Transcript Gahler - Fundamental Neutron Physics at NC State

Neutron Optics and Polarization
R. Gähler; ILL Grenoble
1.
Neutron optics  light optics
2.
Neutron guides
3.
Supermirrors
4.
Polarised neutron beams / Spin flippers
5.
Adiabatic / non-adiabatic spin transport
1. Neutron optics  light optics
scalar matter waves  transversal e.m. waves (diffraction in time)
fermions  bosons (beam correlations)
  k2    k (index of refraction)
The wave equations for light and matter waves in vacuum:
light waves:
Helmholtz equ.
matter waves:
Schrödinger equ.
General case:
Stationary case:
1  2E
E  2 2  0
c t
E  k 2E  0
 
2mi 
0
 t
Light waves: periodic , propagating with c;
matter waves: similar to diffusion equation
except for ‘i’
Optics in time: Different physics;
  k 2   0
E  ;
same patterns;
Optics in space: Same physics;
What is special with optics in time for matter waves?
The Green’s function in the stationary case:
(propagation of a -like excitation in space)
for
  k   0 :
2
 e
G (r ) 

ikr
r
r
The Green’s function in the non-stationary case:
(propagation of a -like excitation in space and time)
for
Spherical waves
 mr 2 
2mi 
1

 
 0 : G(r ,  )  3 expi

2
 t
2





at fixed time :
at fixed position r:
r
A broad spectrum is emitted;
The higher the frequency,
the faster the propagation;
Can be expanded to a plane wave
for a short period in time and space;
t
Example: diffraction of matter waves from a ‘slit in time’
r , t  ?
t
 mr 2 
1

G(r ,  )  3 expi

2
2





r
opening time 2T
  r,t  
T
e
 i 0 t 0
T
Incoming wave
at opening of slit
G(r,  ) dt 0
 e
 i 0 t
scattered wave
at opening of slit
tT
i 
 e 0 G(r, ) d;   t  t 0
t T
Example: diffraction of matter waves from a ‘slit in time’
exponent of integrand
Evaluate the exponent:
mr 2
0  
2 
Minimum for the exponent:
 1/ 
d
mr 2 
 0  
0

d 
2 

 0 
mr 2
2 0
2
0
classical time of propagation
the phase is stationary around 0
0

Diffraction in Time
Significant contributions to the amplitude may only be accumulated
during time periods when the phase is constant: Thus the integral may
only be evaluated around its extremum. ‘Method of stationary phase’
2nd order expansion leads to Fresnel integrals in time, analogue to diffraction in space.
Lateral beam correlations of an extended source of matter waves / e.m. waves
Can we measure a diffraction pattern from two slits (separation xc)?
(for simplicity we consider only one wavelength)
xc
extended incoherent source
e.g. sun; moderator of reactor; light bulb
Plane of detection
Because of incoherence, we sum the intensities of all individual waves
but not add the amplitudes before quadrature.
‘Each wave interferes only with itself.’
Slightly displace all wave fronts in direction of propagation,
so that they are in phase at the two slits.
This does not influence the diffraction pattern, as only the phase difference at both slits is relevant.
If all waves are fairly well in phase at both slits, then one measures a pattern with high contrast.
xc
For this distance xc,
the contrast of the diffraction pattern will be okay.
Plane of detection
The wave field is coherent over the distance xc,
though in reality the waves are not in phase.
xc is called the ‘lateral coherence length’ of the field.
This construction is possible anywhere in the wave field!
With this construction,
the wave field resembles
the pacific ocean
xc
L
2a
From simple geometry: xc = L / ka; k = 2/;
This holds for matter waves and light waves!
xc  100Å for a neutron beam; xc  1m for stars (Michelson);
Longitudinal coherence of a ‘quasi-stationary’ non-monochromatic beam
Can we measure a diffraction pattern (double slit in time) from one slit,
which is at one position at two different times, which are separated by tc?
This slit is here at t0, then it disappears
and is only back here at t0 + tc
Point like sources may emit spherical waves of different wavelengths at arbitrary times
Slightly displace all wave fronts in time (here in direction of propagation),
so that they are in phase at the time t0, when the slit is there.
This does not influence the diffraction pattern, as only the phase difference at both times is relevant.
We will only be able to measure the double slit in time,
if during the time interval t0 to t0+tc the accumulated phases of all waves
will be fairly the same, say within  = 1
Coherence length lc = 2/ 
You may construct this coherence length (wave packet) at any time
and any position along the beam. It remains constant.
The product xc yc lc is the coherence volume Vc
Difference matter waves – light waves, concerning coherence
A) quantitative: N0, the number of particles / coherence volume:
N0 = 10-15 at neutron sources;
N0 = 10-1 at X-ray sources;
B) qualitative: 2 fermions should not be in one coherence volume;
 The Hanbury-Brown / Twiss experiment gives anti-correlations
for neutrons, but correlations for light;
A Det.

