Physics. - Change of the resistance of metals in a magnetic fieLd at Low
temperatures. By W. J. DE HAAS and P. M. VAN ALPHEN. (Com~
munication No. 225a from the KAMERLINGH ONNES Laboratory.
Leiden).
(Communicated at the meeting of March 25. 1933) .
§ 1. Introduction. In previous papers we described the interesting
results of our investigations on the change in resistance of single~crystals
of very pure bismuth 1) and on the simultaneous occurrence of a
dependence of their diamagnetic susceptibility upon the magnetic field 2).
Ta discuss the question thoroughly however we didn't know enough about
the change of the resistance in a magnetic field as a function of the
temperature. Theoretically little can be said. and na wonder. for even
the explication of the influence of the magnetic field on the resistance
merely is entremely difficult. Especially. in trying to explain the order
of magnitude of the longitudinal effect 3) we meet with great difficulties.
We may mention here the detailed investigations of KAPITZA 4). which
furnish data for a very great number of materials. The main subject
however of these investigations is not the change in resistance as a
function of the tempera tu re but rather as a function of the intensity of
his very high magnetic field . The low temperatures (liquid air) were
determined not very accurately and na measurements were made below
77° K.5) .
For this reason we planned a series of measurements not only of the
magnetic properties. but also of the change in resistance. As a matter
of fact our attention was specially directed towards the reg ion of hydrogen
temperatures. though always the effect was also investigated at higher
temperatures.
§ 2. Measuring method. The resistance measurements were made
with the compensation apparatus of DIESSELHORST. as is usuaI in the
I) L . SCHUBNIKOWand W. J. DE HAAS. Proc. Amsterdam 33. p. 130. 363. 1930.
Comm. Leiden NO. 207 .
2) W . J. DE HAAS and P. M. VAN ALPHEN. Proc. Amsterdam 33. p. 680. 1106. 1930.
3S. p. i5i. 1932. Comm. Leiden 208d. 212a. 220d.
3) R. GANS. Ann. der Phys. (i) 20. p. 293. 1906.
A. SOMMERFELD and N. H. FRANCK. Zeit. für Phys. 47. p. I . 1928; Review of Modern
Physics 3. p. I. 1931.
R . PEIERLS. Zelt. für Phys. S3. p. 255. 1929
i) P . KAPITZA. Proc. Roy. Soc. 123. p. 292 . 1929.
5) For gold at i.2° K. see W. MEISSNER and H . SCHEFFERS. Phys. Zeit. 30. p. 827.
1929; Phys. Zelt. 31. p. 574. 1930 ; Naturw. 18. p . 110. 1929.
254
Leyden laboratory. The bifilar resistance wires were wound on a th in
piece of mica , which could be turned over 90° about its longitudinal axis.
The mica was placed alternately with its plane in the direction of the
magnetic lines of force and perpendicular to these. In th is way the
longitudinal and the transverse effect are measured alternately. It is true ,
that in the first case the field and current are not exactly paraUel through
the screw of the windings. This deviation however is smaU and was
afterwards proved to be proportional with cas 2 a. The advantage of th is
form of resistance is, that it enables us to make large resistances, which is
very desirabIe for the measurements at low temperatures.
In order to prevent impurities by soldering, we fuse 4 platinum pieces
to the resistances . The copper wires were then soldered at these pieces.
Five of these resistances were mounted in the cryostat on a glass rod, which
could be moved up and down from the outside by means of a packing tube.
The angle of rotation was read on a divided circ1e in degrees .
As at low temperatures the resistance changes may become very large,
the current of the magnet must be kept very constant. To this purpose
we used the current of a large pile of accumulators .
The value of RIRo was an indication of the degree to which the wires
had remained pure, for the purer ,the materiaL the smaUer the value of
RIRo at hydrogen temperatures . This con trol is of great importance, the
possibility of the introduction of impurities during the fashioning being
very great.
§ 3.
Allays. lnfLuence of impurities. Discussing the conductivity of
alloys , we must always bear in mind , that we may have to do with an
impure metal. When e.g. the compositian is such, that a metallic compound
is formed with a lattice of its own, so that it must be regarded as a new
metaL we must not forget that there will always be a small surplus of one
of the components. This surplus forms an impurity in the new lattice.
