Precise Measurement of a Magnetic Field Generated by the Electromagnetic Flux Compression Technique

0.81 MB·English

by  D. Nakamura

Sample title
Precise Measurement of a Magnetic Field Generated by the Electromagnetic
Flux Compression Technique
D. Nakamura,1,a) H. Sawabe,1 Y. H. Matsuda,1 and S. Takeyama1
Institute for Solid State Physics, University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8581,
Japan
(Dated: 15 January 2013)
Theprecisionofthevaluesofamagneticfieldgeneratedbyelectromagneticfluxcompressionwasinvestigated
inultra-highmagneticfieldsofupto700T.Inanattempttocalibratethemagneticfieldmeasuredbypickup
coils, precise Faraday rotation (FR) measurements were conducted on optical (quartz and crown) glasses. A
3 discernible ”turn-around” phenomenon was observed in the FR signal as well as the pickup coils before the
1 end of a liner implosion. We found that the magnetic field measured by pickup coils should be corrected by
0
takingintoaccountthehigh-frequencyresponseofthe signaltransmissionline. Nearthe peakmagneticfield,
2
however, the pickup coils failed to provide reliable values, leaving the FR measurement as the only method

n to precisely measure an extremely high magnetic fields.
a
J
PACS numbers: 41.20.Jb,07.55.Db,78.20.Ls

3
Keywords: electromagnetic flux compression, ultra-high magnetic field, metrology, Faraday rotation
1

]
t I. INTRODUCTION
e
d
- Theapplicationofhighmagneticfieldsiswidespreadin
s
fields such as physics, chemistry, biology, and medicine.
n
i In particular, in the region of ultra-high magnetic fields,
.
s the Zeemansplitting energyandthe cyclotronresonance
c energy(ofthe quantizedorbitalmotionoffreeelectrons)
i
s canexceedcompetingenergyscalessuchasthermalfluc-
y tuations, and we can access the quantum limit of ma-
h
terials, even at a room temperature. The demand for
p
high magnetic fields in solid state physics applications,
[
FIG.1. (color online) Schematic of theEMFC magnet, com-
is as a result, rapidly growing. An ultra-high magnetic
posedofacopper-linedprimarycoilwithacopperlinerinside,
1 field (above 100 T) can only be achieved by a destruc-
v and a pair of seed field coils.
tive type of magnet due to Maxwell stresses exceeding
5
40,000 kg/cm2.1–3 Pulsed magnetic fields are generated
5
by destruction of magnets over a timescale of microsec-
7
2 onds. The flux compression and single-turn coil tech- method, on the other hand, first developedby Cnare7 in
. niques are currently the only available methods for the the1960’s,ismoresuitableforapplicationtoexperimen-
1
generation of ultra-high magnetic fields. In the single- tal solid state physics. Figure 1 shows a schematic of a
0
3 turn coil technique,4 a magnetic field of up to 300 T coil used for the EMFC method. In the EMFC method,
1 is generated by a mega-ampere current injected into a the liner implosion is caused by an electromagnetic
: single-folded coil. In the flux compression technique, a (repulsive) force from the single-turn (primary) coil
v
seed field is injected into the main coil and compressed at the moment when a huge current is injected into
i
X by the implosion of a metallic cylinder called a ”liner.” this coil. Regardless of the destructive nature of this
r The explosive-driven flux compression technique5,6 uses method, it has the advantage of precise controllability,
a chemical explosives for accelerating the implosive liner, and is more suitable for indoor experiments than any
and can access magnetic fields of around 1000 T. How- chemical explosive methods. Recently, we reported
ever, a single experiment requires substantial prepara- on the generation of magnetic fields of up to 730 T:
tion, and the destructive nature of this technique causes the world’s highest ever magnetic field generated in
difficulties with respect to reproducibility and controlla- an indoor laboratory. This was achieved by a newly
bility. As a result, the application of this technique has designed copper-lined primary coil (shown in Fig. 1),8,9
laggedbehindthesingle-turncoiltechniquewithregards and has already been applied to several solid state
to experimental solid state physics. measurements. For example, interesting results have
The electromagnetic flux compression (EMFC) beenobtainedforthehighTc cupratesuperconductors,10
carbon nanotubes,11 and frustrated magnets.12
RegardlessoftheprogressoftheEMFCtechniquethat
has been made over the past 40 years, less attention has
a)Electronicmail: [email protected] been paid so far to the precision of the measured mag-
Sample title 2
netic field. As the maximum magnetic field and rate of where v is the Verdet constant of the material, L is the
change of the magnetic flux have increased, the accu- sample length, and B is the external magnetic field. By
racy of the measurement of the magnetic field in these measuring the θ of a material whose v is known, if L is
F
experimentshasbecomemoreimportant. Inapulsedop- accurately measured, the magnetic field can be precisely
erationthemagneticfieldisdeterminedfromthevoltage evaluated. In addition, the pickup coil can be calibrated
induced in a pickup coil inserted into the center of the precisely by means of a simultaneous measurement.
magnetcoil,whereasthepickupcoilissetatthecenterof Another candidate for the calibration of ultra-high
implodinglinerinEMFCexperiments. WhentheEMFC magnetic fields is a resonant type of measurement.13,14
technique is used for the purpose of solid state physics For example, Kido et al. used the ESR (Electron Spin
measurements, the pickup coil is the only method avail- Resonance) signal of Ruby, whose ESR peak appears at
able as a magnetic field probe due to the limited space 91.0 T for a laser wavelength of 118.65 µm.14 However,
(thediameterofthefinalspaceinsidetheimplodingliner despite its high precision, this method is only applicable
is less than 6 mm).9 However, a pickup coil is not nec- to a discrete point.
