


Neutrino Oscillations in Structured Matter

Paul M. Fishbane
Physics Dept. and Institute for Nuclear and Particle Physics,
Univ. of Virginia, Charlottesville, VA 22903



Abstract

A layered material structure in a monochromatic neutrino
beam produces interference effects that could be used for
the measurement of features of the neutrino mass matrix.
The phenomenon would be most useful at high energies.






arXiv: v2 1 Jun 2000





1



The MSW effect [1, 2] describes how electron neutrinos in matter propagate
differently from other neutrinos, and from electron neutrinos in vacuum. This effect is an
element in the interpretation of recent experiments [3-6] that have explored the neutrino
mass spectrum. The phenomenon also describes the effects of the presence of boundaries
between different media on neutrino propagation and oscillation. As recent work [7-9]
has shown, the boundaries introduce the possibility of interference between different
amplitudes for neutrino propagation. Indeed, the presence of boundaries within the earth
has implications [7, 8] for the interpretation of the data of reference 3.
Here we point out that an arrangement of layers of materials, containing many
boundaries, can provide another angle on the interference between the propagation
modes. In addition to describing the basic mechanism for a two-family neutrino structure,
we briefly address the question of where this effect could be most profitably employed.

I. Review of MSW and Boundary Interference effects
Neutrino oscillation occurs because the electroweak eigenstates are not the mass
eigenstates. Let us consider two families of neutrinos, with electroweak labels e and 
(generically Greek letters) and mass eigenstate labels 1 and 2, with 1 labeling the lightest
neutrino. The matrix U (the leptonic CKM matrix) connects these states according to
= U

, (1.1)
i i
i 1
= ,2

where U takes the generic form
cos sin
U = . (1.2)
-sin cos
with such that cos(2) is positive.
Starting with a pure beam of, say, , with definite momentum (assumed
throughout), the time evolution is governed by the mass eigenstates and gives after the
neutrino beam has traveled a distance x t a beam containing a mixture of each type of
neutrino in the usual fashion. In particular the probability for the conversion is
P(t) = sin22 sin2, (1.3)
where the angle is determined by the energy difference,
1 m

= (E - E ) 2
x x, (1.4)
2 1
2 4E
with m2 m 2 2
2 - m1 . The approximation refers to the limit in which the mass difference
m is much less than the momentum of the beam, a limit that will be of interest to us.
The oscillation length (E) is the distance that corresponds to a change in by ; that is
(E) = 4E/m2. Finally note that the probability for nonconversion, i.e. that , is
P = cos2 + cos22 sin2 = 1 - P. (1.5)
Propagation in matter. In the presence of matter, each neutrino specie may have a
different potential. (In normal matter it is only the electron neutrinos for which there is a
potential associated with rescattering from electrons in the material.) In particular,
suppose that the potential difference of the two neutrino species is Vk, where k labels the
material. Then the effect of the matter on the propagation parameters is described by




2


m
m
= m
- +
k (cos2 k )2
2 2 2 2
sin 2

2
m

k
= x (1.6)
k 4 k
E
2
sin 2
2
sin 2 =
k k (cos2 - +
k )2 2
sin 2

where
2EV
k
(1.7)
k 2
m


The value of E for which i = cos2 is referred to as the MSW resonance; for that value
the angle 2i goes through /2. for general values of , total conversion in a single
thickness of any medium, including vacuum, is not possible, but it can occur for an
appropriate thickness of medium at MSW resonance.
The expression for i above is equivalent to the form given in Ref. [2], namely
tan 2
tan 2 = (1.8)
i 4 E
1- sec 2
2
) m

0

with the replacement 0 = 2/Vi.
Let us use the label k for the material (carrying information not only on the
composition through the neutrino potentials but on the layer thickness as well). Then the
generic amplitude A {k}
that a -neutrino enters and a -neutrino leaves layer {k} is
given for the two neutrino types and by
{k}
A = cos + i cos 2 sin ,
k k k
{k}
A = cos - i cos 2 sin ,
(1.9)
k k k

{k} {k}
A = A = -i sin 2 sin .
k k

Since we shall mainly be concerned with total conversion it is useful for later
comparison to give here the length Xk for maximum conversion in a single layer of
material k, immediately found from the last of these equations. The maximum possible
conversion is realized for a length Xk such that the factor sink = 1, i.e.,


far below MSW
k Vk

2
4E m

X = = = cot 2 at MSW (1.10)
k 2 k 2
m
2 m
V V
k k k k

far above MSW
V
k
The maximum conversion probability is sin22k, which is sin22 far below MSW, unity at
MSW, and sin22/ 2
k , asymptotically small, far above MSW.





