:L
2003:2
1997:2
1948:0
1944:f
1913:S
1857:Q
1835:0
1831:f
1780:0
1776:r
1754:C
1751:R
1748:L
1724:Q
1657:R
1633:,
1627:4
1622:)
1617:c
1611:0
1607:r
1601:0
1590:(
1580:0
1570:0
1557:6
1543:r
1538:R
1512:0
1508:r
1479:/
1473:0
1469:r
1434:Q
1413:z
1410:H
1407:G
1402:1
1399:=
1394:0
1390:f
1368:m
1365:c
1360:1
1357:=
1352:0
1348:r
1325:0
1321:f
1314:2
1311:=
1306:0
1278:,
1268:0
1258:0
1245:2
1209:,
1198:0
1194:r
1185:Q
1139:,
1121:0
1117:r
1110:2
1092:R
1022:0
1018:r
1011:2
987:.
980:C
977:L
969:R
966:1
958:Q
935:C
932:R
929:L
889:0
879:0
868:/
864:1
861:=
858:c
835:,
827:w
820:t
809:0
805:r
798:2
794:c
789:=
783:C
780:L
776:1
765:2
761:1
751:0
747:f
723:Q
701:0
697:r
670:0
642:,
630:2
625:0
621:r
608:0
600:=
597:L
550:t
530:w
504:0
476:,
470:t
462:w
454:0
446:=
443:C
399:Q
377:0
373:f
351:Q
329:0
325:f
304:Q
282:0
278:f
249:C
246:R
243:L
221:C
218:R
215:L
148:R
125:L
102:C
51:(
43:.
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