Proceedings of ITC/ISHW2007

Consideration of Density Fluctuation Measurement using HeavyIon Beam Probe on Large Helical DeviceHaruhisa NAKANO1) , Takeshi IDO1) , Akihiro SHIMIZU1) , Masaki NISHIURA1) , Shinji KATO1) ,Shinsuke OHSHIMA1) , Akimitsu NISHIZAWA2) , Yasuji HAMADA1) , and LHD group1)1)2)National Institute for Fusion Science, Toki 509-5292, Japan
Tono Geoscience Center, Japan Atomic Energy Agency, Toki, 509-5102, Japan

Heavy ion beam probe (HIBP) can measure density, potential, magnetic field and their fluctuations, simultaneously, in high temperature magnetic confinement plasma. However it is well-known that local density fluctuation is measured with diﬃculty due to a path integral eﬀect. Ionization cross-sections of HIBP beam particles
aﬀects a detected beam current, which informs density. The ionization cross-sections due to not only electron
impact but also proton impact are important for MeV-range beam. Take into account the proton ionization, thepath integral eﬀect becomes twice larger than that only due to electron impact. And the path integral eﬀect candecrease by less than half, when phase diﬀerence between proton and net electron fluctuations is finite.
Keywords: heavy Ion beam probe, fluctuation measurement, path integral eﬀect, proton impact ionization

1Introduction

A fluctuation transport closely relates magnetic plasma
confinement performance. This transport is represented
fluctuations of density, temperature, and electric and magnetic fields. To measure them is one of the most important study in the magnetic plasma confinement. Heavy ion
beam probe (HIBP) is one of powerful diagnostics which
can measure density, potential, magnetic field, and theirfluctuations, simultaneously. However, density, magnetic
field and their fluctuations cannot be measured locally due
to the path integral eﬀects. Measured fluctuation amplitudes are simulated in previous paper [1, 2], and local
density fluctuation amplitude is accomplished to be reconstructed under a specific condition [3].
A HIBP system on a large confinement device needsa few MeV order beam energy, like HIBP system on Large
Helical Device (LHD) [4]. This HIBP system makes adiﬀerence from previous HIBP systems which have a few
hundreds keV in beam energy. In the case of MeV order
HIBP, beam particles ionize by mainly not only electron
impact but also proton impact [5]. In this paper, we discuss
influences of proton impact ionization on density fluctuation measurement, and simulate the path integral eﬀect inthis measurement on some conditions.

2Brief description of density fluctuation measurements of HIBPHIBP system consists of an accelerator, energy analyzer, and beam sweepers. In principle, a singly charged
heavy ion beam ( Rb+ , Cs+ and Au+ , etc ), called primarybeam, is injected into a plasma. In the plasma, the beamions are ionized to doubly (or higher) charged ions throughauthor’s e-mail: nakano@nifs.ac.jp

the collisions with plasma particles. The doubly charged
ions, called secondary beam, come out from the plasma,
and are detected with the energy analyzer.
The detected beam current Id , which has density informations, is expressed in the following form:
Id={}
I0 lsv (r∗ ) ne (r∗ )S e12 + nH+ (r∗ )S H12+∑∫ ()  (1)−ne (r)S ei + nH+ (r)S Hi + dli × exp 

i=1,2where r∗ is an ionization position, I0 is an initial primarybeam current, lsv is a sample volume length, ne and nH+
are electron and proton density, respectively. The termsii
S α12 = ⟨σ12α vthe ⟩M /vb and S α = ⟨σα vthe ⟩M /vb , where α =+(e, H ), and σe and σH+ are ionization cross-sections ofelectron and proton impacts, respectively. The terms vtheand vthi are electron and proton thermal velocity, respectively. And vb is beam particle velocity. Superscript 12of σα means ionization from single charged ion to double charged ion, and superscripts 1 and 2 of σα express
ionization from single and double charged ions to higher
charged ions, respectively. Subscript M of bracket represents a Maxwellian average. And dl1 and dl2 are length elements of primary{ and secondary beam orbits,respectively.
}
1212The term lsv (r∗ ) ne (r∗ )S e + nH+ (r∗ )S H+ and exponential
term are brightness and attenuation of detected beam current, respectively.
By taking variations and square of Eq. (1), the instantaneous normalized measured fluctuation power is represented as2 eId (r∗ )  = T 0 − S C + AC
Id (r∗ )(2)

Proceedings of ITC/ISHW2007
with expression of)2(ene (r∗ )S e12 + enH+ (r∗ )S H12+T0 = ()2 ,ne (r∗ )S e12 + nH+ (r∗ )S H12+()2 ene (r∗ )S e12 + enH+ (r∗ )S H12+SC =ne (r∗ )S e12 + nH+ (r∗ )S H12+)∫∑ (∫e×ne (ri )S ei dli + enH+ (ri )S Hi + dli ,Y1Primary Beamζ
1X(3)Secondary Beam

i=1,2AC)2∫ ∑ (∫ii
ene (ri )S H+ dli  ,ne (ri )S e dli + e= Fig. 1 Beam orbit model.