essarily the ideal probe for an accurate determination of The measurementofmagnetic fieldbythe FRmethod
ultra-high magnetic fields. The difficulties involved with has been previously applied to the flux compression
the evaluation of a precise magnetic field using pickup technique,15–19 however, to the best of our knowledge,
coils are the following: there has been no detailed study of the precise evalua-
tion of the magnetic field above 300 T. In this article,
• Theprecisionofthepickupcoildimensiondegrades we analyzethe results ofFR measurementsin ultra-high
with reducing size. magnetic fields (up to ∼ 700 T) for two types of optical
glass,andcomparethistomeasurementsofthemagnetic
• The electrical insulation breaks down easily for in-
fieldfrompickupcoils. Atthefinalstageofthefluxcom-
duced voltages as high as a few kV.
pression,apeakstructureduetoleakageofthemagnetic
fluxfromtheimplodinglinerisobserved,whichisknown
• The wire used in the pickup coils heats up as a
as the ”turn-around” phenomenon.9 The observation of
result of the current induced from the huge dφ/dt
this phenomenon is regarded as an important indication
(φ: magnetic flux).
of measuring up to the maximum magnetic field. An
optical FR transmission signal was captured up to this
• The high frequency characteristics of the pickup
final stage of the flux compression, as indicated by the
signal are transferred to the measurement instru-
observation of this turn-around structure.
ments.
Inaddition,afictitious signalisoftenobservedwhenthe
pickup coil is damaged or destroyed during the increase II. EXPERIMENTAL METHOD
of magnetic field, which can lead to misinterpretation of
the experimental results. Accordingly, it is important Details of the experimental setup for EMFC are de-
to pursue a precise and reliable calibrationof the pickup scribedinreference9;thecoilwassetinananti-explosion
coilformeasurementofthemagneticfield,bysomeother chamber for explosive experiments. The overall experi-
method. mentalsetupisillustratedinFig. 2. TheFRofanoptical
Todealwiththisissue,wehaveattemptedtoemploya glassrod wasmeasuredsimultaneously to the signalof a
magneto-opticalmethodasaprobeforthemagneticfield, pickup coil in the manner shown in the inset of Fig. 2.
which has some advantages compared with the pickup The sample probe was inserted into the center position
coil. For example, optical measurements do not require of the pickup coil. For FR measurements we used fused
a metal lead wire around the sample, and are thus less quartz and crown glass rods (Kiyohara Optics), with a
influenced by the effect of electromagnetic noise as well diameter of 2 mm. The homogeneity of the magnetic
as also escaping the breakdown of electrical insulation. field along the coil axis within ±1 mm of the center was
Furthermore,complexelectriccomponentssuchasanin- measuredtobeapproximatelywithin0.5%atamoment
tegratorcircuit areavoided,which simplifies the calibra- of the peak field.20 The length of the sample rod was
tion of the magnetic field, and reduces possible sources determined to an accuracy within 1 µm, and the pickup
of error. Therefore,the opticalmeasurementis expected coil was wound around the rod. We used a polyamide-
to be quite useful for the precise evaluation ofultra-high imideenameledcopperwire(AIW wire,TOTOKUElec-
magnetic fields. tric Co.), of diameter 0.06 mm for the pickup coil as it
WeadaptedtheFaradayrotation(FR)methodforop- is known to have a high resistance to insulation break-
ticalmeasurementofthemagneticfield. Opticallytrans- down. Aquartzrodwasfirmlyheldbykaptonandbake-
parent materials generally exhibit a linear response of litetubesinthecenterofthecoil. Asuper-insulationfoil
the FR angle (θ ), with respect to an external magnetic was wound outside the bakelite tube to protect against
F
field, as described by: significant electromagnetic noise and the flushing light
fromtheimplodingliner. Asaresult,theouterdiameter
θ =vLB, (1) of the entire sample holder became about 3.5 mm.
F
Sample title 3
FIG. 2. (color online) Optics setup for the FR measurements with the EMFC coil and the sample holder set inside an anti-
explosion chamber. Inset (bottom left) shows the sample holder, a 2 mm-thick quartz rod with a pickup coil wound around
therod.
V)2 V)2 1 % fluctuation). However, at the final stage of the liner
V+V (sp10 V+V (sp10 Vs,n + Vp,n iamnpalobsriuopnt,wchhaenngtehoeflitnheertarpapnrsomaicthteadncteheocscaumrpreled,hpolodsesri-,
bly arising from disturbance of the optical pass due to
1.5 1.5
R signal (V)10..05 R signal (V)10..05 VsVs,n Vp,n dmVesoc+vreemaVspee,nfstrhooomfwthnaepinpsartmohxpeimleuaphtpoeleldyre4pr2.a.nT7ehlµeso,tfotFthaieglr.terfao3nresst,mawritteteadnnotcroe-
F F malizedtherawFRsignalbydividingthroughbyV +V .
0.0 0.0 Vp Theturn-aroundphenomenonisclearlydemonstrastedapt
(a) 40.0 40.5 41.0 41.5 42.0 42.5 43.0(b) 42.5 42.6 42.7 42.8 42.9 42.8 µs in the FR signal after the normalization proce-
Time (m s) Time (m s)
dure, as shown in Fig. 3(b). The observation of the
turn-around feature is important for reliable calibration
FIG. 3. (color online) (a) The FR signal of the s- and p-
polarized components (Vs, Vp), in experiment #F1. The up- of the pickup coil to the end point of the flux compres-
per panel shows the sum of the two amplitudes of polarized sion. Hereafter, we are only concerned with the normal-
light,Vs+Vp. (b)Aplotclosetotheturn-aroundphenomenon ized data for the FR signals. θF is calculated using Vs
showingboththerawandnormalized data(Vs,Vp,Vs,n,and and Vp as:
Vp,n). Thedashed line in theupperpanel shows Vs,n+Vp,n.
