Neutron Optics and PolarizationR. 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:

General case: Stationary case:

Light waves: periodic , propagating with c;

matter waves: similar to diffusion equationexcept for ‘i’

E ;same patterns;

light waves:Helmholtz equ.

0t

E

c

1E

2

2

2

0EkE 2

0t

mi2

0k 2

matter waves:Schrödinger equ.

Optics in space: Same physics;Optics in time: Different 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)

0t

mi2

0k 2 for

for

r

e)r(G

rki

2

mriexp

1),r(G

2

23

:

:

The Green’s function in the non-stationary case:(propagation of a -like excitation in space and time)

r

r

at fixed time :at fixed position r:

t

A broad spectrum is emitted;

The higher the frequency, the faster the propagation;

Can be expanded to a plane wavefor a short period in time and space;

Spherical waves

Example: diffraction of matter waves from a ‘slit in time’

2

mriexp

1),r(G

2

23

r

t

k 0; 0

?t,r

opening time 2T

0 0 0 0T t T

i t i t i0

T t T

r,t e G(r, ) dt e e G(r, ) d ;

0tt

Incoming wave at opening of slit

scattered wave at opening of slit

Example: diffraction of matter waves from a ‘slit in time’

exp

onen

t of

inte

gran

d

/1

0

Minimum for the exponent:

02

mr

d

d 2

0

02

mr2

0

2

0

classical time of propagation

the phase is stationary around 0

Diffraction in Time

Evaluate the exponent: 2

0mr

2

Significant contributions to the amplitude may only be accumulatedduring 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.

Can we measure a diffraction pattern from two slits (separation xc)?(for simplicity we consider only one wavelength)

extended incoherent sourcee.g. sun; moderator of reactor; light bulb

Plane of detection

Lateral beam correlations of an extended source of matter waves / e.m. waves

Because of incoherence, we sum the intensities of all individual waves but not add the amplitudes before quadrature.

‘Each wave interferes only with itself.’

xc

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.

For this distance xc, the contrast of the diffraction pattern will be okay.

xc

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!

Plane of detection

With this construction, the wave field resembles

the pacific ocean

xc

L2a

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?

Point like sources may emit spherical waves of different wavelengths at arbitrary times

This slit is here at t0, then it disappears and is only back here at t0 + tc

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

A) quantitative: N0, the number of particles / coherence volume: N0 = 10-15 at neutron sources; N0 = 10-1

at X-ray sources;

If, by chance, most waves are in phase at slit A a during the coherence time, then there is a high probability to measurea particle behind A.

Then, if distance AB is smaller than the lateral coherence length, the waves willalso be in phase at B and then the probability to measure a second particlebehind B is high as well.

A

B

Det.

Det.

Correlated det.

Difference matter waves – light waves, concerning coherence

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;

For neutrons, see PRL ? 2006 (is it right?)

Another difference light optics neutron optics

A) Light entering a slab of matter:

c’ < c k’ > k n > 1;

B) Neutrons entering a slab of matter:

For most materials the potential V is positiv.

light: c = /k neutrons: v = k /m

In general, the index n of diffraction is defined as n = k’/k;

V'mv21mv2

1 22

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 :

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 = 2k2/(2m) < V0

(mv = k = h/;

k is the vertical wavevector and the corresponding vertical wavelength)

c

2

0 bNm

2V

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;

c

Potential

The critical angle of reflection c :

k

k

½mv 2 = 2k2/(2m) < V0

k = (4 N bc)½

k is the vertical wavevector

This defines a critical angle of total reflection c k / k = (4 N bc)½ / k = (N bc/ )½

Rule of thumb: For Ni, the critical angle c[degrees] 0.1 wavelength [Å];

c

2

0 bNm

2V

Basic properties of ideal bent guides I

a

i

ax

z

-a

a- i

coordinate system rotates with guide axis

The reflection angle at the outer wall a

is always bigger than at the inner wall 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 .

There are two types of reflections:

• Zig-zag reflections (large a)

• Garland reflections (never touching the inner wall) (small a)

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.

a

Basic properties of ideal bent guides II

transition from Garland- to Zig zag reflections

i

After at least one reflection of all neutrons*, the angular distribution in the guide is well defined. The angles always repeat.

a

Basic properties of ideal bent guides III

transition from Garland- to Zig zag reflections

*after the direct line of sight

a

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

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!