i=1,2

where a tilde expresses fluctuation.
The term enα are statistical variables, hence we willconsider an ensemble of fluctuations enα with ⟨enα ⟩E = 0,
where ⟨⟩E means an ensemble average. If ne = nH+ andene = enH+ , by taking ensemble average of Eq. (2), we obtain

2.0⟨S C ⟩E== -1.5

nn=0=

21.0e0.0-1.0⟨eξ(xi )eξ(y j )⟩E Ŝ i Ŝ j wi (xi )w j (y j )dxi dy j

1.5eAAnn(5)eeA

1.00.5i=1,2⟨AC ⟩E

n-0.5=5=100.5e

A= ⟨e
ξ2 (x∗ )⟩E∑∫=2⟨e
ξ(x∗ )eξ(xi )⟩E Ŝ i wi (xi )dxi
∑ ∑"A

1.519ne [ 10(4)with the expression on a condition of S α12 ≡ S α1 of⟨T 0 ⟩ET-3m ]2.0Te [keV]⟨eη2 (r∗ )⟩E = ⟨T 0 ⟩E − ⟨S C ⟩E + ⟨AC ⟩E

2.5e0.0
r/a

0.5

0.0
1.0Fig. 2 Profiles of electron density (solid lines) and temperature(dashed line). Ane is hollow factor in Eq. (10). A case ofAne = 0 expressed Eq. (8)i=1,2 j=1,2

where the normalized fluctuations eη(r) = e
Id (r)/Id (r) andeξ(r) = ene (r)/ne (r). The normalized ionization rate Ŝ =ne S αi a which depends on beam energy, electron and proton
temperature, and density, where S αi = S ei + S Hi + . The terma is minor radius. And the weight in the integral wi (xi ) =∂lˆi (xi )/∂xi , where lˆi = li /a. The ensemble average of two
fluctuation is represented as⟨e
ξ(x)eξ(y)⟩ = |e
ξ(x)||eξ(y)|γ(x, y)Ψ(x, y),(6)where γ(x, y) and Ψ(x, y) are coherence and phase between
fluctuations at points x and y, respectively. Equation (4)and (5) are the same formula as those of previous HIBP
system which has a few hundreds keV beam energy andbeam particles are mainly ionized by electron impact [1],thus the ionization rate in Eq. (5) is only raised by proton
impact ionization.
There are two kinds of path integral eﬀects; screening
eﬀect S C and accumulation eﬀect AC [3]. The term ⟨S C ⟩E
has negative sign in Eq. (4), and is a local path integral
eﬀect which screens the local fluctuation. The other way,
⟨AC ⟩E has positive sign in Eq. (4), and is a long range path
integral eﬀect which is the same as well-known type eﬀect
in other line integral diagnostics.

3Simulation of Measured Fluctuation

The measured fluctuation is simulated with linear
beam orbit shown in Fig. 1 as the same as previous paper[1]. Using parameter s and t, the primary and secondary
beam orbit are described asl⃗ˆ1 (s) = (0, −s + 1),l⃗ˆ2 = (t, −ζ̂t + ζ̂)

(7)

The
√ corresponding weight factors are w1 = 1, w2 =1 + ζ 2 . In the calculation, minor radius a is assumed 600
mm, and profiles of density and electron temperature T e
are assumed asneTe= ne,0 (1 + ρ4 )2[]1 ( ρ )2= T e,0 exp −,2 0.5(8)(9)

where ne,0 is calculated as the line averaged density is kept
to be 1×1019 m−3 , T e,0 = 2keV (Fig. 2), and ρ = |r/a|. The
primary and secondary beam are selected Au+ and Au2+ ,respectively. Ionization cross-sections of electron and proton impacts are calculated using Lot’z empirical formulaand results in Ref. [5], respectively. Electron temperature
contribution to the path integral eﬀect is assumed to be ignored.
First, we will examine dependence on a position offluctuation peak and coherence. The local fluctuation profile is assumed 0.1 exp[−0.5(ρ − ρ0 )2 /0.22 ], and three patterns; ρ0 = 0.0, 0.5 and 0.9, are simulated. The coherence
γ is assumed γ(x, y) = exp(−|x − y|2 /lˆC2 ), where lˆC = lC /a is
a normalized correlation length expressed with correlation
length lC . And using delta function δ(x, y), the phase difference Ψ is assumed Ψ(x, y) = cos δ(x, y) = 1. The beam