180 V −V
s p
θ[deg.]= ×arccos . (2)
2π V +V
(cid:18) s p(cid:19)
Semiconductorlasers(coherent”CUBE”)withawave-
A pickup coil with a 75 cm long twisted copper wire
length of 404 nm or 638 nm were used as the light
was connected to a 40 m long BNC cable (RG58C/U),
source. Linearlypolarizedlightwastransmittedthrough
in a double-shielded box (as shown in Fig. 2), and mon-
a sample rod and divided into s- and p-polarized light
itored by a transient digital recorder in a shielded room
(V ,V ) by a Wollaston prism. The data acquisition
s p separate from the explosion chamber. As the induced
was performed in an electromagnetically shielded room,
voltage in the pickup coils reached a maximum of a few
which was several meters away from the anti-explosion
kV,thesignalwastransferredintotwobranches: ahand-
chamber. The polarized light was transferred by optical
made-34dBattenuator,andaCRintegrator(withtime
fibers, and transformed to an electric signal by an O/E
constant ∼ 1 ms). After data acquisition, the magnetic
transformer(New Focus,125-MHzPhotoreceiversmodel
field measured by the pickup coil was calculated using
1801), then measured by a transient digital recorder
the following formula:
(SONY Tektronix,RTD710Adigitizer), oranA/Dcon-
verter board (Spectrum, M3i.4142-exp).
The raw and normalized FR signals (Vs, Vp, Vs,n, and B = −gS−1RLR+MRM V3(t)dt : attenuator (3)
Vp,n, respectively) are plotted in Fig. 3. Fig. 3(b) is ( −(SRC)−1RLR+MRRMV4(t) : CR integrator
an enlargedplot about the ”turn-around”point. Except
for the final stage of the liner implosion, an almost con- where, g is a ratio of the attenuator; R and R are
L M
stant optical transmission signal was obtained (within a theresistancesofthe transmissionlineandthe matching
Sample title 4
TABLE I. Experimental conditions for each setup (#F1 to #C2). The material, sample length L, laser wavelength used for
FR measurements λ, total energy,charged voltage, and capacitance of the main condenser bankare given.
FR condition Main bankcondition
Exp. material L [mm] λ [nm] Main energy [MJ] voltage [kV] capacitance [mF]
#F1 fused quartz 2.023 404 3.5 35 5.625 (slow)
#F2 fused quartz 2.023 404 4.0 40 5.0 (fast)
#F3 fused quartz 0.618 638 4.0 40 5.0 (fast)
#C1 crown glass 1.999 404 3.5 35 5.625 (slow)
#C2 crown glass 1.999 404 4.0 40 5.0 (fast)
700 600 3.0
#F2 #C2 #F1
600 500 2.5
#F1
500 400 2.0 F
R
B (T)p 430000 #F3 #C1 B (T)p 320000 Vp Vs 11..50 )V( langiS
200 C = 5.0 mF 100 0.5
100 0 0.0
37 38 39 40 41 42 43
C = 5.625 mF
Time (m s)
0
34 36 38 40 42 44
Time (m s) FIG. 5. (color online) The simultaneous measurement of the
magnetic field bythepickupcoils, and theFR angle of fused
quartz,forexperiment#F1. Thethicklinecorrespondstothe
FIG. 4. (color online) The magnetic field curves obtained by
magneticfieldmeasuredbythepickupcoils,andthethinlines
the pickup coil, plotted as a function of time. The experi-
correspond to the amplitude of the s- (Vs) and p-polarized
mental conditions of #F1 to #C2 are summarized in Table
(Vp) FR signals. The dotted vertical line indicates the time
I.
of the turn-aroundphenomenon observed in theFR signal.
circuit (shownin Fig. 2); RC is the time constantof the
tude of the capacitance.
integratorcircuit;V andV arethemeasuredvoltagesat
3 4
thetransientrecorder(aftertheattenuator),andtheCR
integrator,respectively; andS is the cross-sectionalarea
III. RESULTS AND DISCUSSION
of the pickup coils, calibrated by comparing the induced
voltage by an AC magnetic field (f = 50 kHz) with a
In Figure 4, we summarize the magnetic field calcu-
standard coil. We discuss the response of the electronic
lated by Eq. (3) from the pickup coil signal (B ), in ex-
circuit in more detail later. p
periments #F1 to #C2. A ”time zero” was determined
In Table I we summarize the experimental conditions
by the appearanceof the dischargingtriggernoise in the
employed in this article, where the seed field was 3.8 T.
pickup coil signal. In all experiments, a slowdown of the
The experimental parameters in Table I were chosen for
magnetic field curve was noticed near the peak, which is
comparison in an attempt to clarify the imperfection of
indicative of the turn-around phenomenon. The peak of
measurements using the pickup coils. Different materi-
the magnetic field was reached more slowly in #F1 and
als for the FR measurement were used to check whether
#C1 than in #F2, #F3, and #C2, due to the difference
there were material-dependent contributions. In addi-
in capacitance of the main condenser bank.