Therefore we never can simply compare an aUoy with a pure metal.
Now it seems to be true that impurities have the same influence on the
electric conductivity as a rise of temperature. Both give ri se to an
irregularity in the lattice and th is increases the resistance.
Wishing to compare an impure metal with a pure materiaL we must do
this at high temperatures.
Then the irregularities caused by the impurities are small compared with
those due to the high temperature. When the temperature is lowered we
come in a region, where the influence of the admixtures (which does not
alter with the temperature) becomes great compared with that of the heat
motion. Further cooling down has no influence, the resistance becomes
independent of the temperature and its value depends on the quantity of
the admixture. AU this is duly represented by the rule of MATHIESSEN 1).
I) H . MATTHIESEN, Pogg . Ann. 100, p. 190, 1860 : 122, p. 19. 1869.
255
For the change of the resistance in a magnetic field a similar rule seems
to hold . as will be evident from the nex t paragraph .
TABLE I.
1
Cu
I
Temp.
(236 KG) .
Zn
Cu+Zn
ROT/Ro I 6.R/Roo I 6.R/ROT 1 ROT/Ro 1 6.R/Roo I 6.R/RoT ROT/Roo I 6.RIRoo I6.W/Ro1
290 0 K.
o li8
77 .2
70.i
0.122
0 .094
63 . 8
0 .020
20.4
0.020
17.3
0.020
14.2
1
§ 4.
0 .0000
17
17
17
41
41
41
0 . 000
0.009
O.OH
0.021
0 . 192
0.198
0. 191
0 .00036
0.202 0.0036
0.175 0.00i3
0.H3 0.0053
0.012 0 .0172
0 .009 0 . 0191
0.007 10 .0218
0 .000361
0.018
0.024
0.036
1.377
2.050
3. 103
-
1
0.H9
0.433
0.411
0.298
0 .282
0.280
0 .0039
0 .011
0.011
0.011
0 .011
0.011
0.011
Cu-Zn-Allay.
0.0039
0.022
0 .027
0.025
0. 042
0.Oi7
0.051
As has been mentioned earl ier J) th is alloy has
an abnormally high diamagnetism . which is much stronger than that of its
components. We thought it interesting to investigate the change of
resistance for this material more thoroughly. The results of the measurements are given in table I. At room temperature the change in resistance of
an alloy of 30 % Cu and 70 % Zn is ten times that of pure zinco though
this too shows a rather great effect. No effect of the copper could be
detected with our arrangement. EVidently the change in resistance is much
stronger than in the case of the pure components. Moreover the susceptibility has strongly increased . so that we find in this alloy a new example
of the simultaneousness of a high diamagnetism and a strong change of
resistance. a phenomenon to which in 1914 already the attention was
drawn ~ ). This alloy belongs to the group of metals. for which the hypo thesis of EHRENFEST::) probably holds viz. of the metals with astrong
diamagnetism (bismuth . antimony. gallium and probably also graphite).
The agreement with this group lies in the fact that melting causes a
steep fall of the strong diamagnetism to a much lower value. Moreover. in
contrast with the case of other metals. the conductivity seems to increase
by the melting .
Considering now the change with the temperature. we see that the
impurities begin to play a role. For Zn and Cu the change increases
always ; for the alloy however the change of the resistance in the magnetic
K. GREULICH. Sitz. Ber. Rostock. p. 227. 1915.
K. OVERBECK. Ann . d. Phys. 46. p. 677. 1915.
H. ENDO . Sc . Rep. Tohoku Un o H . p. 479. 1925 ; 16. p. 201. 1927.
2) W . J. DE HAAS. Proc. Amsterdam 16. p. 1110. 19H.
3) P . EHRENPEST. Physica 5. p. 388. 1925 ; Zeit. für Phys. 58. p. 719. 1929.
I)
256
field depends no longer on the temperature below 77° .2 K. That is why at
hydrogen temperatures the zinc shows a stronger effect than this aHoy.
TABLE 2.
H
K Gauss
I
I
1.0
3. 1
5.2
7.8
10 . 3
11 .6
20 . 1
23 . 5
Zinc o
T = 20° .1 K.
