1
KmatrixElements.nb
Calculating the Matrix Elements for 39 K
Determining the
3
3
J’ 2 , I
F ’ 3, m ’ 3 11 J
2
element from the measured lifetime
1
2
3
2
,I
F
2, m
2 matrix
The lifetime of the L 1 excited states in 39 K , , has been measured to be 25.8 ns. All but one of the excited states can
spontaneously decay to several ground states. Because F ’ 3, m’ 3 decays only to F 2, m 2 , the spontaneous
4 k3
2
lifetime is related to
J ’ 32 , I 32 F ’ 3, m’ 3 11 J 12 , I 32 F 2, m 2 by 1sp
.
3
1.054 10 erg s ; k 2 cm ; 766.7 10 3 esu cm 4k 7.46121 10 esu cm.
Generally, is put into Debye (D). 1 D 10
10 27
1
7
s ;
25.8 10 9
3
18
18
18
7.46121
3
2
Calculating the J ’
matrix elements
3
2
1
q
F’
3, m ’
Calculating the reduced matrix element J ’ 32 , I 32
Using JET 92.1:
,I
1
2
J
F’ 3
1
,I
J
3
2
F
12 , I 32
2, m
F 2
j’ 1 j
n’ l’ s j’ 1 n l s j
m’ q m
we can calculate the matrix elements for other transitions between these two F−levels. Knowing the left hand side of the
equation for one transition allows us to calculate the reduced matrix element in the F I J basis. Rewriting 92.1:
F’ 1 F
n’ J ’ I F ’ m’ 1q n J I F m
1 F’ m’
n’ J ’ I F ’ 1 n J I F
m’ q m
The following function calculates the entire coefficient on the right side of the equation:
n’ l’ s j’ m’
1
q
n ls jm
1
j’ m’
threeJcoeff fp_, mp_, q_, f_, m_ :
1 fp mp
ThreeJSymbol fp, mp , 1, q , f, m
2
KmatrixElements.nb
threeJcoeff 3, 3, 1, 2, 2
1 7
F’ 3 , I F’ This gives J ’ 32 , I
determine all of the J ’
3
2
1
3
2
Calculating the J ’ 3
2
32 , I 32
F 2 7
J ,I F
J 12 , I
3, m’ 1q
F ’ 3, m ’
We will express all matrix elements in terms of
and 2 for F 2) performs the calculation:
3
2
1
2
1
q
J
f232 3, 1, 2
1
f232 2, 0, 2
1 3
f232 1,
1, 2
1 15
f232 2, 1, 1
23 f232 1, 0, 1
15
2 2
f232 0,
1, 1
1 5
f232 1, 1, 0
25 12 , I 32
F 2, m matrix elements
as calculated above. The function f232 (2 for the D2 line, 3 for F ’
f232 mp_, q_, m_ : threeJcoeff 3, mp, q, 2, m
Calculations
. Using this result and 92.1 repetitively, we an
2, m matrix elements.
3
2
7
3
3
KmatrixElements.nb
f232 0, 0, 0
35 Results in
!
units
, I F ’ 3, m’ 3 J , I F 2, m 2 1
, I F ’ 3, m’ 2 J , I F 2, m 2 J ’ , I F ’ 3, m’ 1 J , I F 2, m 2 J ’ , I F ’ 3, m’ 2 J , I F 2, m 1 J ’ , I F ’ 3, m’ 1 J , I F 2, m 1 2 J’ ,I
F ’ 3, m’ 0 J , I F 2, m 1 J ’ , I F ’ 3, m’ 1 J , I F 2, m 0 J ’ , I F ’ 3, m’ 0 J , I F 2, m 0 The remaining elements can be determined using the following rule:
F ’, m’ F, m F’ 1 F
1 Fm’’ 1q mF 7 1 m’
q m 7 " 1 3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
J’
J’
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
1
1
1
0
1
1
2
1
2
1
2
1
1
2
1
2
1
1
1
0
1
3
2
3
2
1
3
2
3
2
3
2
1
2
1
2
1
2
1
1
1
0
1
3
3
2
1
15
2
3
1
5
2
5
3
5
3
2
3
2
2
15
1
q
F’ m’
m’ F 1
F’ F 2 m’ 1
for 39 K the 2×m will be even, we can write:
F ’, m’ 1q F, m = 1 F’ F 1 F ’, m’ 1 q F, m
For transitions between these two levels,the quantity
1
The matrix elements can be checked by using the sum−rule:
'&
F ’, m’
m’,m
For q
1
q
F, m
2
F’
1 ( 2 ( 1 ( 2 ) 7 ( 15
3
5
5
3
True
For q
0:
2
8 ( 3 ) 7 13 ( 15
5
3
True
#$ 1 % % 1:
1
1
3
F
2
7
3
F’ F 1
3 2 1
1.
