halbach permanent magnet structures

January 30, 2009 The Future of High Performance Permanent Magnets in the USA – Dejan Trbojevic 11

Halbach Permanent Magnet Structures

•

Introduction•

Halbach segmented multipole magnet

•

Properties of the non-scaling FFAG lattice:–

Basic cell orbits–

radius, magnetic fields, aperture (orbit offsets), betatron functions, energy range, available drift space for cavities and extraction/injection …

•

Proton/carbon therapy very fast growing field–

Is there a reason to compete? Price, circumference, fastest treatment rate, scanning-(treatment length), total amount of steel…

–

Use permanent magnets might reduce the price and make use simpler?

•

Possible design of the combined function dipoles•

Summary


Introduction

•

Present and future use of permanent magnets in accelerators:–

Fermilab –

recycling antiproton ring.–

Wigglers in the light sources, –

Quadrupoles in linac, –

Quadrupoles close to the Collision point in the B-factories…

•

Possibilities in the non-scaling Fixed Field Alternating Gradient (FFAG) machines:–

Future eRHIC (Electron Ion Collider at BNL)–

Proton/carbon therapy accelerator–

Proton gantries



•

Appendix shows more details from Halbach’s publication: [K. HALBACH, “DESIGN OF PERMANENT MULTIPOLE MAGNETS WITH ORIENTED RARE EARTH

COBALT MATERIAL”, NUCLEAR INSTRUMENTS AND METHODS 169

(1980)

1-10.]

r1

r2

β(ϕ) = ϕ (N + 1).Follows directly that the largest possible real bn is obtained by choosing:



We allow discussion of a smaller than maximum possible angular size (2π/M) by making the angular size of the reference piece ε 2π/M.

From eq. (24a) follows for the fundamental harmonic for ε

= 1

(24a)

(34)

For M= 16, r2

/r1

=4 (which is still quite compact) and Br

=0.95 T (which is commercially available), one obtains an aperture field

of 1.34 T.


Halbach segmented multipole magnetJinfang Liuand and Peter DentElectron Energy Corporation


Non-scaling FFAG concept

•

Orbit offsets are proportional to the dispersion function:Δx = Dx

∗ δp/p•

To reduce the orbit offsets to +-5 cm

range, for momentum range of δp/p ~ +-

50 %

the dispersion function Dx

has to be of the order of:Dx

~ 5 cm / 0.5 = 10 cm•

The size and dependence of the dispersion function is best presented in the normalized space and by the H function:

22 χζ

βαβχ

βζ

+=

+′==

H

DDD

x

xxxx

x

x


Basic Properties of the Non-Scaling FFAG

A . Particle orbits

B. Lattice


Basic Properties of the non-scaling FFAG

- Extremely strongfocusing with smalldispersion function.

- large energy acceptance.-

tune dependence on δ- very small orbit offsets-small combined function magnets

•

Concept introduced 1999 at Montauk meeting –Trbojevic, Courant, Garren) using the light source lattice with small emittance minimized H function


Hadron cancer therapy –

fast growing field

–1 in 3 Europeans will confront some form of cancer in their lifetime.

–Cancer is the 2nd most frequent cause of death.

–Hadron therapy [protons, carbon, neutrons] is 2nd only to surgery in its success rates.

–45% of cancer cases can be treated, mainly by surgery and/or radiation therapy.

From Steve Peggs PAC07 talk:


Hadron Cancer Therapy:

Hadron (proton, carbon, neutron) therapy machines today:

synchrotrons, cyclotrons, FFAG’s, ….

Private companies producing them: IBA, Siemens, Varian-ACCEL,

Hitachi, …….

Are there reasons to get involved?

–Price might be to high?

–Size might be to large for a hospital? circumference, magnets?

–Rate for treatment could be faster?

–A total amount of steel could be smaller?

–The energy and intensity modulation could be improved?


Experimental results from: NSRL Laboratory at Brookhaven National Lab -

Adam Rusek

Ion: H+

Peak position: 26.1 cm in high density polyethylene (ρ=0.97 gr/cm3)Kinetic Energy: 205.0 MeV/nLET(in water): 0.4457 KeV/μm

Very similar to the body cell density


New facilities in Heidelberg, Pavia


Fully operating facility: Proton therapy in Massachusetts General Hospital


•

Comparable (synchrotrons ~C=60m) or smaller size (cyclotrons are smaller but definitely require large amount of steel).

