Consideration of magnetic field fluctuation measurements in torus plasma with a heavy ion beam probe

The article proposes a method of magnetic fluctuation measurement using a heavy ion beam probe sHIBPd in an axisymmetric torus configuration. The method is based on the detection of the toroidal position snot velocityd of the secondary beam in the analyzer. However, the method needs careful consideration with respect to path integral fluctuations along the probing beam orbit to evaluate local magnetic fluctuation, similarly to density fluctuation measurements with a HIBP. Here, we present an analytic formula to estimate and calculate the path integral effects for different fluctuation patterns in the profile, the correlation length, the radial wavelength, and the poloidal mode number. As a result, it is found that a large distance between the plasma and the detector lessens the importance of the path integral effect, and that local fluctuation of the magnetic field can be properly detected with a HIBP. © 2005 American Institute of Physics. fDOI: 10.1063/1.1889230 g


I. INTRODUCTION
Heavy ion beam probes ͑HIBPs͒ have been applied to many kinds of plasma confinement devices, such as mirrors ͓Tandem Mirror Experiment ͑TMX͒ 1 , GAMMA10 2 ͔, a bumpy torus ͓Nagoya Bumpy Torus ͑NBT͒ 3 ͔, a reversed field pinch ͓Madison Symmetric Torus ͑MST͒ 4 ͔, tokamaks ͓Impurity Study Experiment ͑ISX-B͒ 5 , Texas Experimental Tokamak ͑TEXT͒ 6 , Japan Institute for Plasma Physics Tokamak ͑JIPP-TIIU͒ 7 , T-10 8 , JAERI's Fusion Torus ͑JFT-2M͒ 9 ͔, and stellarators ͓Advanced Toroidal Facility ͑ATF͒ 10 , Compact Helical System ͑CHS͒ 11 , Large Helical Device ͑LHD͒ 12 ͔.This is because a HIBP has the unique capability of being able to simultaneously sense potential, density, and magnetic fields in the interiors of high temperature plasmas.As well, the highly temporal and spatial resolution with a HIBP allows for the detection of the fluctuations of these physical quantities.Until now, HIBPs have been mainly used to measure density and potential fluctuations 13 and the potential profile, and the results have contributed to the clarification of phenomena associated with plasma confinement, e.g., anomalous transport, barrier formation, 14 bifurcation and, transition. 15However, few attempts have been made to use a HIBP to measure density profile, 16 magnetic field profile, and magnetic field fluctuations, 17,18 even though measurements of these physical quantities would be fascinating.
The measurement of a magnetic field is very attractive, because few diagnostics can measure the magnetic field in plasma.With HIBP measurements, the variation in the structure of a magnetic field is obtained from changes in beam orbits; however, it is difficult to ascertain the full orbit change.On the other hand, it is easy to measure the toroidal beam displacement at the detector, a displacement that is largely caused by magnetic fluctuation localized to the HIBP sample volume in an axisymmetric toroidal device.However, the problem is that some of the beam displacement reflects fluctuation along the beam orbit; a situation that is similar to that of density fluctuations.There are a number of reports regarding this contamination, called the "path integral effect" [19][20][21] ; however, there are few reports regarding the path integral effects of magnetic field fluctuation measurements with a HIBP.
In this article, a model of the path integral effect is presented after a brief description of HIBP diagnostics.The model is used with several assumed profiles of magnetic fluctuations to predict the motion of the HIBP ions at the detector, and to determine the component of the signal due to local magnetic fluctuation at the sample location and the component due to the path effect.It is shown that the sensitivity to local fluctuations relative to the path effect can be improved by choosing a suitable position for the energy analyzer.The feasibility of the method is discussed, in terms of the minimum detectable fluctuation level with this method.plasma.The doubly ionized beam ͑or secondary beam͒ comes out from the plasma and is detected at the energy analyzer.Usually a Proca and Green-type analyzer 22 is used as the energy analyzer, and the beam is detected with a socalled split plate detector ͓see Fig. 1͑b͔͒.The detected beam intensity contains information on the plasma density at an observation point, while the vertical and horizontal movements on the detector plates give information on the potential and the magnetic field, respectively.
If the magnetic configuration is axisymmetric, as it is for an ideal tokamak, then the horizontal movement can be described in an analytic form, since the canonical momentum of the toroidal direction is conserved.The relation of momentum conservation along the primary orbit is written as ͑1͒ Here, m is the beam ion mass, R the major radius, the toroidal angle, dot means the time derivative, q the charge of an electron, and A the toroidal component of the vector potential.The right hand side shows an initial value, and A g , R g are the toroidal component of the vector potential and the major radius at the beam gun ͑or beam injection point͒, respectively.For conciseness, the initial velocity in the toroidal direction is assumed to be zero, i.e., ˙g = 0. On the other hand, the law of conservation along the secondary orbit is expressed as follows: where R s and A s represent the major radius and the vector potential at an observation point, respectively.After ionization, the beam ions experience an increase in the vector potential by qR s A s .The toroidal displacement of the beam at the detector position can be obtained by integrating these two equations.Then, the toroidal angle displacement can be expressed in the following form: where is the magnitude of the velocity of a beam ion, which is assumed to be constant; namely, the effect of the plasma potential is neglected, and t d is the time at which a beam ion reaches the detector position.The integral of time, t, is replaced by an integral about path length, ᐉ, in this equation.The terms ᐉ s and ᐉ d are the path length on the beam orbit from the beam gun to the observation point and from the beam gun to the detector position, respectively.The fluctuated component of the toroidal angle of beam displacement can be written as Therefore, the fluctuation of the beam movement on the detector plate can be estimated using this formula by multiplying the major radius of the detector position R d by the toroidal angle displacement ˜D.