3


The smallest value of energy and hence of k leads to the smallest length Xk. At
the same time, it is easiest to detect energetic neutrinos. It is therefore helpful to have
some idea of the energies that are involved for neutrinos. We note that when we use the
-
number V 9
k = 6  10 cm appropriate for earth [2], then

-
13
k = (E/(mc2)2)  2.5  10 eV

-
For mc2 = O(10 3) eV, k is on the order of 1 for E = O(4 MeV). One can think of this
energy as roughly the dividing line for whether one is below or above MSW, although of
course the precise MSW energy depends not only on but on the correct value of m2 as
well, and the latter number is not yet fully understood [10].

Propagation through layers. When there are layers of matter with differing
densities, then interference is possible. The history of the effects of passage through
repeated layers on mixing is in fact extensive, beginning perhaps with a discussion of
neutron-antineutron oscillations in a nonuniform magnetic field [11], then continuing
with work on neutrinos by Ermilova et al. [12] and later by Akhmedov [13] in a long
series of papers. These latter references describe what the phenomenon of parametric
amplification; in fact the work of refs. [7-9] can be seen as a special case of this treatment
[14]. However, the approaches taken in refs. [7-9] and in this paper are oriented in a
fashion sufficiently different to make them worth independent consideration.
Let us consider two layers oriented perpendicular to a neutrino beam, the first
labeled {1} and the second {2}. Then, as pointed out in [7], the amplitude A for passage
through two successive layers {1} and {2} (what we refer to as a bilayer) contains two
terms, and these terms can interfere:
{ }
1 { }
2 { }
1 { }
2
A = A A + A A

= sin (2 - 2 sin sin (1.11)
2 1 ) 1 2
-i{sin 2 sin cos + sin 2 cos sin
1 1 2 2 1 2}

One can see immediately that the structure of this amplitude is not that of the single layer;
for example, the third of Eqs. (1.9) is purely imaginary. In particular it is possible for this
amplitude to have magnitude onetotal conversionover a wide range of the parameter
space.
The work of reference [7] approaches total conversion for a two-channel problem,
with a beam initially of type passing through a double layer, through the probability
condition |A|2 = 1. In a rather involved calculation it is shown using this condition that
total conversion occurs for layer thicknesses such that
-cos 2
2 2 2
y tan =
1 1 cos 2 cos 2 - 2
1 ( 2 1 )
(1.12)
-cos 2
2 2 1
y tan =
2 2 cos 2 cos 2 - 2
2 ( 2 1 )

in regions of 1 and 2 where the right hand sides are positive. (Note our definition yi
tani.)



4


In the two-channel problem Eq. (1.12) is in fact more simply approached through
the amplitude condition A = 0. This amplitude is given by
{ }
1 { }
2 { }
1 { }
2
A = A A + A A

= cos cos - cos 2 - 2 sin sin (1.13)
1 2 ( 2 1 ) 1 2
-i{cos2 sin cos + cos2 cos sin .
1 1 2 2 1 2 }

This amplitude is zero when both its real and imaginary parts vanish, representing two
conditions for the two angles 1 and 2 and hence for the layer thicknesses if all other
physical parameters are given. One can see immediately from Eq. (1.13) that these
conditions are simply written as conditions for y1 and y2, namely

Real part = 0: 1 - y1y2cos(22 - 21) = 0. (1.14a)

Imaginary part = 0: y1cos21 + y2cos22 = 0. (1.14b)

From these equations, quadratic in the yi, one immediately arrives at the solutions given
in Eqs. (1.12).