tion, as described in the introduction, a difference in the
high frequency characteristics of the pickup coil signal
should be carefully examined. For this reason, we per-
formed the EMFC experiments with different time pro- A. Fused quartz measurements
files by varying the capacitance in the main condenser
bank. Although the implosion speed of the liner is not First, we discuss the FR results of fused quartz for
only a function of capacitance in the main condenser experiments using different values of the capacitance (of
bank, for convenience we separated the dynamics of the the main condenser bank), #F1 and #F2. The simulta-
liner to either ”slow” or ”fast” depending on the magni- neous measurement of the induced voltage in the pickup
Sample title 5
800 T a super-linear tendency was observed. This indicates
an increased deviation of B from that obtained from
)200 p
700 g. the FR angle (B ) on further increase of the magnetic
e FR
d100 field, which does not relate to the estimate of the Verdet
(
600 q F constant for BFR < 200 T.
g.) 00 100 200 BFR was calculated from θF, assuming that Eq. (1)
e
(d 500 Bp (T) still holds at high magnetic fields. Bp (thin lines), and
q F 0.559 deg./mm T cBoFmRpa(trhisiocnk ilnineFsi)g,.ar7e. sFhoorwcnlaarliotyn,gsaidheoroinzeonatnaoltshheirftfoorf
400
1.0 µs was added to the curve corresponding to experi-
fused quartz, 404 nm ment#F3. Forexperiment#F3theVerdetconstantwas
300
#F1
determined to be 0.200 ± 0.007 deg./mm T for a wave-
#F2
lengthofλ=638nm,whichwasalreadyconfirmedinour
200
200 300 400 500 600 previous report.21 A jerky structure in the BFR(t) curve
B (T) (e.g. see #F3 at 39.6 µs), was identified as an artifact.
p
A discontinuity in the θ curve sometimes occurs as V
F s
or V approach zero, however, the overall FR angle was
FIG. 6. (color online) The FR angles θF of fused quartz at p
not affected by this jerky structure in the determination
λ = 404 nm, as afunction of magnetic field measured by the
pickupcoils, Bp. Thedottedlinecorrespondstothelinearfit of BFR.
of θF for magnetic fields less than 200 T (inset). BFR was larger than Bp in all experiments, with
a maximum ratio [max(B )/max(B )] of 1.08, 1.16,
FR p
and 1.18 for #F1, #F2, and #F3, respectively. A
700 + 1 m s
B similar ratio for experiments #F2 (λ = 404 nm) and
FR
600 Bp #F3 (λ = 638 nm) suggests that the wavelength de-
T) BFR #F2 pendence of the refractive index of the material is not
eld ( 500 Bp the main source of this difference. It appears that the
Fi 400 BFR ”max(BFR)/max(Bp)”ratiodepends onthecapacitance
netic 300 Bp #F3 #F1 omfFt)h,eacnodnwdeanssleorwbearnikn #(#FF11c:o5m.6p2a5remdFto; ##FF22,a#nFd3#: F53.0,
g
Ma 200 indicating that the rise time of the magnetic field curve
contributestothedegreeofdiscrepancy. Thesefactssug-
100
gest that the difference between B and B majorly
FR p
0 arises from the high frequency response of the electric
36 37 38 39 40 41 42 43 circuit, which has so far not been accounted for.
Time (m s)
FIG.7. (coloronline)Themeasuredmagneticfieldasafunc-
B. Analysis: The high frequency response of the circuit
tion of time for the induced voltage of the pickup coils (thin
lines,Bp),andθF offusedquartz(thicklines,BFR). Forclar-
ity, the curve corresponding to experiment #F3 was shifted We analyzed the high-frequency response of the elec-
by 1.0 µs. tric circuit used for the pickup coil measurement in our
EMFCsystem. Theequivalentcircuitdiagramofthesig-
naltransmissionlinefromthepickupcoiltothetransient
coilandtheFRangleofthefusedquartz,wasperformed recorder in Fig. 2 is illustrated in Fig. 8, where the cir-
to high precision. Figure 5 shows a typical result of ex- cuitparametersarelistedinthecaption. R dependson
L
periment #F1, where the thick line corresponds to Bp the lengthofthe twistedcopperwireusedforthe pickup
and the thin lines correspondto Vs, and Vp. Note that a coil and is measured for each experiment. The equiva-
turn-around phenomenon was observed in the FR signal lentcircuitis composedofapickupcoilinductance (L ),
c
(dotted line), the first time that a turn-aroundstructure transmissionline(hatchedregionA),aπ-typeattenuator
above 500 T has been clearly observed in an optical sig- for dφ/dt measurements (hatched region B), a matching
nal. resistance(R ),andaCRintegratorforφmeasurements
m
We derived θ by Eq. (2) and show this as a function (hatched region C). R is the input impedance of the
F TR
ofB inFig. 6,wherethesolidlinescorrespondtoθ and transient recorder. The measured matching resistance
p F
the dotted line correspondsto the linear fitfor B < 200 R ,inFig. 2,isthe combinedresistanceofthe attenua-
p M
T (shown in the inset of Fig. 6). Below 200 T, the θ of tor, the matching resistance R , and the CR integrator;
F m
experiments#F1and#F2coincidewellwitheachother, typically R ∼ 50 Ω. The measured quantities in the
M
and respond linearly with respect to B . From the slope EMFCexperimentwereV (t)andV (t),whichwereused
p 3 4
ofthelinearfitinFig. 6,weobtainedaVerdetconstantof to calculate V (t) = −dφ/dt. In the conventional pulse
1
0.559±0.008deg./mmT.Ontheotherhand,above200 magnet experiments, V (t) is calculated from a simple
1
Sample title 6
FIG. 8. (color online) The equivalent electric circuit for a measurement of the magnetic field by the pickup coils. The circuit
parameters were as follows: RL ∼ 10 Ω, RATT1 = 100 Ω, RATT2 = 5 kΩ, RATT3 = 100 Ω, Rm = 100 Ω, RCR = 10 kΩ, CCR
= 100 nF, and RTR = 1 MΩ.
magnetic field (B ), and compare it with B in Fig.