Correlated det.
B
Det.
If, by chance, most waves are in phase
at slit A a during the coherence time, then
there is a high probability to measure
a particle behind A.
Then, if distance AB is smaller than the
lateral coherence length, the waves will
also be in phase at B and then the
probability to measure a second particle
behind B is high as well.
For neutrons, see PRL ? 2006
(is it right?)
Another difference light optics  neutron optics
light: c = /k

neutrons: v = k /m
In general, the index n of diffraction is defined as n = k’/k;
A) Light entering a slab of matter:
c’ < c
 k’ > k  n > 1;
B) Neutrons entering a slab of matter:
1 mv 2  1 mv'2  V
2
2
For most materials the potential V is positiv.
v’ < v
 k’ < k  n < 1;
2. Neutron guides
12 guides from one beam tube at ILL
Analytic calculation of neutron guides
Potential step V0 for of a material surface :
2 2
V0 
N  bc
m
N : number density of atoms
bc : coherent scattering length of the atoms
no Bragg scattering; absorption negligible;
In case of different atoms i, the weighted average <N·bc>i has to be taken.
For positive values of bc, which holds for most isotopes, V0 is positive (n<1), thus
neutrons are totally reflected, if their kinetic energy Ekin perpendicular to the surface is
smaller than V0.
Ekin = ½mv 2 = 2k2/(2m) < V0
(mv =  k = h/;
k is the vertical wavevector and  the corresponding vertical wavelength)
The critical angle of reflection c :
k
k
½mv 2 = 2k2/(2m) < V0
 k = (4 N bc)½
k is the vertical wavevector
c
2 2
N  bc
Potential V0 
m
This defines a critical angle of total reflection c  k / k = (4 N bc)½ / k =  (N bc/ )½
k [Ni] = 1.07 ·10-2 Å-1  m=1; (most guides H1/H2)
Supermirrors of n-guides typically have m =2
 k [SM:m=2] = 2.14 ·10-2 Å-1;
Rule of thumb: For Ni, the critical angle c[degrees]  0.1  wavelength  [Å];
Basic properties of ideal bent guides I
z
a
x
coordinate system
rotates with guide
axis
a
i
 a-  i
All refections are assumed to be specular with
reflectivity 1 up to a well defined critical
angle c and with reflectivity 0 above c .
The reflection angle at the outer wall a
is always bigger than at the inner wall i .
Basic properties of ideal bent guides II
There are two types of reflections:
a
• Zig-zag reflections (large a)
• Garland reflections (never touching
the inner wall) (small a)
i
If the max. reflection angle allows
only Garland reflections near the
outer wall, then the guide is badly
filled.
If high a  i the filling of the guide
will be fairly isotropic.
transition from Garlandto Zig zag reflections
Basic properties of ideal bent guides III
a
After at least one reflection of all
neutrons*, the angular distribution
in the guide is well defined. The
angles always repeat.
*after the direct line of sight
transition from Garlandto Zig zag reflections
Basic properties of bent guides IV
At each point in the long guide, the angular
distribution is symmetric to the actual guide axis,
as there exists always a symmetric path which is
valid. This holds of course also for the guide exit.
a
a
For a given a, the angular width 2 near
the outside is larger than near the inside;
  a