Since
F ’,
m’
1
q
F,
m
4
KmatrixElements.nb
Calculating the other D2 matrix elements
The reduced matrix elements in the D2 line
To calculate the matrix elements between different F−levels of the D2 line, we can use JET
104.1:
J’ I F’
J’ I F’ 1 J I F
1 J’ I F 1
2 F’ 1 2 F 1
J’ 1 J
F 1 J
The function sixJcoeff evaluates the coefficient on the right side of the equation:
** *
+
, , ,
(
(
J’ , I F’ 3 J by evaluating :
+
sixJcoeff jp_, i_, fp_, j_, f_ :
1 jp i f 1
2 fp 1 2 f 1 SixJSymbol
We
J’
know
3
2
1
1
2
3
2
3
2
1
J
, I F
1
2
3
2
jp, i, fp , f, 1, j
2
7
,
so
we
can
calculate
sixJcoeff 3 2, 3 2, 3, 1 2, 2
2 7 This gives J ’
the D2 line.
3
2
1
J
2
1
2
.We can use this result and 104.1 again to calculate other reduced matrix elements in
g2 fp_, f_ : sixJcoeff 3 2, 3 2, fp, 1 2, f
Calculations
g2 3, 2
7
g2 2, 2
- 52 g2 1, 2
1 2
g2 2, 1
52 2
5
KmatrixElements.nb
g2 1, 1
- 52 g2 0, 1
1
Results in
,I
J’ , I J’ , I J’ , I J’ , I J’ , I units
3
2
3
2
3
2
J’
!
3
2
3
2
3
2
3
2
3
2
3
2
F’
F’
F’
3
2
3
2
3
2
3
F’
1
3
2
3
2
3
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
J
1
2
1
, I
J , I
J , I
J , I
J , I
J , I
1
F
F
F
3
2
3
2
3
2
7
' ' 2
2
2
F
5
2
1
F’ 1 1
F 1
1
F’ 0
F 1 1
To check these results, we can use the following sum−rule:
2
2
J’ I F’ 1 J I F
J’ 1 J
2F
F’
For F
2, J
, we have:
1
2
2
7
( 52 2
( 1 )
2
22
2
5
2
1
2
5
2
+ * 1
2J 1
2 2 ( 1 2 1 2 ( 1
True
For F
1, J
, we have:
1
2
5 2
2
( 52 (
2
1
2
)
22
2 1 ( 1 2 1 2 ( 1
True
Calculating the J ’ 32 , I 32
F ’ 2, m ’
, I F’ 2 J , I F 2 and F 2. Defining f222:
With J ’
between F ’
3
2
3
2
1
1
2
3
2
2
1
q
J
. 5
2
f222 mp_, q_, m_ : threeJcoeff 2, mp, q, 2, m
12 , I 32
F 2, m matrix elements
and JET 92.1 we can calculate the matrix elements
52 6
KmatrixElements.nb
Calculations
f222 2, 0, 2
- 1 3
f222 1,
1, 2
- 1 6
f222 2, 1, 1
1 6
f222 1, 0, 1
- 1 2
3
f222 0,
1, 1
- 12 f222 1, 1, 0
12 f222 0, 0, 0
0
Results in
,I
J’ , I J’ , I J’ , I J’ , I J’ , I J’ , I !
units
, I F 2, m 2 . F ’ 2, m’ 1 J , I F 2, m 2 . F ’ 2, m’ 2 J , I F 2, m 1 F ’ 2, m’ 1 J , I F 2, m 1 . F ’ 2, m’ 0 J , I F 2, m 1 F ’ 2, m’ 1 J , I F 2, m 0 F ’ 2, m’ 0 J , I F 2, m 0 0
The # for these two levels is #$ 1 ' 1.