600 #F1 p,c p
CL = 4 nF 9. The results shown are for the case where CL = 4 nF
500 CL = 16nF (dashed lines) and 16 nF (solid lines), for experiments
#F2 #F1 and #F2. The dotted line shows B =B . These
p,c p
T) 400 CCL == 41 6nnFF results highlight the fact that the difference between Bp
B (p,c 300 L a(tnydpiBcap,lclyis1o0f%thefosramCLeo=rd1e6ronfFm).agHnoiwtuedveera,sBthFaRtsohfoBwFeRd
200 a superlinear dependence with respect to Bp in Fig. 6,
whereas B (B ) for constant C exhibits a sublinear
p,c p L
100 dependence in Fig. 9 above 400 T.
Asaresult,wetakethefrequencydependenceofC R
L L
0
0 100 200 300 400 500 600 intoaccount. ThesignalofthepickupcoildV1/dt,iscom-
posed of a wide range of frequencies, therefore to evalu-
B (T)
p
ate the empirical formula for C R of the transmission
L L
line in Fig. 8, the transient response of C R was in-
FIG. 9. (color online) The calibrated magnetic field Bp,c, as vestigated by using the transmission line sepLarLately. We
afunctionofBp forexperiments#F1and#F2,respectively,
applied a triangular wave V (t), to the open end of a 75
whereCL =4nF(dashedlines) and16 nF(solid lines). The 1
cm long twisted copper wire, as shown in the inset of
dotted line indicates Bp,c =Bp.
Fig. 10(a),andmeasuredV (t)withrespecttodV (t)/dt
2 2
through the same BNC cable used in the EMFC ex-
formula [V (t)(R +R )/R , see also Eq. (3)], that periment. In Fig. 10(a), the solid lines correspond to
2 L M M
V (t) and the dotted lines correspond to V (t). Since
only takes the resistanceratiointo account. This simpli- 2 1
ficationneedstoberevised,however,whenV2(t)contains V1(t) ∼ V2(t)+CLRL(dV2(t)/dt) in the hatched region
high frequency components. The difference between the markedAinFig. 8,wecancalculateCLRL asafunction
BFR and Bp discussed in Figs. 6 and 7 should be ac- of dV2(t)/dt as shown in Fig. 10(b). The closed square
counted for well by the high-frequency response of the in Fig. 10(b) is the estimated value (CLRL = 20 ns) at
transmissionlinewithasubstantialcontributionfromCL dV2(t)/dt = 0. By fitting the data of Fig. 10(b), we
andLL. Inthe followingdiscussion,the heating effectof obtained an empirical formula for CLRL of:
the pickup coil during the magnetic field generation is
C R =7×10−10×|dV (t)/dt|0.25+20×10−9. (5)
neglected. L L 2
The detailed expression of V (t) as a function of V (t)
1 3 Note that C R takes a minimum value when
[or V (t)], is summarized in Appendix A. Of this expres- L L
sion,4at high frequencies a dominant term V+(t), is de- dV2(t)/dt = 0. As an example, CLRL and the induced
1 voltage in the pickup coil are compared in the inset of
scribed as follows:
Fig. 10(b).
C R ×R C d2V4(t) By taking into account the frequency-dependent dy-
V1+(t)=( CLLRLL× 1CR+ RRCAARTTTT23dt+2 RRATTRT2 dVd3t(t) (4) nbaramteicditmopceodinancicdeeowfitthheBtFrRan.smUissisnigonEqlisn.e,(4B)panisdc(a5l)i-,
(cid:16) (cid:17) Bp is further calibrated to Bp,c, and plotted in Fig. 11
Since the LL term appears only in higher order terms, for experiments #F1 (a) and #F2 (b), where, Bp, Bp,c,
we ignore it in the following discussion. B , C R correspondto the thin solidline, thick solid
FR L L
At first, we assume that C and R are constant. By line,thick dashedline,andthindottedline,respectively.
L L
adding V+(t) to Eq. (3), we calculated the calibrated Inbothcases,B (t)coincideswellwithB (t),exhibit-
1 p,c FR
Sample title 7
2.5 700 0.6
8.0 6.0 4.0 1.4 0.8 #F1
2.0 600 Bp 0.5
oltage (V) 11..50 Field (T) 500 v = 0 .BB55pF,R1c deg./mm T 0.4 LCR
V 0.5 etic 400 CLRL 0.3 L (sm
n )
ag 300 0.2
M
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
(a) 200 0.1
Time (m s)
70
100 0.0
41 42 43
60 (a)
Time (m s)
50 700 1.0
s)
R (nLL4300 600 # F2 Bp 0.8
C B
p,c
20 T) B
10 Field ( 500 v = 0.56FR6 deg./mm T 0.6 LCR
(b) 00 2 d4V2/dt (V6/m s) 8 10 agnetic 430000 CLRL 0.4 L (s)m
M
FIG.10. (color online) (a) Theoutputresponse ofthetrans- 0.2
200
mission line obtained by the offline measurement to decide
thefrequencydependenceofCLRL. Forreference,eachinput
signal is plotted as a dotted line. The inset shows theexper- 100 0.0
38 39 40
imental setup. (b) Values of CLRL as a function of dV2/dt. (b) Time (m s)
The filled squares are the estimated values under DC condi-
tions; the bold line is the result of fitting. The inset shows a
comparison of CLRL and the induced voltage in the pickup FIG. 11. (color online) The comparison between Bp (solid
coils for experiment #F2. thin lines), Bp,c (solid thick lines), and BFR of fused quartz
(thick dashed lines), for experiments (a) #F1, and (b) #F2.
