Measurements of sub-nT dynamic magnetic field shielding with soft iron and mu-metal for use in linear colliders
C. Gohil, P. N. Burrows, N. Blaskovic Kraljevic, D. Schulte, B. Heilig
SShielding Dynamic Magnetic Fields to Sub-nT Levelsin Linear Colliders
C. Gohil
1, 2 , P. N. Burrows , N. Blaskovic Kraljevic ∗ , D. Schulte , and B.Heilig John Adams Institute, University of Oxford, Oxford, United Kingdom European Organization for Nuclear Research, Geneva, Switzerland Mining and Geological Survey of Hungary, Tihany, Hungary(September 4, 2020)
Abstract
There is an increasing need to shield beams and accelerator elements from straymagnetic fields. The application of magnetic shielding in linear colliders is discussed.The shielding performance of soft iron and mu-metal is measured for magnetic fieldsof varying amplitude and frequency. Special attention is given to characterise theshielding performance for very small-amplitude magnetic fields.
Magnetic fields can influence the operation of an accelerator. This could be a directimpact on the beam or an influence on accelerator elements. Linear colliders (describedbelow) have an unprecedented sensitivity to external dynamic (stray) magnetic fields.
The luminosity of a linear e + e − collider is [1] L = N f rep n b πσ ∗ x σ ∗ y H D , (1)where N is the bunch population, f rep is the repetition frequency, n b is the number ofbunches, σ ∗ x ( σ ∗ y ) is the horizontal (vertical) beam size at the interaction point and H D is a luminosity enhancement factor due to the electromagnetic interaction between thecolliding bunches.To achieve a large luminosity extremely small vertical beam sizes are targeted. Thesmall vertical beam size leads to a sensitivity to stray magnetic fields, which can deflect thebeams and result in a relative offset at collision. The Compact Linear Collider (CLIC) ∗ Present address: European Spallation Source, Lund, Sweden. a r X i v : . [ phy s i c s . acc - ph ] S e p
2, 3] is sensitive to stray magnetic field amplitudes of 0.1 nT [4, 5, 6, 7, 8] and theInternational Linear Collider (ILC) is sensitive to stray magnetic field amplitudes of 1 nT[9]. Ambient magnetic fields in accelerator environments exhibit fluctuations on the levelof 100 nT [8, 10]. Therefore to mitigate performance loss, a mitigation system will beessential.There are two options for such a system: an active compensation device or a passiveshielding system. An active compensation device would measure the magnetic field anduse a set of coils to compensated it. Such a device was demonstrated at an acceleratorfacility in [11]. This system stabilised a magnetic field to fluctuations of less than 10 nT.However, an active compensation device relies on accurately measuring the magnetic field.Measuring magnetic field fluctuations of 0.1 nT is challenging with current commerciallyavailable magnetometers [12]. Therefore, a passive shielding system is preferred. A shield-ing factor of approximately 10 is required to reduce a 100 nT stray magnetic field to thelevel of 0.1 nT. This section describes the current use of magnetic shields in linear colliders and theproperties that affect shielding performance.
The most common use for magnetic shields in linear colliders is for superconducting radio-frequency (SRF) cavities, which are used in the ILC [9]. SRF cavities must be cooled downto superconducting temperatures to operate, usually 2 K. If magnetic flux is trapped insidethe cavity walls during the cool down the quality factor of the cavity is reduced [13]. Amagnetic shield is used to prevent magnetic flux trapping. Studies of potential magneticshields for ILC SRF cavities are presented in [14, 15, 16].In the above application, the magnetic shield is used to shield static magnetic fields.In this work, we look at the use of magnetic shields to shield the beam from dynamicmagnetic fields. In particular, low-frequency small-amplitude magnetic fields.
An overview of magnetic shielding is given in [17]. There are two magnetic shieldingmechanisms, which are shown in Figure 1. On the left is the flux-shunting mechanism,which is effective for shielding static and low-frequency magnetic fields, and on the rightis eddy-current cancellation, which is only effective for high-frequency magnetic fields. InCLIC it is necessary to shield low-frequency magnetic fields, therefore the flux-shuntingmechanism is of interest.The flux-shunting mechanism relies on the material possessing a large permeability todraw the magnetic field away from the shielded region. Ferromagnetic materials [18] arecommonly used for this purpose. The permeability of a ferromagnetic material that isexposed to a dynamic magnetic field is given by µ ( H ) = B ( H ) H , (2)2 hat do not completely separate source and shielded regions. For closed topologies,the only mechanism by which magnetic fields appear in the shielded region ispenetration through the shield, while for open topologies, leakage may also occur.Magnetic fields may leak through seams, holes, or around the edges of the shield aswell as penetrate through it. The extent of the shield is an important factor whenconsidering open shields: the more the shield is extended, the better the shielding.However, if penetration exceeds leakage, an increase in the extent of the shield maybring little improvement in the SE. The extent of the shield plays an important rolealso for closed geometries, as it will be seen later. Besides, the shield thickness isanother key factor; if penetration is the dominant mechanism, a thicker shield resultsin improved shielding.The material parameters of the shield cause two different physical mechanisms inthe shielding of low-frequency magnetic fields: the flux shunting and the eddy-current cancellation.