•

Fast acceleration rate.•

Energy scanning simple: single turn extraction at required energy.

•

No radiation loss (cyclotrons have unavoidable activation due to losses inside of cyclotrons as well as from the raster to allow the required energy range.

•

Easy to operate because of the fixed and linear dependence of the magnetic field.

•

Small orbit offsets –

small aperture.•

Lower price? Permanent magnets.

MOTIVATION to use non-scaling FFAG for the proton/carbon accelerator:


Orbit offsets and dimensions in the cell

Φd=0.1090831

½ Φf = ½ 0.15271631

½ FD

½ QLf=44 cm/2QLd=22 cm

8 cm

38 cm

½ F

½ Φf = ½ 0.15271631

L=1.12 m14.1 cm

8.21

2.6

-2.5

-10.1-6.9


The whole ring with all elements:

r=4.278 m

24 doublets12 cavitiesThree kickers

Circumference = 26.88 m

D=8.56 m


Magnetic Properties:

LBD

= 22 cmLBF

= 30 cmGd

= -14.3 T/mGf

= 8.73 T/mBdo

= 0.804 TBfo

= 0.563 T

Values of the magnetic fields at the maximum orbit offsets:

Bd max-

= 0.804 + (-14.3)⋅(−0.0484) = 1.496 TBd max+

= 0.804 + (-14.2)⋅(0.107) = -0.715 T

Bf max+

= 0.563 + 8.73 ⋅

0.141 = 1.794 TBf max-

= 0.563 + 8.73 ⋅ (−0.102) = -0.327 T

δp/p x0ff (m)50 0.14063840 0.11109730 0.08211420 0.05381910 0.0263760 0.000000

-10 -0.025024-20 -0.048317-30 -0.069370-40 -0.087506-50 -0.101838

Offsets at F

δp/p x0ff (m) 50 0.10735440 0.08358330 0.06073720 0.03901410 0.0186620 0.000000

-10 -0.016560-20 -0.030484-30 -0.041077-40 -0.047447-50 -0.048481

Offsets at D

Minimum horizontal aperture:

Amin =0.140638+0.101838+6σ ∼ 26 cm


Heidelberg’s Gantry:

Heidelberg carbon gantry

13 m diameter

25 m length

630 tons !!


The proton gantry @ PSI

counterweight110 tons


Massachusetts General Hospital: Present Gantry


MOTIVATION: large weight

of the present gantries

•

Large Bρ=6.35 Tm for carbon ions of Ek=400 MeV/n requires large magnetic fields.

•

Presently the beam scanning requires very large magnet at the end of the gantry to accommodate parallel beams

to the patient.•

Results are: very large magnets and large weight of the transfer

line and the whole support (630 /tons). The carbon/proton cancer therapy facilities constraints are very difficult to fulfill with the warm temperature magnets.

•

This leads us to a new concept –

non-scaling light small superconducting gantry (transfer) with the scanning and focusing

above the patient.–

ADVANTAGES:-

Fixed magnetic field for the whole range of treatment-

Could be built with permanent magnets for the proton therapy-

Less expensive, smaller size


Proton Gantry with triplet and scanning magnets

(it could be built with small permanent magnets F=2 cm)

+-10 cm

Magnet dimensions, magnetic fields and gradients:L_BD

= 25 cm, GD =-33.7 T/m, Bmax

= 1.5 T + 33.7 T/m*0.012 mm = 1.9 TL_BF

= 30 cm, GF =+35.5T/m, Bmax

=-0.25 T -+ 35.5 m*0.028 mm = 1.2 T

scanner

magnified

S.A.D.=4.1 mS.A. D - effective source-to-axis distance


-5 mm + 5mm

Simulation of the particle transport through the gantry


Tracking through the gantry x-y

plane @ the end


Adjustable quadrupole

Halbach

design


Halbach segmented multipole magnetJinfang Liuand and Peter Dent

Electron Energy Corporation


V. Kashikhin and D. Harding Fermilab -

Concept:

•

Power the dipole component with permanent magnets–

Compact–

No power issues•

Power the quadrupole component with a (modified) Panofsky coil–

Compatible with rectangular aperture–

Relatively short ends


Summary

•

Permanent magnets could be used in the fast developing field of the proton cancer therapy for the magnets in the gantries as well as for the proton accelerator if it is built by the non-scaling FFAG.