TheCLRL curvesarealsoshown(thindottedlines),asisthe
ingalmostcompleteoverlapupto500Tforthesamepa- characteristic time of CLRL = 0 (vertical dashed line).
rameters,regardlessofany difference intheir timescales.
Interestingly, B starts to deviate from B in both
p,c FR
turn-around phenomenon appeared for B in both ex-
experimentsaboveacharacteristictimecorrespondingto FR
periments, and in Fig. 12(a) we observed a turn-around
C R = 0 (vertical dotted lines in Fig. 11). We discuss
L L
phenomenon for both B and B . The B curve close
this issue further in Sec. III-D. FR p p
totheturn-aroundfeature,however,seemstobe slightly
broadenedcomparedto that ofB , suggestingthat B
FR p
doesnotproperlyreflectthehighfrequencycomponents.
C. Comparative FR measurements using crown glass
The calibrated magnetic field of the pickup coil (thick
solid line, B ), was calculated in the same way as in
p,c
Next, we discuss the results of crown glass as the ma- Sec. III-B. The thin dotted lines of Fig. 12 are the
terial for FR measurements. Crown glass is the generic C R obtained from the analysis using the formula and
L L
name for optical glasses with a relatively low refractive parameters of Sec. III-B. A good coincidence between
indexandwavelengthdispersion. Amongthemanytypes B and B was obtained up to the time where C R
FR p,c L L
of crown glass, in this study we used BK7 because of its = 0 (vertical dotted lines), similar to the case of fused
suitability as a transparentsubstrate. Fig. 12 shows the quartz. These facts indicate that the observed discrep-
results of simultaneous measurement of the pickup coil ancybetweenB andB predominantlyarisefromthe
FR p
and θ of crown glass [(a) #C1, and (b) #C2]. The high frequency contributions and not some difference in
F
thin andthick solid lines correspondto B andB , re- the materials used for FR. Furthermore, an additional
p FR
spectively. The Verdet constants used to calculate B feature observedfor both materials is that B cannot be
FR p
at 404 nm were 0.782 ± 0.008 deg./mm T and 0.739 ± reproduced to the value of B beyond the time where
FR
0.007 deg./mm T in (a) and (b), respectively. A clear C R = 0.
L L
Sample title 8
1.0 1.0
700 #C1 700 #C2
Bp 0.8 Bp 0.8
T) 600 Bp,c T) 600 Bp,c
Field ( 500 v = 0 .B78FR2 deg./mm T 0.6 LCR Field ( 500 v = 0 .B73FR9 deg./mm T 0.6 LCR
gnetic 400 CLRL 0.4 L (s)m gnetic 400 CLRL 0.4 L)( sm
a 300 a 300
M M
0.2 0.2
200 200
100 0.0 100 0.0
41 42 43 38 39 40
(a) (b)
Time (m s) Time (m s)
FIG. 12. (color online) The results of simultaneous measurement of the pickup coil and the FR angle of crown glass for
experiments(a) #C1, and (b) #C2. Thenotations are thesame as for Fig. 11.
D. Precision of the magnetic field measurement deviationof3%. Wethereforesuggestthatthedeviation
oftheVerdetconstantmainlyoriginatesfromanerrorin
We will now consider the precision of measurement of the evaluation of the cross sectional area (i.e. diameter)
the ultra-high magnetic field generated by the EMFC of the pickup coils.
technique. The Verdet constant obtained in this study Thisresultsignifiesonedisadvantageofthepickupcoil
is validated in Appendix B, where it is shown in Table asamagneticfieldprobe. Evenifitwerepossibletocor-
II that the deviation of the Verdet constant (δv/v) was rectly calibrate the induced voltagein the pickupcoilby
approximately 1.5 % for fused quartz and 3 % for crown the analysis of Sec. III-B, there is still some ambigu-
glassat404nm. Inotherwords,theprecisionofthemag- ity about the precision of the cross-sectional area of the
netic fieldevaluatedby θ is 3 % atworst. Onthe other pickupcoil itself. Inparticular,a diagonalcomponentof
F
hand, we can estimate δv using the error in θ , L, and the field withrespectto the axialdirectionofteninduces
F
magnetic field homogeneity. Since the fluctuation of the additionalmagnetic fieldin the pickupcoil. Therefore,a
optical transmissionsignal is typically 1 % (as described precision of less than 3 % of Bp for the flux compression
in Sec. II), the error in θ (δθ), can be estimated using technique is hardly achievable unless the fabrication of
F
Eq. (2) when V =V , as: pickup coils is drastically improved.