The flux-shunting mechanism is determined by two conditionsthat govern the behavior of the magnetic field and the magnetic induction at thesurface of the shield: Ampere’s and Gauss’s laws require the tangential componentof the magnetic field and the normal component of the magnetic induction to becontinuous across material discontinuities. Hence, in order to simultaneously satisfyboth conditions, the magnetic field and the magnetic induction can abruptly changedirection when crossing the interface between two different media. At the interfacebetween air and a ferromagnetic shield material having a large relative permeability,the field and the induction on the air side of the interface are pulled toward theferromagnetic material nearly perpendicular to the surface, whereas on theferromagnetic side of the interface, they are led along the shield nearly tangentialto the surface. The resulting overall effect of the shielding structure is that themagnetic induction produced by a source is diverted into the shield, then shuntedwithin the material in a direction nearly parallel to its surface, and finally releasedback into the air. In Figure B.2 a , the typical behavior of a cylindrical shield placed inan external uniform magnetic field is reported.The field map refers to a structure with internal radius a ¼ : D ¼ : m r ¼
50 at dc ( f ¼ ( a ) ( b ) FIGURE B.2
Magnetic-field distribution for cylindrical shields subjected to a uniformimpressed field: ( a ) ferromagnetic shield; ( b ) highly conductive shield. MAGNETIC SHIELDING that do not completely separate source and shielded regions. For closed topologies,the only mechanism by which magnetic fields appear in the shielded region ispenetration through the shield, while for open topologies, leakage may also occur.Magnetic fields may leak through seams, holes, or around the edges of the shield aswell as penetrate through it. The extent of the shield is an important factor whenconsidering open shields: the more the shield is extended, the better the shielding.However, if penetration exceeds leakage, an increase in the extent of the shield maybring little improvement in the SE. The extent of the shield plays an important rolealso for closed geometries, as it will be seen later. Besides, the shield thickness isanother key factor; if penetration is the dominant mechanism, a thicker shield resultsin improved shielding.The material parameters of the shield cause two different physical mechanisms inthe shielding of low-frequency magnetic fields: the flux shunting and the eddy-current cancellation.
The flux-shunting mechanism is determined by two conditionsthat govern the behavior of the magnetic field and the magnetic induction at thesurface of the shield: Ampere’s and Gauss’s laws require the tangential componentof the magnetic field and the normal component of the magnetic induction to becontinuous across material discontinuities. Hence, in order to simultaneously satisfyboth conditions, the magnetic field and the magnetic induction can abruptly changedirection when crossing the interface between two different media. At the interfacebetween air and a ferromagnetic shield material having a large relative permeability,the field and the induction on the air side of the interface are pulled toward theferromagnetic material nearly perpendicular to the surface, whereas on theferromagnetic side of the interface, they are led along the shield nearly tangentialto the surface. The resulting overall effect of the shielding structure is that themagnetic induction produced by a source is diverted into the shield, then shuntedwithin the material in a direction nearly parallel to its surface, and finally releasedback into the air. In Figure B.2 a , the typical behavior of a cylindrical shield placed inan external uniform magnetic field is reported.The field map refers to a structure with internal radius a ¼ : D ¼ : m r ¼
50 at dc ( f ¼ ( a ) ( b ) FIGURE B.2
Magnetic-field distribution for cylindrical shields subjected to a uniformimpressed field: ( a ) ferromagnetic shield; ( b ) highly conductive shield. MAGNETIC SHIELDING
Figure 1:
Cylindrical shields subject to a uniform magnetic field [19]. Left: flux shunting andright: eddy-current cancellation. where H is the amplitude of the magnetic field variations and B ( H ) is the amplitude ofthe magnetic induction. The permeability is independent of a static offset provided thematerial is not close to saturation.The response of a material to a small-amplitude dynamic magnetic field is governedby Rayleigh’s law, which states the amplitude of the magnetic induction is given by [18] B ( H ) = µ i H + νH , (3)where µ i is the initial permeability and ν is Rayleigh’s constant. The permeability in theRayleigh region is given by µ ( H ) = µ i + νH. (4)In order to effectively shield small-amplitude magnetic fields, the material must possessa sufficiently high initial permeability. Two ferromagnetic materials were characterised: soft iron and a nickel-iron alloy knownas mu-metal. The magnetic shielding performance is measured with a transfer function,which is described below.