The differential flux d /d is constant all
across the guide (Liouville!), however the
maximum angular width 2 is different. 
This width is proportional to the total
intensity at each point. (for  = const.) 
We have to calculate  to get the intensity!
The Maier-Leibnitz guide formula
a
x
x1
x2
-a
 a- 
y


90-a
 y = ·sin(a-  )  ·(a-  );
x1 =  - ·cos(a-  )  /2 ·(a-  )2;
x2 = y ·tan  y·   (a-  ) ;
a
x = x1 + x2 = /2 ·(a2-  2);
 2 = a 2 - 2x/;
The Maier-Leibnitz guide formula
a
x
x1
x2
-a
 a- 
y


y = ·sin(a-  )  ·(a-  );
 x1 =  - ·cos(a-  )  /2 ·(a-  )2;
x2 = y ·sin  y·   (a-  ) ;
a
x = x1 + x2 = /2 ·(a2-  2);
 2 = a 2 - 2x/;
The Maier-Leibnitz guide formula
a
x
x1
x2
-a
 a- 
y


y = ·sin(a-  )  ·(a-  );
x1 =  - ·cos(a-  )  /2 ·(a-  )2;
 x2 = y ·sin  y·   (a-  ) ;
a
x = x1 + x2 = /2 ·(a2-  2);
 2 = a 2 - 2x/;
The intensity distribution in a long curved guide as function of 
To calcukate the max. transmitted divergence, we choose: a = c = k/ k;
Plot of  = (c 2 - 2x/)½ as function
of x for different c as parameter:
For  = c the inner wall just does not get touched.

In this case the divergence is 0 at the inner wall.
The wavelength , corresponding to this c is
called characteristic wavelength *.
parabolas open
to the left
0
a
width of guide
x
The critical parameters of a long curved guide
* = (2a/)½;
* = characteristic angle;
With * = k / k* we get:
k* = k (2a/)-½;
k* = characteristic k-vector;
With * = 2 / k* we get:
* = 2 (2a/)½ / k;
* = characteristic wavelength;
Filling factor F (intensity ratio curved guide/straight guide):

1
1
2x 
F
dx    dx  1 

a c 0
a 0 
   c2 
a*
a*
a* = a
for c  *;
a* =  c 2/2 for c  *;
1
2
F[  = (c 2 - 2x/)½ /  = c ]

c
F = 0.93
c
F = 2/3
c
F = 1/6
0
a/4
F(c= *) = 2/3;
F(c= 2*) = 0.93;
F(c= */2) = 1/6;
a
x
Changes from m=1  m=2 and from a  1.5a
Replacing a Ni guide (m=1) by a SM guide (m=2) doubles k and c.
Increasing the guide width from a  1.5a increases * by 1.5½
and also increases the direct line of sight [Ld = (8a)½] by 1.5½.
New critical angle c
from increase of m
For   * the intensity
increases by a factor 4.

2c
c
New characteristic *
from increase of width
c/2
0
a/4
a
1.5a
x
For   * the intensity increases by more than a
factor 4 from doubling of c. [c/2  c is shown.]
Changes in parameters along a guide:
From here on the final angular
distribution is well defined.
a)
b)
Bad example: Two successive
guide sections of the same *:
*  */m;
a) low m; low *
b) high m; high *
Outgoing beam of high divergence;
loss in phase space density!!
Changes in parameters along a guide:
Example: Split a guide (width a) into two guides (width a/2),
reduce radius of curvature to /2 and keep c2 constant
 = [c2 - 2x/]½ = c [1 - 2x/(c2)]½