L.RIRoo
0 .0003
11
26
13
62
96
111
0. 0172
I
T= 11" . 2 K.
L.RIRoT
L.R/Roo
0 .016
0. 100
0 . 205
0 . 343
0 .188
0 . 753
1.122
1. 377
0 .0002
11
29
50
69
112
177
0 .0220
0.050
.
/
/ t
I
6 R/RoT
0 .010
0.216
0.122
0. 701
1.007
1.617
2. 521
3. 146
u'K '1
.l.
J
op30r ------+------~--~--~----~
0.020
aOA'K .
..I.
o.Ot0t------+---/---~-_C-.f_---~
H
o
40KC
Fig.
§ 5.
Cd-Hg-ALLoy . Beside the pure components we investigated an
aHoy Cd with I % Hg and an aHoy Cd with 30 atom % Hg. This latter
aHoy has the same crystal lattice as the pure cadmium. so that the mercury
is dissolved in the cadmium . By long annealing. this resistance was made
very homogeneous. while in the case of the I % aHoy not much attention
was paid to homogeneity.
From table 111 we see. that at room temperature no difference is seen
257
between the pure cadmium and the cadmium with 1 % admixture of
mercury. At low temperatures however this difference becomes evident.
TABLE 3. (23 .6 (K
Cd
Temp.
Gauls).
Cd 1% Hg
Cd+30%Hg
I
I
ROTIRoo 6.R/Roo \ 6.R/RoT ROT/Roo I 6.RIRoo M ,ROT ROT/Roo I 6.RIRoo I 6.R/RoT
290 o K.
77 . 2
63.8
20 . 4
14 . 2
-
0.2513
0.1919
0 .02297
0 .00811
0 .0022
0 .00954
0.01027
0.02672
0.03345
0.0020
0.0379
0.0533
1.163
4.132
-
0 . 4298
0.3844
0.2627
0.2459
0 .0023
0 .0056
0.0064
0.0144
0.0157
0 . 0022
0.0132
0.0165
0 .0549
0 .0638
1
-
-
-
0.406
0.363
0.232
0.215
0 .003
0.003
0.003
0 .003
0.005
0.008
0.018
0 .019
In teresting is also the difference in the course of 6R/RH"O and 6 R/RHoT
where RHoOO and RHoT are the values of the resistance without magnetic
field at 0° C and at the temperature T for which 6. R was determined
respectively.
Por the cadmium with 30 % Hg 6. RIRHooo is se en to become independent
of the temperature. Here the influence of the admixture predominates by
far over that of the temperature. The change in resistance is much smaller.
In table IV and fig. II the resistance has been given as a function of the
field and as a function of the temperature.
~O .------.------.-----~
300 ~-----+----~~----~
Fig . 2
17
P roceedings Ro ya l A cad . Amsterda m, Vol. XXXVI, 1933.
I
258
T ABLE i . Cadmium.
IH
l ' = 290 0 K.
T= 77°. 2
I
I
T
= 63°. 8
T = 20°. i
T = HO . 2
I
I
I
l:.RIRoo l:.R/RoT l:.R/Roo l:.R/RoT l:.R/Roo l:.R/RoT l:.R/Roo l:.RIRoT M /Roo l:.R/RoT
0.00003 0 .00015 0.00028
1.0
96
0.00010 O. OOOiO
15
78
2.1
183
33
173
30
115
3.1
i7
18i
61
320
i.1
291
298
93
i83
5.2 0 .00007 0.00006
li
389
6H
176
917
7.8
15i
686
iO
2i7
981
H78
10.3
37
28i
986
iH
12.6
351
1396
2150
1278
H9
1783
513
2663
H .6
135
1522
H61
1
166
i281
20.1
15i
li9
29li
825
2236
i788
21.7
839
3333
923
2ii7
23.5 0.00219 0.00203 0 .0095i 0.03787 0.01038 0.05325 0 .02672
0.012 0 .00028
157
i2
27i
80
iOO
122
I
170
526
299
863
HO
1196
557
152i
1820
663
2725
9li
3023
1.065
1.163 0.033i5
0. 036
196
3iO
i95
651
1067
H78
1883
22i9
3367
3.73i
i . 132
§ 6. Hf] with 50 atom % Cd. This alloy crystallizes in the Hg lattice.