B. Calculation model
For normalization and to use nondimensional variables in the calculation, Eq. ͑4͒ is transformed into the following form: The parameters R 0 and B 0 are the major radius and toroidal magnetic field strength that characterize an experimental device, respectively.The function ␣ ˜ is the normalized vector potential, of which the maximum is one, with ␦ 0 being the maximum level of ˜ is a function of the position vector r ˆ; r ˆs and r ˆg are the position vectors at the observation point and at beam gun, respectively.ˆD is the beam angle displacement to the toroidal direction normalized by R 0 ␦ 0 / r L .Then, the fluctuation power is written as FIG. 1. ͑a͒ A schematic view of a HIBP system consisting of a beam injector, an energy analyzer, and beam sweep plates, and the assumed orbits in our calculations.The rectangular termed electric field means a region of a sweep plate to control the primary beam orbit and to change the observation point.͑b͒ A schematic view of a split plate detector set in the energy analyzer to detect the secondary beam.

͑6͒
Here the bracket ͗¯͘ means the ensemble average, and the symbols are In these equations, r ˆ1 , r ˆ1 Ј, r ˆ2 , r ˆ2 Ј, r ˆare the functions of ᐉ ˆ1 , ᐉ ˆ1 Ј, ᐉ ˆ2 , ᐉ ˆ2 Ј, ᐉ ˆ, respectively.A 1 represents the local fluctua- tion power of the magnetic field to be observed, while the other terms are considered contaminations; A 2 is the local fluctuation power at the beam gun, A 3 is the correlation between two local A terms, B i the cross terms between the fluctuation in the orbits and at local points, and C i the cross terms between primary and secondary orbits.G 1 and G 2 are geometrical factors which only depend on the beam orbit.

C. Model of cross correlation terms
In Eqs.͑7͒-͑18͒, the path integral terms can be evaluated if the cross correlations with the vector potentials are given.The correlation term of ͗␣ ˜͑r ˆi͒␣ ˜͑r ˆj͒͘ is assumed as ͗␣ ˜͑r ˆi͒␣ ˜͑r ˆj͒͘ = P͑r ˆi,r ˆj͒␥͑r ˆi,r ˆj͒⌿͑r ˆi,r ˆj͒.͑19͒ Here, P is the product of the amplitude of normalized fluctuations ␣ ˜͑r ˆi͒, and ␣ ˜͑r ˆj͒, ␥ and ⌿ are the coherence and the cosine of the phase difference, respectively, between these fluctuations.A radial profile of amplitude of ␣ ˜͑r ˆi͒, which is written as P ˆi in this article, is assumed to be, exp͑ −͑r ˆ− r ˆ0͒ 2 / r ˆw 2 ͒ in plasma; r is the minor radius, and r 0 is the radial position where the amplitude is a maximum; r w is the width of the amplitude.The hat symbol of these variables means normalization by R 0 .Outside the plasma, the radial profile of amplitude of ␣ ˜ is assumed to be exp͑−͑a − r ˆ0͒ 2 / r ˆw 2 ͒ • ͑a ˆ/ r ˆ͒m , as is seen in Ref. 17, where a ˆis the minor radius of outermost magnetic surface normalized by R 0 , m is the poloidal mode number.
The product of the fluctuation amplitude, P, is expressed as follows: The coherence ␥ and the cosine of the difference in the phase ⌿ between the fluctuations are assumed as Here, ͉r i − r j ͉ is the distance between two points under consideration, ᐉ c the correlation length, m the poloidal mode number, i , j the poloidal angles, r i , r j the minor radii, r the wavelength of fluctuation to the radial direction.The hat of these variables means normalization by R 0 as described above.We change the values of r ˆ0, ᐉ ˆc, m, ˆr and investigate the effect of these parameters on the path integral terms in the function, ˆD.