Considering a magnetic shield exposed to the time-varying magnetic field H e e j πft , where f is the frequency, t is the time, H e is the external magnetic field amplitude and j = √− H i e j (2 πft − φ ) , where φ is a phase shift introducedby the shield and H i is the internal magnetic field amplitude. The transfer function ofthe magnetic shield is given by T ( f ) = H i e − jφ H e , (5)The absolute value of T ( f ) is known as the amplitude response and the phase of T ( f ) isknown as the phase response.For simple geometries, such as an infinitely long cylinder, analytical solutions toMaxwell’s equations exist for the propagation of electromagnetic waves through magneticshields. A method for calculating the shielding factor of cylindrical shields is describedin [20]. 3 .2 Methodology A cylinder of inner diameter 5 cm, thickness 1 mm and length 50 cm was formed from softiron and another cylinder with the same dimensions were formed from mu-metal. Bothcylinders were annealed in their final form. The advertised magnetic properties of eachmaterial (provided by the supplier) are summarised in Table 1.
Property Soft Iron Mu-Metal
Initial permeability 300-500 50,000Maximum relative permeability 3,500-8,000 250,000Magnetic induction at saturation 2.15 T 0.74 TTable 1:
Advertised specifications of each material.
A three-axis Bartington Mag-13 fluxgate magnetometer [21] was used in measure-ments. The noise level of this sensor is low enough to measure magnetic field amplitudesof less than 0.1 nT. A set of Helmholtz coils [12] was used to generate a magnetic fieldexcitation at a precise frequency and amplitude. A Mag-13 sensor was placed at thecentre of the Helmholtz coils.The magnetic field H ( t ) was measured with and without a shield surrounding thesensor. In both measurements the current in the Helmholtz coils I ( t ) was simultaneouslyrecorded. A transfer function that relates the current in the Helmholtz coil to the magneticfield measured by the sensor was calculated: T IH ( f ) = P IH ( f ) P II ( f ) , (6)where P IH ( f ) is the cross power spectral density of I ( t ) and H ( t ) and P II ( f ) is the powerspectral density of I (t). The transfer function for the shield was calculated as T ( f ) = T IH, sh ( f ) T IH, no sh ( f ) , (7)where T IH, sh ( f ) is the transfer function measured with the shield and T IH, no sh is thetransfer function measured without the shield.
The transfer function of a high purity (99.9%) iron cylinder was measured with differentexternal magnetic field amplitudes. The transfer functions are shown in Figure 2. There isa clear dependence on the external magnetic field amplitude, where the shielding improveswith the amplitude. The phase response of the iron cylinder is independent of the externalmagnetic field amplitude.Figure 3 shows the measured amplitude response as a function of external magneticfield. It is clear the amplitude response tends to a constant as the external magnetic fieldis decreased.The model described in [20] can be used to fit a permeability to the transfer functionfor each amplitude. Figure 4 shows the relative permeability as a function of externalmagnetic field amplitude. The initial permeability is extrapolated by fitting a straightline to the relative permeability. An initial permeability of µ i = (204 ±
5) was measuredfor this iron cylinder, which is somewhat below the advertised value of 300-500.4 f [Hz] . . . . . | T ( f ) | µ T4.4 µ T8.8 µ T13.2 µ T 10 f [Hz] − − − − − T ( f ) [ ◦ ] µ T4.4 µ T8.8 µ T13.2 µ T Figure 2:
Transfer function of the iron cylinder for different external magnetic field amplitudes.Left: amplitude response | T ( f ) | vs frequency f . Right: phase response ∠ T ( f ) vs frequency f .Error bars are too small to be seen. − − H e [ µ T] . . . . . . . . . | T ( H e ) |
11 Hz111 Hz211 Hz411 Hz711 Hz1111 Hz
Figure 3:
Amplitude response of the soft iron cylinder | T ( H e ) | vs external magnetic fieldamplitude H e for different frequencies. Error bars too small to be seen. The chemical composition of the mu-metal used was 80% Ni, 15% Fe, 4.5% Mo, 0.4%Mn and 0.1% Si. The transfer function of the mu-metal cylinder measured with differentexternal magnetic field amplitudes is shown in Figure 5. The transfer functions appearto be similar.Figure 6 shows the permeability fitted to each transfer function in Figure 5. There is aclear linear relationship between the permeability and external magnetic field amplitude.The relative change in permeability over the range measured is much smaller for themu-metal compared to the iron. The initial permeability of the mu-metal cylinder is µ i = (55 , ± µ i = 50 , CLIC has stray magnetic field tolerances down to 0.1 nT [4, 5, 6, 7, 8]. Realising this levelrequires a very effective magnetic shield with a sufficiently high initial permeability. A5 H e [ µ T] µ r ( H e ) µ i = (204 ± ν = (4 . ± .