TABLE 5. (23 . 6 K Gauss).
Hg
Temp.
ROT/Roo
77 °. 2 K
63 . 8
20 . i
H.2
0.06626
0 .05317
0.OH65
0.00817
50%Hg
I l:. R/Roo I l:. R/RoT
-
0 . 000109
0.000211
-
-
O.OOli
0 .0268
ROT/Roo
+50%Cd
I l:.R/Roo I l:.RIRoT
0. 305
0 . 290
0 .253
0 . H7
-
-
-
-
0.0018
0.0018
I
0 . 0~
0.0007
As to the Hg we must bear in mind , that it is measured in a glass tube.
At low temperatures it becomes solid and we don 't know, whether it is
then tension free and polycrystalline. For RHoO the value for the liquid Hg
has been taken.
§ 7. Solid solution. Crystal of Ti and Zr 1) . The change in resistance
was rather sm all and the rods were rather thick, so that the resistance
change could not be measured very a ccurately. So not many conclusions
can be drawn . The change is smaller than in the case of Zr. The com position of the alloy was not known.
I) These rods were kindly put at our disposal by Dr. G.
Laboratory, Eindhoven .
DE
BOER of the Philips
I
259
3.S
TABLE 6. Mercury.
6R H T
T = 20° .1 K. T= 11°.2 K.
H
K. Gauss
6 RRoT
Î
oRHT
:I.
2.5
I
5.2
0.0001
8. 1
11
10 .6
20
57
15.9
36
125
23 . 1
78
256
26.2
86
316
29 .8
121
398
i
0.0023
-
I
I
I
2.0
32 . 2
0 .0135
HO
32 .3
0 .0115
0.0159
/
1.5
0.5
/
r.1
1.0
_ .
j/
lY
o
2
!
,
6 R/RoT
'i ·K.
V2j:4·K.
/
/
H
20
10
30Ke
Fig. 3
TABLE 7. (23.6 K. Gauss).
Zr
Ti
Temp.
ROT/Roo
0
n.2K.
63.8
20.1
H.2
0. 2721
0.2H5
0 . H23
0.13904
16 R/Roo 16 R/ROT
-
-
-
-
0.OJ02
0.002
0 .002
I 0.0002
TABLE 8. Zircoon.
T=H~2
H
6 R/ Roo
I
ROT/Roo
I
0.2087
0 . 1566
0.0591
0.0568
K=J
0.00006
0.OJ11
5. 2
13
204
10.3
36
62
H .6
92
159
20.1
151
272
21.7
173
301
0.00206
0.0360
I
16 R/Roo 16 R/ROT
0. 0005
0.0007
0.0020
0.0021
0. 0025
0 . 0017
0 . 0330
0.0370
1
ROT/Roo è:. R/RooI 6 R/ROT
I
0 .369
0 .222
0. 219
-
-
0.0005
0. 0001
0022
0.
0.0018
-
0.0 4 0 . - -- - , - - - - , . - --
--,
6 R/ROT
3. 1
23 .5
Ti+Zr
Fig. 1 .
17*
1
260
§ 8. Transition {rom the transverse ta the langitudinal effect. The
high values of the resistance change at Iow temperatures enabled us to
measure the effect very accurately. In two cases we even investigated
precisely how the resistance change depended upon the angle a (between
the direction of the measuring current and that of the magnetic field). The
two extreme cases are a = 0° and a = 90° giving resp. the Iongitudinal
and the transverse effect.
We investiga ted a pure metaI (Al) and an impure metal (Cd with
! % Hg). Both cases showed a similar behaviour. The resistance change
for an angle a is represented by the formula :
6. R 'J. = 6. Ril cas 2 a + 6. Rl. sin 2 a.
wh ere 6. Ril and 6.Rl. are the values of t:, R in the IongitudinaI and the
transverse case respectively. This expression is the same as that found both
theoretically 1) and experimentally 2) for the conductivity of a monoaxial crystaI at room temperature in a direction making an angle a with the
principaI axis.
In the originally isotropic metaI a direction is fixed by the magnetic field.
In directions parallel and perpendicular to this direction the metaI has
different conductivities. The resistance in an intermediate direction is
found in the same way as for a mono-axial crystaI.