s p
We will now summarize the discussion about the dis-
180δV +δV
s p crepancyofmagneticfieldsfoundinthisstudy. Themys-
δθ = =0.57 deg. (6)
π Vs+Vp tery of the difference between Bp and BFR was almost
solved in the previous section. The B obtained by pre-
p
For the linear fit of Fig. 6, we did not use the data vious analysis using only a resistance ratio is valid up
at Vs = 0 or Vp = 0 due to a divergence of the error, to magnetic fields of 200 T. Above 200 T, Bp should
therefore, we can assume δθ ∼ 0.57 deg. In Fig. 6, we be calibrated using the dynamic impedance of the signal
δmθe/aθsu=red8.1θF4 ×up10t−o4.∼ T70h0e derergo.,r wduheichtowuonucldertraeisnutlyt ionf ttrhaenvsamluisessioonftlhineem. aSginnecteicBfiFeRldiisnaplrweavyiosulsarEgMerFtChaenxpBepr-,
the exact length of the sample rod δL, is only 1 µm, iments were likely underestimated. Fig. 13(a) shows the
which results in δL/L = 0.5×10−3. For the magnetic difference ∆B ≡ B −B , plotted against B . The
1 FR p FR
fieldhomogeneityalongthecoilaxis,wecanassumethat dottedlinesindicatetheestimatederrorsof3%,5%,and
δB/B ∼ 0.5×10−2 within ± 1 mm of the center of the 10 %. As is evident in the inset of Fig. 13(a), the dif-
liner (nearly at the peak field), as described in Sec. II.20 ference ∆B , of experiments with the same capacitance
1
As a result, δv/v is estimated to be: shows almost the same value for different materials. For
example, ∆B (400 T) ∼ 20 T for #F1 and #C1, whilst
1
δv = δθ 2+ δL 2+ δB 2 =0.51×10−2. (7) ∆B1(400 T) ∼ 30 T for #F2 and #C2. Above 200 T
∆B increases suddenly, and for B > 600 T a typical
v s θ L B 1 FR
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ∆B of 10 % is observed.
1
The propagation of errors for each experiment therefore In Fig. 13(b) the difference ∆B ≡ B − B , is
2 FR p,c
results in a total error of the Verdet constant of approx- shown as a function of B . In all experiments ∆B is
FR 2
imately 1 %, which is less than the maximum observed almost zero up to 400 T, and increases gradually there-
Sample title 9
150 150
40 #F1
#F1
#F2 #F1
30 #F2
B = B - B (T)1FRp 15000 DB (T)12110005020025B01F03R 0% 0(T3)50400450 5 ##%CC12 B = B - B (T)2FRp,c 15000 ##CC12 10 % #C1 #F#C22 5 %
3 % 3 %
D D
0 0
0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700
(a) (b)
B (T) B (T)
FR FR
FIG. 13. (color online) Summary of the deviation from BFR for (a) Bp, and (b) Bp,c, plotted against BFR. The dotted lines
indicate an estimated error boundary of 3 %, 5 %, and 10 %. The vertical lines in (b) indicate a position of BFR, where the
induced voltage in the pickup coil reached a maximum value. The inset of (a) shows an enlarged plot between 150 T and 450
T.
after (in a different way for each experiment). The ver- and exhibited time-dependent behavior rather than ma-
tical lines indicate a characteristicvalue of B for each terial dependence (see #F1, #F2 and #C1, #C2). This
FR
experiment, where the induced voltage of the pickup fact implies that f(B) ∼ 0 and that the deviation in
coil takes a maximum value (the point at which C R the linearity of θ in optical glasses does not present a
L L F
approaches zero). Note that the #F2 curve behaves dominantcontributiontothediscrepancyin∆B forthe
2
quitedifferentlytotheothersabove600T,andincreases highest magnetic fields. Furthermore, there is difficulty
rapidly up to ∆B = 150 T. This rapid increase of ∆B indetermininganyreasonforthedeviationfromlinearity
2 2
above 600 T is considered to be due to damage to the at only the ”turn-around” point.
pickup coil at high fields, where the induced voltage was To summarize, the pickup coil method is limited with
extremely high. respect to the precise measurement of the magnetic field
Some other possibilities for the origin of the discrep- in extreme experimental conditions such as the EMFC
ancy in ∆B at very high magnetic fields can also be technique, whichis capableof generatingmagnetic fields
2
considered. The first is an incompleteness of our high above 500 T. Therefore, the evaluation of the magnetic
frequency analysis. The equation used to calibrate the fieldusingFRmeasurementstogetherwithapickupcoil,
induced voltage in the pickup coil was complicated, as is necessary.
shown in Appendix A. For simplicity we ignored the
higher order terms of C and L . The temporal change
L L
in RL due to heating of the pickup coil during the flux IV. CONCLUSION
compression step is also an unknown factor (and is con-
sidered to occur by eddy currents in the metal), and
We succeeded in the precise evaluation of ultra-high
quantitative evaluation is difficult at present.
magneticfieldsofupto700TusingtheFaradayrotation
Secondly,apickupcoilexposedtoahugeinducedvolt- (FR)angleofopticalglass,confirmedbythe observation
age is damaged to some extent, and therefore becomes of a turn-around structure in the FR signal. We com-
incapable of correct measurement of the magnetic field paredthemagneticfieldsmeasuredbyapickupcoilwith
near the turn-around point. Indeed, ∆B2 started to in- that calculated from the FR angle, and found that a de-
crease slightly before CLRL =0 [the vertical line in Fig. viation starts to appear above 200 T. As a result of this
13(b)], where the induced voltage was at its maximum analysis, for the correct evaluation of the magnetic field
value, as shown in the inset of Fig. 10(b). At the mo- measured by a pickup coil, the high frequency response
ment where ∆B2 increases, the induced voltage of the of the signal transmission line must be accounted for in
pickup coil becomes as high as an order of 1 kV. the calibration. However,this is not sufficient above 500
The third possibility is that the assumption used for T.Theprecisemeasurementofultra-highmagneticfields
Eq. (1) does not hold in extremely high magnetic fields. is only possible by the use of FR measurements of fused
Inthiscase,θ canbeexpressedbyaddinganadditional quartzorcrownglass,inwhichthelinearityoftheFRan-
F
term: θ =L(vB+f(B)). ThedeviationofB fromB glewasmaintainedinmagneticfieldsofupto700T.The
F p FR
is given by Lf(B), which should be time-independent. values of the magnetic field evaluated by only a pickup
However, in Fig. 13(b), ∆B shows different behavior, coil in previous EMFC experiments were likely underes-
2
depending on the choice of the main condenser bank, timated.
