P. Blümler
ICG-III, Research Centre Jülich, Germany
SI-Units of Electro-Magnetism and used Sysmbols
Symbol
Derived
Units
Name of Quantity
Unit
Base Units
I
Current
Ampere
A
SI-base unit
q
Electric charge
Coulomb
C
A·s
U
Potential difference
Volt
V
J/C = kg·m2·s−3·A−1
R, Z
Resistance,
Impedance
Ohm
Ω
V/A = kg·m2·s−3·A−2
L
Inductance
Henry
H
Wb/A = V·s/A =
kg·m2·s−2·A−2
C
Capacitance
Farad
F
q/V = kg−1·m−2·A2·s4
ε
Permittivity
Farad/metre
F/m
kg−1·m−3·A2·s4
H
magnetic field
strength
Ampere/
metre
A/m
A·m−1
Φm
Magnetic flux
Weber
Wb
V·s = kg·m2·s−2·A−1
B
magnetic flux density,
magnetic field
Tesla
T
Wb/m2 = kg·s−2·A−1
µ
Permeability
Henry/metre
H/m
kg·m·s−2·A−2
χm
Magnetic
susceptibility
(dimensionless)
-
-
Conversion of magnetic units
Quantity
SI - Units
Magnetic flux
density, B
T (Tesla)
Magnetic field
strength, H
A/m
Max. energy
product , BH
J/m
Magnetic flux,
Φ
Wb
1 T = 10-4
Vs/cm2
3
Permeability of 4π 10-7
free space, µ0 Wb/A/m
1 Wb =
1Vs
CGS - Units
Conversion
G (Gauss)
1 T = 104 G
Oe (Oersted)
1 A/m = 1.2566 10-2Oe
G Oe
J/m3 = 1.256 102 G Oe
1 MGOe = 7.958 kJ/m3
M (Maxwell)
1 Wb = 108 M
unity
Topics
p
•
•
•
•
•
•
•
Magnet Concepts
Properties of permanent magnets
Design Step 1: Dipolar Approximation
Design Step 2: FEM Simulations
Type of Magnets
Construction of Magnets
g
((some hints!))
Gradient/Shim Coils
CMMR6 Educational 6.9.06
ICG-III, FZJ
Magnets
The word magnet originates from the Greek
"magnítis líthos" (μαγνήτης λίθος), which
means "magnesian stone".
(same is true for magnesium
magnesium, the element)
…and Magnesia (Greek: Μαγνησία), deriving from
g
((homeland of
the Macedonian tribe name Magnetes
the mythical heroes Jason, Peleus and his son Achilles)
…legend says that shepherd Magnes got stuck to magnetite
due to
o the
e iron
o nails
a s in his
s sshoes.
oes
Magnetite ferrimagnetic mineral (Spinel: iron(II,III) oxide:
Fe3O4).
) Magnetite is the most magnetic of all the minerals on
Earth (use in as an early form of magnetic compass since
1250 AC).
Crystals of magnetite have been found in some bacteria, the
brains of bees, of termites, of some birds, and of humans
(→ magnetoreception, navigation)
CMMR6 Educational 6.9.06
ICG-III, FZJ
Physics of Magnetism
M
Magnetism
ti
occurs whenever
h
electrically
l t i ll charged
h
d particles
ti l are iin motion.
ti
• movement of electrons in an electric current, resulting in "electromagnetism“
(free currents)
• spin (and orbital motion) of electrons, resulting in "permanent magnets“
(bound currents)
μ
magnetic moment of particle
I
q
μ=
r× p = γ I
mc
macroscopic magnetization
ω
μ
∑
M=
V
Note: Pauli-priciple! Only partially filled e-shells
give para/ferromagnetic properties
CMMR6 Educational 6.9.06
ICG-III, FZJ
Electromagnets
dB
da
r
CMMR6 Educational 6.9.06
I d a × dr
dB =
3
4πμμ0r
ICG-III, FZJ
The Earth-Magnet
Note: earth
earth‘s
s north-pole
north pole is a magnetic south-pole
south pole
Earth magnetic field:
ca. 30 – 60 μT (0.3 -0.6 G)
NMR for 1H:
1.2 – 2.5 kHz
EPR for
f e -:
0.8 – 1.6 MHz
CMMR6 Educational 6.9.06
ICG-III, FZJ
Permanent Magnets
Many materials have unpaired electron spins, but the majority of these materials
are paramagnetic.
