29/07/2014
Accelerator Magnets
Neil Marks,
STFC- ASTeC / U. of Liverpool,
Daresbury Laboratory,
Warrington WA4 4AD,
U.K.
Tel: (44) (0)1925 603191
Fax: (44) (0)1925 603192
n.marks@stfc.ac.uk
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Contents
Theory
•
Maxwell's 2 magneto-static equations;
With no currents or steel present:
•
Solutions in two dimensions with scalar potential (no currents);
•
•
Cylindrical harmonic in two dimensions (trigonometric formulation);
Field lines and potential for dipole, quadrupole, sextupole;
Introduction of steel:
•
Ideal pole shapes for dipole, quad and sextupole;
•
•
Field harmonics-symmetry constraints and significance;
Significance and use of contours of constant vector potential;
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Contents (cont.)
Three dimensional issues:
•
•
Termination of magnet ends and pole sides;
The ‘Rogowski roll-off’
Introduction of currents:
•
Ampere-turns in dipole, quad and sextupole;
•
•
Coil design;
Coil economic optimisation-capital/running costs;
Practical Issues:
•
Backleg and coil geometry- 'C', 'H' and 'window frame' designs;
•
FEA techniques - Modern codes- OPERA 2D and 3D;
•
Judgement of magnet suitability in design.
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Magnets - introd uction
Dipoles to bend the beam :
Quad rupoles to focus it:
Sextupoles to correct
chrom aticity:
We shall establish a formal
approach to describing
these magnets.
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N o cu rrents, no steel:
.B = 0 ;
Maxw ell’s equations:
H=j;
Then we can put:
B = - 
So that:
2 = 0
(Laplace's equation).
Taking the two dimensional case (ie constant in the z
direction) and solving for cylindrical coordinates (r,):
 = (E+F )(G+H ln r) + n=1 (Jn r n cos n +Kn r n sin n
+Ln r -n cos n  + M n r -n sin n  )
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In practical situations:
The scalar potential sim plifies to:
 = n (Jn r n cos n +Kn r n sin n),
w ith n integral and Jn ,Kn a function of geom etry.
Giving components of flux density:
Br = - n (n Jn r n-1 cos n +nKn r n-1 sin n)
B = - n (-n Jn r n-1 sin n +nKn r n-1 cos n)
Then to convert to Cartesian coord inates:
x = r cos ;
y = r sin ;
and
Bx = - / x;
By = - / y
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Significance
This is an infinite series of cylind rical harm onics;
they d efine the allow ed d istributions of B in 2
d im ensions in the absence of currents w ithin the
d om ain of (r,).
Distributions not given by above are not physically
realisable.
Coefficients Jn , Kn are d eterm ined by geom etry (rem ote
iron bound aries and current sources).
N ote that this form ulation can be expressed in term s of
com plex field s and potentials.
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Dipole field n=1:
Cylindrical:
 =J1 r cos  +K1 r sin .
Br = J1 cos  + K1 sin ;
B = -J1 sin  + K1 os ;
Cartesian:
 =J1 x +K1 y
Bx = -J1
By = -K1
So, J1 = 0 gives vertical dipole field:
 = c o n s t.
B
K1 =0 gives
horizontal
dipole field.
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Quad rupole field n=2:
Cylindrical:
 = J2 r 2 cos 2 +K2 r 2 sin 2;
Br = 2 J2 r cos 2 +2K2 r sin 2;
B = -2J2 r sin 2 +2K2 r cos 2;
J2 = 0 gives 'normal' or
‘upright’ quadrupole
field.
Line of constant
scalar potential
K2 = 0 gives 'skew'
quad fields (above
rotated by /4).
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Cartesian:
 = J2 (x2 - y 2)+2K2 xy
Bx = -2 (J2 x +K2 y)
By = -2 (-J2 y +K2 x)
Lines of flux
density
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Sextupole field n=3:
Cylindrical;
 = J3 r 3 cos 3 +K3 r 3 sin 3;
Br = 3 J3r 2 cos 3 +3K3r 2 sin 3;
B= -3J3 r 2 sin 3+3K3 r 2 cos 3;
-C
+C
+C
-C
-C
+C
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Cartesian:
 = J3 (x3-3y 2x)+K3(3yx2-y 3)
Bx = -3J3 (x2-y 2)+2K3yx
By = -3-2 J3 xy +K3(x2-y 2)
J3 = 0 giving ‘upright‘
sextupole field.
Line of constant
scalar potential
Lines of flux
density
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Variation of By on x axis (upright field s).
By
Dip ole;
constant field :
x
Qu ad ru p ole;
linear field
By
x
Sextu p ole:
qu ad ratic variation:
100
By = G Q x;
G Q is quadrupole gradient
(T/m).
By
x
By = G S x2;
G S is sextupole gradient
(T/m 2).
0
-20
0
20
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Alternative notation:
(used in most lattice programs)