II. Passage through multiple layers
The amplitude A for the survival of neutrino type through multiple layers is
developed in a straightforward way from the single-layer amplitudes of Eq. (1.9). In the
two-family problem, this amplitude is an element of a 2  2 matrix resulting from the
multiplication of two primitive (single layer) 2  2 matrices. The generalization to more
than two layers is straightforward. We give here the cases of three and four layers as
examples; in each case we give the conditions for A = 0.
For three layers, the conditions that the real and imaginary parts of A = 0 are,
respectively,
3
1- y y cos
(2 -2 )= 0 (2.1a)
i j i j
i, j 1
=
i< j

3
y cos 2
- y y y cos 2 - 2 + 2 = 0 . (2.1b)
i i 1 2 3 ( 1 2 3 )
i 1
=
For four layers the respective conditions are
4
1- y y cos
(2 -2 )+ y y y y cos 2 -2 +2 -2 = 0 (2.2a)
i j i j 1 2 3 4 ( 1 2 3 4 )
i, j 1
=
i< j

4 4
y cos 2
- y y y cos
(2 -2 +2 )= 0. (2.2b)
i i i j k i j k
i 1
= i, j,k 1
=
i< j<k

These two examples are sufficient to understand the more general cases. The only
important feature to note here is that for more than two layers the two conditions that A
vanish are insufficient to determine uniquely the yi and hence the layer thicknesses.




5


Repeated layers. A solvable case is that of alternating layers with every other
layer identical to its partners. In other words, we have exactly repeating layer pairs or
repeating layer pairs plus a last layer identical to the first. If we label N as the total
number of layers, then these possibilities correspond to N even and N odd, respectively.
The number of bilayers is n = [N/2], where the square bracket indicates the largest integer
in N/2. We found no especially interesting solutions for the odd N case and will make
only passing comments on it.
For even N, we have in mind ultimately a situation in which the first member of a
bilayer is vacuum and the second is a given thickness of a dense material, but we treat the
more general situation of a separate potential difference for each layer. In this case m 2
1 =
m 2 2 2 2
3 = m5 = ...; m2 = m4 =...; 1 = 3 =...; and so forth, so that we have only the
subscripts 1 and 2. The two conditions for real and the imaginary part will now determine
y1 and y2.
We give a series of explicit results for the conditions for total conversion for
multiple bilayers in the appendix. We remark here that the 2  2 matrix A that gives the
amplitude for the passage through n bilayers can be written as a factor (cos1 cos2)n
times a remaining matrix A. Since the conditions refer to the vanishing only of the -
component of A, we derive the (necessary and sufficient) conditions from A = 0. These
are the conditions given in the appendix.
The calculations presented in the appendix reveal two important feature that we
shall assume to be general: First, the imaginary part of the amplitude A contains a
single factor of the combination
F y1cos21 + y2cos22 (2.3)
This will turn out to be quite useful, as we shall see below. Second, aside from this single
factor, the modified mixing angles i appear in A only in the combination defined by
21 - 22. (2.4)

Conditions for total conversion. In the case of the single bilayer, the imaginary
part in particular vanishes only if the factor F defined by Eq. (2.3) vanishes. For more
than one bilayer, either F or its coefficient could vanish. Let us consider the latter
possibility for some low order examples.
For n = 2, the imaginary part of the amplitude is given by Eq. (A.2b), and we want
to consider the possibility that the second term vanishes, i.e. that 1 - y1y2cos = 0. This
-
gives y 1
2 = (y1cos) , and when this is substituted into the real part, Eq. (A.2a), we find
the condition
1 1
2
1+ y + + = 0.
1 2 2 2
cos y cos
1

But each term on the right side of this expression is positive, so we do not have a
solution.
For n = 3, we consider the possibility that imaginary part vanishes because the
expression in curly brackets of Eq. (A.3b) vanishes. But three times the curly bracket in
Eq. (A.3b) + the left side of Eq.(A.3a) (the real part for n = 3) is

y cos +1
-3( y + ) 2 2
2 1
1
1 2 2
y cos
1




6


and, as for its analog in n = 2, this quantity cannot vanish.
Although once again we do not have a general proof, it is reasonable that the only
way for the imaginary part of the amplitude A to be zero is with the condition that F
vanishes. Using this condition we can make an arbitrary n generalization for the form of
the amplitude at the total conversion point. To do so we write the amplitude for passage
through a single bilayer in canonical form, namely
i
e cos i
e sin
B = (2.5)
i i
e sin
e cos
The parameters of this "unit cell" amplitude are determined by comparison to the explicit
result,
B = A{1}A{2}
where the single layer amplitudes A{k} are given by Eq. (1.9). Using Eq. (1.9), we find
that B has the more restrictive form
i i
e cos e sin
B = (2.6)
-i
-e sin -i
e cos
with the three bilayer parameters of this expression, , , and , given in terms of the
single layer parameters by
1
cos = ( 1- y y cos = cos cos 1- y y cos (2.7)
2
y + )
1 ( 2
y +1
1 2 ) ( 1 2 ) 1 2 ( 1 2 )
y cos 2 + y cos 2
1 1 2 2
tan =
(2.8)
1- y y cos
1 2

