Total angular momentum of water molecule and magnetic eld interaction C. H. Zepeda Fern andez12C. A. L opez T ellez2 Y. Flores Orea2 and E. Moreno Barbosa2 1C atedra CONACyT 03940 CdMx Mexico

2025-05-06 2 0 1.08MB 6 页 10玖币
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Total angular momentum of water molecule and magnetic field interaction
C. H. Zepeda Fern´andez1,2,∗C. A. L´opez T´ellez2, Y. Flores Orea2, and E. Moreno Barbosa2
1C´atedra CONACyT, 03940, CdMx Mexico
2Facultad de Ciencias F´ısico Matem´aticas,
Benem´erita Universidad Aut´onoma de Puebla,
Av. San Claudio y 18 Sur,
Ciudad Universitaria 72570, Puebla, Mexico
Abstract
One of the must important non-invasive techniques in medicine is the Magnetic Resonance Imaging
(MRI), it is used to obtain information of the structure of the human body parts using three
dimensional images. The technique to obtain these images is based by the emission of radio waves
produced by the protons of the hydrogen atoms in water molecules when placed in a constant
magnetic field after they interact with a pulsed radio frequency (RF) current, the spin of the protons
are in a spin excited state. When the RF field is turned off, the MRI sensors are able to detect the
energy released (RF waves) as the protons realign their spins with the magnetic field.
We used a three particles model for the water molecule: the two protons from the hydrogen atoms
move around the doubly negatively charged oxygen (unstructured), to describe the total angular
momentum. The energy levels from the water molecule are studied in presence of an uniform external
magnetic field, which interacts with the proton’s spin and orbital angular momentum. The energy
is shifted and the degeneration is lifted. To illustrate the results, we provide numerical results for a
magnetic field strength commonly used in MRI devices.
Keywords: Magnetic resonance imaging, water molecule, total angular momentum, energy levels, quantum
mechanics
I. INTRODUCTION
One of the must important non-invasive techniques
in medicine to obtain information of the structure
and composition of the human body parts using three
dimentional images is the Magnetic Resonance Imaging
(MRI) [1–3]. The images obtained are high resolution
compared to other techniques that are invasive, as the
Computed Tomography (CT) [4–6], which requires using
x-rays [7–9]. The procedure to obtain high resolution
images is constantly development [10–13].
The MRI scanner is a complex device [14], which
needs a strong and constant magnetic field between
0.5 and 7 T. This magnetic field interacts with the
spin of the in the hydrogen atoms that make up the
water molecules, which make up the human body by
80% [15]. Some spins are oriented parallel and some
others anti-parallel with respect to the magnetic field
direction. Then, the radio frequency coils (RFC) [14]
produce a radio frequency field (RF), which reverse the
orientation of the spins (parallel becomes anti-parallel
and vice versa). When, the RF is turned off, the spins
return to their original orientation. In this transition,
the protons produce radio waves that are detected by
the MRI device and finally the image is produced [14].
The protons and neutrons (from other atoms that
make up other molecules in the human body) do not
contribute to the magnetic interaction, because they are
∗Corresponding author, email:hzepeda@fcfm.buap.mx
mostly spin-paired. On the other hand, the spin of the
electrons that are not in spin-paired, also interact with
the magnetic field, however, they radiate in microwaves,
which are not detected by the RFC (see section II of [16]).
We use a simple model for the water molecule,
which consists of a triangle shape at whose vertices
are the two protons and the oxygen atom (doubly
charged negative and unstructured), this shape is due
to the electrostatic interaction, holding the distance
between protons constant (151.05 pm) and the distance
between the protons and the oxygen, also constant
(95.60 pm) [17]. In a previous work [16], we showed
the energy levels for the protons of the hydrogen atoms
in the water molecule, where it was also shown the
breakdown of degeneracy caused by the interaction with
an external constant magnetic field, for the quantum
numbers m1and m2(the magnetic quantum number
for each proton), nevertheless, the degeneration remains
in l1and l2(the orbital angular momentum quantum
number for each proton). In this previous work, the