B (x)
= B  
k
n
n 0
x
n
n!
magnet strengths are specified by the value of kn;
(normalised to the beam rigidity);
order n of k is different to the 'standard' notation:
dipole is
quad is
n = 0;
n = 1; etc.
k0 (dipole)
k1 (quadrupole)
m-1;
m-2;
k has units:
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etc.
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Introd u cing iron yokes and poles.
What is the id eal pole shape?
• Flux is norm al to a ferrom agnetic surface w ith infinite :

curl H = 0
therefore  H .d s = 0;
in steel H = 0;
1
therefore parallel H air = 0
therefore B is norm al to surface.
• Flux is norm al to lines of scalar potential, (B = - );
• So the lines of scalar potential are the id eal pole
shapes!
(but these are infinitely long!)
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Equations of id eal poles
Equations for Id eal (infinite) poles;
(Jn = 0) for ‘upright’ (ie not skew ) field s:
D ipole:
y=  g/ 2;
(g is inter-pole gap).
Quadrupole:
xy= R2/ 2;
Sextupole:
3x2y - y 3 = R3;
R
R is the ‘inscribed radius’ of a multipole magnet.
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‘Pole-tip’ Field
The radial field at pole centre of a m ultipole m agnet:
BPT = G N R(n-1) ;
Quad rupole: BPT = G Q R; sextupole: BPT = G S R2; etc;
Has it any significance?
- a quad rupole R = 50 m m ; G Q = 20 T/ m ; BPT = 1.0 T;
i)
Beam line – round beam r = 40 mm;
pole extends to x = 65 mm; B y (65,0) = 1.3 T; OK  ;
ii)
Synchrotron source – beam ± 50 mm horiz.; ± 10 mm vertical;
pole extends to x = 80 mm; By (80,0)= 1.6 T; perhaps OK ??? ;
iii)
FFAG – beam ± 65 mm horiz.; ± 8 mm vertical;
Pole extends to x = 105 mm; By (105, 0) = 2.1 T .
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Com bined function m agnets
'Com bined Function Magnets' - often d ipole and
quad rupole field com bined (but see later slid e):
A quad rupole m agnet w ith
physical centre shifted from
m agnetic centre.
B

 
n =  
0




 B

 ,
  x 



Characterised by 'field ind ex' n,
+ve or -ve d epend ing
cuervatu
re of
w h eof
r egrad
th e r a d iu s o f c ur visa rad
tu r iu
e so of
f th
beam
a nthe
d B is th e c e n tr a l d ip o l
 is ient;
on d irection
0
beam ;
d o not confuse w ith harm onic n!
Bo is central d ip ole field
I f p h y s ic a l a n d m a g n e tic c e n tr e s a r e s e p a r a te d b y X
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T hen
B
th e r e f o r e
X
0
0
0
 B
 X ;
0
 x 
 
=   / n;
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Typ ical com bined
d ip ole/ qu ad ru p ole
SRS Booster c.f. dipole
‘F’ type -ve n
‘D ’ type +ve n.
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N IN A Com bined function m agnets
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Pole for a com bined d ipole and quad .
Physical
and magnetic
centres
displacmen
t from
Horizontal
are separated
true quad
Then
by
centre
B
X
is
0
0
x'
  B 
 X