y sin 2 + y sin 2
1 1 2 2
tan = -
. (2.9)
y y sin
1 2

This expression simplifies further if we apply the condition that for n bilayers the
factor of Eq. (2.3) is zero at the total conversion point:
= 0 for total conversion (2.10)
(The denominator of Eq. (2.8) is not independently zero except for the single bilayer
case.)
With this condition, the n bilayer amplitude with total conversion of the beam
becomes

cos i
n e sin n
n
B = . (2.11)
-i
-e sin n cos n
In turn, we see immediately that for total conversion
cos(n) = 0. (2.12)
In turn Eq. (2.12) gives (2m+ )1
= ,
m = 0,1,n -1 . (2.13)
2n
The pair of conditions that F and cos(n) each vanish provide us with two
equations for the angles 1 and 2, i.e., for the layer widths x1 and x2. We shall describe
the solution to these equations in the next section.



7



III. Total conversion in a repeated multilayer system

We apply the simultaneous conditions F = 0 and cosn = 0 to determine yi = tani here.
The F = 0 condition determines y1 in terms of y2. The quantity y2 is determined in terms
of the angle by the inversion of Eq. (2.7), which is a quadratic equation for y2 in terms
of cos. Since cosn is an nth order polynomial in cos, there are 2n solutions for y2.
Because N = 2n, this matches the number of solutions coming from the Nth order
polynomial for y2 coming from the original real part equations, as described below Eqs.
(2.10). Thus we find all the solutions in this way. Because we would like to minimize the
thickness of the material layers, we shall be interested in small y2 solutions, and we shall
see that this corresponds to small values of .
If we define
zi yicos2i, (3.1)
then the condition that F vanish reads z1 + z2 = 0. In turn this means that
z 2 2
1 = z2 z2. (3.2)
Equation (2.7) now becomes
1
cos = ( 2
cos 2 cos 2 + z cos (3.3)
1 2 )
( 2 2
z + cos 2 )( 2 2
z + cos 2
1 2 )

One can see quickly that for small , for which cos 1, Eq. (3.3) becomes
homogeneous in z2, and so has solutions at z2 = 0. For more detail, we consider separately
different regimes of the MSW parameter i. In doing so it is simplest to treat our the layer
labeled 1 as a layer of vacuum (1 = 0). It is straightforward to generalize to a bilayer
consisting of two different materials each with nonzero values of i.


2
2 small (below MSW resonance). We treat 2 as a small perturbation, with m2
m2 and 2 . We have
cos 1 - (1/2) 2
2 sin22,
and hence Eq. (3.3) becomes to leading order in 2
2
y ( 2 2
1+ y cos 2 ) 2
tan 2
2 2
2
f 1- cos .
2 ( 2
1+ y2 )2 2

2

With the definition 2
, this equation reads
2 f

y (1+ y cos 2 ) tan 2 = (1+ y )2
2 2 2 2 2 . (3.4)
2 2 2

The right side of this quadratic equation for y 2
2 is O(1) or larger, so there are no solutions
unless = O(1), i.e., f = O( 2
2 ). But by comparison with Eq. (2.13), we see that for small
values of m/n,

1 (2m + )2 2
1
f . (3.5)
2
2 4n
By choosing n large enough, or more particularly m/n small enough, we can imagine
choosing = O(1). The equation for y 2
2 will then have small positive solutions.



8


The formal solution of Eq. (3.4) is

2
sin 2 - 4cos 2 - 2cos 2 + sin 2
2 ( 2 ) 2 2
y = . (3.6)
2  2
2 cos 2 ( 2
1- sin 2 )
We see immediately that there is no (real) solution unless > 4cos22. The suitable (both
small and positive) solutions correspond to the minus branch y 2
2 -, and it is this solution
that we look at henceforth. The solutions are smooth as we pass through the point where
the denominator factor 1 - sin22 = 0 and are simple for greater than or equal to O(1)
for the = 0.7 case that we use below for illustration. Indeed, in this range we can use the
very accurate approximation
2
2 2
cot 2 cot 2
2
y + - +
- ( 2
2 cos 2
2 )

