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On the statistical mechanics in a magnetic field
Physica
XII,
no
I
April
ON THE STATISTICAL IX A MAGNETIC
1946
MECHANICS FIELD
by L. J. I;. BROER Communication
from
thr Zccman
Laboratory
of the University
of Xmstrrdnm
Summary The fundamental relations of the statistical mechanics and thermodynamics of magnetised matter are derived in a systematic way, starting from the most general form of the I, a g r a n g i a n of matter and magnet. The H a m i 1 t o n equations of the matter alone are established by means of the R o u t h function. The well-known fact that the corresponding H a m i 1 t o n function instead of the total energy of the matter must be used in the partition function is rigourously proved. A comparison is made between the various quantities of energy which enter in the processes of magnetisation and electrisation respectively.
Introduction. It has been well asserted that the equations of motion of a system of charged particles in a constant magnetic field can be brought in the H a m i 1 t o n form. In the H a m i 1 t o n function however not all the terms of the total energy containing the coordinates of the particles will occur. When the system is quantisised the characteristic values IV,, of the H am i 1t o n operator therefore will not represent the total energy of the stationary states of the system. Nevertheless one can, as will be generally known, derive the thermodynamical properties of the system from the partition function : Wtl Z = t exp. - kT (1) by applying the usual formalism of statistical mechanics. This formalism has been developed under the assumption that W,, is the total energy of a closed system. Its use in the case of systems in a magnetic field therefore needs some further considerations. The correct way to Physica
XII
49 4*
50
L.
J. F. BROER
handle (1) in this case is current practice since long and seems never to have given rise to uncertainties. We were somewhat surprised therefore in not succeeding to find in litterature a wholly satisfactory discussion of this question. G o r t e r ‘) gives, following some considerations of L o r e n t z 2), a detailed treatment of the energetics involved in the magnetisation process whereas the question had been discussed from a thermodynamical point of view among others by F o k k e r “). These authors did not seek a close connection with the H am i 1 t o n i a n method and with statistical mechanics and confined themselves to homogeneous fields. Furthermore G o rt e r and L o r e n t z do not introduce some hypothetical device to keep the magnetic field constant under fluctuations of the magnetisation. As this field figures as a constant parameter in the H a m i lt o n i a n of the magnetic matter this method is not quite appropriate to our purposes. We will now derive the fundamental formulae of statistical mechanics and thermodynamics starting from the most general form of the L a g r a n g i a n function of matter plus field currents. These formulae will be, strictly spoken, valid for a constant field and therefore concur with the usual H a m i 1 t o n i a n operators. The method employed, viz. the use of the R o u t h function, has been adopted from the work of V a n L e e u w e n 4) on this problem. Her treatment is not general because she uses special kinetical models, furthermore it will appear that the formulation can be made much simpler than that of V a n L e e u w e n. 5 1. The equations of motion 0.t the system. We will consider a system of charged particles described by a set of generalised coordinates ql, . . . . q,, placed in the field of an electromagnet. It is not yet necessary to suppose that the field is hom?geneous. The state of the magnet is defined by the coordinate qO= / Jdt, which indicates
the electrical charge passed through the coiysince the time t, . qOis a cyclical coordinate, only GO= J occurs in the equations of motion. It involves no loss of generality when we neglect the resistance of the coil. We can take a superconducting circuit in which the current can be adjusted by a separate mutual induction. The most general form of the L a g r a n g i a n of the considered system and the magnet will now be : 4 = Q Lh,2 + CO(%, 4, * * ..& ‘II . * . (1,s).
(2)
ON
THE
STATISTICAL
MECHANICS
This will give the equations .. Lqo
a ap ----=
dt agk
IN
of motion
A MAGNETIC
in the following
d ap + a F = Qo, ap 0 (k=l aqk
Equation (3) must be equivalent duction law: Lj
51 .
FIELD
form:
(3) . ..a).
with the electrodynamical
in-
+ cj = v,
(5)
where ‘p is the magnetic flux of the system through the circuit of the magnet, V the electromotive force operating in this circuit. It is seen by comparison of (3) and (5) that:
In (4) we put all force coefficients QR equal to zero. This means that the system is closed, apart from the influence of the magnet. The equations (4) have the form of the equations of motion of a conservative system with the L a g r a n g i a n lo, containing the parameter do = J. When the field current is kept constant the conduct of the system is therefore governed by (4) alone. Solution of these equations will yield ‘p (t), from which we can with (5) calculate the electromotive force required to keep J constant. The work clone in this adjusting of the current is seen to be from (5) A =tfVdqo
=t/?4jo
The total electromagnetic the Hamiltonian:
dt =
dt = J(cp2-
(r,)
(6)
energy of system plus coil is given by
where
The first term of (7) is the energy of the magnet, the second the interaction energy of magnet and field, the third is the H a m i lt o n i a n corresponding to Lo. tie is therefore not, as has been indicated already in the introduction, the total energy of the system. Physica
XII
4
52
L.