0
 x 
 
2
x' y   R
therefore
x' x  X
As
Pole equation
0
R
y  
is
2
2
where
g is the half gap at the physical
rewritten
as
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centre
n 
1  

n x


 
y   g 1 -
or
/ 2
n x
1




1

of the magnet
y

x
  g 1 
B
0

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 B


  x






1
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Other com bined function m agnets:
• d ipole, quad rupole and sextupole;
• d ipole & sextupole (for chrom aticity control);
• d ipole, skew quad , sextupole, octupole;
Generated by
• pole shapes given by sum of correct scalar potentials
- am p litu d es bu ilt into p ole geom etry – not variable!
OR:
• m ultiple coils m ounted on the yoke
- am plitud es ind epend ently varied by coil currents.
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The SRS multipole magnet.
Could d evelop:
• vertical d ipole
• horizontal d ipole;
• upright quad ;
• skew quad ;
• sextupole;
• octupole;
• others.
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The practical pole in 2D
Practically, poles are finite, introducing errors;
these appear as higher harm onics w hich d egrad e the
field d istribution.
H ow ever, the iron geometries have certain
sym m etries that restrict the nature of these errors.
Dipole:
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Quad rupole:
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Possible sym m etries.
Lines of sym m etry:
Dipole:
Quad
Pole orientation
d etermines w hether pole
is upright or skew .
y = 0;
x = 0; y = 0
Ad d itional sym m etry
im posed by pole ed ges.
x = 0;
y=x
The ad d itional constraints im posed by the sym m etrical
pole ed ges lim its the values of n that have non zero
coefficients
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Dipole sym m etries
Type
Symmetry
Constraint
Pole orientation
() = -(-)
all Jn = 0;
Pole ed ges
() = ( -)
Kn non-zero
only for:
n = 1, 3, 5, etc;
+
+
+
-
So, for a fully sym m etric d ipole, only 6, 10, 14 etc pole
errors can be present.
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Quad rupole sym m etries
Type
Symmetry
Constraint
Pole orientation
() = -( -)
All Jn = 0;
() = -( -)
Kn = 0 all od d n;
() = (/ 2 -)
Kn non-zero
only for:
n = 2, 6, 10, etc;
Pole ed ges
So, a fully sym m etric quad rupole, only 12, 20, 28 etc
pole errors can be present.
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Sextupole sym m etries.
Type
Symmetry
Constraint
Pole orientation
() = -( -)
() = -(2/ 3 - )
() = -(4/ 3 - )
All Jn = 0;
Kn = 0 for all n
not m ultiples of 3;
Pole ed ges
() = (/ 3 - )
Kn non-zero only
for: n = 3, 9, 15, etc.
So, a fully sym m etric sextupole, only 18, 30, 42 etc pole
errors can be present.
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Su m m ary
For p erfectly sym m etric m agnets, the ‘allow ed ’ error field s are fu lly
d efined by the sym m etry of scalar p otential  and the trigonom etry:
D ipole:
+
Quadrupole:
-
+
+
+
-
-
D ipole: only dipole, sextupole,
10 pole etc can be present
Sextupole: only sextupole, 18, 30, 42
pole, etc. can be present
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Quadrupole: only quadrupole, 12
pole, 20 pole etc can be present
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Vector p otential in 2 D
We have:
and
Expand ing:
B = cu rl A
(A is vector potential);
d iv A = 0
B = cu rl A =
(A z / y - A y / z) i + (A x/ z - A z / x) j + (A y / x - A x/ y) k;
w here
i, j, k, are u nit vectors in x, y, z.
In 2 dimensions
Bz = 0;
 / z = 0;
So
A x = A y = 0;
and
B = (A z / y ) i - (A z / x) j
A is in the z direction, normal to the 2 D problem.
N ote:
d iv B = 2A z / (x y) - 2A z / (x y) = 0;
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Total flux betw een tw o points  DA
In a tw o dimensional problem the m agnetic flux betw een
tw o points is proportional to the d ifference betw een the
vector potentials at those points.
B
F  (A2 - A1)
A1
A2
F
Proof on next slid e.
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Proof:
Consid er a rectangular closed path, length l in z
d irection at (x 1,y 1) and (x2,y 2); apply Stokes’ theorem :
A
F =   B.dS =   ( cu rl A).dS =  A.ds
Bu t A is exclu sively in the z
d irection, and is constant in this
d irection.
dS
l
So:
 A.d s = l { A(x1,y 1) - A(x2,y 2)};
(x1, y 1)
B
ds
(x2, y 2)
y
z
x
F = l { A(x1,y 1) - A(x2,y 2)};
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Contours of constant A
Therefore:
i) Contou rs of constant vector p otential in 2D give a grap hical
rep resentation of lines of flu x.
ii) These are u sed in 2D FEA analysis to obtain a grap hical im age of
flu x d istribu tion.
iii) The total flu x cu tting the coil allow s the calcu lation of the ind u ctive
voltage p er tu rn in an ac m agnet:
V = - d F/d t;