Wh
When
th spins
the
i
i t
interact
t with
ith each
h other
th in
i such
h a way that
th t the
th spins
i
align
li
spontaneously, the materials are called either ferromagnetic or ferrimagnetic.
Historically, the term "ferromagnet" was used for any material that could exhibit
spontaneous magnetization: a net magnetic moment in the absence of an external
magnetic field.
Ferromagnetic elements: Fe, Co, Ni, Gd*, Dy*
CMMR6 Educational 6.9.06
* @ low T
ICG-III, FZJ
anti/ferro/i - magnets
A material is "ferromagnetic" only if all of its elementary
magnetic moments (spins) add a positive contribution to the net
magnetization (e.g.
magnetization.
(e g Fe,
Fe Co,
Co Ni,
Ni Gd,
Gd Dy,
Dy MnBi
MnBi, MnSb
MnSb, MnAs
MnAs,
CrO2, EuO)
If some spins subtract from the net magnetization (partially antianti
aligned), then the material is "ferrimagnetic". (e.g. FeOFe2O3,
NiOFe2O3, CuOFe2O3, MgOFe2O3, MnOFe2O3, Y3Fe5O12)
If the spins anti-align completely (resulting in zero net
magnetization), despite the magnetic ordering, then it is an
antiferromagnet. (e.g. Cr, simple oxids of ferromagnetic
elements)
All of these alignment effects only occur at temperatures below
a certain critical temperature, called the Curie temperature (for
ferromagnets and ferrimagnets) or the Néel temperature (for
antiferromagnets). This is a second-order phase transition.
CMMR6 Educational 6.9.06
ICG-III, FZJ
Ferromagnets
Heuristics:
H
i ti
T
Two
nearby
b magnetic
ti dipoles
di l will
ill tend
t d to
t align
li
i opposite
in
it directions
di ti
( in
(as
i
antiferro-magnets). Many will be different!
g
theyy tend to align
g in the same direction because of the Pauli p
principle:
p
In a ferromagnet,
two electrons with the same spin state cannot lie at the same position, and thus feel an
effective additional repulsion that lowers their electrostatic energy. This difference in
energy is called the exchange energy (only between Fermions with overlapping wave
wavefunctions) and induces nearby electrons to align.
At long distances (after many thousands atoms), the exchange energy advantage is
overtaken
t k
b the
by
th classical
l
i l tendency
t d
off dipoles
di l
t anti-align.
to
ti li
→equilibriated
ilib i t d (non(
magnetized) ferromagnetic material, the net-magnetization is small.
g
into magnetic
g
domains ((aka Weiss domains,, 0.01 -1 mm)) that are aligned
g
Theyy organize
(magnetized) at short range, but at long range adjacent domains are anti-aligned.
The transition between two domains, where the magnetization flips, is called a Domain
wall (e.g.,
(e g a Bloch/Néel wall,
wall depending upon whether the magnetization rotates
parallel/perpendicular to the domain interface) and is a gradual transition on the atomic
scale (covering a distance of about 300 atoms for iron).
CMMR6 Educational 6.9.06
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Ferromagnetic Material in External M-Field
B, M
Hext
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Hysteresis Loop
Behavior of material in an
external AC
AC-field
field
„magnetic memory“
Hc
Hc
Hard
H
d magnetic
ti material:
t i l
(e.g. permanent magnet)
large hysteresis
large remaining Remanence
Remanence, Br
Br,
hard to demagnetize
CMMR6 Educational 6.9.06
Hext
e t
Hextt
Soft
S
ft magnetic
ti material:
t i l
(e.g. soft iron, transformers)
small hysteresis
i i R
Remanence, B
Br,
smallll remaining
easy to demagnetize
ICG-III, FZJ
Characterization of permanet magnets
Remanenz, Br:
remaining magnetization/flux density after switch the
magnetization field off
Coercivity, BHc: reverse field needed to demagnetize magnet
E-product BHmax: the maximum energy-product is a quality factor, the
higher the more energy is stored (in a smaller volume)
T-drift, ΔT:
reversibe drift of Br per K (usually in % relative to RT)
Curie-Temp TC: loss of magnetization
Curie-Temp,
Isotropic magnets: produced without add. field, can be magnetized in any
direction ine
direction,
inexpensive
pensi e b
butt weak
eak
Anisotropic magnets: produced in presence of external M-field,
ca. 3 fold BHmax
CMMR6 Educational 6.9.06
ICG-III, FZJ
Maximum
Energy Product
Energy-Product
BHmax
CMMR6 Educational 6.9.06
ICG-III, FZJ
Composite permanent magnets
Ceramic, (hard) ferrite: sintered composite of powdered iron oxide and
barium/strontium carbonate ceramic. (inexpensive magnets: for cores in electronics
component)
non-corroding, but brittle
Alnico magnets are made by casting or sintering a combination of aluminium, nickel
and cobalt with iron and small amounts of other elements (sintered or casted)
resist corrosion and better physical properties than ferrite, low Hc at high Br, low ΔT
I j ti molded
Injection
ld d magnets
t are a composite
it off various
i
types
t
off resin
i and
d magnetic
ti
powders
complex shapes, but are generally lower in magnetic strength, flexible
CMMR6 Educational 6.9.06
ICG-III, FZJ
Rare Earth Magnets
'Rare earth' (lanthanoid) elements have a partially occupied f electron shell. The spin
of these electrons can be aligned,
g
, resulting
g in very
y strong
g magnetic
g
fields,, but
expensive
Samarium cobalt (SmCo5 and Sm2Co7) magnets are highly resistant to oxidation, with
higher magnetic strength and temperature resistance than alnico or ceramic materials.