J. F. BROER
When we bring the equations of motion in H a m i 1 t o n i a n form starting from (7), the current J = & must be eliminated and replaced by :
PO=
a&- = L&
f 9.
The condition J = constant can therefore not be expressed in such a simple form when the canonical moments are used instead of the generalised velocities. Yet there is a way to obtain the equation of the system in H a m i 1 t o n i a n form containing the parameter J, viz. the method of R o LI t h. The R o u t h function (see e.g. Sommerfeld5))is:
= ffO(qob, Pl . * .
Pm41 * . f q,,) -
;s L&,
where
From R the equations
of motion
can be derived
in the form:
(8) arfo Pk=K, From
(9)
(5) and (8) follows:
ap
a*
0 2. Quantisation of the system. When we suppose that the current is kept constant by a suitably adjusted V then we have in (9) the equations of motion of the system in Ham i 1 ton form containing i. = J as a parameter. (In reality one cannot adjust the current after each quantum jump of the system, the influence of our supposition in the result will be considered in the appendix I). The quantisation of the system can now be performed in the usual way.FfO will be an operator with characteristic values W,, depending still on J. The flux is represented by a matrix r.pllP,The diagonal elements fpfisp
ON
THE
STATISTICAL
MECHANICS
IN
A MAGNETIC
53
FIELD
.
are the flux of the system through the magnet stationary state. One has from (10) : ‘PPP
=
when it is in the p’”
(11)
-
(See for the last equality K r a n s “) page 72). The total energy in.the state p is from (7) : E, = 4 LJ2 + J’ppp f When the system jumps the energy : EP
-
W,.
from the state p to the state q therefore
E, = J(‘PPP -
PA
+
WP
-
W,)
is set free. From (6) it is seen that J(‘ppp - (pq4)will be absorbed by the magnet, thus W, - IV, has to leave the system as radiation or in some other form. The H a m i 1 t o n i a n M” can therefore be called the spectroscopical energy l). Its characteristic values satisfy the I3 o h r frequency condition. 5 3. Statistical mechanics. With the aid of the results of the former section one can derive the properties of ensembles of systems in a magnetic field by a formal analogy from the statistical mechanics of, closed systems. First we consider a micro-canonical ensemble. This coilGsts of a large number of weakly interacting identical systems. It is not required that the number of degrees of freedom of each system be large. The identity of the systems entails the equality of the magnetic field for all systems. An example is furnished by the magnetic ions in a cristal which is placed in a homogeneous field when their mutual interaction is small enough. When there are ?apsystems in the state p the total energy is :
E = 3 L J2 + J x np ‘ppp+ C n.pW,. P
P
If the distribution of the systems over the levels is varied, J constant we have: E’ = 4 L J’ + J C ni ‘ppp + X ni W,. P
According
to (6) is:
E - E’ = J F ($ - apI ‘ppp, when the ensemble is isolated.
keeping
54
L.
Elimination
J. I;.
BROER
of E yields:
c np wp = I: n; wp = w. (12) P P The spectroscopical energy of the isolated ensemble is therefore constant. As in our case the total number of systems C, np = N is a p. constant too the problem of the most probable distribution is formally the same as in ordinary statistical mechanics, only the spectroscopical energy has to be substitued for the total energy. The most probable distribution is therefore given by:
WP np=cexp-lTkT This result is equivalent with (1). For a canonical ensemble analogous considerations can be given. We imagine a system with a large number of degrees of freedom, e.g. a magnetic crystal with arbitrary interactions between the ions, in contact with a non-magnetic thermostate placed in an electromagnet. The field now does not need to be homogeneous. At a transition from the state fi to q the thermostate absorbs the difference in spectroscopical energy W,, - W,. We can now conclude in the usual way that, provided the validity of the ergodic hypothesis, the time average of some quantity in our system is equal to the average over an ensemble of identical systems distributed with a density exp - W/kT. (1) is therefore valid in this case too. 5 4. Thermodynamics. The total average energy of system magnet in the last discussed case is (cf equation (7)):
and
E=~LJ~+JG+W, where the bar may be interpreted ensemble too. When the current A J the rise in energy is : A&=LJAJ+
as the average over a canonical in the magnet is increased with JA&‘pAJ+Aw.
In the electrical circuit the work A, = L J AJ is done to increase the current at constant i, the work A, = JA.Cp to keep J constant when I;, is varied. The thermostate has therefore to supply the heat : AQ=AE-A,
-A2=Aw+(paJ.