iv) For a sine w ave oscillation frequ ency w:
Vpeak = w (F/2)
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In 3D – pole end s (also pole sid es).
Fringe flu x w ill be p resent at p ole end s
so beam d eflection continu es beyond
m agnet end :
z
1. 2
B0
By
0
0
The magnet’s strength is given by
 By (z) dz
along the
magnet , the integration including the fringe field at each end;
The ‘magnetic length’ is defined as
(1/B0)( By (z) dz )
over the same integration path, where B0 is the field at the azimuthal
centre.
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End Field s and Geom etry.
At high(ish) fields it is necessary to terminate the pole
(transverse OR longitudinal) in a controlled way:;
•to prevent saturation in a sharp corner (see diagram);
•to maintain length constant with x, y;
•to define the length (strength) or preserve quality;
•to prevent flux entering normal
to lamination (ac).
Longitudinally, the end of the
magnet is therefore
'chamfered' to give increasing gap
(or inscribed radius) and lower
fields as the end is approached.
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Classical end or sid e solution
The 'Rogowski' roll-off: Equation:
y = g/2 +(g/a) [exp (ax/g)-1];
g/2 is dipole half gap;
y = 0 is centre line of gap;
a = 1.5
a ( ̴ 1); parameter to control
a = 1.25
the roll off;
a = 1.0
With a = 1, this profile
provides the maximum rate
of increase in gap with a
monotonic decrease in flux
density at the surface;
For a high By magnet this avoids any
additional induced non-linearity
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Introd uction of currents
Now for j  0
H=j;
To expand, use Stoke’s Theorum:
for any vector V and a closed
curve s :
V.ds = curl V.dS
Apply this to:
curl H = j ;
dS
ds
V
then in a magnetic circuit:
 H.ds = N I;
N I (Ampere-turns) is total current cutting S
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Excitation cu rrent in a d ip ole
B is approx constant round the
loop made up of l and g, (but see
below);
But in iron,
and
So
>>1,
Hiron = Hair / ;
l
N I/ 2
g
N I/ 2
  1
Bair = 0 NI / (g + l/);
g, and l/ are the 'reluctance' of the gap and iron.
Approximation ignoring iron reluctance (l/ << g ):
NI = B g /0
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Excitation current in quad & sextupole
For quadrupoles and sextupoles, the required
excitation can be calculated by considering fields and
gap at large x. For example:
Quadrupole:
y
Pole equation:
xy = R2 /2
On x axes
BY = GQx;
where G Q is gradient (T/m).
B
x
The sam e m ethod for a
Sextupole,
( coefficient G S,), gives:
At large x (to give vertical
lines of B):
N I = (GQx) ( R2 /2x)/0
N I = GQ R2 /2 0 (per pole).
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N I = G S R3/ 30 (per pole)
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General solu tion -m agnets ord er n
B = 0 H
B = - 
In air (remote currents! ),
y
Integrating over a lim ited path
(not circular) in air:
N I = (1 – 2)/ o
1, 2 are the scalar potentials at tw o points in air.
Define  = 0 at m agnet centre;
then potential at the pole is:
o N I
 = 0 NI
Apply the general equations for m agnetic
=0
field harm onic ord er n for non -skew
m agnets (all Jn = 0) giving:
N I = (1/ n) (1/ 0) Br / R (n-1) R n
Where:
N I is excitation per pole;
R is the inscribed rad ius (or half gap in a d ipole);
term in brackets  is m agnet grad ient G N in T/ m (n-1).
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x
B
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Coil geom etry
Stand ard d esign is
rectangular copper (or
alum inium ) cond uctor, w ith
cooling w ater tube. Insulation
is glass cloth and epoxy resin.
Am p -turns (N I) are
d eterm ined , but total copper
area (A copper ) and num ber of
turns (N ) are tw o d egrees of
freed om and need to be
d ecid ed .
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Current d ensity:
j = N I/ A copper
Optim um j
d eterm ined from
economic criteria.
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Current d ensity - optim isation
Ad vantages of low j:
• low er pow er loss – pow er bill is d ecreased ;
• low er pow er loss – pow er converter size is d ecreased ;
• less heat d issipated into m agnet tunnel.
15.0
• smaller coils;
• low er capital cost;
• smaller magnets.
Lifetime cost
Ad vantages of high j:
Chosen value of j is an
optim isation of magnet
capital against pow er costs.
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total
capital
running
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Current density j
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N um ber of turns per coil-N
The value of num ber of turns (N ) is chosen to m atch
pow er supply and interconnection imped ances.
Factors d eterm ining choice of N :
Large N (low current)
Small N (high current)