J’
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
F’
2, m’
2
1
0
1
1
1
1
0
1
J
1
2
3
2
1
2
1
1
2
1
2
1
1
2
1
2
1
1
1
0
3
2
3
2
3
2
1
2
3
2
3
2
1
6
3
2
1
2
1
3
1
6
1
2 3
1
2
2 2 1
Checking these results using the sum−rule
For q
'&
1:
m’,m
F ’, m’
1
q
F, m
2
% :
F’
1
3
F
2
7
KmatrixElements.nb
( ( ( ) 1 5 2
2
3
2
2
1
2
1
6
1
2
1
2
2
6
True
For q=0:
1 ( 1 2
2
3
2
3
2
( 0 ) 13 52 2
True
Calculating the J ’ 32 , I 32
F ’ 1, m ’
1
q
J
f212 mp_, q_, m_ : threeJcoeff 1, mp, q, 2, m
12 , I 32
1 2
Calculations
f212 1,
1, 2
1 10
f212 1, 0, 1
- 1 2
5
f212 0,
1, 1
1 2
5
f212 1, 1, 0
1 2
15
f212 0, 0, 0
- 1 15
F 2, m matrix elements
8
KmatrixElements.nb
Results in
!
units
, I F’ J’ , I F’ J’ , I F’ J’ , I F’ J’ , I F’ #$ 1 1
J’
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
1 2 1
1, m’
1, m’
1, m’
1, m’
1, m’
1 0 1 0 1
1
1
0
1
1
, I F 2, m 2 J , I F 2, m 1 . J , I F 2, m 1 J , I F 2, m 0 J , I F 2, m 0 . 1
2
J
3
2
1
2
1
1
1
1
0
1
10
1
2 5
1
2 5
1
2 15
1
15
3
2
1
2
3
2
1
2
1
2
3
2
3
2
1 ( 1 ( 1 ) 1 1 3
2
2
10
2
2
5
2
15
2
2
True
1 ( 1 ) 1 1 3
2
2
2
5
2
15
2
2
True
Calculating the J ’ 32 , I 32
F ’ 2, m ’
1
q
f221 mp_, q_, m_ : threeJcoeff 2, mp, q, 1, m
Calculations
f221 2, 1, 1
1 2
f221 1, 0, 1
12 f221 0,
1, 1
1 2
3
f221 1, 1, 0
12 f221 0, 0, 0
1 3
12 , I 32
J
5 2
F 1, m matrix elements
9
KmatrixElements.nb
Results in terms of
,I
J’ , I J’ , I J’ , I J’ , I #$ 1 J’
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
2 1 1
2, m’
2, m’
2, m’
2, m’
2, m’
F’
F’
F’
F’
F’
!
, I F 1, m 1 1 J , I F 1, m 1 0 J , I F 1, m 1 1 J , I F 1, m 0 0 J , I F 1, m 0 2
1
1
1
0
1
1
2
1
2
J
1
1
2
1
2
1
2
1
1
1
0
3
2
3
2
1
2
3
2
3
2
3
2
1
2
1
2
1
2
3
1
3
1
( ( ) 1 2
3
2
1
2
2
1
2
1
2
3
2
5
2
True
1 ( 1 ) 1 2
3
2
2
2
3
5 2
2
True
Calculating the J ’ 32 , I 32
F ’ 1, m ’
1
q
J
f211 mp_, q_, m_ : threeJcoeff 1, mp, q, 1, m
Calculations
f211 1, 0, 1
- 2 5
3
f211 0,
1, 1
- 2 5
3
f211 1, 1, 0
2 5
3
f211 0, 0, 0
0
12 , I 32
52 F 1, m matrix elements
10
KmatrixElements.nb
!
Results in
units
, I F ’ 1, m’ 1 J , I F 1, m 1 . / / //
J ’ , I F ’ 1, m’ 0 J , I F 1, m 1 . //
J ’ , I F ’ 1, m’ 1 J , I F 1, m 0 J ’ , I F ’ 1, m’ 0 J , I F 1, m 0 0
#$ 1 . 1
1 5 ( 1 5 ) 1 5 2
3
2
3
3
2
J’
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
1 1 1
1
2
1
0
1
1
2
1
2
2
5
3
3
2
1
2
1
2
1
1
1
0
5
3
3
2
2
5
3
3
2
3
2
2
2
2
True
1 5 ( 0 ) 1 5 2
3
3
2
2
2
2
True
32 , I 32 F ’ 0, m ’ 1q J 12 , I 32
: threeJcoeff 0, mp, q, 1, m 1
Calculating the J ’ f201 mp_, q_, m_
Calculations
f201 0,
1, 1
1 3
f201 0, 0, 0
- 1 3
Results in
!