ON
THE
STATISTICAL
MECHANICS
IN
A MAGNETIC
FIELD
55
This is the first law of magnetic thermodynamics. As we excluded the irreversible J o u 1 e heat in the coil from our considerations we are justified in assuming the validity of the second law. Thus we can write : AE = ~-AS - (~AJ. (14) From (13) and (14) the magnetic thermodynamics can be developed in the usual way. It is easier however to obtain the formulae by substituting (r and J for p and v in the ordinary thermodynamical treatment of gases and fluids. The foregoing considerations can be extended without difficulties to the case that the system is subjected to the influence of more than one electromagnet. When these carry the currents J, then go and Lo are now functions of the parameters J,. The flux of the system through the circuit M is:
ap alFE0 %==aJ,=-‘aJ,’ Instead
of (14) we have : A@=
TAS-Z&
a
AJa.
(14’)
We will now consider the important specialisation to homogeneous fields. We suppose we have three electromagnets, giving on the place where the system is located homogeneous fields in the x, j’, P directions. The proportionality factor a between current and field can be taken the same for the three magnets: H, = a Jz etc. @ and I0 can now be expressed as-functions of the field components instead of the currents. We now define the components of the vector of magnetic moment by:
It is sometimes
With form :
convenient
to abbreviate
the aid of this definition
we can bring
A@ = TAS In appendix M is proved.
II the aequivalence
(15) to:
(14’) into the usual
(M . AH). of this and the usual definition
of
56
L.
J. F. BROER
Finally we will point out that the H a m i 1 t o n i a n go represents the total energy of the system minus the interaction with the coil. It is not necessary only to include herein the kinetical and potential energies of the separate electrons and to treat their interaction apart as has been done in G o r t e r’s I) discussion of this problem. The difficulties met in establishing a satisfactory H a m i It o n i a n for a many particle problem presumably induced the choice of this treatment. They belong however entirely to the domain of atomic theory and cannot mar a discussion of the foundation of statistical mechanics. $ 5. Comparison with electric field. It is customary in pure thermodynamical work without statistical considerations to start from the total energy ,!? = w + Ji. Formula (14) then takes the form:
Ai? = TAS f JAG.
(16)
From this expression a set of thermodynamical formulae is derivable which is entirely aequivalent with that indicated in the former section. It can be obtained from it by substituting J, (p for @, -J and interchanging energy and enthalpy, free energy and thermodynamic potential. The second system has the practical advantage that the energy in normal paramagnetic substances is a function of temperature alone, the first however the more theoretical advantage that is connected in the simplest way with statistical mechanics and the B o h r frequency condition. In the case of homogeneous fields (I 6) takes the form: Ai? = TAS + (H. AM). This formula is the exact analogue of that valid for a dielectricum placed between the plates of a condenser: Ai? = TAS + (F. AP), where F is the electric field, P the polarisation. It is perhaps useful to stress that this anology is purely formal and in a certain sense accidental. This will be clear when we compare a magnetic dipole within a coil and an electric dipole in a condenser. The charge K of the condenser is now the analogue of the current J in the coil. There are three important differences as regards the energy. lo. The field energy of the magnetic dipole (interaction energy in
ON
THE
STATISTICAL
MECHANICS
IN
A MAGNETIC
FIELD
57
. (7)) is (H . M), whereas that of an electric dipole is -(F, P). This difference is caused by the course of the lines of force inside the body. 2”. The energy required for an increase of J is not dependent on a permanent magnetic moment in the coil, that is: this moment has no bearing on the inductivity of the coil. When the charge of the condensor is added to when a permanent electrical moment is in the field the extra work AU’, = -- (P . AF) must be done because the dipole attracts the charge on the plates. That is: a permanent moment influences the capacity of a condenser. 3”. On a change of magnetisation the work AW; = (H . AM) has to be done in the coil to correct the current after its decrease by the induction. If the magnetic body is moved in an inhomogeneous field the induction energy is AJVi=A(H. M). The condenser charge is constant on a change of electrical moment. We can now make the following comparison : A body is magnetized
in a coil.
To increase the current in the coil is required the energy L J A J for the field of the magnet only, 0 owing to the presence of a permanent magnetic‘ moment, AkVj = (H . AM) to compensate the induction. The field energy is changed by : Ar/v, = A(H. The supply
M).
body has therefore the energy :
4W,, -
.lWj
AW, = to
-= (M . AH).
The change in magnetisation energy is therefore : 4W,,, = -
A body is electrisized in a condensor. To increase the charge on the plates is required the energy KAK/C for the field of the condensor only, AI+‘! = - (P . AF) owing to the presence of a permanent electric moment, 0 to increase P at constant K. The field energy is changed by :
(M . AH).
This energy consists of two terms, firstly the increase AW
The supply
A(F , P).
body was therefore the energy :
AN’,, -
4bI’, = -
to
(F . AP).
The change in electrisation energy is therelore :
4W, = (F . AP). Thus energy consists of two ternls. firstlv _ the increase AE of
58
L.
