Creating a Theoretical Model

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Nov 15, 2013 (3 years and 11 months ago)

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Creating a Theoretical Model

The magnetic moment is typically described as a difference between the H and B fields

(Lynn 1990, 6)
:







(



)

Because the data was taken with a measured external field H, it is necessary
to remove the B
-
field from
the above equation so that it can be used to model the experiment. In superconductors, The H
-
field

is
defined in terms of the free energy density per unit mass

(Schneider and Singer 2000, 64)
:








Thus, in order to express H in terms of B, the derivative of F must first be expressed in terms of B. The
free energy density in the Shubnikov phase has been evaluated using the London approach by De
Gennes, arriving at the following expressio
n in terms of F (the free energy per unit mass), B (the external
field),
λ

(the London penetration depth), L (the average spacing between flux lines),
ξ

(the coherence
length of the cooper pairs), and b (a constant of order one that depends on the lattice
structure)

(Schneider and Singer 2000, 64)
:


















However, such a relationship must also be converted in terms of constant and the external field. To do
this, the average length between flux lines c
an be expressed as


(Schneider and Singer 2000, 64)
:















The higher critical field H
c2

is the field above which all internal superconductivity ceases. This occurs
when the length over which the field can be screened by a superconducting vortex from a flux quantum
to zero is smaller than the coherence length of the cooper pairs. Based on thi
s principle, scientists have
developed the following relationship, which is then changed to the form needed for this investigation

(Schneider and Singer 2000, 58)
:


































By substitu
ting in these equations for L and

, the free energy density is expressed in terms of only fields
and London penetration depth:






























































































Due to the length of this investigation, it is here necessary to substitute the B
-
field with the H
-
field
value. Although this creates a very small inaccuracy in the model, the approximation allows the
magnetic moment to be expressed in t
erms of B
-
field alone, which is necessary for this investigation.











































Taking the derivative of the above equation in terms of B allows its substitution into the original
equation for
H
-
field:

































(


















)



















Note that the free energy density was per unit mass, so the magnetic moment per unit mass can be
found by substituting this
expression of H into the original equation and simplifying:








(



)








(
(















)


)








(
(















)


)








(













)











(








)



















Now that an equati
on for magnetization has been obtained in terms of H, the variables of
λ

and H
c2

must
be converted in terms of temperature.

It is known that the higher critical field H
c2

that a superconductor can withstand varies with
temperature based on the critical fie
ld at 0
˚
K (H
0
), temperature (T), and critical temperature (T
c
)

(Solymar and Walsh 1979, 355)
:







(


(



)

)


The implications of such a relationship hold under logical scrutiny. When the temperature is
equal to the
critical temperature above which superconductivity cannot exist, the critical field would be
0. As the temperature drops, the critical field a superconductor can withstand increases, in agreement
with the relationship.


The London penetration depth is know
n to vary with temperature with a constant of the
penetration depth at 0
˚
K (
λ
0
)

(Buckel 1991, 118)
:









(



)



Although this relationship is not intuitive, it models the expected behavior. When the
temperature is a
t critical, the London penetration depth would tend to infinity, meaning that the
magnetic flux would permeate the entire sample and there would be no superconducting regions. As
the temperature decreased, the London penetration depth would also decrease a
nd the field would
penetrate the sample less.


By substituting these two relationships, the equation for magnetism can be written in the
desired form necessary to model the data for the experiment:



























(








(



)

)











(


(



)

)







(


(



)

)













(


(



)

)

Now that the formula for the magnetic moment has been converted to a function of temperature, mass,
and H
-
field, it is possible to utilize known constants to model the data.

-



, the flux qua
ntum, has a known value of 2.0
7*10
-
15

Wb

(Lynn 1990, 10)

-




has been studied in other samples of
YBa
2
Cu
3
O
6.91

and has a known value of 140nm

(Weingstock 2000, 302)
.

-




has

a known value of 250T

(Weingstock 2000, 302)
.

-



, the critical temperature for the sample, which is known to be in the realm of approximately
90
˚
K

(Weingstock 2000, 11)
. This is ve
rified by the data of this investigation where one can
observe that the sample enters a weak paramagnetic state at a temperature of 92.5
˚
K.

-

As H approaches and reaches H
0
, the magnetic moment should vanish entirely as
superconductivity ceases when the ex
ternal field is equal to the critical field. In order for this to
occur, the natural log term must become 0 as it is the only part of the equation dependent on
field. A



value of 1 is used to attain this, meaning there is no correction for the lattice st
ructure.








Using these values and inputting the external field and range of temperatures in the experiment into the
equation, a graph for the theoretical magnetic moment is obtained. The results are displayed below:

-



-4.500000E-08
-4.000000E-08
-3.500000E-08
-3.000000E-08
-2.500000E-08
-2.000000E-08
-1.500000E-08
-1.000000E-08
-5.000000E-09
0.000000E+00
5.000000E-09
30
40
50
60
70
80
90
100
Magnetic Moment / A * m
2

Temperature / ˚K

Zero Field Cooling
10 Gauss Field Cooling
Theoretical Magnetization