Proceedings of ITC/ISHW2007
20151050-1.0-0.50.0r/a0.5|ξ|lc = 10mm
30120300600(b)|ξ|
lc = 10mm
30120300
600

Fluctuation Level [%]Fluctuation Level [%]

(a)

15

1050-1.01.0-0.50.0r/a0.520(c)Fluctuation Level [%]201050-1.01.0|ξ|lc = 10mm
30120300600

15-0.50.0
r/a0.5

1.0Fig. 3 Dependence of path integral eﬀect on fluctuation peak position and coherence. Coherence is assumed γ(x, y) = exp(−|x − y|2 /lˆC2 ),where lˆC = lC /a is the local correlation length. No phase diﬀerence is assumed; Ψ(x, y) = 1. Radial profile of the local density
fluctuation and the measured fluctuation are indicated dashed and solid lines, respectively. (a) Edge (pattern A), (b) half of minor
radius (pattern B), and (c) center (pattern C) peaks local density fluctuation profiles.

25

20|ξ|
An = - 1.5

15= 0= 2= 5= 10e

10
50
-1.0

-0.5

0.0
r/a0.525(b)Fluctuation Level [%]Fluctuation Level [%](a)

1.020|ξ|An = - 1.5

15= 0= 2= 5= 10(c)Fluctuation Level [%]25e

1050-1.0-0.50.0
r/a0.51.0|ξ|
An = - 1.520

e= 0= 2= 5= 10

151050-1.0-0.50.0
r/a

0.5

1.0Fig. 4 Dependence of the path integral eﬀect on density profile. The correlation length is 300mm, and other parameters are the same asFig. 3 except density profile (Fig. 2). Radial profile of the local density fluctuation and the measured fluctuation are indicateddashed and solid lines, respectively.

energy is assumed to be 1.5 MeV. Figure 3 shows three profile patterns of local fluctuation eξ with several correlation
lengths; lC = 10mm, 30mm, 120mm, 300mm and 600mm.
The short and long correlation lengths are the cases of drift
wave turbulence and MHD fluctuation, respectively. The
primary beam is injected from right hand side in Fig. 3.
In all three cases, as correlation length becomesshorter, profile distortions of the measured fluctuation amplitudes from original local fluctuation amplitudes become
smaller. The measured fluctuation profiles mostly correspond to the local fluctuation profiles in the cases oflC = 10mm in upstream of the original peak. In pattern
A, the measured fluctuation amplitude in the area of upstream ( r/a < 0.6 ) remains finite amplitude ∼ 2% in thecase of lC = 600mm due to the path integral eﬀects of theplasma edge fluctuation (10%). The accumulating eﬀect islarger than the screening eﬀect in this area; S C /AC < 1.
On the other hand, in pattern C, the peak of measured fluctuation amplitude reduces and shifts to upstream as lC becomes large. The reduction of the measured fluctuation
amplitude is calculated less than half of the original local
fluctuation amplitude in the case of lC = 600mm, and represents that the screening eﬀect is larger than accumulating eﬀect; S C /AC > 1. An amount of the local maximum
shift is r/a = 0.1 in the case of lC = 600mm. The shiftarises as a result of which the accumulating eﬀect along
primary beam orbit is larger than that along the secondary

beam orbit. In pattern B, a upstream peak of the measured
fluctuation is reduced and shifts as the same as pattern A.
An apparent peak, which can be larger amplitude than theoriginal peak, can appear near the center. This is generated as the result of the screening eﬀects around two original peaks and the accumulating eﬀect originated from upstream peak of the local fluctuation. The qualitative property of the measured fluctuation profiles is the same as
HIBP system considered only electron impact ionization[1], this is because model equations, Eq. (4) and (5), arethe same formulae. However, the path integral eﬀect considered electron and proton ionization can be twice larger
than that considered only electron ionization in these case.
The dependence of measured fluctuation amplitude on
density profile is simulated. Hollow and parabolic density
profile are supposed for center heating of ECRH and NBI
plasma, respectively. Here, Eq. (8) is instead of()2 ()ne (ρ) = ne,0 1 + ρ4 1 − Ane ρ2 ,(10)where Ane is hollow factor. Hollow depth becomes deep
as hollow factor increases (Fig. 2). Figure 4 shows amplitude profiles of the measured fluctuations calculated forseveral hollow factors; Ane = −1.5, 0, 2, 5 and 10. The correlation length is assumed 300mm and other parameters
are the same as Fig. 3 except density profiles. If density
is calculated negative, density replaces suﬃciently small
positive value.