For a given a, the angular width 2 nearthe outside is larger than near the inside;

a

Basic properties of bent guides IV

y = ·sin(a- ) ·(a- );

x1 = - ·cos(a- ) /2 ·(a- )2;

x2 = y ·tan y· (a- ) ;

x = x1 + x2 = /2 ·(a2-

2);

2 = a 2 - 2x / ;

- a

a

xa

a-

x2x1

y

The Maier-Leibnitz guide formula

90-a

y = ·sin(a- ) ·(a- );

x1 = - ·cos(a- ) /2 ·(a- )2;

x2 = y ·sin y· (a- ) ;

x = x1 + x2 = /2 ·(a2-

2);

2 = a 2 - 2x / ;

- a

a

xa

a-

x2x1

y

The Maier-Leibnitz guide formula

y = ·sin(a- ) ·(a- );

x1 = - ·cos(a- ) /2 ·(a- )2;

x2 = y ·sin y· (a- ) ;

x = x1 + x2 = /2 ·(a2-

2);

2 = a 2 - 2x / ;

- a

a

xa

a-

x2x1

y

The Maier-Leibnitz guide formula

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 *.

xa0

c

2 = 2a/

width of guide

c2 > 2a/

c

2 < 2a/

parabolas open to the left

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): F[ = (c 2 - 2x/)½ / = c ]

1a* a* 2

2c c0 0

1 1 2xF dx dx 1

a a

F(c= *) = 2/3; F(c= 2*) = 0.93;

F(c= */2) = 1/6;

xa

0

c = *

c = 2*

c =½ *

a/4

F = 0.93

F = 2/3

F = 1/6

c

c

c

a* = a for c *;a* = c 2/2 for c *;

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.

xa

0

old

new

a/4

New characteristic *from increase of width

c

c/2

2c

1.5a

New critical angle c

from increase of mFor * the intensity increases by a factor 4.

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:

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!!

From here on the final angular distribution is well defined.

a)

b)

Example: Split a guide (width a) into two guides (width a/2), reduce radius of curvature to /2 and keep c

2 constant

Changes in parameters along a guide:

c2 = 2a/ = *

xa0

a/2 a/2

The dashed line shows the angular distribution in the original guide

The full lines show the max. possible angular distribution in the split guides

= [c2 - 2x/]½ = c [1 - 2x/(c

2)]½

All neutrons will be transmitted.

Loss in phase space density?

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:

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;

h

waviness:

Let the neutron be reflected under angle + instead of .

For 6 reflections, L = 1.2 L0 ; k = 0.7k*: fw = 1 - w /*;

For w = 10-4, * = 1.7 10-3 [1Å, Ni]: 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

2w

2

2w

exp1

g

I()

c

with waviness

without waviness

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.

Estimate of main n-guide losses

Summary of main guide formulas

= (c 2 - 2x/)½; = angular width at output of curved guide; c = critical angle of total reflection;

* = (2a/)½; * = characteristic angle; conditions: each neutron is at least once reflected at outer surface and reflectivity is step function up to c;

= radius of curvaturea = width of guidex = width from outer surfacek = max. vertical wavevector; = angular width w.r.t. guide axisc k / k = (4 N b)½ / k = (N b/ )½

* = 2 (2a/)½ / k; * = characteristic wavelength;

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 =2k [glass] = 0.63 ·10-2 Å-1; k [Ni-58] = 1.27 ·10-2 Å-1;

dFz

1

dd

d

d

d2

source

2

Flux d/d for given source brilliance d2/dd and distance z from source:

yx

2

dd

d

d

d

Flux d/d in long straight guide for constant source brilliance d2/dd

if angular acceptance in guide x y is smaller than angular emittance of source:

Summary of main guide formulas

ng = L/(2 ) ; nz = L/( (a - i )); or nz L /a; n = number of reflections; ng for Garland; nz for Zig-zag

refl.;

L02= 8a; L0 = direct line of sight of bent guide;

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;

xb= L2/(2); xb = lateral deviation from start direction; L = length of guide;

F = filling factor of guide

1a* a* 2

2c c0 0

1 1 2xF dx dx 1

a a

a* = a for c *;a* = c 2/2 for c *;

= a-(L0/2-dL)2/2R; = width of direct sight of bent guide; dL = missing length to L0

ab4

L

b

1

a

1

2

LV

2c

2c

Loss V in intensity due to gap of length L for a guide of cross scection ab:

3. Supermirrors (polarising)

•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)