The dashed line shows the
angular distribution in the
original guide
The full lines show the max.
possible angular distribution in
the split guides
All neutrons will be transmitted.
c2 = 2a/ = *
Loss in phase space density?
0
a
a/2
a/2
x
Both curves intersect at a,
because c2 is constant
Estimate of main n-guide losses
Reflectivity:
Garland refl.: lg = 2   ;
Zig-zag refl.: lz =  (a - i );
or lz  d/;
Rn = (1 - )n  1 - n ; for  << 1;
for R = 0.97 and L = 1.2 L0; n  3 ( = 2700m;  = 1/200)
 beam transmission fR by reflectivity R: < fR > = 0.9; higher for long SM-guides!
l = mean length for one reflection from side walls;
n = mean number of reflections; R = reflectivity; L0 = length of free sight; L=100m
Alignment errors: Gauss distrib.: f(h) = exp(-h2/hf2) / (-1/2 hf )
Transmission fa = 1 - L/Lp hf -1/2 /a;
For L = 1.2 L0 ; a = 3 cm, Lp = 1m, L = 100m and hf = 20 m:
h
 mean transmission due to alignment errors: fa = 0.96;
For guide of 20x3 cm, the necessary precision on top / bottom is 7 times worse (0.14 mm).
hf = mean alignment error; a = guide width (30 mm); Lp = length of plates;
Estimate of main n-guide losses
waviness:
Let the neutron be reflected under angle  +  instead of .
g  
For 6 reflections, L = 1.2 L0 ; k = 0.7k*: fw = 1 - w /*;
For w = 10-4, * = 1.7 10-3 [1Å, Ni]:
 2 
1
 
exp
2 
2
w 
  w

fw = 0.94;
The outer areas of the intensity distribution I() are more affected than the inner ones.
fw : transmission due to waviness; w = RMS waviness; * = characteristic angle of guide
I()
without waviness
with waviness
c

w = 2·10-4 for the new guides seems acceptable (m=2!)
The angle  between the guide sections can be treated as waviness.  = 1/27000 for H2.
Summary of main guide formulas
 = radius of curvature
a = width of guide
x = width from outer surface
k = max. vertical wavevector;
 = angular width w.r.t. guide axis
c  k / k = (4 N b)½ / k =  (N b/ )½
 = (c 2 - 2x/)½;
 * = (2a/)½;
k [Ni] = 1.07 ·10-2 Å-1  m=1;
 k = 1.07 ·x · 10-2 Å-1; [SM: m=x]
Supermirrors of n-guides typically have m =2
k [glass] = 0.63 ·10-2 Å-1;
k [Ni-58] = 1.27 ·10-2 Å-1;
 = angular width at output of curved guide; c = critical angle of total reflection;
* = characteristic angle;
conditions: each neutron is at least once reflected at outer surface and reflectivity is step function up to c;
* = 2 (2a/)½ / k;
* = characteristic wavelength;
Flux d/d for given source brilliance d2/dd and distance z from source:
Flux d/d in long straight guide for constant source brilliance d 2/dd
if angular acceptance in guide x y is smaller than angular emittance of source:
d
d 2 1
 
dF
d source dd z 2
d d2

 x y
d dd
Summary of main guide formulas

1
1
2x 
F
dx    dx  1 

a c 0
a 0 
   c2 
a*
L02= 8a;
a*
1
2
a* = a
for c  *;
2
a* =  c /2 for c  *;
F = filling factor of guide
L0 = direct line of sight of bent guide;
= a-(L0/2-dL)2/2R;
xb= L2/(2);
 = width of direct sight of bent guide; dL = missing length to L 0
xb = lateral deviation from start direction; L = length of guide;
ng = L/(2  ) ; nz = L/( (a - i )); or nz  L  /a; n = number of reflections; ng for Garland; nz for Zig-zag refl.;
fw = 1 - w /*;
fw = loss due to waviness; w = RMS value of waviness; L = 1.2 ld ;
fa = 1 - L/Lp hf -1/2 /a;
fa = loss due to steps; hf = RMS value of alignment
error;
Loss V in intensity due to gap of length L for a guide of cross scection ab:
V
L   c  1 1  L2   c2
  