brittle and may fracture when subjected to thermal shock. Sm2Co7 has very low T-drift.
Neodymium magnets (Nd2Fe14B) magnets, have the highest magnetic field strength.
inferior to SmCo in resistance to oxidation (coating) and temperature drift, low Curie-T
Typical:
Limited machinable
•Limited
•Brittle (avoid hits, depends on coating)
•Corrosive (no H2O or H2-atmosphere)
•No radioactivityy
CMMR6 Educational 6.9.06
ICG-III, FZJ
Properties of Permanent Magnets
Material
(BH)max
[MGOe]
Br
[T]
Hc
[kOe]
max. op. T
[°C]
ΔT
[%/K]
TC
[°C]
Density
[g/cm3]
Relative
Costs
Ceramic
Ferrite
0.8 – 4
0.2 – 0.41
1.6 – 3.3
250 – 300
0.19
850
4.7 – 5
very low
Alnico
12–9
1.2
0 6 – 1.35
0.6
1 35
0 4 – 1.75
0.4
1 75
450 – 850
6 9 – 7.3
6.9
73
moderate
Bonded
NdFeB
5 – 12
0.5 – 0.76
3.8 – 6.0
110 – 150
312
5.8 – 6.4
high
0.7 – 1.1
8 – 15
250 – 300
0.042
720
8.1 – 8.3
very high
0.98 – 1.1
5 – 15
250 – 350
0.032
827
8.1 – 8.3
very high
0.9 – 0.45
11 – 30
80 – 220
0.08 – 0.12
312
7.3 – 7.6
high
SmCo5
Sm2Co17
Nd2Fe14B
14 – 28
(32.5)*
20 – 28
22 – 53
((64)*
)
* theoretical limit
CMMR6 Educational 6.9.06
ICG-III, FZJ
CMMR6 Educational 6.9.06
ICG-III, FZJ
„Moore
Moore‘s
s Law“
Law
for permanent magnets
1 carbon steel
2 tungsten steel
3 cobalt steel
4 Fe-Ni-Al alloy
5 Ticonal II
6 Ticonal G
7 Ticonal GG
8 Ticonal XX
9 SmCo5
10 (S
(SmPr)Co
P )C 5
11 Sm2(Co0.85F0.11Mn0.04)17
12 Nd2Fe14B
CMMR6 Educational 6.9.06
ICG-III, FZJ
Design Step 1: Dipolar Approximation
B(r ) = ∑
3 ( ri ⋅ mi ) ri − mi
i
r
3
r
Typically the sample is
far enough away to allow
such an simplified
analytical approach
CMMR6 Educational 6.9.06
m
ICG-III, FZJ
Dipole Approximation: when valid?