all the way to small values of . That is because the expansion is in cot22/, something
that follows from the fact that it is tan22 that appears in the original equation. Thus we
can write our solution in the form
cot 2 cot 2
y = ,
2 2n2
where in the last step we have chosen a large n solution with m = 0. Under the assumption
that cot2 is small enough, we can replace y2 by 2, and solve for x2:

x = cot 2 .
2 nV2
The total amount of material nx2 is less than the corresponding amount of a single layer of
material 2 [Eq. (1.10)] only in the circumstance that cot2 is very small.
There is a second problem in this region of 2. Once we have found y2, then y1,
and hence the thickness of layer 1, is determined through the condition
0 = y1cos21 + y2cos22
But in this region 1 2 . We would then require y1 = -y2, or, assuming that y2 is
sufficiently small that 2 tan2 = y2,
-
1
1 tan (-2)
Since the distances x2 must be positive, the only way we can satisfy this condition is to
take 1 2 - 2, or in other words,
8 E
x = - x . (3.7)
1 2 2
m

The first term on the right side of this expression, which is the oscillation length in
vacuum, is not necessarily small. The multiple bilayer arrangement offers no advantages
below MSW.


2 at MCW resonance. We have 1 = and 1 = (m2/4E)x1. For medium 2, 2 =
cos2, and sin22 = 1 or 22 = /2 and cos22 = 0. (In fact we shall assume that we are a
little above MSW resonance, so that cos22 is small and negative. This helps to clarify





9


2 2
m m sin 2
limits.) The angle 2

= x = x . We also have = /2 - 2 and cos =
2 2 2
4E 4E
sin2. In this limit the relation between y2 and of Eq. (3.3) becomes
1
cos = (3.8)
2
y +1
2

The relevant solution to Eq. (3.8) is
y2 = tan. (3.9)
or, since y2 = tan2,
(2m + )
1
= = . (3.10)
2 2n
Since cos22 is small and negative, y1 is satisfactorily positive and is also small:
y1 = y2|cos22|/cos2. (3.11)
When we calculate the layer thicknesses we see why this case is of no special
interest. Recall [Eq. (1.10)] that at MSW resonance a single layer of thickness X2 = (/V2)
cot2 of material 2 gives total conversion. In comparison, Eq. (3.10) shows us that the
minimum value of 2 for an n-bilayer system occurs for m = 0, in which case we have 2
= /(2n), or
4E 1
x = = cot 2 . (3.13)
2 2
2n m n V
2 2

Thus the total amount nx2 of material 2 is exactly the amount needed for the single layer.
Moreover in the MSW limit, cos22 is zero, so that from Eq. (3.11) y1 and hence the total
thickness of vacuum nx1 vanishes. The entire system limits to a single layer of material 2.
We have been able to find no quantity associated with n bilayers that scales to any
experimental advantage in the MSW limit.


2 large (above MSW resonance). Again we assume that the medium labeled 1 is
vacuum, 2
1 = 0. For 2 >> cos and sin, sin222 sin22/2 0, with sin22 positive
and cos22 approximately -1. The fact that sin22 is small means [Eq. (1.9)] that one can
at best have very little conversion in a single layer of material 2. Thus the very possibility
of total conversion makes this limit interesting.
The relation (3.3) between y2 and gives solutions independent of 2 in the large
2 limit, namely
sin cos 2
y =  .
2 2 2
cos 2 - cos

The positive y2 solution is then simply
sin cos 2
y = (3.13)
2 2 2
cos 2 - cos

Before we deal with the issue of many bilayers, let us consider the case of a single
bilayer. We show here that total conversion may not be possible in the single bilayer,
although it will always be possible for n 2. The original total conversion conditions for




10


the single bilayer are given by Eqs. (1.14). If we take y1 from the second of these
equations and substitute into the first, we find an equation for y2, namely
cos 2 cos 2 - 2
2 2 ( 2 )
1+ y = 0 (3.14)
2 cos 2
Well above MSW, 22 = - , and expanding to leading order in gives
1
2
y = - . (3.15)
2 1- tan 2
For this equation to have a valid (positive) solution, one requires that tan2 > 1, and this
will not always hold; indeed it can hold only in a decreasing domain of as E becomes
larger ( 0). This situation is illustrated in the numerical example of the next section. It
is not difficult to show that there will always be a total conversion solution in this limit
for two or more bilayers.
Let us turn next to the case of many bilayers. Given that we are interested in the
case of small , and supposing that cos2 is much larger than sin, we can replace the
denominator in this expression by sin2, and our solution becomes
y2 = cot2 sin. (3.16)
In turn, Eq. (3.11) gives us the (small) value of y1, namely y1 = y2|cos22|/cos2
sin/sin2. (In the numerical example treated in the next section, we chose = 0.7, in
which case cot2 0.17, while sin2 0.98.)
We have m 2
2 m2 2 in this limit. We also choose the minimum value /(2n)
for , and expand for small . Then we compute from our results for y1 and y2 the total
amounts of material 2 and of vacuum space to be, respectively,