A. Path integral effects for fluctuation patterns
Here we calculate path integral effects or contaminations for three fluctuation patterns to show the possibility of magnetic field measurements with a HIBP.The assumed HIBP geometry and orbits for the calculation are shown in Fig. 1͑a͒; The beam injection point ͑R ˆg , Z ˆg͒ is ͑1.0, 0.7͒, the detection point ͑R ˆd , Z ˆd͒ is ͑1.7, 0.05͒ respectively.The cross section of the torus is circular, and the toroidal magnetic field is proportional to 1 / R ˆ.The ratio of the major radius to the Larmor radius, R 0 / L , is 2.3.Figure 2 upper shows assumed fluctuation patterns of vector potential ␣ ˜͑r ˆ͒ in the power function, P ˆi.Here, we define P ˆi ϵ exp͑−͑r ˆ− r ˆ0͒ 2 / r ˆw 2 ͒, and consider three cases for r ˆ0 , r ˆw: ͑A͒ r ˆ0 = 0.05, r ˆw = 0.03, the fluctuation profile has a peak near the center, ͑B͒ r ˆ0 = 0.11, r ˆw = 0.05, a peak at the half radius, ͑C͒ r ˆ0 = 0.17, r ˆw = 0.05, a peak near the edge.In order to give a perspective, the lower part of Fig. 2 shows the calculation results of the angle displacement, ͱ ͗ ˆD 2 ͘, and the local term ͱ A 1 , which are shown as thick and thin lines, respectively.The assumed parameters are m =2, ᐉ ˆc = 0.1, ˆr = 0.1 in this calculation.In case ͑A͒, no significant contamination can be found in either the central or the outer regime.This is because ␣ ˜ Ϸ 0 is valid except in the central regime and the path integral becomes small for the orbit to observe the outer regime of the plasma.However, in case ͑C͒ the path integral effect is large in the central regime.The beam orbit to observe the plasma center passes through the outer regime of the plasma where ␣ ˜ has a significant value.
The sensitivity of the path integral effects is examined with respect to variations of the fluctuation parameters, correlation length, radial wavelength, and poloidal mode numbers.First, the effect of the correlation length ᐉ ˆc is investigated; ᐉ ˆc is changed from 0.01 to 0.2.The other parameters, ˆr, m are fixed to 0.1, and 2, respectively.The calculated angle displacements for the profiles of fluctuation are shown in Fig. 3͑a͒.The square root of the local term, ͱ A 1 , is also expressed as thin solid lines.In case ͑A͒, the difference between the beam angle displacement and the local term is small for any value of ᐉ ˆc.In cases ͑B͒ and ͑C͒, if ᐉ ˆc ഛ 0.01, the beam angle displacement shows good agreement with the local term at the peak of the fluctuation.However, around the center of the plasma in case ͑C͒ the beam angle displacement does not show agreement with the local term.
Second, the results of a scan of ˆr are shown in Fig. 3͑b͒, and are similar to a scan of ᐉ ˆc.In this scan, m and ᐉ ˆc are fixed to 2 and 0.1, respectively.In case ͑A͒, the path integral effect is small; however, in case ͑C͒ it becomes larger.As the radial wavelength becomes smaller, the results show that the path integral effect becomes smaller.The path integral contribution around the center is more sensitive to ˆr than ᐉ ˆc.If ˆr ഛ 0.01, the path integral terms become small in all cases from ͑A͒ to ͑C͒.For ˆr ഛ 0.01, the difference that remains near the plasma center in case ͑C͒ is due to the effect of the local term at the beam gun, A 2 .
Finally, the results of a scan of m are shown in Fig. 3͑c͒.The poloidal number m is changed from 1 to 3. In this case, ᐉ ˆc, ˆr are fixed to 0.1, 0.1, respectively.In case ͑A͒, the difference of the beam angle displacement and the local term is small, and its dependence on m is also very small.However, in case ͑C͒, the difference at the plasma center for m = 1 is very large.The angle between a primary and a secondary beam orbit is about / 4 radian in our case.When m =1, this angle corresponds to only one fourth of the wavelength; therefore, a reduction of the path integral cannot be expected.For m = 2, the angle corresponds to one half of the wavelength, and so a reduction of path integral can be expected.
It is interesting to determine which terms are large in path integral terms.The profiles of these terms are shown in Fig. 4. In this case, the parameters for fluctuation, ᐉ ˆc, ˆr and m are set to 0.1, 0.1 and 2, respectively.The largest term in path integral terms is B 2 , which is the first order path integral term on the secondary beam path.Since the charge number of the secondary beam is twice of that of the primary beam, in this case; therefore B 1 is cancelled out with C 2 .C 1 , C 3 are small but cannot be ignored.Other terms are very small and can be ignored, except A 2 in case ͑C͒.Because the amplitude of the fluctuation is proportional to 1 / r ˆm outside of the plasma, the amplitude at the beam gun does not become sufficiently small in case ͑C͒.In the region near the plasma center, this term is the dominant one for beam angle displacement.