J. F. BROER
of kinetic and potential energy of the particles, secondly the evolved heat or radiation AQ. Thus : AQ = AW + (M. AH). At a constant
field we have:
AQ = AW. The changes of Wi and W, just compensate. W is called the spectroscopical energy because it governs the frequency of radiation according to B 0 h r’s law. W consists of the kinetic and potemptial energy of the particles and is the Ham i 1 t on function for a constant field. It has to be used in the formulae of statistical mechanics. The mechanical work needed for a displacement of a magnetic dipole in an inhomogeneous field at- a constant J is: AA = AW,, + A~v, - A6Yi = AW,,, = - (M . AH), as AW, = AW, = A(H M).
kinetic and potential energy of the particles, secondly the evolved heat or radiation A Q. Thus : AQ=AE-(F.AP) =A{E-(F.P)}+(P.AF). At a constant field we have: AQ = A {E - (F . P)}. WV is not compensated
here.
E -
(F . P) is called the spectroscopical energy bevause it governs the frequency of radiation according to B o h r’s law. It consists of the kinetic and potential energy of the particles, together with the field energy and is the Hamilton function for a constant field. It has to be used in the formulae of statistical mechanics. The mechanical work needed for a displacement of an electric dipole in an inhomogeneous field at a constant K is: AA = AW, + AW, = (F . AP) -(F.P)=-(P.AF).
The increase of the field energy is therefore just supplied by the e.m.f. in the coil. Appendix
for a transition
I. When there is no e.m.f. acting in the circuit we have p + 4 according to (5) :
- L . AJ = (‘P~)J+AJ- hvb
(17)
ON
THE
The difference
STATISTICAL
MECHANICS
IN
59
FIELD
in energy is now:
AE = E, - E, = A(* LJ2) -t (JY,, + = LJAJ + + WJ12 f J{(T~~)~+A~ With
A MAGNETIC
wq)~+A,
-
(11) and (17) we get, neglecting
-
(T&J)
(Jcppp+ q) = + AJ(~P~*)J+A, -C-
+ twq).r+AJ - (W&J. terms from the order
of
@J12. %+J: AE = W, -
W, -
4 A J(cp,, -
where all quantities can be taken at J. When we use a homogeneous field of fixed H = aJ, 9 = aM. From (17) it is seen that:
therefore
ppp), direction
we have
:
The discrepancy is thus from the order of the square of the change in energy of the system divided by the total energy of the coil and can be reduced therefore arbitrarely by increasing the seize of the coil.
Appendix volume
II.
integral
The L a g r a n g i a 11 I0 can be considered of the L a g r a n g i a II density G:
as the
4” =/GdV. G will in general be a function of the vector potential A of ‘the external magnetic field. When this field is homogeneous however A can be eliminated by means of the relation:
A=-&[H.r]. The magnetic
moment
But aG/aA 1s the current tion, thus we have:
is now according
density
M = ?$
(18) to (15) and ( 18) :
S of the system
= 4 /“[S . r] dV.
under considera-
60
ON
THE
STATISTICAL
Both definitions
MECHANICS
IN
A MAGNETIC
FIELD
are therefore aequivalent.
The attention of the author was called to this problem by Prof. Dr. C. J. G o r t e r, to whom he is also indebted for some clearifying discussions. Received
June
2nd,
1944. REFERENCES
I) 2) 3) 4) 5) 5)
C. H. A. J. A. 1~.
J. G or t e r, Arch. du Mu&e Teyler, Ser. III, A. L or e n t z, Handbuch fiir Radiologie VI, D. F o k k er, Physica 6, 791, 1939. ;\I. van Lee u w e n, Thesis, Leiden, 1919. S o n1 m e r f e I d, Vorlesungen tiber theoretische L. Ii r a n s, Thesis, Leiden, 1931.
Vol. 1925.
VII,
183,
Physik
1932.
I, Leipzig,
1943.
ERRATUM
ON THE
THEORY
OF PARAMAGNETIC
RELATION
by L. J. F. BROER (Physica The formulae
on the bottom
M, =
M,U
X, 801, 1943) of page 807 must
- UM, ih
crkl
x
akl)a
on page 813 is corrected I0S.p’ without
Cr alum Fe alum Gdz(S0,),.8H,0
21)
F. W.
. .
. . .
.
d e V r ij e r,
fields
with
V o 1 g e r and
is used.
1OS.p’ tryst. fields
C. J:
G or
t e r,
mistake
109.p’ exp.
1.7 1.0 0.21
0.7 J.
. =kl)
for this and another
2.9 2. I
.
. tak 4
Itl the later calculations this correct expression Formula ( 12) on page 808 must read :
When the table it reads :
read:
1.4 0.9 0.3 Physica,
1’) 1,) a’)
in press.