2 a b
4ab
3. Supermirrors (polarising)
A rule of thumb to estimate reflectivity and number of layers*
•for calculations see e.g.:
• F. Mezei; Commun.Phys.1(1976)81; + Corrigen. : Commun.Phys.2(1977)41; (first paper)
• J. Hayter, A Mook: J. Appl. Cryst.: 22(1989)35; (used for supermirror production)
Mirror reflection on multilayer in 1D
magnetic layers
b2 = bc2  bm;
Plane wave; k0
non-magn. layers:
b1 = bc1
bc2 + bm
For spin : bc2 + bm  bc1
bc2
bc1
For spin : bc2 – bm  bc1; bc1  0
bc2 - bm
bc1
Reflectivity of multilayer in 1D
Plane wave; 0
d
0
magnetic layers
n2 = nc2  nm
 1 + 2nm;
non-magn. layers:
n1 = nc1  1
Reflectivity at normal incidence
at one layer (far beyond total refl.):
 n  n2 
R 1

n

n
2 
 1
 = 0 for reflecting on smaller n;
 =  for reflecting on higher n;
2
Raleigh formulas
R  (2nm)2

Jump in phase at reflection;
 For the proper 0 =2d
the reflected waves from
all boundaries are in phase
R  (2Z2nm)2 for reflection at Z double layers of width d;
Reflectivity of multilayer in 1D
Estimate of number of double layers Z for reflectivity near 1:

R  (2Z2nm)2 for reflection at Z double layers of width d;
With nm = bmZ2/2 = 2:
For R  1:
Z  1/(4
2);
(However, 1. Born approx. is very bad here!
Because of attenuation less layers are needed)
Estimate of the reflected -band:
The phase variation  of the reflected waves from all 2Z boundaries
should be:   /2;
 Reflected wavelength band   1/Z  2 ;
Phase variations  from micro roughness of given r :   1/
 It is very hard to make good high –m supermirors!
Co/Ti polarising supermirrors; K. Andersen, ILL
-6
-2
N.b (10 A )
15
q
10
5
Nb
0
No magnetic
field
substrate
B
-5
Ti Co Ti Co Ti Co air
substrate
-6
-2
N.b (10 A )
15
10
5
N(+p)
0
Nb
B
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
1
N up
N down
-5
0.8
Ti Co Ti Co Ti Co air
-6
-2
N.b (10 A )
R
15
B
10
5
0
Nb
N(-p)
-5
Ti Co Ti Co Ti Co air
2 Nb
n  1
1
2
0.6
0.4
No qc !
0.2
0
0
0.2
0.4
0.6
0.8
q(°)
1
1.2
1.4
1.6
Fe/Si polarising supermirrors; K. Andersen, ILL
-6
-2
N.b (10 A )
15
10
Nb
5
=3Å : labs = 70cm
q
B
0
Si substrate
-5
Si Fe Si Fe Si Fe
Si substrate
-6
-2
N.b (10 A )
15
N(+p)
10
Nb
5
0
-5
Si substrate
-6
-2
N.b (10 A )
15
10
Nb N(-p)
5
0
1
Reflectivity
Si Fe Si Fe Si Fe
0.8
0.6
0.4
No qc !
0.2
0
q
-5
Si Fe Si Fe Si Fe
Si substrate
concept of neutron supermirrors; Swiss Neutronics
neutron reflection at grazing incidence (< ≈2°)
@ smooth surfaces
@ multilayer
@ supermirror
refractive index n < 1
  2d sin q
d2>d
d1
d3>d
1.0
1.0
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.5
0.4
1.0
reflectivity
reflectivity
reflectivity
total external reflection
e.g. Ni qc = 0.1 °/Å
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
=5Å
0.0
0.0
0.2
0.4
0.6
0.8
q [°]
1.0
1.2
1.4
0.1
=5Å
0.0
0.0
0.2
0.4
0.6
0.8
q [°]
1.0
1.2
1.4
=5Å
0.0
0.0
0.2
0.4
0.6
0.8
q [°]
1.0
1.2
1.4
concept of neutron supermirrors; Swiss Neutronics
the m – value
layer sequence of Hayter & Mook1
range of supermirror reflectivity
in units of qc, nat. Ni
/4 layer thickness
overlapp of superlattice Bragg peaks
m - value
0.0
1.0
0.5
1.0
1.5
2.0
2.5
m=2
350
0.9
0.7
0.6
regime of
total reflection
layer thickness [Å]
reflectivity
0.8
regime of
supermirror
0.5
0.4
0.3
0.2
0.1
0.50
0.75
q [°]
1
1.00
1.25
5 6 7 8
300
250
200
150
100
 = 0.985
=5Å
0.25
4
Ni layer
Ti layer
50
0.0
0.00
3
1
10
100
1000
10000
number of layers
J. B. Hayter, H. A. Mook, Discrete Thin-Film Multilayer Design for X-Ray and Neutron Supermirrors, J. Appl. Cryst. 22 (1989) 35
concept of neutron supermirrors; Swiss Neutronics
layer material – contrast of SLD ( · b)
K. Soyama et al., NIMA. 529 (2004) 73
 1  b1   2  b2 
2
reflectivity
bNi = 10.3 fm
bTi = -3.4 fm
 Ni/Ti supermirrors
General goals:
high m - value
high neutron reflectivity
M. Hino et al., NIMA. 529 (2004) 54
 large number of layers, e.g.
m
m
m
m
=
=
=
=
2
3
4
5