B. Effect of diagnostic position on magnetic fluctuation measurement
Equation ͑6͒ includes the geometrical factors G 1 and G 2 .The squares of these factors appear in A 1 and A 2 , and the factors are linear in B 1 , B 2 , B 3 , and B 4 .Hence, as G 1 becomes larger, the term A 1 increases more rapidly than B 1 , B 2 and other terms.Thus, G 1 has the effect of magnifying the contribution of the local term A 1 to the fluctuation amplitude ˆD 2 , and this leads to the relative reduction of path integral effects.The factor G 1 can be enhanced by setting the detector location farther away from the plasma, because the length of the path then becomes longer.Note ␣ ˜ becomes negligibly small outside of the plasma; hence, the integral terms including ␣ ˜ do not significantly contribute.
To investigate this geometrical ͑or HIBP configuration͒ effect, the path integral effect is calculated for several cases in which the major radii of the diagnostics points are different.The calculation results are shown in Figs.5͑a͒ and 5͑b͒.The parameters of fluctuation are, ᐉ c = 0.1, r = 0.1, m =2.We consider two cases, ͑B͒ and ͑C͒, because in case ͑A͒ the path integral effect is already small.In Fig. 5͑a͒ The geometrical factor G 1 is the integral of 1 / R ˆ2; therefore, for a larger value of R ˆan improvement of the path integral effect cannot be expected since the rate of G 1 rising becomes slower.In fact, this tendency is seen for the region of R ˆജ 2.3 in Fig. 5͑b͒.Consequently, R ˆϷ 2.3 is the best detector position for the accurate measurement of magnetic field fluctuation amplitudes.