6) T − Measurement
Figure 4:
Relative permeability of the soft iron cylinder µ r ( H e ) vs external magnetic fieldamplitude H e : measurement (blue) and a straight line fit (orange). The errors bars were derivedfrom fitting the model described in [20] to the transfer functions in Fig. 2. f [Hz] . . . . . . | T ( f ) | µ T4.4 µ T8.8 µ T13.2 µ T 10 f [Hz] − − − − − − T ( f ) [ ◦ ] µ T4.4 µ T8.8 µ T13.2 µ T Figure 5:
Transfer function of the mu-metal cylinder for different external magnetic field am-plitudes. Left: amplitude response | T ( f ) | vs frequency f . Right: phase response ∠ T ( f ) vsfrequency f . Error bars are too small to be seen. µ T. This is shown in Figure 7. The expected amplitude of straymagnetic fields in accelerator environments is up to 100 nT [8, 10], which is an order ofmagnitude less than the excitation used in the measurement shown in Figure 7. Therefore,we can be confident that the stray field amplitude inside a mu-metal shield will be lessthan 0.1 nT for external amplitudes of 100 nT.
Mu-metal is also available in thin foils, typically of thicknesses 0.1-0.5 mm. These foils areannealed and advertised as retaining their magnetic properties after slight deformation.A set of three cylindrical shields of varying diameter D and thickness ∆ were formedfrom a mu-metal foil. The foil had the same chemical composition as the mu-metalcylinder discussed in the previous section. Figure 8 shows the transfer function of eachshield formed from the mu-metal foil. 6 H e [ µ T] µ r ( H e ) µ i = (55955 ± ν = (67 . ± .
9) T − Measurement
Figure 6:
Relative permeability of the mu-metal cylinder µ r ( H e ) vs external magnetic fieldamplitude H e : measurement (blue) and straight line fit (orange). The errors bars were derivedfrom fitting the model described in [20] to the transfer functions in Fig. 5
10 20 30 40 50 60 f [Hz] . . . . . H i ( f ) [ n T ] Figure 7:
Internal magnetic field amplitude H i ( f ) of the mu-metal cylinder with an externalmagnetic field amplitude of 1.1 µ T vs frequency f . Diameter, D [cm] Thickness, ∆ [mm] Relative Permeability, µ r . ± . , ± . ± . , ± . ± . , ± Measured relative permeability of three shields formed from a mu-metal foil.
Table 2 shows the permeability fitted to each amplitude response. The foils have arelative permeability of less than 5,000, which is very poor for mu-metal. It is likely thatthe permeability was damaged by deforming the cylinder when rolling the mu-metal foilto produce the shield, this is discussed in Sec. 4.3.3.A simple model for the shielding factor of a mu-metal shield is presented in [22]. For7 f [Hz] . . . . . . . | T ( f ) | D = 5 . ± . . D = 4 . ± . . D = 4 . ± . . f [Hz] − − − − − T ( f ) [ ◦ ] D = 5 . ± . . D = 4 . ± . . D = 4 . ± . . Figure 8:
Transfer function of three shields formed from a mu-metal foil. Left: amplituderesponse | T ( f ) | vs frequency f . Right: phase response ∠ T ( f ) vs frequency f . An externalmagnetic field amplitude of 1.1 µ T was used. The 0.2 mm thick shield was formed with twolayers of foil. Error bars are too small to be seen. a single layer, the amplitude response is given by T = Dµ r ∆ . (8)This model does not include shielding via the eddy-current cancellation mechanism. Forsmall external magnetic field amplitudes, the relative permeability in Eq. (8) is replacedwith the initial permeability. The measured amplitude response for the different mu-metalfoils is roughly consistent with Eq. (8). This section describes various considerations for using the above materials to shield mag-netic fields in linear colliders. The factors that affect shielding performance are alsodiscussed.