120 layers
400 layers
1200 layers
2400 layers
 interface quality
 internal stress
(R  90%)
(R  80%)
(R ≈ 75%)
(R ≈ 63%)
Ni/Ti supermirrors – reactive sputtering NiNx
0.9
0.8
0.8
0.7
0.7
Reflectivity
Reflectivity
0.9
0.6
0.5
N2 flow rate:
0.4
15 sccm
10 sccm
5 sccm
3 sccm
2 sccm
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.0
1.0
75
70
65
60
55
50
45
40
35
30
25
20
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Limit for stability
0.2
reflectivity R
transmission T
R+T
0.1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.5
m - value
1.0
1.5
N
2
2.5
3.0
flow2.0rate [sccm]
3.5
m - value
increase of reflectivity with increasing content of N2 during sputtering of Ni layers
increase of internal strain  damage of glass substrate
0.0
4.0
Transmission
1.0
1.0
Compressive force on substrate [arb. units]
reactive sputtering of Ni in Ar:N2 atmosphere
Ni/Ti supermirrors – high ‘m’ ; Swiss Neutronics
m=4
1.0
0.9
0.9
0.8
0.8
0.7
0.7
Reflectivity
1.0
0.6
0.5
0.4
0.3
exp. data
simulation
0.2
0.6
0.5
0.4
0.3
exp. data
simulation
0.2
0.1
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.0
0.5
1.0
1.5
m - value
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0
1000
2000
z [nm]
2.0
2.5
3.0
3.5
4.0
m - value
rms [nm]
rms [nm]
Reflectivity
m=3
3000
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0
1000
2000
3000
4000
5000
6000
z [nm]
reflectivity simulation: SimulReflec V1.60, F. Ott, http://www-llb.cea.fr/prism/programs/simulreflec/simulreflec.html, 2005
4.5
Ni/Ti supermirrors – high ‘m’; ; Swiss Neutronics
m=5
1.0
0.9
0.8
Reflectivity
0.7
0.6
0.5
0.4
0.3
0.2
exp. data
0.1
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
m - value
polarizing supermirrors – ‘remanent’ option
concept of ‘remanent’ polarizing supermirrors
magnetic anisotropy
high remanence
1.0
M / Ms
0.5
0.0
-0.5
-1.0
-200
-100
0
100
200
H [Oe]
guide field to maintain neutron polarization ≈ 10 G
switching of polarizer/analyzer magnetization  short field pulse ≈ 300 G
4.
Polarised neutron beams / spin flippers
Emag = n B = 610-8 eV/Tesla;
For thermal/cold neutrons: Emag  Ekin:
 the neutron trajectory is hardly affected by the magnetic interactions;
the spins can easily be turned but not easily be pushed or pulled;
(longitudinal and lateral Stern Gerlach effects are small effects);
dS
  SB
dt
 =  /S;  = gyromagnetic ratio;
Graphical interpretation of
B
dS
  SB
dt
Ssin1
B
Ssin2
dS1
1
S