XII,
no
I
April
ON THE STATISTICAL IX A MAGNETIC
1946
MECHANICS FIELD
by L. J. I;. BROER Communication
from
thr Zccman
Laboratory
of the University
of Xmstrrdnm
Summary The fundamental relations of the statistical mechanics and thermodynamics of magnetised matter are derived in a systematic way, starting from the most general form of the I, a g r a n g i a n of matter and magnet. The H a m i 1 t o n equations of the matter alone are established by means of the R o u t h function. The well-known fact that the corresponding H a m i 1 t o n function instead of the total energy of the matter must be used in the partition function is rigourously proved. A comparison is made between the various quantities of energy which enter in the processes of magnetisation and electrisation respectively.
Introduction. It has been well asserted that the equations of motion of a system of charged particles in a constant magnetic field can be brought in the H a m i 1 t o n form. In the H a m i 1 t o n function however not all the terms of the total energy containing the coordinates of the particles will occur. When the system is quantisised the characteristic values IV,, of the H am i 1t o n operator therefore will not represent the total energy of the stationary states of the system. Nevertheless one can, as will be generally known, derive the thermodynamical properties of the system from the partition function : Wtl Z = t exp. - kT (1) by applying the usual formalism of statistical mechanics. This formalism has been developed under the assumption that W,, is the total energy of a closed system. Its use in the case of systems in a magnetic field therefore needs some further considerations. The correct way to Physica
XII
49 4*
50
L.
J. F. BROER
handle (1) in this case is current practice since long and seems never to have given rise to uncertainties. We were somewhat surprised therefore in not succeeding to find in litterature a wholly satisfactory discussion of this question. G o r t e r ‘) gives, following some considerations of L o r e n t z 2), a detailed treatment of the energetics involved in the magnetisation process whereas the question had been discussed from a thermodynamical point of view among others by F o k k e r “). These authors did not seek a close connection with the H am i 1 t o n i a n method and with statistical mechanics and confined themselves to homogeneous fields. Furthermore G o rt e r and L o r e n t z do not introduce some hypothetical device to keep the magnetic field constant under fluctuations of the magnetisation. As this field figures as a constant parameter in the H a m i lt o n i a n of the magnetic matter this method is not quite appropriate to our purposes. We will now derive the fundamental formulae of statistical mechanics and thermodynamics starting from the most general form of the L a g r a n g i a n function of matter plus field currents. These formulae will be, strictly spoken, valid for a constant field and therefore concur with the usual H a m i 1 t o n i a n operators. The method employed, viz. the use of the R o u t h function, has been adopted from the work of V a n L e e u w e n 4) on this problem. Her treatment is not general because she uses special kinetical models, furthermore it will appear that the formulation can be made much simpler than that of V a n L e e u w e n. 5 1. The equations of motion 0.t the system. We will consider a system of charged particles described by a set of generalised coordinates ql, . . . . q,, placed in the field of an electromagnet. It is not yet necessary to suppose that the field is hom?geneous. The state of the magnet is defined by the coordinate qO= / Jdt, which indicates
the electrical charge passed through the coiysince the time t, . qOis a cyclical coordinate, only GO= J occurs in the equations of motion. It involves no loss of generality when we neglect the resistance of the coil. We can take a superconducting circuit in which the current can be adjusted by a separate mutual induction. The most general form of the L a g r a n g i a n of the considered system and the magnet will now be : 4 = Q Lh,2 + CO(%, 4, * * ..& ‘II . * . (1,s).
(2)
ON
THE
STATISTICAL
MECHANICS
This will give the equations .. Lqo
a ap ----=
dt agk
IN
of motion
A MAGNETIC
in the following
d ap + a F = Qo, ap 0 (k=l aqk
Equation (3) must be equivalent duction law: Lj
51 .
FIELD
form:
(3) . ..a).
with the electrodynamical
in-
+ cj = v,
(5)
where ‘p is the magnetic flux of the system through the circuit of the magnet, V the electromotive force operating in this circuit. It is seen by comparison of (3) and (5) that:
In (4) we put all force coefficients QR equal to zero. This means that the system is closed, apart from the influence of the magnet. The equations (4) have the form of the equations of motion of a conservative system with the L a g r a n g i a n lo, containing the parameter do = J. When the field current is kept constant the conduct of the system is therefore governed by (4) alone. Solution of these equations will yield ‘p (t), from which we can with (5) calculate the electromotive force required to keep J constant. The work clone in this adjusting of the current is seen to be from (5) A =tfVdqo
=t/?4jo
The total electromagnetic the Hamiltonian:
dt =
dt = J(cp2-
(r,)
(6)
energy of system plus coil is given by
where
The first term of (7) is the energy of the magnet, the second the interaction energy of magnet and field, the third is the H a m i lt o n i a n corresponding to Lo. tie is therefore not, as has been indicated already in the introduction, the total energy of the system. Physica
XII
4
52
L.