nx = cot 2 and
nx = csc2 . (3.17)
2 1 2
V V
2 2

These numbers should be compared to the length of material X needed for maximum
conversion in the large 2 limit, namely [Eq. (1.10)] X = /V2. We see that if the amount
of material is the controlling issue one can gain considerably, in that one may have nx2 <<
X. However, nx1 >> X, so that if the total length of the experiment is the controlling issue
this limit is not useful. We should also recall that the maximum conversion in a single
layer of width X is asymptotically, so the very possibility of total conversion is an
attractive feature of the multiple bilayer arrangement.

IV. Numerical example

As indicated by the discussion of the previous section, the most interesting cases to look
at are those for which 2 puts one above the MSW resonance. We present two numerical
illustrations here, each for the arbitrary value of = 0.7, corresponding to a large degree
of mixing. Our bilayer consists of a layer of vacuum followed by a layer of a material 2
-
for which the potential is given by V 9
2 = 6  10 cm [2]. In the first example we assume
the energy and masses are such that one is slightly above MSW and in the second
example one is far above MSW. Our strategy is to first allow the possibility of total
conversion by fixing the thickness x1 of the first layer in terms of the second layer through
the condition that F, as defined in Eq. (2.3), vanish, i.e. through Eq. (3.11). We then plot




11


the probability for nonconversion as a function of the total width X2 = nx2 of the material
layers for various numbers n of bilayers, including the single bilayer.
For the first example, we suppose that we are slightly above MSW resonance, 22
= /2 + 0.02. Figure 1 shows the total length of material versus the probability of
nonconversion (i.e., total conversion is a zero in this plot) for n = 1, 2, 3, and 6. There is
very little dependence on the number of layers. For the parameters used one can directly
locate the first large-n zero [Eq. (3.13)], and it matches the numerical value on the plot
precisely. We have also made a variation on this calculation, in which we have shifted x1
from the zero-determining value, and we have observed the zero fill in as the shift
increases, verifying that the conversion is no longer total.
As a second example we suppose that we are far above MSW, 22 = - 0.02.
Figure 2 again shows the probability of nonconversion for n = 1, 2, 3, and 6 as a function
of the total amount of material used. In this case the single bilayer does not give total
conversion. One can see the zero move to the left (less material) as n increases, with the
overall pattern quite distinctly dependent on n.

V. Comments

We have concentrated here on the possibility of total conversion of neutrinos in
multilayer systems. It would appear that the technique is more interesting at high
energies. If these effects are ever to play a role in experiments it will be important to
understand several features that we have not looked at, including in particular the
implications of a realistic energy spread and, less importantly, the generalization to three
families. The three family calculation in principle has a richer variety of possible
outcomes for conversion experiments.
The neutral K-system presents another well-known case of oscillation. It differs
radically from the neutrino system; among other differences materials in the kaon beam
produce absorption as well as forward scattering. This system may be interesting to think
about from the point of view taken here.
Finally we remark that there is another class of effects that exploits the fact that
the order of layers matters in conversion probabilities. We shall discuss this elsewhere.


Acknowledgements

We particularly want to thank Peter Kaus for helpful suggestions, and Stephen
Gasiorowicz for useful conversations. We also want to thank Dominique Schiff and the
members of the LPTHE at Universit de Paris-Sud for their generous hospitality. This
work is supported in part by the U.S. Department of Energy under grant number
DE-FG02 -97ER41027.