dS1
dS  B;  precession around B;
dS  S;  precession frequency is constant;
dS2
2
S
dS2
L = 2B;
 = 2.9 kHz / G
In both cases: dS  Ssin;
 during dt, the angular change of Ssin around B is constant:
 the precession ‘Larmor’ frequency L does not depend on ;
A neutron entering a static B-field, precesses with L around the B-field;
QM -description of a neutron beam, entering a B-field
Plane wave, entering a static B-field


i k 0y 0t 
x

0  e 
?
B = Bz

y

y=0
Neutrons: s = ½; µ  0; 2 states + and - with different kinetic energies E0  µB
Static case [dB/dt = 0]: no change in total energy ( = 0) but change in k;
2 2
k
2m
µB  Ekin



2 2
k0
2m
 B; 
k
2m
2
2


 k 02  B;

k 2  k 02   k   k 0  2k 0  k   2k 0 ;
v is the classical neutron velocity;
k  
2m B
B

2
2k 0
v
 
 
 
 

ei k  y 0t  


 ei k  y 0t  


 e
i k 0y 0t 
 e i k  y 
  i k  y 
e

Both states have equal amplitudes, as the initial polarization is
perpendicular to the axis of quantization (z-axis);
1
These amplitudes
2 are set to 1 here.
Energy diagram:
E0
+ = 0  ei k y
= 1
1 0
2µB
Ekin
[-Epot]
- = 0  e –i k y
y=0
y
k = µ B/v
Polarization downstream of the field (length y0): the static spin flipper;
1
    0
1
 eiky 0 
 
  
 0
 ik y 0

 
e

2µB
Ekin
[-Epot]
E0
y0
0
y
k = µ B/v;
Setting the polarizer to x-direction:
Ix 
Ix 
1
2

        CC 

1 iky0
e
 e iky 0
2
t0 = y0 /v
 e
 ik y 0
 eiky 0


1
 2B

1  1  e2iky 0  e2iky 0  1  cos  2k  y 0   1  cos 
 y0 
2
 v

I x  1  cos  L t 0  ; I y  1  sin  Lt 0   ; L  2B ;
Larmor precession!
Considering wave packets instead of plane waves:
wavepacket (bandwidth ) of length y and lateral width x = z ;
y  2/ ; x  /(2q); q = beam divergence; typ. values: x, y  100 Å
v
 = v·
y0
Time splitting dt =  of the two wave packets, separated by
the propagation through the field of length y0:
dE 2
 ;   
E
t0
y 0 2B