units
, I F ’ 0, m’ J ’ , I F ’ 0, m’ #$ 1 . 1
1 ) 1 1
J’
3
2
3
2
0
3
2
3
2
1 1 1
0
2
3
True
3
2
1
1
0
1
, I F 1, m 1 J , I F 1, m 0 . 1
2
J
1
2
3
2
3
2
1
3
1
3
F 1, m matrix elements
11
KmatrixElements.nb
1 ) 1 2
1
2
3
3
True
Calculating the D1 matrix elements
12
12 1 l 0, s 12 J 12 matrix element
To get l’ 1, s J ’ l 0, s J and the reduced matrix elements in the D line, we must reduce
the
l’ 1, s J ’ l 0, s J to the l basis usingJET 104.1 again:
l’ s J ’
*
*
*
l’ s J ’ ls J 1
2 J’ + 1 2 J + 1
l’ 1 l 0
J 1 l
Determining the l ’ 1, s 1
2
1
2
3
2
1
2
J’ 1
2
1
1
2
1
1
2
1
2
1
l’’ s J 1
1
1
sixJcoeff 1, 1 2, 3 2, 0, 1 2
2 3
l’ 1 l 0 3 .
Then, in units, l’ 1, s With l’
1
2
1, s
3
2
J’
1
l
0, s
J 2
1
2
1
2
:
1
1
2
J’
1
2
1
l
sixJcoeff 1, 1 2, 1 2, 0, 1 2
-
l’
0, s
J 1
2
1
2
is:
3
2
1
2
1, s
J’
1
2
1
l
0, s
J .
1
2
1
2
2
The reduced matrix elements in the D1 line
With
l’
J’ I F’
1, s
1
1
2
J’
JI F
g1 fp_, f_
Calculations
g1 2, 2
- 52 l 0, s J = 0 2 , the D reduced matrix elements are given by
1 * * * 2 F ’ + 1 2 F + 1 JF’ 1I FJ’ J ’ J using the function g1:
2
: sixJcoeff 1 2, 3 2, fp, 1 2, f 1
2
1
J’ I F 1
1
2
1
2
1
1
12
KmatrixElements.nb
g1 1, 2
52 g1 2, 1
- 52 g1 1, 1
1 2
Results
,I
J’ , I J’ , I J’ , I 1
2
1
2
1
2
1
2
J’
3
2
3
2
3
2
3
2
F’
F’
2
1
1
1
F’ 2 1
F’ 1 1
Checking with the sum−rule:
J’ I F’ 1 J I F
F’
For F
, I
J , I
J , I
J , I
2, J
2
, we have:
1
2
1
2
1
2
1
2
J
3
2
3
2
3
2
3
2
J’
F
F
F
F
1
2
2
1
1
2
5 ) 2
2
2
5
2
5
2
1
2
5
2
2 2F 1
2J 1
J
1
2
5 (
2
' ' ** 2 2 ( 1 2 ( 1
1
2
2
True
For F
1, J
, we have:
1
2
5 ( 1 ) 2
2
2
2
2
2
2 1 ( 1 2 ( 1
1
2
True
Calculating the J ’ 12 , I 32
F ’ 2, m ’
1
q
J
f122 mp_, q_, m_ : threeJcoeff 2, mp, q, 2, m
12 , I 32
52 F 2, m matrix elements
13
KmatrixElements.nb
Calculations
f122 2, 0, 2
- 1 3
f122 1,
1, 2
- 1 6
f122 2, 1, 1
1 6
f122 1, 0, 1
- 1 2
3
f122 0,
1, 1
- 12 f122 1, 1, 0
12 f122 0, 0, 0
0
Results
,I
J’ , I J’ , I J’ , I J’ , I J’ , I J’ , I , I F 2, m 2 . F ’ 2, m’ 1 J , I F 2, m 2 F ’ 2, m’ 2 J , I F 2, m 1 F ’ 2, m’ 1 J , I F 2, m 1 . F ’ 2, m’ 0 J , I F 2, m 1 F ’ 2, m’ 1 J , I F 2, m 0 F ’ 2, m’ 0 J , I F 2, m 0 0
The # for these two levels is #$ 1 ' 1.