dipo
ole
ap
pproxim
mation
mechanically resonable,
easy to handle
size of magnet
magne
etic flux, Bz [T]
1
0,1
0,01
1E-3
1
10
100
distance, z [mm]
CMMR6 Educational 6.9.06
ICG-III, FZJ
Dipole approximation
1
d
deviation
from dipole
0,1
production tolerance (1 - 10%)
0,01
1E-3
1E-4
1E-5
0
20
40
60
80
100
120
140
160
distance, z [mm]
CMMR6 Educational 6.9.06
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Dipolar Approximation
1 dipole
CMMR6 Educational 6.9.06
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Dipolar Approximation
2 dipoles
CMMR6 Educational 6.9.06
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Dipolar Approximation
3 dipoles
CMMR6 Educational 6.9.06
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Dipolar Approximation
4 dipoles
CMMR6 Educational 6.9.06
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FEM-Finite Element Methods
Software:
1)
Define Problem (dimensionality, geometry, materials)
2)
Mesh the problem (different algorithms, adaptive)
3))
Solve p
problem ((typically
yp
y Maxwell‘s or Ampere‘s
p
laws))
4)
Display
Packages: (for an overview see: http://www.emclab.umr.edu/csoft.html)
1) FEMM (only 2D, but freeware from David Meeker, http://femm.berlios.de)
2)
FEMLAB (Comsol, http://www.comsol.com)
3)
OPERA-3D (Vector-Fields, http://www.vectorfields.com)
4)
Amperes (Integrated Engineering Software)
CMMR6 Educational 6.9.06
ICG-III, FZJ
Example: Mandhala-4 with FEMM
Define geometry:
• structure
• boundaries
• properties
ti off domains
d
i
and boundaries
CMMR6 Educational 6.9.06
ICG-III, FZJ
Example: Mandhala-4 with FEMM
Mesh it!
CMMR6 Educational 6.9.06
ICG-III, FZJ
Example: Mandhala-4 with FEMM
Solve it
and analyze it!
CMMR6 Educational 6.9.06
ICG-III, FZJ
Magnet Types
CMMR6 Educational 6.9.06
ICG-III, FZJ
Block Magnet
B
Bx ( x) = r
π
⎡ −1
ac
tan
⎢
2
2
2
2
x
4
x
+
a
+
c
⎣
− tan −1
⎤
⎥
2
4 ( b + x ) + a 2 + c 2 ⎥⎦
ac
2 (b + x )
0.7
0.6
0.5
B
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
x
CMMR6 Educational 6.9.06
ICG-III, FZJ
⎡
⎤
B
l+x
x
c
r ⎢
Bx ( x) =
−
⎥
2
2
2 ⎢ r 2 + ( l + x )2
r +x ⎦
⎣
Cylinder Magnet
07
0.7
0.6
0.5
0.4
B
0.3
0.2
0.1
0
CMMR6 Educational 6.9.06
0
1
2
3
4
5
x
6
7
8
9
10
ICG-III, FZJ
Linear combination: Barrel magnet
Bx ( x) = Bxc ( x, ro ) − Bxc ( x, ri )
0.15
0.1
Careful when using yokes:
For soft iron without saturation:
A rule of thumb:
Double the length of the magnet
if applied in magnetization
direction
B
0.05
0
-0.05
-0.1
0
1
2
3
4
5
6
7
8
9
x
CMMR6 Educational 6.9.06
ICG-III, FZJ
10
Single Magnet: ca. dipole
favoured by the RWTH-Aachen group
CMMR6 Educational 6.9.06
ICG-III, FZJ
z
one magnet
versus
two magnets
CMMR6 Educational 6.9.06
r
ICG-III, FZJ
NMR-MOUSE
Nimbus
Field Designs:
Two Magnets
Jackson
Kleinberg
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NMR-MOUSE
Nimbus
Field Designs:
Two Magnets
Jackson
Kleinberg
CMMR6 Educational 6.9.06
ICG-III, FZJ
U-, C- and Θ- Shapes
7 MHz
core analyzer
CMMR6 Educational 6.9.06
ICG-III, FZJ
STRAFI
(Stray Field Imaging)
A.A. Samoilenko et al. Jept Lett 47 (1988) 417
Review: P.J. McDonald, Prog. NMR Spec. 30 (1997) 69
spatial resol
resolution
tion ca
ca. 10 µm,
m b
butt
CMMR6 Educational 6.9.06
∇×B = 0
ICG-III, FZJ
GARField (Gradient At Right-angles to the Field)
P.M. Glover, P.J. McDonald, et al.J. Magn. Reson. 139 (1999) 90.
Review: J. Mitchell, P. Blümler, and P. J. McDonald, Prog. NMR Spec. 48 (2006) 161
CMMR6 Educational 6.9.06
ICG-III, FZJ
GARField (Gradient At Right-angles to the Field)
P.M. Glover, P.J. McDonald, et al.J. Magn. Reson. 139 (1999) 90.