J. F. BROER
When we bring the equations of motion in H a m i 1 t o n i a n form starting from (7), the current J = & must be eliminated and replaced by :
PO=
a&- = L&
f 9.
The condition J = constant can therefore not be expressed in such a simple form when the canonical moments are used instead of the generalised velocities. Yet there is a way to obtain the equation of the system in H a m i 1 t o n i a n form containing the parameter J, viz. the method of R o LI t h. The R o u t h function (see e.g. Sommerfeld5))is:
= ffO(qob, Pl . * .
Pm41 * . f q,,) -
;s L&,
where
From R the equations
of motion
can be derived
in the form:
(8) arfo Pk=K, From
(9)
(5) and (8) follows:
ap
a*
0 2. Quantisation of the system. When we suppose that the current is kept constant by a suitably adjusted V then we have in (9) the equations of motion of the system in Ham i 1 ton form containing i. = J as a parameter. (In reality one cannot adjust the current after each quantum jump of the system, the influence of our supposition in the result will be considered in the appendix I). The quantisation of the system can now be performed in the usual way.FfO will be an operator with characteristic values W,, depending still on J. The flux is represented by a matrix r.pllP,The diagonal elements fpfisp
ON
THE
STATISTICAL
MECHANICS
IN
A MAGNETIC
53
FIELD
.
are the flux of the system through the magnet stationary state. One has from (10) : ‘PPP
=
when it is in the p’”
(11)
-
(See for the last equality K r a n s “) page 72). The total energy in.the state p is from (7) : E, = 4 LJ2 + J’ppp f When the system jumps the energy : EP
-
W,.
from the state p to the state q therefore
E, = J(‘PPP -
PA
+
WP
-
W,)
is set free. From (6) it is seen that J(‘ppp - (pq4)will be absorbed by the magnet, thus W, - IV, has to leave the system as radiation or in some other form. The H a m i 1 t o n i a n M” can therefore be called the spectroscopical energy l). Its characteristic values satisfy the I3 o h r frequency condition. 5 3. Statistical mechanics. With the aid of the results of the former section one can derive the properties of ensembles of systems in a magnetic field by a formal analogy from the statistical mechanics of, closed systems. First we consider a micro-canonical ensemble. This coilGsts of a large number of weakly interacting identical systems. It is not required that the number of degrees of freedom of each system be large. The identity of the systems entails the equality of the magnetic field for all systems. An example is furnished by the magnetic ions in a cristal which is placed in a homogeneous field when their mutual interaction is small enough. When there are ?apsystems in the state p the total energy is :
E = 3 L J2 + J x np ‘ppp+ C n.pW,. P
P
If the distribution of the systems over the levels is varied, J constant we have: E’ = 4 L J’ + J C ni ‘ppp + X ni W,. P
According
to (6) is:
E - E’ = J F ($ - apI ‘ppp, when the ensemble is isolated.
keeping
54
L.
Elimination
J. I;.
BROER
of E yields:
c np wp = I: n; wp = w. (12) P P The spectroscopical energy of the isolated ensemble is therefore constant. As in our case the total number of systems C, np = N is a p. constant too the problem of the most probable distribution is formally the same as in ordinary statistical mechanics, only the spectroscopical energy has to be substitued for the total energy. The most probable distribution is therefore given by:
WP np=cexp-lTkT This result is equivalent with (1). For a canonical ensemble analogous considerations can be given. We imagine a system with a large number of degrees of freedom, e.g. a magnetic crystal with arbitrary interactions between the ions, in contact with a non-magnetic thermostate placed in an electromagnet. The field now does not need to be homogeneous. At a transition from the state fi to q the thermostate absorbs the difference in spectroscopical energy W,, - W,. We can now conclude in the usual way that, provided the validity of the ergodic hypothesis, the time average of some quantity in our system is equal to the average over an ensemble of identical systems distributed with a density exp - W/kT. (1) is therefore valid in this case too. 5 4. Thermodynamics. The total average energy of system magnet in the last discussed case is (cf equation (7)):
and
E=~LJ~+JG+W, where the bar may be interpreted ensemble too. When the current A J the rise in energy is : A&=LJAJ+
as the average over a canonical in the magnet is increased with JA&‘pAJ+Aw.
In the electrical circuit the work A, = L J AJ is done to increase the current at constant i, the work A, = JA.Cp to keep J constant when I;, is varied. The thermostate has therefore to supply the heat : AQ=AE-A,
-A2=Aw+(paJ.