Sm all, neat term inals.
Large, bu lky term inals
Thin interconnections-hence low
cost and flexible.
Thick, exp ensive connections.
More insulation layers in coil,
hence larger coil volum e and
increased assem bly costs.
H igh percentage of copper in
coil volum e. More efficient use
of space available
H igh voltage pow er supply
-safety problem s.
H igh current pow er supply.
-greater losses.
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Exam ples-turns & cu rrent
From the Diam ond 3 GeV synchrotron source:
Dipole:
N (per m agnet):
I m ax
Volts (circuit):
40;
1500
500
A;
V.
Quad rupole:
N (per pole)
54;
I m ax
200
Volts (per m agnet): 25
A;
V.
N (per pole)
48;
I m ax
100
Volts (per m agnet) 25
A;
V.
Sextupole:
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Magnet geom etry
Dipoles can be ‘C core’ ‘H core’ or ‘Wind ow fram e’
''C' Core:
Ad vantages:
Easy access;
Classic d esign;
Disad vantages:
Pole shim s need ed ;
Asym m etric (sm all);
Less rigid ;
S h im
d e t a il
The ‘shim ’ is a sm all, ad d itional p iece of ferro -m agnetic m aterial
ad d ed on each sid e of the tw o p oles – it com p ensates for the
finite cu t-off of the p ole, and is op tim ised to red u ce the 6, 10,
14...... p ole error harm onics.
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Flux in the gap.
Flux in the yoke includ es the gap flux and stray flux,
w hich extend s (approx) one gap w id th on either sid e of
the gap.
b
Thus, to calculate total flux
in the back-leg of magnet
length l:
g
F =Bgap (b + 2g) l.
Wid th of backleg is chosen
to lim it Byoke and hence
m aintain high .
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g
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Steel- B/ H cu rves
-for typical silicon steel laminations
N ote:
• the relative perm eability is the
grad ient of these curves;
• the low er grad ient close to the
origin - low er perm eability;
• the perm eability is m axim um at
betw een 0.4 and 0.6 T.
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Typical ‘C’ cored Dipole
Cross section
of the
Diam ond
storage ring
d ipole.
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H core and w ind ow -fram e m agnets
''Wind ow Fram e'
Ad vantages:
H igh qu ality field ;
N o p ole shim ;
Sym m etric & rigid ;
Disad vantages:
Major access p roblem s;
Insu lation thickness
‘H core’:
Ad vantages:
Sym m etric;
More rigid ;
Disad vantages:
Still need s shim s;
Access p roblem s.
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Wind ow fram e d ip ole
Provid ing the cond uctor is continuous to the steel ‘w ind ow
fram e’ surfaces (im possible because coil m ust be electrically
insulated ), and the steel has infinite , this m agnet generates
perfect d ipole field .
Provid ing current d ensity J is
uniform in cond uctor:
• H is uniform and vertical up
outer face of cond uctor;
J
• H is uniform , vertical and
H
w ith sam e value in the m id d le
of the gap;
•  perfect d ipole field .
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Practical w ind ow fram e d ip ole.
Insulation ad d ed to coil:
B increases
close to coil
insulation
surface
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B d ecrease
close to coil
insulation
surface
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best
com prom ise
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Open-sid ed Quad rupole
‘Diam ond ’ storage ring
quad rupole.
The yoke support
pieces in the horizontal
plane need to provid e
space for beam -lines
and are not ferrom agnetic.
Error harm onics
includ e n = 4 (octupole)
a finite perm eability
error.
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Typical pole d esigns
To com pensate for the non -infinite pole, shim s are ad d ed
at the pole ed ges. The area and shape of the shim s d eterm ine
the amplitud e of error harmonics w hich w ill be present.
Dipole:
Quad rupole:
A
The d esigner optim ises the
pole by ‘pred icting’ the field
resulting from a given pole
geom etry and then ad justing
it to give the required
quality.
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A
When high field s are p resent,
cham fer angles m u st be sm all,
and tap ering of p oles m ay be
necessary
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Pole-end correction.
As the gap is increased, the size (area) of the shim is
increased, to give some control of the field quality at the
lower field. This is far from perfect!
Transverse adjustment at end of
dipole
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Transverse adjustment at end of
quadrupole
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Assessing p ole d esign
A first assessm ent can be m ad e by just exam ining By (x)
w ithin the required ‘good field ’ region.
N ote that the exp ansion of By (x) y = 0 is a Taylor series:
By (x) =