Review: J. Mitchell, P. Blümler, and P. J. McDonald, Prog. NMR Spec. 48 (2006) 161
CMMR6 Educational 6.9.06
ICG-III, FZJ
GARField (Gradient At Right-angles to the Field)
P.M. Glover, P.J. McDonald, et al.J. Magn. Reson. 139 (1999) 90.
Review: J. Mitchell, P. Blümler, and P. J. McDonald, Prog. NMR Spec. 48 (2006) 161
CMMR6 Educational 6.9.06
ICG-III, FZJ
NMR-MOUSE®
(NMR- Mobile Universal Surface Explorer)
G. Eidmann, R. Savelsberg, P. Blümler, B. Blümich, J. Magn. Reson. Ser. A 122 (1996) 104
Review: J. Mitchell, P. Blümler, and P. J. McDonald, Prog. NMR Spec. 48 (2006) 161
CMMR6 Educational 6.9.06
ICG-III, FZJ
NMR-MOUSE®
(NMR- Mobile Universal Surface Explorer)
G. Eidmann, R. Savelsberg, P. Blümler, B. Blümich, J. Magn. Reson. Ser. A 122 (1996) 104
Review: J. Mitchell, P. Blümler, and P. J. McDonald, Prog. NMR Spec. 48 (2006) 161
CMMR6 Educational 6.9.06
ICG-III, FZJ
High-resolution profiling
with
i h a NMR-MOUSE
NMR MOUSE
slice flatness?
precision = 10 µm
Casanova,
Blümich
RWTH-Aachen
CMMR6 Educational 6.9.06
ICG-III, FZJ
High-resolution profiling
with
i h a NMR-MOUSE
NMR MOUSE
depth profiles
0.5
1.0
Inte
ensity [a.u
u.]
profiler
1.0
0.0
-10
0.5
-5
0
5
depth [mm]
10
slice flatness?
precision = 10 µm
0.0
Casanova,
Blümich
RWTH-Aachen
CMMR6 Educational 6.9.06
0
1
2
depth [mm]
3
4
normal
NMR MOUSE
NMR-MOUSE
ICG-III, FZJ
Klaus Halbach
1925 - 2000
K. Halbach: “Design of Permanent Multipole
M
Magnets
t with
ith O
Oriented
i t d Rare
R
E
Earth
th C
Cobalt
b lt
Materials” Nucl. Instr. Meth. 169 (1980) 1-10.
CMMR6 Educational 6.9.06
ICG-III, FZJ
Why is the field only on one side?
linear combination
CMMR6 Educational 6.9.06
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Extension to 2D,, 3D
CMMR6 Educational 6.9.06
ICG-III, FZJ
⎛M ⎞
⎛ sin θ ⎞
M := ⎜ t ⎟ = M 0 ⎜
⎟
M
cos
θ
⎝
⎠
⎝ n⎠
with θ = 0K 2π
magnetization angle, γ,
in the ith of n magnets
Generalized Idea of Halbach
for Cylinders/Spheres
number and direction of poles
γ i := (1 + k
k ) βi
2π
n
and i = 0, 1, K, n − 1
with k ∈ N, βi = i
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Different forms
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ICG-III, FZJ
Different forms
CMMR6 Educational 6.9.06
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Different forms
CMMR6 Educational 6.9.06
ICG-III, FZJ
Different forms
CMMR6 Educational 6.9.06
ICG-III, FZJ
Construction (some hints)
• Permanent magnets are brittle and may break into
sharp edged pieces
pieces, might cause damage on impact
→ safety glasses, gloves, non-magnetic tools
• They may have enormous attracting or repulsing
forces (storage!)
(
g )
• Caution with recording media (tape, floppy, credit
cards),
d ) metallic
t lli or active
ti iimplants
l t ((pacemakers)
k )
• No use in presence of explosive atmosphere
(sparks!)
CMMR6 Educational 6.9.06
ICG-III, FZJ
Carefully plan the assembly
Storage: e.g. wall-mounted iron plate
covered with wood + rubber, cork
Separators: e.g. styrofoam, wood
Tools: non-magnetic (Al, Ti, stainless, Bronze):
screw drivers, wedges, knives, hammer
CMMR6 Educational 6.9.06
ICG-III, FZJ
always
y careful!
• Create a non-magnetic working area!