ON
THE
STATISTICAL
MECHANICS
IN
A MAGNETIC
FIELD
55
This is the first law of magnetic thermodynamics. As we excluded the irreversible J o u 1 e heat in the coil from our considerations we are justified in assuming the validity of the second law. Thus we can write : AE = ~-AS - (~AJ. (14) From (13) and (14) the magnetic thermodynamics can be developed in the usual way. It is easier however to obtain the formulae by substituting (r and J for p and v in the ordinary thermodynamical treatment of gases and fluids. The foregoing considerations can be extended without difficulties to the case that the system is subjected to the influence of more than one electromagnet. When these carry the currents J, then go and Lo are now functions of the parameters J,. The flux of the system through the circuit M is:
ap alFE0 %==aJ,=-‘aJ,’ Instead
of (14) we have : A@=
TAS-Z&
a
AJa.
(14’)
We will now consider the important specialisation to homogeneous fields. We suppose we have three electromagnets, giving on the place where the system is located homogeneous fields in the x, j’, P directions. The proportionality factor a between current and field can be taken the same for the three magnets: H, = a Jz etc. @ and I0 can now be expressed as-functions of the field components instead of the currents. We now define the components of the vector of magnetic moment by:
It is sometimes
With form :
convenient
to abbreviate
the aid of this definition
we can bring
A@ = TAS In appendix M is proved.
II the aequivalence
(15) to:
(14’) into the usual
(M . AH). of this and the usual definition
of
56
L.
J. F. BROER
Finally we will point out that the H a m i 1 t o n i a n go represents the total energy of the system minus the interaction with the coil. It is not necessary only to include herein the kinetical and potential energies of the separate electrons and to treat their interaction apart as has been done in G o r t e r’s I) discussion of this problem. The difficulties met in establishing a satisfactory H a m i It o n i a n for a many particle problem presumably induced the choice of this treatment. They belong however entirely to the domain of atomic theory and cannot mar a discussion of the foundation of statistical mechanics. $ 5. Comparison with electric field. It is customary in pure thermodynamical work without statistical considerations to start from the total energy ,!? = w + Ji. Formula (14) then takes the form:
Ai? = TAS f JAG.
(16)
From this expression a set of thermodynamical formulae is derivable which is entirely aequivalent with that indicated in the former section. It can be obtained from it by substituting J, (p for @, -J and interchanging energy and enthalpy, free energy and thermodynamic potential. The second system has the practical advantage that the energy in normal paramagnetic substances is a function of temperature alone, the first however the more theoretical advantage that is connected in the simplest way with statistical mechanics and the B o h r frequency condition. In the case of homogeneous fields (I 6) takes the form: Ai? = TAS + (H. AM). This formula is the exact analogue of that valid for a dielectricum placed between the plates of a condenser: Ai? = TAS + (F. AP), where F is the electric field, P the polarisation. It is perhaps useful to stress that this anology is purely formal and in a certain sense accidental. This will be clear when we compare a magnetic dipole within a coil and an electric dipole in a condenser. The charge K of the condenser is now the analogue of the current J in the coil. There are three important differences as regards the energy. lo. The field energy of the magnetic dipole (interaction energy in
ON
THE
STATISTICAL
MECHANICS
IN
A MAGNETIC
FIELD
57
. (7)) is (H . M), whereas that of an electric dipole is -(F, P). This difference is caused by the course of the lines of force inside the body. 2”. The energy required for an increase of J is not dependent on a permanent magnetic moment in the coil, that is: this moment has no bearing on the inductivity of the coil. When the charge of the condensor is added to when a permanent electrical moment is in the field the extra work AU’, = -- (P . AF) must be done because the dipole attracts the charge on the plates. That is: a permanent moment influences the capacity of a condenser. 3”. On a change of magnetisation the work AW; = (H . AM) has to be done in the coil to correct the current after its decrease by the induction. If the magnetic body is moved in an inhomogeneous field the induction energy is AJVi=A(H. M). The condenser charge is constant on a change of electrical moment. We can now make the following comparison : A body is magnetized
in a coil.
To increase the current in the coil is required the energy L J A J for the field of the magnet only, 0 owing to the presence of a permanent magnetic‘ moment, AkVj = (H . AM) to compensate the induction. The field energy is changed by : Ar/v, = A(H. The supply
M).
body has therefore the energy :
4W,, -
.lWj
AW, = to
-= (M . AH).
The change in magnetisation energy is therefore : 4W,,, = -
A body is electrisized in a condensor. To increase the charge on the plates is required the energy KAK/C for the field of the condensor only, AI+‘! = - (P . AF) owing to the presence of a permanent electric moment, 0 to increase P at constant K. The field energy is changed by :
(M . AH).
This energy consists of two terms, firstly the increase AW
The supply
A(F , P).
body was therefore the energy :
AN’,, -
4bI’, = -
to
(F . AP).
The change in electrisation energy is therelore :
4W, = (F . AP). Thus energy consists of two ternls. firstlv _ the increase AE of
58
L.