A beam pipe is used to contain the vacuum in an accelerator. In linear colliders, theytypically consist of a few millimetres of steel and a 10-100 µ m inner copper coating tomitigate long-range wakefields.The impact of stray magnetic fields can be mitigated by preventing them from reachingthe beam. This can be achieved by surrounding the beam with a shield or surroundingthe sources with a shield. The beam pipe is usually the closest component to the beam.Therefore, surrounding the beam pipe is the safest option because it prevents stray fieldsfrom all external sources reaching the beam. To shield the sources, they must first beidentified and the feasibility of surrounding them with a shield must be studied.Mu-metal is a good candidate material to wrap around the beam pipe. Since it couldbe wrapped around the beam pipe if deemed necessary after the accelerator has beenconstructed. Alternatively, a mu-metal layer could be incorporated into the beam pipedesign and the entire beam pipe could be annealed in its final form, which would ensurea good shielding performance. 8 .2 Magnets Beam pipes are typically formed from non-magnetic materials because they run throughthe aperture of magnets. They should not impede the magnetic field generated by amagnet, which is used to guide the beam.The sensitivity to stray magnetic fields in linear colliders comes from the long driftsbetween magnets. Therefore, only the drifts need to be shielded, which avoids the problemof shielding inside the magnets.Large static magnetic fields saturate ferromagnetic materials. Once a material issaturated, it is no longer effective as a magnetic shield. Depending on the requiredinternal field level, this property enables the possibility of replacing the steel in a beampipe with soft iron. Inside a magnet the soft iron beam pipe will be saturated and willnot impede the magnetic field, whereas in the drifts the soft iron beam pipe will shieldthe beam.
Factors that affect the performance of magnetic shields are discussed in [23]. The factorsthat affect the shielding performance of dynamic magnetic fields are summarised below.
Eq. (4) is valid provided a static magnetic field does not saturate the material. if thematerial is saturated, its permeability and shielding performance drops. Using Eq. (8) itis straightforward to show a mu-metal shield will not saturate provided B s > D ∆ H, (9)where B s is the magnetic induction at saturation. The magnetic induction for the mu-metal used in this work is B s = 0 .
74 T (see Table 1). The dominant static magnetic field inan accelerator environment is typically the Earth’s magnetic field, which is approximately20-70 µ T [24]. Assuming 50 µ T for the Earth’s magnetic field, this requires a shieldgeometry that satisfies D/ ∆ < , D/ ∆ between 10 and 1000.Alternatively, an additional outer layer can be included in the shield, which has ahigher magnetic induction at saturation, e.g. a nickel-iron alloy with a lower nickel contentthan mu-metal [25]. The outer layer will attenuate the static magnetic field and ensurean inner mu-metal layer does not saturate. Soft ferromagnetic materials are often annealed in a dry hydrogen environment after beingbent into their final form. This removes impurities from the material and alters the crystalstructure of the material, which allows magnetic domains to move freely [26, 27]. As aresult, the permeability of the material is significantly increased [28, 29, 30].
It is well known that mechanical stress, deformation and shock can significantly reducethe permeability of a ferromagnetic material [31, 32]. The damage can be reversed by re-annealing the shield, which can increase the permeability by an order of magnitude [30].9u-metal requires hydrogen annealing at very high temperatures (above 1000 ◦ C [30])which means re-annealing in the accelerator tunnel impractical. The sample should behandled with care after annealing to avoid performance loss.
It was observed in [15, 30] that the shielding factor of a mu-metal shield degrades at verylow (superconducting) temperatures. This is only a concern for accelerators that operateat superconducting temperatures, such as the ILC. CLIC operates at room temperature,which means the degradation of shielding at low temperatures is not a concern.
The behaviour of the permeability for very small-amplitude magnetic fields (Rayleigh’slaw) has been verified. It is possible to shield extremely small-amplitude magnetic fields,down to the level of 0.1 nT, with mu-metal. Mu-metal is sensitive to permeability lossfrom mechanical stress and deformation. It should be handled with care after annealing.A simple formula (Eq. (8)) was verified for calculating the transfer function of mu-metal.There is an increasing need to shield beams in accelerators from external magneticfields, in particular for future linear colliders. In this paper, we have confirmed experi-mentally that mu-metal is a viable material that can be used to shield dynamic magneticfields to amplitudes of less than 0.1 nT. This is particularly important for CLIC whichrequires the stray fields experienced by the beam do not exceed 0.1 nT. A mu-metal shieldhas been included in the design of CLIC for this purpose [8].
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