A rule of thumb to estimate reflectivity and number of layers*

For spin : bc2 + bm bc1

For spin : bc2 – bm bc1; bc1 0

bc2

bc2 + bm

bc1

bc2 - bm

bc1

magnetic layers

b2 = bc2 bm;

non-magn. layers:

b1 = bc1

Plane wave; k0

Mirror reflection on multilayer in 1D

magnetic layers

n2 = nc2 nm 1 + 2nm;

non-magn. layers:

n1 = nc1 1

Plane wave; 0

Reflectivity at normal incidence at one layer (far beyond total refl.):

2

1 2

1 2

n nR

n n

Raleigh formulas

Jump in phase at reflection;

= 0 for reflecting on smaller n;

= for reflecting on higher n;

For the proper 0 =2d

the reflected waves fromall boundaries are in phase

0

d

R (2nm)2

R (2Z2nm)2 for reflection at Z double layers of width d;

Reflectivity of multilayer in 1D

R (2Z2nm)2 for reflection at Z double layers of width d;

Reflectivity of multilayer in 1D

For R 1: Z 1/(4 2);

With nm = bmZ2/2 = 2:

(However, 1. Born approx. is very bad here!Because of attenuation less layers are needed)

Estimate of the reflected -band:

Estimate of number of double layers Z for reflectivity near 1:

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

No c !

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

N upN down

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

R

(°)

12

12

Nbn

No magnetic field

B

B

-5

0

5

10

15

Ti Co Ti Co Ti Co air

N.b

(10

-6 A

-2)

Nb

-5

0

5

10

15

Ti Co Ti Co Ti Co air

N.b

(10

-6 A

-2)

Nb

N(+p)

-5

0

5

10

15

Ti Co Ti Co Ti Co air

N.b

(10

-6 A

-2)

Nb N(-p)

substrate B

substrate

Fe/Si polarising supermirrors; K. Andersen, ILL

-5

0

5

10

15

Si Fe Si Fe Si Fe Si substrate

N.b

(10

-6 A

-2)

Nb

-5

0

5

10

15

Si Fe Si Fe Si Fe Si substrate

N.b

(10

-6 A

-2)

N(-p)Nb

-5

0

5

10

15

Si Fe Si Fe Si Fe Si substrate

N.b

(10

-6 A

-2)

Nb

N(+p)

0

0.2

0.4

0.6

0.8

1

Ref

lect

ivity

No c !

Si substrateB

=3Å : labs = 70cm

concept of neutron supermirrors; Swiss Neutronics

neutron reflection at grazing incidence (< ≈2°)neutron reflection at grazing incidence (< ≈2°)

refractive index n < 1

total external reflection e.g. Ni c = 0.1 °/Å

@ smooth surfaces@ smooth surfaces @ multilayer@ multilayer @ supermirror@ supermirror

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

= 5 Å

refle

ctiv

ity

[°]0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0d

= 5 Å

refle

ctiv

ity

[°]0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0d

2>d

1

= 5 Å

refle

ctiv

ity

[°]0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

= 5 Å

refle

ctiv

ity

[°]

d3>d

2>d

1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

= 5 Å

refle

ctiv

ity [°]

sin2d

concept of neutron supermirrors; Swiss Neutronics

0.00 0.25 0.50 0.75 1.00 1.250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00.0 0.5 1.0 1.5 2.0 2.5

m - value

= 5 Å

refle

ctiv

ity

[°]

layer sequence of Hayter & Mook1

/4 layer thickness

overlapp of superlattice Bragg peaks

layer sequence of Hayter & Mook1

/4 layer thickness

overlapp of superlattice Bragg peaks

1 J. B. Hayter, H. A. Mook, Discrete Thin-Film Multilayer Design for X-Ray and Neutron Supermirrors, J. Appl. Cryst. 22 (1989) 35

the m – value

range of supermirror reflectivity in units of c, nat. Ni

the m – value

range of supermirror reflectivity in units of c, nat. Ni

regime oftotal reflection

regime ofsupermirror

1 10 100 1000 10000

50

100

150

200

250

300

350 Ni layer Ti layer

876543

= 0.985

laye

r th

ickn

ess

[Å]

number of layers

m = 2

concept of neutron supermirrors; Swiss Neutronics

General goals:

high m - value

high neutron reflectivity

large number of layers, e.g. m = 2 120 layers (R 90%) m = 3 400 layers (R 80%) m = 4 1200 layers (R ≈ 75%) m = 5 2400 layers (R ≈ 63%)

interface quality

internal stress

General goals:

high m - value

high neutron reflectivity

large number of layers, e.g. m = 2 120 layers (R 90%) m = 3 400 layers (R 80%) m = 4 1200 layers (R ≈ 75%) m = 5 2400 layers (R ≈ 63%)

interface quality

internal stress

layer material – contrast of SLD ( · b)

reflectivity

bNi = 10.3 fm

bTi = -3.4 fm

Ni/Ti supermirrors

layer material – contrast of SLD ( · b)

reflectivity

bNi = 10.3 fm

bTi = -3.4 fm

Ni/Ti supermirrors

22211 bb

M. Hino et al., NIMA. 529 (2004) 54

K. Soyama et al., NIMA. 529 (2004) 73

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

N2 flow rate:

15 sccm 10 sccm 5 sccm 3 sccm 2 sccm

Ref

lect

ivity

m - value

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1620

25

30

35

40

45

50

55

60

65

70

75

Com

pres

sive

forc

e on

sub

stra

te [a

rb. u

nits

]

N2 flow rate [sccm]

Ni/Ti supermirrors – reactive sputtering NiNx

reactive sputtering of Ni in Ar:N2 atmospherereactive sputtering of Ni in Ar:N2 atmosphere

increase of reflectivity with increasing content of N2 during sputtering of Ni layers

increase of internal strain damage of glass substrate

increase of reflectivity with increasing content of N2 during sputtering of Ni layers

increase of internal strain damage of glass substrate

Limit for stability

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

reflectivity R transmission T R + T

Tra

nsm

issi

on

Ref

lect

ivity

m - value

Ni/Ti supermirrors – high ‘m’ ; Swiss Neutronics

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

exp. data simulation

Re

flect

ivity

m - value

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

exp. data simulation

Re

flect

ivity

m - value

m = 3m = 3 m = 4m = 4

0 1000 2000 30000.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

rms [n

m]

z [nm]

0 1000 2000 3000 4000 5000 60000.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

rms [n

m]

z [nm]

reflectivity simulation: SimulReflec V1.60, F. Ott, http://www-llb.cea.fr/prism/programs/simulreflec/simulreflec.html, 2005

Ni/Ti supermirrors – high ‘m’; ; Swiss Neutronics

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

exp. data

Re

flect

ivity

m - value

m = 5m = 5

polarizing supermirrors – ‘remanent’ option

-200 -100 0 100 200-1.0

-0.5

0.0

0.5

1.0

M /

Ms

H [Oe]

concept of ‘remanent’ polarizing supermirrors

magnetic anisotropy

high remanence

concept of ‘remanent’ polarizing supermirrors

magnetic anisotropy

high remanence

guide field to maintain neutron polarization ≈ 10 G

switching of polarizer/analyzer magnetization short field pulse ≈ 300 G

guide field to maintain neutron polarization ≈ 10 G

switching of polarizer/analyzer magnetization short field pulse ≈ 300 G

4. Polarised neutron beams / spin flippers

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);

Emag = n B = 610-8 eV/Tesla;

dSS B

dt

= /S; = gyromagnetic ratio;

dSS B

dt

Graphical interpretation of

B

S

dS1

dS1

Ssin1

1

B

S

dS2

dS2

Ssin2

2

dS B; precession around B;

dS S; precession frequency is constant;

In both cases: dS Ssin; during dt, the angular change of Ssin around B is constant: the precession ‘Larmor’ frequency L does not depend on ;

L = 2B;

= 2.9 kHz / G

A neutron entering a static B-field, precesses with L around the B-field;

Plane wave, entering a static B-field

y

QM -description of a neutron beam, entering a B-field

B = Bzx

?

Neutrons: s = ½; µ 0; 2 states + and - with different kinetic energies E0 µB

Static case [dB/dt = 0]: no change in total energy ( = 0) but change in k;

y = 0

2 22 2 2

2 200

kkB; k k B;

2m 2m 2m

0 0i k y t0 e

µB Ekin 2 20 0 0 0k k k k 2k k 2k ; 2

0

2m B Bk

2k v

v is the classical neutron velocity;

E0

Ekin

= 011

+ = 0

ei k y

- = 0

e –i k yk = µ B/v

[-Epot]

2µB

0

0 0

0

i k y t i k yi k y t

i k yi k y t

e ee

ee

Both states have equal amplitudes, as the initial polarization is perpendicular to the axis of quantization (z-axis);

These amplitudes are set to 1 here.1

2

Energy diagram:

yy=0

E0

Ekin

k = µ B/v;

t0 = y0 /v

[-Epot]