J. F. BROER
of kinetic and potential energy of the particles, secondly the evolved heat or radiation AQ. Thus : AQ = AW + (M. AH). At a constant
field we have:
AQ = AW. The changes of Wi and W, just compensate. W is called the spectroscopical energy because it governs the frequency of radiation according to B 0 h r’s law. W consists of the kinetic and potemptial energy of the particles and is the Ham i 1 t on function for a constant field. It has to be used in the formulae of statistical mechanics. The mechanical work needed for a displacement of a magnetic dipole in an inhomogeneous field at- a constant J is: AA = AW,, + A~v, - A6Yi = AW,,, = - (M . AH), as AW, = AW, = A(H M).
kinetic and potential energy of the particles, secondly the evolved heat or radiation A Q. Thus : AQ=AE-(F.AP) =A{E-(F.P)}+(P.AF). At a constant field we have: AQ = A {E - (F . P)}. WV is not compensated
here.
E -
(F . P) is called the spectroscopical energy bevause it governs the frequency of radiation according to B o h r’s law. It consists of the kinetic and potential energy of the particles, together with the field energy and is the Hamilton function for a constant field. It has to be used in the formulae of statistical mechanics. The mechanical work needed for a displacement of an electric dipole in an inhomogeneous field at a constant K is: AA = AW, + AW, = (F . AP) -(F.P)=-(P.AF).
The increase of the field energy is therefore just supplied by the e.m.f. in the coil. Appendix
for a transition
I. When there is no e.m.f. acting in the circuit we have p + 4 according to (5) :
- L . AJ = (‘P~)J+AJ- hvb
(17)
ON
THE
The difference
STATISTICAL
MECHANICS
IN
59
FIELD
in energy is now:
AE = E, - E, = A(* LJ2) -t (JY,, + = LJAJ + + WJ12 f J{(T~~)~+A~ With
A MAGNETIC
wq)~+A,
-
(11) and (17) we get, neglecting
-
(T&J)
(Jcppp+ q) = + AJ(~P~*)J+A, -C-
+ twq).r+AJ - (W&J. terms from the order
of
@J12. %+J: AE = W, -
W, -
4 A J(cp,, -
where all quantities can be taken at J. When we use a homogeneous field of fixed H = aJ, 9 = aM. From (17) it is seen that:
therefore
ppp), direction
we have
:
The discrepancy is thus from the order of the square of the change in energy of the system divided by the total energy of the coil and can be reduced therefore arbitrarely by increasing the seize of the coil.
Appendix volume
II.
integral
The L a g r a n g i a 11 I0 can be considered of the L a g r a n g i a II density G:
as the
4” =/GdV. G will in general be a function of the vector potential A of ‘the external magnetic field. When this field is homogeneous however A can be eliminated by means of the relation:
A=-&[H.r]. The magnetic
moment
But aG/aA 1s the current tion, thus we have:
is now according
density
M = ?$
(18) to (15) and ( 18) :
S of the system
= 4 /“[S . r] dV.
under considera-
60
ON
THE
STATISTICAL
Both definitions
MECHANICS
IN
A MAGNETIC
FIELD
are therefore aequivalent.
The attention of the author was called to this problem by Prof. Dr. C. J. G o r t e r, to whom he is also indebted for some clearifying discussions. Received
June
2nd,
1944. REFERENCES
I) 2) 3) 4) 5) 5)
C. H. A. J. A. 1~.
J. G or t e r, Arch. du Mu&e Teyler, Ser. III, A. L or e n t z, Handbuch fiir Radiologie VI, D. F o k k er, Physica 6, 791, 1939. ;\I. van Lee u w e n, Thesis, Leiden, 1919. S o n1 m e r f e I d, Vorlesungen tiber theoretische L. Ii r a n s, Thesis, Leiden, 1931.
Vol. 1925.
VII,
183,
Physik
1932.
I, Leipzig,
1943.
ERRATUM
ON THE
THEORY
OF PARAMAGNETIC
RELATION
by L. J. F. BROER (Physica The formulae
on the bottom
M, =
M,U
X, 801, 1943) of page 807 must
- UM, ih
crkl
x
akl)a
on page 813 is corrected I0S.p’ without
Cr alum Fe alum Gdz(S0,),.8H,0
21)
F. W.
. .
. . .
.
d e V r ij e r,
fields
with
V o 1 g e r and
is used.
1OS.p’ tryst. fields
C. J:
G or
t e r,
mistake
109.p’ exp.
1.7 1.0 0.21
0.7 J.
. =kl)
for this and another
2.9 2. I
.
. tak 4
Itl the later calculations this correct expression Formula ( 12) on page 808 must read :
When the table it reads :
read:
1.4 0.9 0.3 Physica,
1’) 1,) a’)
in press.