2v 1 2 mv 2

2B  y 0
mv 3

L  t
20
For fields of typ. 1 kG and length of m,  is in the ns range for cold neutrons;
In Spin echo spectroscopy,  is the ‘spin echo time’;
Splitting and polarization after B fields of different lengths
Quantum mech. picture:
v
after ‘short field’
after ‘long field’
y
y  2/ ;
Classical Picture:
Directions of the individual spins of the polychromatic beam
after passage through B fields of different lengths
Complete separation of the two packets implies that no coherent
superposition of both states exists any more  Polarization = 0;
RF-spin flippers
Movement of the spin S in a
static field B0 and a rotating field Brf
If Brf rotates with L,
S will see Brf as a static field.
Movement of S in a system x’,y’, z
rotating with L around the z axis.
z
B0’ =B0 - / L
B0’ = 0 for  =L
Brf ’ = Brf
z
B0
S
S
Brf
y
Brf
x
B0 causes Larmor precessions L around z axis.
Brf , rotating in xy plane, distorts this precession.
x’
y’
For a complete spin flip (Sz  -Sz),
Sz has to make a  turn around the x axis.
Resonance condition:  = L = L=2B/
Amplitude condition: Lt = Brf L/v = 
B0 = Bz
Brf = Bxy
v
L
Sz
-Sz
For a certain v-distribution v,
the spin flip is not complete,
(different times inside Brf);
However <-Sz>  1 – (v/v)2
y
QM of RF-spin flippers
The evolution of the amplitudes of a spin ½ particle in a static field B0 = Bz
and a field Brf, rotating with  in the x,y plane.
  
2
i
 2


t    
y
     B
      i t
    Ae
Propagation
in y direction
Aeit     
 
B     
Feynman lectures III,
eq. 10.23
Off-diagonal elements create
transitions between - and +
For general solution see R. Golub et al.: Am. J. Phys. 62 (1994); 3 tough pages!!
For the classical -flip conditions (L=2B/ and Lt = Brf L/v =  ),
the transition probability is 1, corresponding to a complete exchange of both states.
The solution for -flip and the energy diagram
Epot
L = 2B / ;
 eikL e iL t 
 0;
  
  ikL  iL t 
e
e

k = L / 2v;
0 = exp{ik0y - 0t}
L
+µB
E0
-µB
+
-
- gives L to the RF field
B0
E0
Brf
+
2 L
Etot
+ takes L from the RF field
-
The change of 2 L in Etot
is transferred to a change
in Ekin after the coils
The change in Ekin can be
used for the longitudinal
Stern-Gerlach effect;
Two RF-flippers: Mach-Zehnder interferometer in time;
L1
L2
e
Coherent reversal
of frequency splitting
k + k ;
0 + e
s
+ s
+ e
E0
0 = e
i(kx - 0t)
- e
k0 - k ;

=
detector
0 + (s - e)
e
+i(keL1 - ksL2)
e
-i(keL1 - ksL2)
0 - e
e
-i(s - e)t
e
+i(s - e)t
d
plane of detection
total energy
Coherent
frequency splitting
0 - (s - e)
0
For eL1 = (s - e)L2 , detector gets independent from v.  beats in time with d = (s - e)
The use of a MachZehnder in time
modulation plane
Continuous, divergent,
polychromatic (10%)
unpolarized cold beam
any instrument
I(t)
Green box:
1 polarizer + 2 RF flippers
+ 1 RF-power flipper*
I(t)
I(t)
I0
I0
without green box
t
*not explained here; see Gahler et al PLA (2006)
 = 0 – 10 MHz
with green box
t
Such a system may be used at many
TOF-instruments to enhance time
resolution by an order of magnitude
The fast adiabatic spin flipper
Static gradient field  in z-direction
Beam area
B > B0
B0
P = +1
B < B0
P = -1
BRF with fuzzy edges,
set to  = L = B0
Spin turn in a frame [‘] rotating with L around the z-axis:
B0
BRF
Btotal
z=z’
x’
z=z’
x’
z=z’
x’
z=z’
x’
B’ = B - L/ = B – B0 ;
Bx’ = Brf
z=z’
x’
The fast adiabatic spin flipper
Static gradient field  in z-direction
B > B0
P = +1
B0
Beam area
RF-field, fuzzy edges,
set to  = L = B0
B < B0
P = -1
This flipper is good for complete flipping, however it cannot keep the phase
relations between both states. It flips a wide wavelength band, as due to the
adiabaticity condition, only the min. but not a max. wavelength is set.