IV. DISCUSSION
We have investigated here the feasibility of magnetic field fluctuation measurement using HIBPs by performing the calculation of beam movements for three assumed patterns of magnetic fluctuation.The findings are as follows: ͑1͒ The path integral effect is so small that the fluctuation profile can be deduced, in cases where the magnetic field fluctuation is localized in the plasma core, independently of the fluctuation properties, e.g., ᐉ ˆc, ˆr, m. ͑2͒ However, if the fluctuation is localized in an outer region of the plasma, the path integral effect becomes larger; particularly, a false signal appears in the central regime of the plasma.͑3͒ The path integral effects can be lessened if the detector position is kept as far as possible from the plasma.
The effect of magnetic field shielding due to a vacuum vessel is not considered in our calculations.If this shielding effect is taken into account, the path integral contribution associated with A 2 , the fluctuation of the vector potential at the injection point, should be smaller.Actually in the case of edge-localized fluctuations, the signal distortion in the plasma core comes mainly from this effect.In Fig. 2, the false angle displacement in case ͑C͒ is reduced from 0.13 to 0.052 at the center if the term A 2 is neglected.Thus, the path integral effect is expected to be negligible for a magnetic field fluctuation localized in an outer region.
In HIBP measurements, a split plate detector is used to measure the beam movement in the energy analyzer.A schematic view of the detector is shown in Fig. 1͑b͒.In this detector, the horizontal beam displacement, d, can be expressed as d = ͑⌬w /2͒͑i R − i L ͒ / ͑i R + i L ͒, where i R and i L are the beam currents on the right and left plates, and ⌬w is the horizontal beam width.With this method, the beam movement of ϳ50 m can be detected, with an assumption of ͑i R + i L ͒ = 100 nA, ⌬w = 5 mm, and a minimum detectable current of 1 nA.
From the formula to describe the beam displacement d min = ͱ ͗ ˆD 2 ͘ ϫ ͑R 0 / r L ͒␦ 0,min R d , the minimum detectable fluc- tuation ␦ 0,min (=A ˜ / ͑B 0 R 0 ͒) is 2.6ϫ 10 −5 for the case of R 0 =1 m, r L Ϸ 0.44 m, R d = 1.7 m, ͱ ͗ ˆD 2 ͘Ϸ0.5.This assump- tion corresponds to the case of a 70 keV cesium beam for a magnetic field strength B t = 1 T. Therefore, quite high sensitivity can be attained in magnetic field fluctuation measurements with HIBPs.
In conclusion, we have presented considerations of the path integral effects in magnetic field fluctuation measurements with a HIBP.We show that if the HIBP geometry is appropriately chosen, path integral effects can be negligible for a wide variety of fluctuation patterns; and it is better if the beam injection and detection points are located at a sufficient distance from the plasma.The results detailed are valid for axisymmetric magnetic configurations, and further consideration will be necessary for more realistic cases of nonaxisymmetrical configurations including toroidal ripples or stellarator magnetic configurations.This method, however, is quite promising since it provides a high sensitivity for magnetic fluctuations, less than 10 −4 , with high resolutions of space ͑ϳ mm͒ and time ͑ϳ s͒.

2 FIG. 2 .
FIG. 2. Upper: Profiles of normalized fluctuation amplitude, P ˆi = exp͑−͑r − r ˆ0͒ 2 / r ˆw 2 ͒; r ˆ0 and 0 mean the minor radius normalized by R 0 and the minor radius normalized by a, respectively, where the profile is a maximum.The profiles of three cases in which r ˆ0 , r w = ͑A͒ 0.05, 0.03, ͑B͒ 0.11, 0.05, ͑C͒ 0.17, 0.05, are shown.Lower: Profiles of normalized beam angle displacement to the toroidal direction, ͱ ͗ ˆD 2 ͘ ͑thick solid lines͒ and the square root of the local term A 1 in Eq. ͑7͒ ͑thin solid lines͒.The poloidal mode number m = 2, correlation length ᐉ ˆc = 0.1, radial wave length ˆr = 0.1 are assumed in the calculation.
, the profile of the beam angle displacement ͱ ͗ ˆD 2 ͘, and the local term A 1 0.5 are shown for the major radius of the diagnostic point R = 1.53, 1.87, 2.38.The magnitude of displacement is larger and the local term becomes dominant as the major radius of the detection point increases.Figure 5͑b͒ shows the dependence of individual path integral terms on the detector position.All terms are normalized by the square root of the local term, A 1 0.5 , and their absolute values are shown in the figure.The path integral contribution ͑A 1 − ͗ ˆD 2 ͒͘ 0.5 / A 1 0.5 is shown as a solid line.In case ͑B͒, the values of these terms at = 0.55 are shown, where the fluctuation amplitude is maximum, and in case ͑C͒ at = 0.85.For R ˆജ 1.7, all terms decrease with the increase of R ˆ, since G 1 becomes larger, while B 2 0.5 / A 1 0.5 is increased when R ˆis changed from 1.5 to 1.7.

FIG. 4 .
FIG. 4.An example of individual contributions of path integral components, from B 1 to C 3 , in normalized angle displacement.The beam angle displacement ͗ ˆD 2 ͘ is shown as a thin solid line.The largest contribution in path integral terms comes from B 2 , which is the first order path integral term on the secondary beam path.