2µB

Setting the polarizer to x-direction:

y0

Polarization downstream of the field (length y0): the static spin flipper;

y0

0

0

i k y

0i k y

e

e

0

1

1

0 0 0 0i k y i k y i k y i k yx

1 1I CC e e e e

22

0 02i k y 2i k yx 0 0

1 2 BI 1 1 e e 1 cos 2 k y 1 cos y

2 v

x L 0I 1 cos t ; y L 0 LI 1 sin t ; 2 B ; Larmor precession!

wavepacket (bandwidth ) of length y and lateral width x = z ;

y 2/ ; x /(2); = beam divergence; typ. values: x, y 100 Å

Considering wave packets instead of plane waves:

= v·vy0

Time splitting dt = of the two wave packets, separated by the propagation through the field of length y0:

0 0 L2 31

0 02

y 2 B y tdE 2 2 B;

E t 2v 2mv mv

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

v

Complete separation of the two packets implies that no coherent superposition of both states exists any more Polarization = 0;

y 2/ ;y

Directions of the individual spins of the polychromatic beam after passage through B fields of different lengths

Quantum mech. picture:

Classical Picture:

after ‘short field’ after ‘long field’

RF-spin flippers

x

y

B0

S

Brf

Movement of the spin S in a static field B0 and a rotating field Brf

z

B0 causes Larmor precessions L around z axis.Brf , rotating in xy plane, distorts this precession.

If Brf rotates with L, S will see Brf as a static field.

Movement of S in a system x’,y’, zrotating with L around the z axis.

B0’ =B0 - / L

B0’ = 0 for =L

Brf’ = Brf

x’

y’

S

Brf

z

For a complete spin flip (Sz -Sz), Sz has to make a turn around the x axis.

Resonance condition: = L = L=2B/

Amplitude condition: Lt = Brf L/v =

For a certain v-distribution v, the spin flip is not complete,(different times inside Brf);

However <-Sz> 1 – (v/v)2

-Sz

Sz

yL

B0 = Bz

Brf = Bxy

v

QM of RF-spin flippers

i t2

2 i t

B Aei

t y Ae B

Feynman lectures III, eq. 10.23

Propagation in y direction

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.

Off-diagonal elements createtransitions between - and +

For general solution see R. Golub et al.: Am. J. Phys. 62 (1994); 3 tough pages!!

For the classical -flip conditions (L=2B/ and Lt = 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

Etot

B0

Brf

+

+

-

- + takes L from the RF field

- gives L to the RF field

The change of 2 L in Etot is transferred to a change

in Ekin after the coils

L

2

L

E0

E0

-µB

+µB

L

L

i ti k L

0i ti k L

e e;

e e

L = 2B / ;

k = L / 2v;

0 = exp{ik0y - 0t}

The change in Ekin can be used for the longitudinal

Stern-Gerlach effect;

Coherentfrequency splitting

Coherent reversal of frequency splitting

tot

al e

ner

gy

E0

L1 L2

0

i(kx - 0t)e

k0 - k ; 0 - e

k + k ; 0 + e

+ e

- e

+ s

e s0 + (s - e)

0 - (s - e)

=detector

+i(keL1 - ksL2)e -i(s - e)te

-i(keL1 - ksL2)e +i(s - e)te

pla

ne

of d

etec

tion

For eL1 = (s - e)L2 , gets independent from v. beats in time with d = (s - e) detector

d

0 =

Two RF-flippers: Mach-Zehnder interferometer in time;

Continuous, divergent,polychromatic (10%)

unpolarized cold beam I(t)

Green box:1 polarizer + 2 RF flippers

+ 1 RF-power flipper*

without green box with green box

any instrument

I 0

t

= 0 – 10 MHz

I 0

t

The use of a MachZehnder in time

I(t)I(t)

modulation plane

*not explained here; see Gahler et al PLA (2006)

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-directionBeam area

BRF with fuzzy edges,set to = L = B0

B0B > B0 B < B0

P = +1 P = -1

Spin turn in a frame [‘] rotating with L around the z-axis:

z=z’

x’

z=z’

x’

z=z’

x’

z=z’

x’

z=z’

x’

B0

BRF

Btotal

Bx’ = Brf

B’ = B - L/ = B – B0 ;

The fast adiabatic spin flipper

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.

Static gradient field in z-direction

Beam area RF-field, fuzzy edges,set to = L = B0

B0B > B0 B < B0

P = +1 P = -1