12


References

1. S. P. Mikheyev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985).

2. L. Wolfenstein, Phys. Rev. D17, 2369 (1978).

3. Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Lett. B, 433, 9 (1998); ibid.,
436, 33 (1998); Phys. Rev. Lett. 81, 1562 (1998); ibid., 82, 1810 (1999); ibid., 82, 2430
(1999); E. Kearns, TAUP97, The 5th International Workshop on Topics in Astroparticle
and Underground Physics, Nucl. Phys. Proc. Suppl. 70, 315 (1999); A. Habig for the
Super-Kamiokande Collaboration, ; K. Scholberg for the Super-
Kamiokande Collaboration,  to appear in the Proceedings of the 8th
International Workshop on Neutrino Telescopes (Venice, Italy, 1999); G. L. Fogli, E.
Lisi, A. Marrone, and G. Scioscia,  to appear in the Proceedings of WIN
'99, 17th Annual Workshop on Weak Interactions and Neutrinos (Cape Town, South
Africa, 1999).

4. Kamiokande Collaboration, K. S. Hirata et al., Phys. Lett. B280, 146 (1992); Y.
Fukuda et al., Phys. Lett. B335, 237 (1994); IMB collaboration, R. Becker-Szendy et al.,
Nucl. Phys. B (Proc. Suppl.) 38, 331 (1995); Soudan-2 collaboration, W. W. M. Allison
et al., Phys. Lett. B391, 491 (1997); Kamiokande Collaboration, S. Hatekeyama et al.,
Phys. Rev. Lett. 81, 2016 (1998); MACRO Collaboration, M. Ambrosio et al., Phys. Lett.
B 434, 451 (1998); CHOOZ Collaboration, M. Apollonio et al., Phys. Lett. B420, 397
(1998).

5. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 67, 781 (1995); J. N. Bahcall,
S. Basu, and M. H. Pinsonneault, Phys. Lett. B 433, 1 (1998); J. N. Bahcall, P. I. Krastev,
and A. Yu. Smirnov, Phys. Rev. D58, 096016 (1998); B. T. Cleveland et al., Nucl. Phys.
B (Proc. Suppl.) 38, 47 (1995); Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev.
Lett. 77, 1683 (1996); GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388, 384
(1996); SAGE Collaboration, J. N. Abdurashitov et al., Phys. Rev. Lett. 77, 4708 (1996);
Liquid Scintillator Neutrino Detector (LSND) Collaboration, C. Athanassopoulos et al.,
Phys. Rev. Lett. 75, 2650 (1996); ibid., 77, 3082 (1996); Phys. Rev. C58, 2489 (1998);
Phys. Rev. Lett. 81, 1774 (1998).

6. G. L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D59, 033001 (1999). See
also G. L. Fogli, E. Lisi, and A. Marrone, Phys. Rev. D57, 5893 (1998) and references
therein.

7. M.V. Chizhov and S. T. Petcov, Phys. Rev. Lett. 83, 1096 (1999).

8. M.V. Chizhov and S. T. Petcov, .

9. M.V. Chizhov, .




13


10. See for example P. M. Fishbane and P. Kaus, J. Phys. G: Nucl. Part. Phys. 26, 295
(2000) and references therein.

11. G. D. Pusch, Nuovo Cim. A74, 149 (1983).

12. V. K. Ermilova, V. A. Tsarev, and V. A. Chechin, Kr. Soob. Fiz. [Short Notices of
the Lebedev Institute] 5, 26 (1986).

13. E. Kh. Akhmedov, preprint IAE-4470/1, 1987; E. Kh. Akhmedov, Yad Fiz. 47, 475
(1988) [Sov. J. Nucl. Phys. 47, 301 (1988)]; P. I. Krastev and A. Yu. Smirnov, Phys. Lett.
B 226, 341 (1989); E. Kh. Akhmedov, Nucl. Phys. B538, 25 (1999); E. Kh. Akhmedov,
A. Dighe, P. Lipari, and A. Yu. Smirnov, Nucl Phys. B542, 3 (1999); M. Chizhov, M.
Maris, and S. T. Petcov, ; E. Kh. Akhmedov, ; E. Kh.
Akhmedov, Pramana 54, 47 (2000).

14. E. Kh. Akhmedov and A. Yu. Smirnov, .



Figure Captions

Figure 1. The total length of material, in units of 108 cm, in a multiple bilayer system
consisting of n alternating slices of vacuum and material versus the probability of
nonconversion. Total conversion is a zero in this plot. The material has the density of the
earth, and the relative width of the layers is determined so that the factor F of Eq. (2.3) is
zero, which guarantees the possibility of complete conversion. The energy of the neutrino
beam is such that we are slightly above MSW resonance, 22 = /2 + 0.02. Plots are
drawn for n = 1, 2, 3, and 6.