• Always
Al
mark
k th
the poles
l off your magnets
t and
dh
have a procedure!
d !
• When you feel the force better let go!
• Be decisive, calm and slow!
CMMR6 Educational 6.9.06
ICG-III, FZJ
Example of bad practice / planning
CMMR6 Educational 6.9.06
ICG-III, FZJ
Example:
NMR-Mandhala-16
CMMR6 Educational 6.9.06
ICG-III, FZJ
Constructing a Mandhala
B
CMMR6 Educational 6.9.06
ICG-III, FZJ
Constructing a Mandhala
CMMR6 Educational 6.9.06
ICG-III, FZJ
Constructing a Mandhala
CMMR6 Educational 6.9.06
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Gradient Coils
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ICG-III, FZJ
The B0 field
B0
z
CMMR6 Educational 6.9.06
ICG-III, FZJ
Bz
B0 g
gradients
B0
Bz
∂B
z
z
= const. = Gz
∂z
CMMR6 Educational 6.9.06
ICG-III, FZJ
gradients = spatial encoding
Spectroscopy
MRI
CMMR6 Educational 6.9.06
ICG-III, FZJ
Helmholtz Coil
Bz
radius = distance
z
3
2
⎛ 4 ⎞ μ 0 nI
B=⎜ ⎟
r
⎝5⎠
z
current
= homogeneous
flux / field
CMMR6 Educational 6.9.06
ICG-III, FZJ
Anti-Helmholtz Coil
Bz
radius = distance
z
0
z
current
= linear
g
inhomogeneous
flux / field
= constant gradient
CMMR6 Educational 6.9.06
ICG-III, FZJ
…not quite that simple
every current source produces a spatially (x
(x,y,z)
y z) varying magnetic field
which also has three components (Bx, By, Bz), so one has to work with
⎡ ∂Bx
⎢ ∂x
⎢ ∂B
G =⎢ x
⎢ ∂y
⎢ ∂Bx
⎢ ∂z
⎣
∂By
∂x
∂By
∂y
∂By
∂z
∂Bz ⎤
∂x ⎥
⎥
∂Bz
⎥
∂y ⎥
gradient tensor
g
∂Bz ⎥
∂z ⎦⎥
but when added to a stronger field, the maths is simpler
B( r ) = B 0 + G r
→ only the three components in field
direction, z, have to be observed…
but the others are there
∂Bz ( x, y , z )
= Gx = const.
…and Gi (i = x,y,z) have to be constant over space =
∂x
CMMR6 Educational 6.9.06
ICG-III, FZJ
Standard MRI
axis and field
CMMR6 Educational 6.9.06
ICG-III, FZJ
field
axis
CMMR6 Educational 6.9.06
ICG-III, FZJ
….hmmm
electrical current
+
CMMR6 Educational 6.9.06
ICG-III, FZJ
….hmmm
+
+
CMMR6 Educational 6.9.06
ICG-III, FZJ
Gradient coils for Halbach Magnets
Current distributions
Gz : Iz (θ, x ) = r0 sin(4πθ) D( x )
Gy : I y (θ, x ) = r0 cos(4πθ) D( x )
with damping function D(x)
⎛ πx ⎞
Gx : I x (θ, x ) = r0 cos(2πθ) sin ⎜ ⎟ D( x )
⎝ h ⎠
CMMR6 Educational 6.9.06
ICG-III, FZJ
Coil templates
Gy,z
Gx
CMMR6 Educational 6.9.06
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Gradients: Calculated fields
Bz ( z ) = Gz z
Bz ( x ) = Gx x
Bz ( y ) = Gz y
CMMR6 Educational 6.9.06
ICG-III, FZJ
Gradients: Calculated gradient fields
∂Bz
= Gz = const.
const
∂z
∂Bz
= Gx = const.
const
∂x
∂Bz
= Gy = const.
const
∂y
CMMR6 Educational 6.9.06
ICG-III, FZJ
Measured Gradient Quality
2.60
2.55
measured with Hall probe
average over sphere with radius r
gradiient streng
gth [mT/m
m/A]
2.50
2.45
2.40
Gy
2.35
Gz
2.30
2.25
2.20
Coil diameter = 170 mm
C
Coil height = 200 mm
measured with I = 10 A
resistivity = 0.3
03Ω
2.15
2.10
2 05
2.05
2.00
0
10
20
30
40
50
60
radius [mm]
CMMR6 Educational 6.9.06
ICG-III, FZJ