﻿the 
  Reflexion 
  of 
  the 
  X-rays* 
  2ol 
  

  

  this 
  : 
  Inn 
  it 
  has 
  been 
  suggested 
  by 
  Webster* 
  that 
  this 
  excess 
  

   scattering 
  is 
  due 
  to 
  an 
  irregular 
  attempt 
  at 
  cooperation 
  

   between 
  the 
  disturbances 
  from 
  the 
  different 
  atoms. 
  It' 
  this 
  

   is 
  so 
  no 
  account 
  need 
  be 
  taken 
  here 
  of 
  this 
  excess 
  radiation. 
  

   For 
  glancing 
  angles 
  less 
  than 
  6°, 
  tin 
  4 
  Factor 
  (1 
  -4- 
  cos 
  2 
  #) 
  is 
  

   approximately 
  constant 
  and 
  need 
  not 
  be 
  considered. 
  The 
  

   intensity 
  of 
  reflexion 
  will 
  therefore 
  depend 
  on 
  two 
  factors: 
  

   (1) 
  the 
  energy 
  of 
  the 
  part 
  of 
  the 
  incident 
  radiation 
  which 
  is 
  

   reflected, 
  (2) 
  the 
  number 
  of 
  atoms 
  of 
  which 
  the 
  effects 
  

   cooperate, 
  [f 
  the 
  components 
  with 
  wave-lengths 
  between 
  A, 
  

   and 
  \ 
  + 
  (/\ 
  are 
  alone 
  reflected, 
  their 
  energy 
  may 
  be 
  expressed 
  

   as 
  E 
  x 
  d\. 
  Now 
  consider 
  the 
  number 
  of 
  atoms 
  that 
  can 
  co- 
  

   operate. 
  This 
  is 
  composed 
  ot" 
  two 
  factors 
  A 
  and 
  13. 
  A 
  

   represents 
  the 
  number 
  ot* 
  atoms 
  in 
  one 
  plane 
  which 
  cooperate, 
  

   B 
  the 
  number 
  of 
  cooperating 
  planes. 
  A 
  is 
  evidently 
  pro- 
  

   portional 
  to 
  the 
  area 
  of 
  a 
  Fresnel 
  zone 
  on 
  the 
  face 
  of 
  the 
  

   crystal, 
  that 
  is 
  to 
  \ 
  2 
  cosec0. 
  To 
  obtain 
  13, 
  consider 
  a 
  pulse 
  

   with 
  no 
  definite 
  wave-length 
  coming 
  from 
  a 
  point 
  source 
  and 
  

   limited 
  by 
  a 
  slit, 
  so 
  that 
  it 
  strikes 
  the 
  crystal 
  only 
  at 
  angles 
  

   between 
  and 
  0-\-d0. 
  Considering 
  only 
  first-order 
  reflexion, 
  

   c 
  imponents 
  with 
  wave-lengths 
  between 
  X 
  and 
  A, 
  4 
  d\ 
  will 
  alone 
  

   be 
  reflected. 
  Cooperation 
  between 
  the 
  reflexions 
  from 
  suc- 
  

   e 
  planes 
  will 
  only 
  continue 
  so 
  long 
  as 
  the 
  extreme 
  com- 
  

   ponents 
  X 
  and 
  X 
  4- 
  d\ 
  remain 
  in 
  step. 
  They 
  will 
  be 
  completely 
  

   out 
  of 
  step 
  in 
  n 
  layers 
  where 
  nd\ 
  = 
  ^\ 
  and 
  so 
  the 
  factor 
  13 
  

  

  is 
  proportional 
  to 
  -jr 
  . 
  The 
  intensity 
  of 
  the 
  reflected 
  beam 
  is 
  

  

  therefore 
  proportional 
  to 
  ~E\dX 
  . 
  A, 
  2 
  cosec 
  6 
  >-p- 
  , 
  that 
  is 
  to 
  

  

  ((X 
  

  

  Ex 
  .X 
  2 
  , 
  since 
  by 
  equation 
  (1) 
  \cosec# 
  is 
  constant. 
  

  

  Now 
  ?/,._, 
  the 
  energy 
  between 
  frequencies 
  v 
  and 
  v 
  -f 
  1 
  , 
  is 
  

   proportional 
  to 
  E\.X 
  2 
  . 
  If, 
  therefore, 
  this 
  argument 
  is 
  based 
  

   on 
  correcf 
  assumptions 
  with 
  regard 
  to 
  the 
  scattering, 
  then 
  

   curve 
  If., 
  fig. 
  2, 
  gives 
  the 
  distribution 
  in 
  the 
  spectrum 
  of 
  the 
  

   energy 
  of 
  the 
  X-rays. 
  The 
  maximum 
  value 
  of 
  u 
  v 
  occurs 
  at 
  

  

  a 
  wave-length 
  of 
  about 
  K* 
  X 
  4'9x 
  10" 
  9 
  cm. 
  The 
  potential 
  

   measured 
  by 
  an 
  alternative 
  spark-gap 
  was 
  50,000 
  volts. 
  The 
  

  

  curve 
  does 
  not 
  show 
  this 
  distribution 
  in 
  its 
  usual 
  form, 
  hut 
  

   '/,. 
  is 
  plotted 
  against 
  0, 
  which 
  for 
  these 
  small 
  angles 
  is 
  pro- 
  

   portional 
  to 
  X. 
  The 
  general 
  form 
  of 
  the 
  curve 
  bears 
  a 
  

   striking 
  resemblance 
  to 
  that 
  of 
  the 
  radiation 
  from 
  a 
  black 
  

   body. 
  The 
  rapid 
  fall 
  of 
  energy 
  for 
  high 
  frequencies 
  and 
  the 
  

   more 
  gradual 
  fall 
  for 
  low 
  are 
  very 
  noticeable, 
  but 
  on 
  both 
  

   sides 
  the 
  rise 
  to 
  the 
  maximum 
  is 
  much 
  steeper 
  than 
  it 
  is 
  for 
  

   the 
  radiation 
  from 
  a 
  black 
  body. 
  

  

  * 
  D. 
  L. 
  Webster, 
  Phil. 
  -Mag. 
  xxv. 
  p. 
  284 
  (1918). 
  

  

  