TABLE 9. (23.6 K Gauss)
Cd 1 % Hg.
"
15°
I Measured ICalculated I Difference
0 . 00966
0.00966
0.00000
0
7
0
932
932
15
25
35
959
1015
966
1022
7
1092
1100
8
i5
1181
1166
5
55
1270
1274
6.R/Roo
AL.
I Measured ICa\culated I Difference
0.00223
0.00008
17° I 0 . 00231
{(
I
207
207
202
3
202
202
13
217
202
215
23
2i6
238
8
i
33
272
13
315
6
7
2
0
0
0
2
65
135i
1350
4
iJ
285
321
75
1410
1406
i
53
351
350
I
90
liil
1441
63
391
387
i
75
1409
1406
0
3
73
415
i14
1
65
0.01348
0.01350
2
83
429
HO
I
87
i32
O.OOi22
i32
0
1
I
77
I 0.00i21
The deviations between the calculated and measured values are greatest
I) W. VOlGT. Kristallphysik. Teubner 1910. section 173.
2) P . BRIDGEMAN . Proc. Amer. Acad. 60. p. 305. 192i.
E. GRUNEISEN and E. GOENS. Zeit. für Phys. 26. p. 250. 1924.
261
in the middle. because here an inaccuracy in the adjusting of the angle a
has the greatest influence. As the reading of the angle was accurate to
T ABLE 10. Al Transversal.
H
K. Gauss
°
1.0
2. 1
3.1
i.1
5.2
7. 8
10 . 3
12 . 6
li .6
20 . 1
21.7
23.5
T = 77°.2 K.
T = 20 0 .i K.
T = I4° .2 K.
/'::,RIRoo
I /'::,RIRoT
/'::,R/Roo
I /'::,R/RoT
/'::,RIRoo
-
-
0. 00002
6
0.0002
8
0.00001
7
13
2i
36
75
123
171
217
358
i03
0.00i51
I /'::, RIRoT
I
0 .00003
9
9
12
22
iO
61
77
139
157
0.00177
0. 0002
6
6
8
14
25
39
i8
87
98
0.0110
-
-
33
23
-
-
\03
172
71
117
-
-
208
3104
387
0 .00i32
308
509
5704
0.0610
o.060~-----+------~~--~
o.oso~-----+------If--------i
0.040~-----+-----If--+-----------,
op30~-----+---Jf----i--h",*,,.,,...
o
20
Fig. 5
30Ke
I
0. 0002
11
21
38
56
114
185
258
326
539
605
0.Ofi78
262
10 , the agreement between the calculated and measured values may be
called satisfactory. The behaviour of Al in different fjelds may be found
in table 10.
T ABLE 11 . AI. Longitudinal.
.H
K Gauss
1.0
2. 1
3.1
4. 1
5.2
7.8
10 .3
12.6
11 .6
20 . 1
21.7
23 .5
§ 9.
I
I
T = 77 °.2 K.
6.R/Roo
I
T= 11°. 2 K.
6.R/RoT
6.R/Roo
0 .00001
2
9
0.0001
4
-
-
-
-
-
-
-
-
-
-
0 .00001
4
12
20
28
53
56
0 .00060
T= 20° . 4 K.
0.0003
3
7
12
17
33
35
0 .0038
I
6.R/RoT
-
13
31
51
-
93
160
180
0.00202
16
75
-
138
236
266
0 .0299
6.RIR.o
-
I
M /RoT
-
0.00002
6
10
16
32
51
77
98
166
188
0 .00211
0 .0003
9
16
21
19
80
111
116
218
281
0 .0316
Summary .
1. A high diama gnetism and astrong resistance change go together.
2. Impurities have a greater influence upon the resistance change at
low temperatures than at high temperatures.
3. Alteration of the angle between the directions of the current and of
the field viz. transition from the transversal case to the longitudinal one,
changes the resistance in the same way as the alteration of the angle
between principal a nd secondary axes does for a mono-axial crystal.
For crystals the phenomena are much more complicated.
Finally we w ish to express our thanks to Mr. J. DE BOER for kindly
preparing the Cd-Hg-alloys and to Mr. O. GU1NAU for his valuable help
in the measurements and the calculations of the results.