Figure 2. The total length of material, in units of 106 cm, in a multiple bilayer system
consisting of n alternating slices of vacuum and material versus the probability of
nonconversion. Total conversion is a zero in this plot. The material has the density of the
earth, and the relative width of the layers is determined so that the factor F of Eq. (2.3) is
zero, which guarantees the possibility of complete conversion for n > 1. The energy of the
neutrino beam is such that we are far above MSW resonance, 22 = - 0.02. Plots are
drawn for n = 1, 2, 3, and 6.



Appendix

Here we work through a series of cases of total conversion in n bilayers in order to
develop insight to the most general case. For n = 1, Eqs. (1.14) apply, although it is useful
to repeat them here. Through n = 4 we find the conditions for total conversion (the "a"
and "b" equations refer respectively to the real and imaginary parts)



14



n = 1:
1 - y1y2 cos = 0 (A.1a)

y1cos21 + y2cos22 = 0. (A.1b)

n = 2:
1 - y 2 2 2 2
1 - y2 - 4y1y2cos + y1 y2 cos(2) = 0 (A.2a)

2[y1cos21 + y2cos22][1 - y1y2cos] = 0 (A.2b)

n = 3:
1 - 3y 2 2 2 2 2 2
1 - 3y2 + 3y1 y2 - 3y1y2cos [3 - y1 - y2 ]
+ 6y 2 2 3 3
1 y2 cos(2) - y1 y2 cos(3) = 0 (A.3a)

[y 2 2 2 2
1cos21 + y2cos22]{3 - y1 - y2 + y1 y2 - 8y1y2cos
+ 2y 2 2
1 y2 cos(2)} = 0 (A.3b)

n = 4:
1 - 6y 2 2 4 4 2 2 2 2 2 2
1 - 6y2 + y1 + y2 + 16y1 y2 - 2y1 y2 (y1 + y2 )
- 8y 2 2 2 2 2 2 2 2
1y2cos [2 - 2y1 - 2y2 + y1 y2 ] + 4y1 y2 cos(2) [5 - y1 - y2 ]
- 8 y 3 3 4 4
1 y2 cos(3) + y1 y2 cos(4) = 0 (A.4a)

-2[y 2 2 2 2
1cos21 + y2cos22]{-2 + 2y1 + 2y2 - 4y1 y2
+ y 2 2 2 2 2 2 3 3
1y2cos [10 - 2y1 - 2y2 + y1 y2 ] - 6y1 y2 cos(2) + y1 y2 cos(3)} = 0. (A.4b)

We can also write systematically pieces of terms in A for general n. As
examples, the terms in the real part that are proportional to yN (by yp we mean in general
y q p - q
1 y2 , q positive) take the form

(y1y2)ncos(n), (A.5)

while the terms in the real part that are proportional to yN - 2 are


2 2
+ -
- n y y n
n sin sin 1
2
n ( y y y y + . (A.6)
1 2 ) ( ) 1 2 ( )
1 2 sin 2 sin


The terms proportional to yN - 1 in the imaginary part are

(y1y2)n[y1cos21 + y2cos22] (A.7)






15


These terms are the largest powers of y possible in both the real and imaginary parts. The
real part contains even powers only, with the largest power yN; the imaginary part
contains odd powers only, with the largest power yN - 1.
Finally, we can also develop systematically low powers of y in the imaginary and
real parts for arbitrary (large) n. A few examples are

Real part, constant terms: 1 (A.8)

Real part, y2 terms: (n/2)[2ny 2 2
1y2cos + (n - 1)(y1 + y2 )] (A.9)

Real part, y4 terms:


( - )2n(n+ ) 2 2
1 y y cos 2 + 4n (n - 2) y y y + y
n n 1 ( 2 2 cos
1 2 1 2 1 2 )

(A.10)
4! (n 2)(n 3)
(y y )2
2 2 2 (n 3) 2 2
y y
+ - - + + +
1 2 1 2


Imaginary part, y1 terms:
n[y1cos21 + y2cos22] (A.11)

Imaginary part, y3 terms:

n (n - )
1
[y cos2 + y cos2 ]2(n+ )1 y y cos +(n-2)
( 2 2
y + y
1 1 2 2 1 2 1 2 )
3 (A.12)

We have worked out such terms all the way through the y8 terms. We have not, however,
found a way to generalize every term.





16








17








18