interaction between the spin of the protons and the
constant external magnetic field was not considered,
only the interaction with the orbital angular momentum
was considered.
In the present work, we describe the interaction between
the total angular momentum (ˆ
J=ˆ
L+ˆ
S) of the water
molecule with the MRI magnetic field. The model can
be thought as a hydrogen atom, then, we make this
analysis analogous to the anomalous Zeeman effect. The
work is organized as follows: In Section II we show the
arXiv:2210.01867v1 [physics.med-ph] 4 Oct 2022
2
states of the total angular momentum. In Section III
we introduce a constant magnetic field to obtain the
fine structure for our water molecule model and we
find the total energy considering the total angular
momentum. Finally, in Section IV we discuss our results
and conclude.
II. TOTAL ANGULAR MOMENTUM OF
WATER MOLECULE
As already described the protons of the hydrogen atom
of the water molecule, are the particles of our analysis,
due to their interaction with the magnetic field in the
MRI study. It is well known that their spin is 1/2, then,
the total angular momentum follows the formalism of
angular momentum addition: the sum of both spins and
the orbital angular momentum l1and l2. As we know,
the addition of angular momentum, it is carried out two
by two. We first add liand their respective spin 1/2,
for both protons, i= 1,2. For this work, we use ito
refer to proton 1 or proton 2. The case li= 0 implies
that there is no angular momentum and the total angular
momentum is purely spin angular momentum. For the
case of li>0, the resulting angular momentum ji, only
has two possibilities ji=li±1/2. Then, the states are
giving by:
|ji;mi>li=li±1
2;miEli
=
sli∓mi+1
2
2li+ 1 li,1
2;mi+1
2,−1
2E±
sli±mi+1
2
2li+ 1 li,1
2;mi−1
2,1
2E,
(1)
where, mi=−ji,−ji+ 1..., ji−1, jiand |li,1
2;mi±
1
2,∓1
2>=|li;mi±1
2>|1
2;∓1
2>is the tensor product of
the spatial part and the spin part. The spin part is
1
2;1
2E=1
0,
1
2;−1
2E=0
1(2)
Then, we can write Eq 1 as follows:
Φji=li±1
2,mi(θi, φi) = 1
√2li+ 1×
±qli±mi+1
2Yli,mi−1
2(θi, φi)
qli∓mi−1
2Yli,mi+1
2(θi, φi)
.
(3)
Where the Yli,mi±1
2(θi, φi) are the spatial functions
(spherical harmonics) for the water molecule [16].
Finally, we add j1and j2, where, it is obtained by four
possible values of the total angular momentum j, giving
by the combinations:
1. j1=l1−1/2 and j2=l2−1/2
2. j1=l1−1/2 and j2=l2+ 1/2
3. j1=l1+ 1/2 and j2=l2−1/2
4. j1=l1+ 1/2 and j2=l2+ 1/2
The total function is represented by |j;m >(l1,l2)
(j1,j2)and
m=−j, j + 1..., j −1, j. We can express the state
|j;m >(l1,l2)
(j1,j2)in terms of the Clebsh-Gordan coefficients
(l1,l2< j1, j2;m1, m2|j;m >) as:
|j;m >(l1,l2)
(j1,j2)=X
m1X
m2
l1,l2< j1, j2;m1, m2|j;m > ×
|j1, j2;m1, m2>l1,l2.
(4)
Where
|j1, j2;m1, m2>l1,l2=|j1;m1>l1⊗|j2;m2>l2(5)
represents the Kronecker product. The wave function
is giving by
Ψl1,l2;m1,m2;1
2,±1
2(θ1, φ1, θ2, φ2) =
Φl1±1
2,m1(θ1, φ1)Φl2±1
2,m2(θ2, φ2).(6)
A. Ground state of water molecule
Fot the Clebsh-Gordan coefficients, it is well known
that < j1, j2;m1, m2|j;m >6= 0 only when m=m1+
m2. Other wise, when m6=m1+m2we have <
j1, j2;m1, m2|j;m >= 0, then, following this rules it can
be obtained the states |j;m > for a giving j1and j2.
Note, that the water molecule model is a fermion sys-
tem, then, the protons follow the Pauli exclusion prin-
ciple. For the ground state, we have l1=l2= 0 and
m1=m2= 0, therefore, one of the spin protons must
have +1/2 value and the other the −1/2 value. The total
angular momentum takes the value j= 0,1 and the only
two states allowed are
|0; 0 >(0,0)
(1
2,1
2)=
1
2,1
2;−1
2,1
2E00
|1; 0 >(0,0)
(1
2,1
2)=
1
2,1
2;1
2,−1
2E00
(7)
B. First state of water molecule
The first excited state must occur with li= 1 and the
other remains in li= 0. For this case all combinations
of the quantum numbers are allowed. To exemplify, we
take l1= 1, then, j1=1
2,3
2; which the six states can be
obtained by Eq. 1. The quantum numbers for the second
proton are l2= 0 and s2=1
2, then j2=1
2. We describe
the two sub-spaces below.
摘要：

Totalangularmomentumofwatermoleculeandmagnetic eldinteractionC.H.ZepedaFernandez1;2,C.A.LopezTellez2,Y.FloresOrea2,andE.MorenoBarbosa21CatedraCONACyT,03940,CdMxMexico2FacultaddeCienciasFsicoMatematicas,BenemeritaUniversidadAutonomadePuebla,Av.SanClaudioy18Sur,CiudadUniversitaria72570,Puebl...

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