n =1
{b n x (n-1)}
= b 1 + b 2x + b 3x2 + … … …
d ip ole
qu ad
sextu p ole
Also note:
 By (x) /  x = b 2 + 2 b 3x + … … ..
So quad grad ient G  b 2 =  By (x) /  x in a quad
But sext. grad ient G s  b 3 = ½ 2 By (x) /  x2 in a sext.
So coefficients are not equal to d ifferentials for n = 3 etc.
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Is it ‘fit for purpose’?
A simple jud gement of field quality is given by plotting:
• D ipole:
• Quad:
• 6poles:
{By (x) - By (0)}/ BY (0)
d By (x)/ d x
d 2By (x)/ d x2
(DB(x)/ B(0))
(Dg(x)/ g(0))
(Dg 2(x)/ g 2(0))
‘Typical’ acceptable variation insid e ‘good field ’ region:
DB(x)/ B(0) 
Dg(x)/ g(0)

Dg 2(x)/ g 2(0) 
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0.01%
0.1%
1.0%
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Design com puter cod es.
Com puter cod es are now used ; eg the Vector
Field s cod es -‘OPERA 2D’ and 3D.
These have:
•
•
•
•
finite elements w ith variable triangular mesh;
m ultiple iterations to sim ulate steel non -linearity;
extensive pre and post processors;
com patibility w ith m any platform s and P.C. o.s.
Technique is iterative:
• calculate flux generated by a d efined geom etry;
• ad just the geom etry until required d istribution is
achieved .
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Design Proced u res – OPERA 2D
Pre-p rocessor:
The m od el is set-up in 2D using a GUI (graphics
user’s interface) to d efine ‘regions’:
• steel regions;
• coils (includ ing current d ensity);
• a ‘background ’ region w hich d efines the physical
extent of the m od el;
• the sym m etry constraints on the bound aries;
• the perm eability for the steel (or use the preprogram m ed curve);
• m esh is generated and d ata saved .
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Mod el of Diam ond storage ring d ipole
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With m esh ad d ed
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Close-u p of p ole region.
Pole profile, show ing shim and Rogow ski sid e roll-off
for Diam ond 1.4 T d ipole.:
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Diam ond qu ad ru p ole: a
sim p lified m od el
N ote – one eighth
of
Qu ad ru p ole cou ld
be u sed w ith
op p osite sym m etries
d efined on
horizontal and y = x
axis.
Bu t I have often
exp erienced
d iscontinu ities at the
origin w ith su ch1/ 8
th geom etry.
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Calculation of end effects using 2D cod es
FEA model in longitudinal plane, with correct end
geometry (including coil), but 'idealised' return yoke:
+
-
This will establish the end distribution; a numerical integration will give the
'B' length.
Provided dBY/dz is not too large, single 'slices' in the transverse plane can be
used to calculated the radial distribution as the gap increases. Again,
numerical integration will give  B.dl as a function of x.
This technique is less satisfactory with a quadrupole, but end effects are less
critical with a quad.
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Calcu lation.
Data Processor:
either:
• linear - w hich uses a pred efined constant perm eability
for a single calculation, or
• non-linear - w hich is iterative w ith steel perm eability
set accord ing to B in steel calculated on previous
iteration.
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Data Disp lay – OPERA