Proceedings of ITC/ISHW2007
The measured fluctuation profiles change as density
profile, even though in the same line average density (Fig.
4). In pattern A, contaminations of the measured fluctuation amplitudes in center region due to edge fluctuation
amplitudes become large as the hollow factor increases.
This is because the density around local fluctuation peakis relatively higher, then the path integral eﬀect becomelarge. For the same reason, in pattern C, distortions of measured fluctuation profiles become small as hollow factor
increases. In pattern B, the distortion is medium between
pattern A and C, and variation of the distortion is small
for hollow factor. In the three case, although the amountsof the distortions are diﬀerent, a tendency of distortion ofmeasured fluctuation profile is the same as the simulationof Fig. 3.

65

200.5(a)-0.5

Te [keV]81-0.51042
10001219

ne [103-3

4xm ]

100.3

0.25

0.15

0.05
0.1

Te [keV]60.2(b)8424Discussion

In a case of low frequency range fluctuation, like MHD
fluctuation, net electron fluctuation is the same phase as
proton fluctuation. However, in the case of higher frequency range fluctuation or the plasma including impurities, the net electron fluctuation cannot be the same phaseas proton fluctuation.
We consider the two cases that net electron fluctuations are in-phase and anti-phase to ion fluctuations,as a extreme case. Assuming infinitesimal correlation in a homogeneous plasma, γ(x, y) = δ(x, y) anddxi dyi = δ(xi − yi )dxi dyi . In the case of the net electron and proton fluctuation are in-phase, the ensemble]
2[averaged Eq. (4) is deformed, η2 = ξ T 0 − S C + AC ,{}√∑iwhere T 0 = 1, S C = 2 2πlˆC ne i=1,2 S eii + S Lossand{}√∑2i2i
2AC =2πlˆC ne i=1,2 (S ei ) + (S Loss ) L̂i . Overline
expresses constant in the homogeneous plasma. Theterms Lˆ1 and Lˆ2 are normalized lengths of primaryand secondary beam, respectively.
A path integral
coeﬃcient Φ is defined as Φ ≡ (S C + AC )/T 0 , andfurthermore, a distortion coeﬃcient Dist is defined as
Dist ≡ (−S C + AC )/T 0 . The distortion ratio indicates themeasured fluctuation distorts the local fluctuation. In thecase of anti-phase on the same assumptions, the ensemble]
2[averaged Eq. (4) is deformed, η2 = ξ T ′ 0 − S ′ C + A′ C ,()2()212
where T ′ 0 = S ei12 /S 12 + S Loss
/S 12 , S ′ C ={}√∑i)2 /S 12 and A′ C=2 2πlˆC ne i=1,2 (S eii )2 + (S Loss{}√∑2i2i22πlˆC ne i=1,2 (S ei ) + (S Loss ) L̂i .
A path integral
coeﬃcient in the case of anti-phase Φ′ is defined asΦ′ ≡ (S ′ C + A′ C )/T ′ 0 .
Figure 5 (a) shows Dist with Lˆ1 = Lˆ2 = 1 andlC = 120mm. The distortion coeﬃcient Dist = 0 means thatthe measured fluctuation amplitude can be the same as thelocal fluctuation amplitude, as a result of the screening andaccumulating eﬀects. Positive or negative Dist represent
that the measured fluctuation amplitude can increase or de-0.35

001

219-3
ne [10 x m ]34Fig. 5 (a) Contours of the distortion coeﬃcient Dist with lC =120mm and (b) the ratio of the path integral coeﬃcientsΦ′ /Φ.
crease from the local fluctuation amplitude. In the case ofne = 1 × 1019 m−3 and T e = 2keV, the distortion ratio isnegative ( Dist = −0.8 ), which is consistent with the measured fluctuation amplitude reduced at center in the caseFig. 3 (c). The ratio Φ′ /Φ shown in Fig. 5 (b) increases
as the density increase, and is 0.2 at ne = 1 × 1019 m−3
and T e = 2keV. This means that the path integral eﬀect
in the case of the anti-phase is 20 % of that in the caseof in-phase. Hence, the case of anti-phase is more local
measurement than that of in-phase.

5Summary

The path integral eﬀect on the density fluctuation measurement by use of MeV-range HIBP system is estimated.
When net electron and proton fluctuations are in in-phase,the equation of the measured fluctuation is the same formula as a few keV-range HIBP system. And, the path integral eﬀect of MeV-range HIBP system is much larger due
to the proton ionization. If the net electron and proton fluctuations are in anti-phase, the fluctuation can be measure
more locally.
[1] A. Fujisawa, H. Iguchi, S. Lee, and Y. Hamada, Rev. Sci.
Instrum. 68, 3393 (1997).
[2] A. Shimizu, A. Fujisawa, S. Ohshima, and H. Nakano, Rev.
Sci. Instrum. 76, 043504 (2005).
[3] H. Nakano, et al, Rev. Sci. Instrum. 78, 063502 (2007).
[4] T. Ido, et al, Rev. Sci. Instrum. 77, 10F523 (2006).
[5] M. Nishiura, et al, J. Plasma Fusion Research (2007) inpress.