1 ( 1 ( 1 ( 1 ) 1 5 2
2
3
2
J’
1
2
1
2
1
2
1
2
1
2
1
2
1
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
F’
2, m’
2
1
0
1
1
1
1
0
1
J
1
2
3
2
1
2
1
3
2
1
2
1
2
1
1
2
1
2
1
1
1
0
3
2
3
2
1
2
3
2
3
2
1
6
3
2
1
2
2 2 1
2
6
True
2
6
2
2
2
1
3
1
6
1
2 3
1
2
14
KmatrixElements.nb
1 ( 1 2
2
3
2
2
3
(
0
2
) 1 5 3
2
2
True
12 , I 32
Calculating the J ’ F ’ 1, m ’
1
q
J
f112 mp_, q_, m_ : threeJcoeff 1, mp, q, 2, m
Calculations
f112 1,
12 , I 32
5 2
1, 2
1 2
f112 1, 0, 1
- 12 f112 0,
1, 1
12 f112 1, 1, 0
1 2
3
f112 0, 0, 0
- 1 3
Results
, I F’ J’ , I F’ J’ , I F’ J’ , I F’ J’ , I F’ #$ 1 1
J’
1
2
1
2
1
2
1
2
1
2
3
2
3
2
3
2
3
2
3
2
1 2 1
1, m’
1, m’
1, m’
1, m’
1, m’
1 0 1 0 1
1
1
0
1
1
, I F 2, m 2 J , I F 2, m 1 . J , I F 2, m 1 J , I F 2, m 0 J , I F 2, m 0 . 1
2
J
1
2
1
1
1
1
0
1
2
1
2
1
2
( ( ) 1 2
3
1
2
True
2
1
2
2
1
2
3
3
2
3
2
3
2
3
2
1
2
1
2
1
2
1
2 3
1
3
3
2
5
2
2
F 2, m matrix elements
15
KmatrixElements.nb
( ) 1 2
3
2
1
2
2
2
1
5
2
3
True
12 , I 32
Calculating the J ’ F ’ 2, m ’
1
q
J
12 , I 32
Calculations
f121 mp_, q_, m_ :
threeJcoeff 2, mp, q, 1, m
52 f121 2, 1, 1
- 1 2
f121 1, 0, 1
- 12 f121 0,
1, 1
- 1 2
3
f121 1, 1, 0
- 12 f121 0, 0, 0
- 1 3
Results
,I
J’ , I J’ , I J’ , I J’ , I #$ 1 J’
1
2
1
2
1
2
1
2
1
2
3
2
3
2
3
2
3
2
3
2
2 1 1
2, m’
2, m’
2, m’
2, m’
2, m’
F’
F’
F’
F’
F’
1
, I F 1, m 1 . 1 J , I F 1, m 1 . 0 J , I F 1, m 1 . 1 J , I F 1, m 0 . 0 J , I F 1, m 0 . 2
1
1
1
0
1
1
1
1
0
J
1
2
1
2
1
1
2
1
2
1
2
3
2
3
2
3
2
3
2
3
2
1
2
1
2
1
2
1
2
1
3
3
F 1, m matrix elements
16
KmatrixElements.nb
( 1 ( 1 ) 1 5 2
3
2
2
1
2
2
2
2
2
3
True
1 ( 1 ) 1 5 2
3
2
2
2
2
2
3
True
12 , I 32
Calculating the J ’ F ’ 1, m ’
1
q
12 , I 32
J
f111 mp_, q_, m_ : threeJcoeff 1, mp, q, 1, m
1 2
Calculations
f111 1, 0, 1
1 2
3
f111 0,
1, 1
1 2
3
f111 1, 1, 0
- 1 2
3
f111 0, 0, 0
0
Results
,I
J’ , I J’ , I J’ , I #$ 1 J’
1
2
1
2
1
2
1
2
, I F 1, m 1 F ’ 1, m’ 0 J , I F 1, m 1 F ’ 1, m’ 1 J , I F 1, m 0 . F ’ 1, m’ 0 J , I F 1, m 0 0
. 1
1 ( 1 ) 1 1 3
2
3
2
3
2
3
2
1 1 1
F’
1, m’
1
True
1
0
1
J
3
1
2
3
2
1
2
1
3
2
3
2
2
2
3
2
3
1
3
2
1
2
1
2
1
1
1
0
2
2
2
2
3
1
2
3
1
2 3
F 1, m matrix elements
17
KmatrixElements.nb
1 (
2
2
2
True
3
0
2
) 1 1 3
2
2