Sample title 10
V1(t)= RCRCCR+RL RA−T1T-TRRCRCCR+CCR(Rm−1RCR+1) dVd4t(t) (A1)
+R(cid:8)L" RL−1+RA−T1T(cid:2)-TR (cid:18)1+ RRCTRR(cid:19)+(cid:18)R1m + Rm−1RRTCRR+(cid:3)(cid:9)1(cid:19)#V4(t)
(cid:0) (cid:1)
+LL("RA−T1T-TR(cid:18)1+ RRTCRR(cid:19)+(cid:18)R1m + Rm−1RRTCRR+1(cid:19)#dVd4t(t) + RA−T1T-TRRCRCCR+CCR(Rm−1RCR+1) d2dVt42(t))
(cid:2) (cid:3)
R dV (t) d2V (t)
+CLRL"(cid:18)1+ RCTRR(cid:19) d4t +RCRCCR dt42 #
R d2V (t) d3V (t)
+CLLL"(cid:18)1+ RCTRR(cid:19) dt42 +RCRCCR dt43 #,
ACKNOWLEDGMENTS
TABLEII.Theparametersusedforthewavelengthdispersion
formula[Eq. (B1)]oftheVerdetconstantinopticalglass,and
We thank Dr. A. Miyata for his support with the ex- the calculated and experimental Verdet constant (vcal, vexp)
perimental setup. at 404 and 638 nm.
fused quartz crown glass
Appendix A: FORMULIZATION OF THE EQUIVALENT a [10−9 /T] 400.7822,23, 401.3917,23 489.9223
CIRCUIT b [10−20 m2/T] 12.90022,23, 13.67117,23 21.75123
λ [10−9 m] 0.92622,23, 16.03317,23 101.023
0
For the circuit in Fig. 8, the induced voltage in the vcal (404 nm) 0.53122,23, 0.55217,23 0.75023
pickup coil V (t), can be described as a function of V (t) vexp (404 nm) 0.559 ± 0.008 0.761 ± 0.022
by Eq. (A1)1. In Eq. (A1), R is the combi4ned vcal (638 nm) 0.20322,23, 0.20717,23 0.26123
resistance of the attenuator andAtThTe-TtrRansient recorder: vexp (638 nm) 0.200 ± 0.007 -
R + R + RATT3RTR
R−1 = ATT1 ATT2 RATT3+RTR . (A2)
ATT-TR R ×(cid:16)R + RATT3RTR (cid:17) thedashedcurvesarethecurvesfittedbyEq. (B1). The
ATT1 ATT2 RATT3+RTR
solid curve is the calculation for crown glass using the
(cid:16) (cid:17)
AlthoughitismorecomplextoexpressV (t)asafunc- fitting parameters given in Table II. The vertical dotted
1
tion of V (t), we can describe it in a similar way by Eq. linesshowthewavelengthsinvestigatedinthisstudy: 404
3
(A3). InEqs. (A1)and(A3),thefirsttermisthefirstap- nm and 638 nm. In Table II, we listed a, b, and λ
0
proximationofthe circuitresponse,whichhasbeenused for each material, and the calculated and experimental
before. The remaining terms (which contain LL and/or Verdetconstant(vcal, vexp),at404nmand638nm. The
C ), become important at high frequencies. The domi- fitting parameters used for fused quartz (a, b, λ ), are
L 0
nant term V+(t), as referenced in section III-B, appears the result of fitting by Garn et al.17 and Ramaseshan22
1
in the fourth term of Eqs. (A1) and (A3). by Eq. (6). The open symbols in Fig. 14 show vexp
determined in this study, which coincides well (within
error) with values reported earlier.
Appendix B: THE WAVELENGTH DISPERSION OF THE
VERDET CONSTANT
1N. Miura and F. Herlach, in Pulsed and Ultrastrong Magnetic
Fields, Springer Topics inAppliedPhysicsVol. 57, editedbyF.
WediscussheretheconsistencyoftheVerdetconstant Herlach(Springer,Berlin,1985), p.247.
used to calculate B . For the wavelength dispersion of 2F.Herlach,Rep.Prog.Phys.62,859(1999).
FR
the Verdet constant in an optical glass, the experimen- 3N. Miura and F. Herlach, in High Magnetic Fields: Science
and Technology Volume 1: Magnet Technology and Experimen-
tal data is known to be well described by the following
tal Techniques, edited by F. Herlach and N. Miura (Singapore,
empirical formula:23
WorldScientific,2003)p.235,andreferencetherein.
4N.Miura,Y.H.Matsuda,K.Uchida,S.Todo,H.Mitamura,T.
π b Osada,andE.Ohmichi,PhysicaB294-295,562(2001).
v = a+ , (B1)
λ λ2−λ2 5C.M.Fowler, W. B. Garn, and R.S. Caird,J. Appl. Phys. 31,
(cid:18) 0(cid:19) 588(1960).
where a, b, and λ are fitting parameters. Fig. 14 shows 6A.I.Bykov,M.I.Dolotenko,N.P.Kolokolchikov,V.D.Selemir,
0 andO.M.Tatsenko, PhysicaB294-295574(2001).
the v(λ) of fused quartz and crown glass in the visible
7E.C.Cnare,J. Appl. Phys.37,3812(1966).
light region. The closed symbols are the data taken for 8S. Takeyama, H. Sawabe, and E. Kojima, J. Low Temp. Phys.
fused quartz by Garn et al.17 and Ramaseshan,22 and 159,328-331(2010).