•

A variation of the Halbach segmented design to produce the required combined function magnet looks like the most promising for the proton gantry application.

•

Future large electron accelerator at Brookhaven National Laboratory eRHIC (Relativistic Heavy Ion Collider) could use the

permanent magnets if magnetic field variations due to temperature variation, radiation effect, or time could be compensated.

•

There is a very good possibility that the additional gantry for the proton therapy facility in Massachusetts General Hospital will be built by using permanent magnets.


APPENDIX -

Halbach Design details

K. HALBACH, “DESIGN OF PERMANENT MULTIPOLE MAGNETS WITH ORIENTEDRARE EARTH COBALT MATERIAL”, NUCLEAR INSTRUMENTS AND METHODS 169(1980)

1-10. •

Basic equations, notation•

Halbach concept–

An example of a quadrupole

)1(

)1(

:1, 2

byV

xAB

axV

yAB

vacuuminieriyxz

y

x

i

∂∂

−=∂∂

−=

∂∂

−=∂∂

=

−==+= ϕ

yx BiBBiVAzF

+=+=)(

zdFdiBB =→ *r

Complex numbers -

Cartesian coordinate system:In vacuum two dimensional magnetic field components Bx

and By

(or Bx

and By ) can be derived from the scalar V or vector potential that only needs a component A in the z direction:

The complex potential ͢F(͢z)

The two dimensional vector B is:


APPENDIX Halbach

design –

basics:

zziIzB

o

oo −

=1

2)(*

πμ

AmVs

o7104 −= πμ

nnno

nno

no

nno

anibzbzB

zazF

==

=

−

=

=

∑

∑;)(

)(

1

1

*1

The field at location ͢zo

generated by a current filament I at location ͢z is:

(3)

The coefficients of the Taylor expansion of ͢B* and ͢F are:

The same expansions, but for n<0 is used to describe fields in region outside the magnets with:


APPENDIX Halbach

design –

basics:

• The B (H) relation of RECThe relationship between B|| and H||

in direction parallel to the easy axis is shown in Fig.1 (“easy axis”

–

alignment of the powder -

grains in the direction of the strong magnetic field).

B|| μο μ|| Η|| +Br (5a)or with γ=1/μ:H||

=

γ||

B||

/ μο

∗ Hc (5b)

In the direction perpendicular to the easy axis, the relationship between B and H :

B = μο H + Br (H /HA)μ =1/γ =1 + Br/ μο HA

B = μο μ H (6)


APPENDIX Description of the rare earth cobalt (REC) magnets in magneto static equations:

From equations 5a and 6:B μο μ∗H +Br (7a)where is the vector with magnitude of the remanent field in the direction of the easy axis and μ∗H

= μ H + μ|| H||.

From equations 5b and 6:

H

= γ ∗ B/ μο

– Hc . (7b)

If we derive H from a scalar potential, we have to satisfy div B

= 0, yielding with eq. (7a) div (μο

μ∗H) = ρ = -div Br

. (8a)If we derive B similarly from a vector potential, we get from eq. (7b) and Amperes law:

curl γ

∗ B/ μο

= j = curl Hc

. (8b)The anisotropy of the material shows up in two different ways: in the inhomogeneous terms on the right sides of eqs. (8a, b), and in the slight anisotropy associated with the weak differential permeability of REC. Because the permeabilities are so close to one, we assume, unless stated otherwise, that μ|| = μ = 1. This very good approximation, together with the assumption of constant Hc and Br, means that the material can be treated as vacuum with either an imprinted charge density div Br or an imprinted current density curl Hc.


integral that contains the magnetization itself, andnot a combination of its spatial derivatives.

APPENDIX


For a homogeneously magnetized piece of REC, Br can be taken outside the integral in eq. (15). Integrating first over x, one obtains:

APPENDIX


APPENDIX


APPENDIX: Halbach-segmented multipole magnet

halbach permanent magnet structures

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