2D
Post-processor:
uses pre-processor mod el for many options for
d isplaying field am plitud e and quality:
•
•
•
•
•
field lines;
graphs;
contours;
grad ients;
harm onics (from a Fourier analysis around a pred efined circle).
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2 D Dipole field hom ogeneity on x axis
Diam ond s.r. d ip ole: DB/ B = {By(x)- B(0,0)}/ B(0,0);
typ ically  1:104 w ithin the ‘good field region’ of -12m m  x  +12 m m ..
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2 D Flux d ensity d istribution in a d ipole
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2 D Dipole field hom ogeneity in gap
Transverse
(x,y) p lane in
Diam ond s.r.
d ip ole;
contou rs are
0.01%
requ ired
good field
region:
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H arm onics ind icate magnet quality
The am plitud e and phase of the integrated
harm onic com ponents in a m agnet provid e an
assessm ent:
• w hen accelerator physicists are calculating beam
behaviour in a lattice;
• w hen d esigns are jud ged for suitability;
• w hen the m anufactured m agnet is m easured ;
• to jud ge acceptability of a m anufactured m agnet.
Measurem ent of a m agnet after m anufacture w ill be
d iscussed in the section on m easurem ents.
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End geom etries - d ip ole
Sim pler geom etries can be used in som e cases.
The Diam ond d ipoles have a Rogaw ski roll-off at the end s
(as w ell as Rogaw ski roll-offs at each sid e of the pole).
See photographs to follow .
This give sm all negative sextupole field in the end s w hich
w ill be com pensated by ad justm ents of the strengths in
ad jacent sextupole m agnets – this is possible because each
sextupole w ill have its ow n ind ivid ual pow er supply.
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OPERA 3D m od el of Diam ond d ipole.
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Diam ond d ipole poles
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2 D Assessm ent of quad rupole grad ient quality
0 .1
D iamond
WM
quadrupole:
D d H y /d x (% )
graph is
percentage
variation in
dBy/dx vs x
at different
values of y.
0 .0 5
0
-0 .0 5
Gradient
quality is to
be  0.1 % or
better to x =
36 mm.
-0 .1
0
4
8
12
16
20
24
28
32
36
x (m m )
y= 0
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y = 4 mm
y = 8 mm
y = 1 2 mm
y = 1 6 mm
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End cham fering - Diam ond ‘W’ quad
Tosca
resu lts d ifferent
d ep ths 45
end
cham fers
on Dg/ g 0
integrated
throu gh
m agnet
and end
fringe field
(0.4 m long
WM
qu ad ).
Fractional
deviation
0.002
0.001
X
5
10
15
20
25
30
35
-0.001
-0.002
8
7
6
4
mm Cut
mm Cut
mm Cut
mm Cut
No Cut
Thanks to Chris Bailey (D LS) w ho performed this w orking using OPERA 3D .
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Sim plified end geom etries - quad rupole
Diam ond
quad rupoles
have an
angular cut at
the end ;
d epth and
angle w ere
ad justed
using 3D
cod es to give
optim um
integrated
grad ient.
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Sextupole end s
It is not
usually
necessary
to chamfer
sextupole
end s (in a
d .c.
m agnet).
Diam ond
sextupole
end :
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‘Artistic’ Diamond Sextupoles
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